Quantum cross-resonator spectrometer

Abstract

The present disclosure provides a contact-less detection using a cross-cavity device where a small dielectric sample is placed at its center. The optical properties of the sample, such as the Kerr and Faraday rotation, or polarizability, manifest in the coupling between the cavities' electromagnetic modes and in the shift of their resonant frequencies. By calculating the dynamics of what are referred to herein as geometrical photonic states, a measuring protocol is formulated based on the quantum metric that maximizes the Fisher information and isolates the individual components of the complex dielectric function.

Description

QUANTUM CROSS-RESONATOR SPECTROMETERCROSS-REFERENCE TO RELATED APPLICATION(S)

[0001] This application claims the benefit of priority to U.S. Provisional Application No. 63 / 488,720, filed March 6, 2023, which is hereby incorporated by reference in its entirety.STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with Government support under N00014-20-1-2481 awarded by the Office of Naval Research, 1944085 awarded by the National Science Foundation, DE-AC05-00OR22725 awarded by the U.S. Dept, of Energy, and 4000187220 awarded by Oak Ridge National Laboratory (ORNL). The Government has certain rights to this invention.BACKGROUND

[0003] Embodiments of the present disclosure relate to determining optical properties of a dielectric sample, and more specifically, to a quantum cross-resonator spectrometer.BRIEF SUMMARY

[0004] According to embodiments of the present disclosure, methods and apparatus for determining optical properties of sample are provided.

[0005] In a first example embodiment, the present invention is an apparatus, comprising: a first electromagnetic resonator; a second electromagnetic resonator intersecting the first resonator, thereby defining an intersection area, the first and the second resonator each comprising at least one interface; a source configured to generate an electromagnetic source signal; at least a first detector; at least one input / output means in electromagnetic communication with the source, the at least one interface of the first resonator and the at least one interface of the second resonator. The at least one input / output means is configured to: based on a source signal, generate a first input signal and a second input signal; apply to the first and the second signals an input amplitude ratio and an input relative phase shift, thereby generating a first transformed input signal and a second transformed input signal; transmit the first transformed input signal to the at least one interface of the first resonator, thereby generating a first output signal therefrom; and transmit the second transformed input signal to the at least one interface of the second resonator, thereby generating a second output signaltherefrom. The at least one input / output means is further configured to: receive the first output signal from the first resonator and the second output signal from the second resonator; apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a sign-reverse of the input phase shift, thereby generating a first transformed output signal and a second transformed output signal; and transmit at least the first transformed output signal to the at least the first detector, the at least first detector configured to generate at least a first result signal.

[0006] In a second example embodiment, the present invention is an apparatus, comprising: a first electromagnetic resonator; a second electromagnetic resonator intersecting the first resonator, thereby defining an intersection area, the first and the second resonator each comprising an input interface and an output interface; a source configured to generate an electromagnetic source signal; a first detector and a second detector; an input means in electromagnetic communication with the source, the input interface of the first resonator and the input interface of the second resonator; and an output means in electromagnetic communication with the output interface of the first resonator, the output interface of the second resonator, and the first and the second detectors. The input means is configured to: based on a source signal, generate a first input signal and a second input signal; apply to the first and the second signals an input amplitude ratio and an input relative phase shift, thereby generating a first transformed input signal and a second transformed input signal; transmit the first transformed input signal to the input interface of the first resonator, thereby generating a first output signal therefrom; and transmit the second transformed input signal to the input interface of the second resonator, thereby generating a second output signal therefrom. The output means is configured to: receive the first output signal from the first resonator and the second output signal from the second resonator; apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a signreverse of the input phase shift, thereby generating a first transformed output signal and a second transformed output signal; and transmit the first transformed output signal to the first detector, and transmit the second transformed output signal to the second detector, the first and the second detectors configured to generate a first result signal and a second result signal.

[0007] In a third example embodiment, the present invention is a method of determining an index of refraction of a sample, the method comprising: providing the apparatus of the first example embodiment, wherein the intersection area is adapted to receive the sample andevanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part; disposing the sample at the intersection area; causing the source to generate the source signal; causing the at least one input / output means to: based on the source signal, generate a plurality of input signal pairs, each input signal pair comprising a first and a second input signals; for each input signal pair, apply to the first and the second input signals an input amplitude ratio and an input relative phase shift, thereby generating a transformed input signal pair, the transformed input signal pair comprising a first and a second transformed input signals; and for each transformed input signal pair, transmit the first transformed input signal to the at least one interface of the first resonator, and transmit the second transformed input signal to the at least one interface of the second resonator, thereby generating an output signal pair, the output signal pair comprising a first and a second output signal; causing the at least one input / output means to: for each output signal pair, receive the first output signal from the first resonator and the second output signal from the second resonator; for each output signal pair, apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a sign-reverse of the input phase shift, thereby generating a transformed output signal pair, each transformed output signal pair comprising a first transformed output signal and a second transformed output signal; and for each transformed output signal pair, transmit at least the first transformed output signal to the at least the first detector; causing the at least the first detector, for each transformed output signal pair, to detect at least the first transformed output signal, thereby generating a result signal; thereby generating a plurality of result signals based on the plurality of input signal pairs; and based on the plurality of result signals, computing the real and the imaginary parts of the index of refraction of the sample.

[0008] In a fourth example embodiment, the present invention is a method of determining an index of refraction of a sample, the method comprising: providing the apparatus of the second example embodoment, wherein the intersection area is adapted to receive the sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part; disposing the sample at the intersection area; causing the source to generate the source signal; causing the input means to: based on the source signal, generate a plurality of input signal pairs, each input signal pair comprising a first and a second input signals; for each input signal pair, apply to the first and the second input signals an input amplitude ratio and an input relative phase shift, thereby generating a transformed input signal pair, the transformed input signal pair comprising a first and a secondtransformed input signals; and for each transformed input signal pair, transmit the first transformed input signal to the input interface of the first resonator, and transmit the second transformed input signal to the input interface of the second resonator, thereby generating an output signal pair, the output signal pair comprising a first and a second output signal; causing the output means to: for each output signal pair, receive the first output signal from the first resonator and the second output signal from the second resonator; for each output signal pair, apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a sign-reverse of the input phase shift, thereby generating a transformed output signal pair, each transformed output signal pair comprising a first transformed output signal and a second transformed output signal; and for each transformed output signal pair, transmit the first transformed output signal to the first detector and the second transformed output signal to the second detector; causing the first and the second detectors, for each transformed output signal pair, to detect the first transformed output signal and the second transformed output signal, respectively, thereby generating a result signal pair, the result signal pair comprising a first and a second result signals; thereby generating a plurality of result signal pairs based on the plurality of input signal pairs; and based on the plurality of result signal pairs, computing the real and the imaginary parts of the index of refraction of the sample.

[0009] In a fifth example embodiment, the present invention is an apparatus, comprising: a first superconducting electromagnetic resonator; a second superconducting electromagnetic resonator intersecting the first resonator, thereby defining an intersection area; a transmon capacitively tunably coupled to the first resonator and the second resonator; a microwave (MW) source operatively coupled to the transmon, the MW source configured to generate one or more drive signals; a transmon readout means operatively coupled to the transmon; and wherein the transmon is configured to: based on the one or more drive signal, be placed into a first quantum state, thereby generating an input signal; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, transmit the input signal into the first resonator and the second resonator, thereby generating a first resonator signal and the second resonator signal, wherein the first resonator signal and the second resonator signal have an input amplitude ratio; reduce coupling to the first and the second resonators; based on the one or more drive signal, be placed in a second quantum state, thereby imparting a first relative phase shift to the first and the second resonator signals; based on the one or more drive signal, be placed in a third quantum state, thereby imparting a second relative phaseshift to the first and the second resonator signals, wherein the second relative phase shift is a sign-reverse of the first relative phase shift; based on the one or more drive signal, be placed in a fourth quantum state; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, apply an output amplitude ratio to the first resonator signal and the second resonator signal, wherein output amplitude ratio that is an inverse of the input amplitude ratio, and transmit the first and the second resonator signals to the transmon; and wherein the transmon readout means is configured to read out at least the first resonator signal from the transmon.

