Optical nonreciprocal response and conversion in a Tavis-Cummings coupling optomechanical system

In recent years, cavity optomechanics Aspelmeyer2014 ; Physics.2.40 ; Science.321.1172 (COM) has attracted a lot of attention due to its theoretical and experimental rapid development, which has produced many interesting phenomena, e.g., the detection of gravitational waves 2018AAPPS , mechanical squeezing Bai2019a ; 2008APL92133102 ; Han2019 ; 2008NP4785 ; Bai2019 ; 2016SR638559 , quantum entanglement PRL10 ; PRA16 , mechanical cooling Liu2018 ; PRL21 ; Wang2018a , nonclassical correlations between single photon and phonon N22 , photon blockade Wang2019 , and coherent wavelength conversion S23 ; N24 . The common optomechanical systems study the two-body interaction between the cavity field and mechanical resonator. A novel interaction has been studied lately, which is the three-body interaction among the cavity field, mechanical resonator, and atom. Such hybrid systems are called Tavis-Cummings coupling optomechanical system, which has attracted wide attention. In the hybrid system, the cavity field interacts with mechanical resonator and few-level system simultaneously, which can lead to quantum interference effects and amazing optomechanical phenomena, for instance, EIT and EIA 2019OE277344 ; 2020OE28580 ; 2015OE2311508 , sensing 2019PR71440 ; 2019QE1e10 , quantum repeater 2012PRA85062311 , interaction with an atomic ensemble may be used to produce entanglement, optomechanical cooling, backaction evading measurements of mechanical motion 2007IJTP462550 ; Dantan2014 ; Q73 ; 2007PRA75022312 .

On the other hand, COM-based optical nonreciprocal phenomena have also attracted significant interest in the decades, which means that the transmission of signals in two opposite directions exhibits different characteristics. Optical nonreciprocity has been realized in various optomechanical structures SR43 ; OE44 ; F64 , such as magneto-optical crystals APL45 ; PRL46 ; Li2016 , optical nonlinear systems OL50 ; S51 ; QEe112019 , spatial-symmetry-breaking structures APL52 ; OE53 ; OE54 , parity-time-symmetric structures PRA55 ; Guo2014 ; 2014NP105394 . Nonreciprocal devices based on nonreciprocity, e.g., isolator, directional amplifier, circulator, etc, play significantly important roles in quantum information processing and communication. Specifically, optical isolator is a device that blocks light in one direction but allows light to pass in the opposite direction, which has already been studied in both theory 2017AO562991 and experiment OE32 ; NP27 . The fabrication of low-loss and high Q optical microcavities has achieved great progress using chemo-mechanical polish lithography 2019QE1e9 . The directional amplification has also been studied in a double-cavity optomechanical system with mechanical gain PRA33 , a superconducting microwave circuit with parametric pumps PRX34 ; PRA35 , a triple-cavity optomechanical system with optical gain PRA36 , and a three-mode optomechanical system with an additional mechanical drive OE37 . It has also been realized in an optomechanical circuit via synthetic magnetism and reservoir engineering in experiment NP38 . Circulator has been investigated in three-mode optomechanical systems PRA39 and this technique can be applied to a circuit-QED architecture PRA40 ; Qi2018 and phonon devices NJP41 .

As we all know, the nonreciprocal effect relies strongly on the relative phase between two different paths, which can be usually reduced to the phase at a single path in most theoretical researches. When the phase satisfies the condition of destructive interference, the nonreciprocal phenomenon occurs, which is similar to the optomechanically induced transparency Liu2015 ; AP2018 ; Zheng2019 . The optical nonreciprocal effect can be achieved by the momentum difference between the forward and backward moving light beams in an optomechanical system PRL58 . Li $et~{}al$. proposed a multimode system consisting of two vibrating membranes suspended inside two cavities, which can produce the tunable optical nonreciprocity and be used as a phonon-photon router PRA59 . Hafezi $et~{}al$. pointed out that the nonreciprocal transmission can be achieved via using an unidirectional optical pump in a microring resonator OE60 . We also note that the optical nonreciprocity is studied in coupled spinning optomechanical resonators OE63 . Furthermore, nonreciprocal response has been realized in experiments until now, for instance, silicon chip PRL72 , strong dispersive coupling of a high-finesse cavity to a micromechanical membrane N65 , a dispersive optomechanical system consisting of a membrane inside a cavity NJP66 , and nonlinear coupling in a low-loss system NP67 . Meanwhile, some other studies related to nonreciprocity have been reported such as fast-slow light effects OE61 and nonreciprocity photon blockade PRL71 . On the other hand, the hybrid phonon-photon systems have been studied extensively for the interesting nature due to their potential applications in information processing network 2014PRA90023817 ; Xing2018 ; 2019PRA99013804 . It has been shown that the conversion between phonons and photons can be realized and the optomechanical circulator can be engineered in the optomechanical systems NJP1 .

