Method and apparatus for characterising the nuclear spin environment around a paramagnetic centre

Abstract

Disclosed is a method for characterising a sample (E), said method comprising magnetically coupling the sample to a microwave resonator (RHS) such that the coupling dominates the relaxation dynamics of an electron spin (SEL) of a paramagnetic centre of the sample, and determining, by nuclear-magnetic-resonance spectroscopy, coefficients Cij of coupling between a nuclear spin of the sample, called the probe nuclear spin (SNi), and other nuclear spins (SNj, SNk) of the sample, said method employing indirect detection of a state of said probe nuclear spin (SNi), through counting of microwave photons, of respective noise signals (SBS) produced by the return to equilibrium of said electron spin (SEL). An apparatus for implementing such a method is also disclosed.

Description

DESCRIPTION Title of the invention: Method and apparatus for characterizing the nuclear spin environment of a paramagnetic center

[0001] The invention lies in the field of magnetic resonance spectroscopy, and more particularly in paramagnetic resonance spectroscopy (PRE). It is particularly applicable to nuclear spin imaging.

[0002] Paramagnetic resonance spectroscopy aims to characterize the paramagnetic species present in a sample, with numerous applications in chemistry, biochemistry, materials physics, archaeology, and quantum computing. It exploits the presence, in the sample, of "paramagnetic centers" (PCs), a term that can refer to all kinds of systems exhibiting a paramagnetic response, including: impurities in crystalline or amorphous matrices, and in particular metallic ions (transition metals or rare earths), metallic ions embedded in molecules or proteins, organic radicals, etc.A paramagnetic center is characterized by the presence of at least one unpaired electron whose spin, in the presence of an external magnetic field, can switch from a ground state to an excited state and vice versa by absorbing or emitting, respectively, a photon at a resonance frequency (Larmor frequency), generally in the microwave range (radio frequency refers to frequencies between 1 MHz and 1 GHz and microwave frequency to frequencies between 1 GHz and 100 GHz).

[0003] One of the important goals of EPR spectroscopy is to characterize the environment of nuclear spins around paramagnetic centers. This involves measuring the resonance frequency of nuclear spins near the paramagnetic centers, as well as their coupling via hyperfine interactions; in doing so, chemical and spatial information about the environment of the atoms around the center is obtained. Ideally, it is possible to achieve nuclear spin imaging, that is, their three-dimensional localization on an angstrom scale; see, for example, [Abobeih] and [van de Stolpe].

[0004] A major difficulty arises from the fact that, in conventional methods of inductive detection EPR spectroscopy, the detected signal comes from a large number of paramagnetic centers, which complicates the characterization of the environment of nuclear spins around an individual paramagnetic center.

[0005] In the aforementioned articles [Abobeih] and [van de Stolpe], the paramagnetic center used is a nitrogen-vacancy (NV) center in a diamond, whose electronic spin exhibits an optical transition that allows the measurement of an optical fluorescence signal. The number of photons emitted by fluorescence depends on the state of this electronic spin. This is known as optical magnetic resonance detection (ODMR). The detection of an individual NV center is possible using ODMR. This NV center can also be used to probe its nuclear spin environment by high-resolution magnetic resonance spectroscopy.

[0006] The ODMR technique is only applicable to paramagnetic centers that can be measured by optical detection, which represents a small fraction of the paramagnetic centers of interest. In particular, this method does not appear to be applicable to metalloproteins.

[0007] The invention aims to overcome, in whole or in part, the aforementioned limitations of the prior art. More specifically, it aims to enable the characterization of the nuclear spin environment of an individual paramagnetic center by overcoming the limitations of the ODMR technique.

[0008] According to the invention, this objective is achieved by using a technique for detecting a spin noise signal—that is, the incoherent signal produced by the return of a spin to its equilibrium state—by counting microwave photons. This technique was proposed by the working group of the present inventors and is described in particular in [Wang] and [Albertinale] and in international application WO 2021 / 191119.

[0009] In this method, a paramagnetic center is magnetically coupled to a microwave electromagnetic resonator and immersed in a stationary magnetic field, the stationary magnetic field being perpendicular to the magnetic component of the microwave field in the resonator. By varying the intensity of the stationary magnetic field, it is possible to tune the Larmor frequency of the electron spin of the paramagnetic center to the resonant frequency of the resonator. Under these conditions, the coupling of the spin to the resonator This increases the probability that the spin, initially assumed to be in its excited state (orientation opposite to that of the stationary magnetic field), will relax to its ground state by emitting a microwave photon (Purcell effect). This photon is then detected by a single-microwave photon detector (SMPD). Since the relaxation microwave signal is incoherent, it is also called "spin noise."

[0010] The method is general and offers high spectral resolution. It applies in principle to all types of paramagnetic centers provided that their non-radiative relaxation time TNR, by coupling to the vibrations of the lattice, is longer than the radiative relaxation time accelerated by the Purcell effect TR (typically on the order of 1 ms).

[0011] The invention allows, for example, the determination of the structure of molecules containing a paramagnetic impurity, such as a protein containing a paramagnetic metal ion like Fe(III). More particularly, it allows the precise determination of the position and chemical environment of at least certain atoms of such a molecule.

