High-resolution spin measurements using a single-spin quantum sensor

Abstract

The invention relates to measurement protocols for performing measurements on a nuclear-spin system using a single-spin quantum sensor disposed in a host material The invention further relates to an imaging method for constructing an image of a nuclear-spin system. The invention further relates to a measurement system configured to perform at least one of the measurement protocols and / or the imaging method. The invention enables improved measurements of nuclear-spin systems based on a single-spin quantum sensor and has among others applications in magnetic imaging, spectroscopy, quantum simulation, quantum sensing, quantum cryptography, distributed quantum computation.

Description

[0001] High-resolution spin measurements using a single-spin quantum sensor

[0002] Field of the invention

[0003] The invention relates to measurement protocols for performing measurements on a nuclear-spin system using a single-spin quantum sensor disposed in a host material. The invention further relates to an imaging method for constructing an image of a nuclear-spin system. The invention further relates to a measurement system configured to perform at least one of the measurement protocols and / or the imaging method.

[0004] Background art

[0005] Nuclear magnetic resonance (NMR) imaging, or magnetic resonance imaging (MRI), provides spectral and / or spatial information on an ensemble of spins comprised in a target sample. Conventional MRI methods provide limited spatial resolution and include averaging over an ensemble of nuclear spins.

[0006] There is a vast interest in nano-MRI technology for enabling magnetic imaging of spin systems with higher spatial resolution. To this end, researchers are investigating measurement systems based on single-spin quantum sensors, such as spectroscopy based on nitrogen-vacancy (NV) spin sensors, so as to build a platform for imaging single molecules, proteins, viruses, spin-networks in solid-state materials, etc., with nano-scale resolution.

[0007] However, developing and improving measurement systems that employ a single-spin quantum sensor is an active field of research. There is a need for new methods and systems enabling improved measurements of nuclear-spin systems based on a single-spin quantum sensor.

[0008] Summary of the invention

[0009] A general object of the invention is to enable improved measurements of nuclear-spin systems based on a single-spin quantum sensor.

[0010] Therefore, according to a first aspect of the invention, a method of measuring, via a single-spin quantum sensor disposed in a host material, a hyperfine-shift frequency related to a spin-spin interaction between the single-spin quantum sensor and a nuclear spin, the nuclear spin being part of a nuclear-spin system, as defined in appended claim 1 is provided.

[0011] According to a second aspect of the invention, a method of measuring, via a single-spin quantum sensor disposed in a host material, a coupling strength of a spin-spin interaction between a nuclear spin and a target nuclear spin, the nuclear spin and the target nuclear spin being part of a nuclear-spin system, as defined in appended claim 19 is provided. According to a third aspect of the invention, a method of measuring, via a single-spin quantum sensor disposed in a host material, at least one of one or more hyperfine-shift frequencies related to respective one or more nuclear spins and / or one or more coupling strengths related to spin-spin interactions between the respective one or more nuclear spins and a further nuclear spin, the one or more nuclear spins and the further nuclear spin being part of a nuclear-spin system, as defined in appended claim 31 is provided.

[0012] According to a fourth aspect of the invention, a method of transferring, via a spin-chain sequence involving a k-tuple of nuclear spins, a signal of a nuclear spin via the k-tuple of nuclear spins to a particular one of the nuclear spins of the k-tuple of nuclear spins, the nuclear spin and the k-tuple of nuclear spins being part of a nuclear-spin system, wherein k > 1, as defined in appended claim 41 is provided.

[0013] According to a fifth aspect of the invention, a method of imaging a nuclear-spin system, as defined in appended claim 45 is provided.

[0014] According to a sixth aspect of the invention, a measurement system as defined in appended claim 46 is provided.

[0015] The above-referenced aspects of the invention enable improved measurements of nuclear-spin systems based on a single-spin quantum sensor. The invention can have, among others, applications in magnetic imaging, spectroscopy, quantum simulation, quantum sensing, quantum cryptography, distributed quantum computation. Further more detailed advantages are discussed further below in the detailed description.

[0016] Brief description of the drawings

[0017] Embodiments of the present disclosure will be described herein below with reference to the accompanying drawings. However, the embodiments of the present disclosure are not limited to the specific embodiments and should be construed as including all modifications, changes, equivalent devices and methods, and / or alternative embodiments of the present disclosure.

[0018] The terms “have,” “may have,” “include,” and “may include” as used herein indicate the presence of corresponding features (for example, elements such as numerical values, functions, operations, or parts), and do not preclude the presence of additional features.

[0019] The terms “A or B,” “at least one of A or / and B,” or “one or more of A or / and B” as used herein include all possible combinations of items enumerated with them. For example, “A or B,” “at least one of A and B,” or “at least one of A or B” means (1) including at least one A, (2) including at least one B, or (3) including both at least one A and at least one B. The terms such as “first” and “second” as used herein may modify various elements regardless of an order and / or importance of the corresponding elements, and do not limit the corresponding elements. These terms may be used for the purpose of distinguishing one element from another element. For example, a first element may be referred to as a second element without departing from the scope the present invention, and similarly, a second element may be referred to as a first element.

[0020] It will be understood that, when an element (for example, a first element) is “(operatively or communicatively) coupled with / to” or “connected to” another element (for example, a second element), the element may be directly coupled with / to another element, and there may be an intervening element (for example, a third element) between the element and another element. To the contrary, it will be understood that, when an element (for example, a first element) is “directly coupled with / to” or “directly connected to” another element (for example, a second element), there is no intervening element (for example, a third element) between the element and another element.

[0021] The expression “configured to (or set to)” as used herein may be used interchangeably with “suitable for” “having the capacity to” “designed to” “adapted to” “made to,” or “capable of’ according to a context. The term “configured to (set to)” does not necessarily mean “specifically designed to” in a hardware level. Instead, the expression “apparatus configured to...” may mean that the apparatus is “capable of...” along with other devices or parts in a certain context.

[0022] The terms used in describing the various embodiments of the present disclosure are for the purpose of describing particular embodiments and are not intended to limit the present disclosure. As used herein, the singular forms are intended to include the plural forms as well, unless the context clearly indicates otherwise. All of the terms used herein including technical or scientific terms have the same meanings as those generally understood by an ordinary skilled person in the related art unless they are defined otherwise. The terms defined in a generally used dictionary should be interpreted as having the same or similar meanings as the contextual meanings of the relevant technology and should not be interpreted as having ideal or exaggerated meanings unless they are clearly defined herein. According to circumstances, even the terms defined in this disclosure should not be interpreted as excluding the embodiments of the present disclosure.

[0023] The person skilled in the art will understand that the features described above and / or below may be combined in any way deemed useful. The drawings of the present disclosure show examples / embodiments of the invention, which will be described in detail hereinafter. It is to be understood that one or more of elements / components shown and / or described in one or more of these examples / embodiments and not in others may be used in those others too unless mechanical or other limitations prevent such an implementation. Moreover, describing features of different examples / embodiments in a single passage does not automatically mean that those features are inextricably linked. They may be applied separately from one another.

[0024] The present invention is discussed in more detail below, with reference to the attached drawings, in which:

[0025] Fig. la-lb illustrate a single-spin quantum sensor disposed in a host material.

[0026] Fig. 2a-2c illustrate examples of a nuclear-spin system located in proximity to a single-spin quantum sensor.

[0027] Fig- 3 illustrates a control system / measurement device / measurement system that can be used to control / access the single-spin quantum sensor and / or nuclear spins of the nuclear-spin system. Fig. 4a-4c illustrate Bloch sphere illustrations of a nuclear spin.

[0028] Fig. 4d-4i illustrate the precession of a nuclear spin due to interactions when a standard Ramsey -type RF pulse sequence is applied to the nuclear spin.

[0029] Fig- 5 illustrates a conceptual graph / network of a nuclear-spin system.

[0030] Fig- 6 illustrates an overview of actions comprised in methods of the present disclosure.

[0031] Fig. 7a-7e illustrate a first method for measuring hyperfine shifts between the single-spin quantum sensor and individual nuclear spins.

[0032] Fig. 8 illustrates an example of a pulse-sequence in the context of an NV center.

[0033] Fig. 9a-9b illustrate double-resonance sequences including one or more n-pulses and one or more spin-transition pulses.

[0034] Fig. 10 shows a power spectral density, PSD.

[0035] Fig. lla-lld illustrate spin chains in the context of the first method.

[0036] Fig. 12 illustrates a double-resonance sequence for transferring a signal from a nuclear spin to another nuclear spin.

[0037] Fig. 13a-13g illustrate an example scenario in which a second method of the present disclosure may be used.

[0038] Fig. 14a-14e illustrate an example situation in which a third method of the present disclosure may be used.

[0039] Fig. 15a-15h show a comparison between experimental and numerical data relating to the first and third methods of the present disclosure.

[0040] Fig. 16a-16b show a graph obtained based on measurement results involving methods of the present disclosure. Fig. 17a shows an image of a spatial distribution of nuclear spins surround an NV center that was used as a quantum sensor.

[0041] Fig. 17b-17c shows a corresponding measured coupling matrix and a predicted coupling matrix.

[0042] Detailed description

[0043] Spins associated to optically accessible solid-state defects have emerged as a versatile platform for exploring quantum simulation, quantum sensing, quantum cryptography and distributed quantum computation. Pioneering experiments have shown the detection, imaging, and control of multiple nuclear-spin qubits surrounding a single electron-spin defect that is used as a single-spin quantum sensor. However, the accessible size and complexity of these spin networks has been constrained by the spectral resolution of current methods.

[0044] The inventors of the present invention have developed sensing protocols capable of providing higher resolution. The present invention enables, among others, mapping a network of a large number of coupled spins through high-resolution correlated sensing schemes via using a single NV center in diamond as a single-spin quantum sensor.

[0045] The present invention provides new opportunities for quantum networks and quantum simulations of many-body physics by increasing the number of available spin qubits. Additionally, the methods developed herein may find applications in nano- and atomic-scale imaging of complex spin systems external to a host crystal hosting the quantum sensor.

[0046] Single-spin quantum sensors

[0047] Fig. la illustrates a single-spin quantum sensor (el) disposed in a host material (1). Typically, the host material (1) is a crystal with a solid-state defect (lattice defect) that is used as the single-spin quantum sensor (el). For example, the solid-state defect can be sensitive to magnetic fields and therefore can be used as a magnetometer. The solid-state defect may host a spin-optical interface, so that the solid-state defect can be controlled and / or read out via for example applied magnetic fields and lasers.

[0048] Fig. lb illustrates an example of a host material hosting a single-spin quantum sensor (el). The host material in this example is a diamond crystal. In Fig. lb, black bullets (C) illustrate carbon nuclei that form a lattice. The lattice has a lattice defect (a solid-state defect) called an nitrogen-vacancy (NV) center. The NV center is a nitrogen-vacancy pair located in the diamond in which one carbon site of the lattice is replaced by a nitrogen atom and a neighbouring carbon site is replaced by a vacancy site; i.e., a pair of two neighbouring carbon nuclei is substituted by a nitrogen atom and a vacancy site. In Fig. lb,14N illustrates a nitrogen site that is located next to a vacancy site hosting one or more electrons. A diamond can generally comprise one or more NV centers, but a single NV center is referred to hereinafter for simplicity. The NV center can be in a negatively charged state (NV-) in which it typically hosts, at its vacancy, two electrons that are spatially localized in a volume of ~1nm3. At this location, a spin-triplet (effective spin-1 particle) with three spin states corresponding to quantum numbers ms,0=0, ms,-=-1, and ms,+=+1 can be observed. The spin-1 quantum state of the NV center may be understood in terms of the spin states of the two electrons: the spin states ms,+ / - may correspond to both electron-spins being aligned, respectively, while the spin state ms,0may correspond to both electron-spins being anti-aligned. The value of ms is associated with the spin-component parallel to a quantization axis referred to as the z-axis. At the example of a NV diamond, the z-axis or quantization axis can refer to a symmetry axis formed along the nitrogen site and the vacancy site. More information on NV centers can be found in [Barry et al, 2020]. The NV center is but one example of a single-spin quantum sensor (el). Other types of host materials for hosting a single-spin quantum sensor (el) as a defect-spin are known, such as silicon carbide [Nagy et al, 2019; Bourassa el al, 2020; Lukin et al, 2020], silicon [Durand et al, 2021], and hexagonal boron nitride (hBN) [Gao et al, 2022]. In general, the single-spin quantum sensor (el) admits a plurality of spin states associated with quantum numbers ms. For example, the single-spin quantum sensor (el) may correspond to a spin-1 / 2 particle with two spin states ms,+ / -= (+ / -) ½, but may also be a spin- triplet with three spin states as in the example of the NV-electron spin. A silicon-vacancy (Vsi) center in diamond is an example of a spin ½ quantum sensor.4H-silicon-carbide (4H-SiC) is an example of a spin 3 / 2 quantum sensor. In general, the number of spin states and their realization may depend on the type of the single-spin quantum sensor (el), e.g., depending on which type of host material is used or which property of the host material is used for realizing a single-spin quantum sensor (el). The detailed description hereinafter is set out, in examples, in more detail in the context of the single-spin quantum sensor (el) corresponding to a NV center, for reasons of intelligibility. In this context, the single-spin quantum sensor (el) is also referred to as an NV- electron spin, or electron spin, or electronic spin, etc. The reference sign (el) reflects that typically electrons are involved in forming the single-spin quantum sensor (el). However, the present invention is not limited to using a NV center but is applicable to other realizations of a quantum sensor as well. The single-spin quantum sensor (el) is for simplicity also referred to as a quantum sensor (el). Nuclear spins S1,…, SN Figs. 2a-2c illustrate examples of a nuclear-spin system SI, ... ,SN that is located in proximity to the single-spin quantum sensor (el). The single-spin quantum sensor (el) may be sensitive to spin-spin interactions with nuclear spins of a nuclear-spin system and may therefore by used as a sensor to infer information about the nuclear-spin system.

[0049] As illustrated in Fig. 2a, the nuclear-spin system Si, . . . ,SN can be part of a target sample that is positioned in proximity of the single-spin quantum sensor (el). The target sample may contain multiple nuclei, each of which exhibits a spin which is referred to as a nuclear spin (Si). Examples of target samples are molecules, protein structures, viruses, etc. As also illustrated in Fig. 2a, a solid-state defect such as a NV center can be created / implanted to be located close to the surface of its host material so as to function as a quantum sensor (el) that can engage with nuclear spins of the target sample outside the host material such as a host crystal.

[0050] As illustrated in Fig. 2b, the nuclear-spin system SI, ... ,SN can be part of a nuclear spin network surrounding the single-spin quantum sensor (el). For example, as shown in Fig. 2c, an NV center forming the single-spin quantum sensor (el) may be surrounded by carbon nuclei including13C isotopes. In this case, the13C isotopes admit an interacting spin ’A states while12C carbon nuclei do not. The nuclear-spin system SI, ... ,SN may thus correspond to a spinnetwork of13C nuclei surrounding the single-spin quantum sensor (el). Example samples of manufactured diamonds with one or more NV centers exhibit a natural abundance of about 1.1% of13C nuclei, i.e. about 1.1% of the carbon nuclei constituting the NV diamond correspond to spin ’A particles. The amount of13C nuclei may generally vary in between samples and may depend on manufacturing methods / conditions.

[0051] Controlling the single-spin quantum sensor (el) and nuclear spins SI,...,SN

[0052] Fig. 3 shows a control system (200) that may be used to control / access the single-spin quantum sensor (el) and / or nuclear spins of the nuclear-spin system SI, ... ,SN.

[0053] The control system (200) may comprise one or more of the following components: one or more magnetic-pulse generators (201), an external-magnetic-field generator (202), one or more lasers (203), one or more detectors (204), one or more temperature generators (205), a processor (206), and / or a memory (207). The components may be for providing one or more control paths to control / access the quantum sensor (el) and / or the nuclear spins SI, ... ,SN.

[0054] The one or more magnetic-pulse generators (201) may be configured to generate oscillating magnetic fields with an oscillation frequency within a microwave (MW) frequency and / or a radio-frequency (RF). For example, a magnetic-pulse generator may be implemented by a waveform generator combined with an RF source and a gold stripline. The gold stripline may be used to guide pulses close to the quantum sensor such as an NV center. Electric currents generated on the gold stripline can generate magnetic fields that are oscillating within certain frequency ranges, preferably including MW and / or RF ranges. MW pulses may be used to control the single-spin quantum sensor (el), e.g. to change its spin state, while RF pulses may be used to control nuclear spins, e.g. to change spin states of nuclear spins. For example, for controlling NV centers, MW pulses are typically in the GHz range, and for controlling a Vsi defect (quantum sensor in silicon carbide), in the MHz range. RF pulses are typically in a range from 100kHz to 10 MHz.