In this paper, we study the nonreciprocal response and conversion effects in a Tavis-Cummings coupling optomechanical system, which is composed of a trapped flexible membrane embedded an ensemble of two-level quantum emitters in an optical cavity. Resorting to the general linearization technique and the Fourier transform, we calculate the transmission matrix analytically and study the nonreciprocal response of the fluctuation signal in the frequency domain. In the present proposal, due to the introduction of the Tavis-Cummings coupling, we find that the phases of the two different paths are correlated each other and further we derive their relation analytically, which is greatly different from the previous studies. By selecting the system parameters appropriately, the prefect nonreciprocal response can be achieved. We also show that the nonreciprocity induces the feasibility to perform signal conversion among the optical mode, mechanical mode, and dopant mode, which implies that the system can be applied as a phonon-photon transducer and an optomechanical circulator. These interesting phenomena indicate that the Tavis-Cummings coupling model has potential applications in experiments, such as integrating the nonreciprocal devices.

The organization of the paper is as follows. In Sec. II, we present the Hamiltonian and dynamical equations of the Tavis-Cummings coupling optomechanical system. In Sec. III, we discuss the effect of the system parameters on nonreciprocal response in detail and show that the system can be applied as a phonon-photon transducer and an optomechanical circulator. Finally, we draw the conclusion in Sec. IV.

As depicted in Fig. 1(a), we consider a Tavis-cummings coupling optomechanical system, where an optical cavity mode interacts with a single vibrational mode of a flexible membrane with an embedded ensemble of two-level quantum emitters. Generally speaking, the operator of two-level quantum emitters is represented by the Pauli operator, here we represent the Pauli operator as a bosonic operator by a bosonic transformation. Under the conditions of sufficiently large two-level quantum emitters number $N$ and the limit of weak excitation, the lowering operators (raising operators) $\sigma_{-}~{}(\sigma_{+})$ of the ensemble can be transformed to a collective bosonic operator $a~{}(a^{\dagger})$ in the Holstein-Primakoff representation,

where operators $a$ and $a^{{\dagger}}$ obey the bosonic commutation relation [$a,a^{{\dagger}}$] = 1 Dantan2014 ; Wang2016 ; 2015PRA92033841 . The total Hamiltonian of the system is written as ($\hbar$ = 1)

with

where $H_{0}$ describes the free Hamiltonian of the system, $H_{\rm{int}}$ represents the interactions of the cavity field with the mechanical membrane via radiation pressure and with the dopant (the embedded ensemble of two-level quantum emitters) via a mechanically modulated Tavis-Cummings coupling, and $H_{\rm{dr}}$ denotes the interaction between the driving field and the cavity field. Here, $c$ ($c^{{\dagger}}$) is the annihilation (creation) operator of the cavity mode with frequency $\omega_{c}$, $a$ ($a^{{\dagger}}$) is the annihilation (creation) operator of the dopant mode with frequency $\omega_{a}$, and $b$ ($b^{{\dagger}}$) is the annihilation (creation) operator of the mechanical mode with frequency $\omega_{m}$. $g_{0}$ is the optomechanical coupling strength between the cavity mode and the mechanical mode, $J_{0}$ is the coupling strength between the cavity mode and the dopant mode, and $\mu_{0}$ is the Tavis-Cummings coupling strength among the cavity mode, mechanical mode, and dopant mode. $\varepsilon$ and $\omega_{d}$ are the amplitude and frequency of the driving field, respectively. In Fig. 1(b), we give the explicit schematic for describing the interaction of the proposed Tavis-Cummings coupling optomechanical system.

In the rotation frame with $H_{r}=\omega_{d}(c^{{\dagger}}c+a^{{\dagger}}a)$, according to the Heisenberg equations of motion, we can get the quantum Langevin equations (QLEs)

where $\Delta_{i}=\omega_{i}-\omega_{d}$ is the detuning of the respective mode $(i=c,a)$ from the driving frequency. $\kappa$ ($\gamma$) is the decay rate of cavity mode $c$ (dopant mode $a$) and $\gamma_{m}$ is the mechanical damping rate. $c_{\rm{in}}$, $a_{\rm{in}}$, and $b_{\rm{in}}$ are the input quantum noises with zero mean values. To solve the nonlinear QLEs in Eq. (14), we rewrite the operators as the sum of the mean values and the small quantum fluctuation terms, i.e. $c=\xi+\delta c$, $a=\alpha+\delta a$, and $b=\beta+\delta b$. Subsequently, the classical mean values and quantum fluctuations can be treated separately. The equations about steady mean values are given by

where $\Delta_{c}^{\prime}=\Delta_{c}+g_{0}(\beta^{*}+\beta)$ is the effective detuning including the frequency shift caused by the mechanical interaction. The solution on $\alpha$ is

While the linearized fluctuations equations around the steady state can be obtained

and the linearized effective interactions of the system are clearly depicted in Fig. 1(c). Hereafter, for simplicity, we define $J=J_{0}+\mu_{0}(\beta^{*}+\beta)$, $g=g_{0}\xi,~{}\mu=\mu_{0}\xi$, $G=g-iJ\mu/(i\Delta_{a}+\gamma)$, and specify the operators $\delta c\rightarrow c$, $\delta a\rightarrow a$, $\delta b\rightarrow b$.

