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PHYSICAL REVIEW A 90, 023622 (2014)

Nonlinear atom-photon-interaction-induced population inversion and inverted quantum phase transition of Bose-Einstein condensate in an optical cavity Xiuqin Zhao,1,2 Ni Liu,3 and J.-Q. Liang1,* 1

Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China Department of Physics, Taiyuan Normal University, Taiyuan, Shanxi 030001, China 3 School of Physics and Electronic Engineering, Shanxi University, Taiyuan, Shanxi 030006, China (Received 5 May 2014; published 15 August 2014) 2

In this paper we explore the rich structure of macroscopic many-particle quantum states for a Bose-Einstein condensate in an optical cavity with a tunable nonlinear atom-photon interaction [K. Baumann et al., Nature (London) 464, 1301 (2010)]. Population inversion, bistable normal phases, and the coexistence of normalsuperradiant phases are revealed by adjusting the experimentally realizable interaction strength and pump-laser frequency. For the negative (effective) cavity frequency we observe, remarkably, an inverted quantum phase transition (QPT) from the superradiant to the normal phases with an increase in atom-field coupling, which is just opposite the QPT in the normal Dicke model. Bistable macroscopic states are derived analytically in terms of the spin-coherent-state variational method by taking into account both normal and inverted pseudospin states. DOI: 10.1103/PhysRevA.90.023622

PACS number(s): 03.75.Mn, 71.15.Mb, 67.85.Pq

I. INTRODUCTION

Quantum phase transition (QPT), which exhibits the properties of quantum correlations, has become an exciting research field in many-body physics. The Dicke model (DM) [1], which shows collective phenomena in a light-matter system [2,3], is of particular interest for the study of the fascinating QPT. This is because it exhibits a second-order phase transition from a normal phase (NP) to the superradiant phase (SP) [4,5], which has a broad range of applications [6], especially in quantum information processing. The collective effects give rise to intriguing many-body phenomena such as the existence of a coherent SP at zero temperature [7]. Although the model itself is quite simple, it displays a rich variety of the unique aspects of many-body quantum theory. The DM Hamiltonian for the interaction of an ensemble of N identical two-level atoms with a single mode of the electromagnetic field is given by [7,8] g HD = ωf a † a + ωa Jz + √ (a † + a)(J+ + J− ), (1) 2 N with = 1, where ωa is the frequency difference between the two atomic levels, ωf is the frequency of the cavity-field mode, and g is the atom-field dipole coupling strength. The boson operators a and a † are the annihilation and creation operators for the field, and the pseudospin Ji (i = z, ±) is the collective atomic operator satisfying the angular momentum commutation relation: [J± ,Jz ] = ∓J± , [J+ ,J− ] = 2Jz , with the spin length j = N/2. The DM, being a classic problem in quantum optics, continually provides a fascinating avenue of research in a variety of contexts, since it is a striking example of the macroscopic many-particle quantum state (MMQS), which can be solved rigorously. The QPT occurs √ at the critical coupling strength gc = ωf ωa and the system enters an SP [7] when g > gc . A significant achievement is the experimental study of the quantum behaviors of Bose-Einstein condensates (BECs) in ultrahigh-finesse optical

*

[email protected]

1050-2947/2014/90(2)/023622(7)

cavities [2,9]. More recently experiments were performed in an open cavity [10,11], which led to theoretical interpretations of nonequilibrium QPTs [12–15]. It is believed that the QPT can take place only if the collective atom-photon coupling strength is of the same order as the energy separation between the two atomic levels. This has long been considered a challenging transition condition. This condition was recently shown to be accessible in the strongly coupled regime of cavity quantum electrodynamics (QED) with the pump laser [8]. For a BEC in a high-finesse optical cavity, the energy separation between two levels can be adjusted to be small enough and the QPT, namely, the superradiance transition, has been observed experimentally [9]. This is achieved by introducing two optical Raman transitions in a four-level atomic ensemble along with the controlling of the pump laser power [12]. It is shown that the theoretical model Hamiltonian in relation to this experiment possesses a nonlinear atom-photon interaction resulting from the dispersive shift of the cavity frequency [11]. For a weak nonlinear interaction, the onset of self-organization for ultracold atoms can be used to detect the normal-superradiant QPT in the blue detuning of the cavity frequency [11,16]. This system of BEC in a high-finesse optical cavity has been regarded as a promising platform to explore the exotic many-body phenomena from atomic physics to quantum optics in a well-controlled way [2,9,17–27] . Since the magnitude of this nonlinear interaction can be tuned to the same order as those of the detuning of the cavity frequency and the collective coupling strength, the QPT from the NP to the SP has been observed successfully [11]. Besides its applications in experiments the model with a nonlinear atom-photon interaction itself is of theoretical interest. A natural question is whether or not the nonlinear interaction can lead to new MMQSs compared with the standard DM of Eq. (1). It has been shown that the nonlinear interaction, indeed, results in a dynamically unstable phase [15]. Based on the numerical simulation of zero points of the energy functional, most recently the coexistence of an NP and an SP was found in the nonequilibrium QPT with time-dependent atom-field coupling [15]. In order to understand the mechanism of multiphase

