Quantum correlations of quadratic optomechanical oscillator

2390

J. Opt. Soc. Am. B / Vol. 31, No. 10 / October 2014

S. K. Singh and C. H. Raymond Ooi

Quantum correlations of quadratic optomechanical oscillator S. K. Singh and C. H. Raymond Ooi* Department of Physics, University of Malaya, 50603 Kuala Lumpur, Malaysia *Corresponding author: [email protected] Received June 4, 2014; revised July 31, 2014; accepted August 5, 2014; posted August 18, 2014 (Doc. ID 213449); published September 19, 2014 The quantum optomechanical system serves as an interface for coupling between photons, excitons, and mechanical oscillations. We use the quantum Langevin approach to study a hybrid optomechanical system that contains a single undoped semiconductor quantum well in a cavity, where one of its mirrors is a thin dielectric membrane having quadratic response to the cavity fields. A decorrelation method is employed to solve for a large number of coupled equations. Transient mean numbers of cavity photons, moving membranes, and excitons that provide dynamical behavior are computed. We obtain the two-boson second-order correlation functions for the cavity field and the membrane oscillator, and their cross correlations that provide nonclassical quantum statistical properties and useful insights into the quadratic optomechanical system. © 2014 Optical Society of America OCIS codes: (200.4880) Optomechanics; (270.0270) Quantum optics; (120.4880) Optomechanics. http://dx.doi.org/10.1364/JOSAB.31.002390

1. INTRODUCTION Recent experiments in cavity quantum electrodynamics (cQED) have explored the interaction of light with atoms as well as semiconductor nanostructures inside a cavity [1]. A single atom-cavity system described by the well-known Jaynes–Cumming model [2] has been a basic theoretical foundation for research toward quantum computation. Recent quantum optomechanical works have been able to couple cavity photons to solid-state mechanical systems containing a large number of atoms [3,4]. In these systems, there is an optical cavity, with a movable mirror in one end [5,6] or a micromechanical membrane with mechanical effects caused by light through radiation pressure [7]. So, cavity quantum optomechanics has emerged as a very interesting area for the study of quantum features at the mesoscale, where it is possible to control the quantum state of mechanical oscillators by their coupling to the light field [8]. Recent advances in this area include the realization of quantum-coherent coupling of a mechanical oscillator with an optical cavity [9], where the coupling rate exceeds both the cavity and mechanical motion decoherence rate and laser cooling of a nanomechanical oscillator to its ground state [10]. Furthermore, new experimental works open up enormous possibilities in the design of hybrid quantum systems whose elementary building blocks are physically implemented by systems of different nature. In this context, a superconducting flux qubit coherently coupled to a spin ensemble has been reported in Ref. [11], while interaction between ultracold atoms and mechanical systems has been studied in Ref. [12], and strong coupling between a Bose–Einstein condensate (BEC) and an optical cavity was experimentally reported in Refs. [13,14], where the mechanical system is embodied by quantum photonic waves of the BEC. Recently, the photon blockade effect has been reported in quadratically coupled optomechanical systems [15]. 0740-3224/14/102390-09$15.00/0

Here, we study a hybrid system that contains a single undoped semiconductor quantum well (QW) placed inside a cavity with one of its mirrors being a thin dielectric membrane with quadratic response to the cavity fields. Coupled Heisenberg–Langevin equations are obtained, and solutions up to second-order correlation operators have been developed in Section 2. Here, we also present the theory for solving the transient dynamics of the system. Section 3 discusses the results with nonclassical photon statistics of the cavity mode as well as the mechanical oscillation mode. We conclude our results in Section 4.

2. QUANTUM LANGEVIN FOR OPTOMECHANICS We consider an optomechanical resonator formed by a micropillar with moveable Bragg reflectors, recently realized experimentally and coupled to an undoped QW cˆ placed at the antinode of the resonator. We also have in this setup a thin dielectric membrane at the node of the resonator. The membrane experiences mechanical displacement quadratically ˆ 2 when coupled to the cavity photon number haˆ † ai. ˆ bˆ † b Considering a sample with comparably high mirror quality factor, strong coupling between cavity photons and QW excitons is possible. In addition, we have a monochromatic laser field with frequency ωL applied to drive the cavity. The corresponding Hamiltonian for the scheme is given by ℏ 1, ˆ H ˆ0H ˆ R; H

(1)

ˆ 0 and H ˆ R are defined by Eqs. (2) and (3) below. In our where H ˆ 0 contains the free field terms including the Jaynes– model, H Cumming, the optomechanical interaction terms, as well as ˆ R includes interactions of the driven cavity interaction. H ˆ phonons bˆ (due to mechanical the cavity photons a, motion of the membrane), and QW excitons cˆ with their © 2014 Optical Society of America


corresponding reservoirs composed of collections of harmonic oscillators [16]:

Vol. 31, No. 10 / October 2014 / J. Opt. Soc. Am. B

2391

X d ˆ ˆ bˆ − bˆ † − i gk nˆ k ; b −iωM bˆ − igopt aˆ † a2 dt k

(7)

d ˆ nˆ −iωk nˆ k − igk b; dt k

(8)

X d cˆ −iΔexciton cˆ − igaˆ − i gk fˆ k ; dt k

(9)

d ˆ f −iωk fˆ k − igk cˆ : dt k

(10)

