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

Xiuqin Zhao; Ni Liu; J.-Q. Liang

doi:10.1364/OE.25.008123

1. 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 [2–4]. 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 [5–7].

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 high-finesse optical cavity [2–4]. Thus the cavity BEC has been regarded as a promising platform to explore the exotic many-body phenomena [8–21].

It was recently revealed that an extended DM with multi-mode cavity fields [22, 23] exhibits interesting phenomena, which have important applications in quantum information and simulation [24–33]. Moreover both abelian and non-abelian gauge potentials [34] are generated in the two-mode DM, from which the spin-orbit-induced anomalous Hall effect [35] is produced as well. With spatial variation of the atom-photon coupling strength various quantum phases have been predicted such as the crystallization, spin frustration [36], spin glass [37–39] and Nambu-Goldstone mode [40]. It is shown that the strong-coupling [41] may lead to the revival of atomic inversion in a time scale associated with the cavity-field period [42]. The optomechanical DM has been also proposed in order to detect the extremely weak forces [43–48].

Recently the dynamics induced by atom-pair tunneling [49–51] was revealed. The QPT was investigated [52, 53] in two-component BECs by means of the semiclassical approximation. It was demonstrated that coupled two-component BECs in an optical cavity [54] display optical [54], fluid [55], multi-stabilities and capillary instability [55–58]. Substantial many-particle entanglement is also possible in a two-component condensate with spin degree of freedom [59–61] and interference between two BECs has been observed [62]. Particularly, variety of topological excitations is admitted in multi-component and spinor BECs such as domain walls, abelian and non-abelian vortices, monopoles, skyrmions, knots, and D-brane solitons [63].

The QPT in DM has been extensively studied [1–3, 14, 21, 40, 64–67] based on variational method with the help of Holstein-Primakoff transformation [7,14,21,40,64,65,68] to convert the pseudospin operators into a one-mode bosonic operator in the thermodynamic limit. The ground-state properties were also revealed in terms of the catastrophe formalism [69], the dynamic approach [70, 71], and the spin coherent-state variational method [48, 66, 67, 72–74], in which both the normal (⇓) and inverted (⇑) pseudospin [68, 75, 76] can be taken into account giving rise to the multi-stable macroscopic quantum states.

In the present paper, we investigate macroscopic (or collective) quantum states for two-component BECs in a single-mode optical cavity by means of the spin coherent state 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.

2. 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 ħ = 1,

\begin{matrix} H = ω a^{†} a + \sum_{l = 1, 2} ω_{l} J_{l z} \\ + \sum_{l = 1, 2} \frac{g l}{\sqrt{N_{l}}} (a^{†} + a) (J_{l +} + J_{l -}) . \end{matrix}

Where J_lz (J_l_± = J_lx ±iJ_ly, l = 1, 2) is the collective pseudospin operator with spin quantum-number 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.

Fig. 1 Schematic diagram for two ensembles of ultracold atoms (blue and green) with transition frequencies ω₁, ω₂ in an optical cavity of frequency ω.

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3. Spin coherent-state variational method

In this paper we provide analytic solutions for the macroscopic quantum state (MQS) for the spin-boson 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,

H_{s p} (α) = 〈 α | H | α 〉 = ω α^{*} α + \sum_{l = 1, 2} ω_{l} J_{l z} + \sum_{l = 1, 2} \frac{g_{l}}{\sqrt{N_{l}}} (α^{*} + α) (J_{l +} + J_{l -}),

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

| \pm n_{l} 〉 = R (n_{l}) {| s, \pm 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, ΔJ₊ΔJ₋ = ⟨J_z⟩/2 so that |±n⟩ are called the MQSs. The unitary operator is explicitly given by

R (n_{l}) = e^{\frac{θ_{l}}{2} (J_{l +} J e^{- i φ_{l}} - J_{l -} e^{i φ_{l}})} .

Since pseudospin operators for two components of atoms commute each other, the entire trail-wave-function is the direct product of two-component spin coherent states

| ψ_{s} 〉 = | \pm n_{1} 〉 | \pm n_{2} 〉,

which is required to be the energy eigenstate of the effective Hamiltonian of pseudospin operator such that

H_{s p} (α) | ψ_{s} 〉 = E (α) | ψ_{s} 〉 .