[0055] The external-magnetic-field generator (202) may be configured to generate an external magnetic field B, e.g. along a symmetry axis (z-axis) of an NV center. For example, an external- magnetic-field generator may be implemented as permanent neodymium magnet.

[0056] The one or more lasers (203) may be configured to generate optical radiation, e.g. for operations such as initializing / resetting and reading out the single-spin quantum sensor (el). Optical radiation may be used for example to drive transitions between ground and excited state energy levels of the quantum sensor (el). Details on lasers suitable for accessing e.g. a NV center can be found in [Barry et al, 2020],

[0057] The one or more detectors (204) may be configured to detect photons, e.g. when detecting a spin state via the quantum sensor (el), the detecting involving photon detection. A solid immersion lens (SIL) may be included in a detector to enhance photon collection efficiency.

[0058] The one or more temperature generators (205) may be configured to control an operating temperature. For example, a room temperature may be configured. As another example, a temperature below room temperature may be configured. For example, a temperature in the range of 0-30 Kelvin may be configured, such as 3-4 Kelvin. The operating temperature may be chosen depending on the type of host material and sample to be measured.

[0059] The processor (206) may be configured to control components (201-205) that are included in the control system (200). The memory (207) may store instructions that, when executed by the processor (206), cause the control system (200) to execute one or more or all of the methods discussed hereinbelow in the present disclosure.

[0060] The control system (200) may be implemented as a measurement device (200) or as a measurement system (200). Although not illustrated in Fig. 3, a control system (200) / measurement device (200) / measurement system (200) according to the present disclosure may additionally comprise a single-spin quantum sensor (el) disposed in a host material (1). For example, a control system (200) / measurement device (200) / measurement system (200) may be implemented as a spectrometer (200) / spectroscope (200) including a single-spin quantum sensor (el) disposed in a host material (1).

[0061] The type of components included in a control system (200) / measurement device (200) / measurement system (200) may depend on the circumstances, e.g., which type of host material is employed, which type of nuclear-spin system is supposed to be measured, etc. For example, the control system (200) / measurement device (200) / measurement system (200) can comprises one or more magnetic-pulse generators (201) configured to generate oscillating magnetic fields; one or more lasers (203) configured to generate optical radiation; one or more detectors (204) configured to detect photons; a processor (206); and a memory (207), wherein the memory (207) stores instructions that, when executed by the processor (206), cause the control system (200) / measurement device (200) / measurement system (200) to execute one or more of the methods discussed hereinbelow in the present disclosure. For intelligibility, the following parts refer to a control system (200).

[0062] An exemplary control system (200) and control operations thereof are described and referenced hereinbelow in the context of a diamond sample with an NV center used as the single-spin quantum sensor (el).

[0063] In an example, experiments were performed on a naturally occurring NV center at a temperature of 3.7K (Montana S50 Cryostation), using a home-built confocal microscopy setup. The diamond sample was homoepitaxially grown using chemical vapor deposition and cleaved along the (111) crystal direction. The sample had a natural abundance of13C (1.1 %). The NV center was selected on the absence of couplings to13C stronger than ~ 500 kHz. No selection was made on other properties of the13C nuclei distribution. A solid immersion lens (SIL) that enhances photon collection efficiency was fabricated around the NV center. A gold stripline was deposited close (e.g., <1 z / m) to the edge of the SIL for applying microwave (MW) pulses and radio-frequency (RF) pulses. An external magnetic field of Bz=403.553G was applied along the symmetry axis (z-axis) of the NV center, using a (temperature-stabilized) permanent neodymium magnet mounted on a piezo stage outside the cryostat [Bradley et al, 2019], The field was aligned to within 0.1 degrees using a thermal echo sequence [Abobeih et al, 2019],

[0064] The NV center admits spin quantum states associated with ms,o=O, ms,-=-l, and ms,+=+I. Typically, the external magnetic field B is applied along the z-axis (symmetry axis of the NV center) to act as a bias magnetic field that splits the energy levels of the NV center associated with ms= + / - 1. Spin transitions for changing a spin state of the NV center can be driven by applying MW pulses, preferably Hermite-shaped pulses. For the NV center, usually only direct spin transitions between 0 and -1 and between 0 and +1 can be controlled. In an example, spin transitions between 0 and -1 were driven at a MW frequency of 1.746666 GHz, and spin transitions between 0 and +1 were driven at a MW frequency of 4.008650 GHz. For driving spin transitions between -1 and +1, two consecutive MW pulses may be applied, e.g. starting with a -1 to 0 MW pulse and followed by a 0 to +1 MW pulse, the MW pulses being spaced by a certain waiting time. In an example, a waiting time of 3 / / s was used. The NV center can be initialized into the state ms=0 via optical pumping and can be read out in a single shot through spin-selective resonant excitation. Details on such operations to access an NV center via optical radiation and MW pulses can be found in [Robledo et al, 2011], Further details on energy levels of an NV center can be found in [Barry et al, 2020],

[0065] Nuclear spins of the nuclear-spin system SI, ... ,SN can also be controlled and sensed via the quantum sensor (el) and via the control system (200). Such operations, including polarization of nuclear spins, read out of nuclear spin states, and control of nuclear spins are described and referenced next.

[0066] Polarizing a nuclear spin Si may mean achieving a certain degree of polarization pi of the nuclear spin Si, for example |pi | > 0.05. Polarizing one or more nuclear spins of a nuclear- spin system may be part of initializing the system with an initial density matrix p0as wherein (omitting hats from the notation) pe 0is the initial density matrix of the NV-electronic spin / single-spin quantum sensor (el), ® is the identity matrix, G [—1, +1] is the degree of polarization of nuclear spin Si, is the spin-operator corresponding to the nuclear spin Si with respect to a quantization axis (z-axis). The spin-operator may be represented by the z- Pauli matrix. In an NV diamond, each nuclear spin Si corresponding to a13C nucleus is an spinx / i particle with two spin eigenstates + / - Yi.

[0067] Polarizing one or more nuclear spins can be performed in a number of different and known ways, which are not described in detail here but are referenced. For example, a dynamical nuclear polarization via a PulsePol ([Schwartz et al, 2018]) may be employed to polarize the nuclear-spin system. Via a PulsePol, polarization can be transferred from the quantum sensor (el) to the nuclear spins so as to polarize the nuclear-spin system. The number of repetitions of the PulsePol sequence can depend on the specific polarisation dynamics of the nuclear spins being used in the given experiment [Randall et al, 2021] and can depend on how well the quantum sensor can be spin-pumped / initialized, but preferably ranges from 500-10000. For example, nuclear spins may in addition and / or alternatively also be polarized on an individual level via SWAP sequences with the NV center / single-spin quantum sensor (el) [Bradley et al, 2019], In general, a signal-to-noise ratio (SNR) may be increased with increased polarization. A low SNR may still be acceptable for practical purposes at the cost of obtaining more measurement points and more averaging. At the example of a NV center in diamond, a polarization of 0.05 may be acceptable.

[0068] A nuclear spin may be controlled via RF pulses that are applied within a frequency range including a resonating frequency of the nuclear spin. For example, oscillating magnetic fields / RF-pulses can be applied in a direction substantially perpendicular to the direction of the external magnetic field B that is applied along a z-axis. The RF pulses can cause rotations of the nuclear spin around an axis perpendicular to the z-axis.

[0069] Fig. 4a shows a Bloch-sphere illustration of a z-component of a nuclear spin (e.g., Si, i=l,...,N) corresponding to a spin ’A particle with associated two spin eigenstates denoted as |0) and | 1), respectively, e.g. | 1) corresponding to the +1 / 2 and |0) to the -1 / 2 eigenstate. The z-component may refer to a preferred direction such as a symmetry axis of an NV center.

[0070] Fig. 4b shows a Bloch vector (arrow pointing upward) of the nuclear spin being initialized in its + A spin eigenstate with respect to the z-axis and being rotated by an angle of n / 2 around an axis perpendicular to the z-axis so as to arrive at a configuration shown in Fig. 4c. The curved arrow illustrates that via a suitable RF-pulse resonant with the nuclear spin, a spin state of the nuclear spin may be changed to a superposition state between the + / - A spin eigenstates. Such a RF pulse rotating around an axis by an angle of ti / 2 is also referred to as a n / 2-pulse. A RF pulse for swapping spin states, e.g. transferring from the + A spin state to the - A spin state or rotating around an axis by an angle of n, is also referred to as a n-pulse. In the Bloch-sphere illustration, a ti / 2 pulse or a n-pulse may correspond to rotating a spin-state vector around an axis perpendicular to the z-axis by an angle of ti / 2 or n, respectively.

[0071] Nuclear spins are interacting with each other and with the quantum sensor (el) and are generally subjected to environment noise. As a result of such interactions, the Bloch vector of a nuclear spin can precess in the Bloch-sphere representation around the z-axis with a certain speed that is a result of the various interactions the nuclear spin is undergoing.

[0072] Figs. 4d-4h illustrate the precession of a nuclear spin due to such various interactions when a standard Ramsey-type RF pulse sequence is applied to the nuclear spin. For illustration purposes, we assume an initial polarization of pi=l of a nuclear spin along the z-axis and that a pure state is maintained over time, i.e. the Bloch vector length does not shrink over time. Fig. 4d shows a standard Ramsey type RF pulse sequence including an initial ti / 2 pulse and a final ti / 2 pulse that are separated by a free-evolution time period of duration T. The final n / 2-pulse is followed by a read out operation.

[0073] Fig. 4e shows the nuclear spin being driven by a n / 2-pulse around an axis perpendicular to the z-axis, which changes the initial spin state of the nuclear spin to a superposition state with equal probabilities of observing spin + / - Yi. The initial rotation axis is typically referred to as an x-axis, but may also be referred to as a y-axis.

[0074] Fig. 4f shows an evolution of the nuclear spin during the free-evolution time period of duration T. Interactions of the nuclear spin, e.g. with the quantum sensor (el) and / or with other nuclear spins, cause the nuclear spin to rotate around the z-axis. That is, the nuclear spin picks up a phase. The picked up phase depends on the duration T. For example, the picked-up phase may be proportional to the duration T. The phase is accumulated due to the various interactions and may thus depend on the position of the nuclear spin relative to its surrounding nuclear spins and the quantum sensor (el).

[0075] Fig. 4g shows a position of the nuclear spin after having accumulated the phase during the free-evolution time window of duration T. After the duration T, a final ti / 2 pulse is applied that rotates again around the same initial axis, e.g., around the x-axis.

[0076] Fig. 4h shows the nuclear spin in a final state after the final n / 2-rotation. In Fig. 4h, after the final IT / 2 -rot at ion, the nuclear spin is in a spin state having a certain expectation value / z-component with respect to the quantization axis / z-axis. The z-component of the nuclear spin state can be read out (RO) via the quantum sensor (el). For example, to obtain a projective spin state, i.e. an expectation value of the z spin component, the Ramsey sequence can be repeated multiple times and obtained read outs (measurement points) can be averaged so as to obtain the expectation value.

[0077] In general, a z-component of a spin state (expectation value with respect to an axis) of a nuclear spin can be read out via the quantum sensor (el). Read out of nuclear spins via the quantum sensor (el) can be performed by detecting sequences applied to the quantum sensor (el), including phase-sensitive dynamical decoupling (DD) sequences or DDRF sequences as detailed for example in [Bradley et al, 2019], For example, by repeating measurements involving a sensing protocol such as the Ramsey sequence and obtaining a measurement point for each measurement and averaging over measurement points, the z-component of a nuclear spin may be obtained. Such operations related to read out and obtaining the z-component are known and are not described in detail here. Fig. 4i illustrates an oscillating signal that is obtained when repeating a Ramsey-type experiment a plurality of times and sweeping the duration τ. As can be seen, the nuclear spin is oscillating as a function of the duration τ. In this case, the oscillating signal may be used to extract a magnetic field strength in the z-direction at the location of the nuclear spin, as the oscillating frequency may be related to the magnetic field strength. The oscillating signal illustrated in Fig. 4i is decaying as the nuclear spin’s initial polarization decays towards an equilibrium due to environmental noise and interactions with other nuclear spins of the nuclear- spin system. In equilibrium, the nuclear spin state is typically in a mixed state again---depending on factors such as e.g. temperature and an applied magnetic field---in which the probability of observing a spin + ½ or - ½ is equal, i.e. pi=0. The decaying is not shown in the Bloch-sphere illustrations for illustration purposes. The spectral resolution of a sensing protocol using the standard Ramsey-type sequence---for measuring a magnetic field strength in the z-direction at the location of the nuclear spin---is limited by the dephasing time ^∗ ^, as different nuclear spins experience different phase accumulation. In a sample diamond with nuclear spins corresponding to 13C nuclei, ^∗ ^ is ~5ms. The resolution of the Ramsey measurement is limited by quasi-static noise sources, such as the surrounding nuclear spins and slow external field drifts, that fluctuate over the course of multiple experimental repetitions and can alter the local field at the position of the nucleus. Sensing sequences for mapping the nuclear-spin system The single-spin quantum sensor (el) is interacting with nuclear spins of the nuclear-spin system, and nuclear spins are interacting with each other. As illustrated above with reference to Figs.4a-4i, interactions cause spin states of nuclear spins to evolve over time. An effective dynamics can be expressed by the Hamiltonian , wherein the parameters N, Ai, Cij, are as follows. ^ ≥ 1 is an integer denoting the total number of nuclear spins in the considered nuclear- spin system S1,…,SN. The value of N may depend on the specific case considered. Examples are N=50, 60, 70, 80, 90, 100, 200. denotes the nuclear-spin operator of the nuclear spin Si. If the nuclear spins are spin ½ particles, e.g. 13C nuclei surrounding an NV center, then is the Pauli spin ½ operator for the nuclear spin Si, which can be represented by the z-Pauli matrix. Ai denotes a nuclear-spin frequency of a nuclear spin Si, which may be understood as a precession frequency associated with the nuclear spin Si. The nuclear-spin frequency Ai may result from a combination of different effects, among them interactions with an external magnetic field B that is applied to the spin system, interactions with the spin system, with the quantum sensor, etc. Nuclear-spin frequencies Ai of spin Si and Aj of spin Sj may differ due to various circumstances including differences in species (e.g., leading to a different gyromagnetic ratio), differences in a local magnetic field of an external magnetic field that is applied to the system, a certain spin environment / distribution of nuclear spins, a different coupling to the single-spin quantum sensor (el), etc. The nuclear-spin frequencies Ai may also depend on a spin state msof the single-spin quantum sensor (el). Cij denotes a coupling strength between nuclear spins Si and Sj. The coupling strength may be regarded as a measure of how strongly two spins are interacting with each other. Furthermore, the coupling strengths may be related to relative coordinates of the nuclear spins, i.e., the distances between the nuclear spins. For example, Euclidian coordinates (xi,yi,zi) of N nuclear spins Si, i=1,..,N, may be related to the coupling constants Cij via wherein Δ^^^denotes the Euclidian distance and α^^= μ^γ^γ^ℏ / 4π with μ^the permeability of free space, γ^the gyromagnetic ratio of spin Siand γ^the gyromagnetic ratio of Sj, and ℏ the reduced Planck constant. Further details on relating the relative coordinates with couplings strengths may be found for example in [Abobeih et al, 2019], e.g., in Sections IV and V therein. By suitable addressing of the quantum sensor (el) and the nuclear spins via certain pulse sequences, the quantum sensor (el) can be used to extract information about the system such as connectivity information. A general objective of the present invention is to provide improved measurement methods capable of providing more information and / or providing higher control of the nuclear- spin system S1,…,SNand the single-spin quantum sensor (el). Fig. 5 and marked areas a-d therein illustrate a graph / network of a nuclear-spin system in terms of frequencies Ai and couplings strengths Cij. The graph is conceptual with vertices organized in frequency space and illustrates aspects of a nuclear-spin system as identified by the inventors. In the graph, larger vertices denote spin precession frequencies Ai and edges denote coupling strengths Cij. Multiple smaller vertices within a larger vertex denotes spins having a similar spin precession frequency (nuclear-spin frequency) ~Ai. For example, in area b, two nuclear spins have a similar nuclear-spin frequency A2. Similarly in area d with A23. In general, for illustration purposes, a spin Si may be represented / illustrated in the figures by its respective nuclear-spin frequency Ai.