[0010] In a sixth example embodiment, the present invention is a method of determining an index of refraction of a sample, the method comprising: providing the apparatus of the fifth example embodiment, wherein the intersection area is adapted to receive the sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part; disposing the sample at the intersection area; causing the MW source to generate one or more drive signals; causing the transmon to: based on the one or more drive signal, be placed in a first quantum state, thereby generating an input signal; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, transmit the input signal into the first resonator and the second resonator, thereby generating a first resonator signal and the second resonator signal, wherein the first resonator signal and the second resonator signal have an input amplitude ratio; reduce coupling to the first and the second resonators; based on the one or more drive signal, be placed in a second quantum state, thereby imparting a first relative phase shift to the first and the second resonator signals; based on the one or more drive signal, be placed in a third quantum state, thereby imparting a second relative phase shift to the first and the second resonator signals, wherein the second relative phase shift is a sign-reverse of the first relative phase shift; based on the one or more drive signal, be placed in a fourth quantum state; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, apply an output amplitude ratio to the first resonator signal and the second resonator signal, wherein output amplitude ratio that is an inverse of the input amplitude ratio, and transmit the first and the second resonator signals to the transmon; causing the transmon readout means to read out at least the first resonator signal from the transmon; and based on at least the first resonator signal, computing the real and the imaginary parts of the index of refraction of the sample.BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0011] Fig. 1A is a schematic view of a cross-cavity device according to embodiments of the present disclosure.

[0012] Fig. IB is a level diagram illustrating spectrum of the two cavities according to embodiments of the present disclosure.

[0013] Fig. 1C is a schematic view of an apparatus comprising the cross-cavity device according to embodiments of the present disclosure.

[0014] Fig. ID is a schematic view of an additional embodiment of an apparatus comprising the cross-cavity device according to the present disclosure.

[0015] Fig. 2A is a polar plot of a quantum metric according to embodiments of the present disclosure.

[0016] Fig. 2B is a density plot of the quantum metric and precession trajectories according to embodiments of the present disclosure.

[0017] Fig. 3 is a schematic view of a measuring sequence according to embodiments of the present disclosure.

[0018] Figs. 4A-D are graphs illustrating the evolution of excitation manifolds according to embodiments of the present disclosure.

[0019] Fig. 5 is a schematic view of an exemplary cross-cavity device utilizing a transmon according to embodiments of the present disclosure.DETAILED DESCRIPTION

[0020] The complex dielectric function is a macroscopic property of a material that provides crucial information about the microscopic behavior of electrons. However, experimentally accessing its components has proven to be challenging, especially when the associated effects are minute.

[0021] To address this challenge, the present disclosure provides a contact-less detection using a cross-cavity device where a small dielectric sample is placed at its center. The optical properties of the sample, such as the Kerr and Faraday rotation, or polarizability, manifest in the coupling between the cavities’ electromagnetic modes and in the shift of their resonant frequencies. By calculating the dynamics of what are referred to herein as geometrical photonic states, a measuring protocol is formulated based on the quantum metric that maximizes the Fisher information and isolates the individual components of the complex dielectric function. In this way, the quantum mechanical Cramer-Rao bound on the variance of the associated parameter estimation is reached. These effects are expected to be presentacross a broad range of experimental platforms including microwave and ultrafast THz devices.

[0022] Reciprocal and nonreciprocal effects in dielectric and magnetic materials provide crucial information about the microscopic properties of electrons. However, experimentally distinguishing between reciprocal and nonreciprocal effects has proven to be challenging, especially when the associated effects are extremely small. To this end, a contact-less detection using a cross-cavity device where a material of interest is placed at its center is disclosed. The optical properties of the material, such as Kerr and Faraday rotation, or, birefringence, manifest in the coupling between the cavities’ electromagnetic modes and in the shift of their resonant frequencies. By calculating the dynamics of a geometrical photonic state, a measurement protocol is formulated based on the quantum metric and quantum process tomography that isolates the individual components of the material’s complex refractive index and minimizes the quantum mechanical Cramer-Rao bound on the variance of the associated parameter estimation. This approach may be applicable across a broad spectrum of experimental platforms including Fock states in optical cavities, or coherent states in microwave and THz resonators.

[0023] Quantum materials offer new technological opportunities while posing key challenges for existing characterization methods. Phenomena such as superconductivity, topology, magnetism, and collective motion are all manifestations of quantum effects in solid-state systems, which can in turn offer potentially novel electronic device functionalities. A commonality between these examples is the way that different symmetries are broken, and the manifestation of these broken symmetries in macroscopic electrodynamic response. One of the most fundamental of such symmetries is time-reversal symmetry (TRS), which when present ensures that material responses are reciprocal, as seen from Onsager’s famous relations. The breaking of TRS then allows for nonreciprocal material responses, which are of practical importance for the design of optical and microwave components such as photon routers and circulators. In topological insulators and semimetals, TRS breaking is expected to induce interesting nonreciprocal responses manifesting as a nonzero Hall conductivity, and in correlated insulators is often associated with the onset of magnetic order such as ferromagnetism or antiferromagnetism.

[0024] There has been an immense interest in unconventional superconductors which spontaneously break TRS as they may be candidates for the highly sought after chiral topological superconductivity. Signatures of reciprocity breaking can, hence, provide insights to the underlying pairing mechanism, as well as elucidate the coexistence ofsuperconductivity and magnetism. Beyond the conventional U(l) gauge symmetry, unconventional superconductors may spontaneously break additional symmetries, such as orbital or spin rotation symmetries. In general, these effects are often orders of magnitude smaller than what can be measured with conventional optical measurements. Therefore, estimating the degree by which a material breaks reciprocity requires sensitive apparatuses.

[0025] Most frequently, high precision measurements of nonreciprocity are reported through muon spin relaxation (pSR) where spin polarized muons precess depending on the complex refractive index and decay in spin-dependent trajectories. Magneto-optical Kerr probes have also been used to directly demonstrate nonreciprocity at the onset of superconductivity below the critical temperature by measuring the rotation of light polarization with a sophisticated zero-area Sagnac interferometer; in this way reciprocal effects, such as birefringence, are explicitly canceled. These techniques have proven to be very powerful for measuring single crystals and superconducting / ferromagnetic hybrid materials. However, the discovery of unconventional superconducting states in van der Waals (vdW) superconductors, such as magic-angle twisted bilayer graphene and monolayer WTe2, requires reimagining probes that reach a high level of precision and overcome small sample mode volumes or low densities.

[0026] Here, an alternative platform is disclosed to measure the components of the complex refractive index in parallel and provide a detection protocol for disentangling distinct symmetry classifications. Specifically, two cross-aligned, single-mode cavities are used where a sample placed at the intersection is evanescently coupled to the electromagnetic fields. The evolution of photonic states are calculated and the photon occupation number to the quantum metric characterizing the space of states are interrelated. As the latter is determined by the sample’s susceptibility and conductivity, the induced quantum geometry is used to separate reciprocal and nonreciprocal effects, in addition to minimizing the quantum uncertainty of the measured parameters. Finally, an optimized detection protocol using the minimum number of sampling points to extract the sample’s optical properties is disclosed herein. Notably, the contact-free spectroscopic probe is particularly useful for studying materials where obtaining reliable electrical contacts can be challenging, such as in vdW 2D materials. Furthermore, embodiments of the present disclosure can be generalized for both coherent and Fock states, allowing for various implementations across the optical, terahertz, and microwave regimes.

[0027] In various embodiments of the present disclosure, two orthogonal arms of a microwave cross-resonator are joined at the sample location, such that the electric fields are polarized in non-collinear directions across the bulk of the sample. The electric fields arethen suited to probe the full polarizability tensor of the sample, including most importantly the off-diagonal Hall conductivity. Each arm functions as a high-quality single-mode microwave cavity which can host a quantize photon mode. To probe the Hall conductivity, a quantum state of N total photons is partitioned into each of the two modes, indicated as x and y corresponding to the polarization of the electric field for that mode at the sample location.