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©2014 American Physical Society

XIUQIN ZHAO, NI LIU, AND J.-Q. LIANG


coexistence we revisit the generalized DM of BEC-cavity experiments [13,16] to reveal the bistable NPs as well as the coexistence of the NP and the SP. The Holstein-Primakof transformation, which converts the pseudospin Ji (i.e., the collective atomic operators) into a one-mode bosonic operator, is the starting point for the most theoretical analysis of the QPT in relation to the DM. In the thermodynamic limit (N → ∞) the DM reduces to a two-mode boson Hamiltonian, the ground state of which can be obtained in terms of the bosonic-coherent-state variational method [7,16,28–31]. In this paper we adopt the direct product of optical and spin coherent states (SCSs) [32–34] as a trial wave function first proposed in Ref. [35] to achieve the energy functional. Based on the SCS variational method we are able to obtain the analytical expressions of MMQSs, energy spectra, the atomic population, and the photon number distribution as well. A full phase diagram with the multistable MMQSs is presented in the whole region of experimental parameters. II. HAMILTONIAN FOR A BEC IN AN OPTICAL CAVITY WITH NONLINEAR INTERACTION AND ANALYTIC SOLUTIONS

Following Refs. [27] and [36], we consider the system of a four-level atomic ensemble in a high-finesse optical cavity with transverse pumping of frequency ωp as depicted in Fig. 1. The ultracold atoms coherently scatter pump light into the cavity mode with a position-dependent phase. Two excited states can be eliminated adiabatically in the large detuning as explained in Appendix A and thus we have an effective two-level system. In an optical cavity all ultracold atoms are assumed to couple identically to the single-mode field and the system reduces to an extended DM given by [16] g H = ωa † a + ωa Jz + √ (a † + a)(J+ + J− ) 2 N U + Jz a † a, N

= ωf − ωp being the pump-cavity field detuning and β an experimental constant [36] (see Appendix B for a complete description of the constant). The nonlinear atom-photon interaction U arising from the dispersive shift in cavity frequency [11,13] can have both positive and negative values. The collective coupling strength g is tunable in experiments by varying the pump laser power. Since the effective frequency ω can be turned from positive to negative regions by the pump-cavity field detuning and the atom-photon interaction strength U , much richer phases arise compared with the ordinary DM. We investigate the MMQSs and related QPT based on the SCS variational method, which has the advantage that both the normal (⇓) and the inverted (⇑) pseudospin states can be taken into account. The inverted pseudospin state was first revealed in a dynamic study [12] in order to see the multiple steady states in the nonequilibrium QPT [12,13,15]. Moreover, an energy functional with one parameter only can be obtained and thus the stability of MMQSs is justified rigorously. A. Spin-coherent-state variational method

We begin with the average of Hamiltonian (2) in the optical coherent state |α, Hsp (α) = α|H |α = ωγ 2 + ωa Jz + +

U Jz γ 2 N

gγ cos η (J+ + J− ), √ N

(4)

where α is the complex eigenvalue of the photon annihilation operator a such that a|α = α|α and can generally be expressed as α = γ eiη .