[0078] Area a illustrates that if all spin frequencies were unique and sufficiently spectrally distinguished, i.e. one spin is in each vertex, then the network could be mapped by measurements that reveal pairwise interactions (e.g., C12) between two spins with spin frequencies (e.g., Ai, A2). Mapping a network may mean extracting parameters such as Ai and Cij and / or other parameters from the network which can be used to reconstruct the network. In a diamond with an NV center, a part of a nuclear-spin system relatively close to the NV center has been successfully mapped using specific pairwise measurements that are inspired by Ramsey interferometry used in atomic physics [Abobeih et al, 2019], The pairwise measurements aim at isolating individual nuclear-nuclear interactions (e.g., between spins with frequencies Ai and A2) through spin-echo double resonance (SEDOR) sequences, to obtain a signal of a nuclear spin that can be read out via the single-spin quantum sensor (el). The signal is oscillating with a frequency related to the corresponding coupling strength (e.g., C12 between the spins involved), and the coupling strength may be retrieved from the signal.

[0079] However, the inventors have identified that certain parts of the nuclear-spin system are difficult to be mapped with such state-of-the-art methods. Such situations identified by the inventors are illustrated in areas b-d.

[0080] Area b illustrates a situation identified by the inventors, in which spins spectrally overlap at frequency A2. Even if a number of pairwise measurements involving A2 would be suitable for extracting the coupling strengths Ci2,C32, and C42, it would still remain unclear to which spin at A2 they couple to. Thus, pairwise measurements alone typically provide insufficient information in such cases to accurately reconstruct the spin network associated with the nuclear-spin system, and the ambiguity when assigning measured interactions to specific spins at A2 cannot be resolved with the state-of-the-art pairwise measurements.

[0081] Area c illustrates another situation identified by the inventors, in which spins further away from Ai (e.g., A4) may be difficult to access, e.g., due to spectral crowding (e.g., multiple spins at A4) and / or due to negligible coupling strengths (e.g., between Ai and A4). State-of-the- art methods may not be suitable for providing information on C34 and / or A4. For example, information on a specific spin at frequency A4 such as its exact spin frequency or couplings to nearby nuclear spins (C4i) cannot be provided by the state-of-the-art methods, so that the specific spin cannot be characterized or mapped within the spin network because one has no access to / control over the specific nuclear spin.

[0082] Area d illustrates yet another situation identified by the inventors, in which different nuclear spins have a similar frequency A23 and additionally couple differently to other nuclear spins. When multiple nuclear spins have a similar frequency A23, i.e. overlap for example due to fluctuations on the frequency A23, their individual nuclear-spin frequencies may not be distinguished by state-of-the-art methods. The inventors found that in such cases, multiple nuclear spins can still be distinguished. Furthermore, even when the multiple nuclear spins couple in addition differently to other nuclear spins, the multiple nuclear spins and their different couplings can also be distinguished. For example, when applying, by the control system (200), an external magnetic field B along the z-axis (e.g., symmetry axis of the NV center), nuclear-spin frequencies of nuclear spins Si (i=l, .., N) may be approximated = coL+ msAj, with a global Larmor frequency coLwith respect the z-axis, the spin state msof the single-spin quantum sensor (el) and a hyperfine-shift frequency A£, which is a local shift due to a hyperfine interaction between the single-spin quantum sensor (el) and the nuclear spin Si. The inventors found that multiple nuclear spins can be distinguished by their respective hyperfine shifts. Such situations are detailed further below in the context of methods (l)-(4).

[0083] The inventors have identified sensing protocols / measurement methods that can provide higher resolution information on the nuclear-spin system than state-of-the-art methods:

[0084] (1) High-resolution electron-nuclear double resonance sequence. A method for measuring hyperfine shifts Ai between the single-spin quantum sensor (el) and individual nuclear spins Si. The method is particularly suitable for distinguishing nuclear spins having a relatively similar nuclear-spin frequency. The method can also be used when multiple single-spin quantum sensors are involved.

[0085] (2) High-resolution nuclear-nuclear double resonance sequence. A method for measuring couplings strengths Cij between nuclear spins Si and Sj. The method is particularly suitable for spins typically further away from the single-spin quantum sensor (el).

[0086] (3) Combination of electron-nuclear and nuclear-nuclear double resonance sequences. A method that correlates hyperfine shifts with nuclear-nuclear couplings. The method is particularly suitable for distinguishing multiple nuclear spins with similar frequencies and coupling differently to other nuclear spins.

[0087] (4) Spin-chains. A method for transferring a signal of a nuclear spin to another nuclear spin. In particular, signals of nuclear spins further away from the single-spin quantum sensor (el) can be transferred to nuclear spins closer to the single-spin quantum sensor (el) so as to enable reading out the signal. Furthermore, multiple nuclear spins may be correlated with each other, thereby yielding information on the network connectivity. For example, spin-chains may be combined with methods (l)-(3) to yield connectivity information. The spin-chain method can also have applications in quantum computing / simulations.

[0088] (5) 3 D-r econstruction of a nuclear spin system. A method for reconstructing a graph model of the nuclear spin system, based on iteratively measuring out components and the connectivity of the nuclear-spin system by using spin chains, sensing sequences, etc.

[0089] In general, methods (l)-(3) relate to measuring, via a single-spin quantum sensor (el) disposed in a host material (1), at least one of one or more hyperfine-shift frequencies (Ai Aj) related to respective one or more nuclear spins (Si,Sj) and / or one or more coupling strengths (Cij) related to spin-spin interactions between the respective one or more nuclear spins (Si,Sj), the one or more nuclear spins (Si,Sj) being part of a nuclear-spin system (SI, ... ,SN) and having respective nuclear-spin frequencies (AI, ... ,AN).

[0090] Fig. 6 illustrates an overview of actions Al, A2, A3 comprised in methods (l)-(3). The actions are related to controlling the single-spin quantum sensor (el) and / or nuclear spins so as to obtain a measurement point.

[0091] Action Al (INIT) relates to polarizing one or more nuclear spins.

[0092] Action A2 (DR) relates to applying one or more double-resonance (DR) sequences involving the one or more nuclear spins.

[0093] Action A3 (DETECT) relates to obtaining a measurement point by reading out at least one of the one or more nuclear spins via the single-spin quantum sensor (el).

[0094] Actions Al -A3 are repeated at least one time to obtain a plurality of measurement points by sweeping a parameter over a parameter domain. The parameter may include one or more time parameters and / or one or more frequency parameters involved in the one or more DR sequences. The obtained plurality of measurement points can be used to determine at least one of one or more hyperfine-shift frequencies (Ai Aj), one or more nuclear-spin frequencies (Ai,Aj) and / or one or more coupling strengths (Cij).

[0095] In general, the methods according to the present invention provide more information for efficiently magnetically imaging a nuclear-spin system SI, ... ,SN that is surrounding the quantum sensor (el), e.g. an NV center. Action A2 (DR) may additionally involve the singespin quantum sensor. Furthermore, in general, one or more single-spin quantum sensors may be employed, as also outlined further below.

[0096] Methods (l)-(5) are discussed next in more detail.

[0097] Method (1): High-resolution electron-nuclear double resonance sequence

[0098] Fig. 7a illustrates an example situation for method (1), which involves a single-spin quantum sensor (el) that is used to sense a hyperfine-shift frequency (e.g., Ai) of a nuclear spin (e.g., Si) having a nuclear-spin frequency (e.g., Ai). Below, one or more reference signs relating to Fig. 7a are used for illustration purposes but not to limit method (1) to the illustration.

[0099] Method (1) is for measuring, via a single-spin quantum sensor (el) disposed in a host material (1), a hyperfine-shift frequency (Ax) related to a spin-spin interaction between the single-spin quantum sensor (el) and a nuclear spin (Si), the nuclear spin (Si) being part of a nuclear-spin system (SI, ... ,SN). The method comprises: action Al (INIT): polarizing the nuclear spin (Si); action A2 (DR): applying a double-resonance sequence (XXJ,) involving a free- evolution time (t) and including: a first pulse-sequence, applied within a first frequency range including a nuclear- spin frequency (Ai) of the nuclear spin (Si), and configured to decouple the nuclear spin (Si) from environment noise and from the rest of the nuclear-spin system (SI, ... ,SN), and a second pulse-sequence, applied within a second frequency range including a spin-transition frequency of the single-spin quantum sensor (el), and configured to change a spin state (ms) of the single-spin quantum sensor (el) so as to recouple the single-spin quantum sensor (el) to the nuclear spin (Si) (in other words, to prevent a decoupling of the nuclear spin from the single-spin quantum sensor); action A3 (DETECT): obtaining a measurement point by applying a detectingsequence (XY'T, RO) configured to read out the nuclear spin (Si) via the single-spin quantum sensor (el), the detecting-sequence comprising a sensing-sequence applied to the single-spin quantum sensor (el); repeating the actions at least one time to obtain a plurality of measurement points by sweeping the free-evolution time (t) over a range of chosen free-evolution times, each of the plurality of measurement points being associated with one of the range of chosen free- evolution times; and determining, based on the plurality of measurement points, the hyperfine-shift frequency (Ax).

[0100] The above-recited first method provides a way of measuring the hyperfine-shift with unprecedented resolution, which is detailed next with reference to an example.

[0101] Performing action Al (INIT) achieves initializing a nuclear spin of interest, such as Si, to a certain degree of polarization.

[0102] Performing action A2 (DR) achieves creating a signal on the nuclear spin of interest, the signal being characteristic of the electron-nuclear type spin-spin interaction between the quantum sensor (el) and the nuclear spin of interest. For example, the signal can carry a phase difference of the nuclear spin of interest, the phase difference being determined by the involved free-evolution time (t) and the hyperfine shift that is imparted on the nuclear spin by the recoupling of the quantum sensor (el). In general, “to recouple the single-spin quantum sensor to the nuclear spin” can also be rephrased as “to prevent a decoupling of the nuclear spin from the single-spin quantum sensor”.

[0103] Performing action A3 (DETECT) achieves extracting the signal on the nuclear spin via the quantum sensor (el). Certain aspects are discussed hereinbelow with reference to the figures.

[0104] Action Al (INIT). As discussed further above under Controlling the single-spin quantum sensor (el) and nuclear spins Si, ... ,SN, by polarizing the nuclear spin Si, the nuclear spin Si is initialized with a certain degree of polarization to an initial polarization pi(0) in [- 1 ,+l ] . Polarizing the nuclear spin of interest Si can be performed in a number of different ways. For example, a dynamical nuclear polarization via a PulsePol may be employed to polarize the nuclear-spin system [Schwartz et al, 2018], The nuclear spin Si may also be polarized on an individual level via SWAP sequences with the single-spin quantum sensor Se[Bradley et al, 2019], Various degrees of polarization pi in [-1,+1] may be achieved / chosen. Preferably, a polarization of IpJ > 0.05 is achieved. More preferably, a polarization IpJ > 0.7 is achieved.

[0105] Action A2 (DR). By applying the double-resonance sequence (DR), the spin-spin interaction between the nuclear spin (Si) and the quantum sensor (el) is isolated from environmental noise and from the rest of the nuclear-spin system. The first pulse-sequence can extend the nuclear coherence time by decoupling the nuclear spin Si, while the second pulse- sequence recouples the quantum sensor Seto the nuclear spin Si. Specifically, by controlling the spin state of the single-spin quantum sensor (el), the hyperfine frequency shift Axcan be recoupled to the nuclear spin Si. In combination, the first pulse-sequence and the second pulsesequence create a signal on the nuclear spin Si that captures the spin-spin interaction and retains the interaction of interest ArAn example of a double-resonance sequence is discussed further below with reference to Fig. 7b. In general, “to recouple the single-spin quantum sensor to the nuclear spin” can also be rephrased as “to prevent a decoupling of the nuclear spin from the single-spin quantum sensor”.

[0106] Action A3 (DETECT). The nuclear spin Si can be read out via the quantum sensor (el) by detecting sequences that comprise a sensing-sequence applied to the quantum sensor [e.g., Bradley et al, 2019], When using an NV center as a quantum sensor (el), the sensing-sequence may include MW pulses applied to the NV center within a frequency range targeting specifically a spin transition (e.g., the 0 / -1 spin transition) to read out the nuclear spin Si directly via the quantum sensor (el). By such a sensing-sequence, a spin projection state / a signal on the nuclear spin Si is transferred to the quantum sensor (el), which in turn can be readout via optical radiation using a readout laser. By reading out a nuclear spin via suitably manipulating and measuring the quantum sensor (el), a z-component / spin projection of the nuclear spin can be obtained, i.e. its spin component in the z-direction or its z expectation value, as DD sequences or DDRF sequences can copy a spin projection of a nuclear spin to the singlespin quantum sensor (el). When reading out the nuclear spin Si directly via the quantum sensor (el), the detecting sequence may also comprise RF pulses applied within the first frequency range including the nuclear-spin frequency Ai, as in the example of DDRF. For example, by repeating actions Al -A3 with the same free-evolution time and obtaining a measurement point for each repetition, an average value of a z-component / expectation value of the nuclear spin Si can be obtained. Such operations related to read out and obtaining a projective spin state via the quantum sensor (el) are known and are not described in detail here.

[0107] Preferably, the first pulse-sequence is a Hahn spin-echo type sequence comprising a n- pulse separating two free-evolution time windows from each other, i.e. in between a first free- evolution time window and a second free-evolution time window. The second pulse-sequence then preferably includes a spin-transition pulse sequence that is applied synchronously with the iT-pulse so as to change the spin state (ms) of the single-spin quantum sensor (el) to be different during the first and second free-evolution time windows. For example, the second pulsesequence may comprise one or more MW pulses configured to cause spin transitions. By manipulating the spin state of the quantum sensor (el) to be different during the respective free- evolution time windows, the interaction of interest Axcan be recoupled to the nuclear spin Si. In general, “to recouple the single-spin quantum sensor (el) to the nuclear spin (Si)” can also be rephrased as “to prevent a decoupling of the nuclear spin from the single-spin quantum sensor”.

[0108] Fig. 7b illustrates actions A1-A3 of method (1) at an example including a Hahn spinecho type sequence. In Fig. 7b, the bottom horizontal line denotes a first frequency range including the nuclear-spin frequency Ai, and the top horizontal line refers to a second frequency range including a spin-transition frequency of the single-spin quantum sensor (el). At the example of an NV center, the spin-transition frequency can be in the MW-frequency range. The block Init in Fig. 7b refers to polarizing the nuclear spin (Si) as part of action Al (INIT). As discussed above, the polarizing can be performed in various ways. The block xx in Fig. 7b refers to a double-resonance sequence that is part of action A2 (DR). The block RO in Fig. 7b refers to a detecting sequence used to readout (RO) the nuclear spin Si via the quantum sensor (el).

[0109] In Fig. 7b, the double-resonance sequence in this example comprises a Hahn-spin echo type sequence applied to the nuclear spin Si and a spin-transition pulse sequence applied to the quantum sensor (el), the spin-transition pulse sequence being timed together with a n-pulse of the Hahn-spin echo type sequence. In the first and second time windows, each of duration t / 2, no RF pulse is applied to the nuclear spin Si so as to let the nuclear spin Si evolve freely in time. The xx block (double-resonance sequence) includes specifically changing the spin state in between the two free-evolution time windows, so that for example a first spin state (-1) during the first free-evolution time window is different from a second spin state (+1) during the second free-evolution time window.

[0110] By specifically changing the spin state of the quantum sensor (el) itself, the hyperfine interaction Axcan make a distinguished contribution to a phase accumulated by the nuclear spin Si, whereas phase contributions of the environment noise and the rest of the nuclear-spin system can be cancelled due to the Hahn-spin echo type sequence applied to the nuclear spin Si. The signal that can be read out from the nuclear spin Si via the quantum sensor (el) is then determined / characterized by the distinguished contribution to the phase, from which the hyperfine frequency Axcan be determined. For example, the distinguished phase contribution may depend on the hyperfine shift Axand the involved (total) free-evolution time t. By sweeping the free-evolution time t and repeating actions Al -A3, one may thus obtain a plurality of measurement points from which the hyperfine shift Axcan be determined. Fig. 7c illustrates a signal using a state-of-the-art method [Abobeih et al, 2019], showing a relatively broad resonance (135 Hz FWHM) that is limited by the nuclear T2* -time and does not achieve the spectral resolution of method (1).