[0028] The state prepared is formed by first applying a beam-splitter operation in order to generate a Fock-state superposition between the two arms of controlled angle, 6. Then a second phase-shift operation is applied to differentially shift the phase of the photons in one arm with respect to the other arm, by an angle 0. This allows for the generation of a spin- coherent state of spin N using the two arms to implement the Schwinger boson representation. This allows for spectroscopy of the sample using the dynamics of the spin- coherent state, which in particular is sensitive to the breaking of time-reversal symmetry in the dynamics. This then enables the measurement of the Hall conductivity. The known phase angles 6 and are then unimplemented, such that the state would return to the initial N-photon state in absence of the dynamics induced by the sample cross-coupling. Finally, the counting statistics of the outgoing photons are measured and used to characterize the dynamics induced by the sample through repetition at different trial angles 6 and 0.

[0029] Accordingly, in a 1stexample embodiment, the present invention is an apparatus. The apparatus comprises: a first electromagnetic resonator; a second electromagnetic resonator intersecting the first resonator, thereby defining an intersection area, the first and the second resonator each comprising at least one interface; a source configured to generate an electromagnetic source signal; at least a first detector; at least one input / output means in electromagnetic communication with the source, the at least one interface of the first resonator and the at least one interface of the second resonator; and wherein the at least one input / output means is configured to: based on a source signal, generate a first input signal and a second input signal;apply to the first and the second signals an input amplitude ratio and an input relative phase shift, thereby generating a first transformed input signal and a second transformed input signal; transmit the first transformed input signal to the at least one interface of the first resonator, thereby generating a first output signal therefrom; and transmit the second transformed input signal to the at least one interface of the second resonator, thereby generating a second output signal therefrom, wherein the at least one input / output means is further configured to: receive the first output signal from the first resonator and the second output signal from the second resonator; apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a signreverse of the input phase shift, thereby generating a first transformed output signal and a second transformed output signal; and transmit at least the first transformed output signal to the at least the first detector, the at least first detector configured to generate at least a first result signal.

[0030] In a 1staspect of the 1stexample embodiment, the intersection area is adapted to receive a sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part.

[0031] In a 2ndaspect of the 1stexample embedment, the apparatus further comprises a controller configured to: cause the source to generate the source signal; and based on the at least the first result signal, compute the real and the imaginary parts of the index of refraction of the sample.

[0032] In a 3rdaspect of the 1stexample embedment, the source is an optical source, and first and the second resonators are each an optical resonator. The remainder of the features and example features of the 1stexample embedment are as described above with respect to its various aspects.

[0033] In a 4thaspect of the 1stexample embedment, the optical source is a single photon source, and the first and second detectors are each a single photon detector. The remainder of the features and example features of the 1stexample embedment are as described above with respect to its various aspects.

[0034] In a 5thaspect of the 1stexample embedment, the optical source is a coherent photon source. The remainder of the features and example features of the 1stexample embedment are as described above with respect to its various aspects.

[0035] In a 6thaspect of the 1stexample embedment, the at least one input / output means comprises an adjustable beam splitter and at least one adjustable phase shifter. The remainder of the features and example features of the 1stexample embedment are as described above with respect to its various aspects.

[0036] In a 7thaspect of the 1stexample embedment, the first and the second optical resonators each comprises a Fabry-Perot cavity.

[0037] In a 2ndexample embedment, the present invention is an apparatus. The apparatus comprises a first electromagnetic resonator; a second electromagnetic resonator intersecting the first resonator, thereby defining an intersection area, the first and the second resonator each comprising an input interface and an output interface; a source configured to generate an electromagnetic source signal; a first detector and a second detector; an input means in electromagnetic communication with the source, the input interface of the first resonator and the input interface of the second resonator; and an output means in electromagnetic communication with the output interface of the first resonator, the output interface of the second resonator, and the first and the second detectors, wherein the input means is configured to: based on a source signal, generate a first input signal and a second input signal; apply to the first and the second signals an input amplitude ratio and an input relative phase shift, thereby generating a first transformed input signal and a second transformed input signal; transmit the first transformed input signal to the input interface of the first resonator, thereby generating a first output signal therefrom; and transmit the second transformed input signal to the input interface of the second resonator, thereby generating a second output signal therefrom, and wherein the output means is configured to:receive the first output signal from the first resonator and the second output signal from the second resonator; apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a signreverse of the input phase shift, thereby generating a first transformed output signal and a second transformed output signal; and transmit the first transformed output signal to the first detector, and transmit the second transformed output signal to the second detector, the first and the second detectors configured to generate a first result signal and a second result signal.

[0038] In a 1staspect of the 2ndexample embedment, the intersection area is adapted to receive a sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part.

[0039] In a 2ndaspect of the 2ndexample embedment, the apparatus further comprises a controller configured to: cause the source to generate the source signal; and based on the first result signal and the second result signal, compute the real and the imaginary parts of the index of refraction of the sample. The remainder of the features and example features of the 2ndexample embedment are as described above with respect to its various aspects.

[0040] In a 3rdaspect of the 2ndexample embedment, the input means and the output means each comprises an adjustable beam splitter and at least one adjustable phase shifter. The remainder of the features and example features of the 2ndexample embedment are as described above with respect to its various aspects.

[0041] In a 3rdexample embedment, the present invention is a method of determining an index of refraction of a sample. The method comprises:- providing the apparatus of the 1stexample embodiment, wherein the intersection area is adapted to receive the sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part;- disposing the sample at the intersection area;- causing the source to generate the source signal;- causing the at least one input / output means to: based on the source signal, generate a plurality of input signal pairs, each input signal pair comprising a first and a second input signals;for each input signal pair, apply to the first and the second input signals an input amplitude ratio and an input relative phase shift, thereby generating a transformed input signal pair, the transformed input signal pair comprising a first and a second transformed input signals; and for each transformed input signal pair, transmit the first transformed input signal to the at least one interface of the first resonator, and transmit the second transformed input signal to the at least one interface of the second resonator, thereby generating an output signal pair, the output signal pair comprising a first and a second output signal;- causing the at least one input / output means to: for each output signal pair, receive the first output signal from the first resonator and the second output signal from the second resonator; for each output signal pair, apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a sign-reverse of the input phase shift, thereby generating a transformed output signal pair, each transformed output signal pair comprising a first transformed output signal and a second transformed output signal; and for each transformed output signal pair, transmit at least the first transformed output signal to the at least the first detector;- causing the at least the first detector, for each transformed output signal pair, to detect at least the first transformed output signal, thereby generating a result signal;- thereby generating a plurality of result signals based on the plurality of input signal pairs; and- based on the plurality of result signals, computing the real and the imaginary parts of the index of refraction of the sample.

[0042] In a 1staspect of the 3rdexample embodiment, the source is an optical source, and first and the second resonators are each an optical resonator. The remainder of the features and example features of the 3rdexample embedment are as described above with respect to its various aspects.

[0043] In a 2ndaspect of the 3rdexample embedment, the optical source is a single photon source, and the source signal comprises a sequence of single photons, and the method further comprises: selecting a plurality of time delay intervals, each time delay interval corresponding to one photon in the sequence, for each photon in the sequence: generating the source signal at a first time point; and generating the corresponding result signal at a second time point, wherein the first and the second time points are separated by the time delay interval corresponding to the photon.

[0044] The remainder of the features and example features of the 3rdexample embedment are as described above with respect to its various aspects.

[0045] In a 3rdaspect of the 3rdexample embedment, the optical source is a coherent photon source, and the method further comprises: selecting a time duration interval; and generating the source signal for the time duration interval.

[0046] The remainder of the features and example features of the 3rdexample embedment are as described above with respect to its various aspects.