(2)

where ω = + βU

is the effective cavity frequency, with

(3)

The effective spin Hamiltonian Hsp (α) possesses two macroscopic eigenstates, namely, the SCSs |n∓ of the south and north pole gauges, respectively, which correspond to the normal (⇓) and inverted (⇑) pseudospin states in the dynamics of the nonequilibrium DM [12]. The SCSs can be generated from the maximum Dicke states |j,±j (Jz |j,±j = ±j |j,±j ) with the SCS transformation [14,37], such that |n± = R(n)|j,±j ,

(5)

where the unitary operator is explicitly given by θ

R(n) = e 2 (J+ e

FIG. 1. (Color online) Experimental setup for a trapped BEC in an optical cavity with a transverse pumping laser to control the cavity frequency.

iφ

−J− e−iφ )

.

(6)

As a matter of fact the SCSs of the north and south pole gauges are actually the eigenstates of the spin projection operator J · n|n± = ±j |n± , where n=(sin θ cos φ, sin θ sin φ, cos θ ) is the unit vector with the directional angles θ and φ. In the SCS the spin operators satisfy the minimum uncertainty relation, for example, J+ J− = Jz /2, and therefore the SCSs |n± are called the macroscopic quantum states. The two macroscopic eigenstates of the north and south pole gauges are orthogonal, i.e., n+ |n− = 0. It is a key point to take into account both macroscopic eigenstates |n± to reveal the multistable phases. Using the unitary transformations R(n) for the spin operators

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NONLINEAR ATOM-PHOTON-INTERACTION-INDUCED . . .

such as Jz = R † (n)Jz R(n), etc., the effective spin Hamiltonian Hsp (α) is diagonalized under the conditions θ θ gγ cos2 − e2iφ sin2 cos η = 0,

sin θ eiφ + √ 2 2 N (7) gγ −iφ 2 θ −2iφ 2 θ cos

sin θ e −e cos η = 0, +√ sin 2 2 N U 2 with = ω2a + 2N γ . From the above conditions, Eq. (7), the angle parameters θ and φ can be determined in principle. Thus we obtain the energy functional for the normal (⇓) and inverted (⇑) states, respectively:

E∓ (α) = n∓ |Hsp |n∓ = ωγ 2 ∓

N A(α,θ,φ), 2

(8)

where

2g U 2 γ cos θ − √ sin θ cos η cos φ. A(α,θ,φ) = ωa + 2N N The desired MMQSs, |ψ∓ = |α|n∓ , and corresponding energies are found as local minima of the energy functional E∓ (α). B. Average energy, atomic population, and mean photon numbers

Using the diagonalization conditions, Eq. (7), to eliminate the parameters cos η and cos φ, we obtain, after tedious algebra, the energy functional of one variable, i.e., γ = |α|, N (9) E∓ (γ ) = ωγ 2 ∓ A(γ ), 2 where the function A(α,θ,φ) in the energy functional, Eq. (8), becomes a one-parameter function, 2 U2 A(γ ) = ωa2 + (ωa U + 2g 2 )γ 2 + 2 γ 4 . N N The MMQS solutions are found from the usual extremum condition of the energy function: U 2 ωa + N γ U + 2g 2 ∂E∓ = γ 2ω ∓ = 0. (10) ∂γ A(γ ) The extremum condition, Eq. (10), always possesses a zerophoton-number solution γ = 0, which gives rise to the NP (γn∓ = 0) only if it is stable with a positive second-order derivative, namely, ∂ 2 E∓ (γn∓ = 0)/∂γ 2 > 0. We denote the NP states γn∓ = 0 by N∓ , respectively. The nonzero-photonnumber solutions are found as N 4g|ω|

2 2 ξς , (11) γs∓ = 2 −(2g + U ωa ) ± U ς


The SPs denoted S∓ are realized from both the positive photon number, Eq. (11), and the second-order derivative, Eq. (12). Substituting the nonzero photon solutions, Eq. (11), back to the extremum condition, Eq. (10), it is easy to find the necessary conditions,

ς = 4ω2 − U 2 .