[0012] An object of the invention is a method for characterizing a sample comprising a paramagnetic center containing an electronic spin and a plurality of atomic nuclei having respective nuclear spins coupled to said electronic spin, said method comprising the following steps: A) placing said sample in a stationary magnetic field, thereby obtaining a polarization of said electronic spin; B) selecting, by paramagnetic resonance spectroscopy, said paramagnetic center from among a plurality of paramagnetic centers contained in the sample; C) determining, by paramagnetic resonance, the values ​​of the hyperfine coupling coefficients Aj between each said nuclear spin and the electronic spin of said paramagnetic center;in which: - said step A) includes the magnetic coupling of the sample to a microwave resonator having a resonance frequency ω0 / 2π equal to the Larmor frequency of said electronic spin in the stationary magnetic field, the coupling constant and the quality factor of the resonator being sufficiently; raised so that the coupling with the resonator (RHS) dominates the relaxation dynamics of the electronic spin; and - said steps B) and C) implement the acquisition, by counting microwave photons, of noise signals produced by the return to equilibrium of said electronic spin.

[0013] According to specific implementations of such a process:

[0014] - A step D) may also be provided, consisting of determining, by nuclear magnetic resonance spectroscopy, coupling coefficients Cij between a said nuclear spin, said probe nuclear spin, and the other said nuclear spins; said step B) then comprising a substep of indirect detection of a state of said probe nuclear spin by means of said electronic spin; said indirect detection substep comprising the application to the sample of a plurality of paramagnetic resonance pulse sequences adapted to excite said electronic spin with a probability which is a function of the state of the probe nuclear spin, and the acquisition, by counting microwave photons, of respective noise signals produced by the return to equilibrium of said electronic spin.

[0015] - Said step D may include the application to the sample, in order: d1) of a sequence of paramagnetic resonance microwave pulses adapted to transfer the electron spin polarization from the paramagnetic center to the probe nuclear spin; d2) of a sequence of nuclear magnetic resonance radio frequency pulses adapted to modify the state of the probe nuclear spin as a function of its coupling coefficients Cij with said other nuclear spins (SNj, SNk); and d3) of said sequence of paramagnetic resonance microwave pulses adapted to excite the electron spin of the paramagnetic center with a probability that is a function of the state of the probe nuclear spin.

[0016] - The nuclear magnetic resonance radio frequency (IRF) pulse sequence of substep d2) can be a spin echo double resonance sequence, or SEDOR.

[0017] - A step F) can also be provided, consisting of determining the relative positions of said atomic nuclei from the values ​​of the hyperfine coupling coefficients Aj.

[0018] - A step E) may also be provided, also including a step E) consisting of determining the relative positions of said atomic nuclei from the coupling coefficients Cij.

[0019] Another object of the invention is an apparatus for implementing such a method comprising: - a superconducting microwave resonator including a sub-micron electromagnetic field confinement zone adapted to accommodate a sample to be characterized; - a system for generating a stationary magnetic field having at least one component perpendicular to an orientation of a magnetic component of said electromagnetic field in said confinement zone; - a Bragg mirror tuned to a resonance frequency of said superconducting microwave resonator, coupling said resonator to a signal input and output port; and - an excitation and measurement system connected to said signal input and output port and configured to inject into said superconducting microwave resonator a microwave excitation signal at said resonance frequency and to detect a microwave signal emitted by said sample;characterized in that said excitation and measurement system - is also configured to inject a radio frequency signal into said resonator; and - comprises a microwave photon counter for detecting said microwave signal emitted by said sample.

[0020] According to specific embodiments of such a device:

[0021] - Said excitation and measurement system can also be configured to inject into said superconducting microwave resonator a suitable direct current signal to temporarily modify its resonant frequency.

[0022] - Said superconducting microwave resonator can be made using planar technology on an insulating substrate.

[0023] - Said superconducting microwave resonator may have a quality factor greater than or equal to 10 2 and preferably greater than or equal to 10 4 .

[0024] - Said superconducting microwave resonator can have a resonance frequency between 5 GHz and 100 GHz.

[0025] - Said stationary magnetic field generation system may include a superconducting coil configured to generate said stationary magnetic field with an amplitude between 0.1T and 10T.

[0026] Other features, details and advantages of the invention will become apparent from the description provided with reference to the accompanying drawings given by way of example, which represent, respectively:

[0027] [Fig.1], a flowchart of a process according to an embodiment of the invention;

[0028] [Fig.2], a functional diagram of a device according to an embodiment of the invention;

[0029] [Fig.3], a plan view of a Bragg mirror assembly – resonator of a device according to an embodiment of the invention;

[0030] [Fig.4], a detailed view of [Fig.3], showing the positioning of two samples in the electromagnetic field confinement zone of the resonator;

[0031] [Fig.5A] and [Fig.5B], experimental results demonstrating the possibility of measuring the hyperfine coupling coefficients between the electronic spin of a paramagnetic center and the surrounding nuclear spins by counting microwave photons;

[0032] [Fig.6A] and [Fig.6B], experimental results demonstrating the possibility of detecting the state of two nuclear spins by counting microwave photons;

[0033] [Fig. 7], experimental results demonstrating the possibility of determining the coupling coefficients of nuclear spins from the counts of [Fig. 6A] and [Fig. 6B]; and

[0034] [Fig.8], a flowchart of a process according to an alternative embodiment of the invention.