[0047] In a 4thexample embodiment, the present invention is a method of determining an index of refraction of a sample. The method comprises:- providing the apparatus of the 2ndexample embodiment, wherein the intersection area is adapted to receive the sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part;- disposing the sample at the intersection area;- causing the source to generate the source signal;- causing the input means to: based on the source signal, generate a plurality of input signal pairs, each input signal pair comprising a first and a second input signals; for each input signal pair, apply to the first and the second input signals an input amplitude ratio and an input relative phase shift, thereby generating a transformed input signal pair, the transformed input signal pair comprising a first and a second transformed input signals; andfor each transformed input signal pair, transmit the first transformed input signal to the input interface of the first resonator, and transmit the second transformed input signal to the input interface of the second resonator, thereby generating an output signal pair, the output signal pair comprising a first and a second output signal;- causing the output means to: for each output signal pair, receive the first output signal from the first resonator and the second output signal from the second resonator; for each output signal pair, apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a sign-reverse of the input phase shift, thereby generating a transformed output signal pair, each transformed output signal pair comprising a first transformed output signal and a second transformed output signal; and for each transformed output signal pair, transmit the first transformed output signal to the first detector and the second transformed output signal to the second detector;- causing the first and the second detectors, for each transformed output signal pair, to detect the first transformed output signal and the second transformed output signal, respectively, thereby generating a result signal pair, the result signal pair comprising a first and a second result signals;- thereby generating a plurality of result signal pairs based on the plurality of input signal pairs; and- based on the plurality of result signal pairs, computing the real and the imaginary parts of the index of refraction of the sample.

[0048] In a 5thexample embodiment, the present invention is an apparatus. The apparatus comprises: a first superconducting electromagnetic resonator; a second superconducting electromagnetic resonator intersecting the first resonator, thereby defining an intersection area; a transmon capacitively tunably coupled to the first resonator and the second resonator;a microwave (MW) source operatively coupled to the transmon, the MW source configured to generate one or more drive signals; a transmon readout means operatively coupled to the transmon; and wherein the transmon is configured to: based on the one or more drive signal, be placed into a first quantum state, thereby generating an input signal; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, transmit the input signal into the first resonator and the second resonator, thereby generating a first resonator signal and the second resonator signal, wherein the first resonator signal and the second resonator signal have an input amplitude ratio; reduce coupling to the first and the second resonators; based on the one or more drive signal, be placed in a second quantum state, thereby imparting a first relative phase shift to the first and the second resonator signals; based on the one or more drive signal, be placed in a third quantum state, thereby imparting a second relative phase shift to the first and the second resonator signals, wherein the second relative phase shift is a sign-reverse of the first relative phase shift; based on the one or more drive signal, be placed in a fourth quantum state; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, apply an output amplitude ratio to the first resonator signal and the second resonator signal, wherein output amplitude ratio that is an inverse of the input amplitude ratio, and transmit the first and the second resonator signals to the transmon; and wherein the transmon readout means is configured to read out at least the first resonator signal from the transmon.

[0049] In a 1staspect of the 5thexample embodiment, the transmon is a tunable transmon.

[0050] In a 2ndaspect of the 5thexample embodiment, the apparatus further comprises a flux bias means operatively coupled to the transmon, the flux bias means configured to change a magnetic flux through the transmon, thereby adjusting the transmon frequency. The remainder of the features and example features of the 5thexample embedment are as described above with respect to its various aspects.

[0051] In a 3rdaspect of the 5thexample embedment, the intersection area is adapted to receive a sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part. The remainder of the features and example features of the 5thexample embedment are as described above with respect to its various aspects.

[0052] In a 4thaspect of the 5thexample embedment, the apparatus further comprises a controller configured to: cause the MW source to generate the one or more drive signals; and based on at least the first resonator signal, compute the real and the imaginary parts of the index of refraction of the sample. The remainder of the features and example features of the 5thexample embedment are as described above with respect to its various aspects.

[0053] In a 6thexample embedment, the present invention is a method of determining an index of refraction of a sample. The method comprises:- providing the apparatus of Claim 18, wherein the intersection area is adapted to receive the sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part;- disposing the sample at the intersection area;- causing the MW source to generate one or more drive signals;- causing the transmon to: based on the one or more drive signal, be placed in a first quantum state, thereby generating an input signal; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, transmit the input signal into the first resonator and the second resonator, thereby generating a first resonator signal and the second resonator signal, wherein the first resonator signal and the second resonator signal have an input amplitude ratio; reduce coupling to the first and the second resonators; based on the one or more drive signal, be placed in a second quantum state, thereby imparting a first relative phase shift to the first and the second resonator signals; based on the one or more drive signal, be placed in a third quantum state, thereby imparting a second relative phase shift to the first and the second resonator signals, wherein the second relative phase shift is a sign-reverse of the first relative phase shift;based on the one or more drive signal, be placed in a fourth quantum state; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, apply an output amplitude ratio to the first resonator signal and the second resonator signal, wherein output amplitude ratio that is an inverse of the input amplitude ratio, and transmit the first and the second resonator signals to the transmon;- causing the transmon readout means to read out at least the first resonator signal from the transmon; and- based on at least the first resonator signal, computing the real and the imaginary parts of the index of refraction of the sample.

[0054] EXEMPLIFICATION

[0055] A. Cross-Cavity Model

[0056] In some embodiments, a cross-cavity device in a planar geometry with a small dielectric sample at the intersection, as shown in Fig. 1A. Fig. 1A illustrates a realization of the cross-cavity device 10 using rectangular resonators 12 fabricated on a substrate 16. A sample 14 is mounted at the intersection of the two resonators 12 and couples evanescently to the electric field modes. The small dielectric sample is well described by a 6(r) distribution, susceptibility tensor / ^. and a conductivity tensor tr^- = £lJaH. where crHis the Hall conductivity and elJis the Levi-Civita tensor. For simplicity, the diagonal conductivity is set to zero and its effect is incorporated in the coherence time of the device. The cavities are characterized by a conductivity tensor cr0and a susceptibility tensor %0which define the “reference vacuum” for the electric field. Without loss of generality, a trivial reference vacuum is chosen with zero conductivity cr0= 0 and an isotropic susceptibility tensor %0~ additional contributions from nontrivial vacua can be equally treated by absorbing them into the definitions of 8(r), x and cr. Furthermore, it can be assumed that each cavity can support a single mode with the electric field sufficiently permeating into free space such that it evanescently couples to the sample.

[0057] Before introducing quantum mechanical effects, it is instructive to showcase the behaviour of the device in its classical limit. Solutions to Maxwell’s equations can be obtained perturbatively using classical electromagnetic fields with their evolution computed using a standard Green’s function approach. As a result, the action of the sample becomesequivalent to a beam splitter where an incident electromagnetic field scatters to the available channels; in this geometry, the associated split ratio is determined by the magnitude of the off-diagonal component of the complex refractive index, while a relative phase shift between the two arms of the device will only occur when there is a finite imaginary off-diagonal element. Importantly, the classical treatment of the disclosed method is only perturbatively valid with leading-order corrections proportional to should be noted that innatural units, conductivity and frequency both have units of energy). Hence, detecting nonreciprocal effects may be challenging due to background radiation or experimental uncertainties.

[0058] In some embodiments, the system quantum is treated mechanically in the case of closed dynamics, i.e., in the absence of any coupling to the environment; nonunitary process, e.g., losses, may be introduced by replacing the unitary evolution operator with a completely positive map. The relevant quantised Hamiltonian in the rotating wave approximation is given by: (1)where 2 AS = Sx— Syis the difference of the cavities’ resonance frequencies, with S} = a>i + 8a>i as the sum of the bare resonant frequency of the cavity in the / th direction(which may be assumed to be almost equal in the two directions) and the shift 8a>i due to the diagonal terms of the sample’s susceptibilit The hybridization between the twocavity modes s determined by the real coupling gxinduced by a finite off-diagonal susceptibility xxy, and the imaginary coupling gainduced by a finite Hall conductivity crH. Hamiltonian (1) can be interpreted as the effective dynamics of a spin in a magnetic field 5, namely (2) where define the elements of the SU(2) algebra and are given in terms of thecreation and annihilation operators d;and d?, and (3)is a vector determined by the complex coupling g between the two cavities and their relative frequency difference AS. While both the Bxand Bzcomponents are expected due to the polarizability of the sample, mode splitting, or even due to geometrical effects of the device’s shape and impurities, a nonzero Bycomponent arises only when time-reversal symmetry is broken.