ˆ 0 ωc aˆ † aˆ ωexciton cˆ † cˆ ωM bˆ † bˆ gaˆ † cˆ cˆ † a ˆ H ˆ 2 Ωe−iωL t aˆ † eiωL t a; ˆ bˆ † b ˆ gopt aˆ † a ˆR H

(2)

X X ˆ †k aˆ aˆ † m ˆ †k m ˆk ˆ k ωk m gk m k k | {z } Cavity mode

X X ωk0 nˆ †k0 nˆ k0 gk0 nˆ †k0 bˆ bˆ † nˆ k0 0

0

k k |{z} Moving membrane

X X ωk00 fˆ †k00 fˆ k00 gk00 fˆ †k00 cˆ cˆ † fˆ k00 ; k00

(3)

k00

|{z} Exciton

ˆ and cˆ correspond to field operators for cavity phoˆ b, where a, tons, phonons (due to mechanical motion of the membrane), and QW excitons, with frequencies ωc , ωM , and ωexciton , respectively. The fourth term in Eq. (2) describes the exciton–photon coupling with strength g. The fifth term in Eq. (2) describes the quadratic optomechanical coupling with strength gopt between the cavity field and the mechanical motion of the membrane. The last term in Eq. (2) describes the driving process of the cavity field where Ω is the Rabi frequency of the driving field. ˆ R given by Eq. (3), the first two terms In the expression for H represents the damping of the cavity mode through the radiˆ k m ˆ †k . The ation reservoir as harmonic oscillator operators m third and fourth terms are responsible for damping of the membrane through the harmonic oscillators nˆ k and nˆ †k . The fifth and sixth terms give rise to excitonic decay through harmonic oscillators fˆ k fˆ †k . In the frame rotating with the ˆ 0 reduces to the driving frequency ωL , the Hamiltonian H following: ˆ 0 Δc aˆ † aˆ Δexciton cˆ † cˆ ωM bˆ † bˆ gaˆ † cˆ cˆ † a ˆ H ˆ 2 Ωaˆ † a; ˆ ˆ bˆ † b gopt aˆ † a

(4)

Now the operators for different reservoirs in Eqs. (5), (7), and (9) can be further removed by replacing the solutions of the reservoir operators from Eqs. (6), (8), and (10). For example, Eq. (6) gives the solution ˆ k 0e−iωk t − igk ˆ k t m m

d ˆ bˆ †2 bˆ 2 2bˆ † bˆ 1 aˆ −iΔc aˆ − igˆc − igopt a dt X ˆ k; − iΩ − i gk m

(5)

k

d ˆ −iωk m ˆ k − igk a; ˆ m dt k

(6)

t 0

0

ˆ 0 e−iωk t−t : dt0 at

(11)

In Eq. (11), the first term represents the free evolution of the reservoir modes, whereas the second term arises from their ˆ k t interaction with the cavity field. The reservoir modes m ˆ k t. We have in Eq. (5) can be eliminated by substituting m d ˆ bˆ †2 bˆ 2 2bˆ † bˆ 1 − iΩ aˆ −iΔc aˆ − igˆc − igopt a dt X Zt 0 ˆ 0 e−iωk t−t Fˆ a t; g2k dt0 at − 0

k

(12)

P ˆ k 0e−iωk t is the noise operator for the where Fˆ a t −i k gk m cavity, which depends on the reservoir variables. Using Markov approximation the term containing the integral is simplified to X k

where Δc ωc − ωL and Δexciton ωexciton − ωL are the corresponding detunings from the driving laser frequency ωL . In our model Hamiltonian, we have considered low density excitons in the QW, and exciton–exciton Coulomb scattering is negligible. Similarly, the density-dependent relaxation effect as well as many-body effects are negligible in our model Hamiltonian. The corresponding Heisenberg equations of motion for the operators of different subsystems (exciton, cavity, membrane) are given by Eqs. (5), (7), and (9), whereas the corresponding equations of motion for reservoir operators are given by Eqs. (6), (8), and (10),

Z

g2k

Z 0

t

1 0 ˆ 0 e−iωk t−t ≃ Γa at; ˆ dt0 at 2

(13)

where Γa is the damping constant. So, Eq. (12) becomes d ˆ bˆ †2 bˆ 2 2bˆ † bˆ 1 aˆ −iΔc aˆ − igˆc − igopt a dt 1 − iΩ − Γa aˆ Fˆ a t: 2

(14)

Similarly, we have equations for the exciton and moving membrane, respectively, as d ˆ 1 ˆ bˆ bˆ † − Γb bˆ Fˆ b t; b −iωM bˆ − 2igopt aˆ † a dt 2 d 1 cˆ −iΔexciton cˆ − igaˆ − Γc cˆ Fˆ c t; dt 2

(15)

(16)

where Fˆ b t and Fˆ c t are the noise operators for the moving membrane and exciton, respectively, given by

2392


X Fˆ b t −i gk nˆ k 0e−iωk t ;


X Fˆ c t −i gk fˆ k 0e−iωk t :

k

k

(17) Similarly, Γb and Γc are damping constants for the moving membrane and exciton, respectively, and are given by X Zt ˆ 0 e−iωk t−t0 ≃ 1 Γb bt; ˆ g2k dt0 bt 2 0 k X Zt 1 0 g2k dt0 cˆ t0 e−iωk t−t ≃ Γc cˆ t: 2 0 k

