Under the classical (strong) pump condition, the Hamiltonian involving the signal and the idler modes of a non-degenerate parametric oscillator is exhibited. Without using the usual rotating wave approximation (RWA), the analytical solutions of the field operators are used to investigate the antibunching of photons of the input radiation field coupled to the non-degenerate parametric oscillator. By using the symbolic calculation, the antibunching of photons for both the signal and idler modes are investigated. In particular, the effects of the inclusion of rotating wave approximated terms on the antibunching of photons are clearly indicated. To substantiate the analytical results, the temporal evolution of signal photon and idler photons, and the antibunching effects of the signal and idler modes are investigated numerically by using the QuTip 3.1.0. The exact numerical results obtained by QuTip 3.1.0 matches extremely well with those of the analytical results. The present article and hence the analytical method might be of use for investigating the situations of having ultra-strongly and deep-strongly coupled systems where the possibilities of using RWA is completely ruled out.
Under the classical (strong) pump condition, the Hamiltonian involving the signal and the idler modes of a non-degenerate parametric oscillator is exhibited. Without using the usual rotating wave approximation (RWA), the analytical solutions of the field operators are used to investigate the antibunching of photons of the input radiation field coupled to the non-degenerate parametric oscillator. By using the symbolic calculation, the antibunching of photons for both the signal and idler modes are investigated. In particular, the effects of the inclusion of rotating wave approximated terms on the antibunching of photons are clearly indicated. To substantiate the analytical results, the temporal evolution of signal photon and idler photons, and the antibunching effects of the signal and idler modes are investigated numerically by using the QuTip 3.1.0. The exact numerical results obtained by QuTip 3.1.0 matches extremely well with those of the analytical results. The present article and hence the analytical method might be of use for investigating the situations of having ultra-strongly and deep-strongly coupled systems where the possibilities of using RWA is completely ruled out.
Our purpose in this paper is to investigate the response of a well-known nonlinear quantum optical medium, i.e. non-degenerate parametric amplifier (NDPA), to the pair coherent and two-mode squeezed states, distinctly, both analytically and numerically. To achieve the aim, we start from the NDPA quantum Hamiltonian in the rotating wave approximation (RWA) and via solving the Heisenberg equations of motion of signal and idler modes of the dynamical system is derived, by which we can achieve the time evolution of the NDPA response to the above-mentioned input field states. Accordingly, we are able to evaluate two important nonclassical properties, i.e. photonic entanglement and photon statistics (second-order correlation function and antibunching effect) after the interaction with NDPA for the output emitted fields, and compare the results with the associated quantities for the input field states. Summing up, our numerical simulations show that the amplification (and in some cases the appearance) of the considered nonclassical properties of the radiation field in the output states is clearly confirmed. The results can be beneficial for various quantum science and technology protocols, i.e. quantum communication, quantum metrology, quantum teleportation and quantum computing.
We consider two coupled quantum harmonic oscillators with different free frequencies. Here the interaction between the two modes does not involve the Rotating Wave Approximation (RWA). The Heisenberg equations of motion are solved analytically using an approximate technique and the solutions are used to measure nonclassicalities associated with the system. The nonclassicality criteria chosen in this study are experimentally measurable. The analytical solutions are matched with numerical simulations with the help of a numerical toolbox. It is evident from the time developments of the operators that nonclasities, namely squeezing and quantum entanglement are present in the system. The counting statistics for the oscillator with lower free energy and the coupled mode are shown to be sub-Poissonian for a particular set of parameters. Consequently, the system may exhibit quantum mechanical antibunching.
