Electromagnetic-field sensor and method

Abstract

A sensor includes a quantum system, one or more electromagnetic cavities, and one or more coupling elements. The quantum system interacts with an external electromagnetic field to form a quantum state that varies with the external electromagnetic field. Each of the electromagnetic cavities has a cavity mode. Each of the coupling elements couple the electromagnetic cavities to each other. At least one of the coupling elements is an asymmetric coupling element with gain and one of the coupling elements includes a transmission line. The quantum system is located adjacent to the transmission line. When the external electromagnetic field is present: the quantum state of the quantum system couples with a transverse mode of the transmission line to induce, in the transmission line, a phase shift based on the external electromagnetic field; and the sensor self-oscillates at a frequency based on the phase shift.

Description

ELECTROMAGNETIC-FIELD SEN SOR AND METHODCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. Provisional Application No. 63 / 534,760, filed on 25 August 2023, the disclosure of which is incorporated herein by reference in its entirety.BACKGROUND

[0002] Quantum sensing, which exploits the laws of quantum mechanics to measure small changes in quantities like magnetic fields, electric fields, strain, temperature, etc., is a promising application of quantum technologies. However, most of the sensitivity enhancements that the laws of quantum mechanics provide need entanglement or suitable nonlinearities as a resource, both of which can be challenging to realize in experiments because of decoherence and other constraints.SUMMARY

[0003] The present embodiments include a class of sensors that draw upon the principles of non-Hermitian lattices and magnetic- field sensitive devices like yttrium- iron-garnet (YIG) spheres and flux-tunable transmon qubits. Lattice structures utilizing high-Q microwave cavities and non- reciprocal hopping through directional amplifiers or microwave isolators may be used to realize non-Hermitian physics. These high-Q, three- dimensional (3D) microwave cavities interface with YIG spheres or transmon qubits to introduce magnetic field sensitivity that is enhanced by exceptional points that emerge from the non-Hermitian physics of the system.

[0004] Some of the present embodiments leverage experimental techniques from superconducting quantum computers, which are adapted to create unique non- Hermitian sensors. However, many of the present embodiments may be constructed using high-Q 3D microwave cavities and YIG spheres that are mechanically robust and can function across a wide temperature range. This marks a significant evolution from traditional quantum sensors that need cryogenic operation and entanglement. Moreover, the non-Hermitian exceptional points utilized by the present embodiments may be tailored to different sensing targets, such as electric fields and strain.

[0005] By merging insights and expertise from quantum sensing and quantum computing, the present embodiments include robust, versatile sensors that exceed current precision levels by utilizing exceptional points and non-Hermitian physics. These sensors may be used, for example, for high-bandwidth sensing of the Earth's magnetic field, enabling navigation for aircraft and spacecraft in GPS-compromised environments.

[0006] In a first aspect, a sensor includes a quantum system, a plurality7of electromagnetic cavities, and a plurality7of coupling elements. The quantum system is configured to interact with an external electromagnetic field to form a quantum state that varies with the external electromagnetic field. The plurality of electromagnetic cavities include (i) a hybridization cavity configured to interact with the quantum system and (ii) two or more non-hybridization cavities configured to not interact with the quantum system. Each of the plurality of electromagnetic cavities has a cavity mode. The plurality of coupling elements that couple the plurality of electromagnetic cavities to each other. At least one of the plurality of coupling elements being an asymmetric coupling element. When the external electromagnetic field is present, the quantum state of the quantum system hybridizes with the cavity mode of the hybridization cavity to form a hybridized quantum state: and the sensor forms a composite quantum state based on the hybridized quantum state and the cavity mode of each of the two or more non-hybridization cavities. The composite quantum state is located near an exceptional point of the sensor.

[0007] In a second aspect, a sensor includes a quantum system, one or more electromagnetic cavities, and one or more coupling elements. The quantum system is configured to interact with an external electromagnetic field to form a quantum state that varies with the external electromagnetic field. Each of the one or more electromagnetic cavities has a cavity mode. Each of the one or more coupling elements couple the one or more electromagnetic cavities to each other. At least one of the one or more coupling elements is an asymmetric coupling element with gain. One of the one or more coupling elements includes a transmission line. The quantum system is located adjacent to the transmission line. When the external electromagnetic field is present: the quantum state of the quantum system couples with a transverse mode of the transmission line to induce, in the transmission line, a phase shift based on the external electromagnetic field; and the sensor self-oscillates at a frequency based on the phase shift.

[0008] In a third aspect, a method includes coupling a probe electromagnetic wave into the sensor of the first aspect The method also includes coupling a transmittedelectromagnetic wave out of the sensor; and detecting the transmitted electromagnetic wave to measure an eigenenergy of the composite quantum state.BRIEF DESCRIPTION OF THE FIGURES

[0009] FIG. 1 is a functional block diagram of a first embodiment of an electromagnetic (EM) field sensor, in an embodiment.

[0010] FIG. 2 is a functional block diagram of a second embodiment of an EM field sensor that includes coupling elements having sufficient gain for the sensor to selfoscillate.

[0011] FIG. 3 shows example spectra of a resonance created when a quantum system couples to a transmission line, each of which is part of the EM field sensor of FIG, 2.

[0012] FIG. 4 illustrates a sensor that is an example of the EM field sensor of FIG. 2.

[0013] FIG. 5 includes a plot illustrating coupling of magnons to a high-Q 3D microwave cavity, which is an example of a cavity of the sensor of FIG. 4.

[0014] FIG. 6 includes a transmission plot, a characteristic polynomial plot, and eigenvalue plot for room-temperature operation of an embodiment of the sensor of FIGs. 1 and 2.

[0015] FIG. 7 shows transmission plots for cryogenic operation of an embodiment ofthe sensor of FIGs. 1 and 2.

[0016] FIG. 8 and FIG. 9 are flowcharts illustrating respective methods for sensing an electromagnetic field.DETAILED DESCRIPTION

[0017] FIG. 1 is a functional diagram of a sensor 100 that may be used to measure the magnitude of an external electromagnetic field 115. For example, external electromagnetic field 115 may be generated by a sample 105 that is located proximate to sensor 100. External electromagnetic field 115 may be static, i.e., externa] electromagnetic field 115 does not change significantly (in direction, magnitude, or both) over the temporal duration of the measurement For clarity herein, it is assumed that external electromagnetic field 115 is a magnetic field. However, external electromagnetic field 115 may alternatively be an electric field.