(18)

d ˆ hai ˆ −iΔc hai ˆ − ighˆci − igopt haˆ bˆ †2 i haˆ bˆ 2 i 2haˆ bˆ † bi ˆ hai dt 1 ˆ (19) − iΩ − Γa hai; 2

d 1 ˆ − Γc hˆci; hˆci −iΔexciton hˆci − ighai dt 2 d † ˆ −ighaˆ † cˆ i − haˆ ˆ c† i − iΩhaˆ † i haˆ ai dt ˆ − Γa haˆ † ai ˆ Γa n¯ ath ; − hai

(20)

(21)

(22)

d ˆ† ˆ ˆ Γb n¯ b ; hb bi −2igopt haˆ † aˆ bˆ †2 i − haˆ † aˆ bˆ 2 i − Γb hbˆ † bi th dt (23)

d † ˆ c† i − haˆ † cˆ i − Γc hˆc† cˆ i Γc n¯ cth ; hˆc cˆ i −ighaˆ dt

(24)

d ˆ† 1 haˆ b i iωM − Δc haˆ bˆ † i − ighbˆ † cˆ i − iΩhbˆ † i − Γa Γb haˆ bˆ † i dt 2 ˆ − haˆ bˆ † bˆ 2 i − haˆ bˆ †3 i igopt 2haˆ † aˆ 2 bi ˆ − haˆ bˆ † i; 2haˆ † aˆ 2 bˆ † i − haˆ bˆ †2 bi

ˆ haˆ † bi; ˆ 2haˆ † bˆ † bˆ 2 i − haˆ †2 aˆ bi

(26)

d ˆ −iΔc ωM haˆ bi ˆ − ighbˆ cˆ i − iΩhbi ˆ − 1 Γa Γb haˆ bi ˆ haˆ bi dt 2 ˆ 2haˆ † aˆ 2 bˆ † i haˆ bˆ 3 i − igopt 2haˆ bˆ † i haˆ bˆ †2 bi

A. Quantum Correlations We have obtained the coupled equations involving the mean of ˆ cˆ , their adjoints, and their odd and even ˆ b, the operators a, products. Since all three thermal reservoirs are big as well as always in thermal equilibrium, we have hFˆ a ti ˆ hFˆ b ti hFˆ c ti 0. We also have hFˆ †a tbti † † ˆ ˆ ˆ ˆ ˆ ˆ hF b tati hF a tb ti hF b tati 0. Similarly, quantum correlations between any subsystem (cavity, exciton, moving mirror) operators with noise operators of different reservoirs vanish altogether. The details are shown in Appendix A. The following set of equations has been obtained by using the Heisenberg–Langevin approach given in [20]:

d ˆ ˆ − 2igopt haˆ † aˆ bi ˆ ˆ haˆ † aˆ bˆ † i − 1 Γb hbi; hbi −iωM hbi dt 2

d †ˆ ˆ − 1 Γa Γb haˆ † bi ˆ ighbˆ ˆ c† i iΩhbi ˆ haˆ bi iΔc − ωM haˆ † bi dt 2 ˆ haˆ † bˆ 3 i − 2haˆ †2 aˆ bˆ † i igopt haˆ † bˆ †2 bi

(25)

ˆ haˆ bˆ † bˆ 2 i haˆ † aˆ 2 bi ˆ haˆ bi; ˆ 2haˆ bi

(27)

d † ˆ† haˆ b i iΔc ωM haˆ † bˆ † i ighbˆ † cˆ † i dt 1 iΩhbˆ † i − Γa Γb haˆ † bˆ † i 2 ˆ haˆ † bˆ †3 i ˆ haˆ † bˆ † bˆ 2 i 2haˆ †2 aˆ bi igopt 2haˆ † bi ˆ haˆ †2 aˆ bˆ † i haˆ † bˆ † i; 2haˆ † bˆ † i haˆ † bˆ †2 bi

(28)

d † ˆ haˆ cˆ i iΔc − Δexciton haˆ † cˆ i ighˆc† cˆ i − haˆ † ai dt 1 iΩhˆci − Γa Γc haˆ † cˆ i igopt haˆ † bˆ †2 cˆ i 2 haˆ † bˆ 2 cˆ i 2haˆ † bˆ † bˆ cˆ i haˆ † cˆ i;

(29)

d ˆ c† i iΔexciton − Δc haˆ ˆ c† i ighaˆ † ai ˆ − hˆc† cˆ i haˆ dt 1 ˆ c† i − iΩhˆc† i − Γa Γc haˆ 2 ˆ c† i haˆ ˆ c† i; − igopt haˆ bˆ †2 cˆ † i haˆ bˆ 2 cˆ † i 2haˆ bˆ † bˆ

(30)

d haˆ cˆ i −iΔexciton Δc haˆ cˆ i − ighaˆ 2 i hˆc2 i dt 1 − iΩhˆci − Γa Γc haˆ cˆ i 2 − igopt haˆ bˆ †2 cˆ i haˆ bˆ 2 cˆ i 2haˆ bˆ † bˆ cˆ i haˆ cˆ i;

(31)

d † † haˆ cˆ i iΔexciton Δc haˆ † cˆ † i ighaˆ †2 i hˆc†2 i dt 1 iΩhˆc† i − Γa Γc haˆ † cˆ † i 2 igopt haˆ † bˆ †2 cˆ † i haˆ † bˆ 2 cˆ † i ˆ c† i haˆ † cˆ † i; 2haˆ † bˆ † bˆ

