Abstract In this paper we present a general theoretical model for the interaction between a number of two-level atoms constituting Bose-Einstein condensate (BEC) and a single-mode quantized field. In addition to the usual interacting terms, we take into account interatom as well as higher-order atom-field interactions. To simplify the Hamiltonian of system, after using the Bogoliubov approximation we proceed to calculate the transformed operators of atoms and field. Then, to quantify the spontaneous emission, we get analytical expressions for the expectation value of J ^ z as the atomic population inversion (API), in the cases of number and coherent states for the atomic subsystem. Our results show that the above-mentioned model interaction leads to the appearance of collapse-revival phenomenon in API. In more detail, the revival time may be tuned by adjusting the interatom interaction constant. Also, the damping process lowers the amplitude of API, but does not change the CR times for weak damping. Moreover, increasing the damping may decrease the number of CRs in a given interval of time such that no revival occurs. Briefly, it may be concluded that in the resonant case the revival times are insensitive to the change of the higher-order atom-field interaction constant and are affected only by the interatom interactions. Finally, we express that, how we can find a practical procedure to measure the quantum states of atoms in BEC .
Abstract In this paper we present a general theoretical model for the interaction between a number of two-level atoms constituting Bose-Einstein condensate (BEC) and a single-mode quantized field. In addition to the usual interacting terms, we take into account interatom as well as higher-order atom-field interactions. To simplify the Hamiltonian of system, after using the Bogoliubov approximation we proceed to calculate the transformed operators of atoms and field. Then, to quantify the spontaneous emission, we get analytical expressions for the expectation value of J ^ z as the atomic population inversion (API), in the cases of number and coherent states for the atomic subsystem. Our results show that the above-mentioned model interaction leads to the appearance of collapse-revival phenomenon in API. In more detail, the revival time may be tuned by adjusting the interatom interaction constant. Also, the damping process lowers the amplitude of API, but does not change the CR times for weak damping. Moreover, increasing the damping may decrease the number of CRs in a given interval of time such that no revival occurs. Briefly, it may be concluded that in the resonant case the revival times are insensitive to the change of the higher-order atom-field interaction constant and are affected only by the interatom interactions. Finally, we express that, how we can find a practical procedure to measure the quantum states of atoms in BEC .
Abstract In this paper we consider a system consisting of a number of two-level atoms in a Bose–Einstein condensate (BEC) and a single-mode quantized field, which interact with each other in the presence of two different damping sources, i.e. cavity and atomic reservoirs. The reservoirs which we consider here are thermal and squeezed vacuum ones corresponding to field and atom modes. Strictly speaking, by considering both types of reservoirs for each of the atom and field modes, we investigate the quantum dynamics of the interacting bosons in the system. Then, via solving the quantum Langevin equations for such a dissipative BEC system, we obtain analytical expressions for the time dependence of atomic population inversion, mean atom as well as photon number and quadrature squeezing in the field and atom modes. Our investigations demonstrate that for modeling the real physical systems, considering the dissipation effects is essential. Also, numerical calculations which are presented show that the atomic population inversion, the mean number of atoms in the BEC and the photons in the cavity possess damped oscillatory behavior due to the presence of reservoirs. In addition, non-classical squeezing effects in the field quadrature can be observed especially when squeezed vacuum reservoirs are taken into account. As an outstanding property of this model, we may refer to the fact that one can extract the atom–field coupling constant from the frequency of oscillations in the mentioned quantities such as atomic population inversion.
In this paper we study the interaction between two two-level atoms with a two-mode quantized field in the presence of damping. Dipole–dipole interaction between the two atoms and the correlation between the two modes of field are also taken into account. To solve the model, using appropriate transformations, we reduce the considered model to a well-known Jaynes–Cummings model. After finding the analytical solution for the atom–field system, the effects of damping, field–field correlation and atomic dipole–dipole interaction on the entanglement between atoms and population inversion are investigated, numerically. It is observed that the dynamical behavior of the degree of entanglement for damped systems, in relatively large domains of time, takes a low but constant value adequately far from the beginning of the interaction. In addition, it is found that the value of population inversion after the initial oscillations takes negative values for damped systems and eventually vanishes by increasing time. Also, it is seen that simultaneous presence of both dipole–dipole interaction and field–field correlation provides typical collapse–revival phenomenon in the time-behavior of atomic inversion.