Where

| ψ_{s} 〉 = U {| \pm s 〉}_{1} {| \pm s 〉}_{2},

with

U = R (n_{1}) R (n_{2}),

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^†(n₂)R^†(n₁) to the energy eigenequation Eq. (3) we have

{\tilde{H}}_{s p} (α) {| \pm s 〉}_{1} {| \pm s 〉}_{2} = E (α) {| \pm s 〉}_{1} {| \pm s 〉}_{2},

where

{\tilde{H}}_{s p} (α) = U^{†} H_{s p} (α) U .

Under the spin coherent-state transformation the spin operators J_lz, J_l₊, J_l₋ (l = 1, 2) become [80]

\begin{array}{l} {\tilde{J}}_{l z} = J_{l z} \cos θ_{l} + \frac{1}{2} \sin θ_{l} (J_{l +} e^{- i φ_{l}} + J_{l -} e^{i φ_{l}}), \\ {\tilde{J}}_{l +} = J_{l +} \cos^{2} \frac{θ_{l}}{2} - J_{l -} e^{2 i φ_{l}} \sin^{2} \frac{θ_{l}}{2} - J_{l z} e^{i φ_{l}} \sin θ_{l}, \\ {\tilde{J}}_{l -} = J_{l -} \cos^{2} \frac{θ_{l}}{2} - J_{l +} e^{- 2 i φ_{l}} \sin^{2} \frac{θ_{l}}{2} - J_{l z} e^{- i φ_{l}} \sin φ_{l} . \end{array}

Then the effective spin Hamiltonian can be diagonalized under the conditions

\begin{array}{l} \frac{ω_{l}}{2} e^{- i φ_{l}} \sin θ_{l} + {\tilde{g}}_{l α} (\cos^{2} \frac{θ_{l}}{2} - e^{- 2 i φ_{l}} \sin^{2} \frac{θ_{l}}{2}) = 0, \\ \frac{ω_{l}}{2} e^{i φ_{l}} \sin θ_{l} + {\tilde{g}}_{l α} (\cos^{2} \frac{θ_{l}}{2} - e^{2 i φ_{l}} \sin^{2} \frac{θ_{l}}{2}) = 0, \end{array}

from which the angle parameters θ_l, φ_l are determined in principle. Thus we obtain the energy function

E (α) = ω | α |^{2} \pm \sum_{l = 1, 2} \frac{N_{l}}{2} A_{l} (α, θ_{l}, φ_{l}),

where

A_{l} (α, θ_{l}, φ_{l}) = ω_{l} \cos θ_{l} - {\tilde{g}}_{l α} (e^{i φ_{l}} + e^{- i φ_{l}}) \sin θ_{l},

with

{\tilde{g}}_{l α} = \frac{g_{l}}{\sqrt{N_{l}}} (α^{*} + α)

. The total trial-wave-function is

| ψ 〉 = | α 〉 | ψ_{s} 〉,

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

α = γ e^{i ϕ} .

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

\frac{E}{ω} (γ) = γ^{2} \pm \sum_{l = 1, 2,} \frac{N_{l}}{2} \sqrt{{(\frac{ω_{l}}{ω})}^{2} + \frac{16 γ^{2}}{N_{l}} {(\frac{g_{l}}{ω})}^{2}} .

The local minima of energy function Eq. (8) can be determined in terms of the variation with respect to the parameter γ.

4. 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, second-spin inverted and vas versa). For the configuration of both normal spins the dimensionless energy is

\frac{E_{⇊} (γ)}{ω} = γ^{2} - \sum_{l = 1, 2} \frac{N_{l}}{2} \sqrt{{(\frac{ω_{l}}{ω})}^{2} + \frac{16 γ^{2}}{N_{l}} {(\frac{g_{l}}{ω})}^{2}} .

In the following evaluations we assume the equal atom numbers for the two components that N₁ = N₂ = N/2. The atomic frequencies are parametrized according to the cavity frequency ω and atom-field detuning Δ

ω_{1} = ω - Δ, ω_{2} = ω + Δ .

The ground-state is obtained from the variation of average energy

ε_{⇊} = \frac{E_{⇊} (γ)}{N ω}

with respect to the variational parameter γ. The energy extremum condition is found as

\frac{\partial ε_{⇊}}{\partial γ} = 2 γ_{⇊} p_{⇊} (γ_{⇊}) = 0,

where

p_{⇊} (γ_{⇊}) = 1 - \sum_{l = 1, 2} \frac{4 g_{l}^{2}}{ω^{2} F_{l} (γ_{⇊})},

and

F_{l} (γ_{⇊}) = \sqrt{{(\frac{ω_{l}}{ω})}^{2} + 32 {(\frac{g_{l}}{ω})}^{2} \frac{γ_{⇊}^{2}}{N}} .