[0018] Sensor 100 includes three or more electromagnetic cavities 110. For clarity in the figures, each of cavities 110 is shown as a node of a graph. In the example of FIG. 1, sensor 100 has a first cavity 110(1), a second cavity 110(2), and a third cavity 110(3).However, sensor 100 may alternatively have a different integer number of cavities 110 (e.g., four or more) without departing from the scope hereof Each of cavities 110 may be a microwave cavity that forms at least one mode that resonates ata microwave frequency. While the term “microwave” typically refers to frequencies between 300 MHz and 300 GHz, the term "microwave” is used herein to include radio waves with frequencies less than 300 MHz (e.g., VHF, LF, etc.) and infrared radiation with frequencies greater than 300 GHz (e.g., far-infrared, terahertz radiation, etc.). Accordingly, each of cavities 110 need not be a microwave cavity.

[0019] Sensor 100 also includes two or more coupling elements 120 that couple electromagnetic cavities 110 together. Electromagnetic cavities 110 and coupling elements 120 may form a non- reciprocal lattice. For clarity in the figures, each of coupling elements 120 is a two-port component shown as a vertex connecting two nodes (i.e., two cavities 110) of the graph. For example, in FIG. 1 sensor 100 has a first coupling element 120(1) that couples first cavity 110(1) and second cavity 110(2), a second coupling element 120(2) that couples second cavity 110(2) and third cavity 110(3), and a third coupling element 120(3) that couples third cavity 110(3) and first cavity 110(1). Sensor 100 may alternatively have only two coupling elements 120 or four or more coupling elements 120.

[0020] In embodiments, first coupling element 120(1) is a first amplifier 122(1) having a first input connected to first cavity 110(1) and a first output connected to second cavity 110(2). Furthermore, second coupling element 120 (2) is a second amplifier 122(2) having a second input connected to second cavity 110(2) and a second output connected to third cavity 110(3). First amplifier 122(1) and second amplifier 122(2) may have the same gain or different gains.

[0021] At least one of coupling elements 120 is asymmetric, i.e., it has a forward coupling strength in the forward direction that is greater than its reverse coupling strength in the reverse direction. For clarity in the figures, the vertex points in the forward direction, thereby making the graph directed. For example, in FIG. 1 first coupling element 120(1) includes a first amplifier 122(1) whose input connects to first cavity 110(1) and whose output connects to second cavity7110(2). In this case, the forward coupling strength going from first cavity7110(1) to second cavity 110(2) is greater than the reverse coupling strength from second cavity 110(2) to first cavity 110(2). Therefore, first coupling element 120(1) is shown as a vertex that points towardsecond cavity 110(2]. The same argument applies to second coupling element 120(2]. In some embodiments, cavities 110 and coupling elements 120 form a non-reciprocal lattice.

[0022] In embodiments, one of cavities 110 is a hybridization cavity that contains a quantum system 140 that interacts with external electromagnetic field 115 to form a quantum state that varies with external electromagnetic field 115. In embodiments, the hybridization cavity has, or functions as, one or both of an input port and an output port.

[0023] Quantum system 140 also interacts with a cavity mode of the hybridization cavity to form a hybridized quantum state. In the example of FIG. 1, quantum system 140 is located within third cavity7110(3), therefore making third cavity7110(3) the hybridization cavity. All of the other cavities 110 are referred to as non-hybridization cavities since they do not contain any such quantum system. Sensor 100 may include a field source 107 that generates a bias field 116, which hybridizes the quantum state of quantum system 140 with the cavity mode of the hybridization cavity (e.g., third cavity 110(3)). Field source 107 may be a magnetic field source, such as a magnet. Bias field 116 may be a magnetic field, an electric field, an electromagnetic field, or a combination thereof

[0024] In embodiments of sensor 100, cavities 110 form a lattice of coupled cavities, where cavities 110 alternate between non-hybridization cavities and hybridization cavities. Thatis, each pair of coupled cavities ofthe lattice Includes one non-hybridization cavity and one hybridization cavity7. Sensor 100 may include two or more cavities 110 that are non-hybridization cavities, wherein at least one of said cavities may have, or function as, an input port and / or an output port

[0025] FIG. 1 shows quantum system 140 as a Y1G sphere whose energy varies with external electromagnetic field 115. However, without departing from the scope hereof, quantum system 140 may alternatively be another type of quantum system that couples to both the cavity mode of the hybridization cavity and external electromagnetic field 115 to form a hybridized quantum state. Examples of such alternative quantum systems include, but are not limited to, molecular defects and color centers (e.g., nitrogen-vacancy centers in diamond, detects in SIN, etc, ), atoms (e.g., optically trapped cold neutral atoms, cold trapped ions, transition-metal atoms embedded in a thin film, etc.), molecules, and superconducting qubits (e.g., transmons).

[0026] For hybridization to occur, the quantum state of the quantum system should have an energy near that of the cavity mode of the hybridization cavity (e.g., within ten times the linewidth of the cavity mode). The quantum state has a quantum-state eigenfrequency. The cavity mode of each electromagnetic cavity 110 has a cavity-mode eigenfrequency. In embodiments, the quantum-state eigenfrequency and each cavitymode eigenfrequency are both in either (i) the radio-frequency, (ii) the microwave, or (iii) the millimeter- wave region of the electromagnetic spectrum.

[0027] Bias field 116 may be applied to vary the quantum state and its energy, Sensor 100 forms a composite quantum state that is based on the hybridized quantum state and the cavity mode of each of the two or more non-hybridization cavities. The composite quantum state is located near an exceptional point of sensor 100. In this regime (i.e,, when the sensor 100 operates near the exceptional point), small (i.e,, perturbative) changes in external electromagnetic field 115 advantageously change the energy' of the composite quantum state less than linearly, thereby making sensor 100 more sensitive (i.e., having a steeper slope) as compared to sensors that do not operate near exceptional points (e.g., diabolic points),

[0028] The energy of the composite quantum state may be measured using a transmission-type measurement. As shown in FIG. 1, an excitation signal 112, such as an electromagnetic wave, may be coupled into one of cavities 110. A transmission signal 114, such as an electromagnetic wave, maybe coupled out of one of cavities 110, where it may be processed to determine the energy of the composite quantum state (e.g., by measuring the frequency of the transmission signal 114). While FIG. 1 shows the excitation signal 112 being coupled into first cavity 110(1) (e.g., through an input port of first cavity 110(1)), the excitation signal 112 may alternatively be coupled into any of cavities 110, Similarly, while FIG. 1 shows the transmission signal 114 being coupled out of third cavity 110(3), the transmission signal 114 may alternatively be coupled out of any of cavities 110.

