The interaction between atomic dipole moments and the vacuum state of a quantized field leads to spontaneous emission (SE). Experimental results show that the SE from excited atoms can be either enhanced or suppressed.^[1–2] The recent progress of Bose-Einstein condensate (BEC) in magnetic traps has created a new state of matter where all atoms share a single macroscopic quantum state.^[3] This leads to great opportunities to explore and test new phenomena related to macroscopic quantum coherence. In recent years authors have theoretically studied SE in a trapped BEC. In this respect, the continuous center-of-mass momentum distribution leads to an increase of SE.^[4–5] In this line of research, different physical aspects and various usefulness of BEC systems have been recently considered and discussed.^[6–9] It should be noted that, the dynamics of BEC systems and their corresponding interesting topics have been mostly investigated, as a few one may refer to the interference effects between two BECs,^[10] superfluidity and coherence in BECs^[11] and many of fundamental challenging papers have been published elsewhere.^[12] The process of collective SE in the Dicke model has been investigated in the literature. An exact solution for the SE of a multi-photon single-atom Dicke model has been obtained.^[13] Also, the SE in the presence of N atoms was analytically presented for the case in which only one atom is initially excited.^[14–15] The collective SE from a system of two atoms with multi-photon transitions in a cavity has been studied in Ref. [16]. Collective SE from a BEC in the framework of a multi-photon q-deformed Dicke model has been investigated in Ref. [17] The effect of atomic collisions on the collective spontaneous emission from an f-deformed BEC has been studied.^[18] Lately, controlling SE noise in measurement-based feedback cooling of a BEC has been reported.^[19] A general theoretical model of the interaction between a single-mode cavity field and the excitons, which leads to collapses and revivals of exciton emission in a semiconductor microcavity has been presented.^[20] Recently, we investigated the quantum dynamics of a system consisting of atomic BEC and a single-mode quantized field in the presence of interatom collisions, which leads to clear collapse-revival (CR) phenomenon specially in the atomic population inversion.^[21–22]

In the present paper we first give a general theoretical model of the interaction between a single-mode cavity field and a BEC system consisting of N two-level atoms in the presence of interatom collisions and higher-order (nonlinear) interaction of photons, where the atoms are in their excited states. We consider the vacuum state as the initial state of field, therefore, we are able to evaluate the SE in our considered system. Then, to quantify the SE, the expectation values of as atomic population inversion (API) in two cases of number and coherent states of atomic subsystem are evaluated. As will be revealed, the collapse-revival (CR) quantum phenomena can be observed. The effects of the number of atoms in BEC, interatom as well as higher-order atom-field couplings and damping parameter on the discussed quantities are studied in detail. At last, we show that, one can find a practical procedure to measure the quantum state of atoms in BEC.

This paper is organized as follows. In the next section, the model and the analytical equations of transformed operators of field and atoms are presented. In Sec. 3 we get the analytical expressions for the API () with considering some particular initial conditions for BEC states. We follow our studies in this section with presenting the numerical results. The last section contains a summary and concluding remarks.

According to the experimental results of Mewes et al. in 1997,^[23] short pulses of rf radiation can be used to create a BEC in a superposition of trapped and untrapped states. In this respect, we consider a system, which consists of a BEC of N two-level atoms in a magnetic trap coupled to an rf field. In fact, the atomic subsystem is a BEC consisting of N atoms with two internal state (|1⟩, |2⟩) in a magnetic trap, which coupled to a single-mode rf field. The second quantization of the Hamiltonian model is given by

In order to study the dynamical properties of the system, we use Bogoliubov approximation, in which the slow change of the atom number in the ground state can be neglected in the process of interacting with quantized field when the number of BEC atoms at initial moment is enough large. Therefore, one can replace the ground state operators of BEC with a c-number where N₁ is the mean number of atoms in the ground state.^[17,26–28] It is worth to note that the ground state operators of BEC atoms is not a constant of motion and therefore does not commute with the Hamiltonian, however, considering the Bogoliubov approximation it can be considered as a fixed number. Using this approximation, we can replace with . For simplicity, also let () and . Then, the Hamiltonian (1) can be simplified as follows