[0059] Since the Hamiltonian (2) conserves the total photon number operatorN dxax+ e Fock space is diagonal with respect to the total number of photons in thecross-cavity system. In the absence of any coupling, i.e., when \B\ = 0, the spectrum of the system is given by the tensor product of equispaced energy levels corresponding to each cavity, as shown in Fig. IB.

[0060] Fig. IB illustrates the spectrum 20 of two cavities with equispaced energy levels separated by the resonant frequency toxor a>y. The complex coupling g = gx+ igabetween the energy levels is determined by the off-diagonal components of the susceptibility tensor % and Hall conductivity aH. The energy levels in each Fock subspace are split according to the magnitude of 5, as shown in spectrum 20. A finite coupling between the modes or a nonzero energy difference between the cavities’ resonant frequencies lifts the degeneracy in each Fock subspace and forms an effective spin-# 2 Schwinger boson in a magnetic field B with the irreducible representations of the SU(2) algebra characterized by the total number of photons.

[0061] Unitary Evolution

[0062] According to embodiments of the present disclosure, a measuring protocol can be based on observing the dynamics of the N-photon geometrical Fock state:where |0, 0) is the vacuum state with zero photons in both cavities, and |n, m) = |0, 0) represents the (w / wj-photon eigenstate with n (m) photons in thex (y) cavity. The proposed geometrical state |ip0(0, < / >)) can be prepared by a pre-processing optical setup of beam splitters and phase shifters, see Fig. 1C.

[0063] Fig. 1C is a schematic of an exemplary embodiment of a setup 30. A photon source 32 is used to excite one arm of the cross cavity device 44. The output electric fields from the two cavities are collected and sent to their corresponding photo detector 36. In addition, a pre- and post-processing setup of beam splitters 42 and phase shifters 40 is used to rotate the frame of reference to obtain the geometrical state of Eq. (4). Specifically, a photon source 32 initializes the system in the |N, 0) Fock state of N photons in the x cavity. The beam splitter 42 rotates the state according to the operator with 0 defined by the split ratio.Finally, a phase shifter is used to further rotate the state leading to the desired geometrical state

[0064] FIG. ID is a schematic of an exemplary alternative apparatus in which the input and output beam paths are shared. An optical circulator is used to route an output beam to the detector.

[0065] The geometrical photonic state in the device will evolve according to:where t / (t) = eltHis the evolution operator; consequently, the state undergoes precession around the vector B with frequency proportional to |B|. However, the direction and magnitude of B are a priori unknowns, therefore, the precision in estimating the angular change is limited by the quantum metric g(0, ( / >) which determines the distance between adjacent quantum states, namely=and serves as ameasure of their distinguishability. For embodiments of the described system, the quantum metric is given byG [0, 1] takes values between zero and one, as shown in Fig. 2A. Fig. 2A is a polar plot 50 of the quantum metric g(6, cp) as a function of 6 and 0. The quantum metric vanishes along the vector B. chosen randomly anormalized vector and (7)is a unit vector on the 2-sphere defined by the expectation value of the spin operators S with respect to the geometrical state of Eq. (4). For example, when S is perpendicular to B, the quantum metric is maximized and the state undergoes precession around the equator defined by B, see Fig. 2B.

[0066] Fig. 2B is a density plot 60 of the quantum metric and the precession trajectories of the state vector S. The inset shows the trajectories when B is along they axis. On the contrary, when the initial state vector S is parallel to B the quantum metric is equal to zero and the state will not undergo precession.

[0067] In some embodiments, the quantum metric is extracted by performing a projective measurement of the final state after a postprocessing setup of a beam splitter Y_gand phase shifters Z_^ shown in Fig. 1C. The measuring protocol, shown in Fig. 3, is based on thealgebraic properties of the SU(2) group which, ultimately, are related to the geometry of the photonic states.

[0068] Fig. 3 illustrates an exemplary embodiment of the measuring sequence 70. The initial state is represented by a vector pointing at the north pole of the Bloch sphere. The first beam splitter induces a rotation around they-axis by an angle 0. The phase shifters further rotate the state vector around the z-axis by an angle 0. The state precesses around B according to the evolution operator U (t). Next, phase shifters are used to rotate the state around the z-axis by an angle -< / >. The last beam splitter induces a rotation around they-axis by an angleFinally, the state is projectively measured by the photo-detector. The change between the initial and final state vectors is determined by the quantum metric 72.

[0069] The probability distribution of simultaneously measuring N - n photons in the x cavity and n photons in they cavity after evolving the state for time t is given by: (8) wher is the evolution operator in the rotated basis induced by thepre- and postprocessing optical setup. The mean photon number in each cavity Nt=and the variance by &N2= Ny— ^N2for both i = x andy. Depending on the quantum metric, the average occupation of each cavity will oscillate with frequency [B|; these oscillations are maximized (minimized) when the initial state vector S is perpendicular (parallel) to B. The direction and magnitude of B can, therefore, be determined by observing the precession of the geometrical state for different values of 6 and 0. When the initial state vector S is prepared to be perpendicular to B. the Fock states |7V - n, ri) of the N-excitation submanifold will perform oscillations with frequency |B| and the entire photon population will be transferred between the two cavities, as shown in Figs. 4A-B.

[0070] In Fig. 4A, the average photon number in each cavity as a function of time for S as shown in graph 80 almost perpendicular to B. i.e.,« 1. The width of each line indicates the variance A / V2. The average photon number is transferred entirely between the two cavities with a vanishing variance at period

[0071] Fig. 4B illustrates the probability distribution of each state associated to Fig.4A in graph 90. The photon population oscillates between the entire set of states with frequency |B|. In addition, the precession path (dashed line) associated to the probability evolution is shown on the Bloch sphere. The shaded region is a plane perpendicular to B.

[0072] Notably, the variance of the mean photon number becomes zero every half oscillation period. On the other hand, when the state vector S lays almost parallel to / T the photons remain primarily in the x cavity, as shown in Figs. 4C-D.

[0073] Fig. 4C is a graph 100 illustrating the average photon number in each cavity as a function of time for S almost parallel to / T i.e., g(0, (p) = 0.1. The width of each line indicates the variance A / V2i . In this case, the average photon number stays primarily in the x cavity throughout the cycle.

[0074] In Fig. 4D, graph 110 illustrates the probability distribution PN(n) associated to Fig. 4C. The photonic states oscillate between only a certain subset with frequency [B|. The precession path (dashed line) on the Bloch sphere associated to the probability evolution is now located closer to the position of the vector B.

[0075] Process Tomography of Nonunitary Evolution

[0076] In any experimental setup, the system will decohere through various decay channels due to its coupling to environment and due to Ohmic dissipation (finite conductivity). Hence, the unitary evolution is replaced by a completely positive map Et that evolves the initial density matrix p0of a Fock state according to p0(<p, &) = Et[p0(< / >, 0)L F°rprocess tomography of a noisy implementation of the unitary gate U(t) there is a total of sixteen free parameters that have to be uniquely determined. Here, it is assumed that the dominant contributions to Etare well captured by a photon leakage out of the device with coherence time r that results in a nonunitary evolution towards the center of the Bloch sphere. Such process has four unknowns that can be extracted using three states and a set of positive operator valued measure (POVM) that consists of two elements{Nx, Ny}. The states are chosenassociated photon numbers {NVx,N®y. The three components of B. and the coherence time r can be found from a minimization routine of the relations:where 9B(f>B) is the polar (azimuthial) angle of B.