2 The second-order derivative in the solutions γs∓

analytically:

2 2 ∂ 2 E∓ γs∓ ς γs∓ = ±ς . 2 N ∂γ ξ g

is also derived

(12)

ς >0

ω < 0,

ς < 0,

(13)

and (14) 2 γs−

and to be fulfilled for the normal state (⇓) solution 2 the inverted state (⇑) solution γs+ , respectively. It may be 2 worthwhile to remark that the solution γs+ for the inverted state (⇑) is only possible when U 0 and thus is induced by the nonlinear interaction. The critical lines can be fixed from the equations ∂E∓ (γ = 0) = 0, ∂γ which lead to the phase boundaries U gc∓ = ±ω − ωa . 2

(15)

Substituting the photon number given in Eq. (11) into Eq. (9), we obtain the mean energies per atom in the SP: ξ ξ Es∓ 1 ω∓g εs∓ = = 2 −(2g 2 + U ωa ) ± 4gω . N U ς ς (16) The energies become well-known values, 2 ωa =0 =∓ , εs∓ γs∓ 2 at the critical lines gc∓ for the normal (⇓) and inverted (⇑) states, respectively. The mean photon number of the SP in the wave functions |ψ∓ is obviously ψ∓ |a † a|ψ∓ 2 = γs∓ , N while the atomic population imbalance becomes 1 g ς ψ∓ |Jz |ψ∓ = −|ω| ± , na∓ = N U 2 ξ np∓ =

(17)

(18)

which reduces to the well-known values 2 na∓ γs∓ = 0 = ∓ 12 at the critical lines gc∓ . A complete atomic population inversion, i.e., na = 1/2, is found in the inverted state (⇑). III. BISTABILITY AND ATOMIC POPULATION INVERSION

where ξ = g 2 + U ωa ,

ω > 0,

We have presented in the previous section the general formula and now show the particular MMQSs in the experimental parameter values. Following a previous experiment [11] the nonlinear interaction value spreads in a wide region, U ∈ [−80,80] MHz (the unit of energy and frequency megahertz throughout the paper), with the atomic frequency ωa = 1 and

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FIG. 2. (Color online) Phase diagram in blue detuning with (a) = −20 and (b) = −10. NPbi (N− ,N+ ) indicates bistable NPs, and SPco (S− ,N+ ) means the SP of S− coexisting with N+ . The single NP of N+ with complete population inversion is located between the line of U = 30 (a) and the phase boundary gc+ . The unstable macroscopic vacuum (UMV) is found under the curve gc+ .

the collective atom-field coupling strength g ∈ [0,10]. We first consider the blue detuning of the pump-cavity field = −20. With the experimental constant [36] β = 7/6 (see Appendix B for a complete description of the value and its meaning in the experiment), the effective frequency becomes 7U . (19) ω = −20 + 6 The phase boundary for the normal state (⇓) is found from Eq. (15), U , (20) gc− = 2 −10 + 3 which increases with the nonlinear interaction U in exact agreement with the previous observation [36]. Below gc− we have only the phase N− shown in Fig. 2. The SP of S− exists in the region g > gc− and U > 30 [the value can also be evaluated from the necessary condition, Eq. (13), i.e., ς > 0]. The phase boundary for the inverted state (⇑) obtained from Eq. (15) is U , (21) gc+ = 5 4 − 3 which increases with decreasing U as shown in Fig. 2. The NP of N+ for the inverted state (⇑) always exists above the boundary curve gc+ , so that we have the bistable NPs denoted NPbi (N− , N+ ) in Fig. 2, in which N− , with the lower energy, is the ground state. Correspondingly, the notation SPco (S− , N+ ) means that the SP of S− coexists with N+ since S− is the lower energy state. The bistable MMQSs observed in this paper agree with the dynamic study of nonequilibrium QPTs [12,15]. In the region (yellow) between the line of U = 30 (a) and the curve gc+ we have only the single NP of N+ with complete population inversion, which, induced by the nonlinear interaction, is a new observation. The SP of S+ for the inverted state (⇑) does not exist, since the necessary condition, Eq. (14), cannot be fulfilled with the blue detuning. Below the phase boundary gc+ the zero-photon-number solution (γ = 0)

FIG. 3. (Color online) Variations of the (a) average photon number np , (b) population imbalance na , and (c) average energy ε with respect to the coupling constant g for U = 35 in the blue detuning of = −20, with solid black lines for the the normal state (⇓) and dot-dashed (red) lines for the inverted state (⇑). The √ critical point of QPT from the NP of N− to the SP of S− is gc− = 10/3 in the given U value.

