Menghan Chen,Yue Chang and Tao Shi

1 CAS Key Laboratory of Theoretical Physics,Institute of Theoretical Physics,Chinese Academy of Sciences,Beijing 100190,China

2 School of Physical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

3 Beijing Automation Control Equipment Institute,Beijing 100074,China

4 CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences,Beijing 100049,China

Abstract We study the correlation statistics of phonon radiations in a weakly driven optomechanical system.Three dominated scattering processes are identified by the scattering theory analytically and the master equation numerically,whose interplay determines the phonon statistical properties.Our results show that for the large detuning,the driving field off-resonant with the system induces a small emission rate of two anti-bunched phonons.For the resonant driving field,there is a relatively large emission rate of two bunched phonons.

Keywords:quantum optics,optomechanics,quantum information

1.Introduction

Optomechanical systems(OMS)have raised a rapidly growing interest,since it not only produces intriguing phenomena such as mode conversion[1–3],optomechanically induced transparency[4,5]and optomechanical entanglement[6–9],but have extensive applications ranging from optical feedback cooling[10–12],noise squeezing of the light beam[13–15]to quantum information processing[16–18].Recently,the unprecedented experimental progress like mechanical squeezing control[19],single-phonon level quantum control[20–23]and fabrication of entanglement between light and mechanics[7]has greatly promoted the development in OMS,such as the increasing coupling strengths between light and mechanical motion in optomechanical setups,which have improved by orders of magnitude[24–26].There are also several theoretical studies on OMS in strong-coupling regimes,for example,the optical nonlinearities like photon blockade[27–30]and nonlinear dynamics[31]in OMS have been studied,and another study found that the steady state mechanical Wigner density contains strong negative parts at strong optomechanical coupling,signaling stable nonclassical states[32].

One of the most fundamental quantum properties of emitting light is the statistics,which can be characterized by correlation functions[33,34]measured via a Hanbury–Brown–Twiss experiment[35].Similarly,the emitting phonons can also exhibit some quantum statistical properties,e.g.phonon anti-bunching behaviors and phonon blockade effects[36,37],which can also be described by the phonon correlation functions.The statistics of photons in OMS have been extensively studied[27,38,39],however,to our best knowledge,there have been few studies on that of phonons.For example,the phonon sub-Poissonian statistics and phonon blockade effects in quadratically coupled OMS have been studied by solving the master equation numerically[40–43]or analytically via the phase space method[44].Although the phonon statistical properties are explored,a comprehensive understanding of different scattering processes is required to reveal the mechanism of phonon(anti-)bunching behaviors.

Figure 1.Schematic of OMS in which the optical cavity contains a photon mode,whose creation(annihilation)operator is a?(a),coupling to a phonon mode,whose creation(annihilation)operator is b?(b).The two cavity modes decay into the reservoirs respectively.

In this paper,we study the quantum statistics of phonon radiations in linearly coupled OMS at low temperatures,with the aim to establish the relation of the phonon(anti-)bunching behavior on relevant system parameters.In addition to providing the numerical results based on the standard master equation approach,we also calculate the second-order correlation functions analytically using the scattering formalism in the weak driving and coupling limit.With the analytical result,we can identify three dominant scattering processes,from which an unambiguous mechanism of the phonon pair generation naturally arises.Both the numerical and analytical results show that the anti-bunching behavior of two phonons requires a large detuning between the frequencies of the incident laser and the cavity.

The paper is organized as follows.In section 2,we illustrate OMS with the Hamiltonian,and derive the formal expression of correlation functions.In section 3,we evaluate correlation functions numerically using the master equation,which agree with that from perturbative expansion with high accuracy.Inspired by the correspondence between the scattering theory and the master equation,in section 4,we also use the field theoretical approach to obtain the correlation functions analytically,which allows us to find out the condition of phonon(anti-)bunching behaviors and provide an intuitive picture of phonon pair generations.Finally,the results are summarized with an outlook in section 5.

2.Model

3.Quantum statistical character

Figure 2.The numerical results of the normalized correlation function The parameters are ωb=1,Ω=10-7 and(a)g=10-5,γa=0.13,γb=10-5;(b)g=10-5,γa=0.13,γb=0.1;(c)g=10-5,γa=1,γb=10-5;(d)g=0.1,γa=0.13,γb=10-5.

The approximate result in the weak driving intensity limit can be obtained analytically via the Dyson expansion

describe three different processes respectively:(a)the photon is emitted before the emissions of two phonons,(b)the photon is emitted between the emissions of two phonons and(c)the photon is emitted after the emissions of two phonons.The analytical results of equations(9)–(11)are discussed in the next section via the scattering theory.