[0077] It should be highlighted that embodiments of the disclosed protocol can be implemented in optical Fabry-Perot cavities, allowing access to complex dielectric properties at THz and optical frequencies. It is noted that experimentally, embodiments of the present disclosure have been realized, where microwave cavity devices drive polarization selective transitions. Accordingly, there may be an implementation in the micro wave regime, aiming to maximize the sensitivity of the technique. An approach involves utilizing high-quality factor superconducting resonators, either in a coplanar waveguide geometry or in 3D cavities, which can achieve large Q factors ranging from 10A7 to 10A12. The initial Fock state can be prepared using a coupler transmon that dispersively couples to the two cavity modes, enabling quantum state transfer between the qubit and the cavity. The coupler transmon functions as a beamsplitter that uses the nonlinearity of the Josephson junction to drive parametric conversion. Furthermore, by controlling the phase of the microwave drive tones applied to the transmon, the phase between the photonic states can also be manipulated. In addition, the possibility of using highly entangled states as optimal probes provides a promising route to extract the complex dielectric properties of the material by attaining the Heisenberg limit of precision. For such a microwave device at finite temperature, the uncertainty associated to the POVM is bounded by thermal noise. For a thermal coherent state, the experimental error in measuring the By component from the photon expectation value Ny is given by the variance:where F = — ( ] is the Fisher information, N is the mean number of photons in the h2\dBy)1coherent state and nth. « l / (eP“° — 1) is the mean number of thermal photons, with P is the inverse of temperature multiplied by the Boltzmann factor. In reality, the period of precession T = 2n\B\~1will be dominated by AS due experimental challenges in engineering identical cavities; typical values that can be achieved in superconducting cavities operating at GHz frequencies can be as small as ~ 1 MHz, leading to a precession period on the order of a few ps. which is well below the device’s lifetime r ~ Is. Assuming a coherentstate with a typical mean number of photons N = 10 and a thermal photon number nth. = 1.6 at lOOmK, the optimal mean-squared error can be estimated to be of order y / rLBy «300Hz / VHz or in terms of unitless Hall conductivity / VHz , wherer| is a geometrical factor proportional to the ratio of the volume of the sample over the cavity mode volume. In materials that exhibit notably small Kerr signals, such as SnRuCh, low frequency Hall conductivity is reported as I ()2(c2 / h). which is well within the sensitivity of the embodiments disclosed herein.

[0078] One of the ingredients that may be crucial for certain embodiments of the protocol is the coupling of the sample to the evanescent modes of the electric field in the cross-cavity device. However, such coupling can be hindered by contact imperfections that can arbitrarily change the complex susceptibility or generate stray fields. Therefore, introducing an insulating layer between the sample and the cavities can prevent any build-up of surface effects.

[0079] In embodiments of the present disclosure, it is demonstrated how oscillations between cavity modes in a cross-aligned geometry can be used to detect the relative complex dielectric function of a sample placed at the intersection. The Hamiltonian dynamics describing the photonic states in the cavities is derived, where the sample’s Hall conductivity oH and susceptibility tensor % are shown to induce a complex coupling between the two cavity modes, as well as shift their resonant frequencies. By considering the JV-photon excitation subspace, the evolution of a geometrical quantum state prepared by a pre- and post-optical setup of beam splitters and phase shifters is determined. It is shown that the oscillations of the photon population in each cavity depend on the quantum metric defined by both the shift of the resonant frequencies induced by the diagonal terms of the susceptibility %xx and / yy. as well as the complex coupling between the modes induced by the off-diagonal elements of the susceptibility Xxyar|d Hall conductivity aH. Finally, a measuring protocol to uniquely determine the dielectric properties of the sample using a minimal number of sampling points is described.

[0080] METHODS

[0081] Classical Treatment

[0082] The equation of motion for the vector potential in the cross-cavity geometry up to linear order in the complex susceptibility is given by the Helmholtz equation:where n2=+ 8n2is determined by the refractive index of the cavityand the sample . Without loss of generalityis taken forsimplicity and work in natural units where conductivity has units of energy and susceptibility is dimensionless. The incident vector potential in each waveguide is classically described by A\nc=where a; is a complex coefficient, a>t is the frequency and / i (r) is the mode profile. From the equation (12) the function fi(r) satisfies (where the index i is omitted for simplicity):as well as The total vector potential can be found bydecomposing the solutions into incident and scattered fields, i.e.,and solving Eq. (12) using a Green’s function approach. In the regime where % and o can be treated perturbatively, the total vector potential at the output of the device is given by: (14) where is the transfer matrix,related to the complex susceptibility, and G(r, r') is the Green’s function of the homogenous equations of motion: (15) with

[0083] Quantum Treatment

[0084] The LaGrangian density corresponding to the differential equations (12) is given by:with the corresponding Hamiltonian given by:where II = and in the second line surface terms have been neglected, but which vanish inthe limit of large volume. It is understood that in principle both n2and o are tensors characterizing the susceptibility and Hall conductivity, respectively, and the transpose acts on the vectorial indices, which in particular will change sign(cr4)T= — ATa due to the antisymmetric nature of the Hall conductivity. The Hamiltonian (17) is quantized by defining the vector potential as an operator (taking h = 1):where ag and fi(r) are the frequency and spatial profile of the cavity mode in the ith direction, respectively. Using the relations (13) and the normalization condition J d3r \f\2= 1, the Hamiltonian operator in the rotating wave-approximation is given by:where N = d^xax+ d^yayis the total photon number operator, <n0» AS, \g\ are the central frequency, mode splitting and hybridizations, respectively, with g = gx+ igaa complex coupling between the two modes. The central frequency 2<n0= Sx+ Syand splittingare determined by the resonant frequencies of the cavities +as the bare frequency and 28a)i = gatiX11the frequency shift due tothe diagonal susceptibility %11of the sample. The rea and imaginary ga=upling between the modes arise due to an off-diagonal susceptibility xxyandfinite Hall conductivity crH, respectively. The geometrical factor inthe above expressions is determined by the cavities’ refractive index, mode profile, and sample shape. When the cavities’ bare frequencies are equal, i.e., <nx= o)y= <n0, the mode splitting and hybridization are simplified torespectively.

[0085] Equation (19) describes two bosonic oscillators with commutation relations and resonant frequenciesSxand Sy. that interact via a complex coupling g. Itcan be readily recast into Eq. (2) by defining the elements of the SU(2) algebra as:with commutation relations

[0086] Probability Evolution

[0087] The evolution of the geometrical ground state, c.f, Eq. (4), is obtained from the operator relations:

[0088] The probability distribution of photon states after evolving with U(t) is given by:where C” is the binomial coefficient and:

[0089] In the specific case of n0= 0, i.e., when the initial state has all the photons in the x cavity, Eq. (23) is reduced to: (26)

[0090] Thermal Noise

[0091] It is assumed that a microwave resonator device with intrinsic loss rate ki, driven by a thermal coherent state with external coupling rate kext. In the density matrix representation the state operator of a thermal coherent state is given by:1 — etp 1j is the normalization factor, and D(a) = eaax~a axis the displacement operator.

[0092] The expectation value of the photon number in they cavity is given by: (28)where the trace is all over Fock states, N = |a|2is the mean photon number of the coherent state is a probability amplitude (see Eq. (25)) and nth=kedetermined by the thermal distribution of the states. Similarly, the variance of the photon number is given by:

[0093] Assuming that the mean photon number of the input field is much larger than the mean photon number of thermal noise, i.e., N » nth, the variance is approximated a

[0094] The precision in estimating they component of the vector B. which is related to the Hall conductivity, is given by the mean-squared error: rAfi2>where T is the coherence time of the device and zm~ \B\1is the duration of a single measurement which is determined by the period of oscillations. The Fisher information is given by F = and at the working time t = 7i|B |1it can be as high as F~|B|2

[0095] Exemplary Transmon Embodiments

[0096] Referring to Fig. 5, an exemplary cross-cavity device utilizing a transmon is illustrated. Device 500 includes waveguides Rx (501) and Ry(502). An ancilla transmon Q (503) is connected to waveguides 501, 502 and is used for loading Fock states in R(x / y). Flux bias 504 is used for tuning the frequency of transmon 503. RF ports 505, 506, 507 are used for direct readout and drive of the resonators 501, 502 and transmon 503.

[0097] As shown, an exemplary circuit consists of a cross resonator which is composed of A two overlapping - waveguide superconducting resonators 501, 502. In some embodiments, the resonators are coplanar. Each resonator is capacitively coupled to a transmon 503 (implementing an ancilla qubit). The transmon 503 is additionally capacitively coupled to a 50(1 transmission line port 507 to directly drive with microwave pulses. There is a flux bias line 504 inductively coupled to the SQUID loop of the transmon 503 allowing for frequency tunablility. The two resonators 501, 502 are also each capacitively coupled to 50(1 transmission line port 505, 506 for direct addressability.