[0029] While FIG. 1 shows sensor 100 with two asymmetric coupling elements, sensor 100 may alternatively have only one asymmetric coupling element (e.g., only first amplifier 122(1) or only second amplifier 122(2)) or more than twro asymmetric coupling elements. While FIG. 1 shows each asymmetric coupling element as an amplifier, each asymmetric coupling element may alternatively be or include a directional coupler, acirculator, a phase shifter, a combination thereof and / or another kind of component known in the art.

[0030] Each of coupling elements 120 includes all of the components in between the pair of cavities 110 it couples. Such components may include transmission Sines, active components (e.g., amplifiers), passive components (e.g., attenuators, circulators), and the like. In the example of FIG. 1, cavities 110 and coupling elements 120 create a loop, and the asymmetric coupling elements establish a directionality 118 indicating how energy flows through the loop. It is not necessary that such a loop be formed. Nor is it necessary that a clear directionality (e.g., through a loop or a linearly coupled chain of cavities 110) be established.

[0031] In sensor 100, the number of coupling elements 120 may be less than the number of electromagnetic cavities 110 such that at least one electromagnetic cavity 110 is directly coupled to only one other electromagnetic cavity 110. In sensor 100, the number of coupling elements 120 may be greater than the number of electromagnetic cavities 110 such that at least one electromagnetic cavity 110 is directly coupled to at least two other cavities 110, In sensor 100, the number of coupling elements 120 may be equal to the number of electromagnetic cavities 110 such that each electromagnetic cavity 110 is directly coupled to only two other cavities 110,

[0032] FIG, 2 is a functional diagram of a sensor 200 which includes at least one cavity 110, at least one coupling element 220, and quantum system 140. Coupling elements 220 couple electromagnetic cavities 110 to each other. At least one coupling element 220 is an asymmetric coupling element with gain. This asymmetric coupling may include an amplifier, such as a tunable amplifier. One of coupling elements 220 includes a transmission line 222, and quantum system 140 is located adjacentto transmission line 222, One of cavities 110 may include or function as an output port,

[0033] Quantum system 140 is configured to interact with external electromagnetic field 115 to form a quantum state that varies with external electromagnetic field 115, When external electromagnetic field 115 is present: (i) the quantum state of quantum system 140 couples with a transverse mode of transmission line 222 line to induce, in transmission line 222, a phase shift based on external electromagnetic field 115, and (ii) sensor 200 self-oscillates at a frequency based on the phase shift

[0034] In embodiments, sensor 200 includes several cavities 110 and several coupling elements 220. The number of coupling elements 220 may be equal to thenumber of cavities 110 such that each cavity 110 is directly coupled to only two other cavities 110. The several cavities 110 and several coupling elements 220 may form a loop.

[0035] Sensor 200 may include cavities 110(1) and 110(2) in which each of coupling elements 220(1) and 220(2) couples cavities 110(1) and 110(2) to each other, as shown in FIG, 2. In such embodiments, coupling element 220(1) includes first amplifier 122(1), which has a first input connected to first cavity 110(1) and a first output connected to second cavity 110(2). Similarly, in such embodiments, coupling element 220(2) includes second amplifier 122(2), which has a second input connected to second cavity 110(2) and a second output connected to first cavity 110(1).

[0036] Sensor 200 may include magnetic field source 107. In sensor 200, field source 107 generates bias field 116 that hybridizes the quantum state of quantum system 140 with the transverse mode of transmission line 222.

[0037] Coupling element 220 is an example of coupling element 120, in which coupling element 220 has sufficient gain for sensor 200 to self-oscillate. This selfoscillation is shown in FIG. 2 as an internal oscillation 212 tbatpropagat.es through sensor 200 in a loop. Part of internal oscillation 212 is coupled out of one of cavities 110 to form an output signal 214 that may be measured to determine the magnitude of external electromagnetic field 115. Since sensor 200 self-oscillates, it is not necessary to couple an excitation signal (e.g., the excitation signal 112 of FIG. 1) into sensor 200. Sensor 200 also differs from embodiments of sensor 100 of FIG. 1, as sensor 200 may have only one cavity 110 or only two cavities 110. In general, sensor 200 has one or more cavities 110 and one or more coupling elements 220, as described in more detail below.

[0038] First coupling element 220(1) includes a field-sensitive phase shifter 230 that phase shifts internal oscillation 212 propagating along first coupling element 220(1). This phase shift is based on external electromagnetic field 115. Phase shifter 230 includes quantum system 140. However, instead of hybridizing with the cavity mode of one of cavities 110, quantum system 140 is placed adjacent to transmission line 222 so that quantum system 140 hybridizes with one of the transverse (i.e., propagation) modes of the transmission line 222. While FIG. 1 shows transmission line 222 as a coplanar waveguide, transmission line 222 may alternatively be another type of planar waveguide (e.g., a stripline, or a microstrip line) or transmission line (e.g., a planar transmission line) whose modes can couple with quantum system 140. As shown in FIG. 2, quantum system 140 is a YIG sphere, in which case a magnon couples to transmission line 222.

[0039] In embodiments, sensor 200 operates at room temperature and at least one of cavities 110 and coupling elements 220 introduces loss. Cavities 110(1) and 110(2) may be modeled as harmonic oscillators. When the gain is turned very low, the harmonic oscillators decouple, resulting in a very small signal. This low-gain regime may be described by an effective Hamiltonian H :

[0040] In an example mode of operation, the value of gain reaches the point where it compensates for the loss in sensor 200 (e.g., in cables connecting cavities 110). In such scenarios, the gain is balanced between the two paths and the Hamiltonian H may be described by Eqn. 2, where t0is the coupling resulting from when the cavities are coupled by the same ports and no losses exist:In between these two regimes, off-diagonal coupling teffcan be tuned with amplifier gain between 0 and t0. Importantly, off-diagonal coupling can be tuned for each direction of photon transport individually.

[0041] With the introduction of the phase shifter in coupling elements 220, the Hamiltonian H is given bywhere e‘ ^ represents the tunable phase of the photons. The model of Eqn. 3 produces effective Hermitian dynamics when v although the constraintthat the off-diagonal elements be equal in magnitude does not produce interesting non- Hermitian physics.

