Collective atomic-population-inversion and stimulated radiation for two-componentBose-Einstein condensate in an optical cavity

In this paper we investigate the ground-state properties and related quantum phase transitions for the two-component Bose-Einstein condensate in a single-mode optical cavity. Apart from the usual normal and superradiant phases multi-stable macroscopic quantum states are realized by means of the spin-coherent-state variational method. We demonstrate analytically the stimulated radiation from collective state of atomic population inversion, which does not exist in the normal Dicke model with single-component atoms. It is also revealed that the stimulated radiation can be generated only from one component of atoms and the other remains in the ordinary superradiant state. However the order of superradiant and stimulatedradiation states is interchangeable between two components of atoms by tuning the relative atom-field couplings and the frequency detuning as well.


Introduction
The Dicke model (DM), which describes an ensemble of two-level atoms interacting with a single-mode quantized field [1], plays a important role in the study of Bose-Einstein condensate (BEC) trapped in an optical cavity [2][3][4]. It successfully illustrates the collective and coherent radiations [1]. A second-order phase transition from a normal phase (NP) to a superradiant phase (SP) was revealed long ago by increase of the atom-field coupling from weak to strong regime [5][6][7].
In order to realize experimentally the quantum phase transition (QPT) predicted in the DM the collective atom-photon coupling strength ought to be in the same order of magnitude as the energy level-space of atoms. This condition is far beyond atom-field coupling region in the conventional atom-cavity system. Recently the QPT was achieved with a BEC trapped in a highfinesse optical cavity [2][3][4]. Thus the cavity BEC has been regarded as a promising platform to explore the exotic many-body phenomena [8][9][10][11][12][13][14][15][16][17][18][19][20][21].
In the present paper, we investigate macroscopic (or collective) quantum states for twocomponent BECs in a single-mode optical cavity by means of the spin coherent variational method in order to reveal the rich structure of phase diagrams and the related QPTs. Particularly the collective state of atomic population inversion, namely the inverted pseudospin (⇑), is demonstrated along with the stimulated radiation, which does not exists in the usual DM.

Collective population inversion and stimulated radiation beyond the normal and superradiant phases
We consider two ensembles of ultracold atoms, which are coupled simultaneously to an optical cavity mode of frequency ω as depicted in Fig. 1. Effective Hamiltonian of the system has the form [77] of two-component DM in the unit convention = 1, Where J lz (J l± = J lx ± iJ ly , l = 1, 2) is the collective pseudospin operator with spin quantumnumber s l = N l /2. N l denotes the atom number of l-th component and ω l is the atomic frequency. a † (a) is the photon creation (annihilation) operator and g l is the atom-field coupling strength.

Spin coherent-state variational method
In this paper we provide analytic solutions for the macroscopic quantum state (MQS) for the spinboson system in terms of the recently developed spin coherent variational method [48,73,78,79]. The meaning of MQS in the present paper is that the variational wave function is considered as a product of boson and spin coherent states seen in the followings. We begin with the partial average of the system Hamiltonian in the trial wave function |α , which is assumed as the boson coherent state of cavity mode such that a |α = α |α . After the average in the boson coherent state we obtain an effective Hamiltonian of the pseudospin operators only, which is going to be diagonalized in terms of spin coherent-state transformation. A spin coherent state can be generated from the maximum Dicke states |s, ±s (J z |s, ±s = ±s |s, ±s ) with a spin coherent-state transformation [74,80]. For the l-th component pseudospin operator we have two orthogonal coherent states defined by |±n l = R(n l ) |s, ±s l , which are called north-and south-pole gauges respectively. As a matter of fact the spin coherent states are actually the eigenstates of the spin projection operator J l · n l |±n l = ± j |±n l , where n l = (sin θ l cos ϕ l , sin θ l sin ϕ l , cos θ l ) is the unit vector with the directional angles θ l and ϕ l . In the spin coherent states the spin operators satisfy the minimum uncertainty relation, for example, ∆J + ∆J − = J z /2 so that |±n are called the MQSs. The unitary operator is explicitly given by Since pseudospin operators for two components of atoms commute each other, the entire trailwave-function is the direct product of two-component spin coherent states which is required to be the energy eigenstate of the effective Hamiltonian of pseudospin operator such that Where with being the total unitary operator of spin coherent-state transformation. It is a key point to take into account of both spin coherent states |±n for revealing the multi-stable MQSs. Applying the unitary transformation U † = R † (n 2 )R † (n 1 ) to the energy eigenequation Eq. (3) we have Under the spin coherent-state transformation the spin operators Then the effective spin Hamiltonian can be diagonalized under the conditions from which the angle parameters θ l , ϕ l are determined in principle. Thus we obtain the energy function The total trial-wave-function is and corresponding energies are found as local minima of the energy function E (α), in which the complex eigenvalue of boson coherent state is parametrized as By solving the Eq. (6) and eliminating the angle parameters θ l , ϕ l , φ we derive the scaled-energy as a function of one variational-parameter γ only The local minima of energy function Eq. (8) can be determined in terms of the variation with respect to the parameter γ.