Recalling that counter rotating terms (CRTs) in atom–field interaction are not always ignorable, particularly in the strong coupling regimes of atom–field systems, in this paper we are going to study the interaction between two qubits (two-level atoms) with a two-mode field in a high quality cavity by considering CRTs while the dipole–dipole (D–D) interaction is also taken into account. Indeed, by performing an appropriate dynamical Hamiltonian associated with the Jaynes–Cummings and anti-Jaynes–Cummings terms, as well as the D–D interaction for the system, we solve the corresponding time-dependent Schrödinger equation with specific initial conditions for atoms and field. In the continuation, concurrence, as an appropriate measure of the degree of entanglement of the two qubits, and atomic population inversion of the qubits, as the rate of energy exchange between atoms and field, are evaluated. Meanwhile, we investigate the influence of CRTs and D–D interaction (as well as other involved parameters in the model interaction) on the mentioned quantities. Our numerical results indicate that collapse-revival phenomenon can be observed in the absence of D–D interaction with or without rotating wave approximation, however, CRTs cause an increase or decrease in the number of collapse-revivals in a finite interval of time, depending on the chosen parameters. In fact, by entering the D–D interaction term, this phenomenon is destroyed and the time behaviour of the quantities experiences irregular oscillations. Noticing that the initial field is considered in the vacuum state, the occurrence of collapse-revival in the population inversion of the two qubits is an interesting feature of this work. As is shown, the degree of entanglement and atomic population inversion of the qubits can generally be tuned, and the death of entanglement can be prevented, both by adjusting the evolved parameters in the model.
In this paper, we investigate the spontaneous emission of an f -deformed Bose–Einstein condensate of a gas with N identical two-level atoms immersed in a single-mode ideal cavity with s atoms initially excited. We apply an f -deformed quantum model in which Gardiner's phonon operators are deformed by an operator-valued function of the particle number operator . We consider the collisions between the atoms as a special kind of f -deformation where the collision rate κ is regarded as a corresponding deformation parameter. The time evolution of the expectation value of the atomic inversion is presented, the phenomenon of collective collapses and revivals is shown and the effects of deformation on the cooperative behaviour of the system are discussed.
A single atom coupled to a quantized mode of an electromagnetic cavity, driven by an external coherent field, is described by the Hamiltonian H djc = ig(a†σ− − aσ+) + iε(a† − a), where (a†, a) and (σ+, σ) are the raising and lowering operators for the cavity mode and atom, respectively. This is the single atom version of the Hamiltonian used in the study of optical bistability. It describes the evolution of the atom-cavity system on time scales much smaller than the inverse spontaneous emission and cavity decay rates. For 2ε/g < 1, HDJC possesses a discrete set of interaction eigenstates1 with eigenvalues E n ± = ± g n [ 1 − ( 2 ε / g ) 2 ] 3 / 4 . These form a renormalized version of the usual Jaynes-Cummings eigenstates2, which correctly take into account the effect of the driving field to all orders of magnitude. We investigate the collapse and revival nature of the dynamical evolution of the single atom in a cavity, with and without dissipation and above and below the threshold 2ε/g = 1 of the driven Jaynes-Cummings Hamiltonian.
In the preceding chapters concerning the interaction of a radiation field with matter, we assumed the field to be classical. In many situations this assumption is valid. There are, however, many instances where a classical field fails to explain experimentally observed results and a quantized description of the field is required. This is, for example, true of spontaneous emission in an atomic system which was described phenomenologically in Chapter 5. For a rigorous treatment of the atomic level decay in free space, we need to consider the interaction of the atom with the vacuum modes of the universe. Even in the simplest system involving the interaction of a single-mode radiation field with a single two-level atom, the predictions for the dynamics of the atom are quite different in the semiclassical theory and the fully quantum theory. In the absence of the decay process, the semiclassical theory predicts Rabi oscillations for the atomic inversion whereas the quantum theory predicts certain collapse and revival phenomena due to the quantum aspects of the field. These interesting quantum field theoretical predictions have been experimentally verified. In this chapter we discuss the interaction of the quantized radiation field with the two-level atomic system described by a Hamiltonian in the dipole and the rotating-wave approximations. For a single-mode field it reduces to a particularly simple form. This is a very interesting Hamiltonian in quantum optics for several reasons. First, it can be solved exactly for arbitrary coupling constants and exhibits some true quantum dynamical effects such as collapse followed by periodic revivals of the atomic inversion.