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,

\frac{\partial^{2} (ε_{_{⇊}} (γ_{_{⇊}}^{2} = 0)}{\partial γ^{2}} = 2 [1 - \frac{4}{ω} (\frac{g_{1}^{2}}{ω_{1}} + \frac{g_{2}^{2}}{ω_{2}})],

is positive. Therefore a phase boundary is determined from

\partial^{2} (ε_{⇊} (γ_{⇊}^{2} = 0) / \partial γ^{2} = 0

, which gives rise to the relation of two critical coupling values

\frac{g_{1, c}^{2}}{ω_{1}} + \frac{g_{2, c}^{2}}{ω_{2}} = \frac{ω}{4} .

When

\frac{g_{1}^{2}}{ω_{1}} + \frac{g_{2}^{2}}{ω_{2}} < \frac{ω}{4},

we have a stable zero photon-number solution, which we call the NP denoted by N_⇊. The energy function for the configuration ⇵ is

ε_{↓ ↑} = \frac{γ^{2}}{N} - \frac{1}{4} [F_{1} (γ_{↓ ↑}) - F_{2} (γ_{↓ ↑})]

The energy extremum condition ∂ε_⇵/∂γ = 2γ_⇵ p_⇵ (γ_⇵) = 0 with

p_{↓ ↑} (γ_{↓ ↑}) = 1 - \frac{4}{ω^{2}} [\frac{g_{1}^{2}}{F_{1} (γ_{↓ ↑})} - \frac{g_{2}^{2}}{F_{2} (γ_{↓ ↑})}],

has the zero photon-number solution, which is stable when the second-order derivative

\frac{\partial^{2} (ε_{↓ ↑} (γ_{↓ ↑}^{2} = 0))}{\partial γ^{2}} = \frac{2}{N} [1 - \frac{4}{ω} (\frac{g_{1}^{2}}{ω_{1}} - \frac{g_{2}^{2}}{ω_{2}})]

is positive. Thus we have the NP (denoted by N_⇵) region when

\frac{g_{1}^{2}}{ω_{1}} - \frac{g_{2}^{2}}{ω_{2}} < \frac{ω}{4} .

Correspondingly for the configuration ⇅ the energy function is

ε_{↑ ↓} = \frac{γ^{2}}{N} + \frac{1}{4} [F_{1} (γ) - F_{2} (γ)] .

The energy extremum condition is ∂ε_⇅/∂γ = γ_⇅p_⇅(γ_⇅) = 0 with

p_{↑ ↓} (γ_{↑ ↓}) = 1 + \frac{4}{ω^{2}} (\frac{g_{1}^{2}}{F_{1} (γ_{↑ ↓})} - \frac{g_{2}^{2}}{F_{2} (γ_{↑ ↓})}) .

Again the stable zero photon-number solution denoted by N_⇅ requires

\frac{g_{2}^{2}}{ω_{2}} - \frac{g_{1}^{2}}{ω_{1}} < \frac{ω}{4} .

The energy function for the configuration ⇈ is

ε_{↑ ↑} = \frac{E_{↑ ↑} (γ)}{ω N} = \frac{γ^{2}}{N} + \frac{1}{4} \sum_{l = 1, 2} F_{l} (γ) .

The extremum condition is

\frac{\partial (ε_{_{⇈}})}{\partial γ} = 2 γ_{_{⇈}} p_{_{⇈}} (γ_{_{⇈}}) = 0,

with

p_{_{⇈}} (γ_{_{⇈}}) = 1 + \sum_{l = 1, 2} \frac{4 g_{l}^{2}}{ω^{2} F_{l} (γ_{_{⇈}})} .

The zero photon-number solution is stable denoted by N_⇈ since the second-order derivative

\frac{\partial^{2} ε_{⇈} (γ_{⇈}^{2} = 0)}{\partial γ^{2}} = \frac{2}{N} [1 + \frac{4}{ω} (\frac{g_{1}^{2}}{ω_{1}} + \frac{g_{2}^{2}}{ω_{2}})] > 0,

is always positive. The nonzero-photon solution can be obtained from the extremum condition.