The first-order interaction between the atom and the field is described by the linear operators of atom and field,^[30] therefore, the term contains the nonlinear operators may be referred to as higher-order interaction term. Also, the higher-order atom-field interaction represents the phase-space filling effects as is mentioned in Ref. [31]. It should be noticed that, we have neglected a constant term, which contains the average number of the condensate atoms in the state |1⟩ (), which does not affect the dynamics of the system. Also, it is worthwhile to mention that, the Hamiltonian (2) has been used to investigate the exciton-induced squeezed state of light in semiconductors^[29] and the exciton emission of a quantum well embedded in a semiconductor microcavity.^[20] To obtain the dynamical evolution of the system, we introduce the following canonical transformations,

To achieve the time variation of API, we introduce the Schwinger angular momentum operators as

The spontaneous emission (SE) arises from the interaction between atomic dipole moments and the vacuum of the radiation field. To evaluate this quantity, we proceed to investigate the API as a measure for the rate of SE in our considered BEC atomic system. Since, we have represented the BEC operators in terms of the angular momentum operators, therefore, we define API as the expectation value of operator as follows:^[17]

3.1 API with Number State as the Initial State of BEC System

The initial state vector of the system, which is the direct product of the two initial BEC and field states can be written in terms of the eigenstates of and as |ψ(0)⟩ = |j,j⟩ with j = N/2. Therefore, one arrives at

where the following definitions have been utilized:^[32]

It should be noted that if some realistic conditions are taken into account, it requires High-precision evaluation of Wigner’s d matrix and some numerical calculations are needed,^[33] however, for the chosen resonance case (θ = π/4) one can use the above closed form of the Wigner’s d matrix to obtain analytical results. Substituting Eq. (17) into Eq. (13) and after some lengthy calculations, we get the API at time t in the number state case

In the resonance regime (δ = 0), i.e., θ = π/4, the API reduces to

Figure 1 shows the time evolution of API in resonance condition for various number of atoms in BEC with fixing the other parameters in terms of the dimensionless time τ = gt. We can see that increasing the number of atoms in BEC leads to an ‘increase in the amplitudes of API’ as well as ‘the number of revivals (peaks)’ in a fixed interval of time. It should be noted that, due to the damping effect the amplitude of API gradually decreases, irrespective of the value of N. The CR phenomena as a pure quantum mechanical feature in the API are related to the envelope term [cos(g₁t/2)]^N−1 in the expression of (Eq. (20)). In this case, the revival periods of the oscillations are 2π/g₁.

Fig. 1

	Figure Option ViewDownloadNew Window
	Fig. 1 The evolution of API for γ1 = γ2 = 0.01G, g1 = g2 = 0.5G against the dimensionless time τ = gt in the resonance regime. (a) N = 10, (b) N = 20, (c) N = 50, (d) N = 100.

Figure 2 shows the influence of damping process on API for the fixed number of atoms N = 20. In the presence of damping, the amplitude of API decreases with time, but the collapse time intervals remain constant. In Fig. 2(a), the amplitude of API is constant, because of the absence of decay process, while in the other plots of Fig. 2 the rate of reducing of the amplitude of API, increases by increasing the decay constant such that for enough large value of γ no revival may be observed after the first collapse (Fig. 2(d)). In Fig. 3 the effect of interatom interaction on API has been investigated. It can be seen from the figure that, increasing the strength of interatom coupling leads to an increase in the number of revivals considering a fixed interval of time. This is due to the fact that the latter parameter exists in the envelope term of .

Fig. 2

	Figure Option ViewDownloadNew Window
	Fig. 2 The influence of dissipation on the time evolution of API for N = 20 and g1 = g2 = 0.5G in the resonance regime. (a) γ1 = γ2 = 0.0, (b) γ1 = γ2 = 0.01G, (c) γ1 = γ2 = 0.1G, (d) γ1 = γ2 = 1.0G.

Fig. 3

	Figure Option ViewDownloadNew Window
	Fig. 3 The effect of interatoms interaction on the time dependence of API for N = 20, γ1 = γ2 = 0.01G and g2 = 0.5G in the resonance regime. (a) g1 = 0.25G, (b) g1 = 0.5G, (c) g1 = 0.75G, (d) g1 = G.