(32)

d ˆ† 1 hb cˆ i iωM − Δexciton hbˆ † cˆ i − ighaˆ bˆ † i − Γb Γc hbˆ † cˆ i dt 2 2igopt haˆ † aˆ bˆ cˆ i haˆ † aˆ bˆ † cˆ i; (33) d ˆ † ˆ c† i ighaˆ † bi ˆ c† i ˆ − 1 Γb Γc hbˆ hbˆc i iΔexciton − ωM hbˆ dt 2 − 2igopt haˆ † aˆ bˆ † cˆ † i haˆ † aˆ bˆ cˆ † i; (34)



d ˆ ˆ − 1 Γb Γc hbˆ cˆ i hb cˆ i −iΔexciton ωM hbˆ cˆ i − ighaˆ bi dt 2 − 2igopt haˆ † aˆ bˆ † cˆ i haˆ † aˆ bˆ cˆ i; (35) d ˆ† † 1 hb cˆ i iωM Δexciton hbˆ † cˆ † i ighaˆ † bˆ † i − Γb Γc hbˆ † cˆ † i dt 2 † † † † † ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2igopt ha a b c i ha ab c i; (36) d 2 ˆ − Γa haˆ 2 i haˆ i −2iΔc haˆ 2 i − 2ighaˆ cˆ i − 2iΩhai dt ˆ haˆ 2 i; − 2igopt haˆ 2 bˆ †2 i haˆ 2 bˆ 2 i 2haˆ 2 bˆ † bi

(37)

d ˆ2 hb i −2iωM hbˆ 2 i − Γb hbˆ 2 i dt ˆ 2haˆ † aˆ bˆ 2 i; ˆ 2haˆ † aˆ bˆ † bi − 2igopt haˆ † ai

(38)

d 2 hˆc i −2iΔexciton hˆc2 i − 2ighaˆ cˆ i − Γc hˆc2 i; dt n¯ ath ,

n¯ bth ,

† ˆ aˆ † ai ˆ haˆ †2 ihaˆ 2 i haˆ †2 aˆ 2 i 2haˆ aih ; ≈ ˆ 2 ˆ 2 haˆ † ai haˆ † ai

(42)

g2 b

ˆ† ˆ ˆ† ˆ hbˆ †2 bˆ 2 i 2hb bihb bi hbˆ †2 ihbˆ 2 i ; ≈ ˆ 2 ˆ 2 hbˆ † bi hbˆ † bi

(43)

† ˆ ˆ† ˆ aˆ † ai ˆ ˆ hbˆ † bih ˆ haˆ † bˆ † ihbˆ ai ˆ haˆ † bˆ † bˆ ai haˆ bihb ai ≈ : ˆ aˆ † ai ˆ aˆ † ai ˆ ˆ hbˆ † bih hbˆ † bih

2 2 2 If g2 a 0, gb 0, and gab 0 satisfy the inequality gX 0 < 1; X a; b, ab, then the statistics of the bosonic systems are referred to as sub-Poissonian. Statistics with g2 X 0 1 and g2 X 0 > 1 are referred to as Poissonian and superPoissonian, respectively. Here, we have studied the temporal dynamics of second-order correlation functions, namely the self-correlation and cross correlation for cavity and oscillating membrane modes.

(39)

3. RESULTS AND DISCUSSION

and are the thermal mean numbers of where the cavity photons, membrane vibrations, and QW excitons, respectively. B. Coupled Equations and Decorrelation of HigherOrder Operators The above set of equations are not closed, constitute higherorder operator products, and need to be solved numerically with approximations. We proceed to decorrelate all the higher-order correlations in the above equations. The above set of equations will then be closed up to second order when we apply the decorrelation method. We proceed to decorrelate the higher-(third- and fourth-)order quantum correlations present in the above equations just like our previous work [21], which studied the correlation of photon pairs from a double Raman amplifier. This approach corresponds to truncation of higher-order operator products in order to solve for all the second-order correlation functions. A similar kind of approximation has also been used in Ref. [22] to study the dynamics of a two-mode BEC beyond mean field theory: (40)

and hABCDi ≈ hABihCDi hACihBDi hADihBCi:

g2 a

(44)

n¯ cth

hABCi ≈ hAihBCi hABihCi hACihBi

g2 ab

2393

(41)

After applying the decorrelation method, the above set of equations reduced to a closed set of coupled equations given in Appendix B. These equations contain all possible first-order operator averages and second-order correlation functions. The closed set of equations, given in Appendix B, is further solved numerically for the transient dynamics as well as nonclassical photon statistics. C. Decorrelation of Higher-Order Operator Products ˆ and hˆc† cˆ i, we ˆ hbˆ † bi, In addition to the mean numbers haˆ † ai, compute the normalized two-boson correlation functions 2 for the cavity field g2 a and dielectric membrane gb , and 2 the cross correlations between them gab , using the relations