p_{k} (γ_{s k}) = 0

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(b) for k =⇊ (black line), ⇵ (olive line), ⇅ (blue line) respectively. For the dimensionless coupling g₂/ω = 0.2 [Figs. 2(a1) and 2(b1)] both solutions γ_s_⇊ and γ_s_⇵ of the extremum equation are stable with a positive sloop [Fig. 2(a1)], namely a positive second-order derivative of the energy function with respect the variation parameter γ. The corresponding energies are local minima [Fig. 2(b1)]. γ_s_⇵ indicates the solution of stimulated radiation from the state of atomic population inversion for the second-component of atoms. Increasing the coupling strength to g₂/ω = 0.4 [Figs. 2(a2) and 2(b2)] and 0.7 we have only one stable solution γ_s_⇊. While the two stable solutions appear again for g₂/ω = 0.9 [Figs. 2(a4) and 2(b4)]. It is interesting to see a fact that the the stimulated radiation becomes the first-component of atoms i.e. γ_s_⇅. The superradiant states are denoted respectively by S_⇊, S_⇅ and S_⇵ in the following phase diagrams. 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 g₁–g₂ plane with the resonance condition ω₁ = ω₂ = ω. The phase boundaries g_c_⇊ g_c⇵ g_c_⇅ are determined from the following three relations respectively

\begin{array}{l} g_{2} = \frac{1}{2} \sqrt{1 - {(\frac{2 g_{1}}{ω})}^{2}}, \\ g_{2} = \frac{1}{2} \sqrt{{(\frac{2 g_{1}}{ω})}^{2} + 1}, \\ g_{2} = \frac{1}{2} \sqrt{{(\frac{2 g_{1}}{ω})}^{2} - 1} . \end{array}

In the region denoted by NP_ts (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 g₂/g₁ < 1 (yellow region) to g₂/g₁ > 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 g₂/g₁ = 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].

Fig. 2 Graphical solutions of the extremum equation pk (γ_sk) = 0 for k = ⇊ (black line), k_⇵ (olive line), and k =⇅ (blue line) with g₁/ω = 0.6 and g₂/ω = 0.2 (a1), 0.4 (a2), 0.7 (a3), 0.9 (a4). The corresponding average energy curves ε are plotted in the lower panel (b1–b4). ${\bar{γ}}^{2} = γ^{2} / N$ denotes the mean photon number.

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Fig. 3 Phase diagram in the resonance condition ω₁ = ω₂ = ω. The notations NP_ts(N_⇊, N_⇅,N_⇊) and NP_ts (N_⇊, N_⇵,N_⇊) mean the NP with triple states, in which N_⇊ is the ground state. SP_co (S_⇊, N_⇅, N_⇈) [SP_co (S_⇊, N_⇵,N_⇈)] means the SP characterized by the ground state S_⇊, which coexists with N_⇵ (N_⇅) and N_⇈. SP_co (S_⇊, S_⇅,N_⇈) [SP_co (S_⇊, S_⇵,N_⇈)] is also the coexisting SP, in which the first excited-state is a superradiant state S_⇅ (S_⇵).

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We now consider the phase diagram for the atom-field detuning ω₁ = ω − Δ and ω₂ = ω + Δ with Δ ∈ [−0.9, 0.9] and the atom-field coupling imbalance parameter δ given by

g_{1} = g, g_{2} = (1 + δ) g .

Substituting atom-field coupling Eq. (15) into the corresponding ground-state energy function we obtain the phase diagram of g-Δ 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)

g_{c ↓ ↓} = \frac{1}{2} \sqrt{\frac{(ω^{2} - Δ^{2})}{ω [2 ω + (ω - Δ) (2 δ + δ^{2})]}},

The phase diagram for δ = 0 as depicted in Fig. 4(a) is symmetric with respect to the horizontal line Δ = 0. The triple-state NP region denoted by NP_ts (N_⇊, N_⇵, N_⇈) (yellow) and NP_ts (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 |Δ| 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 ground-state S_⇊ by the variation of atom-field coupling g is the standard DM type for the fixed atom-field detuning Δ. The phase boundary lines g_c_⇵, g_c_⇅, which separate the states S_⇵ and S_⇅, are respectively determined from Eqs. (12, 13)

g_{c ↓ ↑} = \frac{1}{2} \sqrt{\frac{(ω^{2} - Δ^{2})}{ω [(ω + Δ) - (ω - Δ) {(1 + δ)}^{2}]}} = \frac{1}{2} \sqrt{\frac{(ω^{2} - Δ^{2})}{ω [2 Δ - (ω - Δ) (2 δ + δ^{2})]}},

and

g_{c ↑ ↓} = \frac{1}{2} \sqrt{\frac{(ω^{2} - Δ^{2})}{ω [(ω - Δ) {(1 + δ)}^{2} - (ω + Δ)]}} = \frac{1}{2} \sqrt{\frac{(ω^{2} - Δ^{2})}{ω [(ω - Δ) (2 δ + δ^{2}) - 2 Δ]}} .