[0035] With reference to the flowchart in [Fig. 1], the first step A of a process according to one embodiment of the invention consists of depositing a ECH sample in an electromagnetic field concentration zone (ZCC) of a microwave resonator (RHS). As illustrated in [Fig. 3], the microwave resonator is fabricated using planar technology by depositing a superconducting thin film (SCM) onto an insulating silicon (SI) substrate and etching said thin film. For example, the resonator can take the form of two interdigitated electrodes, forming a capacitor, connected by a very thin track, micrometer or sub-micrometer wide (50–1000 nm, as illustrated in [Fig. 4], and preferably on the order of 100 nm), which serves as both the inductance and the concentration zone (ZCC). The sample can, for example, take the form of a molecular microcrystal (CM), a microdroplet of frozen solution (SG), or MDD molecules directly deposited on the ZCC track.It is understood that the magnetic component of the oscillating electromagnetic field in the resonator has, in correspondence with the sample, an x ​​orientation parallel to the xz plane of the CMS layer and perpendicular to the longitudinal z direction of the ZCC track.

[0036] The RHS resonator exhibits a resonant frequency ω0 / 2π in the microwave range, typically between 5 GHz and 200 GHz. This resonant frequency can be adjusted by modifying the dynamic inductance of the superconducting track ZCC by injecting a direct current IDC (see [Fig. 2], where GDC refers to the generator of the current IDC).

[0037] Its quality factor Q must be sufficiently high for the Purcell effect to dominate the spin relaxation dynamics: Γ1 ≈ ΓP where Γ1 is the energy relaxation rate of the spins and ΓP the Purcell factor given by Γ^ = 4g^ / κ (1) where κ=ω ^ / Q is the energy dissipation rate in the resonator and g is the coupling coefficient of the sample to the resonator, when the spin resonance frequency is tuned to that of the resonator. Typically, this quality factor will be greater than or equal to 10 2 , preferably greater than or equal to 10 4 To maintain a high quality factor while allowing signal injection and extraction, the RHS resonator is connected to a PES signal input / output port via a Bragg MB mirror (or grating) tuned to the frequency ω0 / 2π. In the example in [Fig. 3], the Bragg MB mirror consists of a coplanar transmission line, produced by etching the SMT layer, whose center conductor width, and therefore characteristic impedance, varies in a periodic, preferably by square waves (see [Sigillito]). The Bragg mirror effectively confines microwave signals close to the resonance frequency ω0 / 2π, thus allowing a sufficient quality factor to be achieved, while allowing the passage of direct current IDC or radio frequency signals.

[0038] A coil, typically superconducting (reference BS in [Fig. 2]), generates a stationary magnetic field B0 oriented perpendicularly to the magnetic component of the oscillating electromagnetic field in the resonator. More specifically, in the illustrated embodiment, the stationary magnetic field B0 is oriented along the z-direction, parallel to the plane of the resonator.

[0039] Under the influence of the magnetic field B0, the spins present in the ECH sample become polarized. Figure 2 shows that the ECH sample comprises a paramagnetic center CP containing an unpaired electron, and therefore an electron spin SEL, and three nuclei Ni, Nj, and Nk with non-zero nuclear spins SNi, SNj, and SNk, respectively, located near this paramagnetic center. The Larmor frequencies of these spins are given by the product of the magnetic field strength B0 and their gyromagnetic ratio. It should be noted that the gyromagnetic ratio of an electron is approximately three orders of magnitude greater than that of an atomic nucleus; thus, the Larmor frequency of the electron spin SEL will be approximately three orders of magnitude greater than the Larmor frequencies of the nuclear spins SNi, SNj, and SNk.For example, for a magnetic field B0 on ​​the order of 0.1T to 10T, the Larmor frequency of the electron spin SEL will be in the microwave (GHz) range and that of a nuclear spin SNi, SNj, SNk will be in the radio frequency (MHz) range. The intensity of the magnetic field B0 can be adjusted so that the Larmor frequency of the electron spin SEL corresponds to the resonance frequency ω0 / 2π of the resonator (hereafter, for simplicity, we will call ω0 the resonance "frequency"; similarly, angular frequencies designated by "ω" will simply be called "frequencies").

[0040] The assembly is placed in a cryostat and cooled to a cryogenic temperature T, both to ensure the superconducting transition of the CMS layer, to reduce microwave noise, and to increase the polarization of the spins—electronic as well as nuclear—of the sample. When ℏω^ ≫ kT, the polarization of the electron spin in its ground state can be considered substantially complete at thermal equilibrium. For example, we can take T=10mK and ω0 / 2π greater than or equal to 5 GHz.