Multi-stable states and phase diagram
In our formalism both the normal (⇓) and inverted (⇑) pseudospin states [68,75] are taken into account to reveal the multiple stable states. Thus there exist four combinations of two-spin states labeled by ↓↓ (both normal spins), ↑↑ (both inverted spins), ↓↑ and ↑↓ (first-spin normal, secondspin inverted and vas versa). For the configuration of both normal spins the dimensionless energy is In the following evaluations we assume the equal atom numbers for the two components that N 1 = N 2 = N/2. The atomic frequencies are parametrized according to the cavity frequency ω and atom-field detuning ∆ The ground-state is obtained from the variation of average energy with respect to the variational parameter γ. The energy extremum condition is found as where The extremum condition Eq. (10) possesses always a zero photon-number solution γ ↓↓ = 0, which is stable if the second-order derivative of energy function, is positive. Therefore a phase boundary is determined from ∂ 2 (ε ↓↓ (γ 2 ↓↓ = 0)/∂γ 2 = 0, which gives rise to the relation of two critical coupling values When we have a stable zero photon-number solution, which we call the NP denoted by N ↓↓ . The energy function for the configuration ↓↑ is The energy extremum condition has the zero photon-number solution, which is stable when the second-order derivative is positive. Thus we have the NP (denoted by N ↓↑ ) region when Correspondingly for the configuration ↑↓ the energy function is The energy extremum condition is ∂ε ↑↓ /∂γ = γ ↑↓ p ↑↓ (γ ↑↓ ) = 0 with Again the stable zero photon-number solution denoted by N ↑↓ requires The energy function for the configuration ↑↑ is The extremum condition is The zero photon-number solution is stable denoted by N ↑↑ since the second-order derivative is always positive. The nonzero-photon solution can be obtained from the extremum condition.
for the four configurations k =↓↓, ↓↑, ↑↓, ↑↑. The extremum condition Eq. (14) is able to be solved numerically. We display in Fig. 2(a) the stable nonzero photon solutions γ sk , which are called the superradiant states, and the corresponding energies ε(γ sk ) as shown in Fig. 2 The new observation with the spin coherent-state variational-method is that besides the ground states we also obtained the stable MQSs of higher energies. Fig. 3 depicts the phase diagram in g 1 -g 2 plane with the resonance condition ω 1 = ω 2 = ω. The phase boundaries g c↓↓ g c↓↑ g c↑↓ are determined from the following three relations respectively   In the region denoted by N P t s (bounded by the critical line g c↓↓ ) there exist triple zero-photon states, in which N ↓↓ with lowest energy is the ground state. This region is separated into two areas (pink and yellow) with only one state difference that the state N ↓↑ in one area is replaced by N ↑↓ in the other. We see the simultaneous spin-flip from the state N ↓↑ to N ↑↓ by adjusting the ratio of two coupling constants from g 2 /g 1 < 1 (yellow region) to g 2 /g 1 > 1 (pink region). The notation, for example, SP co (S ↓↓ , N ↑↓ , N ↑↑ ) (cyan area) means the SP region characterized by the superradiant ground-state S ↓↓ coexisting with the first (N ↑↓ ) and second (N ↑↑ ) excited states of zero photons). The phase diagram is symmetric with respect to the line g 2 /g 1 = 1, which separates the SP region to two areas. Below the symmetric line (green area) only the first excited state is changed to N ↓↑ by the coupling-variation induced spin flip. The critical line g c↑↓ is a boundary, above which the first excited state becomes surperradiant state S ↑↓ (cyan region) in the upper area of the symmetric line. While g c↓↑ is the corresponding boundary for the first excited states N ↓↑ and S ↓↑ (olive area). The superradiant states S ↑↓ , S ↓↑ , which are new observation for the two-component BECs, are seen to be the stimulated radiation from the higher-energy atomic levels. The stable population inversion state N ↑↑ for both components exists in the whole region. The multi-stable MQSs observed in this paper agree with the dynamic study of nonequilibrium QPTs [68,75]. We now consider the phase diagram for the atom-field detuning ω 1 = ω − ∆ and ω 2 = ω + ∆ with ∆ ∈ [−0.9, 0.9] and the atom-field coupling imbalance parameter δ given by Substituting atom-field coupling Eq. (15) into the corresponding ground-state energy function we obtain the phase diagram of g-∆ space displayed in Fig. 4 for the imbalance parameter δ = 0 [ Fig. 4(a)], 0.5 [ Fig. 4(b)], −0.5 [ Fig. 4(c)]. The phase boundary line g c↓↓ for the normal state N ↓↓ is found from Eq. (11) The phase diagram for δ = 0 as depicted in Fig. 4(a) is symmetric with respect to the horizontal line ∆ = 0. The triple-state NP region denoted by N P t s (N ↓↓ , N ↓↑ , N ↑↑ ) (yellow) and N P t s (N ↓↓ , N ↑↓ , N ↑↑ ) (pink) is located on the left-hand side of the critical line g c↓↓ , which shifts towards the lower value direction of the atom-field coupling g [21,68] with the increase of absolute value of detuning |∆| seen from Fig. 4(a). SP co S ↓↓ , N ↓↑ ,N ↑↑ (green region) and SP co S ↓↓ , N ↑↓ ,N ↑↑ (cyan) denote the SP characterized by the ground-state S ↓↓ coexisting with the normal states N ↓↑ , N ↑↓ and N ↑↑ respectively. The QPT from the NP of ground-state N ↓↓ to the SP of ground-state S ↓↓ by the variation of atom-field coupling g is the standard DM type for the fixed atom-field detuning ∆. The phase boundary lines g c↓↑ , g c↑↓ , which separate the states S ↓↑ and S ↑↓ , are respectively determined from Eqs. (12, 13) and The superradiant region denoted by SP co S ↓↓ , S ↓↑ ,N ↑↑ (olive area) is above the the critical line g c↓↑ , while SP co S ↓↓ , S ↑↓ ,N ↑↑ (blue) is located below the critical line g c↑↓ . We see that the second excited-state varies from the normal state N ↓↑ to the superradiant state S ↓↑ by the increase of detuning ∆. The difference of upper and lower half-plane of the phase diagram is made only by the first excited-states N ↓↑ , S ↓↑ and N ↑↓ , S ↑↓ with the interchange of spin polarizations between two components. This boundary line, which separates the regions with different first excited-states, moves upward and downward respectively for δ = 0.5 [ Fig. 4(b)], −0.5 [ Fig. 4(c)].