[0098] An exemplary protocol for measurement comprises: 1. State initialization; 2.Beamsplitting; 3. Phase shifting; 4. Time evolution; 5. Phase shifting; 6. Beam splitting; and 7. State readout. The below explanation focuses on an implementation with the 1 -photon geometrical fock state, as defined in Eq (4) above.

[0099] To perform state initialization a signal is transmitted into resonators 501, 502, thereby preparing the cross resonator is in the state | l,0)x y, where x,y represent the modes of the respective cross resonators (denoted Rx and Ryor horizontal and vertical). In particular, to prepare the state, the transmon 503 is excited into |e). This may be referred to as a n -pulse for the qubit state embodied in the transmon. The n -pulse consists of a microwave pulse at the frequency a>geof the transmon tranisiton \g) -» |e). The amplitude and duration of the pulse are calibrated to achieve high fidelity. Once the transmon is excited, it is brought into the strong coupling regime with mode x of the cross resonator 501. The resonant interaction between resonator x and the transmon 503 are described by the Jaynes-CummingsHamiltonian and a quantum state transfer between ) can be performed with aSWAP pulse. The cross resonator is now in the staty. It will be appreciatedthat there are multiple ways to perform SWAP pulses. In various embodiments, a flux bias line 504 is used to move the frequency of the transmon into resonance with the cavity. Other techniques do not require a tunable transmon but have more complicated pulse sequences. Accordingly, a SWAP gate may be applied according to various alternative implementations. It will be further be appreciated that state initialization step provides a mechanism by which the transmon injects a photon into the cavity system, similar to a quantum emitter in optics.

[0100] In various embodiments, beamsplitting (Yg) is implemented through an engineered time-dependent coupling between mode x and mode y. The transmon 503 is driven by two microwave pulses with amplitudes and frequencies respectively. The strengthof the time-dependent interaction is determined by the dispersive coupling of the transmon to each mode of the resonator and the amplitudes of the pulses. The frequencies of the pulses are chosen such thayAs a result, this four-wave mixing drive pulse drives the interaction HintThe interaction Hamiltonian describes a beamsplitter where the length of the interaction determines the beamsplitter ratio. For example, a- pulse will prepare the state into |ip) = 10, 1) + 11,0), acting as a 50:50 4 beamsplitter.

[0101] In various embodiments, phase shifting (Z^) is implemented by driving the transmon503 into a superposition of the ground and excited state with pulse.

[0102] The dispersive interaction between the transmon 503 and the cavities are given by Hdtsp =Since the values#=x^ethe y mode will evolve at a different rate than the x mode. The magnitude of the phase shift < / > =where t is the interaction time between the superposed qubit state and each of the resonator modes.

[0103] After the interaction time, another pulse is applied to the transmon to return to the ground state, switching the interaction off.

[0104] In various embodiments, evolution (t / (t)) is implemented as follows. After state initlaization, beam splitting, and phase shifting as described above, the state | ip0(6, < / >)) = 10,0) has been prepared with arbitrary 0 and (p. Waitingfor time zmallows the state will evolve according to Eq (5) provided above.

[0105] In various embodiments, phase shifting (Z_cp) is implemented according to the method provided above for. In order to reverse the operation, the transmon is prepared with an additional n pulse.

[0106] In various embodiments, beamsplitting (Y_g) is implemented according to the method provided above for Ye. The operation is reversed with a pulse.

[0107] In various embodiments, the final state of the transmon is determined with standard homodyne detection of the qubit state. If all is reciprocal in the circuit, the same photon state should be measured in 11,0). Since the qubit is dispersively coupled to both of the resonators modes, the populations in each mode can be directly measured by applying a n pulse to the transmon at frequencies determined by the dispersive shift. For example, to measure the state of |nx, ny) one can apply a n pulse at u>nx,ny= (q- nyxy3- nxxy3

[0108] In various embodiments, the state of the transmon is read out by applying a microwave signal directly to the transmon. The transmitted signal then goes through a readout circuit which consists of a circulator and an amplifier. This signal is then mixed with an additional microwave signal coming from the same microwave source. The second microwave signal can be tuned in phase to provide constructive and destructive interference with the MW signal transmitted from the transmon. The amplitude and phase modulation of the MW signal from the qubit can then be determined.

[0109] Various embodiments employ quantum non-demolition readout of the transmon. This mechanism requires an additional readout resonator coupled to the transmon. The state can be read out with this method without destroying the transmon state. The readout resonator is going to have a resonance frequency which depends on the state of the transmon. The state is determined by sending a microwave signal into the transmon and measuring the microwave photons transmitted through the resonator.

[0110] In certain embodiments, the invention is described herein according to the following numbered embodiments.1. An apparatus for measuring the optical properties of a sample, the apparatus comprising: a first waveguide and a second waveguide, the first and second waveguides configured to cross at a crossing point; a photon source optically coupled to at least the first or second waveguide; a first photodetector optically coupled to the first waveguide;a second photodetector optically coupled to the second waveguide, wherein the photon source is configured to excite the at least first or second waveguide, thereby evanescently coupling a sample placed at the crossing point with electric field modes of the first and second waveguides, and the first and second photodetectors are configured to measure output electric fields of the first and second waveguides.2. The apparatus described above in numbered embodiment 1, further comprising: a phase shifter interposed between the photon source and the at least first or second waveguide and configured to apply a time-varying phase shift to the photon source.[oni] The descriptions of the various embodiments of the present disclosure have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application or technical improvement over technologies found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.

Claims

CLAIMSWhat is claimed is:

1. An apparatus, comprising: a first electromagnetic resonator; a second electromagnetic resonator intersecting the first resonator, thereby defining an intersection area, the first and the second resonator each comprising at least one interface; a source configured to generate an electromagnetic source signal; at least a first detector; at least one input / output means in electromagnetic communication with the source, the at least one interface of the first resonator and the at least one interface of the second resonator; and wherein the at least one input / output means is configured to: based on a source signal, generate a first input signal and a second input signal; apply to the first and the second signals an input amplitude ratio and an input relative phase shift, thereby generating a first transformed input signal and a second transformed input signal; transmit the first transformed input signal to the at least one interface of the first resonator, thereby generating a first output signal therefrom; and transmit the second transformed input signal to the at least one interface of the second resonator, thereby generating a second output signal therefrom, wherein the at least one input / output means is further configured to: receive the first output signal from the first resonator and the second output signal from the second resonator; apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a signreverse of the input phase shift, thereby generating a first transformed output signal and a second transformed output signal; and transmit at least the first transformed output signal to the at least the first detector,the at least first detector configured to generate at least a first result signal.

2. The apparatus of Claim 1, wherein the intersection area is adapted to receive a sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part.

3. The apparatus of Claim 2, further comprising a controller configured to: cause the source to generate the source signal; and based on the at least the first result signal, compute the real and the imaginary parts of the index of refraction of the sample.

4. The apparatus of any one of Claims 1-3, wherein the source is an optical source, and first and the second resonators are each an optical resonator.

5. The apparatus of Claim 4, wherein the optical source is a single photon source, and the first and second detectors are each a single photon detector.

6. The apparatus of Claim 4, wherein the optical source is a coherent photon source.

7. The apparatus of any one of Claims 4-6, wherein the at least one input / output means comprises an adjustable beam splitter and at least one adjustable phase shifter.

8. The apparatus of any one of Claims 4-7, wherein the first and the second optical resonators each comprises a Fabry-Perot cavity.