[0041] The operations following this preparation step A require exposing the ECH sample to sequences of high-frequency (HF) microwave pulses (to act on the electron spin SEL) and high-frequency (RF) pulses (to act on the nuclear spins), and detecting, by photon counting, high-frequency signals (SBS) emitted by the electron spin SEL. For this reason, the apparatus illustrated in [Fig. 2] includes, in addition to the high-frequency resonator (HFR), the Bragg mirror (MB), and the superconducting coil (BS), a superconducting excitation and measurement (SEM) system connected to the HFR via the PES port and the Bragg mirror (MB). The SEM system is configured to generate the HF and IRF pulses, detect the SEL signals, and process them to extract information of interest (e.g., the relative positions of the atomic nuclei Ni, Nj, Nk).It can also control the intensity of the stationary magnetic field B0 (via the current flowing in the superconducting coil BS) to determine the value of the electronic Larmor frequency and drive the DC generator GDC to change the resonant frequency of the RHS resonator.

[0042] As illustrated in [Fig. 2], the SEM excitation and measurement system includes a GHF microwave generator, configured to generate IHF microwave pulses. The GHF generator is connected to the PES port of the MB Bragg mirror via a CIR circulator and a DX diplexer. A CPH microwave photon counter, designed to detect the SBS spin noise signal, is also connected to the CIR circulator. The CPH counter can, for example, be based on a transmon-type superconducting qubit; see [Lescanne]. A GRF radio frequency generator, configured to generate GRF radio frequency pulses, and the aforementioned GDC direct current generator, are also connected to the PES port via the DX diplexer.

[0043] A SECT data control and processing system, for example a computer, drives the GHF, GRF, and GDC generators and controls the current flowing through the superconducting coil BS (and therefore the strength of the stationary magnetic field B0). In addition, it receives as input the photon counting signals from the CPH detector and processes them to produce DS output data containing the information of interest.

[0044] The first step – B – following sample preparation consists of performing paramagnetic resonance spectroscopy analysis to identify and select a paramagnetic center (CP) in the ECH sample. To implement this step, an IHF microwave pulse at the resonance frequency of the RHS resonator ω is applied to the ECH sample. ^ and the number of photons is counted using the CPH detector. <c>of the spin noise signal SBS re-emitted by the sample following excitation by this pulse. By finely sweeping the stationary magnetic field B0, a series of peaks is obtained, indicating the resonance of individual paramagnetic centers. A CP center is selected by choosing the stationary magnetic field B ^ corresponding to one of these peaks. We denote γ^ = ω^ / B^ its gyromagnetic ratio. The magnetic field B0 will be maintained at this value throughout the entire process.

[0045] The next step – C – consists of determining the hyperfine coupling coefficients of the nuclear spins SNi, SNj, SNk with the electronic spin SEL of the selected paramagnetic center CP. This step is also implemented by paramagnetic resonance spectroscopy with microwave photon counting.

[0046] The Hamiltonian of the interaction between the paramagnetic center CP and a generic nuclear spin SNj denoted by the subscript ^ is where ^ ^!^,^ is the spin operator of spin ^, and ^ ^ (resp. ^ ^ ) is the isotropic (resp. anisotropic) hyperfine coupling constant. The exact value of the constants ^ ^ and ^ ^ depends on the relative location of the paramagnetic center and the nuclear spin. The nuclear spin resonance frequency therefore depends on the spin state of the paramagnetic center, defined by the value of ms, and is equal to où )* = 0 ^ depending on the electronic spin state SEL of the paramagnetic center CP, et = 1#^^ is the Larmor frequency of the nuclear spin. This formula can be approximated by when ≫ ^^ , ^^, which is generally the case when the magnetic field B0 is sufficiently high, for example on the order of 1T or more. Therefore, as a first approximation, only the coefficients Aj can be taken into consideration.

[0047] We see that the Larmor frequency of each nuclear spin depends on its interaction with the paramagnetic center CP; this is what allows us to excite and measure each nuclear spin individually. In turn, the resonance frequency of the electron spin SEL depends on the spin state of each nuclear spin and is equal to (always within the limit) where ) is the magnetic spin quantum number of the nucleus identified by #,^ the index j.

[0048] Several approaches are possible to measure the Aj coefficients.

[0049] To measure a coefficient Aj with a value greater than the linewidth of the paramagnetic center, Γ∗ ∗ ∗^ = 1 / 6^, 6^ being the coherence time of the paramagnetic center CP measured by the Ramsey fringe method, it is possible to apply Ramsey pulse sequences, that is, pairs of IHF pulses at the frequency " of type (that is, capable of inducing a ^ ^ 7. Electron spin flip by an angle), separated by a time interval τ ^ variable. Several values ​​of the interval τ are considered, and for each of them a plurality of Ramsey sequences is applied. The resulting signal consists of an average number of photon counts for each value of τ, 〈C〉(τ). Since applying a Ramsey sequence produces either 0 photons or 1 photon, 〈C〉(τ)<1 represents an electron spin excitation probability. The Fourier transform of 〈C〉(τ) directly gives the coefficients Aj. ∗

[0050] Une autre possibilité, toujours dans le cas A > Γ , est d’appliquer à< ^ the ECH sample of IHF microwave pulses of a length * approximately equal to T and of variable frequency (but always within ^ of the resonator's bandwidth (which is inversely proportional to its quality factor), and to perform a microwave photon count for each value of ". The spectrum 〈C〉$ω' shows the Larmor frequency of the electron spin SEL of the parametric center CP at the time of the spectrum measurement. By repeating this measurement a large number of times, we observe this frequency of Larmor jumps between the different possible values, determined by equation (5). This allows us to measure the coefficients A > .