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 Δ. 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)].

Fig. 4 Phase diagram in g-Δ space with the atom-photon coupling parameter δ = 0 (a), δ = 0.5 (b), and δ = −0.5 (c). The boundary line, which separates the regions with different first-excited-states (N_⇵, S_⇵ and N_⇅, S_⇅), moves upward and downward respectively for δ = 0.5 (b), −0.5 (c).

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5. 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. (7) with spin-state |ψ_s (−s, −s)〉 = U|−s〉₁ |−s〉₂ given in Eq. (4). The result is obviously

n_{p} (↓ ↓) = \frac{〈 α | a^{†} a | α 〉}{N} = {\begin{matrix} 0, & g < g_{c ↓ ↓}, \\ \frac{γ_{↓ ↓}^{2}}{N}, & g > g_{c ↓ ↓} . \end{matrix} .

While the atomic population imbalance becomes

Δ n_{a} (↓ ↓) = \frac{〈 ψ_{s} (- s, - s) | (J_{1 z} + J_{2 z}) | ψ_{s} (- s, - s) 〉}{N} = - \frac{1}{4} \sum_{l = 1, 2} \frac{ω_{l}}{ω F_{l} (γ_{↓ ↓})},

which reduces to the well-known standard Dicke-model value

Δ n_{a} (↓ ↓) = - \frac{1}{2},

at the critical line g_c_⇊ and also the NP state N_⇊. The average energy in ground states N_⇊ and S_⇊ is given by

ε_{↓ ↓} = {\begin{matrix} - 0.5, g < g_{c ↓ ↓}, \\ \frac{γ_{↓ ↓}^{2}}{N} - \frac{1}{4} \sum_{l = 1, 2} F_{l} (γ_{↓ ↓}), g > g_{c ↓ ↓} . \end{matrix} .

For the states N_k and S_k with opposite spin-polarizations k =⇵,⇅ the average photon number is

n_{p} (N_{k}) = 0; n_{p} (S_{k}) = \frac{γ_{k}^{2}}{N} .

The atomic population imbalance becomes

Δ 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

\begin{array}{l} Δ n_{a} (S_{↓ ↑}) = \frac{1}{4 ω} [- \frac{ω_{1}}{F_{1} (γ_{↓ ↑})} + \frac{ω_{2}}{F_{2} (γ_{↓ ↑})}], \\ Δ n_{a} (S_{↑ ↓}) = \frac{1}{4 ω} [\frac{ω_{1}}{F_{1} (γ_{↑ ↓})} - \frac{ω_{2}}{F_{2} (γ_{↑ ↓})}] . \end{array}

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 Δn_a(N_⇈) = 0.5 and the average energy is found as

ε (N_{↑ ↑}) = \frac{1}{4 ω} (ω_{1} + ω_{2})

The stable nonzero-photon state does not exists for this configuration of both inverted spins. The average photon number n_p, atomic population imbalance Δn_a, and the average energy ε are plotted in Fig. 5 as functions of the atom-field coupling strength g in the red and blue detuning Δ = ±0.6 with δ = 0. Below the critical point g_c_⇊ we have triple stable (zero-photon) states denoted by NP_ts (N_⇊, N_⇵, N_⇈) [or NP_ts (N_⇊, N_⇵, N_⇈)], in which N_⇊ (black line) is the ground state with lowest energy. Between the critical points g_c_⇊ and g_c_⇵ (or g_c_⇵) the superradiant ground-state S_⇊ (black line) coexists with the states N_⇵ [olive lines in Figs. 5(a1)–5(c1)], or N_⇅ [blue lines in Figs. 5(a2)–5(c2)], and N_⇈ (red lines). The QPT from the NP (N_⇊) to the SP (S_⇊) is the standard DM type, which takes place at the critical point g_c_⇊. From Fig. 5(c1) we see that the states N_⇵ and S_⇵ (olive lines) of opposite spin-polarizations are the first excited-states in the case Δ = 6. While the states N_⇅ and S_⇅ with interchange of the spin polarizations between two components become the first excited states seen from Fig. 5(c2) (blue lines) for the negative detuning Δ = −6. We observe for the first time the phase transition at the critical point g_c_⇵ (g_c_⇅) from the normal state N_⇅ (N_⇅) to the suprradiant state S_⇅ (S_⇅), which is the stimulated radiation from the collective states of atomic population inversion for one component of BECs seen from Figs. 5 and 6. The ground state does not change at the critical point g_c_⇵ (or g_c_⇅), which separates the normal state N_⇅ (or N_⇅) and the superradiant one S_⇵ (or S_⇅), which are the collective excited-states of the system. For the given frequency detuning Δ = ±0.6 (Fig. 5) the critical points can be evaluated precisely for Eq. (16, 17, 18),