We have studied the temporal evolutions of the mean number ˆ and exciton ˆ moving membrane hbˆ † bi, of cavity photons haˆ † ai, hˆc† cˆ i in the resolved sideband regime where ωM > Γa , Γb , and Γc . Figure 1(a) shows the temporal evolutions of the mean ˆ moving numbers of excitations for the cavity photons haˆ † ai, ˆ and exciton hˆc† cˆ i for several chosen sets of membrane hbˆ † bi, parameters. These numbers’ oscillations are more strongly damped for larger decays Γa , Γb , Γc [Fig. 1(b)]; i.e., all three oscillations show stronger damping with time. Careful inspection reveals that the number of oscillations for the membrane is half the number of oscillations for the photons and excitons. This is due to the quadratic response of the membrane to photonic excitations. For larger detunings [Fig. 1(c)], the amplitudes of the oscillations are reduced, with the presence of suboscillations within an oscillation. When both dampings and detunings are large, both features of damped oscillations and suboscillations are found [Fig. 1(d)]. For the larger detuning as well as decay, all three oscillations are strongly damped with time. ˆ shows The mean number of cavity photon excitations haˆ † ai asymmetric splitting due to hybrid resonances as gopt is not very small as compared to g. In this regime, the phonon density due to mechanical motion is comparable to the photon density, and hence hybrid resonances dominate over just optomechanical resonances. Furthermore, the mechanical ˆ also shows asymmetric shape motion of membrane hbˆ † bi within the oscillations, but without hybrid resonances. As the detuning increases [in Fig. 1(c)], the mean number of ˆ shows more asymmetric splitting cavity excitations haˆ † ai due to hybrid resonances, whereas the shape of oscillations ˆ becomes more regular and symmetric. for hbˆ † bi ˆ We have also studied the temporal dynamics for haˆ † ai, ˆ and hˆc† cˆ i for a finite thermal phonon number as shown hbˆ † bi, in Fig. 2 for small detuning as well as large detuning. The mean ˆ increases with time due to thermal phonon number hbˆ † bi noise (for finite n¯ bth ) even for the strong decay regime of the cavity mode, the exciton mode, as well as the moving membrane mode. So, a finite thermal noise due to phonons increases the mean number of excitations for the dielectric membrane.

2394



ˆ (green dashed line), and exciton hˆc† cˆ i (red dotted line) for ˆ (blue solid line), moving membrane hbˆ † bi Fig. 1. Mean number of cavity photons haˆ † ai chosen set of parameters. (a) Δc ∕ωM 5.0; Δexciton ∕ωM 6.0; g∕ωM 2.0; gopt ∕ωM 0.3, Γa ∕ωM Γc ∕ωM 0.01; Γb ∕ωM 0.001; and Ω∕ωM 0.8, where ωM is the frequency of mechanical motion of the membrane chosen as 1 in our numerical simulations. (b) For larger decays Γa ∕ωM Γc ∕ωM 0.1; Γb ∕ωM 0.1. Other parameters are the same as in (a). (c) For larger detuning parameters Δc ∕ωM 20.0; Δexciton ∕ωM 20.0. Other parameters are the same as in (a). (d) For larger decays as well as larger detunings compared to (a) Δc ∕ωM 20.0; Δexciton ∕ωM 20.0; g∕ωM 2.0; gopt ∕ωM 0.3, Γa ∕ωM Γc ∕ωM 0.1; Γb ∕ωM 0.1; and Ω∕ωM 0.8.

ˆ (blue solid line), and Fig. 2. Effects of thermal vibrations of the membrane n¯ bth 3 (n¯ ath n¯ cth 0) on the mean number of cavity photons haˆ † ai ˆ (green dashed line) and exciton hˆc† cˆ i (red dotted line) for chosen set of parameters in strong decay regime g∕ωM 2.0; moving membrane hbˆ † bi gopt ∕ωM 0.3, Γa ∕ωM Γc ∕ωM 0.1; Γb ∕ωM 0.1; and Ω∕ωM 0.8. (a) For lower detunings Δc ∕ωM 5.0; Δexciton ∕ωM 6.0; (b) for larger detunings Δc ∕ωM 20.0; Δexciton ∕ωM 20.0.



2395

2 Fig. 3. Second-order autocorrelations g2 a 0 (black solid line) for cavity mode, gb 0 for moving dielectric membrane (red dashed line), and cross 0 (blue dotted line) for chosen set of parameters Δ ∕ω 5.0; Δexciton ∕ωM 6.0; g∕ωM 2.0; Γa ∕ωM Γc ∕ωM 0.01; correlation g2 c M ab Γb ∕ωM 0.001; and Ω∕ωM 0.8 with varying optomechanical coupling strength. (a) gopt ∕ωM 0.1; (b) gopt ∕ωM 0.7; (c) gopt ∕ωM 1.2.

The second-order correlation functions, namely the selfcorrelation and cross correlation for photonic and membrane 2 vibration modes, i.e., ga2 0, g2 b 0, and gab 0 for different optomechanical strengths gopt , are shown in Fig. 3. It can be seen that g2 a 0 oscillates between super-Poissonian to sub-Poissonian regimes over the time interval and g2 b 0 always follows super-Poissonian photon statistics for a small value of gopt as shown in Fig. 3(a). Furthermore, for a small value of gopt , cross correlation g2 ab 0 always follows subPoissonian photon statistics except at some point where it becomes Poissonian. As the optomechanical coupling strength gopt increases, g2 b 0 also starts to oscillate between subPoissonian and super-Poissonian photon statistics over time, but in an irregular manner, as shown in Figs. 3(b) and 3(c). For larger gopt , the cross correlation g2 ab 0 in Fig. 3(c) shows stronger sub-Poissonian photon statistics as shown by the smaller values of g2 ab 0 at certain ranges of time.

