We propose a simple method for generating spin squeezing of atomic ensembles in a Floquet cavity subject to a weak, detuned two-photon driving. We demonstrate that the weak squeezing of light inside the cavity can, counterintuitively, induce strong spin squeezing. This is achieved by exploiting the anti-Stokes scattering process of a photon pair interacting with an atom. Specifically, one photon of the photon pair is scattered into the cavity resonance by absorbing partially the energy of the other photon whose remaining energy excites the atom. The scattering, combined with a Floquet sideband, provides an alternative mechanism to implement Heisenberg-limited spin squeezing. Our proposal does not need multiple classical and cavity-photon drivings applied to atoms in ensembles, and therefore its experimental feasibility is greatly improved compared to other cavity-based schemes. As an example, we demonstrate a possible implementation with a superconducting resonator coupled to a nitrogen-vacancy electronic-spin ensemble.
In analogy to squeezed states of light, spin squeezing in atomic ensembles , , ,  describes the reduction of quantum fluctuation noise in one component of a collective pseudospin, at the expense of increased quantum fluctuation noise in the other component. This property is an essential ingredient for high-precision quantum metrology and also enables various quantum-information applications , . For this reason, significant effort has been devoted to generating spin squeezing; such effort includes exploiting atom–atom collisions in Bose–Einstein condensates , , , , , , , , , and atom–light interactions in atomic ensembles , , , , , , . In particular, cavity quantum electrodynamics , , which can strongly couple atoms to cavity photons, is considered as an ideal platform for spin squeezing implementations , , , , , , , , , , . Here, we propose a fundamentally different approach to prepare atomic spin-squeezed states in cavities and demonstrate that the weak squeezing of the cavity field can induce strong spin squeezing.
One-axis twisting (OAT) and two-axis twisting (TAT) are two basic mechanisms to generate spin-squeezed states , . In high-precision measurements, TAT is considered to be superior to OAT  because TAT can reduce quantum fluctuation noise to the fundamental Heisenberg limit , lower than the OAT-allowed limit . Here, N refers to the number of atoms in an ensemble. Note that both mechanisms depend on controlled unitary dynamics, such that they are extremely fragile to dissipation and also require high-precision control for time evolution. Alternatively, dissipation, when treated as a resource , , , , , has also been exploited to implement Heisenberg-limited squeezing , , , . In dissipative protocols, atomic ensembles can be driven to a spin-squeezed steady state. However, these TAT and dissipative schemes have not been experimentally demonstrated because of their high complexity. This is partially attributed to the need for multiple classical and cavity-photon drivings applied to atoms. For example, various approaches for spin squeezing in cavities rely on a double off-resonant Raman transition (i.e., the double-Λ transition) , , , , , , , . It is generally difficult to realize such a transition for each atom in ensembles for spin squeezing.
In this manuscript, we propose a simplification by introducing a weak and detuned two-photon driving for a Floquet cavity and demonstrate the dissipative preparation of steady-state spin squeezing (SSSS), with Heisenberg scaling. Remarkably, light squeezing inside the cavity in our proposal is very weak and can be understood as a seed for strong spin squeezing. This is essentially different from the process that directly transfers squeezing from light to atomic ensembles , , , , . Such weak squeezing of light avoids two-photon correlation noise and thermal noise, which can give rise to the so-called 3 dB limit in degenerate parametric amplification processes  and can greatly limit spin squeezing.
Furthermore, in contrast to other cavity-based proposals for Heisenberg-limited spin squeezing, our method does not require multiple classical and cavity-photon drivings on atoms, thus significantly reducing the experimental complexity. The key element underlying our method is the absorption of a detuned-driving photon pair: one of these photons is absorbed by the cavity and the other one by an atom. This process can be understood as anti-Stokes scattering, of one photon of the driving photon pair, into the cavity resonance by absorbing part of the energy of the other photon, which excites the atom with its remaining energy. As opposed to typical Raman scattering , the scattered photon in the description above absorbs the energy of another photon, rather than the excitation of matter, e.g., atoms, molecules, or mechanics.
2 Physical model
We consider an ensemble consisting of N two-level atoms in a single-mode cavity of frequency ωc, as shown in Figure 1. For simplicity, these atoms are assumed to be identical, such that they have the same transition frequency ωq and their transitions from the ground state to the excited state are driven by the same coupling g to the cavity photon. This atomic ensemble can be described using collective spin operators , where (α = x, y, z) are the Pauli matrices for the jth atom. The cavity mode is driven by a weak, detuned two-photon driving, e.g., with amplitude Ω, frequency ωL, and phase θL. Such a parametric driving can produce photon pairs at ωL/2 and induce a squeezing sideband at ωL − ωc [see Figure 2(a)]. If this sideband is tuned to the atomic resonance ωq (i.e., ωq ≈ ωL − ωc), one photon of the driving photon pair is then scattered into the cavity resonance by absorbing a small part of the energy of the other photon; at the same time the main part of the absorbed-photon energy resonantly excites an atom [see Figure 2(b)]. We further assume that the cavity frequency ωc is periodically modulated with amplitude Am and frequency ωm and ensure that ωq ≈ ωc − ωm. In this case, a detuned atom can emit a photon into the cavity resonance via a Floquet sideband at ωc − ωm [see Figure 2(a)]. The above dynamics demonstrates that the cavity-photon creation gives rise to a competition between the atomic excitation and deexcitation.
To be specific, we consider the Hamiltonian
Here, and . In addition to the driving Ω, we have also assumed another two-photon driving, which has the same frequency and phase as the driving Ω, but with a time-dependent amplitude . The use of such a driving is to suppress an undesired two-photon driving of the cavity mode, which is induced by the periodic modulation of the cavity frequency and can destroy the dynamics of generating SSSS.