[0111] Figs. 7d-7e illustrate determining the hyperfine-shift frequency Axbased on a plurality of measurement points that are obtained by sweeping the free-evolution time t.

[0112] Fig. 7d shows a signal P[0] that may be obtained by sweeping the free-evolution time (t) over a range of chosen free-evolution times (t), each of the plurality of measurement points being associated with one of the range of chosen free-evolution times. Obtaining the signal P[0] in this example includes averaging the values that are obtained for each chosen free-evolution time so as to obtain an approximation for the z-expectation value of the nuclear spin Si. Fig. 7d shows undersampled data of a time domain signal of Ai = 14549.91(5) Hz, with a coherence time of 0.36(2) s, that is fitted for illustration purposes by a sinusoid with Gaussian decay.

[0113] Fig. 7e shows a power spectral density (PSD) of the signal in Fig. 7d obtained involving a Fourier transform. The PSD shows a spectral linewidth of 1.8Hz, which means an ~75 times improved linewidth as comparted to state-of-the-art methods as shown in Fig. 7c.

[0114] The hyperfine shift frequency may in general be retrieved from undersampled data, which may be advantageous as less data points need to be obtained. To retrieve the frequency from such undersampled data, one may take at least two data sets with different bandwidths and sampling rates [Boss et al, 2017], and correct for aliasing by minimizing the mean squared error between the multiple measurements to select the most likely alias. The hyperfine-shift frequency Axmay also be obtained by taking the Fourier transform of the signal P[0], e.g. when obtaining sufficiently many measurement points. Signal processing methods known to the person skilled in the art may be employed to analyse the plurality of measurement points. For example, a conventional SEDOR as shown in Fig. 7c may be performed to obtain an estimate of a frequency, which may be used to determine a sufficient sample rate such as a Nyquist rate. Such detailed known processing methods are not described in details here.

[0115] In general, a frequency range of the respective pulses used in the first and second pulse sequence, respectively, may depend on the type of single-spin quantum sensor and on nuclear- spin frequencies. Preferably, the first pulse-sequence comprises RF pulses, and, preferably, the second pulse-sequence comprises MW pulses. For example, if the quantum sensor corresponds to a NV center, then spin-transition frequencies are typically in the MW range. For example, if the nuclear-spin system corresponds to13C nuclei surrounding the NV center, the corresponding nuclear-spin frequencies are typically in the RF frequency range. At the example of the NV center, the second pulse sequence applied on the NV center corresponds to changing electronic spin levels, while the first pulse sequence applied within a nuclear-spin frequency range is resonant with a nuclear spin.

[0116] Fig. 8 shows a yet further detailed example of a second pulse-sequence in the context of an NV center. In this example, MW pulses (indicated by TTX) are applied synchronously (are timed) with a n-pulse of the first pulse-sequence, so as to trigger a spin transition of the NV center from the -1 spin state via the 0 spin state to the +1 spin state. The first MW pulse nxwith respect to the horizontal axes is for transitioning between -1 and 0, the second MW pulse nxis for transitioning between 0 and +1. As shown in Fig. 8, further MW pulses may be applied at the end so as to transition back from the +1 spin state via the 0 spin state to the -1 spin state.

[0117] Preferably, the first pulse-sequence is a Hahn spin-echo type sequence that comprises one or more n-pulses. Including more n-pulses may further extend the decoherence time and can enhance the spectral resolution. The second pulse- sequence then preferably includes one or more spin-transition pulses being applied synchronously with the one or more n-pulses of the first pulse-sequence so as to change the spin state of the single-spin quantum sensor Seto flip back and fourth between two spin states and to be different in any two consecutive free- evolution time windows.

[0118] Figs. 9a and 9b illustrate double-resonance sequences including one or more n-pulses and one or more spin-transition pulses applied synchronously / in accordance thereto. Fig. 9a shows an example in which the Hahn-spin echo type sequence includes two n-pulses and in which spin-transition pulses are applied accordingly in coordination so as to flip the quantum sensor’s (el) spin state (ms) back and fourth between two interacting spin states (in the example -1 and +1). Fig. 9b shows an example in which the Hahn-spin echo type sequence includes three n-pulses and in which spin-transition pulses are applied accordingly. The number of n-pulses may also be 3, 4, . . ., etc. As an example, 256 n-pulses may be employed as well. However, the method is not limited to a specific number of n-pulses. Preferably, a maximum number of 1000 n-pulses is used.

[0119] In general, preferably the quantum sensor (el) can spend an equal time in its two involved interacting spin states. In the example of Fig. 7b, a first time (t / 2) spent before the n- pulse of the first pulse-sequence is equal to a second time (t / 2) spent after said n-pulse. In the example of Fig. 9a, a first total time spent in a first interacting spin state (-1) is t / 4 + t / 4 = t / 2 and is equal to a second total time spent in a second interacting spin state (+1) which is t / 2. In the example of Fig. 9b, a first total time spent in a first interacting spin state (-1) is t / 4 + t / 4 = t / 2 and is equal to a second total time spent in a second interacting spin state (+1) which is t / 4 + 1 / 4 = t / 2. In general, n-pulses are preferably chosen and accordingly timed with spin-transition pulses so as to achieve an equal division of the total free-evolution time in two groups of time windows associated with the respective involved first and second interacting spin states.

[0120] Further examples of double-resonance sequences may be considered. For example, a final n / 2-pulse of a Hahn spin echo type sequence may by configured to rotate around -x axis instead of the +x-axis. Such a rotation direction may be non-essential, as a frequency may still be determined from the plurality of measurement points. Furthermore, instead of using a Hahn- spin echo type sequence and a spin-transition pulse timed thereto, a nuclear spin of interest and the quantum sensor (el) may also be continuously driven by pulses so as to specifically isolate their interaction from the environment and the rest of the nuclear-spin system. E.g., a nuclear spin may be continuously rotated while the quantum sensor (el) may be continuously driven to rotate between two spin states as well and in coordination with the rotating of the nuclear spin, thereby decoupling the nuclear spin while recoupling the quantum sensor (el).

[0121] Preferably, the nuclear spin Si is polarized by polarizing the nuclear-spin system Si, . . . ,SN. For example, the single-spin quantum sensor (el) may be used to polarize the nuclear- spin system, thereby also polarizing the nuclear spin Si. Polarizing the nuclear-spin system may enhance a signal quality and may reduce a number of measurements used for averaging.

[0122] Preferably, action Al (INIT) comprises initializing the single-spin quantum sensor (el) in an interacting spin state ms, i.e., a non-zero spin state ms=£0. By initializing to an interacting spin state, nuclear-spin flip-flop interactions can be prevented. A flip-flop interaction typically occurs between nuclear spins having similar nuclear-spin frequencies. For example, when the hyperfine interaction A£due to spin-spin interactions between the quantum sensor (el) and nuclear spins is about three orders of magnitude larger than nuclear-nuclear couplings Cij, then preparing the quantum sensor (el) in an interacting spin state may prevent / suppress / freeze flipflop interactions. For example, an NV center may be initialized in one of its interacting spin states ms=+ / - 1 and not in its non-interacting spin state ms=0. A quantum sensor (el) with spin-x / i spin states is typically already in an interacting spin state, but may also be initialized to one particular interacting spin state. In a diamond including an NV center (el), initializing the spin state mse.g. to -1 or +1 may also be preferred so as to better isolate the nuclear-spin frequencies Ai, which may be dominated by their interaction with the NV center (el).

[0123] The first frequency range may comprise further nuclear-spin frequencies. For example, as illustrated in Fig. 5, area d, a first frequency range A23 may comprise nuclear-spin frequencies of two nuclear spins S2 and S3 with hyperfine shifts A2and A3, respectively; i.e., their corresponding nuclear-spin frequencies A2 and A3 may both be included in the first frequency range A23.

[0124] In such a case, action Al (INIT) may comprise polarizing both nuclear spins, e.g., nuclear spin S2 corresponding to nuclear spin Si in method (1) above and a further nuclear spin S3. In action A2 (DR), the first pulse-sequence may be further configured to decouple, in addition to the nuclear spin S2, the further nuclear spin S3 from the environment noise and from the rest of the nuclear-spin system (SI, ... ,SN). In this case, the determining may further comprise determining, based on the plurality of measurement points, a further hyperfine-shift frequency (A3) of a further spin-spin interaction between the sensor spin (el) and the further nuclear spin S3.

[0125] Fig. 10 illustrates a PSD for such a case of a further nuclear-spin frequency being involved. The PSD may reveal two hyperfine-shift frequencies corresponding to A2and A3. The solid curve in Fig. 10 is a four-frequency fit to the data. In the example data of Fig. 10, four peaks occur instead of two due to an additional coupling strength C23 between the involved nuclear spins S2 and S3. Further details related thereto are discussed in the context of method (3).

[0126] More general than illustrated in Fig. 10, if multiple nuclear spins are included in the first frequency range, then multiple corresponding shift frequencies A£can be determined.

[0127] A nuclear spin may be read out directly via the quantum sensor. Preferably, the detecting sequence used for reading out the nuclear spin includes phase-sensitive dynamical decoupling (DD) sequences or DDRF sequences as detailed in [Bradley et al, 2019],

[0128] However, reading out the nuclear spin directly may be difficult. For example, the nuclear spin may not couple sufficiently strongly to the quantum sensor, rendering direct read out with DD or DDRD difficult and / or not feasible or even impossible. The inventors found that it is still possible in such cases to read out the nuclear spin, by transferring a signal of the nuclear spin via a spin-chain sequence to another nuclear spin that in turn can be read out directly via the quantum sensor (el). Typically, the other nuclear spin that can be read out directly is located closer to the quantum sensor (el). The spin-chain sequence is described next with reference to nuclear spin Si for illustration purposes only.

[0129] Preferably, the detecting sequence in action A3 (DETECT) further comprises a spinchain sequence configured to transfer a signal of the nuclear spin Si via a k-tuple of nuclear spins (Si;Ch,...,Sk,ch) of the nuclear-spin system (Si, ..., SN) to a particular one (Si,ch) of the nuclear spins of the k-tuple of nuclear spins (Si,ch,...,Sk,ch), k > 1, wherein the sensing- sequence is configured to read out the particular nuclear spin (Si,ch) directly via the single-spin quantum sensor (el).

[0130] In this example, via the spin-chain sequence, the signal obtained on the nuclear spin (Si) via the double-resonance sequence is first mapped to a particular nuclear spin (Si,ch) before being read out directly via the quantum sensor (el). The signal of the nuclear spin Si obtained after the applying of the double-resonance sequence of action A2 (DR) may correspond to a projective spin state of the nuclear spin Si.

[0131] As illustrated in Fig. 5, area a, and Fig. 10, in case one or more further nuclear-spin frequencies are involved (i.e., multiple nuclear spins are probed by the first pulse-sequence), then a spin-chain sequence may also be used to transfer a signal obtained on a nuclear spin Si back to the quantum sensor. In such a case, the signal corresponds to a sum of polarizations carrying information about the one or more further hyperfine-shift frequencies.

[0132] Preferably, action Al (INIT) further comprises polarizing the k-tuple of nuclear spins (Si;Ch,...,Sk,ch) of the nuclear-spin system (Si, ..., SN), wherein k > 1. Alternatively and / or in addition, one or more of the k-tuple of nuclear spins (Si,ch,...,Sk,ch) may be polarized on an individual level before being involved in the spin-chain sequence. Preferably, the particular one nuclear spin (Sch,i) is reinitialized before being involved in the spin-chain sequence and before being read out directly via the single-spin quantum sensor Se. By reinitializing, a polarization of the particular one nuclear spin (Sch,i) may be maximized, thereby enhancing a signal quality.

[0133] Figs. 1 la-d illustrate spin chains according to different examples of method (1).

[0134] Fig. I la illustrates a sensing protocol as shown in Fig. 7b, in which a double-resonance sequence decouples a nuclear spin Si while recoupling the quantum sensor (el), thereby creating a signal on the nuclear spin Si that contains information about the electron-nuclear interaction between the quantum sensor (el) and the nuclear spin (Si). The signal created on the nuclear spin Si can then be read-out directly / optically via the quantum sensor (el). In this example, a spin-chain sequence may be omitted.

[0135] Fig. 1 lb illustrates a spin-chain sequence of length k=l . This example involves a nuclear spin S2 with nuclear-spin frequency A2 on which a signal is created and a 1 -tuple of a nuclear spin Si with nuclear-spin frequency Ai, which is used as a spin chain (Sch,i=Si). First, a signal is created on nuclear spin S2 by action Al including polarizing (Init) the nuclear spin S2 and action A2 including a double-resonance sequence (XXJ,) that decouples the nuclear spin S2 while recoupling the quantum sensor (el) to the nuclear spin S2. The signal obtained on S2 is then transferred, e.g. via a double-resonance sequence (XY'T), to the nuclear spin Si, which in turn can be read out (RO) directly / optically via the quantum sensor (el). Fig. 11c illustrates a spin-chain sequence of length k=2. This example involves a nuclear spin S3 with nuclear-spin frequency A3 on which a signal is created and a 2-tuple of nuclear spins Si and S2 with respective nuclear-spin frequencies Ai and A2, which are used as a spin chain (Sch,i=Si, Sch,2=S2). Similarly, first a signal is created on nuclear spin S3 by action Al polarizing (Init) the nuclear spin S3 and action A2 including a double-resonance sequence (XXJ,) that decouples the nuclear spin S3 while recoupling the quantum sensor (el) to the nuclear spin S3. The obtained signal is then transferred via the spin-chain sequence, e.g. via doubleresonance sequences (XY'T), to the nuclear spin Si, first from S3 to S2 and then from S2 to Si. The nuclear spin Si can then be read out (RO) directly / optically.

[0136] Fig. l id illustrates including a spin-chain sequence of length k=3. This example involves a nuclear spin S4 with nuclear-spin frequency A4 on which a signal is created and involving a 3-tuple of three nuclear spins Si, S2, S3 with respective nuclear-spin frequencies Ai, A2, A3, which are used as a spin-chain (Sch,i=Si, Sch.2=S2, Sch,3=S3). Similarly, first a signal is created on nuclear spin S4 by action Al polarizing (Init) the nuclear spin S4 and action A2 including a double-resonance sequence (XXJ,) that decouples the nuclear spin S4 while recoupling the quantum sensor (el) to the nuclear spin S4. The obtained signal is then transferred via the spin-chain sequence, e.g. via double-resonance sequences (XY'T), to the nuclear spin Si, first from S4 to S3, then from S3 to S2, and finally from S2 to Si. The nuclear spin Si can then be read out (RO) directly.

[0137] In general, preferably, the spin-chain sequence in action A3 (DETECT) comprises applying consecutively k double-resonance sequences (XY'T) of a chain of double-resonance sequences (XY$), the chain being configured to transfer a signal of a nuclear spin Si (e.g. Si = S2~ A2 in Fig. 1 lb; Si = S3 ~ A3 in Fig. 11c; Si = S4 ~ A4 in Fig. 1 Id) to the particular one (Si,ch) of the nuclear spins, wherein k > 1, by: applying an initial double-resonance sequence of the chain to an initial pair (Si,Sk,ch) including the nuclear spin, applying, if k > 3, each of one or more subsequent double-resonance sequences (XY$) to a respective pair (Si,ch,Si-i,ch) of the k- tuple of nuclear spins (Si,ch,...,Sk,ch), and applying a final double-resonance sequence of the chain to a final pair (S2,ch,Si,ch) including the particular one (Si,ch) of the nuclear spins, wherein if k = 1, the final pair is equal to the initial pair (Si,Sk,ch).

[0138] Double-resonance sequences (XY$) may be particularly suitable for transferring a signal from the nuclear spin Si (e.g. Si = S2 ~ A2 in Fig. 1 lb; Si = S3 ~ A3 in Fig. 11c; Si = S4 ~ A4 in Fig. l id) via a chain to the particular one (Sch,i) of the nuclear spins.

[0139] Fig. 12 illustrates a double-resonance sequence (XY'T) for transferring a signal from a nuclear spin (e.g., labelled Sj) to another nuclear spin (e.g., labelled Sj-i). In Fig. 12, the top horizontal line indicates a frequency range including a nuclear-spin frequency Aj-i of nuclear spin Sj-i. The bottom horizontal line indicates a further frequency range including a nuclear- spin frequency Aj of nuclear spin Sj.