g_{c}_{↓ ↓} = \sqrt{2} / 5 = 0.2828

and

g_{c ↓ ↑} = g_{c ↑ ↓} = \sqrt{2} / 15 = 0.365148

. The normal state N_⇈ (red line) of atomic population inversion for both components does not involve in radiation process.

Fig. 5 Variations of the average photon number n_p (a), atom population imbalance Δn_a (b), and average energy ε (c) with respect to the coupling constant g = g₁ = g₂ in the atom-field frequency detuning Δ = 0.6 (1) and Δ = −0.6 (2).

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We display the variation curves of average photon-number n_p as shown in Figs. 6(a1) and 6(a2), atom population imbalance Δn_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 Δ = 0. The QPT from normal state N_⇊ to the superradiant state S_⇊ takes place at the critical point $g_{c ↓ ↓} = \sqrt{5} / 5 = 0.447214$ . 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 $g_{c ↓ ↑} = \sqrt{3} / 3 = 0.577350$ . This transition is from the normal state of atomic population inversion to the superradiant state for the second component realized from atom population imbalance and the energy in Figs. 5(b1) and 5(c1). By adjusting the imbalance parameter to δ = 0.5 as shown Figs. 6(a2)–6(c2), the transition becomes from N_⇅ to S_⇅ for the first component. The collective stimulated-radiation shifts to the first component, which has lower atom-field coupling than the second component in this case. The transition critical point is found as $g_{c ↑ ↓} = \sqrt{5} / 5 = 0.447214$ .

Fig. 6 The average photon number n_p (a), atomic population Δn_a (b), and average energy ε (c) curves for the imbalance parameter δ = −0.5 (1), δ = 0.5 (2) in the resonance condition Δ = 0.0. The stimulated radiation shifts from one component to the other by adjusting the relative coupling constants.

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6. Conclusion and discussion

In summary, multiple MQSs are derived analytically for two-component BECs in a single-mode cavity by means of the spin coherent-state variational method. The rich phase diagrams are presented with the variation of atom-field coupling imbalance between two components and the atom-field frequency detuning. Indeed the ground states display a typical Dicke-model QPT from the NP to SP for both components in the normal spin-states. When the atom-field coupling imbalance between two components increases the normal spin-state with relatively lower coupling-value flips to the inverted spin-state, the radiation from this state is the stimulated radiation from atomic population-inversion levels. The stimulated radiation can be also generated from manipulation of atom-field frequency detuning. In the specific cases when one of the coupling constants vanishes or two couplings are equal the ground-states and related QPT reduce to that of an ordinary Dicke model. The controllable stimulated radiation may have technical applications in the laser physics. The spin coherent-state variational method is a powerful tool in the study of macroscopic quantum properties for the atom-ensemble and cavity-field system, since it takes into account both the normal and inverted pseudospins, which result in multiple MQSs in agreement with the semiclassical dynamics of nonequilibrium QPT in the Dicke model [76]. In addition a one-parameter variational energy-function is able to be derived in this formalism, so that one can evaluate the second-order derivative to determine rigorously the local minima of energy functions in consistence with the numerical simulation [48, 68, 73, 75, 77].

Funding

National Natural Science Foundation of China (Grant Nos. 11275118, 11404198, 91430109, 61505100); the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi Province (STIP) (Grant No. 2014102); the Launch of the Scientific Research of Shanxi University (Grant No. 011151801004); the National Fundamental Fund of Personnel Training (Grant No. J1103210); Natural Science Foundation of Shanxi Province(Grant No. 2015011008).

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

Abstract

1. Introduction

2. Collective population inversion and stimulated radiation beyond the normal and superradiant phases

3. Spin coherent-state variational method

4. Multi-stable states and phase diagram

5. Mean photon number, atomic population and average energy from viewpoint of phase transition

6. Conclusion and discussion

Funding

References and links

Cited By

Figures (6)

Equations (51)

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