To describe the dissipative dynamics, we use the Lindblad dissipator, given by . Thus, corresponds to cavity loss at a rate κ, and , where , describes atomic spontaneous emission at a rate γ. It follows, on taking the Fourier transformation , that , indicating that the collective spin operators are related only to the zero momentum mode , , . Consequently, we have because different momentum modes are uncoupled and nonzero momentum modes only decay. The full dynamics of the system is therefore determined by the master equation
We begin by restricting our discussion to the limits and . In such a case, the squeezing sideband resulting from the driving Ω enables a coupling in the form
with strength . The coupling becomes resonant when ωq ≈ ωL − ωc. Such a coupling can be understood from the interaction between a driving photon pair and a single atom, as shown in Figure 2(c). The ground state is driven to a virtual excited state via the two-photon driving Ω with detuning ≈ 2∆c and then is resonantly coupled to the state via the atom-cavity coupling g. Here, the number in the ket refers to the cavity-photon number. This mechanism is responsible for anti-Stokes scattering of correlated photon pairs mentioned above. Furthermore, for ωq ≈ ωc − ωm, the coupling,
is also made resonant via a first-order Floquet sideband but its strength becomes . As we demonstrate in more detail in Appendix A, these two resonant couplings lead to an effective Hamiltonian
where and . Here, we have set and a phase factor i has been absorbed into a. The dynamics driven by Heff describes two distinct atomic transitions, which can cause the spin-squeezed state to become a dark state , , , . In particular, in the optimal case of , assuming G+ to be very close to G_, it yields the maximally spin-squeezed state corresponding to the Heisenberg-limited noise reduction . In Figure 3(a) we plot the spin Husimi distribution using H(t). Here, , where refers to a coherent-spin state with all the atoms in the excited state, and is a rotation operator, which rotates by an angle θ about the axis of the collective Bloch sphere. We find, as predicted by Heff, that quantum noise is reduced along the x direction, at the expense of increased quantum noise along the y direction.
where is the total spin operator, and is the minimum spin fluctuation in the direction perpendicular to the mean spin . Spin-squeezed states, where quantum fluctuation in one quadrature is reduced below the standard quantum limit, exhibit . We find from Figure 3(b) that a strong loss of a weakly and parametrically driven Floquet cavity can enable to be ≪1 in the steady state. In contrast, atomic spontaneous emission carries away information about spin-squeezed states, and hence limits spin squeezing, as plotted in the inset of Figure 3(b). In Figure 3(c), we plot the steady-state , labeled , versus the number N of atoms. The enhancement of spin squeezing by increasing N has a lower bound which, as demonstrated below, is determined by the ratio in the limit of .
3 Spin-wave approximation
We now consider the case of , so that the dynamics of the collective spin can be mapped to a bosonic mode b, i.e., . Here, we have assumed that the number of excited atoms is much smaller than the total number N, i.e., , and have made the spin-wave approximation. The effective Hamiltonian is correspondingly transformed to
where , and , with , describes a squeezed mode of the collective spin. The cavity loss thus can drive the mode β to its vacuum, which corresponds to a squeezed vacuum state of the mode b. Under the spin-wave approximation, the parameter is likewise transformed to
This implies that the two-atom correlation, , characterizes a key signature of spin squeezing.
In order to achieve , we have neglected the off-resonant coupling to the zero-order Floquet sideband, which lowers the degree of spin squeezing [see Figure 3(b) and (c)]. Let us now consider this off-resonant coupling. In the limit , such a coupling shifts the cavity and atomic resonances , and as a result it causes an additional detuning between cavity and atoms. To avoid this undesired effect, the modulating frequency ωm needs to be modified to compensate δ, such that (see Appendix B). With such a modification, we directly calculate the parameter and the correlation obtained using the effective and full Hamiltonians under the spin-wave approximation. We find from Figure 4(a) that after compensating the detuning δ, the full dynamics are in excellent agreement with the desired effective dynamics. This allows us to investigate stronger spin squeezing, according to such an effective Hamiltonian.
Based on , we derive the steady-state and , yielding
where . Here, is the collective cooperativity. Having r ≥ 1 gives , and therefore a strong spin-squeezed state is achieved if . More specifically, we consider the steady-state expressed as
This demonstrates that if , then the parameter r and, thus, spin squeezing increases. However, as , the effective coupling, , between modes a and β tends to zero (i.e., ), which suppresses the cooling of the mode β. The optimal SSSS therefore results from a tradeoff between these two processes , , . Furthermore, we find that for a spin-squeezed steady state, the number of excited atoms scales as , but at the same time, the spin-wave approximation requires . To demonstrate the squeezing scaling, we assume that in the steady state, , where 0 < μ < 1. In this case, , and consequently , is justified even for , as long as N is sufficiently large. Hence, our approach can, in principle, enable spin squeezing to be far below the standard quantum limit, and approach the Heisenberg limit in a large ensemble.
To consider the squeezing time, we adiabatically eliminate the cavity mode (see Appendix C), yielding
where ρspin describes the reduced density matrix of the collective spin, and represents the cavity-induced atomic decay. According to this adiabatic master equation, and evolve as
where , , and Xini refers to the initial X. We therefore find that the atomic ensemble can be driven into a spin-squeezed state from any initial state in the spin- manifold. Under time evolution, is given by
Here, we have assumed, for simplicity, that . This expression predicts that time evolution leads to an exponential squeezing with a rate γc + γ, as plotted in Figure 4(b). For a realistic setup, e.g., a nitrogen-vacancy (NV) spin ensemble coupled to a superconducting resonator (see below), a negligibly small spin decay rate and a typical collective coupling MHz could result in a spin-squeezed steady state of ≈ −20 dB in a squeezing time ≈ 8 µs. This allows us to neglect spin decoherence because the coherence time in ensembles of NV centers can experimentally reach the order of ms  or even ∼1 s .
4 Proposed experimental implementation
As an example, we now consider a hybrid quantum system , , , where a superconducting transmission line (STL), terminated by a superconducting quantum interference device (SQUID), is magnetically coupled to an NV spin ensemble in diamond (see Appendix D for details). The coherent coupling of an STL cavity to an NV spin ensemble has already been widely implemented in experiments , , , , , , . In particular, the studies by Kubo et al. , ,  used a SQUID to control the cavity frequency. Therefore to achieve a parametrically driven Floquet cavity, we connect a SQUID to one end of the STL. We then assume the driving phase f(t) across the SQUID loop to be
Here, the components f1 and f2(t) result in the drivings Ω and Ω1(t), respectively, while the component f3 is to modulate the cavity frequency ωc. Moreover, the electronic ground state of NV centers is a spin triplet, whose ms = 0 and ms = ±1 sublevels are labeled by and . There exists a zero-field splitting ≈ 2.87 GHz between state and states . In the presence of an external magnetic field, the states are further split through the Zeeman effect, which enables a two-level atom with as the ground state and (or ) as the excited state. When the diamond containing an NV spin ensemble is placed on top of the STL, the cavity photon can drive the transition (or ) via a magnetic coupling.