4. CONCLUSION We have studied a quadratically coupled optomechanical system containing a single undoped semiconductor QW through the Heisenberg–Langevin approach. We obtained coupled equations up to the second order without any approximation. So, nonlinearity due to optomechanical coupling has been included up to second order. We decorrelate higher-order correlation functions in the coupled equations to get a closed set of equations. This enables the study of temporal dynamics in the hybrid system as well two-boson correlation functions under different regimes of decay rates, cavity detuning, optomechanical coupling strength, and thermal noise of phonons. In the resolved sideband regime, cavity excitation as well as mechanical excitation show hybrid resonances, whereas the semiconductor QW shows symmetric shape within the oscillation. We also looked at the effects due to thermal noise of phonons on the mechanical excitation for different detunings. Furthermore, we analyzed the temporal dynamics of the second-order correlation functions, namely the self- and cross-correlation of the cavity and mechanical modes under different optomechanical coupling strengths. Our study is useful for coherent control of photon statistics as well as photon and phonon correlations in quadratically coupled hybrid optomechanical systems.

APPENDIX A: DETAILED CALCULATIONS From Eq. (14) we get d ˆ bˆ †2 bˆ 2 2bˆ † bˆ 1 aˆ −iΔc aˆ − igˆc − igopt a dt 1 − iΩ − Γa aˆ Fˆ a t; 2

(A1)

where Fˆ a t is the noise operator for the cavity mode and Γa is the damping constant. Similarly, we have equations for the exciton and moving dielectric membrane as follows: d ˆ 1 ˆ bˆ bˆ † − Γb bˆ Fˆ b t; b −iωM bˆ − 2igopt aˆ † a dt 2 d 1 cˆ −iΔexciton cˆ − igaˆ − Γc cˆ Fˆ c t; dt 2

(A2)

(A3)

where the noise operators for the dielectric membrane and exciton are given, respectively, by X Fˆ b t −i gk nˆ k 0e−iωk t ;

X Fˆ c t −i gk fˆ k 0e−iωk t :

k

k

(A4) Similarly, Γb and Γc are damping constants for the membrane and exciton and are respectively defined by X

g2k

k

X k

g2k

Z

t

ˆ 0 e−iωk t−t0 ≃ 1 Γb bt; ˆ dt0 bt 2

t

1 0 dt0 cˆ t0 e−iωk t−t ≃ Γc cˆ t: 2

0

Z

0

(A5)

We have the coupled equations for the single and paired averages of operators in our system Hamiltonian as follows: d ˆ −iΔc hai ˆ − ighˆci − igopt ha ˆ bˆ †2 bˆ 2 2bˆ † bˆ 1i hai dt 1 ˆ − iΩ − Γa hai; 2

(A6)

2396


d ˆ 1 ˆ − 2igopt haˆ † a ˆ ˆ bˆ bˆ † i − Γb hbi; hbi −iωM hbi dt 2 d 1 ˆ − Γc hˆci; hˆci −iΔexciton hˆci − ighai dt 2


Thus, Eq. (A.12) becomes (A7)

(A8)

ˆ − haˆ bˆ † i 2haˆ † aˆ 2 bˆ † i − haˆ bˆ †2 bi

d † ˆ ˆ −ighaˆ † cˆ i − haˆ ˆ c† i − iΩhaˆ † i − hai haˆ ai dt ˆ Γa n¯ ath ; − Γa haˆ † ai

1 − Γa Γb haˆ bˆ † i: 2

d † ˆ c† i − haˆ † cˆ i − Γc hˆc† cˆ i Γc n¯ cth : hˆc cˆ i −ighaˆ dt

(A11)

Since all three reservoirs are big and always in thermal equilibrium, we have hFˆ a ti hFˆ b ti hFˆ c ti 0. The equaˆ haˆ bˆ † i are tions for cross-correlation terms such as haˆ bi, given as follows:

(A12)

k

P ˆ k bˆ † can be expressed as The average of the term −i k gk m X X ˆ k tbˆ † −i gk fm ˆ k 0bˆ † te−iωk t g − i gk m k

k

−

X k

g2k

Z 0

t

d 1 ˆ −iΔc hai ˆ − ighˆci − iΩ − Γa hai ˆ hai dt 2 ˆ bˆ †2 i 2haˆ bˆ † ihbˆ † ig − igopt fhaih ˆ big ˆ 2fhaih ˆ ˆ bˆ † bi ˆ bˆ 2 i 2haˆ bih fhaih ˆ haˆ bih ˆ bˆ † ig hai; ˆ haˆ bˆ † ihbi

(B1)

(B2)

ˆ†

− haˆ b bi − haˆ b i X X ˆ k bˆ † i gk nˆ †k aˆ . − i gk m k

APPENDIX B: DECORRELATED AND CLOSED SET OF COUPLED EQUATIONS

d ˆ ˆ − 1 Γb hbi ˆ − 2igopt haˆ † ifhaˆ bi ˆ haˆ bˆ † ig hbi −iωM hbi dt 2 ˆ hbˆ † ig haifh ˆ haˆ † bˆ † ig; ˆ bi ˆ aˆ † bi haˆ † aifh

d ˆ† haˆ b i iωM − Δc haˆ bˆ † i − ighbˆ † cˆ i − iΩhbˆ † i dt ˆ − haˆ bˆ † bˆ 2 i − haˆ bˆ †3 i igopt 2haˆ † aˆ 2 bi ˆ †2 ˆ

(A16)

(A9)

d ˆ† ˆ ˆ Γb n¯ b ; hb bi −2igopt haˆ † aˆ bˆ †2 i − haˆ † aˆ bˆ 2 i − Γb hbˆ † bi th dt (A10)

2haˆ † aˆ 2 bˆ † i

d ˆ† haˆ b i −iΔc iωM haˆ bˆ † i − ighbˆ † cˆ i − iΩhbˆ † i dt ˆ − haˆ bˆ † bˆ 2 i − haˆ bˆ †3 i igopt 2haˆ † aˆ 2 bi