is unstable, with a negative second-order derivative, and we call it the unstable macroscopic vacuum. The QPT in the DM is characterized by the average photon number np (or γ ), which serves as an order parameter, with np > 0 for the SP and np = 0 for the NP. As a comparison the phase diagram for detuning = −10 is plotted in Fig. 2(b). The two phase diagrams are qualitatively the same, while the two phase boundary lines gc∓ tend to get closer to each other with a decrease in the absolute value of the detuning. In Fig. 3 we present the average photon number np [Fig. 3(a)], atomic population imbalance na [Fig. 3(b)] between the two atom levels, and the average energy ε [Fig. 3(c)] as a function of the atom-field coupling strength g in the blue detuning of = −20, where solid black lines are for the normal state (⇓) and dot-dashed (red) lines are for the inverted state (⇑) throughout the paper. The QPT from the NP of N− to the SP of S− is the standard DM type, while the nonlinear interaction U only shifts the critical point gc− toward the higher value direction of the atomfield coupling g [36]. For the given value of U = 35 in Fig. 3 the critical point√of the QPT can be evaluated precisely for Eq. (20), gc− = 10/3. The NP of N+ remains unchanged through the critical point gc− , so that it is the bistable NPbi (N− , N+ ) below gc− but the coexistence phase of the SPco (S− , N+ ). IV. INVERTED QUANTUM PHASE TRANSITION

It is an interesting aspect of the nonlinear interaction to see whether or not the SP of S+ for the inverted state can be realized in experiments. To this end we now turn to the red detuning (ωp < ωf ) with, for example, = 20. The phase boundary lines are, respectively, found from Eq. (15) as U gc− = 2 10 + (22) 3

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NONLINEAR ATOM-PHOTON-INTERACTION-INDUCED . . .


FIG. 6. (Color online) The QPT among NPs of different types showing, by the variation of atomic population imbalance, na for U = −14 and = 20. FIG. 4. (Color online) Phase diagram in red detuning with (a) = 20 and (b) = 10. The SP of S+ for the inverted state (⇑) appears when the effective frequency becomes negative, ω < 0. (1) The single NP of N− ; (2) NPco (N− ,S+ ).

and

gc+ =

U , −5 4 + 3

(23)

shown in Fig. 4(a). The two lines cross at the point gc− = gc+ (U = −120/7 ), which appears as a critical point of multiple 2 phases. The SP of S− with γs− > 0 coexists with the NP of N+ in the region g > gc− and U > −12 determined from the necessary condition, Eq. (13) (cyan region in Fig. 4). The bistable NPbi (N− , N+ ) is located in the area for g gc− (pink region in Fig. 4). The single NP of N+ with complete atomic population inversion is bounded by the line for U = −12 [Fig. 4(a)] and the critical curves gc− and gc+ (yellow region in Fig. 4). Label (1) indicates a small region of the single 2 NP of N− . The SP of S+ with the stable solution γs+ > 0 is found in the region U > −30 and U < −17 [Fig. 4(a)] when g < gc+ , which is below the critical boundary line gc+ , just opposite to the normal-state (⇓) case. The notation NPco (N− , S+ ), labeled (2) in Fig. (4), means that the NP of N− coexists with S+ since N− is the lower energy state. The phase diagram

for = 10 is presented in Fig. 4(b) for comparison. The phase diagram is also qualitatively the same as that in Fig. 4(a). However, the overlap region between the two phase boundary lines gc− and gc+ is suppressed with a decrease in . The QPT from the NP to the SP for the standard DM is along the increasing direction of the coupling constant g, while the QPT in the inverted state (⇑) would be in the direction opposite that shown in the phase diagram in Fig. 4, where the SP of S+ is located on the left-hand side of the critical line gc+ . Figure 5 displays the curves for the average photon number np [Fig. 5(a)], atomic population imbalance na [Fig. 5(b)], and energy ε [Fig. 5(c)] for U = −20, = 20. The critical point gc− = 2 5/3 separates the coexistence phase NPco (N− , S+ ) and the single SP of S+ . Then with an increase in g the SP of√S+ transitions to the NP of N+ at the critical point gc+ = 2 10/3. The QPT from the SP of S+ to the NP of N+ is just in the inverted direction compared with the standard DM. One should not be surprised by this inverted QPT, since the effective frequency is negative (ω < 0) in the region where the SP of S+ exists. By adjusting the nonlinear interaction constant, for example, U = −14, the QPT among NPs of different types can also be realized, which is displayed in Fig. 6 for = 20. The transition from the NP of N− to the √ bistable NPbi (N− , N+ ) takes place at the critical point gc+ = 10/3 [determined from Eq. (23)] and then the transition from the bistable NPbi√(N− , N+ ) to the NP of N+ follows at the critical point gc− = 4 2/3 [obtained from Eq. (22)]. Even though the order parameter is 0 (np∓ = 0) on both sides of the critical point, the ground-state structure changes. V. CONCLUSION AND DISCUSSION