[0051] This second method is demonstrated by [Fig. 5A] and [Fig. 5B] in the case of a paramagnetic center Er3+:CaWO4 and considering 2 nuclear spins of 183 W – and therefore 4 frequencies corresponding to the configurations |↑↑^, |↑↓^, |↓↑^, |↓↓^. More specifically, [Fig. 5A] is a count density map as a function of time and frequency. [Fig. 5B] shows the average number of photon counts as a function of frequency. The four peaks corresponding to the four aforementioned spin configurations are clearly visible. By measuring the frequency difference between these peaks, we obtain E ^F = 35kHz, and DI ^F = 21kHz. ∗

[0052] Lorsque A< < Γ^, les méthodes précédentes ne sont pas applicables car les The peaks of 〈C〉$ω' are not well resolved. However, other methods are suitable for this case:

[0053] – ENDOR-pulse method (or Mims-ENDOR) – see [Mims], where ENDOR is an acronym for the English expression "double electronic-nuclear resonance". The method consists of applying stimulated echo sequences. 7 7 7 7 ^ − M − ^ − 6 − ^ − M − ^ where « ^ » denotes an IHF pulse at the frequency » capable of inducing a flip of the electron spin by an angle 7 M is the duration ^ ^ of two intervals during which the spins evolve freely without the application of microwave or radiofrequency pulses, and T is the duration of an interval during which an IRF radiofrequency pulse of variable frequency ω is applied. A microwave echo signal, emitted by the electron spin, is detected by counting microwave photons after each Mims-ENDOR sequence. It can be shown that the amplitude of this signal is reduced when ω corresponds to one of the frequencies, which are thus determined and provide access coefficients Aj.

[0054] Other possible approaches for determining the hyperfine coupling coefficients in the case A < Γ are the radio frequency dynamic decoupling< ^(DdRF) known from (Bradley) and the method of repeated weak measurements disclosed by (Cujia).

[0055] The values ​​of the hyperfine coupling coefficients – or, equivalently, the frequencies – constitute, in themselves, useful information, for example on the chemical environment of atomic nuclei. However, within the framework of the embodiment of the invention considered here, their main interest is to enable the implementation of step B, which consists of determining the coupling coefficients Cij of the nuclear spins with respect to one of these spins – SNi – chosen arbitrarily as a reference (the “probe” spin). Indeed, nuclear spins interact with each other through the magnetic dipole interaction. This interaction is described to zero order by the Hamiltonian^ = Σ>Σ^QRSR^^^,R^^,^ (6) where the coefficients with ]R^ = ^^1R1^ℏ / 4_ (1R and 1^ being the respective gyromagnetic ratios of nuclear spins ` and ^) and represent the secular part of magnetic dipolar couplings between nuclear spins.

[0056] A more complete model could take into account corrections due to hyperfine couplings at the paramagnetic center, and to non-secular terms of the interaction between nuclear spins.

[0057] One of the reasons that makes the determination of the Cij coefficients interesting is that equation (8) allows us to deduce the relative positions of the atomic nuclei carrying said nuclear spins.

[0058] Step D for determining the coupling coefficients Cij comprises three sub-steps, repeated iteratively:

[0059] - First, a substep d1 which consists of transferring the polarization of the electronic spin SEL from the paramagnetic center CP to the nuclear spins Sni, SNj, SNk. Indeed, in a given magnetic field, the degree of polarization of the electronic spins is much greater than that of the nuclear spins due to the difference between the gyromagnetic ratios.

[0060] Next, a substep d2 involves applying microwave and radio frequency pulse sequences to modify the state of the SNi probe's nuclear spin. More specifically, these pulse sequences leave the probe's nuclear spin in its ground state with a probability that depends on the values ​​of the Cij coefficients to be measured.

[0061] Finally, a substep d3 of detection of the probe spin state by photon-counting paramagnetic resonance spectroscopy.

[0062] The polarization transfer substep d1 can be carried out in several different ways. For example, and not exhaustively:

[0063] - In the case where the nuclear spin frequency, is greater than the width of the electronic spin transition, Γ ^ ∗ It is possible to subject the ECH sample to IHF preparation microwave pulses whose frequency is swept around the frequency or the frequency for a period of several times 6 f , the radiative relaxation time of the paramagnetic center CP (for reference, " * is given by equation (5) and by equation (4); their values ​​are determined using the measurements from step C).