4.Analytical results by scattering theory

4.1.Field theoretical approach

Figure 3.The relative differences between the results in figure 2 and those based on the scattering theory.The parameters are the same as those in figure 2.

4.2.Results

The intuitive picture of the quantum statistical characters is shown in the following.G1(x)displays the bunching behavior,since it represents the two-phonon cascade decays after the emission of one photon; G2(x)displays the anti-bunching behavior since it represents the emissions of the two phonons are separated by the emission of the single photon.G3(x)displays the bunching behavior for small |Δ| and the antibunching behavior for large |Δ|.The intriguing behavior of G3(x)can be interpreted as follows.In the process represented by G3(x),two phonons are emitted before the emission of the photon.As result,the presence of the photon in the cavity induces the destructive interference of two phonons at x=0 for large |Δ|.

Figure 4.Representative transitions among the energy levels of the system corresponding to the three scattering processes,where the subfigures(a)–(c)correspond to processes(a)–(c)respectively.We have omitted the higher-order transitions of g due to the weak coupling condition.

Figure 5.The relative proportion of to their sum.The proportion of process(a)–(c)are given byrespectively.The parameters are ωb=1,g=10-5,γa=0.13,γb=10-5.

We note that the larger |Δ| is,the larger proportion of processes(b)and(c)occupy among(a)–(c).To illustrate it,and show the ratios ri=gi/∑jgjin figure 5,where r1reaches its peak while r2and r3reach their minima at Δ ≈ωb.

Because of the large proportion of the process(b)in the large|Δ|limit,the two emitting phonons show anti-bunching behavior,and the large |Δ| also indicates an inresonant scattering of the incident photon with a small emission rate(lower than 10-6)of two phonons.

For small |Δ|,the large proportion of the process(a)indicates the bunching behavior of two emitting phonons.Besides,due to the small detuning,the driving field resonates with the system,making the emission rate of two phonons relatively larger but still not high(lower than 10-3).

If the weak coupling condition is released,e.g.|g/(ω-iγb/2)|～1,it turns out that the emissivity of two bunched phonons can be enhanced by one to two orders of magnitudes,i.e.10-2–10-1,while the emission rate of two anti-bunched phonons can reach 10-4.we define

5.Conclusion and outlook

In summary,we have identified the quantum statistical behavior of two phonons in the weak coupled OMS under a weak driving field.We calculate the second-order correlation functions by using the standard master equation approach and the scattering formalism.We obtain the analytic results for the correlation functions by identifying three dominant physical processes.Our results show that the small detuning induces the bunching behavior of phonons with a relatively large emission rate.The anti-bunching behavior of phonons requires large detuning,which leads to a small two-phonon emissivity.To make the manifest anti-bunching phenomenon two feasible schemes,increasing the coupling strength and the driving intensity are proposed,which require more studies on multiphonon and multiphoton scatterings in future work.

Acknowledgments

We are grateful to Junqiao Pan,Yuqi Wang and Yaoqi Tian for useful discussions.T.Shi acknowledges the support from NSFC Grant No.11 974 363.

Appendix A.Mapping of operators to superspace

Consider a formula in the form of AρB where A,B are operators and ρ is a density matrix.Make the following mappings

and

So the master equation(5)can be mapped to

where He=Hc+Hd-iγaa?a/2-iγbb?b/2.Equation(A3)enables us to evaluate the master equation numerically.

Appendix B.Numerical results with nonexperimental parameters

We have shown the numerical results of the normalized correlation functions under experimental parameters in figure 2,and here we give some other results under nonexperimental parameters in figure B1 in order to show the validity of our conclusion in a wider range of parameters.In figure B1 we can see that the phonons also show an antibunching behavior as |Δ| increases to greater than other parameters.

Figure B1.Some other numerical results of the normalized correlation functionwith nonexperimental parameters.The parameters are ωb=1,Ω=10-7 and(a)g=0.01,γa=0.02,γb=0.002;(b)g=0.3,γa=20,γb=5;(c)g=0.05,γa=1,γb=0.05;(d)g=0.3,γa=2,γb=5.

Appendix C.Scattering amplitude by quantum regression theorem

We introduce another method by using the quantum regression theorem to calculate the scattering amplitude equation(13).Let’s start with equation(17),which can be written in another form

is the driving term.Substituting equations(14)and(C1)into(16)and coming back from the rotating frame with respect to Hr,we can write the scattering amplitude as