[0140] As shown in Fig. 12, preferably the double-resonance sequence (XYp includes a Hahn spin-echo type sequence applied at a frequency range that includes the nuclear-spin frequency Aj-i of nuclear spin Sj-i, wherein the Hahn spin-echo type sequence comprises an initial ti / 2- pulse followed by a n-pulse and a final ti / 2 pulse. The initial and final ti / 2 pulses are preferably applied with a phase difference of substantially 90 degrees. By including such a phase difference, a Bloch vector of the nuclear spin Si-i is effectively rotating around different rotation axes (y-axis and x-axis). Preferably, the double-resonance sequence (XYp includes another n- pulse applied at the further frequency range including the nuclear-spin frequency Aj of nuclear spin Sj, the another n-pulse being applied synchronously with the respective n-pulse of the Hahn spin-echo type sequence. While the Hahn spin-echo type sequence decouples nuclear spin Sj-i from environment noise and the rest of the nuclear-spin system, specifically applying the n-pulses synchronously recouples nuclear spin Sj to nuclear spin Sj-i in a way that a signal is created on nuclear spin Sj-i that includes signal information of nuclear spin Sj. Thereby, a signal may be transferred from Sj to Sj-i. For example, by including the phase difference of substantially 90 degrees, a polarization value may be transferred from Sj to Sj-i .

[0141] Preferably, a free-evolution time ti,i- 1 of a double-resonance sequence (XYp of the chain involving a first nuclear spin Si and a second nuclear spin Si-i is chosen as wherein is the coupling strength between the involved first and second nuclear spins. By this choice, a signal transfer from the first to the second nuclear spin may be improved. For example, a signal amplitude may be enhanced, thereby enhancing spectral resolution.

[0142] Instead of using a spin-chain, the sensing-sequence in action A3 (DETECT) may be configured to directly read out the nuclear spin (Si), as illustrated in Fig. 7b and Fig. I la.

[0143] Method (1) and its various preferably variations described above may be applied in combination with one or more of the following methods (2)-(5), and similarly, aspects described in the context of method (1) may be combined with aspects described in the context of methods (2)-(5) in any way deemed useful, and vice versa. For example, details of chain-sequences that may be used in method (1) are further disclosed in the context of method (4) about spin chains. For example, details of combining method (1) with method (2) are further disclosed in the context of method (3) about combining electron-nuclear and nuclear-nuclear measurements and method (5) about mapping a spin network / nuclear-spin system based on methods (l)-(4). Method (2): High-resolution nuclear-nuclear double resonance sequence.

[0144] Method (2) is for measuring, via a single-spin quantum sensor (el) disposed in a host material, a coupling strength (C45) of a spin-spin interaction between a nuclear spin (S4) and a target nuclear spin (S5), the nuclear spin (S4) and the target nuclear spin (S5) being part of a nuclear- spin system (Si, ..., SN). The method comprises: action Al (INIT): polarizing the nuclear spin (S4) and a k-tuple of nuclear spins (81,82,83) of the nuclear-spin system, wherein k > 1; action A2 (DR): applying a double-resonance sequence (XX$) involving a free- evolution time (t4s) and including: a first pulse-sequence, applied within a first frequency range including a first nuclear-spin frequency (A4) of the nuclear spin (S4), and configured to decouple the nuclear spin (S4) from environment noise and from the rest of the nuclear-spin system (SI, ... ,SN), and a second pulse-sequence, applied within a second frequency range including a second nuclear-spin frequency (As) of the target nuclear spin (Ss), and configured to recouple the target nuclear spin (Ss) to the nuclear spin (S4); action A3 (DETECT): obtaining a measurement point by applying a detecting-sequence (XY'T, RO) configured to read out the nuclear spin (S4) via the single-spin quantum sensor (el), the detecting-sequence comprising a sensing-sequence applied to the single-spin quantum sensor (el); repeating the actions at least one time to obtain a plurality of measurement points by sweeping the free-evolution time (t) over a range of chosen free-evolution times, each of the plurality of measurement points being associated with one of the range of chosen free-evolution times; and determining, based on the plurality of measurement points, the coupling strength (C45), wherein the detecting sequence in action A3 (DETECT) further comprises a spin-chain sequence configured to transfer a signal of the nuclear spin (S4) via the k-tuple of the nuclear spins (81,82,83) of the nuclear-spin system (Si, ..., SN) to a particular one (Si) of the nuclear spins of the k-tuple of nuclear spins (81, 82, 83), wherein the sensing-sequence is configured to read out the particular one (Si) of the nuclear spins directly via the single-spin quantum sensor (el). Nuclear spins further away from the quantum sensor (el), labelled in an example by S4 and S5, may be challenging to access directly via the quantum sensor (el). Thereby, it can be difficult to measure a coupling strength between such nuclear spins, as a direct read out via the quantum sensor (el) may be not effective or feasible. For example, pairwise measurements alone can only provide information on a correlated list of three frequencies {Ai,Cij,Aj} when the spins involved are spectrally isolated, but provide insufficient spectral resolution for distinguishing between nuclear spins with similar nuclear frequencies. For example, nuclear spins that are further away from the NV center tend to exhibit similar nuclear frequencies.

[0145] Method (2) enables a read out of such nuclear spins via mapping a signal that is created on one of the involved spins, in an example S4, via a chain to a nuclear spin that can be read out directly via the quantum sensor (el). Thereby, method (2) generally enables measuring couplings strengths Cij between nuclear spins Si and Sj that are typically further away from the quantum sensor (el).

[0146] Figs. 13a-13g illustrate a typical scenario in which method (2) may be used to measure coupling strengths for spins typically further away from the single-spin quantum sensor (el).

[0147] Fig. 13a shows nuclear spins S1-S5 with respective nuclear-spin frequencies A1-A5 and a single-spin quantum sensor (el). Two nuclear spins Si and Si-i are coupled via a respective coupling strength Ci,i-i. In this example scenario, method (2) enables to measure the coupling strengths, wherein nuclear spin Si is used as a particular nuclear spin that can be read out directly via the quantum sensor (el).

[0148] Fig. 13b shows an example of measuring the coupling strength C12. After polarizing (Init) nuclear spin Si, a double-resonance sequence (XX$) applied to nuclear spins Si and S2 with respective nuclear-spin frequencies Ai and A2 creates / imparts a signal on Si that can be read out directly (RO) via the quantum sensor (el). For example, spin-echo double resonance (SEDOR) type sequences (XX'T) as discussed in [Abobeih et al, 2019] may be used to obtain C12 when reading out Si directly via the quantum sensor (el). A SEDOR block (XX$) is illustrated in Fig. 13b, which shows a Hahn-spin echo type sequence applied to nuclear spin Si and a timed n-pulse to nuclear spin S2, which decouples Si while recoupling S2 to Si.

[0149] Fig. 13c shows an example of measuring the coupling strength C45. In Fig. 13c, the first double-resonance sequence (XX$) imparts a signal on spin S4 that is characteristic of the coupling strength C45, as S4 is decoupled while S5 is recoupled. The signal created on S4 is then transferred to spin Si via a chain of double-resonance sequences (XYp and is then measured via the quantum sensor (el). Preferably, each double-resonance sequence (XYp in the chain includes a Hahn spinecho type sequence with an initial n / 2-pulse followed by a n-pulse and a final n / 2 pulse, wherein the initial and final n / 2 pulses are applied with a phase difference of substantially 90 degrees so as to effectively rotate a Bloch vector of the nuclear spin Si is around different rotation axes, which may be labelled y-axis and x-axis. Preferably, the Hahn-spin echo type sequence is applied to the respective nuclear spin at which a signal is to be transferred, while the double-resonance sequence (XYp includes another n-pulse applied to another nuclear spin whose signal is to be transferred, the another n-pulse being applied synchronously with the respective n-pulse of the Hahn spin-echo type sequence.

[0150] Figs. 13d-g show measurement results of method (2) applied to the example scenario.

[0151] The left graphs in Figs. 13d-g show experimental data when choosing a fixed firee- evolution time and sweeping a frequency used in a double-resonance sequence so as to sense candidates for nuclear spin frequencies Ai. For example, sweeping the second frequency range while fixing the corresponding free evolution time used in the corresponding double-resonance sequence may be used to obtain the left graph in Fig. 13d. Highlighted parts in the left graphs denote frequency ranges in which a candidate nuclear-spin frequency (A2, A3, A4, A5) is detected in a signal P[0], A candidate may be identified e.g. by detecting a drop in the signal P[0], The signal P[0] may be obtained by reading out the quantum sensor optically (el).

[0152] The right graphs in Figs. 13d-g show experimental data when sweeping the respective free evolution time tjj+i involved in the double-resonance sequence in method (2), j=l,2,3,4. The signals P[0] in the right graphs correspond to respective pluralities of measurement points obtained by the sweeping of the respective free-evolution times. E.g., in Fig. 13g, nuclear spin S5 with nuclear-spin frequency As is targeted as its coupling strength to nuclear spin S4 is measured, C45. Solid lines in the right graphs are fits to the measurement data for illustration purposes. The measured signal P[0] relating to C45 (Fig. Xg) is inverted relative to the other measured signals P[0] (Figs. 13d- 13f), which indicates a negative coupling C45.

[0153] Method (2) and its various preferably variations described above may be applied in combination with one or more of methods (1) and (3)-(5), and similarly, aspects described in the context of method (2) may be combined with aspects described in the context of methods (1) and (3)-(5) in any way deemed useful, and vice versa.

[0154] For example, a spin state (ms) of the quantum sensor (el) may be initialized to an interacting spins state so as to supress nuclear-nuclear flip-flop interactions, as detailed further above. As another example, preferably a free-evolution time ti,i-i of a double-resonance sequence (XYp of the chain involving a first nuclear spin Si and a second nuclear spin Si-i is chosen asi-1= Ciyil1 / 2, wherein is the coupling strength between the involved first and second nuclear spins. Furthermore, for the determining, one may obtain multiple measurement points for each chosen time point in the range of free-evolution times and average the multiple measurement points so as to obtain an average value associate with each time point.

[0155] For example, starting from measuring C12, one may iteratively expand chain sequences to unlock / measure more nuclear spins and coupling strengths. Each expanded chain sequence may include measurement information obtained from previous measurements, such as obtained coupling strength(s). Examples of iteratively exploring a nuclear-spin system are detailed further below in the context of method (5).

[0156] Another method may be for measuring, via a single-spin quantum sensor (el) disposed in a host material and via a nuclear spin (S4), a target nuclear-spin frequency (A5) of a target nuclear spin (S5), the nuclear spin (S4) and the target nuclear spin (S5) being part of a nuclear- spin system (Si, ..., SN). The method comprises: action Al (INIT): polarizing the nuclear spin (S4) and a k-tuple of nuclear spins (81,82,83) of the nuclear-spin system, wherein k > 1; action A2 (DR): applying a double-resonance sequence (XX$) involving a free- evolution time (t4s) and including: a first pulse-sequence, applied within a first frequency range including a first nuclear-spin frequency (A4) of the nuclear spin (S4), and configured to decouple the nuclear spin (S4) from environment noise and from the rest of the nuclear-spin system (SI, ... ,SN), and a second pulse-sequence, applied within a second frequency range (RF5) including the target nuclear-spin frequency (As) of the target nuclear spin (Ss), and configured to recouple the target nuclear spin (Ss) to the nuclear spin (S4); action A3 (DETECT): obtaining a measurement point by applying a detecting-sequence (XY'T, RO) configured to read out the nuclear spin (S4) via the single-spin quantum sensor (el), the detecting-sequence comprising a sensing-sequence applied to the single-spin quantum sensor (el); repeating the actions at least one time to obtain a plurality of measurement points by sweeping the second frequency range (RFs) over a range of chosen second frequency ranges, each of the plurality of measurement points being associated with one of the range of chosen second frequency ranges; and determining, based on the plurality of measurement points, the target nuclear-spin frequency (As), wherein the detecting sequence in action A3 (DETECT) further comprises a spin-chain sequence configured to transfer a signal of the nuclear spin (S4) via the k-tuple of the nuclear spins (81,82,83) of the nuclear-spin system (Si, ..., SN) to a particular one (Si) of the nuclear spins of the k-tuple of nuclear spins (81,82,83), wherein the sensing-sequence is configured to read out the particular one (Si) of the nuclear spins directly via the single-spin quantum sensor (el).

[0157] This method is illustrated in the left graph of Fig. 13g with a free-evolution time t45=50ms and sweeping the second frequency range (RF5) over a range of ~428-442kHz. This method enables finding candidates for nuclear-spin frequencies such as As. After finding a suitable candidate, method (2) may for example be employed to measure a coupling strength involving the found candidate nuclear-spin frequency.

[0158] A preferred combination of methods (1) and (2) is discussed next.

[0159] Method (3): Combination of electron-nuclear and nuclear-nuclear double resonance sequences.

[0160] Method (3) is for measuring, via a single-spin quantum sensor (el) disposed in a host material, at least one of one or more hyperfine-shift frequencies (A2, A3) related to respective one or more nuclear spins (S2, S3) and / or one or more coupling strengths (C24.C34) related to spin-spin interactions between the respective one or more nuclear spins (82,83) and a further nuclear spin (S4), the one or more nuclear spins (82,83) and the further nuclear spin (S4) being part of a nuclear-spin system (SI, ... ,SN). The method is illustrated in an example situation in Fig. 14. The method comprises: action Al (INIT): polarizing the one or more nuclear spins (82,83); action A2 (DR): applying a first double-resonance sequence (XXJ,) involving a first free-evolution time (ti) and including: a first pulse-sequence, applied at a first frequency-range including one or more nuclear-spin frequencies (A2, A3A23) corresponding to the respective one or more nuclear spins (82,83), and configured to decouple the respective one or more nuclear spins (82,83) from environment noise and from the rest of the interacting nuclear-spin system (Si, . . . ,SN); a second pulse-sequence, applied within a second frequency range including a spin-transition frequency of the single-spin quantum sensor (el), and configured to change a spin state (ms) of the single-spin quantum sensor (el) so as to recouple the single-spin quantum sensor (el) to the respective one or more nuclear spins (82,83) (in other words, to prevent a decoupling of the one or more nuclear spins from the single-spin quantum sensor); and applying a second double-resonance sequence (XX$) involving a second firee- evolution time (t?) and including: a third pulse-sequence, applied within the first frequency range, and configured to decouple the respective one or more nuclear spins (82,83) from environment noise and from the rest of the interacting nuclear-spin system (Si,. . . ,SN); a fourth pulse-sequence, applied within a third frequency range including a further nuclear-spin frequency (A4) of the further nuclear spin (S4), and configured to recouple the further nuclear spin (S4) to the respective one or more nuclear spins (82,83) (in other words, to prevent a decoupling of the one or more nuclear spins from the further nuclear spin); action A3 (DETECT): obtaining a measurement point by applying a detectingsequence (XY'T, RO) configured to read out the respective one or more nuclear spins (82,83) via the single-spin quantum sensor (el), the detecting-sequence comprising a sensingsequence applied to the single-spin quantum sensor (el); repeating the actions at least one time to obtain a plurality of measurement points by sweeping the involved first and second free-evolution times (ti,t2) over a domain of chosen free- evolution times, each of the plurality of measurement points being associated with one of the domain of chosen free-evolution times; and determining, based on the plurality of measurement points, at least one of the one or more hyperfine-shift frequencies (A2,A3) and / or the one or more coupling strengths (C24.C34).

[0161] By applying both the first and second double-resonance sequences in action A2 (DR), the one or more nuclear spins (e.g., 82,83) carry one or more respective signals that capture information about their respective one or more hyperfine shifts (e.g., A2, A3) due to the quantum sensor (el) and about their respective couplings (e.g., C24.C34) to the further nuclear spin (e.g., S4). The quantum sensor (el) is recoupled in the first double-resonance sequence, while the further nuclear spin (S4) is recoupled in the second double-resonance sequence. For example, the applying of the first and second double-resonance sequences can create one or more signals on the respective one or more nuclear spins (e.g., 82,83), the one or more signals comprising at least one of the involved one or more respective free-evolution times (ti,t2), the one or more hyperfine-shift frequencies (e.g. A2,A3) and / or the one or more coupling strengths (e.g., C24.C34).

[0162] The plurality of measurement points, obtained by sweeping both free-evolution times ti and t2 involved in the respective double-resonance sequences, can therefore capture sufficient information that allows to determine at least one of the one or more hyperfine-shift frequencies (e.g., A2,A3) and / or the one or more coupling strengths (e.g., C24.C34).