With the help of second-order nonlinear interactions and hence Hamiltonian, we construct the equations of motion corresponding to pump and signal (second harmonic) fields in a second harmonic generation. The corresponding coupled differential equations involving the non-commuting field operators are not solvable in closed analytical forms. With the help of a perturbation method, we obtain analytical solutions of these field operators up to cubic orders in the interaction constant. In an appropriate limit (truncating the solution up to the second order in the interaction constant), these solutions lead to the existing solutions under the short-time approximation. The present analytical solutions are exploited to investigate the squeezing and the antibunching of photons of the input coherent light coupled to the said second-order nonlinear medium. We report the squeezing and antibunching effects of the pump field even for solutions up to the second order and hence the present results are consistent with earlier results. However, for the second harmonic mode, we report the squeezing involving the leading cubic term in the interaction constant.
Recalling that the rotating wave approximation (RWA) is only valid in the weak coupling regimes, the purpose of this paper is to study the Hamiltonian dynamics describing the full quantum mechanical approach of the interaction between various configurations of three-level atoms Lambda, V and ladder distinctly with a two-mode radiation field, while the RWA is not considered; the counter-rotating terms (CRTs) are taken into account. Generally, the presence of CRTs in the Hamiltonian prevents one to achieve an analytical solution. Moreover, as we will show in the present work, using the perturbation theory, analytical solvable Hamiltonians can be successfully obtained. According to our calculations, the contribution of CRTs within the ordinary Hamiltonian is equivalent to the replacement of the constant detuning with a specific intensity dependent detuning in the first order, and the constant atom field coupling with a particular intensity dependent (f deformed) coupling in the second order of the associated perturbation parameter. Moreover, noticing that according to the initial expression of the Hamiltonian, each mode of the field interacts only with a specific pair of the allowed transitions of each type of the three-level atom, it is surprisingly observed that via applying the mentioned approach, the obtained intensity dependent coupling functions depend on both modes of the field, f(na,nb). In this way, it is seen that the CRTs are removed with the price of arriving at some intermixed intensity dependent atom field coupling functions of the two modes of the field. In this way, the obtained final Hamiltonians are analytically solvable. At last, by determining the time evolution of the atom field wave function, we study the effects of CRTs on a few nonclassical properties of the state of the system, including the atomic population inversion and photon statistics.
An initially prepared coherent state coupled to a second-order nonlinear medium is responsible for stimulated and spontaneous hyper Raman processes. By using an intuitive approach based on perturbation theory, the Hamiltonian corresponding to the hyper Raman processes is analytically solved to obtain the temporal development of the field operators. It is true that these analytical solutions are valid for small coupling constants. However, the interesting part is that these solutions are valid for reasonably large time. Hence, the present analytical solutions are quite general and are fresh compared to those solutions under short-time approximations. By exploiting the analytical solutions of field operators for various modes, we investigate the squeezing, photon antibunching and nonclassical photon statistics for pure modes of the input coherent light responsible for hyper Raman processes. At least in one instance (stimulated hyper Raman processes for vibration phonon mode), we report the simultaneous appearance of classical (photon bunching) and nonclassical (squeezing) effects of the radiation field responsible for hyper Raman processes.
Abstract We apply a simple analytical method based on a unitary transformation to calculate the ground state, its excitation spectrum and quantum dynamic evolution of physical quantities for the double-photon quantum Rabi Hamiltonian over the wide coupling-strength range. The concise analytical method possesses the same mathematical simplicity as the approach of the rotating wave approximation (RWA). By quantitative comparison with the numerically exact result obtained by matrix diagonalization, we confirm that our calculated results obtained by transformed rotating-wave method are not only accurate in the weak coupling regime but also correct in intermediate strong-coupling case. In the intermediate ultrastrong-coupling regime, the calculated values of the ground state and lower lying excited states are nearly the same as the exact ones. It turns out that our calculation for the energy spectrum is beyond the ordinary-RWA. Meanwhile, we demonstrate the signatures resulting from the counter-rotating wave terms by monitoring the population, the coherence, the squeezing of the photon under the ultra-strong conditions. In particular, we find that when the frequency of the photon is much larger than the transition frequency of the system, the lineshape of the time evolution becomes complicated with the increase of the coupling strength, which may be verified experimentally.