0

ˆ 0 e−iωk t−t dt0 at

bˆ † t :

d 1 ˆ − Γc hˆci; hˆci −iΔexciton hˆci − ighai dt 2

d † ˆ −ighaˆ † cˆ i − haˆ ˆ c† i − iΩhaˆ † i haˆ ai dt ˆ − Γa haˆ † ai ˆ Γa n¯ ath ; − hai

(B3)

(B4)

(A13) ˆ k t. If we consider Here, we have used the solutions for m ˆ k 0 and bˆ † t to be statistically independent, we may write m ˆ k 0bˆ † ti as hm ˆ k 0ihbˆ † ti. As we have considered that all hm three reservoirs are at thermal equilibrium, this gives ˆ k 0i 0,Rand the first term on the right-hand side vanishes. hm P 0 ˆ 0 e−iωk t−t ≃ 12 Γa at, ˆ Using k g2k 0t dt0 at we get X 1 ˆ k tbˆ † Γa haˆ bˆ † i; i gk m 2 k

ˆ Γb n¯ b ; − Γb hbˆ † bi th d † ˆ c† i − haˆ † cˆ i − Γc hˆc† cˆ i Γc n¯ cth ; hˆc cˆ i −ighaˆ dt

(B5)

(B6)

(A14)

as found in [20] in the example of an atom in a damped cavity. ˆ k 0 and bˆ † t are statistically independent; We note that m i.e., there is no interaction term in the Hamiltonian, which shows that the moving membrane interacts with the harmonic oscillator reservoir modes that are coupled to the cavity. ˆ k 0 and bˆ † t are at two different times. Moreover, both m Similarly, we have X 1 1 ˆ − Γb haˆ bˆ † i: i gk nˆ †k aˆ − Γb hbˆ † ai 2 2 k

d ˆ† ˆ ˆ bˆ †2 i 2haˆ † bˆ † ihaˆ bˆ † ig hb bi −2igopt fhaˆ † aih dt ˆ aˆ big ˆ ˆ bˆ 2 i 2haˆ † bih − fhaˆ † aih

(A15)

d ˆ† haˆ b i iωM − Δc haˆ bˆ † i − ighbˆ † cˆ i − iΩhbˆ † i dt 1 − Γa Γb haˆ bˆ † i 2 ˆ haˆ † bih ˆ aˆ 2 ig ˆ aˆ bi igopt 2f2haˆ † aih ˆ bˆ † big ˆ − fhaˆ bˆ † ihbˆ 2 i 2haˆ bih ˆ aˆ bˆ † i haˆ † bˆ † ihaˆ 2 i 2f2haˆ † aih ˆ bˆ †2 ig ˆ aˆ bˆ † i haˆ bih − 2hbˆ † bih − haˆ bˆ † i3hbˆ †2 i 1;

(B7)



d †ˆ ˆ − 1 Γa Γb haˆ † bi ˆ ighbˆ ˆ c† i iΩhbi ˆ haˆ bi iΔc − ωM haˆ † bi dt 2 ˆ bˆ †2 i 2haˆ † bˆ † ihbˆ † big ˆ igopt fhaˆ † bih ˆ aˆ † bˆ † ig − 2fhaˆ †2 ihaˆ bˆ † i 2haˆ † aih ˆ bˆ † bi ˆ 2fhaˆ † bˆ † ihbˆ 2 i 2haˆ † bih ˆ haˆ † bi3h bˆ 2 i 1;

(B8)

d ˆ −iΔc ωM haˆ bi ˆ − ighbˆ cˆ i − iΩhbi ˆ − 1 Γa Γb haˆ bi ˆ haˆ bi dt 2 ˆ bˆ 2 i 1 − igopt 2haˆ bˆ † i 3haˆ bih ˆ bˆ †2 i 2haˆ bˆ † ihbˆ † bi ˆ − igopt haˆ bih

ˆ haˆ † bih ˆ aˆ 2 i; ˆ aˆ bi 2haˆ † aih

2igopt

ˆ haˆ bˆ † i; haˆ † cˆ ihaˆ bi

(B15)

d ˆ † ˆ c† i ighaˆ † bi ˆ c† i ˆ − 1 Γb Γc hbˆ hbˆc i iΔexciton − ωM hbˆ dt 2 ˆ c† i haˆ ˆ haˆ † bˆ † i ˆ bˆ † cˆ † i hbˆ ˆ c† ihaˆ † bi − 2igopt haˆ † aih

ˆ ˆ aˆ † bi 2haˆ † aih

ˆ haˆ bˆ † i; haˆ † cˆ † ihaˆ bi

ˆ 2haˆ † bˆ † ihbˆ † bi ˆ hbˆ †2 ihaˆ † bi (B10)

d † ˆ iΩhˆci haˆ cˆ i iΔc − Δexciton haˆ † cˆ i ighˆc† cˆ i − haˆ † ai dt 1 − Γa Γc haˆ † cˆ i 2 igopt haˆ † cˆ ihbˆ †2 i 2haˆ † bˆ † ihbˆ † cˆ i ˆ bˆ cˆ i haˆ † cˆ ihbˆ 2 i 2haˆ † bih

(B11)