[0163] The order of applying the first and second double-resonance sequences is not fixed to the order of their mentioning. E.g., the second double-resonance sequence may be applied prior to the first double-resonance sequence. The adjectives “first” and “second” are merely names given to the respective double-resonance sequences so as to distinguish between them. Applying both the first and second double-resonance sequences correlates hyperfine shifts with nuclear-nuclear couplings.

[0164] In general, the one or more nuclear spins (e.g., 82,83) can be read out directly via the quantum sensor (el), or can be read out by first transferring a signal of the one or more nuclear spins (e.g., 82,83) via a spin-chain sequence to another nuclear spin which in turn can be read out directly via the quantum sensor (el). Including a spin-chain sequence is described above in the context of methods (1) and (2) but applies similarly to method (3), which is not repeated in detail for brevity of the description.

[0165] Fig. 14a illustrates an example situation in which method (3) may be employed. In this example situation, two nuclear spins labelled S2 and S3 have similar nuclear-spin frequencies A2 and A3 that are both within a relatively narrow frequency range A23, due to slightly different hyperfine shifts A2and A3, respectively. Both nuclear spins S2 and S3 are coupled to a further nuclear spin S4 via respective coupling strengths C24 and C34, respectively. Nuclear spins S2 and S3 are coupled to each other with a coupling strength C23 and couple to a nuclear spin Si with a coupling strength C~Ci,23.

[0166] Such a situation in which two nuclear spins have similar nuclear-spin frequencies may be discovered in different ways. For example, as illustrated in Fig. 14b, a double-resonance involving a nuclear-spin frequency Ai of a directly accessible nuclear spin Si and the frequency range A23 may be employed, and a coupling strength C=Ci,23 may be obtained by sweeping a respective involved free evolution time ti,23, which may result in a relatively quick decay in the obtained signal P(0). The relatively quick decay may indicate that two nuclear spins have spin frequencies in A23that cannot be spectrally distinguished by state-of-the-art methods. As a further example, as shown in Fig. 10, employing method (1) may reveal hyperfine shift frequencies Δ^and Δ^, thereby providing a measurement result that consistent with two nuclear spins. At the example of Fig.10, the respective frequencies were measured to be Δ^= 8019.5(2) Hz and Δ^= 7695.2(1) Hz, split by an internal coupling strength of C23= 7.6(1) Hz. Fig. 14c illustrates actions A1-A3 of method (3) for the example situation. The illustrated pulse sequence combines electron-nuclear resonance (first double-resonance sequence, XX↓ block in Fig. 14c) and nuclear-nuclear double resonance (second double- resonance sequence, XX↑ in Fig.14c). Thereby, signals can be created on nuclear spins S2and S3, which in the example are transferred via a 1-chain to nuclear spin S1. The nuclear spin S1 can then be read out (RO) directly via the quantum sensor (el). Transferring via spin chains is described in the context of methods (1) and (2) and can be applied similarly. A detailed description of the spin-chain is therefore not repeated. A plurality of measurement points can be obtained by sweeping the involved first and second free-evolution times (t1,t2) over a domain of chosen free-evolution times, each of the plurality of measurement points being associated with one of the domain of chosen free- evolution times. For example, the domain may include first and second ranges of free-evolution times. For example, for every t1 point ranging e.g. from 1-300 ms, 20 t2 points are acquired, ranging from 10 to 260 ms. For each chosen 2d time point in the domain, one may obtain multiple measurement points and average the multiple measurement points so as to obtain an average value associate with each 2d time point. Other examples may be considered, such as acquiring 10, 15, 25, 30 t2points for every t1point. Furthermore, the t1range may vary as well. How many measurement points are obtained may vary depending on the context and on factors such as the SNR. Preferably, the first range includes a plurality of first free-evolution times (t1) and the second range includes a plurality of second free-evolution times (t2). Preferably, the obtaining of the plurality of measurement points involves obtaining one or more samples for each point (t1,t2) of the domain and averaging over the obtained one or more samples so a to obtain an averaged measurement point associated with each point (t1,t2). Fig.14e shows an example of determining hyperfine frequency shifts from an obtained plurality of measurement points. Fig. 14e shows a PSD signal obtained via method (3), thus including two frequency domains which are labelled by f1 and f2, respectively. The measurement data depicted in Fig. 14e shows two distinct frequency regions (indicated by dotted lines) along the f1-axis, corresponding to Δ2=8019.5(2) Hz and Δ3=7695.2(1) Hz, each being split by an internal coupling strength of C23=7.6(1) Hz. Fig. 14f shows an example of determining the coupling strengths C24 and C34 from an obtained plurality of measurement points. Fig. 14f shows fz signals at the two frequency regions obtained as a binned line cut of the frequency data shown in Fig. 14e along the t'2-axis at frequencies A2 and A3. Positions of the averaged peaks indicate couplings to the nuclear spin S4. E.g., the data in the left graph indicates a coupling strength of C34=-0.2(5) Hz, and the data in the right graph indicates a coupling strength of C24=-l 1.8(2) Hz. In both graphs, split peaks are observed due to the internal coupling strength C23. Solid lines in Fig. 14f are each a fit of two Gaussians that can be used to extract the respective coupling strengths.

[0167] Figs. 15a-h shows a comparison between experimental and numerical data relating to methods (1) and (3), wherein Figs. 15a-d correspond to the experimental data and Figs. 15e-h correspond to the numerical data. The numerical data is generated by evaluating the Hamiltonian taking into account two spins at A23, a spin at Ai and the single-spin quantum sensor (el). The numerical data shows agreement with the experimental data, reconfirming that the methods presented herein can provide consistent spectroscopy data.

[0168] Fig. 15a shows time domain data corresponding to an experiment involving method (1) or method (3) when targeting two nuclear spins with frequencies in A23. Fig. 15b shows the corresponding PSD, similar as shown in Fig. 10. Fig. 15c shows 2d measurement data obtained via method (3) when sweeping ti and t2 over a 2d time domain. Fig. 15d shows the corresponding 2d Fourier transform / PSD. Fig. 15e illustrates a beating signal / an interference pattern indicative of two nuclear spins being targeted, in this case simulated via the Hamiltonian. Fig. 15f shows the corresponding PSD / Fourier transform. Figs. 15g-h show 2d time and frequency data numerically simulated to validate method (3). As can be appreciated by comparing, the numerical data shown in Figs. 15g-h validates the experimental data shown in Figs. 15c-d.

[0169] Hence, as illustrated in Figs. 14-15, by adding the second double-resonance sequence (e.g., a nuclear-nuclear block XX $ as illustrated in Fig. 14c) and sweeping both free-evolution times ti and t2, a correlation between the involved hyperfine shifts Ai and spin-spin couplings Cij can be revealed.

[0170] Method (4): Spin-chains.

[0171] Method (4) is for transferring, via a spin-chain sequence involving a k-tuple of nuclear spins (81,82,83), a signal of the nuclear spin (S4) via the k-tuple of nuclear spins (81,82,83) to a particular one (Si) of the nuclear spins of the k-tuple of nuclear spins (81,82,83), the nuclear spin (S4) and the k-tuple of nuclear spins (81,82,83) being part of a nuclear-spin system (Si, . . ., SN), wherein k > 1. The method comprises: polarizing the k-tuple of nuclear spins (81, 82, 83); applying consecutively k double-resonance sequences (XY'T) of a chain of doubleresonance sequences (XY'T), the chain being configured to transfer the signal of the nuclear spin (S4) to the particular one (Si) of the nuclear spins (81, 82, 83), by: applying an initial double-resonance sequence (XY'T) of the chain to an initial pair (84,83) including the nuclear spin (S4), applying, if k > 3, each of one or more subsequent double-resonance sequences (XY'T) to a respective pair (83, 82; 82,81) of the k-tuple of nuclear spins (81, 82, 83), and applying a final double-resonance sequence (XY$) of the chain to a final pair (82,81) including the particular one (Si) of the nuclear spins, wherein if k = 1, the final pair is equal to the initial pair.

[0172] As demonstrated in methods (l)-(3), a spin chain can be used in various situations so as to transfer a signal of one nuclear spin to another nuclear spin in the nuclear-spin system. In methods (l)-(3), spin chains may be included so as to transfer a signal to a nuclear spin that can more easily be directly accessed by the quantum sensor (el).

[0173] However, spin-chain sequences are not limited to the context of characterizing / mapping a spin network and extracting connectivity information about the spin network. For example, spin-chain sequences can enable read out of nuclear spins typically located further away from the quantum sensor (el) also in other contexts such as quantum computing. For example, a diamond with an NV center (el) surrounded by13C nuclei can be used as a quantum-computing device in which nuclear spins Si corresponding to the13C nuclei may function as a quantum register, and nuclear spins Si may provide qubits for quantum memory and can be used in a quantum computations and / or quantum simulations. Examples of extracting connectivity information is discussed further below in the context of method (5) about efficiently combining methods (l)-(4).

[0174] Method (4) may include aspects of spin-chains as disclosed with reference to methods (l)-(3). These aspects are not repeated here, but examples are provided below.

[0175] Preferably, a double-resonance sequence (XY$) of the chain comprises a Hahn spinecho type sequence including an initial n / 2 pulse and a final n / 2 pulse that are applied with a relative phase difference of substantially 90 degrees, the Hahn spin-echo type sequence being applied at a frequency range including a nuclear-spin frequency of one nuclear spin (Si-i,ch) on which a signal is transferred.

[0176] Preferably, for one or more double-resonance sequences (XYp of the chain, involving a respective first nuclear-spin frequency (Ai) and a respective second nuclear-spin frequency (Am) and a respective free-evolution time ti,i-i, the respective free-evolution time ti,i-i is chosen as wherein is a respective coupling strength corresponding to the involved respective first and second nuclear-spin frequencies (Ai,Ai-i).

[0177] Preferably, each one of the k double-resonance sequences (XYp of the chain, acting on a corresponding pair of nuclear spins (Sj,ch,Sj-i,ch) comprising a first nuclear spin (Sj,ch) and a second nuclear spin (Sj-i,ch), is configured to transfer a signal from the first nuclear spin (Sj,ch) to the second nuclear spin (Sj-i,ch).

[0178] As illustrated in Fig. 5, area a, and Fig. 10, in case one or more further nuclear-spin frequencies are involved (i.e., multiple nuclear spins are probed by the first pulse-sequence), then a spin-chain sequence may also be used to transfer a signal obtained on a nuclear spin back to the quantum sensor. In such a case, the signal corresponds to a sum of polarizations carrying information about the one or more further hyperfine-shift frequencies.

[0179] Spin-chain sequences can provide connectivity information, as discussed below in the context of method (5).

[0180] Method (5): 3 D-r econstruction of a nuclear spin system.

[0181] Method (5) is for imaging a nuclear-spin system (SI, ... ,SN). The method comprises: performing a plurality of measurements based on at least one of any one of the preceding methods (l)-(4), the performing including obtaining at least one of one or more hyperfine-shift frequencies (Ai Aj) related to respective one or more nuclear spins (Si, Sj) and a single-spin quantum sensor (el) disposed in a host material, one or more coupling strengths (Cij) related to spin-spin interactions between the respective one or more nuclear spins (Si,Sj) and / or one or more nuclear-spin frequencies (Ai,Aj) of the respective one or more nuclear spins (Si,Sj), the one or more nuclear spins (Si,Sj) being part of a nuclear-spin system (S- I, . . . ,SN); and constructing an image of the nuclear-spin system (SI, ... ,SN), based on the plurality of measurements. The constructing of an image of the nuclear-spin system can be effectively based on measurements performed via at least one of methods (l)-(4), including one or more aspects described in the context of methods ( 1 )-(4).

[0182] For example, methods (l)-(4) may reveal more information regarding connectivity than state-of-the-art methods. At the example of method (2), as illustrated in Fig. 5, area b, two nuclear spins may have a similar nuclear-spin frequency ~A2, so that pairwise measurements for obtaining lists (Ai,Ci2,A2), (A2,C23,As) and (A2,C24,A4) may not be sufficient to determine which spins in A2 the respective spins in Ai, A3 and A4 couple to.

[0183] However, with method (2) including a spin-chain, correlated lists may be obtained to determine the connectivity. For example, measurements with method (2) are suitable for obtaining a first correlated list (Ai,Ci2,A2,C24,A4) and a second correlated list (Ai,Ci2,A2,C23,A3). The first correlated list may be obtained via a first measurement using method (2), with a first double-resonance sequence involving nuclear-spin frequencies A2 and A4 to create a signal on a nuclear spin in A2, and a chain configured to transfer a signal from A2 to Ai. The second correlated list may be obtained via second measurement using method (2), with a first double-resonance sequence involving nuclear-spin frequencies A2 and A3 to create a signal on a nuclear spin in A2, and a chain configured to transfer a signal from A2 to Ai. The first measurement may reveal no correlation while the second measurement may well reveal a correlation. E.g., the first measurement may lead to obtaining no signal e.g. by the plurality of obtained measurement points showing no correlation. The first measurement result may thus indicate that a nuclear spin at A4 is not coupling to a nuclear spin at A2 which connects to Ai, but to a different nuclear spin at A2. E.g., the second measurement may instead well lead to obtaining a signal, thereby indicating that a nuclear spin at A3 connects to a nuclear spin at A2 that does connect to Ai. In this example connectivity information is obtained by involving spinchain measurements according to method (2). Similarly, connectivity information may also be obtained by involving spin-chain measurements in methods (1) and (3).

[0184] Methods (l)-(4) may be applied in combination and iteratively so as to unravel an underlying network connectivity step by step. A method for reconstructing the network from such measurements may thus include iteratively exploring the network by expanding spin chains, executing sensing sequences, and / or adding newly detected spins to a graph of the network. The reconstructing may include at least one of distinguishing nuclear spins by their high-resolution / hyperfine-shift frequency, distinguishing nuclear spins by the spin-chain to which they are connected, distinguishing nuclear spins via their computed spatial positions ([Abobeih et al, 2019]). By systematically exploring the spin network, parameters of a Hamiltonian, such as coupling strengths Cij, corresponding to the spin network may be efficiently obtained.

[0185] For example, Figs. 1 la-1 Id illustrate that a first measurement involving a first frequency range including a nuclear-spin frequency Ai may be suitable for measuring a hyperfine-shift frequency Ai, thereby determining the nuclear-spin frequency Ai of a nuclear spin Si with higher resolution. Then, said measured nuclear spin Si may be used in a 1 -chain as shown in Fig. 1 lb to measure a further hyperfine shift frequency A2 of a further nuclear spin S2, so that a corresponding further nuclear-spin frequency A2 may be obtained. As shown in Figs, l lc-l ld, nuclear spins that have been unlocked / measured already may participate in a spin chain so as to enable measuring yet further nuclear spins.

[0186] For example, as shown in Figs. 13a-g, nuclear-spin frequencies Ai and couplings strengths Cij may be iteratively unlocked / measured. A first measurement using method (1) or a state-of-the-art method may be performed to determine Ai and A2, e.g. via sweeping a nuclear- spin frequency used in a double-resonance sequence to obtain data as shown in the left graph in Fig. 13d. A second measurement may be used to determine C12, e.g. via using a SEDOR and sweeping an involved free-evolution time ti2, as shown in the right graph in Fig. 13d. Another SEDOR involving A2 as a probe nuclear spin can be used to obtain a nuclear-spin frequency A3, e.g. by fixing a free-evolution time t23 (e.g. to 50ms) sweeping the second frequency range to obtain measurement data as shown in the left graph in Fig. 13e. Then using the obtained nuclear-spin frequency A3, a coupling strength C23 may be measured via method (2) involving sweeping the free-evolution time t23. The procedure may be continued as shown in Figs. 13f-g.

[0187] By iteratively measuring out components and the connectivity of the nuclear-spin system using methods (l)-(4) (e.g., electron-nuclear measurements, nuclear-nuclear measurements, combinations of electron-nuclear and nuclear-nuclear measurements, combinations involving spin chains, etc.), a graph may be obtained based on obtained measurement results. Method (5) can be used for constructing for example an image / a graph model showing a spatial relation between nuclear spins, or an image / a graph model showing a frequency and / or coupling relation between nuclear spins.

[0188] Figs. 16a-16b show such a graph obtained based on measurement results involving methods (l)-(4). The graph shows a nuclear-spin system (spin network) in terms of coupling relations and nuclear-spin frequencies.