We have introduced an experimentally feasible method for how to implement Heisenberg-limited SSSS of atomic ensembles in a weakly and parametrically driven Floquet cavity. This method demonstrates a counterintuitive phenomenon: the weak squeezing of light can induce strong spin squeezing. This approach does not require multiple actions on atoms, thus greatly reducing the experimental complexity. We have also shown an anti-Stokes scattering process, induced by an atom, of a correlated photon pair, where one photon of the photon pair is scattered into a higher-energy mode by absorbing a fraction of the energy of the other photon, and the remaining energy of the absorbed photon excites the atom. If the scattered photon is further absorbed by another atom before being lost, then such a scattering process can also generate an atom-pair excitation and, as a consequence, can enable TAT spin squeezing. The two distinct atomic transitions demonstrated are functionally similar to, but experimentally simpler than, the double off-resonant Raman transition in multilevel atoms widely used for generating spin squeezing , . Thus, we could expect that our method can provide a universal building block for implementing spin-squeezed states and simulating ultrastrong light–matter interaction ,  and quantum many-body phase transition .
Funding source: Japan Society for the Promotion of Science (JSPS)
Award Identifier / Grant number: P19028
Funding source: Polish National Science Centre (NCN)
Award Identifier / Grant number: DEC-2019/34/A/ST2/00081
Funding source: Army Research Office (ARO)
Award Identifier / Grant number: W911NF-18-1-0358
Funding source: Japan Science and Technology Agency (JST)
Award Identifier / Grant number: CREST No. JPMJCR1676
Funding source: the Foundational Questions Institute Fund (FQXi)
Award Identifier / Grant number: FQXi-IAF19-06
Funding source: NTT Research
Award Identifier / Grant number: none
Funding source: Japan Science and Technology Agency (JST)
Award Identifier / Grant number: Q-LEAP
Funding source: Japan Society for the Promotion of Science (JSPS)
Award Identifier / Grant number: KAKENHI No. JP20H00134
Funding source: Japan Society for the Promotion of Science (JSPS)
Award Identifier / Grant number: JSPS-RFBR No. JPJSBP120194828
Funding source: Asian Office of Aerospace Research and Development (AOARD)
The authors thank Fabrizio Minganti, Nathan Shammah, and Vincenzo Macrì for their valuable discussions. Y.-H.C. is supported by the Japan Society for the Promotion of Science (JSPS) Foreign Postdoctoral Fellowship No. P19028. A.M. is supported by the Polish National Science Centre (NCN) under the Maestro Grant No. DEC-2019/34/A/ST2/00081. F.N. is supported in part by: NTT Research, Army Research Office (ARO) (Grant No. W911NF-18-1-0358), Japan Science and Technology Agency (JST) (via the Q-LEAP program and the CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (via the KAKENHI Grant No. JP20H00134, and the JSPS-RFBR Grant No. JPJSBP120194828), and the Grant No. FQXi-IAF19-06 from the Foundational Questions Institute Fund (FQXi), a donor advised fund of the Silicon Valley Community Foundation.
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: The authors thank Fabrizio Minganti, Nathan Shammah, and Vincenzo Macrì for their valuable discussions. Y.-H.C. is supported by the Japan Society for the Promotion of Science (JSPS) Foreign Postdoctoral Fellowship No. P19028. A.M. is supported by the Polish National Science Centre (NCN) under the Maestro Grant No. DEC-2019/34/A/ST2/00081. F.N. is supported in part by: NTT Research, Army Research Office (ARO) (Grant No. W911NF-18-1-0358), Japan Science and Technology Agency (JST) (via the Q-LEAP program and the CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (via the KAKENHI Grant No. JP20H00134 and the JSPS-RFBR Grant No. JPJSBP120194828), the Asian Office of Aerospace Research and Development (AOARD), and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
Appendix A: Effective Hamiltonian and decay of the collective spin
Let us first derive the effective Hamiltonian Heff. We begin with the full Hamiltonian in a rotating frame,
Here, , where ωc is the cavity frequency, ωq is the atomic transition frequency, and ωL is the frequency of the two-photon driving. The cavity mode a is dressed by the detuned two-photon driving Ω and becomes a squeezed mode α. This squeezing operation can be described by the Bogoliubov transformation,
determines the degree of squeezing of the cavity field. It then follows that
where is the squeezed-mode frequency. It is seen from Eqs. (A4) and (A6) that, inside the cavity, there exist an upper squeezing sideband at and a lower squeezing sideband at . The Hamiltonian H(t), when expressed in terms of the mode α, is transformed to
where . In Eq. (A7), we have assumed that , such that an undesired parametric driving of the mode α can be eliminated. The last two terms of Eq. (A7) describe two distinct spin-cavity couplings, which are associated with the upper and lower squeezing sidebands, respectively.
We now focus our discussion on the limit , where light squeezing inside the cavity is very weak. Such weak squeezing can avoid two-photon correlation noise and thermal noise, which are generally considered detrimental in strong-squeezing processes , . In this limit, we have
which, in turn, gives
Consequently, the squeezed mode α can, according to the Bogoliubov transformation in Eq. (A4), be approximated by the bare mode a, i.e.,
The Hamiltonian H(t) is therefore approximated by
Note that, in the limit of , the upper squeezing sideband becomes the cavity resonance due to , and the lower squeezing sideband is likewise shifted to ωL − ωc (i.e., ).
Upon introducing a unitary transformation
with , in Eq. (A11) is then transformed to
where we have used the Jacobi–Anger identity
with being the nth-order Bessel function of the first kind.
We find that, when (i.e., ), the last sum in Eq. (A13) contains a resonant coupling of the form
with strength . Such a coupling, which originates from the lower squeezing sideband at (ωL − ωc), describes the anti-Stokes scattering process of a driving photon pair interacting with an atom. Specifically, one photon of the photon pair is scattered into the cavity resonance by absorbing part of the energy of the other photon and simultaneously the remaining energy of the absorbed photon excites the atom. When we further choose 2ωs = ωm (i.e., ωq ≈ ωc − ωm), the first sum in Eq. (A13) also contains a resonant coupling of the form
with strength . This coupling, which is mediated via a first-order Floquet sideband at (ωc − ωm), describes that a detuned atom can emit a photon into the cavity resonance. Under the assumptions, and (i.e., ), off-resonant couplings can be neglected, and thus the system dynamics is determined by the following effective Hamiltonian
where and . Here, we have set and a phase factor i has been absorbed into a.