[0042] Loss and drive terms may be introduced to these effective models (i.e., Eqns. 1-3), after which a mean-field approximation may be applied to predict the classical photon numbersandin the first cavity 110(1) and second cavity 110(2), respectively. With a drive e on the first site at frequencyand on-site dissipation K, the first time derivatives of the photon numbers may be expressed as:,

[0043] FIG. 3 shows a resonance created when quantum system 140 couples to transmission line 222. Specifically, the top of FIG. 3 is a plot showing transmission ofinterna] oscillation 212 through first coupling element 220(1) as a function of frequency. The bottom of FIG. 3 is a plot showing the induced phase-shift as a function of frequency . FIG. 3 may be thought of as a transfer function of phase shifter 230. The resonance in FIG. 3 is centered near the magnon frequency (i.e., the frequency of the quantum state) of quantum system 140, and therefore varies with external electromagnetic field 115. As external electromagnetic field 115 slightly varies (e.g., by approximately ±1 linewidth of the resonance), the induced phase shift rapidly varies between +n / 2 and —n / 2. This rapid and dramatic variation in the induced phase shift changes the feedback condition of sensor 200, and therefore the self-oscillation frequency (i.e., the frequency of internal oscillation 212). Accordingly, a measured change in the frequency of output signal 214 may be used to determine the change in external electromagnetic field 115.

[0044] To establish a nominal frequency of the resonance in FIG. 3, bias field 116 may be applied to the quantum system (e.g., an external magnetic bias field when quantum system 140 is a YIG sphere). Furthermore, one or both of amplifiers 122(1) and 122(2) may be tunable. A tunable amplifier has both a gain and phase-shift that varies with its tuning. The phase-shift imparted by the tunable amplifier contributes to the overall phase shift of first coupling element 220(1), and therefore contributes to the feedback condition that determines the frequency of internal oscillation 212. A variable phase shifter (e.g., a voltage-controlled phase shifter) may also be added to first tuning element 220(1) and adjusted to establish oscillation.

[0045] FIG. 2 shows each of coupling elements 220(1) and 220(2) with an amplifier, and therefore as having gain and being asymmetric. However, it is not necessary that all of coupling elements 220 be asymmetric and have gain. For example, in FIG. 2 second amplifier 122(2) may be replaced with a directional coupler that has no gain. The gain of first amplifier 122(1) may then be increased to ensure that sensor 200 self-oscillates. Bias field 116) may additionally or alternatively be changed to vary the transmission and phase shift of first coupling element 220(1) to ensure that sensor 200 self-oscillates.[0046s] Similar to FIG, 1, each of coupling elements 220 in FIG. 2 includes all of the components in between the pair of cavities 110 it couples. Such components may include transmission lines and other components that affect the transmission and phase shift of the coupling element, and therefore affect the self-oscillation frequency. In the example of FIG. 2, cavities 110 and coupling elements 120 create a loop, and the asymmetric coupling elements establish a directionality 118 indicating how energy7flow's through theloop. It is not necessary that such a loop be formed. Nor is it necessary that a clear directionality (e.g., through a loop or a linearly coupled chain of cavities 110) be established.

[0047] While FIG. 2 shows sensor 200 with two cavities 110, sensor 200 may alternatively have only one cavity'' 110. In this embodiment, sensor 200 has one coupling element 220 that couples from one port of the one cavity' 110 to another port of the same one cavity 110. The one coupling element is asymmetric and may be configured similarly to first coupling element 220(1) shown in FIG. 2 (e.g., with the field-sensitive phase shifter 230). When represented as a graph, this embodiment forms a loop.

[0048] In some embodiments, self-oscillating sensor 200 of FIG. 2 and the "passive" sensor 100 of FIG. 1 are combined into a hybrid sensor. For example, the hybrid sensor may' have a self-oscillating section and a passive section. The seif-oscillating section generates internal oscillation 212, which is then transmitted to the passive section via a coupling element between one of cavities 110 of the self-oscillating section and one of cavities 110 of the passive section. A transmission signal (e.g., the transmission signal 114) may' then be coupled out of one of the cavities of the passive section and processed to determine the magnitude of the external magnitude field. The hybrid sensor may have more than one self-oscillating section, more than one passive section, or both.

[0049] Embodiments disclosed herein include sensors based on non-reciprocal lattices that utilize exceptional points to achieve more precise sensing. The system may be assembled from high-quality seamless 3D micro wave cavities with photon lifetimes beyond 1 ms, spins in YIG spheres that have high quality factors (Q) even at room temperature, and superconducting transmon qubits that can have lifetimes of >200 us and provide the ability to manipulate the states of photons in the 3D cavities. Once assembled in a lattice, the system may be used for magnetic-field sensing with «1 nT precision and high bandwidth, outperforming the precision of current technologies like nitrogen-vacancy (NV) centers in diamond. In addition, the present embodiments are mechanically' robust and may be used at various temperatures, potentially increasing the possible applications compared to platforms like NV centers or trapped atoms that require sensitive optics.

[0050] To realize exceptional points using non- Hermitian, non-reciprocal lattices built from high-Q 3D micro wave cavities, YIG spheres or flux-tunable superconducting qubits may' be strongly coupled to these lattices to introduce magnetic field sensitivity.This approach builds upon expertise in large-scale superconducting circuit lattices and a recent insight that existing commercial components can introduce non-reciprocal photon hopping elements, enabling non-Hermitian system dynamics. By coupling to YIG spins or flux-tunable transmon qubits, we can traverse exceptional points by tuning a magnetic field, creating a novel non-reciprocal lattice sensor. Due to the high Q of the constituent components, the present embodiments exhibit high-precision magnetic field sensing beyond current techniques based on NV centers in diamond, which are limited by spin dephasing.

[0051] FIG. 4 illustrates a sensor 400 which is an example of sensor 200. Sensor 400 includes microwave cavities 410(1) and 410(2) and a coupling element 420 that couples cavity 410(1) to cavity 410(2). Coupling element 420 functions as a non-reciprocal hopping element that only allows photons to hop in one direction, and may include one or more of a directional microwave amplifier, a microwave isolator, or a microwave circulator. Sensor 400 also includes a quantum system 440 coupled to cavity 410(2) to introduce magnetic field sensitivity. Cavities 410 are examples of cavities 110 of FIG. 1. Similarly, coupling element 420 is an example of coupling element 122 of FIG. 1. Quantum system 440 may include YIG spheres or flux-tunable transmon qubits. Sensor 400 enables the realization of magnetic-field-dependent tight-binding Hamiltonians that are non- Hermitian and realize exceptional points that are sensitive to small changes in the magnetic field.Coupling Magnetic Fields to High-Q Microwave Cavities

[0052] FIG. 5 includes a plot 520 illustrating coupling of magnons to a high-Q 3D microwave cavity 510, which is an example of cavity 410(2) of FIG. 4. Cavity 510 may be tuned using an external magnet. Plot 520 shows transmission as a function of the magnetic field.