[0189] Fig. 16a(l) illustrates that separate high-resolution chains (correlated lists) can be obtained via methods (l)-(4) to obtain chains through the spin network. Fig. 16a(2) illustrates merging chains that share a common section of the network. If two chains share a common section, they can be merged together to create a larger map. Fig. 16a(3) illustrates that optionally, an algorithm as introduced e.g. in [Abobeih et al, 2019] can be used to estimate most likely nuclear-spin positions. The algorithm may predict unmeasured nuclear-nuclear couplings (e.g., dotted lines in Fig. 16a(3)) and may provide a validation for the assignment and merging shown in Fig. 16a(2). Fig. 16b shows a graph of a 50-spin network surrounding an NV center. Edges in the graph indicate spin-spin interactions above 2 Hz and vertices denote spin frequencies. The graph / image is constructed based on measurements involving methods (1)- (4). Limited sized chains may be sufficient because couplings may be highly non-uniform, so that a few overlapping vertices and edges enable fusing chains with high confidence.

[0190] Fig. 17a shows an image of a spatial distribution of nuclear spins surround an NV center that was used as a quantum sensor (el). The origin denotes the NV center, bullets denote nuclear spins. Coupling strengths larger than 3 Hz are visualised by connections / links. The spatial distribution shown in Fig. 17a may be reconstructed from measurement data via a positioning algorithm as shown e.g. in [Abobeih et al, 2019], However, as compared to [Abobeih et al, 2019], methods according to the present invention provide additional information on the connectivity and coupling signs, thereby reducing the complexity of the numerical reconstruction. Additionally, positions of nuclear spins can be further constrained using the measured hyperfine shifts Ai. Since the problem of reconstructions may be highly overdetermined, the fact that a spatial solution is found that closely matches the measured frequencies and assignments validates the obtained network connectivity. Additionally, the reconstruction yields a spatial image of the spin network and predicts the remaining unmeasured 976 spin-spin interactions, most of which were weak (< 1Hz). Fig. 17b shows a measured coupling matrix including the coupling strengths Cij, showing absolute value of coupling strengths, |Cij | . Matrix elements that were not measured are left non-highlighted. Fig. 17c shows for comparison a predicted coupling matrix.

[0191] The below table shows sequences used to obtain nuclear-spin frequencies Ai of nuclear spins Si (Ci in the table under “Paper label”) with i=l, .. .,50. Column “initialisation” indicates which type of sequences were used to polarize nuclear spins. In this example, PulsePol and SWAP sequences were used. Column “Readout” indicates whether a nuclear spin was read out directly via the quantum sensor (el), indicated by “Direct”, or whether a signal on a nuclear spin was first transferred via a spin chain of length k to another nuclear spin and then read out directly, indicated by “Chain-k” with k=l, 2, 3. Examples of “Direct” and “Chain-k” (k=l,2,3) are shown in Figs. 1 la-1 Id. Colum “A” shows obtained nuclear-spin frequencies Ai and column “Ai” shows obtained hyperfine-shift frequencies Ai, assigned to associated nuclear spins Si. Columns “x”, “y”, “z” show spatial coordinates of nuclear spins Relative to one nuclear spin (C1). The relative spatial positions are illustrated in Fig.17a. A particularly effective way of constructing an image of the nuclear-spin system is described in the following passages, in which a network mapping procedure (network reconstruction algorithm) more detailed than method (5), as disclosed above, is outlined. The inventors have identified that mapping a nuclear-spin system can resemble a graph search, wherein the quantum sensor (el) is used as a root. Methods of graph theory (e.g., [Gross et al, 2004]) may thus be used when constructing the image based on measurements involving methods (1)-(4). Below, an imaging protocol based on a breadth-first like search is shown. The imaging protocol may yield a spanning tree as output, and enables completely characterising a network corresponding to the nuclear-spin system. The following pseudo-code may describe aspects of the imaging protocol: Thus, new vertices that are detected by chained measurements can be iteratively added, once it has been verified that a vertex was not characterised before (e.g., when the vertex has a duplicate in the spanning tree T). The function MeasureCoupling(v, f) performs a spin-echo double resonance sequence between vertex v and a frequency f, (a spin chain of length I – 1 may be used to access v) and checks whether a single, resolvable coupling is present. The check may be stored in the Boolean variable ‘singlecoupling’. In the case that v is the quantum sensor (el), an electron-nuclear DD(RF) sequence is performed. The function CheckVertex(w, T) instructs performing a number of spin-chain and electron-nuclear double resonance measurements, comparing the vertex w and its position in the network with that of the (possibly duplicate) vertex k. An example of the function CheckVertex(w, T) is discussed further below.

[0192] If one of these measurements is not consistent with previously obtained data relating to k, w may be concluded to be a unique vertex and may be added to the spanning tree T. If all measurements coincide with previously obtained data relating to k, one may conclude it is the same vertex and merge w and k. If the CheckVertex(w, T) is inconclusive (e.g. due to limited measurement resolution), one may not add w to the spanning tree T.

[0193] The imaging protocol outlined above is platform-independent, but can be complemented by logic based on the 3D spatial structure of the system. For example, when the CheckVertex(w, T) function is inconclusive, one can sometimes still conclude that w is unique (or vice versa equal to k), based on the restricted number of possible physical positions of these two spins in 3D space. One may also alternate the graph search procedure with calls to a positioning algorithm, which continuously checks whether the spanning tree T is physical and may aide in the identification of possible duplicates. Further aspects related to graph construction may also be found in [Abobeih et al, 2019],

[0194] Next, an example of the function Check Vertex(w, T) is discussed. The following is a pseudo-code for a procedure of checking whether vertex w has already been mapped in the spanning tree T.

[0195] The function CheckVertex(w, T) starts by comparing a (^∗ ^-limited) frequency Awof vertex w to frequencies of all mapped vertices in T. If the frequencies coincide within the measurement uncertainty (e.g., σ^∼ 100 Hz) for a vertex k, a procedure is initiated to check whether w is that vertex. The procedure may be executed by comparing information already obtained on vertex k and its surroundings to specific measurements taken from w. In particular, one may measure the spanning tree Twwith root w, and compare it to the spanning tree Tkwith root k (for some maximum depth). If all couplings (Cij, up to some threshold ‘equaledges’) and hyperfine frequency shifts (Δi) of the two spanning trees are the same, it may be concluded that w and k are the same vertex. Measuring a single deviation can mean that w is unique. If the procedure is inconclusive, for example because the uncertainty of all measurements is large, ‘duplicate’ remains None.

[0196] In any of the above-discussed aspects in methods (1 )-(5), the single-spin quantum sensor (el), the host material (1), and / or the nuclear-spin system may be as follows relating to a NV center in diamond.

[0197] Preferably, the host material (1) is a diamond.

[0198] Preferably, the single-spin quantum sensor (el) is a nitrogen-vacancy, NV, center of the diamond, the NV center exhibiting three spin states (ms=0, ms ±= ±1).

[0199] Preferably, action Al (INIT) in one any of the above-discussed methods further comprises initializing the spin state (ms) of the NV center to one of its two non-zero spin states (ms=+l; ms=— 1). By initializing to one such spin state, nuclear-nuclear flip-flop interactions can be supressed.

[0200] Preferably, the nuclear-spin system (Si, ..., SN) is exposed to an external magnetic field (B) that is applied along a symmetry axis (z-axis) of the NV center. By the external magnetic field, energy levels of the spin states ms=+ / - 1 split and spin transitions can be controlled via MW pulses.

[0201] Preferably, the nuclear-spin system (Si, ..., SN) corresponds to carbon nuclei (13C) surrounding the NV center.

[0202] Preferably, the nuclear-spin system (Si, . . ., SN) is part of a target sample positioned in proximity to the single-spin quantum sensor (el).

[0203] Preferably, pulse-sequences for manipulating nuclear spins comprise radiofrequency, RF, pulses, and the pulse-sequences for manipulating spin states of the NV center comprise microwave, MW, pulses.

[0204] List of references

[0205] [Abobeih et al, 2019] Abobeih, M. H. et al\ Atomic-scale imaging of a 27-nuclear-spin cluster using a quantum sensor. Nature 576, 411-415 (2019).

[0206] [Barry et al, 2020] Sensitivity optimization for NV-diamond magnetometry. John F. Barry, et al. Rev. Mod. Phys. 92, 015004 (2020).

[0207] [Boss et al, 2017] Boss, J. M., Cujia, K. S., Zopes, J. & Degen, C. L. Quantum sensing with arbitrary frequency resolution. Science 356, 837-840 (2017).

[0208] [Bourassa el al, 2020] Bourassa, A. et al. Entanglement and control of single nuclear spins in isotopically engineered silicon carbide. Nature Materials 19, 1319-1325 (2020). [Bradley et al, 2019] Bradley, C. et al. A Ten-Qubit Solid-State Spin Register with Quantum Memory up to One Minute. Phys. Rev. X 9, 031045 (2019).

[0209] [Durand et al, 2021] Durand, A. et al. Broad Diversity of Near-Infrared Single-Photon Emitters in Silicon. Phys. Rev. Lett. 126, 083602 (2021).

[0210] [Gao et al, 2022] Gao, X. et al. Nuclear spin polarization and control in hexagonal boron nitride. Nat. Mater. 21, 1024-1028 (2022).

[0211] [Gross et al, 2004] Gross, J. L. & Yellen, J. (eds.) Handbook of Graph Theory. Discrete Mathematics and Its Applications (CRC Press, Boca Raton, 2004).

[0212] [Lukin et al, 2020] Lukin, D. M., Guidry, M. A. & Vuckovic, J. Integrated Quantum Photonics with Silicon Carbide: Challenges and Prospects. PRX Quantum 1, 020102 (2020).

[0213] [Nagy et al, 2019] Nagy, R. et al. High-fidelity spin and optical control of single silicon- vacancy centres in silicon carbide. Nature Communications 10, 1954 (2019).

[0214] [Randall et al, 2021] Randall, J. et al. Many-body-localized discrete time crystal with a programmable spin-based quantum simulator. Science 374, 1474-1478 (2021).

[0215] [Robledo et al, 2011] Robledo, L. et al. High-fidelity projective read-out of a solid-state spin quantum register. Nature 477, 574-578 (2011).

[0216] [Schwartz et al, 2018] Schwartz, I. etal. Robust optical polarization of nuclear spin baths using Hamiltonian engineering of nitrogen-vacancy center quantum dynamics. Sci. Adv. 4, eaat8978 (2018).

Claims

What is claimed is:

1. A method of measuring, via a single-spin quantum sensor (el) disposed in a host material (1), a hyperfine-shift frequency (Ax) related to a spin-spin interaction between the single-spin quantum sensor (el) and a nuclear spin (Si), the nuclear spin (Si) being part of a nuclear-spin system (Si,. . . ,SN), the method comprising: action Al (INIT): polarizing the nuclear spin (Si); action A2 (DR): applying a double-resonance sequence (XXJ,) involving a free- evolution time (t) and including: a first pulse-sequence, applied within a first frequency range including a nuclear- spin frequency (Ai) of the nuclear spin (Si), and configured to decouple the nuclear spin (Si) from environment noise and from the rest of the nuclear-spin system (SI, ... ,SN), and a second pulse-sequence, applied within a second frequency range including a spin-transition frequency of the single-spin quantum sensor (el), and configured to change a spin state (ms) of the single-spin quantum sensor (el) so as to recouple the single-spin quantum sensor (el) to the nuclear spin (Si); action A3 (DETECT): obtaining a measurement point by applying a detecting-sequence (XY'T, RO) configured to read out the nuclear spin (Si) via the single-spin quantum sensor (el), the detecting-sequence comprising a sensing-sequence applied to the single-spin quantum sensor (el); repeating the actions at least one time to obtain a plurality of measurement points by sweeping the free-evolution time (t) over a range of chosen free-evolution times, each of the plurality of measurement points being associated with one of the range of chosen free-evolution times; and determining, based on the plurality of measurement points, the hyperfine-shift frequency (Ax).

2. The method of claim 1, wherein the first pulse-sequence is a Hahn spin-echo type sequence comprising a n-pulse separating a first free-evolution time window and a second free-evolution time window, wherein the second pulse-sequence comprises one or more spin-transition pulses applied synchronously with the n-pulse so as to change the spin state (ms) of the single-spin quantum sensor (el) to be different during the first and second free-evolution time windows.

3. The method of claim 1, wherein the first pulse-sequence is a Hahn spin-echo type sequence comprising one or more n-pulses, wherein the second pulse-sequence comprises one or more spin-transition pulses applied synchronously with the one or more n-pulses so as to change the spin state (ms) of the single-spin quantum sensor (el) to flip back and fourth between two spin states and to be different in any two consecutive free-evolution time windows.

4. The method of any one of the preceding claims, wherein the detecting sequence includes at least one of phase-sensitive dynamical decoupling, DD, sequences or DDRF sequences.

5. The method of any one of the preceding claims, wherein action Al (INIT) comprises initializing the single-spin quantum sensor (el) to an interacting spin state (ms=£0).

6. The method of any one of the preceding claims, wherein action Al (INIT) further comprises polarizing a further nuclear spin (S3) of the nuclear-spin system (Si, ..., SN); wherein in action A2 (DR), the first frequency range of the first pulse-sequence further includes a further nuclear-spin frequency (A3) of the further nuclear spin (S3), the first pulsesequence being further configured to decouple the further nuclear spin (S3) from the environment noise and from the rest of the nuclear-spin system (Si,. . . ,SN), wherein the determining further comprises determining, based on the plurality of measurement points, a further hyperfine-shift frequency (A3) of a further spin-spin interaction between the sensor spin (el) and the further nuclear spin (S3).

7. The method of any one of the preceding claims, wherein the detecting sequence in action A3 (DETECT) further comprises a spin-chain sequence configured to transfer a signal of the nuclear spin (Si) via a k-tuple of nuclear spins (Si;Ch,...,Sk,ch) of the nuclear-spin system (Si, ..., SN) to a particular one (Si,ch) of the nuclear spins of the k-tuple of nuclear spins (Si,ch,.. -,Sk,ch), wherein k > 1, wherein the sensing-sequence is configured to read out the particular nuclear spin (Si,ch) directly via the single-spin quantum sensor (el).

8. The method of claim 7, wherein action Al (INIT) further comprises polarizing the k-tuple of nuclear spins (Si,ch,...,Sk,ch) of the nuclear-spin system (Si, . . ., SN), wherein k > 1; andwherein the spin-chain sequence in action A3 (DETECT) comprises applying consecutively k double-resonance sequences (XY'T) of a chain of double-resonance sequences (XY'T), the chain being configured to transfer a signal of the nuclear spin (Si) to the particular one (Si,ch) of the nuclear spins, by: applying an initial double-resonance sequence of the chain to an initial pair (Si,Sk,ch) including the nuclear spin (Si), applying, if k > 3, each of one or more subsequent double-resonance sequences (XY'T) to a respective pair (Si,ch,Si-i,ch) of the k-tuple of nuclear spins (Si,ch, . . - ,Sk,ch), and applying a final double-resonance sequence of the chain to a final pair (S2,ch,Si,ch) including the particular one (Si,ch) of the nuclear spins, wherein if k = 1, the final pair is equal to the initial pair (Si,Sk,ch).

9. The method of claim 8, wherein a double-resonance sequence (XY'T) of the chain comprises a Hahn spin-echo type sequence including an initial n / 2 pulse and a final n / 2 pulse that are applied with a relative phase difference of substantially 90 degrees, the Hahn spin-echo type sequence being applied at a frequency range including a nuclear-spin frequency of a nuclear spin (Sj-i,ch) on which a signal is to be transferred.

10. The method of any one of claims 8-9, wherein for one or more double-resonance sequences (XY'T) of the chain, involving a respective first nuclear-spin frequency (Ai) and a respective second nuclear-spin frequency (AM) and a respective free-evolution time ti,i-i, the respective free-evolution time ti,i-i is chosen as= Ciyil1 / 2, wherein C£is a respective coupling strength corresponding to the involved respective first and second nuclear-spin frequencies (Ai,Ai.i).

11. The method of any one of claims 8-10, wherein each one of the k double-resonance sequences (XY$) of the chain, acting on a corresponding pair of nuclear spins (Sj,ch,Sj-i,ch) comprising a first nuclear spin (Sj,ch) and a second nuclear spin (Sj-i,ch), is configured to transfer a signal from the first nuclear spin (Sj,ch) to the second nuclear spin (Sj-i,ch).