We now consider the dissipative dynamics of the system. The dissipative dynamics can be described with the Lindblad operator
such that corresponds to cavity loss, and to atomic spontaneous emission. It is, in general, very difficult to perform numerical simulations for a large ensemble because the Hilbert space of the ensemble grows as 2N. In order to reduce the dimension of this Hilbert space, we follow the method in a study by Gelhausen et al. , Shammah et al. , and Macrì et al.  and perform a Fourier transformation,
It then follows, using , that
where the first and second terms on the right-hand side describe the dissipative processes of the zero and nonzero momentum modes, respectively. It is seen, from the full Hamiltonian H(t) in Eq. (A1) or the effective Hamiltonian Heff in Eq. (A17), that the coherent dynamics only involves the zero (k = 0) momentum mode. This implies that we can only focus on the zero momentum mode; that is,
This is valid in the steady-state limit or the long-time limit because the nonzero momentum modes in Eq. (A20) only decay. In particular, such a reduction can exactly describe the dissipative dynamics of an atomic ensemble initially in the ground state. Therefore, the dynamics of the system is driven by the following master equation
where can be taken to be H(t) for the full dynamics or to be Heff for the effective dynamics.
In Figure A1, we numerically integrated the master equation in Eq. (A22), with the full Hamiltonian H(t) and the effective Hamiltonian Heff. Specifically, we plot the spin squeezing parameter versus the scaled evolution time in Figure A1(a) and versus the ratio in Figure A1(b). The result in this figure reveals that Heff can describe well the dynamics of the system. The divergence between them mainly arises from neglecting an off-resonant coupling to the zero-order Floquet sideband. In the next section, we discuss how to remove the detrimental effect induced by such an off-resonant coupling under the spin-wave approximation.
Appendix B: Detuning arising from non-resonant couplings
Under the spin-wave approximation (i.e., ), the Hamiltonian in Eq. (A13) becomes
where represents a collective coupling. It is seen that, when and , the off-resonant coupling to the zero-order (n = 0) Floquet sideband, given by
with , dominates other off-resonant couplings, due to the property that for . Therefore, we may drop these counter-rotating terms for n ≠ 0.
As demonstrated above, two resonant couplings in lead to the effective Hamiltonian
Here, we have defined a squeezed mode, , of the collective spin, with and .
Furthermore, after time averaging , the effective dynamics of the coupling is determined by
This implies that the coupling shifts the cavity resonance frequency and the atomic transition frequency by and , respectively. This, in turn, enables an additional detuning of between cavity and atoms. For the effective Hamiltonian , the detuning δ has no effect on the coupling of the form , but it causes the coupling to become far off-resonant if gcol is comparable to Ω. As a result, the degree of spin squeezing decreases, and even the desired dynamics is destroyed. To remove such a detrimental effect, we need to modify the resonant condition (i.e., ) to be
which compensates the detuning δ. In Figure A2, we use the full Hamiltonian H(t) by compensating the detuning δ to numerically calculate the excited-atom number , the two-atom correlation , and the spin squeezing parameter . We then compare them with the predictions of the effective Hamiltonian . Note that the full Hamiltonian H(t) has been obtained under the spin-wave approximation. We see from Figure A2 that, when the detuning δ is compensated, the full dynamics is in excellent agreement with the desired effective dynamics.
Appendix C: Adiabatic elimination of the cavity mode
We now discuss how to adiabatically eliminate the cavity mode. To begin, we consider the master equation with the effective Hamiltonian ,
As mentioned already, we work within the limit , and the squeezing of the cavity field is very weak. In this case, the occupation of the cavity mode is very low, such that we can only consider the vacuum state and the single-photon state of the cavity mode. The density matrix, ρ, of the system can therefore be expanded as
Upon substituting this expression into the master equation in Eq. (C1), we obtain
and . It then follows, on setting , that
Here, we have assumed . This assumption is generally valid because, for a typical atomic ensemble, e.g., an NV spin ensemble, the atomic decay rate γ is negligible compared to the cavity loss rate κ. Then, substituting Eq. (C6) into Eqs. (C3) and (C4) leads to the following adiabatic master equation
where is the reduced density matrix of the collective spin, and represents the cavity-induced atomic decay. We analytically find, according to Eq. (C.7), that
Here, is the initial excited-atom number, is the initial two-atom correlation, and the corresponding steady-state values are
where . It follows, using , that
where, for simplicity, we have assumed .
In Figure A3, we compare the analytical in Eq. (C11) with the exact numerical simulations of the full Hamiltonian H(t) in Eq. (A1). This figure shows a good agreement, in particular, for the steady-state behavior (yellow regions). The oscillation of red solid curves results from the reversible energy exchange between cavity and atoms (i.e., Rabi oscillation). However, this Rabi oscillation vanishes in the limit , as shown in Figure A3. This is because the coupling, Ggcol, in the effective Hamiltonian becomes smaller when G+ approaches G−. Thus, Eqs. (C10) and (C11) may be used to analytically predict stronger SSSS.
Appendix D: Proposed experimental implementation with hybrid quantum systems and its feasibility
In this section, we consider a hybrid system, where a superconducting transmission line (STL) is terminated by a superconducting quantum interference device (SQUID) and is magnetically coupled to an NV spin ensemble in diamond. The strong coupling between the STL cavity and the NV spin ensemble has already been widely implemented experimentally , , , , , , . In particular, in the studies by Kubo et al. , , , a SQUID has already been used to tune the cavity frequency.