FIG. 5. (Color online) (a) np , (b) na , and (c) ε curves in the red detuning of = 20 for U = −20. The QPT from the NP of N+ to the SP of S+ is in the inverted direction.

The system of a BEC in an optical cavity provides a marvelous model for study of the fascinating QPT in the strongly coupled regime of cavity QED. The new predictions of population inversion, bistable MMQSs, and an inverted QPT can be detected experimentally by tuning the frequency of the pump laser and the atom-photon interaction strength. The SP of S+ for the inverted pseudospin state (⇑) exists only when the effective frequency becomes negative, ω < 0, along with the nonzero atom-photon interaction. We remark that the SCS variational method has an advantage in the theoretical investigation of macroscopic quantum properties, since both the normal and the inverted pseudospin states can be taken into account to reveal the bistable phases. Our observations of bistable phases are in agreement with the semiclassical dynamics of the nonequilibrium DM [12]. Coexistence of

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governed by the pump laser of frequency ωp , with p being the pump Rabi frequency. ωA denotes the atomic transition frequency. It has been demonstrated that [9] the exited states |±k,0 and |0,±k can be adiabatically eliminated in the large detuning A = ωp − ωA . Thus we have the effective two-level model with splitting energy ωa = 2k 2 /2m, which is twice the recoil energy. The two-level states may be simplified as |0 = |0,0 and |1 = |±k,±k. The cavity mode of frequency ω´ = + U0 /2 induces the transition of balanced Raman channels [9] between state |0 and state |1, where U0 = g02 /A and = ωf − ωp is the the pump-cavity field detuning as defined in the model Hamiltonian given in Eq. (2). APPENDIX B: MEANING OF THE EXPERIMENTAL CONSTANTβ

FIG. 7. Two-level model realization in the large detuning A .

the NP and SP was also realized recently by Liu, Li, and Liang in a time-driving nonequilibrium model, where multiple local minima of the energy functional are found by numerical simulation [15].

According to Ref. [9], the atom-photon interaction exists only in atoms of excited state |1 with nonzero momentum such that Hnl =

1 U0 Mn1 a † a, N

(B1)

APPENDIX A: EXPERIMENTAL REALIZATION OF THE TWO-LEVEL MODEL

with M = 3/4 being a matrix element [9]. The coupling constant in the extended DM Hamiltonian given in Eq. (2) is U = MU0 . n1 denotes the collective atom number operator in excited state |1. Using the relation n1 + n0 = N and Jz = (n1 − n0 )/2 the original nonlinear term can be written as 1 Jz † + a a. Hnl = U (B2) 2 N

The four-level atomic transitions are depicted schematically in Fig. 7, following Ref. [9]. The cavity photon mode of frequency ωf induces the transitions |0,0 ↔ |±k,0, |±k,±k ↔ |0,±k with the single-photon Rabi frequency g0 , where |0,0, |±k,±k denote atomic momentum states of px = py = 0 and px = py = ±k, respectively [9,38–41]. The transitions |0,0 ↔ |0,±k and |±k,±k ↔ |±k,0 are

The first term is obviously an induced cavity mode and can be combined with the cavity mode of frequency ω . The second term is the nonlinear interaction given in Eq. (2). Finally, the effective mode frequency of the extended DM is then ω = ω + U/2. Thus we have β = (1 + 1/M)/2 = 7/6, which characterizes the transition of balanced Raman channels and the nonlinear interaction induced cavity mode as well.

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ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant No. 11275118) and the Research Training Program for Undergraduates of Shanxi University (Grant No. 2014012174).

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