12. The method of claim 1, wherein the sensing-sequence in action A3 (DETECT) is configured to directly read out the nuclear spin (Si) via the single-spin quantum sensor (el).

13. The method of any one of the preceding claims, wherein the host material (1) is a diamond, wherein the single-spin quantum sensor (el) is a nitrogen-vacancy, NV, center of the diamond, the NV center exhibiting three spin states (ms=0, ms ±= ±1).

14. The method of claim 13, wherein action Al (INIT) further comprises initializing the spin state (ms) of the NV center to one of its two non-zero spin states (ms=+l; ms=— 1).

15. The method of any of claims 13-14, wherein the nuclear-spin system (Si, . . ., SN) is exposed to an external magnetic field (B) that is applied along a symmetry axis (z-axis) of the NV center.

16. The method of any one of claims 13-15, wherein the nuclear-spin system (Si, ..., SN) corresponds to carbon nuclei (13C) surrounding the NV center.

17. The method of any one of claims 1-15, wherein the nuclear-spin system (Si, . . ., SN) is part of a target sample positioned in proximity to the single-spin quantum sensor (el).

18. The method of any one of the preceding claims, wherein the first pulse-sequence comprises radio-frequency, RF, pulses, and wherein the second pulse-sequence comprises microwave, MW, pulses.

19. A method of measuring, via a single-spin quantum sensor (el) disposed in a host material (1), a coupling strength (C45) of a spin-spin interaction between a nuclear spin (S4) and a target nuclear spin (S5), the nuclear spin (S4) and the target nuclear spin (S5) being part of a nuclear- spin system (Si, . . ., SN), the method comprising: action Al (INIT): polarizing the nuclear spin (S4) and a k-tuple of nuclear spins (81,82,83) of the nuclear-spin system (Si, ... , SN), wherein k > 1; action A2 (DR): applying a double-resonance sequence (XX$) involving a free- evolution time (t4s) and including: a first pulse-sequence, applied within a first frequency range including a first nuclear-spin frequency (A4) of the nuclear spin (S4), and configured to decouple the nuclear spin (S4) from environment noise and from the rest of the nuclear-spin system (SI, ... ,SN), anda second pulse-sequence, applied within a second frequency range including a second nuclear-spin frequency (As) of the target nuclear spin (Ss), and configured to recouple the target nuclear spin (Ss) to the nuclear spin (S4); action A3 (DETECT): obtaining a measurement point by applying a detecting-sequence (XY'T, RO) configured to read out the nuclear spin (S4) via the single-spin quantum sensor (el), the detecting-sequence comprising a sensing-sequence applied to the single-spin quantum sensor (el); repeating the actions at least one time to obtain a plurality of measurement points by sweeping the free-evolution time (t) over a range of chosen free-evolution times, each of the plurality of measurement points being associated with one of the range of chosen free-evolution times; and determining, based on the plurality of measurement points, the coupling strength (C45), wherein the detecting sequence in action A3 (DETECT) further comprises a spin-chain sequence configured to transfer a signal of the nuclear spin (S4) via the k-tuple of nuclear spins(81.82.83) to a particular one (Si) of the nuclear spins of the k-tuple of nuclear spins (81, 82, 83), wherein k > 1, wherein the sensing-sequence is configured to read out the particular one (Si) of the nuclear spins directly via the single-spin quantum sensor (el).

20. The method of claim 19, wherein the spin-chain sequence in action A3 (DETECT) comprises applying consecutively k double-resonance sequences (XY'T) of a chain of double-resonance sequences (XY'T), the chain being configured to transfer the signal of the nuclear spin (S4) to the particular one (Si) of the nuclear spins, by: applying an initial double-resonance sequence (XY$) of the chain to an initial pair(84.83) including the nuclear spin (S4), applying, if k > 3, each of one or more subsequent double-resonance sequences (XY'T) to a respective pair (83, 82; 82,81) of the k-tuple of nuclear spins (81, 82, 83), and applying a final double-resonance sequence (XY$) of the chain to a final pair (82,81) including the particular one (Si) of the nuclear spins, wherein if k = 1, the final pair is equal to the initial pair.

21. The method of claim 20, wherein a double-resonance sequence (XY$) of the chain comprises a Hahn spin-echo type sequence including an initial n / 2 pulse and a final n / 2pulse that are applied with a relative phase difference of substantially 90 degrees, the Hahn spin-echo type sequence being applied at a frequency range including a nuclear-spin frequency of one nuclear spin (Si-i,ch) on which a signal is transferred.

22. The method of any one of claims 20-21, wherein for one or more double-resonance sequences (XYp of the chain, involving a respective first nuclear-spin frequency (Ai) and a respective second nuclear-spin frequency (AM) and a respective free-evolution time ti,i-i, the respective free-evolution time ti,i-i is chosen aswhereinis a respective coupling strength corresponding to the involved respective first and second nuclear-spin frequencies (Ai,Ai-i).

23. The method of any one of claims 20-22, wherein each one of the k double-resonance sequences (XY'T) of the chain, acting on a corresponding pair of nuclear spins (Sj,ch,Sj-i,ch) comprising a first nuclear spin (Sj,ch) and a second nuclear spin (Sj-i,ch), is configured to transfer a signal from the first nuclear spin (Sj,ch) to the second nuclear spin (Sj-i,ch).

24. The method of any one of claims 19-23, wherein the host material (1) is a diamond, wherein the single-spin quantum sensor (el) is a nitrogen-vacancy, NV, center of the diamond, the NV center exhibiting three spin states (ms=0, ms ±= ±1).

25. The method of claim 24, wherein action Al (INIT) further comprises initializing the spin state (ms) of the NV center to one of its two non-zero spin states (ms=+l; ms=— 1).

26. The method of any of claims 24-25, wherein the nuclear-spin system (Si, ..., SN) is exposed to an external magnetic field (B) that is applied along a symmetry axis (z-axis) of the NV center.

27. The method of any one of claims 24-26, wherein the nuclear-spin system (Si, ..., SN) corresponds to carbon nuclei (13C) surrounding the NV center.

28. The method of any one of claims 19-26, wherein the nuclear-spin system (Si, . . . , SN) is part of a target sample positioned in proximity to the single-spin quantum sensor (el).

29. The method of any one of claims 19-27, wherein the first and second pulse-sequences comprise radio-frequency, RF, pulses.

30. The method of any one of claims 19-28, wherein the detecting sequence includes at least one of phase-sensitive dynamical decoupling, DD, sequences or DDRF sequences.

31. A method of measuring, via a single-spin quantum sensor (el) disposed in a host material (1), at least one of one or more hyperfine-shift frequencies (A2, A3) related to respective one or more nuclear spins (S2, S3) and / or one or more coupling strengths (C24.C34) related to spinspin interactions between the respective one or more nuclear spins (82,83) and a further nuclear spin (S4), the one or more nuclear spins (82,83) and the further nuclear spin (S4) being part of a nuclear-spin system (Si,. . . ,SN), the method comprising: action Al (INIT): polarizing the one or more nuclear spins (82,83); action A2 (DR): applying a first double-resonance sequence (XXJ,) involving a first free-evolution time (ti) and including: a first pulse-sequence, applied at a first frequency-range (A23) including one or more nuclear-spin frequencies (A2,As) corresponding to the respective one or more nuclear spins (82,83), and configured to decouple the respective one or more nuclear spins (82,83) from environment noise and from the rest of the interacting nuclear-spin system (Si, . . . ,SN); a second pulse-sequence, applied within a second frequency range including a spin-transition frequency of the single-spin quantum sensor (el), and configured to change a spin state (ms) of the single-spin quantum sensor (el) so as to recouple the single-spin quantum sensor (el) to the respective one or more nuclear spins (82,83); and applying a second double-resonance sequence (XX$) involving a second free- evolution time (t2) and including: a third pulse-sequence, applied within the first frequency range, and configured to decouple the respective one or more nuclear spins (82,83) from environment noise and from the rest of the interacting nuclear-spin system (Si,. . . ,SN);a fourth pulse-sequence, applied within a third frequency range including a further nuclear-spin frequency (A4) of the further nuclear spin (S4), and configured to recouple the further nuclear spin (S4) to the respective one or more nuclear spins (82,83); action A3 (DETECT): obtaining a measurement point by applying a detectingsequence (XY'T, RO) configured to read out the respective one or more nuclear spins (82,83) via the single-spin quantum sensor (el), the detecting-sequence comprising a sensingsequence applied to the single-spin quantum sensor (el); repeating the actions at least one time to obtain a plurality of measurement points by sweeping the involved first and second free-evolution times (ti,t2) over a domain of chosen free- evolution times, each of the plurality of measurement points being associated with one of the domain of chosen free-evolution times; and determining, based on the plurality of measurement points, at least one of the one or more hyperfine-shift frequencies (A2,A3) and / or the one or more coupling strengths (C24,C34).

32. The method of claim 31, wherein the first pulse-sequence is a Hahn spin-echo type sequence comprising a n- pulse separating a first free-evolution time window and a second free-evolution time window, wherein the second pulse-sequence comprises one or more spin-transition pulses applied synchronously with the n-pulse so as to change the spin state (ms) of the single-spin quantum sensor (el) to be different during the first and second free-evolution time windows.

33. The method of claim 31, wherein the first pulse-sequence is a Hahn spin-echo type sequence comprising one or more n-pulses, wherein the second pulse-sequence comprises one or more spin-transition pulses applied synchronously with the one or more n-pulses so as to change the spin state (ms) of the single-spin quantum sensor (el) to flip back and fourth between two spin states and to be different in any two consecutive free-evolution time windows.

34. The method of any one of claims 31-33, wherein the detecting sequence includes at least one of phase-sensitive dynamical decoupling, DD, sequences or DDRF sequences.

35. The method of any one of claims 31-34, wherein action Al (INIT) comprises initializing the single-spin quantum sensor (el) to an interacting spin state (ms=£0).

36. The method of any one of claims 31-35, wherein the detecting sequence in action A3 (DETECT) further comprises a spinchain sequence configured to transfer one or more signals of the one or more nuclear spins (S2, S3) via a k-tuple of nuclear spins (Si,ch,...,Sk,ch) of the nuclear-spin system (Si, . . ., SN) to a particular one (Si,ch) of the nuclear spins of the k-tuple of nuclear spins (Si,ch,...,Sk,ch), wherein k > 1, wherein the sensing-sequence is configured to read out the particular nuclear spin (Si,ch) directly via the single-spin quantum sensor (el).

37. The method of claim 36, wherein action Al (INIT) further comprises polarizing the k-tuple of nuclear spins (Si, ch,. . .,Sk,ch) of the nuclear-spin system (Si, . . ., SN), wherein k > 1; and wherein the spin-chain sequence in action A3 (DETECT) comprises applying consecutively k double-resonance sequences (XY'T) of a chain of double-resonance sequences (XY'T), the chain being configured to transfer the one or more signals of the one or more nuclear spins (S2, S3) to the particular one (Si, ch) of the nuclear spins, by: applying an initial double-resonance sequence of the chain to an initial pair ((S2,S3);Sk,ch) and involving frequency ranges (A23,Ak,ch) including the first frequency -range (A23) that comprises the one or more nuclear-spin frequencies (A2, A3) of the respective one or more nuclear spins (82,83), applying, if k > 3, each of one or more subsequent double-resonance sequences (XY'T) to a respective pair (Si,ch,Si-i,ch) of the k-tuple of nuclear spins (Si, ch,...,Sk,ch), and applying a final double-resonance sequence of the chain to a final pair (S2, ch, Si, ch) including the particular one (Si, ch) of the nuclear spins, wherein if k = 1, the final pair is equal to the initial pair.

38. The method of claim 37, wherein a double-resonance sequence (XY$) of the chain comprises a Hahn spin-echo type sequence including an initial n / 2 pulse and a final n / 2 pulse that are applied with a relative phase difference of substantially 90 degrees, the Hahn spin-echo type sequence being applied at a frequency range including a nuclear-spin frequency of a nuclear spin (Sj-i,ch) on which a signal is to be transferred.

39. The method of any one of claims 37-38, wherein for one or more double-resonance sequences (XYp of the chain, involving a respective first nuclear-spin frequency (Ai) and a respective second nuclear-spin frequency (AM) and a respective free-evolution time ti,i-i, the respective free-evolution time ti,i-i is chosen aswhereinis a respective coupling strength corresponding to the involved respective first and second nuclear-spin frequencies (Ai,Ai-i).

40. The method of any one of claims 37-39, wherein each one of the k double-resonance sequences (XY'T) of the chain, involving the respective first nuclear-spin frequency (Ai) and the respective second nuclear-spin frequency (Am), is configured to transfer a signal from a respective first nuclear spin (Sj,ch) with the respective first nuclear-spin frequency (Ai) to a respective second nuclear spin (Sj-i,ch) with the respective second nuclear-spin frequency (AM).

41. A method of transferring, via a spin-chain sequence involving a k-tuple of nuclear spins(51.82.83), a signal of a nuclear spin (S4) via the k-tuple of nuclear spins (Si, 82,83) to a particular one (Si) of the nuclear spins of the k-tuple of nuclear spins (81,82,83), the nuclear spin (S4) and the k-tuple of nuclear spins (81,82,83) being part of a nuclear-spin system (Si, . . . , SN), wherein k > 1, the method comprising: polarizing the k-tuple of nuclear spins (81,82,83); applying consecutively k double-resonance sequences (XY'T) of a chain of doubleresonance sequences (XY'T), the chain being configured to transfer the signal of the nuclear spin (S4) to the particular one (Si) of the nuclear spins (81,82,83), by: applying an initial double-resonance sequence (XY$) of the chain to an initial pair(84.83) including the nuclear spin (S4), applying, if k > 3, each of one or more subsequent double-resonance sequences (XY'T) to a respective pair (83,82; S2,Si) of the k-tuple of nuclear spins (81,82,83), and applying a final double-resonance sequence (XY$) of the chain to a final pair (S2,Si) including the particular one (Si) of the nuclear spins, wherein if k = 1, the final pair is equal to the initial pair.

42. The method of claim 41, wherein a double-resonance sequence (XY$) of the chain comprises a Hahn spin-echo type sequence including an initial n / 2 pulse and a final n / 2 pulse that are applied with a relative phase difference of substantially 90 degrees, the Hahnspin-echo type sequence being applied at a frequency range including a nuclear-spin frequency of one nuclear spin (Si-i,ch) on which a signal is transferred.

43. The method of any one of claims 41-42, wherein for one or more double-resonance sequences (XYp of the chain, involving a respective first nuclear-spin frequency (Ai) and a respective second nuclear-spin frequency (AM) and a respective free-evolution time ti,i-i, the respective free-evolution time ti,i-i is chosen aswhereinis a respective coupling strength corresponding to the involved respective first and second nuclear-spin frequencies (Ai,Ai-i).

44. The method of any one of claims 41-43, wherein each one of the k double-resonance sequences (XY'T) of the chain, acting on a corresponding pair of nuclear spins (Sj,ch,Sj-i,ch) comprising a first nuclear spin (Sj,ch) and a second nuclear spin (Sj-i,ch), is configured to transfer a signal from the first nuclear spin (Sj,ch) to the second nuclear spin (Sj-i,ch).

45. A method of imaging a nuclear-spin system (Si, . . . ,SN), the method comprising: performing a plurality of measurements based on at least one of any one of the preceding methods, the performing including obtaining at least one of one or more hyperfine-shift frequencies (Ai Aj) related to respective one or more nuclear spins (Si, Sj) and a single-spin quantum sensor (el) disposed in a host material, one or more coupling strengths (Cij) related to spin-spin interactions between the respective one or more nuclear spins (Si,Sj) and / or one or more nuclear-spin frequencies (Ai,Aj) of the respective one or more nuclear spins (Si,Sj) according to any one of the preceding methods, the one or more nuclear spins (Si,Sj) being part of a nuclear-spin system (Si,. . . ,SN); and constructing an image of the nuclear-spin system (SI, ... ,SN), based on the plurality of measurements.

46. A measurement system (200) comprising: one or more magnetic-pulse generators (201) configured to generate oscillating magnetic fields; one or more lasers (203) configured to generate optical radiation; one or more detectors (204) configured to detect photons; a processor (206); anda memory (207), wherein the memory (207) stores instructions that, when executed by the processor (206), cause the measurement system (200) to execute one or more of the preceding methods.

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