3.2 API with Coherent State as the Initial State of BEC System

To achieve more insights into the effect of initial state of BEC atoms on the API, we consider another initial state for the system. In this respect, we assume that at t = 0 the cavity field is in the ground state and the BEC atoms are in the coherent state. Consequently, the initial state of the total system reads as

where ⟨N⟩ = |α|² is the average number of atoms in BEC. We can get the final result for the time evolution of API in the coherent state representation by substituting Eq. (21) into Eqs. (13)–(14) leading to

In the resonance regime (δ = 0, i.e., θ = π/4), Eq. (22) is reduced to

Interestingly, the analytical expressions of API in this case are very similar to the results of Ref. [34] in which the authors investigated ‘macroscopic quantum self-trapping and atomic tunneling in two-species Bose-Einstein condensates’ with a different approach. In the coherent state case, the CRs can be occurred even for due to the quantum superposition properties of the atomic coherent state in BEC. Again, Fig. 4 shows that the revival amplitudes of API increases by increasing the number of atoms and the decay process decreases the amplitudes successively in each plot of this figure. In this case, the envelope term in Eq. (23) leads to the CRs.

Fig. 4

	Figure Option ViewDownloadNew Window
	Fig. 4 The time dependence of API for different number of BEC atoms with γ1 = γ2 = 0.01 G, g1 = g2 = 0.5 G in terms of the dimensionless time τ = gt in the resonance regime. (a) N = 10, (b) N = 20, (c) N = 50, (d) N = 100.

The revival periods of oscillation in the coherent state may be determined as 4π/g₁. Also, we found that increasing the interatoms interaction strength in the coherent state case leads to an increment in the number of revivals in a fixed interval of time, but does not significantly affect the amplitude of API (Fig. 5). It should be pointed out that the changes in the number of revivals occur due to the interatom interactions in the considered system (similar to the number state case). Comparing the numerical results for API in the number and coherent state as initial states of BEC shows that: (i) Similarities: in both cases the same behavior can be seen by changing the number of atoms in BEC and atomic coupling strength. The amplitude of API decreases in the two considered cases due to damping effect. (ii) Differences: as mentioned earlier the revival periods of the oscillations in the coherent state is two times of this parameter for the number state, which can be seen in Figs. 1 and 4 (see also Figs. 3 and 5). Therefore, the number of revivals in a fixed interval of time with the same selected initial values is larger in the number state comparing with coherent state case. In the weak nonlinearity condition, that is, Ng₂ ≪ G, the effect of g₂ can be neglected. Therefore, from Eqs. (20) and (23), the oscillation period of API in both number and coherent states can be calculated as π/2G, which is the period of the oscillatory term of API (cos([2G+g₂(N−1)]t)), when g₂(N − 1) is very small (weak nonlinearity regime). Therefore, in both cases the oscillation period can be adjusted by tuning the number of atoms in the BEC as well as the atom-field coupling constant .

Fig. 5

	Figure Option ViewDownloadNew Window
	Fig. 5 The effect of interatoms interaction on the time evolution of API for N = 20, γ1 = γ2 = 0.01G and g2 = 0.5 G in the resonance regime. (a) g1 = 0.25G, (b) g1 = 0.5G, (c) g1 = 0.75G, (d) g1 = G.

The observation of CRs in nonlinear systems such as the nonlinear directional coupler,^[35] the relative phase between two superfluids or superconductors,^[36] exciton-exciton interaction in a semiconductor microcavity,^[32] and the population imbalance of a two-mode BEC^[37–40] have been investigated. In this line of research we also showed that the nonlinear interactions in the our presented model can lead to CR phenomenon.

Summing up, we outlined a theoretical model for the investigation of quantum dynamic of BEC interacting with a single-mode quantized field in the presence of interatom collisions and higher order atom-field interaction terms. In this line and to achieve the purpose of paper, at first, via solving the equation of motion the transformed operators of system have been analytically obtained. In the continuation, the API () have been evaluated when two different initial states of the BEC system are considered (number state and coherent state). Our numerical results show that this quantity is very sensitive to the variation of number of atoms in BEC and interatom coupling constant, while the higher-order atom-field interaction does not significantly affect the API. In addition, the collapse-revival is an observable quantum phenomenon in the API. In this regard, it may be expected that with the help of our theoretical model and the appearance of collapses-revival phenomenon in the API, the quantum states of atoms in the BEC and photons in the cavity can be measured in practical experiments.

We briefly present our concluding remarks in what follows: (i)

Number of atoms: by fixing all parameters except the number of atoms in BEC, it is found that increasing the number of BEC atoms leads to the occurrence of more revivals in a definite interval of time, so apparently the intervals of collapse time are considerably decreased.