D1 Proposed experimental implementation
We first show how to use an STL terminated by a SQUID to implement a parametrically driven Floquet cavity. The equivalent circuit for this setup is schematically illustrated in Figure A4. The STL of length d can be divided into N segments of equal length , and then this can be modeled as a series of LC circuits each with a capacitance and an inductance . Here, C0 and L0 are the characteristic capacitance and inductance per unit length, respectively. The Lagrangian for the STL is therefore given by , , :
where ϕi is the node phase, and is the speed of light in the STL. In the continuum limit , we have , and . As a result, becomes
The Lagrangian for the SQUID is
Here, EJ,i, CJ,i, and ϕJ,i are, respectively, the Josephson energy, capacitance, and phase of the ith component Josephson junction in the SQUID loop. The phases ϕJ,i of the Josephson junctions depend on the external magnetic flux, such that is determined by a driving phase f(t) across the SQUID, yielding . We assume that the SQUID is symmetric, i.e., CJ,1 = CJ,2 = CJ and EJ,1 = EJ,2 = EJ. The Lagrangian is reduced to
where we have assumed that an effective phase of the SQUID, , is equal to the boundary phase of the STL, ϕd = ϕ(d, t). The cavity Lagrangian, including the STL and SQUID Lagrangians, is
We now discuss how to quantize the system. We begin with the massless scalar Klein–Gordon equation ,
which results from the Lagrangian . This wave equation is complemented with two boundary conditions at the open end of the STL, and
at the end connected to the SQUID. We tune the driving phase f(t) to be
where f0, f1 and f3 are time-independent, but f2(t) is time-dependent. We restrict our discussion to the case where f1, f2(t), and f3 are much weaker than f0. As we demonstrate below, f1 corresponds to the two-photon driving with a time-independent amplitude, f2(t) to another two-photon driving with a time-dependent amplitude, and f3 to the cavity-frequency modulation. Following the procedure in a study by Wustmann et al. , the solution of the wave function in Eq. (D6) is given by
and the cavity Lagrangian , accordingly, becomes
Here, Mn is an effective mass, defined as
and V is a nonlinear potential, defined as
Consequently, the canonical conjugate variable of qn is
thereby resulting in the cavity Hamiltonian
with a free Hamiltonian
We find that H0 describes a collection of independent harmonic oscillators, but V can provide either linear or nonlinear interactions between them.
Following the standard quantization procedure, we replace the c-numbers qn and pn by operators, which obey the canonical commutation relation . We then introduce the annihilation and creation operators an and
where is the zero-point fluctuation of the variable qn. Here, an and obey the canonical commutation relation . With these definitions, the free Hamiltonian H0 is transformed to
We find that the quantized STL contains infinitely many modes, but the existence of the driving phase f(t) enables us to selectively excite a desired mode, e.g., the fundamental mode a0 (see below). The nonlinear potential V can be approximated as
by assuming that and . According to the solution in Eq. (D9), the quadratic potential V can be expressed, in terms of the modes an, as
This means that the potential can excite or couple different modes. To select the fundamental mode a0, we further assume that ωL1 = ωL2 ≈ 2ω0 and . In this case, we can only focus on the a0 mode and other modes can be neglected, yielding
Here, ωL = ωL1 = ωL2, ωm = ωL3, θL = θL1 = θL2, and . Moreover, we have defined
In a frame rotating at , the cavity Hamiltonian becomes (hereafter, we set )
where we have written . The Hamiltonian in Eq. (D23) describes a parametrically driven Floquet cavity.
Below let us consider the coupling of such a cavity to an NV spin ensemble in diamond. The electronic ground state of a single NV center is a long-lived spin triplet, whose ms = 0 and ms = ±1 sublevels we label by and , respectively. The level structure is shown in Figure A5. If there is no external magnetic field, the states are degenerate, and due to the spin–spin interaction, they are separated from the state by the zero-field splitting D ≈ 2.87 GHz. In the presence of an external magnetic field B, the Zeeman splitting, which depends on the magnetic field strength, appears between the states . This yields a two-level atom or a qubit, with as the ground state and either or as the excited state. Here, we focus on, e.g., the transition, and the transition can be neglected due to large detuning. When a diamond containing an NV spin ensemble is placed on top of an STL, the STL mode a can magnetically couple to the transition. Therefore, the collective spin-cavity coupling can be described by the following Hamiltonian
where is the lowering operator for the jth spin qubit, , gj is the single spin-cavity coupling strength, and N is the total number of spins. Such a spin ensemble can also be described with collective spin operators
Here, . The Hamiltonian Hint is accordingly transformed into
Furthermore, we assume, for simplicity but without loss of generality, that gj is a constant, such that gj = g, yielding . Combined with the cavity Hamiltonian in Eq. (D23), the full Hamiltonian for the system becomes
It is seen that the Hamiltonian H in Eq. (D27) is exactly the one applied by us in the main article.
2 Experimental feasibility
In Table A1, we list some relevant parameters reported in recent experiments demonstrating the coupling between an NV spin ensemble and an STL cavity. In addition to these parameters listed in Table I, the coherence time of NV spin ensembles, with spin-echo sequences, has experimentally reached the order of ms (i.e., kHz)  and harnessing dynamical-decoupling sequences can further make this coherence time close to 1 s (i.e., Hz) .
|||2.87⋆||∼1.9 × 103||∼1.5||∼1012||∼11||∼3||–|
|||2.701||∼3.2 × 103||∼0.8||∼1012||∼10||–||∼0.004|
|||2.88⋆||∼1.8 × 103||∼1.6||∼1012||∼11||∼5.3||–|
|||2.6899||∼3.0 × 103||∼0.8||∼1012||∼9||∼5.2||–|
|||2.7491||∼4.3 × 103||∼0.6||–||∼10||–||–|
Note that the studies by Kubo et al. , ,  used a SQUID to tune the resonance frequency of an STL cavity coupled to an NV spin ensemble. This setup is similar to the one we have already proposed for a possible implementation of our proposal.
The analytical in Eq. (C11) predicts that, for typical parameters MHz, MHz, and in Table I, a spin-squeezed steady state of ≈−12 dB can be achieved for a squeezing time ≈0.8 µs, or ≈−20 dB for ≈8 µs. This justifies neglecting spin decoherence, which, as described above, could be made much slower. We also find, according to an exponential squeezing given in Eq. (C11), that by properly increasing γc, we can achieve a shorter squeezing time.
Moreover, in addition to the NV spin ensembles, ion spin ensembles , ,  and P1 center ensembles  can also couple to an STL cavity. In a recent experiment , the coupling of an ensemble of 87Rb atoms to an STL cavity has already been reported.
Hence, we expect that our proposal could be realized with current technologies.