[0053] To create high-Q lattices with magnetic field sensitivity, the present embodiments may use a strongly coupled flux-tunable superconducting qubit or a macroscopic spin known as a magnon in YIG spheres to impart a magnetic-field- dependent dispersive shift to the cavities (e.g., cavity 510). The YIG spins may be tuned by an electromagnet. Due to the small mode volumes of the microwave cavities compared to the YIG spheres, strong coupling can be achieved between the photons in the cavities and the magnon spins in the YIG spheres. Furthermore, YIG spins are remarkably high-Qat microwave frequencies (meaning their transition linewidths are narrow) and have spin transitions that depend on magnetic fields. By coupling YlGs to microwave cavities, a magnetic-field dependence may be introduced to the modes of the system that can then be probed spectroscopically (plot 520).

[0054] In contrast to other spin systems, YIG spins are shielded by large volumes of high-conductivity (and superconducting at cryogenic temperatures) metal, effectively shielding the YIG spins from other spins and deleterious effects of cosmic rays that lead to dephasing of the spins. This has the effect of broadening resonance linewidths. Although the best noise performance of the present embodiments will likely occur at cryogenic temperatures, where the 3D cavities become superconducting and thermal noise is minimized, the versatility of hardware involving YlGs opens new applications inaccessible to other hardware platforms, such as for extreme environments like airplanes or spacecraft.

[0055] Flux-tunable transmon qubits may be coupled to the cavities instead of YlGs. This approach may require cryogenic operation, butthe lattices will inherit the enhanced sensitivity of superconducting quantum interference devices (SQUIDs). Additionally, the introduction of superconducting qubits provides nonlinear control over the photonic modes, something used by all bosonic superconducting quantum computers.Nou-Reciprocal Lattices Using One-Directional Circuit Elements

[0056] The present embodiments leverage two unique properties of 3D microwave cavities: (1) they are incredibly high-Q (i.e., narrow linewidths), and (2) they may be integrated into lattices to realize different topologies. Prior work showed that lattices of superconducting microwave cavities realize effective photonic materials governed by tight-binding Hamiltonians that are uniquely determined by the cavity frequencies toc, the photon hopping rates t between different cavities, and the lattice connectivity or topology.

[0057] FIG. 6 includes a transmission plot 610, a characteristic polynomial plot 620, and an eigenvalue plot 630 for a room-temperature non-Hermitian sensor which is an example of sensor 100 (see FIG. 1) or sensor 200 (see FIG. 2). Plot 610 shows simulated transmission S21fin dB) through a non-reciprocal lattice with reasonable room temperature values (g=t=150 MHz, <uc=10 GHz, and ic~ 10 MHz) as a function of magnetic field strength B (in mT), Dashed line 619 indicates the YIG frequency(B). Plot 620shows characteristic polynomials 622, 624, and 626 at respective vertical dashed lines 612, 614, and 616 in piot 610, corresponding to different magnetic field strengths B, showing that the characteristic polynomials transition from having two eigenvalues that are complex conjugates (see characteristic polynomial 622) through a critical magnetic field Bc= 356.2 mT (see characteristic polynomial 624) to a regime with two real roots (see characteristic polynomial 626). Plot 630 shows the real and imaginary components of the resulting eigenvalues λ, where the gray level indicates Re[2] and shows thedependence shown in plot 610. Note that two other real roots exist but lie outside the frequency range plotted in FIG. 6.

[0058] In one embodiment of a non-reciprocal sensor, a lattice has three cavities in a ring with non-reciprocal hopping along two edges and a YIG spin that interacts with one of the cavities through a Jaynes-Cummings (see inset 611 of plot 610). Simulations of the device (see FIG. 6) show that an exceptional point exists in this triangular lattice, which may be used for magnetic-field sensing. This magnetic-field sensing may be performing by tracking homodyne transmission at various frequencies through the lattice near the exceptional point (B=356.2 mT). It should be noted that a simple dimer with onedirectional hopping and coupling to a YIG does not produce exceptional points in this system.Enhanced Sensing Using Non-Hermitian Sensors

[0059] Cryogenic temperatures significantly decrease the linewidths of 3D microwave cavities. FIG. 7 shows transmission plots 710 and 720 of a non- Hermitian sensor in cryogenic operation. This sensor is an example of sensor 100 of FIG. 1 or sensor 200 of FIG. 2. Plot 710 is similar to plot 610 but with expected cryogenic parameters of 0=700 MHz, t=5MHz, a.>c=10 GHz, and k=3.5 kHz. Plot 720 is a line-cut of theoretical transmission S21(in dB) versus magnetic field strength B (in nT) for the non-Hermitian sensor (curve 722). Plot 720 also shows the change in a simulated Lorentzian resonance at the YIG frequencyas the magnetic field strength B is changed (see curve 724), showing significant advantages for the non-Hermitian sensor.

[0060] Preliminary simulations predict that cryogenic operation will realize a sensor where the transmission (S21, plot 720) changes significantly with magnetic field changes significantly less than one nanotesla (see plot 720). The sensor may also be used to measure the AC magnetic fields.

[0061] FIG. 8 and FIG. 9 are flowcharts illustrating respective methods 800 and 900 for sensing an electromagnetic field. In embodiments, methods 800 and 900 are implemented within one or more aspects of sensor 100 of FIG. 1 and sensor 200 of FIG. 2, respectively. Method 800 includes atleastone of steps 820, 830, and 840. Method 800 may also include at least one of steps 810 and 850. Method 900 includes at least one of steps 910, 920, and 950.

[0062] The following description of methods 800 and 900 include parenthetical numbers following terms used in a method step. The parenthetical number indicates that the element associated with the number in parenthesis is an example of the term. For example, the description of step 820 below recites "a probe electromagnetic wave (112)," which means that excitation signal 112 of FIG. 1 is an example of the probe electromagnetic wave introduced in step 820.

[0063] Step 810 includes applying a bias electromagnetic field (116) to a hybridization cavity (110(3)) of a sensor (100) such that the quantum state of the quantum system (140) of the sensor hybridizes with the cavity mode of the hybridization cavity.

[0064] Step 820 includes coupling a probe electromagnetic wave (112) into the sensor. Step 830 includes coupling a transmitted electromagnetic wave (114) out of the sensor. Step 840 includes detecting the transmitted electromagnetic wave to measure an eigenenergy of a composite quantum state formed by the sensor. Step 850 includes processing the measured eigenenergy to determine a magnitude of the external electromagnetic field (115). Method 800 may also include at least one of tuning an asymmetric coupling element (122) of the sensor and cryogenically cooling the sensor.