[0064] Figure 6A illustrates how the polarization of two nuclear spins of 183 W changes as a function of the number of preparation pulses. The triangles represent the probability of finding the system in its ground state |↓↓^, while the dots represent the probability of finding the system in the excited state |↑↓^ (the |↓↑^ state follows a similar evolution, while the probability of the |↑↑^ state decreases even more rapidly). It can be seen that beyond 15 pulses, the probability of the |↓↓^ state ^ is close to 1, which means that the polarization transfer is practically total.

[0065] Figure 6B illustrates the detection of the state of two nuclear spins of 183 W by microwave photon counting. The figure shows photon counting histograms for the four measurement frequencies corresponding to the four spin states |↑↑^, |↑↓^, |↓↑^, |↓↓^. More specifically, each histogram represents the number of occurrences (y-axis) of the quantity C (x-axis) when the same experiment is repeated a large number of times. For each prepared state, the corresponding histogram is clearly distinct from the other three, proving the possibility of detecting this state by microwave photon counting.

[0066] Other possible approaches are the PulsePol method disclosed by [Schwartz], the NOVEL method of [Henstra], and the SWAP method of [Taminiau] followed by a wait of time 36 f .

[0067] Regardless of the method used, at the end of substep d1 both the probe nuclear spin SNi and the electron spin SEL of the paramagnetic center CP are in their ground state.

[0068] The next substep, d2, consists of applying radiofrequency pulses to the ECH sample to excite the probe's nuclear spin with a probability that depends on the couplings Cij. For example, one can use 7 SEDOR sequences (see [Abobeih]) having the following structure 7 7 ^ − _ − où ^ and _ denote radio frequency pulses at the frequency " #,R adapted to cause a tilting of an angle ^ and _ respectively of the nuclear spin probe, and M denotes the duration – variable – of two periods of free spin evolution. Another radiofrequency pulses at a frequency " ^ The variable is applied simultaneously with the impulse. The frequency is swept. ^ , and the deadline Mr. When " ^ is resonant with a nuclear spin SNj close to the probe spin SNi, the final probability of finding the probe spin SNi in its ground state is a sinusoidal function of time M, sin $S R^ M / 2'. For greater resolution, a number of pulses π greater than 1 can be used, as demonstrated in reference [Abobeih].

[0069] The duration of the SEDOR sequence determines the spectral resolution that can be achieved, and the maximum attainable duration is fixed by the coherence time of the nuclear spins. This coherence time is limited by the duration for which the paramagnetic center CP remains in its ground state. This time is given by 60 / jk, where jk = 0 ≪ 1 is the average number of excitations at equilibrium. thermal in a frequency mode " ^ and T1 is the energy relaxation time of the CP, given by 6[0 = 6[0 + 6[00 f qf (for reference, TR and TNR are the radiative and non-radiative relaxation times, respectively, of the paramagnetic center; at resonance, 6[0f = Γr). In order to increase T1, it is desirable to transiently increase TR during the SEDOR sequence. To this end, a variant consists of applying the direct current IDC during the SEDOR sequence in order to modify the kinetic inductance of the RHS resonator and thus change its frequency of resonance of a few MHz. Thus, the maximum duration of the SEDOR sequence (and therefore the ultimate spectral resolution of the method) is given by v k which can reach 10 3 TNR or even more, on the order of 1 to 100 seconds. Thus, a high spectral resolution is possible, which translates into a high spatial resolution of nuclear spin imaging.

[0070] Next, in substep d3, the state of the probe nuclear spin SNi is detected indirectly, via measurements performed on the electron spin SEL of the paramagnetic center CP. More specifically, microwave pulses are applied to the ECH sample to excite the electron spin SEL with a probability that depends on the state of the probe nuclear spin SNi (and therefore, indirectly, on the coefficients Cij); then, the relaxation signal of said electron spin is detected by counting microwave photons.

[0071] This sub-step can also be implemented in several different ways.

[0072] Dans le cas où ^R ≫ Γ∗ The state of the nuclear spin probe is detected by applying a pulse resolved at the frequency "* − ^R / 2, and counting the microwave photons emitted during a time 6 f following this, and by repeating this sequence a sufficient number of times to obtain a signal-to-noise ratio greater than 1. Thus, we obtain a reading of the state of the nuclear spin probe SNi.

[0073] Alternativement, dans le cas où ^R < Γ∗ We can use the SWAP pulse sequence described in [Taminiau], which exchanges the state of the paramagnetic center CP and the state of the nuclear spin probe SNi. The state of the CP is finally measured by counting the number S of microwave photons that follow the SWAP sequence.

[0074] Substeps d1, d2 and d3 are repeated for several values ​​of and M, and, for each pair of values ​​of these parameters, several times to obtain an average photon count, 〈S〉, which is proportional to sin $S R^ M / 2'. The values ​​of the coefficients Cij are deduced from this signal.

[0075] Figure 7 illustrates the measurement of the coupling constant between two nuclear spins of 183W in a CaWO4 crystal, by mutual coupling to an Er ion. 3+ The curve on the right shows the spin echo signal of the nuclear spin SNi probe detected by the fluorescence of the Er ion 3+ The curve on the left shows the same spin echo, in the middle of which a pulse has been applied to another nuclear spin SNj. The frequency difference between the two oscillations directly gives C >< / 4.