 D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, “Spin squeezing and reduced quantum noise in spectroscopy,” Phys. Rev. A, vol. 46, p. R6797, 1992. https://doi.org/10.1103/physreva.46.r6797.Search in Google Scholar PubMed
 D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A, vol. 50, p. 67, 1994. https://doi.org/10.1103/physreva.50.67.Search in Google Scholar PubMed
 L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, “Quantum metrology with nonclassical states of atomic ensembles,” Rev. Mod. Phys., vol. 90, p. 035005, 2018,. https://doi.org/10.1103/revmodphys.90.035005.Search in Google Scholar
 A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, “Many-particle entanglement with Bose–Einstein condensates,” Nature, vol. 409, p. 63, 2001. https://doi.org/10.1038/35051038.Search in Google Scholar PubMed
 C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, and M. A. Kasevich, “Squeezed states in a Bose–Einstein condensate,” Science, vol. 291, pp. 2386–2389, 2001. https://doi.org/10.1126/science.1058149.Search in Google Scholar PubMed
 J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, “Squeezing and entanglement in a Bose–Einstein condensate,” Nature, vol. 455, p. 1216, 2008. https://doi.org/10.1038/nature07332.Search in Google Scholar PubMed
 M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature, vol. 464, p. 1170, 2010. https://doi.org/10.1038/nature08988.Search in Google Scholar PubMed
 C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature, vol. 464, p. 1165, 2010. https://doi.org/10.1038/nature08919.Search in Google Scholar PubMed
 B. Lücke, M. Scherer, J. Kruse, et al., “Twin matter waves for interferometry beyond the classical limit,” Science, vol. 334, pp. 773–776, 2011. https://doi.org/10.1126/science.1208798.Search in Google Scholar PubMed
 L. Yu, J. Fan, S. Zhu, G. Chen, S. Jia, and F. Nori, “Creating a tunable spin squeezing via a time-dependent collective atom-photon coupling,” Phys. Rev. A, vol. 89, p. 023838, 2014. https://doi.org/10.1103/physreva.89.023838.Search in Google Scholar
 X.-Y. Luo, Y.-Q. Zou, L.-N. Wu, et al., “Deterministic entanglement generation from driving through quantum phase transitions,” Science, vol. 355, pp. 620–623, 2017. https://doi.org/10.1126/science.aag1106.Search in Google Scholar PubMed
 M. Fadel, T. Zibold, B. Décamps, and P. Treutlein, “Spatial entanglement patterns and Einstein–Podolsky–Rosen steering in Bose–Einstein condensates,” Science, vol. 360, pp. 409–413, 2018. https://doi.org/10.1126/science.aao1850.Search in Google Scholar PubMed
 A. Kuzmich, K. Mølmer, and E. S. Polzik, “Spin squeezing in an ensemble of atoms illuminated with squeezed light,” Phys. Rev. Lett., vol. 79, p. 4782, 1997. https://doi.org/10.1103/physrevlett.79.4782.Search in Google Scholar
 J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, “Spin squeezed atoms: a macroscopic entangled ensemble created by light,” Phys. Rev. Lett., vol. 83, p. 1319, 1999. https://doi.org/10.1103/physrevlett.83.1319.Search in Google Scholar
 B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature, vol. 413, p. 400, 2001. https://doi.org/10.1038/35096524.Search in Google Scholar PubMed
 A. Kuzmich, L. Mandel, and N. P. Bigelow, “Generation of spin squeezing via continuous quantum nondemolition measurement,” Phys. Rev. Lett., vol. 85, p. 1594, 2000. https://doi.org/10.1103/physrevlett.85.1594.Search in Google Scholar
 M. Koschorreck, M. Napolitano, B. Dubost, and M. W. Mitchell, “Quantum nondemolition measurement of large-spin ensembles by dynamical decoupling,” Phys. Rev. Lett., vol. 105, p. 093602, 2010. https://doi.org/10.1103/physrevlett.105.093602.Search in Google Scholar
 T. Chalopin, C. Bouazza, A. Evrard, et al., “Quantum-enhanced sensing using non-classical spin states of a highly magnetic atom,” Nat. Commun., vol. 9, p. 4955, 2018. https://doi.org/10.1038/s41467-018-07433-1.Search in Google Scholar PubMed PubMed Central
 A. Evrard, V. Makhalov, T. Chalopin, et al., “Enhanced magnetic sensitivity with non-Gaussian quantum fluctuations,” Phys. Rev. Lett., vol. 122, p. 173601, 2019. https://doi.org/10.1103/physrevlett.122.173601.Search in Google Scholar PubMed
 X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep., vols 718–719, pp. 1–102, 2017. https://doi.org/10.1016/j.physrep.2017.10.002.Search in Google Scholar
 A. Banerjee, “Generation of atomic-squeezed states in an optical cavity with an injected squeezed vacuum,” Phys. Rev. A, vol. 54, p. 5327, 1996. https://doi.org/10.1103/physreva.54.5327.Search in Google Scholar PubMed
 I. D. Leroux, M. H. Schleier-Smith, and V. Vuletić, “Implementation of cavity squeezing of a collective atomic spin,” Phys. Rev. Lett., vol. 104, p. 073602, 2010. https://doi.org/10.1103/physrevlett.104.073602.Search in Google Scholar
 M. H. Schleier-Smith, I. D. Leroux, and V. Vuletić, “States of an ensemble of two-level atoms with reduced quantum uncertainty,” Phys. Rev. Lett., vol. 104, p. 073604, 2010. https://doi.org/10.1103/physrevlett.104.073604.Search in Google Scholar PubMed
 J. G. Bohnet, Ke. C. Cox, M. A. Norcia, J. M. Weiner, Z. Chen, and J. K. Thompson, “Reduced spin measurement back-action for a phase sensitivity ten times beyond the standard quantum limit,” Nat. Photonics, vol. 8, p. 731, 2014. https://doi.org/10.1038/nphoton.2014.151.Search in Google Scholar
 O. Hosten, N. J. Engelsen, R. Krishnakumar, and M. A. Kasevich, “Measurement noise 100 times lower than the quantum-projection limit using entangled atoms,” Nature, vol. 529, p. 505, 2016. https://doi.org/10.1038/nature16176.Search in Google Scholar PubMed
 K. C. Cox, G. P. Greve, J. M. Weiner, and J. K. Thompson, “Deterministic squeezed states with collective measurements and feedback,” Phys. Rev. Lett., vol. 116, p. 093602, 2016. https://doi.org/10.1103/physrevlett.116.093602.Search in Google Scholar PubMed