[0065] In method 900, step 910 includes applying a bias electromagnetic field (116) such that a quantum state of a quantum system (140) of a sensor (200) hybridizes with the transverse mode of a transmission line of the sensor. Step 920 includes detecting an oscillation (212) of the sensor. Step 950 includes processing the detected oscillation to determine a magnitude of the external electromagnetic field (115). Step 950 may include measuring a frequency7of the oscillation. Method 900 may also include at least one of tuning an asymmetric coupling element (220) of the sensor and cryogenically cooling the sensor.Combinations of Features

[0066] Features described above, as well as those claimed below, may be combined in various ways without departing from the scope hereof. The following enumerated examples illustrate some possible, non-limiting combinations.

[0067] Embodiment 1. A sensor includes a quantum system, a plurality of electromagnetic cavities, and a plurality of coupling elements. The quantum system is configured to interact with an external electromagnetic field to form a quantum state that varies with the external electromagnetic field. The plurality of electromagnetic cavities include (i) a hybridization cavity configured to interact with the quantum system and (ii) two or more non- hybridization cavities configured to not interact with the quantum system. Each of the plurality of electromagnetic cavities has a cavity mode. The plurality of coupling elements that couple the plurality of electromagnetic cavities to each other. At least one of the plurality of coupling elements being an asymmetric coupling element. When the external electromagnetic field is present, the quantum state of the quantum system hybridizes with the cavity mode of the hybridization cavity to form a hybridized quantum state; and the sensor forms a composite quantum state based on the hybridized quantum state and the cavity mode of each of the two or more non-hybridization cavities. The composite quantum state is located near an exceptional point of the sensor.

[0068] Embodiment 2. The sensor of embodiment 1, the number of the plurality of coupling elements being less than the number of the plurality of electromagnetic cavities such that at least one of the plurality of electromagnetic cavities is directly coupled to only one other of the plurality of electromagnetic cavities.

[0069] Embodiments. The sensor of embodiment 1 or 2, the number of the plurality of coupling elements being greater than the number of the plurality of electromagnetic cavities such at least one of the plurality of electromagnetic cavities is directly coupled to at least two others of the plurality of electromagnetic cavities.

[0070] Embodiment 4, The sensor of any one of embodiments 1-3, the number of the plurality of coupling elements being equal to the number of the plurality of electromagnetic cavities such that each of the plurality of electromagnetic cavities is directly coupled to only two others of the plurality of electromagnetic cavities.

[0071] Embodiment 5. The sensor of any one of embodiments 1-3, the plurality of electromagnetic cavities and the plurality of coupling elements forming a non-reciprocal lattice.

[0072] Embodiment 6. The sensor of any one of embodiments 1-3, the external electromagnetic field being a magnetic field.

[0073] Embodiment 7. The sensor of embodiment 6, further comprising a magnetic field source configured to generate a bias magnetic field that hybridizes the quantum state of the quantum system with the cavity mode of the hybridization cavity.

[0074] Embodiment 8. The sensor of embodiment 7, the magnetic field source comprising a magnet

[0075] Embodiment 9. The sensor of any one of embodiments 1~8, the external electromagnetic field being an electric field.

[0076] Embodiment 10. The sensor of any one of embodiments 1-9, wherein: the quantum state has a quantum-state eigenfrequency that lies in the radio-frequency, microwave, or millimeter- wave regions of the electromagnetic spectrum; and the cavity mode of each of the plurality of electromagnetic cavities has a cavity-mode eigenfrequency that lies in the radio-frequency, microwave, or millimeter- wave regions of the electromagne tic spectrum.

[0077] Embodiment 11, The sensor of any one of embodiments 1-10, the quantum system being a Y1G sphere.

[0078] Embodiment 12, The sensor of any one of embodiments 1-10, the quantum system being a superconducting qubit,

[0079] Embodiment 13. The sensor of embodiment 12, the superconducting qubit being a transmon.

[0080] Embodiment 14. The sensor of any one of embodiments 1-113, the asymmetric coupling element comprising an amplifier.

[0081] Embodiment 15. The sensor of embodiment 14, the amplifier being a tunable amplifier.

[0082] Embodiment 16. The sensor of any one of embodiments 1-15, the asymmetric coupling element comprising a directional coupler or circulator.

[0083] Embodiment 17. The sensor of any one of embodiments 1-16, the asymmetric coupling element including a phase shifter.

[0084] Embodiment 18, The sensor of any one of embodiments 1-10, one of the plurality of electromagnetic cavities forming an input port.

[0085] Embodiment 19. The sensor of embodiment 18, said one of the plurality of electromagnetic cavities being the hybridization cavity.

[0086] Embodiment 20. The sensor of embodiment 18 or 19, said one ofthe plurality of electromagnetic cavities being one ofthe two or more non-hybridization cavities.

[0087] Embodiment 21. The sensor any one of embodiments 1-20, one of the plurality of electromagnetic cavities forming an output port

[0088] Embodiment 22. The sensor of embodiment 21, said one of the plurality of electromagnetic cavities being the hybridization cavity.

[0089] Embodiment 23. The sensor of either of embodiment 21 or 22, said one of the plurality of electromagnetic cavities being one of the two or more non-hybridization cavities.

[0090] Embodiment 24. The sensor of any one of embodiments 21-23, said one of the plurality of electromagnetic cavities also forming an input port.

[0091] Embodiment 25. The sensor of any one of embodiments 1-24, wherein: the plurality of electromagnetic cavities comprises a first cavity, a second cavity, and a third cavity; and the plurality of coupling elements comprises a first coupling element that couples the first and second cavities, a second coupling element that couples the second and third cavities, and a third coupling element that couples the third and first cavities.

[0092] Embodiment 26. The sensor of embodiment 25, wherein: the first coupling element is a first amplifier having a first input connected to the first cavity7and a first output connected to the second cavity; and the second coupling element is a second amplifier having a second input connected to the second cavity and a second output connected to the third cavity.

[0093] Embodiment 27. The sensor of either one of embodiments 26 and 27, the first and second amplifiers having the same gain.

[0094] Embodiment 28. A method includes coupling a probe electromagnetic wave into the sensor of embodiment 1; coupling a transmitted electromagnetic wave out of the sensor; and detecting the transmitted electromagnetic wave to measure an eigenenergy of the composite quantum state.

[0095] Embodiment 29. The method of embodiment 28, further comprising processing the measured eigenenergy to determine a magnitude of the external electromagnetic field.

[0096] Embodiment 30. The method of either one of embodiment 28 or 29, further comprising cryogenically cooling at least part of the sensor.