9. An apparatus, comprising: a first electromagnetic resonator; a second electromagnetic resonator intersecting the first resonator, thereby defining an intersection area, the first and the second resonator each comprising an input interface and an output interface; a source configured to generate an electromagnetic source signal; a first detector and a second detector; an input means in electromagnetic communication with the source, the input interface of the first resonator and the input interface of the second resonator; andan output means in electromagnetic communication with the output interface of the first resonator, the output interface of the second resonator, and the first and the second detectors, wherein the input means is configured to: based on a source signal, generate a first input signal and a second input signal; apply to the first and the second signals an input amplitude ratio and an input relative phase shift, thereby generating a first transformed input signal and a second transformed input signal; transmit the first transformed input signal to the input interface of the first resonator, thereby generating a first output signal therefrom; and transmit the second transformed input signal to the input interface of the second resonator, thereby generating a second output signal therefrom, and wherein the output means is configured to: receive the first output signal from the first resonator and the second output signal from the second resonator; apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a signreverse of the input phase shift, thereby generating a first transformed output signal and a second transformed output signal; and transmit the first transformed output signal to the first detector, and transmit the second transformed output signal to the second detector, the first and the second detectors configured to generate a first result signal and a second result signal.

10. The apparatus of Claim 9, wherein the intersection area is adapted to receive a sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part.

11. The apparatus of Claim 10, further comprising a controller configured to: cause the source to generate the source signal; and based on the first result signal and the second result signal, compute the real and the imaginary parts of the index of refraction of the sample.

12. The apparatus of any one of Claims 9-11, wherein the input means and the output means each comprises an adjustable beam splitter and at least one adjustable phase shifter.

13. A method of determining an index of refraction of a sample, the method comprising:- providing the apparatus of Claim 1, wherein the intersection area is adapted to receive the sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part;- disposing the sample at the intersection area;- causing the source to generate the source signal;- causing the at least one input / output means to: based on the source signal, generate a plurality of input signal pairs, each input signal pair comprising a first and a second input signals; for each input signal pair, apply to the first and the second input signals an input amplitude ratio and an input relative phase shift, thereby generating a transformed input signal pair, the transformed input signal pair comprising a first and a second transformed input signals; and for each transformed input signal pair, transmit the first transformed input signal to the at least one interface of the first resonator, and transmit the second transformed input signal to the at least one interface of the second resonator, thereby generating an output signal pair, the output signal pair comprising a first and a second output signal;- causing the at least one input / output means to: for each output signal pair, receive the first output signal from the first resonator and the second output signal from the second resonator; for each output signal pair, apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a sign-reverse of the input phase shift, thereby generating a transformed output signal pair, each transformed output signal pair comprising a first transformed output signal and a second transformed output signal; and for each transformed output signal pair, transmit at least the first transformed output signal to the at least the first detector;- causing the at least the first detector, for each transformed output signal pair, to detect at least the first transformed output signal, thereby generating a result signal;- thereby generating a plurality of result signals based on the plurality of input signal pairs; and- based on the plurality of result signals, computing the real and the imaginary parts of the index of refraction of the sample.

14. The method of Claim 13, wherein the source is an optical source, and first and the second resonators are each an optical resonator.

15. The method of Claim 14, wherein the optical source is a single photon source, and the source signal comprises a sequence of single photons, the method further comprising: selecting a plurality of time delay intervals, each time delay interval corresponding to one photon in the sequence, for each photon in the sequence: generating the source signal at a first time point; and generating the corresponding result signal at a second time point, wherein the first and the second time points are separated by the time delay interval corresponding to the photon.

16. The method of Claim 15, wherein the optical source is a coherent photon source, the method further comprising: selecting a time duration interval; and generating the source signal for the time duration interval.

17. A method of determining an index of refraction of a sample, the method comprising:- providing the apparatus of Claim 9, wherein the intersection area is adapted to receive the sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part;- disposing the sample at the intersection area;- causing the source to generate the source signal;- causing the input means to:based on the source signal, generate a plurality of input signal pairs, each input signal pair comprising a first and a second input signals; for each input signal pair, apply to the first and the second input signals an input amplitude ratio and an input relative phase shift, thereby generating a transformed input signal pair, the transformed input signal pair comprising a first and a second transformed input signals; and for each transformed input signal pair, transmit the first transformed input signal to the input interface of the first resonator, and transmit the second transformed input signal to the input interface of the second resonator, thereby generating an output signal pair, the output signal pair comprising a first and a second output signal;- causing the output means to: for each output signal pair, receive the first output signal from the first resonator and the second output signal from the second resonator; for each output signal pair, apply to the first and the second output signals an output amplitude ratio and an output relative phase shift, wherein the output amplitude ratio is an inverse of the input amplitude ratio and the output phase shift is a sign-reverse of the input phase shift, thereby generating a transformed output signal pair, each transformed output signal pair comprising a first transformed output signal and a second transformed output signal; and for each transformed output signal pair, transmit the first transformed output signal to the first detector and the second transformed output signal to the second detector;- causing the first and the second detectors, for each transformed output signal pair, to detect the first transformed output signal and the second transformed output signal, respectively, thereby generating a result signal pair, the result signal pair comprising a first and a second result signals;- thereby generating a plurality of result signal pairs based on the plurality of input signal pairs; and- based on the plurality of result signal pairs, computing the real and the imaginary parts of the index of refraction of the sample.

8. An apparatus, comprising: a first superconducting electromagnetic resonator; a second superconducting electromagnetic resonator intersecting the first resonator, thereby defining an intersection area; a transmon capacitively tunably coupled to the first resonator and the second resonator; a microwave (MW) source operatively coupled to the transmon, the MW source configured to generate one or more drive signals; a transmon readout means operatively coupled to the transmon; and wherein the transmon is configured to: based on the one or more drive signal, be placed into a first quantum state, thereby generating an input signal; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, transmit the input signal into the first resonator and the second resonator, thereby generating a first resonator signal and the second resonator signal, wherein the first resonator signal and the second resonator signal have an input amplitude ratio; reduce coupling to the first and the second resonators; based on the one or more drive signal, be placed in a second quantum state, thereby imparting a first relative phase shift to the first and the second resonator signals; based on the one or more drive signal, be placed in a third quantum state, thereby imparting a second relative phase shift to the first and the second resonator signals, wherein the second relative phase shift is a sign-reverse of the first relative phase shift; based on the one or more drive signal, be placed in a fourth quantum state; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, apply an output amplitude ratio to the first resonator signal and the second resonator signal, wherein output amplitude ratio that is an inverse of the input amplitude ratio, and transmit the first and the second resonator signals to the transmon; and wherein the transmon readout means is configured to read out at least the first resonator signal from the transmon.

19. The apparatus of Claim 18, wherein the transmon is a tunable transmon.

20. The apparatus of Claim 19, wherein the apparatus further comprises a flux bias means operatively coupled to the transmon, the flux bias means configured to change a magnetic flux through the transmon, thereby adjusting the transmon frequency.

21. The apparatus of any one of Claims 18-20, wherein the intersection area is adapted to receive a sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part.

22. The apparatus of Claim 19, further comprising a controller configured to: cause the MW source to generate the one or more drive signals; and based on at least the first resonator signal, compute the real and the imaginary parts of the index of refraction of the sample.

23. A method of determining an index of refraction of a sample, the method comprising:- providing the apparatus of Claim 18, wherein the intersection area is adapted to receive the sample and evanescently couple thereto, the sample having a complex index of refraction defined by a real part and an imaginary part;- disposing the sample at the intersection area;- causing the MW source to generate one or more drive signals;- causing the transmon to: based on the one or more drive signal, be placed in a first quantum state, thereby generating an input signal; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, transmit the input signal into the first resonator and the second resonator, thereby generating a first resonator signal and the second resonator signal, wherein the first resonator signal and the second resonator signal have an input amplitude ratio; reduce coupling to the first and the second resonators; based on the one or more drive signal, be placed in a second quantum state, thereby imparting a first relative phase shift to the first and the second resonator signals;based on the one or more drive signal, be placed in a third quantum state, thereby imparting a second relative phase shift to the first and the second resonator signals, wherein the second relative phase shift is a sign-reverse of the first relative phase shift; based on the one or more drive signal, be placed in a fourth quantum state; strengthen coupling to the first and the second resonators, and, based on the one or more drive signal, apply an output amplitude ratio to the first resonator signal and the second resonator signal, wherein output amplitude ratio that is an inverse of the input amplitude ratio, and transmit the first and the second resonator signals to the transmon;- causing the transmon readout means to read out at least the first resonator signal from the transmon; and- based on at least the first resonator signal, computing the real and the imaginary parts of the index of refraction of the sample.

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