 Y.-C. Zhang, X.-F. Zhou, X. Zhou, G.-C. Guo, and Z.-W. Zhou, “Cavity-assisted single-mode and two-mode spin-squeezed states via phase-locked atom-photon coupling,” Phys. Rev. Lett., vol. 118, p. 083604, 2017. https://doi.org/10.1103/physrevlett.118.083604.Search in Google Scholar
 R. J. Lewis-Swan, M. A. Norcia, J. R. K. Cline, J. K. Thompson, and A. M. Rey, “Robust spin squeezing via photon-mediated interactions on an optical clock transition,” Phys. Rev. Lett., vol. 121, p. 070403, 2018. https://doi.org/10.1103/physrevlett.121.070403.Search in Google Scholar
 B. Braverman, A. Kawasaki, E. Pedrozo-Peñafiel, et al., “Near-unitary spin squeezing in Y171b,” Phys. Rev. Lett., vol. 122, p. 223203, 2019. https://doi.org/10.1103/physrevlett.122.223203.Search in Google Scholar
 C. Song, K. Xu, H. Li, et al., “Generation of multicomponent atomic Schrödinger cat states of up to 20 qubits,” Science, vol. 365, pp. 574–577, 2019. https://doi.org/10.1126/science.aay0600.Search in Google Scholar PubMed
 F. Verstraete, M. M. Wolf, and J. I. Cirac, “Quantum computation and quantum-state engineering driven by dissipation,” Nat. Phys., vol. 5, pp. 633–636, 2009. https://doi.org/10.1038/nphys1342.Search in Google Scholar
 H. Krauter, C. A. Muschik, K. Jensen, et al., “Entanglement gnerated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett., vol. 107, p. 080503, 2011. https://doi.org/10.1103/physrevlett.107.080503.Search in Google Scholar
 Y. Lin, J. P. Gaebler, F. Reiter, et al., “Dissipative production of a maximally entangled steady state of two quantum bits,” Nature, vol. 504, p. 415, 2013. https://doi.org/10.1038/nature12801.Search in Google Scholar PubMed
 W. Qin, X. Wang, A. Miranowicz, Z. Zhong, and F. Nori, “Heralded quantum controlled-phase gates with dissipative dynamics in macroscopically distant resonators,” Phys. Rev. A, vol. 96, p. 012315, 2017. https://doi.org/10.1103/physreva.96.012315.Search in Google Scholar
 W. Qin, A. Miranowicz, P.-B. Li, X.-Y. Lü, J. Q. You, and F. Nori, “Exponentially enhanced light–matter interaction, cooperativities, and steady-state entanglement using parametric amplification,” Phys. Rev. Lett., vol. 120, p. 093601, 2018. https://doi.org/10.1103/physrevlett.120.093601.Search in Google Scholar
 A. S. Parkins, E. Solano, and J. I. Cirac, “Unconditional two-mode squeezing of separated atomic ensembles,” Phys. Rev. Lett., vol. 96, p. 053602, 2006. https://doi.org/10.1103/physrevlett.96.053602.Search in Google Scholar PubMed
 S.-B. Zheng, “Generation of atomic and field squeezing by adiabatic passage and symmetry breaking,” Phys. Rev. A, vol. 86, p. 013828, 2012. https://doi.org/10.1103/physreva.86.013828.Search in Google Scholar
 E. G. Dalla Torre, J. Otterbach, E. Demler, V. Vuletic, and M. D. Lukin, “Dissipative preparation of spin squeezed atomic ensembles in a steady state,” Phys. Rev. Lett., vol. 110, p. 120402, 2013. https://doi.org/10.1103/physrevlett.110.120402.Search in Google Scholar
 S.-L. Ma, P.-B. Li, A.-P. Fang, S.-Y. Gao, and F.-L. Li, “Dissipation-assisted generation of steady-state single-mode squeezing of collective excitations in a solid-state spin ensemble,” Phys. Rev. A, vol. 88, p. 013837, 2013. https://doi.org/10.1103/physreva.88.013837.Search in Google Scholar
 J. Borregaard, E. J. Davis, G. S. Bentsen, M. H. Schleier-Smith, and A. S. Sørensen, “One-and two-axis squeezing of atomic ensembles in optical cavities,” New J. Phys., vol. 19, p. 093021, 2017. https://doi.org/10.1088/1367-2630/aa8438.Search in Google Scholar
 G. Liu, Y.-N. Wang, L.-F. Yan, N.-Q. Jiang, W. Xiong, and M.-F. Wang, “Spin squeezing via one-and two-axis twisting induced by a single off-resonance stimulated Raman scattering in a cavity,” Phys. Rev. A, vol. 99, p. 043840, 2019. https://doi.org/10.1103/physreva.99.043840.Search in Google Scholar
 K. Jensen, W. Wasilewski, H. Krauter, et al., “Quantum memory for entangled continuous-variable states,” Nat. Phys., vol. 7, p. 13, 2011. https://doi.org/10.1038/nphys1819.Search in Google Scholar
 Z. Yan, L. Wu, X. Jia, et al., “Establishing and storing of deterministic quantum entanglement among three distant atomic ensembles,” Nat. Commun., vol. 8, p. 718, 2017. https://doi.org/10.1038/s41467-017-00809-9.Search in Google Scholar
 G. Milburn and D. F. Walls, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun., vol. 39, pp. 401–404, 1981. https://doi.org/10.1016/0030-4018(81)90232-7.Search in Google Scholar
 R. W. Boyd, Nonlinear Optics, North Holland, Elsevier, 2003.Search in Google Scholar
 J. Gelhausen, M. Buchhold, and P. Strack, “Many-body quantum optics with decaying atomic spin states: (γ, κ) Dicke model,” Phys. Rev. A, vol. 95, p. 063824, 2017. https://doi.org/10.1103/physreva.95.063824.Search in Google Scholar
 N. Shammah, S. Ahmed, N. Lambert, S. De Liberato, and F. Nori, “Open quantum systems with local and collective incoherent processes: efficient numerical simulations using permutational invariance,” Phys. Rev. A, vol. 98, p. 063815, 2018. https://doi.org/10.1103/physreva.98.063815.Search in Google Scholar
 V. Macrì, F. Nori, S. Savasta, and D. Zueco, “Spin squeezing by one-photon–two-atom excitation processes in atomic ensembles,” Phys. Rev. A, vol. 101, p. 053818, 2020. https://doi.org/10.1103/physreva.101.053818.Search in Google Scholar
 O. Gamel and D. F. V. James, “Time-averaged quantum dynamics and the validity of the effective Hamiltonian model,” Phys. Rev. A, vol. 82, p. 052106, 2010. https://doi.org/10.1103/physreva.82.052106.Search in Google Scholar
 X. Wang, H.-R. Li, P.-B. Li, C.-W. Jiang, H. Gao, and F.-L. Li, “Preparing ground states and squeezed states of nanomechanical cantilevers by fast dissipation,” Phys. Rev. A, vol. 90, p. 013838, 2014. https://doi.org/10.1103/physreva.90.013838.Search in Google Scholar