Mean photon number, atomic population and average energy from viewpoint of phase transition
The mean photon numbers in the states N ↓↓ and S ↓↓ can be evaluated directly from the average of photon number-operator in the corresponding wave functions |ψ = |α |ψ s in Eq. While the atomic population imbalance becomes which reduces to the well-known standard Dicke-model value at the critical line g c↓↓ and also the NP state N ↓↓ . The average energy in ground states N ↓↓ and S ↓↓ is given by For the states N k and S k with opposite spin-polarizations k =↓↑,↑↓ the average photon number is The atomic population imbalance becomes ∆n a (N k ) = 0 for the zero-photon states N k . While the atomic population imbalance for the superradiant states S k is seen to be ∆n a (S ↑↓ ) = 1 4ω The average energies ε k (S k ) of the superradiant states S k for k =↓↑,↑↓ can be obtained from the energy functions with the corresponding solutions γ k , which lead to ε k (N k ) = 0. For the inverted-spin state of zero photon the atomic population imbalance is ∆n a (N ↑↑ ) = 0.5 and the average energy is found as The stable nonzero-photon state does not exists for this configuration of both inverted spins. The average photon number n p , atomic population imbalance ∆n a , and the average energy ε are plotted in Fig. 5  We display the variation curves of average photon-number n p as shown in Figs. 6(a1) and 6(a2), atom population imbalance ∆n a as shown in Figs. 6(b1) and 6(b2), and energy ε as depicted in Figs. 6(c1) and 6(c2) with respect the coupling constant g for the imbalance parameter δ = ±0.5 at the resonance condition ∆ = 0. The QPT from normal state N ↓↓ to the superradiant state S ↓↓ takes place at the critical point g c↓↓ = √ 5/5 = 0.447214. In the case δ = −5 as depicted in Figs. 6(a1)-6(c1), namely the second component has lower coupling value, an additional transition appears between the collective excited-states N ↓↑ and S ↓↑ at the critical point g c↓↑ = √ 3/3 = 0.577350.