[0076] Once the Cij coefficients are determined, a step E for determining the relative positions of the atomic nuclei SNi, SNj, SNk is implemented by inverting equations (7) and (8). This step is implemented by the SECT data processing and control system and does not involve any physical intervention on the ECH sample.

[0077] According to another embodiment of the invention, schematically illustrated in [Fig. 8], step D of determining the coupling coefficients Cij can be omitted. In this case, the relative positions of the atomic nuclei can be calculated directly from the hyperfine coupling coefficients Ai, as explained in [Cujia 2022] – step F in the flowchart of [Fig. 8], which replaces step E of the embodiment of [Fig. 1]. This embodiment is simpler but gives less precise results.

[0078] The invention has been described with reference to particular embodiments, but variations are possible. For example:

[0079] Microwave resonators with geometries different from that of [Fig. 3], and not necessarily planar, can be used; for example, conductive cavities surrounding the sample on all sides. Similarly, the Bragg mirror need not be made using coplanar technology but can be based, for example, on a microstrip transmission line. Furthermore, the capacitor in the resonant circuit can be made not with an interdigitated geometry but, for example, with parallel plates separated by a thin insulating layer, which can increase the coupling constant ω ^ and therefore the rate of radiative relaxation Γ r .

[0080] The architecture of the excitation and measurement system may differ from that illustrated in [Fig. 2], provided that it can perform all the required functions (the generation of the direct current IDC is not essential). The various generators GHF, GRF, GDC (if present) and the CPH detector can be implemented using any known technology.

[0081] The SECT data control and processing system can be a single device, such as a computer, or comprise a plurality of separate devices, possibly networked, such as computers, electronic boards, microprocessors, digital, analog or hybrid integrated circuits, application-specific or programmable.

[0082] Step B of selecting an individual paramagnetic center can be omitted if, for example, it is known that the sample has only one paramagnetic center, or its paramagnetic centers have been previously characterized.

[0083] Furthermore, although several implementation methods for step C have been proposed, the list is not exhaustive. The same applies to the various sub-steps of step D.

[0084] The E / F step of determining the relative positions of the atomic nuclei is not essential, the process can be completed by measuring the coupling coefficients Aj or Cij.

[0085] One difficulty with the method according to the invention is that, when the coupling coefficient Aj between the electron spin and a nuclear spin SNi is high compared to the resonator bandwidth, a change in the state of said nuclear spin can bring the resonance frequency of the paramagnetic center CP outside said bandwidth, causing a loss of the signal until the nuclear spin relaxes. This situation applies to a fairly large number of systems, and in particular to organometallic enzymes, which are of particular interest to the invention; moreover, the change in the state of nuclear spins is favored by repeated excitations of the paramagnetic center CP. One method for recovering the signal consists of irradiating the sample with a radiofrequency signal whose frequency is swept around the resonance of the nuclear spin at the very beginning of the sequence.This induces a change in the state of the nuclear spin, and thus makes it possible for the electron spin resonance frequency to return within the resonator band. Another option is to irradiate the paramagnetic CP center at the frequency "* ± "#", or with a frequency sweep around this frequency, which induces an effective relaxation of the nuclear spin and thus promotes the return of the paramagnetic CP center within the resonator bandwidth.

[0086] References

[0087] [Abobeih] M. Abobeih et al. Nature 576, 411 (2019)

[0088] [Albertinale] Emanuele Albertinale, Léo Balembois, Eric Billaud, Vishal Ranjan, Daniel Flanigan, Thomas Schenkel, Daniel Estève, Denis Vion, Patrice Bertet, Emmanuel Flurin, Nature 600, 434 (2021).

[0089] [Bradley] CE Bradley et al., Physical Review X 9 (3), 031045 (2019)

[0090] [Cujia] Cujia et al., Nature volume 571, pages 230–233 (2019)

[0091] [Cujia 2022] Cujia et al., Nature Comm.13, 1260 (2022)

[0092] [Henstra] A. Henstra et al., Journal of Magnetic Resonance 77, 389(1988).

[0093] [Lescanne] R. Lescanne et al., Physical Review X, 10, 021038 (2020).

[0094] [Mims] WB Mims, Proc. Roy. Soc. Vol.283, n° 1395, p.452 (1965)

[0095] [Schwartz] I. Schwartz et al., Sci. Adv.4, eaat8978 (2018)

[0096] [Sigillito] A. Sigillito et al., Nature Nanotechnology volume 12, pages 958–962 (2017)

[0097] [Taminiau] TH Taminiau et al., Nature Nanotechnology volume 9, pages 171–176 (2014)

[0098] [van de Stolpe] GL van de Stolpe et al., arxiv :2307.06939

[0099] [Wang] Z. Wang, L. Balembois, M. Rančić, E. Billaud, M. Le Dantec, A. Ferrier, P. Goldner, S. Bertaina, T. Chanelière, D. Esteve, D. Vion, P. Bertet & E. Flurin, Nature 619, 276 (2023)< / c>