[0097] Embodiment 31. The method of any one of embodiments 28-30, further comprising tuning the asymmetric coupling element

[0098] Embodiment 32. The method of any one of embodiment 28-31, further comprising applying a bias electromagnetic field to the hybridization cavity such that the quantum state of the quantum system hybridizes with the cavity mode of the hybridization cavity7.

[0099] Embodiment 33. A sensor includes a quantum system, one or more electromagnetic cavities, and one or more coupling elements. The quantum system is configured to interact with an external electromagnetic field to form a quantum state that varies with the external electromagnetic field. Each of the one or more electromagnetic cavities has a cavity mode. Each of the one or more coupling elements couple the one or more electromagnetic cavities to each other,. At least one of the one or more coupling elements is an asymmetric coupling element with gain. One of the one or more coupling elements includes a transmission line. The quantum system is located adjacent to the transmission line. When the external electromagnetic field is present: the quantum state of the quantum system couples with a transverse mode of the transmission line to induce, in the transmission line, a phase shift based on the external electromagnetic field: and the sensor self-oscillates at a frequency that is based on the phase shift such that the frequency determines a magnitude of the external electromagnetic field or a relative change of the external electromagnetic field with respect to a reference point.

[0100] Embodiment 34. The sensor of embodiment 33, wherein: the one or more electromagnetic cavities comprises a plurality of electromagnetic cavities; the one or more coupling elements comprises a plurality of coupling elements; the number of the plurality of coupling elements being equal to the number of the plurality7of electromagnetic cavities such that each of the plurality of electromagnetic cavities is directly coupled to only two others of the plurality of electromagnetic cavities.

[0101] Embodiment 35. The sensor of either one of embodiments 33 and 34, wherein: the one or more electromagnetic cavities comprises a plurality of electromagnetic cavities; the one or more coupling elements comprises a plurality7of coupling elements; the plurality7of electromagnetic cavities and the plurality of coupling elements forming a loop.

[0102] Embodiment 36. The sensor of any one of embodiments 33-35, the external electromagnetic field being a magnetic field.

[0103] Embodiment 37. The sensor of embodiment 36, further comprising a magnetic field source configured to generate a bias magnetic field that hybridizes the quantum state of the quantum system with the transverse mode of the transmission line.

[0104] Embodiment 38, The sensor of embodiment 37, the magnetic field source comprising a magnet.

[0105] Embodiment 39. The sensor of any one of embodiments 33-38, the external electromagnetic field being an electric field.

[0106] Embodiment 40. The sensor of any one of embodiments 33-39, wherein: the quantum state has a quantum-state eigenfrequency that lies in the radio-frequency, microwave, or millimeter- wave regions of the electromagnetic spectrum; and the cavity mode of each of the one or more electromagnetic cavities has a cavity-mode eigenfrequency that lies in the radio-frequency, microwave, or millimeter-wave regions of the electromagnetic spectrum.

[0107] Embodiment 41. The sensor of any one of embodiments 33-40, the quantum system being a Y1G sphere,

[0108] Embodiment 42. The sensor of any one of embodiments 33-41, the quantum system being a superconducting qubit.

[0109] Embodiment 43. The sensor of embodiment 42, the superconducting qubit being a transmon.

[0110] Embodiment 44. The sensor of any one of embodiments 33-43, the asymmetric coupling element comprising an amplifier.

[0111] Embodiment 45. The sensor of any one of embodiments 33-44, the amplifier being a tunable amplifier.

[0112] Embodiment 46. The sensor of any one of embodiments 33-45, the transmission line comprising a planar transmission line,

[0113] Embodiment 47. The sensor of embodiment 46, the planar transmission line comprising a coplanar waveguide or a microstrip line.

[0114] Embodiment 48. The sensor of any one of embodiments 33-47, one of the one or more electromagnetic cavities forming an output port,

[0115] Embodiment 49. The sensor of any one of embodiments 33-48, wherein: the one or more electromagnetic cavities comprises a first cavity and a second cavity; and the one or more coupling elements comprises a first coupling element that couples the firstand second cavities and a second coupling element that couples the first and second cavities.

[0116] Embodiment 50. The sensor of embodiment 49, wherein: the first coupling element is a first amplifier having a first input connected to the first cavity and a first output connected to the second cavity; and the second coupling element is a second amplifier having a second input connected to the second cavity and a second output connected to the first cavity,

[0117] Embodiment 51. A method, comprising detecting an oscillation of the sensor of embodiment 33.

[0118] Embodiment 52. The method of embodiment 51, further comprising processing the detected oscillation to determine a magnitude of the external electromagnetic field.

[0119] Embodiment 53. The method of either one of embodiments 52 or 53, wherein said processing includes measuring a frequency of the oscillation.

[0120] Embodiment 54. The method of any one of embodiments 51-53, further comprising cryogenically cooling at least part of the sensor.

[0121] Embodiment 55. The method of any one of embodiments 51-54, further comprising tuning the asymmetric coupling element

[0122] Embodiment 56. The method of any one of embodiments 51-55, further comprising applying a bias electromagnetic field such that the quantum state of the quantum system hybridizes with the transverse mode of the transmission line.* * *

[0123] Changes may be made in the above methods and systems without departing from the scope of the present embodiments. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. Herein, and unless otherwise indicated the phrase "in embodiments" is equivalent to the phrase "in certain embodiments," and does not refer to all embodiments.

[0124] Regarding instances of the terms "and / or” and "at least one of,” for example, in the cases of "A and / or B,” "at least one of A and B,” and "at least one of A or B," such phrasing encompasses the selection of (i) A only, or (ii) B only, or (iii) both A and B. In the cases of “A, B, and / or C, " "at least one of A, B, and C,” and "at least one of A, B, or C,” such phrasing encompasses the selection of (i) A only, or ( ii) B only, or (iii) C only, or (iv) A andB only, or (v) A and C only, or (vi) B and C only, or (vii) each of A and B and C. This may be extended for as many items as are listed.

[0125] The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.

Claims

CLAIMSWhat is claimed is:

1. A sensor comprising: a quantum system configured to interact with an external electromagnetic field to form a quantum state that varies with the external electromagnetic field; a plurality of electromagnetic cavities including (i) a hybridization cavity7configured to interact with the quantum system and (ii) two or more non-hybridization cavities configured to not interact with the quantum system, each of the plurality of electromagnetic cavities having a cavity mode; and a plurality of coupling elements that couple the plurality7of electromagnetic cavities to each other, at least one of the plurality of coupling elements being an asymmetric coupling element; wherein, when the external electromagnetic field is present: the quantum state of the quantum system hybridizes with the cavity mode of the hybridization cavity to form a hybridized quantum state; and the sensor forms a composite quantum state based on the hybridized quantum state and the cavity mode of each of the two or more nonhybridization cavities, the composite quantum state being located near an exceptional point of the sensor.