 P. L. Stanwix, L. M. Pham, J. R. Maze, et al., “Coherence of nitrogen-vacancy electronic spin ensembles in diamond,” Phys. Rev. B, vol. 82, no. R, p. 201201, 2010. https://doi.org/10.1103/physrevb.82.201201.Search in Google Scholar
 N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nat. Commun., vol. 4, p. 1743, 2013. https://doi.org/10.1038/ncomms2771.Search in Google Scholar PubMed
 Z.-L. Xiang, X.-Y. Lü, T.-F. Li, J. Q. You, and F. Nori, “Hybrid quantum circuit consisting of a superconducting flux qubit coupled to a spin ensemble and a transmission-line resonator,” Phys. Rev. B, vol. 87, p. 144516, 2013. https://doi.org/10.1103/physrevb.87.144516.Search in Google Scholar
 Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys., vol. 85, pp. 623–653, 2013. https://doi.org/10.1103/revmodphys.85.623.Search in Google Scholar
 P.-B. Li, Z.-L. Xiang, P. Rabl, and F. Nori, “Hybrid quantum device with nitrogen-vacancy centers in diamond coupled to carbon nanotubes,” Phys. Rev. Lett., vol. 117, p. 015502, 2016. https://doi.org/10.1103/physrevlett.117.015502.Search in Google Scholar PubMed
 Y. Kubo, F. R. Ong, P. Bertet, et al., “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett., vol. 105, p. 140502, 2010. https://doi.org/10.1103/physrevlett.105.140502.Search in Google Scholar PubMed
 R. Amsüss, C. Koller, T. Nöbauer, et al., “Cavity QED with magnetically coupled collective spin states,” Phys. Rev. Lett., vol. 107, p. 060502, 2011. https://doi.org/10.1103/physrevlett.107.060502.Search in Google Scholar
 Y. Kubo, C. Grezes, A. Dewes, et al., “Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble,” Phys. Rev. Lett., vol. 107, p. 220501, 2011. https://doi.org/10.1103/physrevlett.107.220501.Search in Google Scholar PubMed
 Y. Kubo, I. Diniz, A. Dewes, et al., “Storage and retrieval of a microwave field in a spin ensemble,” Phys. Rev. A, vol. 85, p. 012333, 2012. https://doi.org/10.1103/physreva.85.012333.Search in Google Scholar
 S. Putz, D. O. Krimer, R. Amsuess, et al., “Protecting a spin ensemble against decoherence in the strong-coupling regime of cavity QED,” Nat. Phys., vol. 10, p. 720, 2014. https://doi.org/10.1038/nphys3050.Search in Google Scholar
 C. Grezes, B. Julsgaard, Y. Kubo, et al., “Multimode storage and retrieval of microwave fields in a spin ensemble,” Phys. Rev. X, vol. 4, p. 021049, 2014. https://doi.org/10.1103/physrevx.4.021049.Search in Google Scholar
 T. Astner, S. Nevlacsil, N. Peterschofsky, et al., “Coherent coupling of remote spin ensembles via a cavity Bus,” Phys. Rev. Lett., vol. 118, p. 140502, 2017. https://doi.org/10.1103/physrevlett.118.140502.Search in Google Scholar PubMed
 A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, “Ultrastrong coupling between light and matter,” Nat. Rev. Phys., vol. 1, pp. 19–40, 2019. https://doi.org/10.1038/s42254-019-0046-2.Search in Google Scholar
 P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, “Ultrastrong coupling regimes of light-matter interaction,” Rev. Mod. Phys., vol. 91, p. 025005, 2019. https://doi.org/10.1103/revmodphys.91.025005.Search in Google Scholar
 P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, “Introduction to the Dicke model: from equilibrium to nonequilibrium, and vice versa,” Adv. Quantum Technol., vol. 2, p. 1800043, 2019. https://doi.org/10.1002/qute.201800043.Search in Google Scholar
 X.-Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett., vol. 114, p. 093602, 2015. https://doi.org/10.1103/physrevlett.114.093602.Search in Google Scholar
 M. Wallquist, V. S. Shumeiko, and G. Wendin, “Selective coupling of superconducting charge qubits mediated by a tunable stripline cavity,” Phys. Rev. B, vol. 74, p. 224506, 2006. https://doi.org/10.1103/physrevb.74.224506.Search in Google Scholar
 J. R. Johansson, G. Johansson, C. M. Wilson, and F. Nori, “Dynamical Casimir effect in superconducting microwave circuits,” Phys. Rev. A, vol. 82, p. 052509, 2010. https://doi.org/10.1103/physreva.82.052509.Search in Google Scholar
 W. Wustmann and V. Shumeiko, “Parametric resonance in tunable superconducting cavities,” Phys. Rev. B, vol. 87, p. 184501, 2013. https://doi.org/10.1103/physrevb.87.184501.Search in Google Scholar
 K. Y. Bliokh and F. Nori, “Klein–Gordon representation of acoustic waves and topological origin of surface acoustic modes,” Phys. Rev. Lett., vol. 123, p. 054301, 2019. https://doi.org/10.1103/physrevlett.123.054301.Search in Google Scholar PubMed
 D. I. Schuster, A. P. Sears, E. Ginossar, et al., “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett., vol. 105, p. 140501, 2010. https://doi.org/10.1103/physrevlett.105.140501.Search in Google Scholar
 S. Probst, H. Rotzinger, S. Wünsch, et al., “Anisotropic rare-earth spin ensemble strongly coupled to a superconducting resonator,” Phys. Rev. Lett., vol. 110, p. 157001, 2013. https://doi.org/10.1103/physrevlett.110.157001.Search in Google Scholar PubMed
 I. Wisby, S. E. de Graaf, R. Gwilliam, et al., “Coupling of a locally implanted rare-earth ion ensemble to a superconducting micro-resonator,” Appl. Phys. Lett., vol. 105, p. 102601, 2014. https://doi.org/10.1063/1.4894455.Search in Google Scholar
 V. Ranjan, G. de Lange, R. Schutjens, et al., “Probing dynamics of an electron-spin ensemble via a superconducting resonator,” Phys. Rev. Lett., vol. 110, p. 067004, 2013. https://doi.org/10.1103/physrevlett.110.067004.Search in Google Scholar PubMed
 H. Hattermann, D. Bothner, L. Y. Ley, et al., “Coupling ultracold atoms to a superconducting coplanar waveguide resonator,” Nat. Commun., vol. 8, pp. 1–7, 2017. https://doi.org/10.1038/s41467-017-02439-7.Search in Google Scholar PubMed PubMed Central
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