Claims

CLAIMS

1. A method for characterizing a sample (ECH) comprising a paramagnetic center (CP) containing an electronic spin (SEL) and a plurality of atomic nuclei (Ni, Nj, Nk) having respective nuclear spins (SNi, SNj, SNk) coupled to said electronic spin (SEL), said method comprising the following steps: A) placing said sample in a stationary magnetic field (B0), thereby obtaining a polarization of said electronic spin (SEL); B) selecting, by paramagnetic resonance spectroscopy, said paramagnetic center (CP) from among a plurality of paramagnetic centers contained in the sample (ECH); C) determining, by paramagnetic resonance, the values ​​of the hyperfine coupling coefficients Aj between each said nuclear spin (SNi, SNj, SNk) and the electronic spin (SEL) of said paramagnetic center (CP);wherein: - said step A) comprises the magnetic coupling of the sample to a microwave resonator (MHR) having a resonance frequency ω0 / 2π equal to the Larmor frequency of said electron spin (ES) in the stationary magnetic field (B0), the coupling constant and the quality factor of the resonator being sufficiently high such that the coupling with the resonator (MHR) dominates the relaxation dynamics of the electron spin (ES); and - said steps B) and C) implement the acquisition, by microwave photon counting, of noise signals (BS) produced by the return to equilibrium of said electron spin (ES).

2. A method according to claim 1 further comprising a step D) of determining, by nuclear magnetic resonance spectroscopy, coupling coefficients Cij between said nuclear spin, said probe nuclear spin (SNi), and the other said nuclear spins (SNj, SNk);wherein said step B) comprises a substep of indirect detection of a state of said probe nuclear spin (SNi) by means of said electron spin (SEL); said indirect detection substep comprising the application to the sample; of a plurality of paramagnetic resonance pulse sequences (IHF) adapted to excite said electron spin with a probability that is a function of the state of the probe nuclear spin, and the acquisition, by microwave photon counting, of respective noise signals (SBS) produced by the return to equilibrium of said electron spin (SEL).

3. A method according to claim 2 wherein step D comprises applying to the sample (ECH), in the following order: d1) a sequence of microwave pulses (IHF) of paramagnetic resonance adapted to transfer the electron spin polarization (SEL) of the paramagnetic center (CP) to the probe nuclear spin (Sni); d2) a sequence of radio frequency pulses (IRF) of nuclear magnetic resonance adapted to modify the state of the probe nuclear spin (Sni) as a function of its coupling coefficients Cij with said other nuclear spins (SNj, SNk); and d3) said sequence of microwave pulses (IHF) of paramagnetic resonance adapted to excite the electron spin (SEL) of the paramagnetic center (CP) with a probability that is a function of the state of the probe nuclear spin ((Sni).

4. A method according to claim 3 wherein the nuclear magnetic resonance radio frequency (IRF) pulse sequence of substep d2) is a spin echo double resonance (SEDOR) sequence.

5. A method according to claim 1 further comprising a step F) of determining the relative positions of said atomic nuclei (Ni, Nj, Nk) from the values ​​of the hyperfine coupling coefficients Aj.

6. A method according to any one of claims 2 to 4 further comprising a step E) of determining the relative positions of said atomic nuclei (Ni, Nj, Nk) from the coupling coefficients Cij.

7. An apparatus for carrying out a method according to any one of the preceding claims comprising: - a superconducting microwave resonator (SMR) comprising a sub-micron electromagnetic field confinement zone (CFZ) adapted to accommodate a sample (SEM) to be characterized; - a stationary magnetic field generation system (SFS) (SMF) having at least one component perpendicular to an orientation (y) of a magnetic component of said electromagnetic field in said confinement zone; - a Bragg mirror (BR) tuned to a resonance frequency of said superconducting microwave resonator (SMR), coupling said resonator to a signal input and output port (SIP); and - an excitation and measurement system (EMS) connected to said signal input and output port (SIP) and configured to inject into said superconducting microwave resonator (SMR) an excitation microwave signal (IMF) at said resonance frequency and to detect a microwave signal (SBS) emitted by said sample (SEM);characterized in that said excitation and measurement system (EMS) is also configured to inject a radio frequency (RF) signal into said resonator; and comprises a microwave photon counter (MPC) for detecting said microwave signal (BSS) emitted by said sample (SHE).

8. Apparatus according to claim 7 in which said excitation and measurement system (EMS) is also configured to inject a matched direct current (DDC) signal into said superconducting microwave resonator (SMR) to temporarily modify its resonant frequency.

9. Apparatus according to any one of claims 7 or 8 in which said superconducting microwave resonator (SMR) is fabricated using planar technology on an insulating substrate (IS).

10. Apparatus according to any one of claims 7 to 9 in which said superconducting microwave resonator (SMR) has a quality factor greater than or equal to 10; 2and preferably greater than or equal to 10 4 .

11. An apparatus according to any one of claims 7 to 9, wherein said superconducting microwave resonator (SMR) has a resonant frequency between 5 GHz and 100 GHz.

12. An apparatus according to any one of claims 7 to 9, wherein said stationary magnetic field generation system (B0) comprises a superconducting coil (BS) configured to generate said magnetic field.

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