ˆ haˆ bˆ † i; haˆ † cˆ ihaˆ bi

(B17)

d ˆ† † 1 hb cˆ i iωM Δexciton hbˆ † cˆ † i ighaˆ † bˆ † i − Γb Γc hbˆ † cˆ † i dt 2 ˆ c† i haˆ ˆ haˆ † bˆ † i ˆ bˆ † cˆ † i hbˆ ˆ c† ihaˆ † bi 2igopt haˆ † aih (B18)

d 2 ˆ − Γa haˆ 2 i − 2igopt haˆ 2 i haˆ i −2iΔc haˆ 2 i − 2ighaˆ cˆ i − 2iΩhai dt ˆ − 2igopt haˆ 2 ihbˆ †2 i hbˆ 2 i 2hbˆ † bi ˆ 2 2haˆ bih ˆ aˆ bˆ † i; 2haˆ bˆ † i2 haˆ bi

d ˆ c† i iΔexciton − Δc haˆ ˆ c† i ighaˆ † ai ˆ haˆ dt 1 ˆ c† i − hˆc† cˆ i − iΩhˆc† i − Γa Γc haˆ 2 ˆ bˆ ˆ c† i ˆ c† ihbˆ 2 i 2haˆ bih − igopt haˆ

(B16)

d ˆ ˆ − 1 Γb Γc hbˆ cˆ i hb cˆ i −iΔexciton ωM hbˆ cˆ i − ighaˆ bi dt 2 † † ˆ ˆ ˆ haˆ † bˆ † i ˆ ˆ ˆ ˆ ˆ ˆ − 2igopt ha aihb ci hb ci ha cihaˆ † bi

ˆ haˆ bˆ † i; haˆ † cˆ † ihaˆ bi

ˆ bˆ † cˆ i 2haˆ † bˆ † ihbˆ cˆ i haˆ † bih ˆ haˆ † cˆ i; haˆ † cˆ ihbˆ † bi

(B14) d ˆ† 1 hb cˆ i iωM − Δexciton hbˆ † cˆ i − ighaˆ bˆ † i − Γb Γc hbˆ † cˆ i dt 2 ˆ haˆ † bˆ † i ˆ bˆ cˆ i hbˆ † cˆ i haˆ cˆ ihaˆ † bi 2igopt haˆ † aih

ˆ bˆ † bi ˆ 2haˆ † bih

ˆ aˆ † bˆ † i haˆ bˆ † ihaˆ †2 i; 2haˆ † aih

d † † haˆ cˆ i iΔexciton Δc haˆ † cˆ † i ighaˆ †2 i hˆc†2 i dt 1 iΩhˆc† i − Γa Γc haˆ † cˆ † i 2 ˆ bˆ ˆ c† i igopt haˆ † cˆ † ihbˆ 2 i 2haˆ † bih

(B9)

d † ˆ† haˆ b i iΔc ωM haˆ † bˆ † i ighbˆ † cˆ † i iΩhbˆ † i dt 1 − Γa Γb haˆ † bˆ † i 2 ˆ 3haˆ † bˆ † ihbˆ †2 i 1 igopt 2haˆ † bi

(B13)

ˆ c† i haˆ † bih ˆ bˆ † cˆ † i haˆ † cˆ † ihbˆ † bi ˆ haˆ † cˆ † i; 2haˆ † bˆ † ihbˆ

ˆ bˆ † bi ˆ haˆ bˆ † ihbˆ 2 i 2haˆ bih

ˆ haˆ †2 ihaˆ bi

ˆ bˆ † cˆ i haˆ bˆ † ihbˆ cˆ i haˆ cˆ ihbˆ † bi ˆ haˆ cˆ i; 2haˆ bih

haˆ † cˆ † ihbˆ †2 i 2haˆ † bˆ † ihbˆ † cˆ † i

ˆ aˆ bˆ † i haˆ † bˆ † ihaˆ 2 i − 2igopt 2haˆ † aih

igopt

d haˆ cˆ i −iΔexciton Δc haˆ cˆ i − ighaˆ 2 i hˆc2 i dt 1 − iΩhˆci − Γa Γc haˆ cˆ i 2 ˆ bˆ cˆ i − igopt haˆ cˆ ihbˆ 2 i 2haˆ bih haˆ cˆ ihbˆ †2 i 2haˆ bˆ † ihbˆ † cˆ i

ˆ 2haˆ † bih ˆ aˆ † aig ˆ − haˆ †2 ihaˆ bi

haˆ † bˆ † ihbˆ 2 i

2397

(B19)

d ˆ2 ˆ hb i −2iωM hbˆ 2 i − Γb hbˆ 2 i − 2igopt haˆ † ai dt ˆ haˆ bih ˆ aˆ † bˆ † i ˆ bˆ 2 i hbˆ † bi − 4igopt haˆ † aih

ˆ c† ihbˆ †2 i 2haˆ bˆ † ihbˆ † cˆ † i haˆ

ˆ haˆ † bih ˆ aˆ bˆ † i; 2haˆ † bi

(B20)

ˆ c† i ˆ bˆ † cˆ † i haˆ bˆ † ihbˆ 2haˆ bih ˆ haˆ ˆ c† ihbˆ † bi ˆ c† i; haˆ

(B12)

d 2 hˆc i −2iΔexciton hˆc2 i − 2ighaˆ cˆ i − Γc hˆc2 i: dt

(B21)

2398


ACKNOWLEDGMENTS This research is supported by High Impact Research MoE Grant UM.C/625/1/HIR/MoE/CHAN/04 from the Ministry of Education Malaysia.

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