2. The sensor of claim 1, the number of the plurality of coupling elements being less than the number of the plurality of electromagnetic cavities such that at least one of the plurality of electromagnetic cavities is directly coupled to only one other of the plurality of electromagnetic cavities.

3. The sensor of claim 1, the number of the plurality of coupling elements being greater than the number of the plurality of electromagnetic cavities such at least one of the plurality of electromagnetic cavities is directly coupled to at least two others of the plurality of electromagnetic cavities.

4. The sensor of claim 1, the number of the plurality of coupling elements being equal to the number of the plurality of electromagnetic cavities such that each of the plurality of electromagnetic cavities is directly coupled to only two others of the plurality of electromagnetic cavities.

5. The sensor of claim 1, the plurality of electromagnetic cavities and the plurality of coupling elements forming a non- reciprocal lattice.

6. The sensor of claim 1, the external electromagne tic field being a magnetic field.

7. The sensor of claim 6, further comprising a magnetic field source configured to generate a bias magnetic field that hybridizes the quantum state of the quantum system with the cavity mode of the hybridization cavity.

8. The sensor of claim 1, wherein: the quantum state has a quantum-state eigenfrequency that lies in the radiofrequency, microwave, or millimeter-wave regions of the electromagnetic spectrum: and the cavity7mode of each of the plurality7of electromagnetic cavities has a cavity-mode eigenfrequency that lies in the radio-frequency, microwave, or millimeterwave regions of the electromagnetic spectrum.

9. The sensor of claim 1, the quantum system being a YIG sphere.

10. The sensor of claim 1, the quantum system being a superconducting qubit11. The sensor of claim 1, the asymmetric coupling element comprising an amplifier.

12. The sensor of claim 11, the amplifier being a tunable amplifier.

13. The sensor of claim 1, the asymmetric coupling element comprising a directional coupler or circulator.

14. The sensor of claim 1, the asymmetric coupling element including a phase shifter.

15. The sensor of claim 1, one of the plurality of electromagnetic cavities forming an input port.

16. The sensor of claim 15, said one of the plurality of electromagnetic cavities being the hybridization cavity.

17. The sensor of claim 15, said one of the plurality of electromagnetic cavities being one of the two or more non-hybridization cavities.

18. The sensor of claim 1, one of the plurality of electromagnetic ca vities forming an output port19. The sensor of claim 18, said one of the plurality of electromagnetic cavities being the hybridization cavity.

20. The sensor of claim 18, said one of the plurality of electromagnetic cavities being one of the two or more non-hybridization cavities.

21. The sensor of claim 1, wherein: the plurality of electromagnetic cavities comprises a first cavity, a second cavity, and a third cavity; and the plurality of coupling elements comprises a first coupling element that couples the first and second cavities, a second coupling element that couples the second and third cavities, and a third coupling element that couples the third and first cavities.

22. The sensor of claim 21, wherein: the first coupling element is a first amplifier having a first input connected to the first cavity and a first output connected to the second cavity; and the second coupling element is a second amplifier having a second input connected to the second cavity and a second output connected to the third cavity.

23. The sensor of claim 22, the firstand second amplifiers having the same gain.

24. A sensor comprising: a quantum system configured to interact with an external electromagnetic field to form a quantum state that varies with the external electromagnetic field;one or more electromagnetic cavities, each of the one or more electromagnetic cavities having a cavity mode; and one or more coupling elements that couple the one or more electromagnetic cavities to each other, at least one of the one or more coupling elements being an asymmetric coupling element with gain, one of the one or more coupling elements including a transmission line, the quantum system being located adjacent to the transmission line; wherein, when the external electromagnetic field is present: the quantum state of the quantum system couples with a transverse mode of the transmission line to induce, in the transmission line, a phase shift based on the externa] electromagnetic field; and the sensor self- oscillates at a frequency that is based on the phase shift such that the frequency determines a magnitude of the external electromagnetic field or a relative change of the external electromagnetic field with respect to a reference point25. The sensor of claim 24, wherein: the one or more electromagnetic cavities comprises a plurality of electromagnetic cavities; the one or more coupling elements comprises a plurality of coupling elements; the number of the plurality of coupling elements being equal to the number of the plurality of electromagnetic cavities such that each of the plurality of electromagnetic cavities is directly coupled to only two others of the plurality of electromagnetic cavities.

26. The sensor of claim 24, wherein: the one or more electromagnetic cavities comprises a plurality of electromagnetic cavities; the one or more coupling elements comprises a plurality of coupling elements; the plurality of electromagnetic cavities and the plurality of coupling elements forming a loop.

27. The sensor of claim 24, the externa] electromagnetic field being a magnetic field.

28. The sensor of claim 27, farther comprising a magnetic field source configured to generate a bias magnetic field that hybridizes the quantum state of the quantum system with the transverse mode of the transmission line.

29. The sensor of claim 24, wherein: the quantum state has a quantum-state eigenfrequency that lies in the radiofrequency, microwave, or millimeter- wave regions of the electromagnetic spectrum; and the cavity mode of each of the one or more electromagnetic cavities has a cavitymode eigenfrequency that lies in the radio-frequency, microwave, or millimeter-wave regions of the electromagnetic spectrum.

30. The sensor of claim 24, the quantum system being a Y1G sphere.

31. The sensor of claim 24, the quantum system being a superconducting qubit.

32. The sensor of claim 24, the asymmetric coupling element comprising an amplifier.

33. The sensor of claim 24, one of the one or more electromagnetic cavities forming an output port34. The sensor of claim 24, wherein: the one or more electromagnetic cavities comprises a first cavity and a second cavity; and the one or more coupling elements comprises a first coupling element that couples the first and second cavities and a second coupling element that couples the first and second cavities.

35. The sensor of claim 34, wherein: the first coupling element is a first amplifier having a first input connected to the first cavity and a first output connected to the second cavity: and the second coupling element is a second amplifier having a second input connected to the second cavity and a second output connected to the first cavity.

36. A method comprising: coupling a probe electromagnetic wave into the sensor of claim 1; coupling a transmitted electromagnetic wave out of the sensor; and detecting the transmitted electromagnetic wave to measure an eigenenergy of the composite quantum state.

37. The method of claim 36, further comprising applying a bias electromagnetic field to the hybridization cavity such that the quantum state of the quantum system hybridizes with the cavity mode of the hybridization cavity.

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