For convenience, the linearized QLEs in Eq. (27) can be written in the matrix form

where the vectors $F=[c,a,b,c^{{\dagger}},a^{{\dagger}},b^{{\dagger}}]^{T}$ and $F_{\rm{in}}=[c_{\rm{in}},a_{\rm{in}},b_{\rm{in}},c_{\rm{in}}^{{\dagger}},a_{\rm{in}}^{{\dagger}},b_{\rm{in}}^{{\dagger}}]^{T}$ ($T$ represents the transpose operator), the matrix $\Gamma$= Diag $[\sqrt{2\kappa},\sqrt{2\gamma},\sqrt{2\gamma_{m}},\sqrt{2\kappa},\sqrt{2\gamma},\sqrt{2\gamma_{m}},]$, and the coefficient matrix

The system is stable only if the real parts of all the eigenvalues of matrix $M$ are negative. The stability condition can be derived by applying the Routh-Hurwitz criterion PRA74 , however, whose concrete form is too cumbersome to give here. In the following, all the parameters in the present work have been chosen to make sure the stability of the system.

Generally, it is more convenient to reveal the nonreciprocity in the frequency domain. To this end, we perform the Fourier transformations of the operators

and then Eq. (34) in the frequency domain becomes

where $I$ represents the identity matrix. By substituting Eq. (39) into the standard input-output relation $o_{\rm{in}}+o_{\rm{out}}=\sqrt{\kappa_{o}}o$, where $o_{\rm{in}}$, $o_{\rm{out}}$, and $\kappa_{o}$ denote input operators, output operators, and their corresponding damping rates, respectively. Then the output field vector in the frequency domain is given by

where the output field vector $F_{\rm{out}}(\omega)=[c_{\rm{out}},a_{\rm{out}},b_{\rm{out}},c_{\rm{out}}^{{\dagger}},a_{\rm{out}}^{{\dagger}},b_{\rm{out}}^{{\dagger}}]^{T}$ and the coefficient matrix $U(\omega)$ is given by

The spectrum of the output field is defined by

Accordingly, the vector of the output field $S_{\rm{out}}(\omega)$ is written in the form of $S_{\rm{out}}(\omega)=[s_{c}^{\rm{out}}(\omega),s_{a}^{\rm{out}}(\omega),s_{b}^{\rm{out}}(\omega)]^{T}$. Substituting Eq. (40) into Eq. (42) and making use of correlation functions $\langle o_{\rm{in}}^{{\dagger}}(\omega^{\prime})o_{\rm{in}}(\omega)\rangle=s_{o}^{\rm{in}}(\omega)\delta(\omega+\omega^{\prime})$ and $\langle o_{\rm{in}}(\omega^{\prime})o_{\rm{in}}^{{\dagger}}(\omega)\rangle=[s_{o}^{\rm{in}}(\omega)+1]\delta(\omega+\omega^{\prime})$, we can obtain

where the vector of the input field $S_{\rm{in}}(\omega)=[s_{c}^{\rm{in}}(\omega),s_{a}^{\rm{in}}(\omega),s_{b}^{\rm{in}}(\omega)]^{T}$ and the transmission matrix

The element of $T(\omega)$ denotes as $T_{ij}(\omega)$ $(i,j=c,a,b)$, which indicates the scattering probability from the input mode $j$ to the output mode $i$ and their respective forms are

where $U_{vw}(\omega)$ ($v,w=1,...,6$) is the element in the $v$-th row and the $w$-th column of the matrix $U(\omega)$ in Eq. (41). And $S_{\rm{v}}(\omega)=[s_{c}^{\rm{v}}(\omega),s_{a}^{\rm{v}}(\omega),s_{b}^{\rm{v}}(\omega)]^{T}$ in Eq. (43) is the output spectrum caused by the input vacuum field and its elements are

In conclusion, we have studied the nonreciprocal response effect and signal conversion in a Tavis-Cummings coupling optomechanical system. Using the general linearization technique and the Fourier transform, we analytically calculate the transmission matrix of the quantum input signal in the frequency domain. Different from the previous schemes, we find that, due to the introduction of the Tavis-Cummings interaction, both the effective optomechanical coupling strength and the interaction strength between the dopant mode and the mechanical mode are complex and their phases are correlated each other. To obtain the optimal nonreciprocal response, we analytically get the relation of the two phases and derive the relative phase between the two different paths. By selecting the suitable system parameters, especially the relative phase, the optimal nonreciprocal response can be effectively achieved. Furthermore, based on this interesting property, we further discuss the nonreciprocal conversion phenomena among the optical mode, mechanical mode, and dopant mode, which can be used as a phonon-photon transducer and an optomechanical circulator. We hope that the proposed proposal would have potential applications in quantum information processing network.

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61822114, 61465013, and 11465020.