We reconsider the problem of a two-level system interacting with a radiation field in order to study some new features suggested by the actual experimental conditions. Pulse shape and duration effects are included in the formalism and the counter-rotating terms are retained. The criterion of validity of the rotating wave approximation (RWA) for pulsed fields is investigated; generalizing results well known in RWA, we establish some new formal results, including non-RWA contributions to all orders and for any pulse shape. The analysis is then carried out for fluctuating fields, by developing a method based on the theory of multiplicative stochastic differential equations. For short pulses, the non-Markovian character of the stochastic field parameters is expected to affect the process and, for broadband fields, the RWA has to be checked ab initio. Thus, a formalism is proposed which allows the study of both features. A wide range of numerical calculations are presented, clarifying the role of both intensity fluctuations and spectrum bandwidth in the case of a chaotic field, as well as their interplay with the pulse features.
We reconsider the problem of a two-level system interacting with a radiation field in order to study some new features suggested by the actual experimental conditions. Pulse shape and duration effects are included in the formalism and the counter-rotating terms are retained. The criterion of validity of the rotating wave approximation (RWA) for pulsed fields is investigated; generalizing results well known in RWA, we establish some new formal results, including non-RWA contributions to all orders and for any pulse shape. The analysis is then carried out for fluctuating fields, by developing a method based on the theory of multiplicative stochastic differential equations. For short pulses, the non-Markovian character of the stochastic field parameters is expected to affect the process and, for broadband fields, the RWA has to be checked ab initio. Thus, a formalism is proposed which allows the study of both features. A wide range of numerical calculations are presented, clarifying the role of both intensity fluctuations and spectrum bandwidth in the case of a chaotic field, as well as their interplay with the pulse features.
The exact adiabatic representation scheme of Nguyen-Dang [J. Chem. Phys. 90, 2657 (1989)] is applied to time evolution of the general laser-driven two- and three-level systems. It leads to exact formal results that do not assume a specific shape of the pulse and do not invoke the rotating-wave approximation (RWA). These results are a generalization of the corresponding RWA results because the former depend on an effective pulse area that varies with the detuning and involves a Rabi frequency modulated by a phase cphi(t). In addition, the exact adiabatic representation scheme is applied to N-level systems exhibiting various dynamic symmetries and, as in the previous cases, leads to results that do not invoke the RWA and depend on an effective pulse area. Time-reversal analysis on two- and three-level systems leads to quantization of the effective pulse area for time-symmetric pulses. This quantization leads to localized asymptotic population distributions.
The excitation of a two-level system by a chirped laser pulse is analyzed using an analytical approach (in the perturbation-theory limit) that involves the modified rotating-wave approximation and the numerical solution to the Bloch equation. The dependence of the population of the upper energy level on the chirp is studied for various radiation intensities and pulse durations. An exact solution is compared with the result obtained using the modified rotating-wave approximation and the perturbation theory. It is demonstrated that, for a certain range of parameters, the excitation probability of a two-level system can be effectively controlled via a variation in the chirp.
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.
The interaction is studied of a Gaussian pulsed or a continuous wave (CW) electric field with a molecule that has a nonzero difference d between the permanent dipoles of the ground and excited states involved in a transition. The unusual spectral effects associated with the transition with d ≠ 0, relative to the often studied case with d = 0, have been investigated previously in the two-level approximation and for continuous wave lasers. The observation of these effects, including markedly increased periods of excited state populations and negative Bloch-Siegart shifts, will often require the intense fields associated with pulsed rather than purely continuous wave electric fields. Transitions induced by continuous wave and pulsed laser-‘permanent dipole molecule’ interactions, for high field intensities and varying pulse durations, are investigated with model calculations carried out using exact Floquet and symmetry-adapted Riemann product integral methods. These calculations illustrate how the increased period of the excited state populations for dipolar molecules, and the associated molecule-field couplings, can be probed by sampling the time-dependent populations of the excited states using ultrashort to short pulses. The discussion illustrates the predictive and interpretive importance of the analytical expression for the molecule-EMF coupling provided by the rotating wave approximation (RWA) with d ≠ 0. The examples considered show, for example, temporal beating effects between the field and the RWA or Rabi frequencies and their dependence on d.
We study the stochastic dynamics of a two-level quantum system interacting with a stochastic magnetic field, and a single frequency electromagnetic field, with and without making the rotating wave approximation (RWA). The transformation to the rotating frame does not commute with the stochastic Hamiltonian if the stochastic field has nonvanishing components in the transverse direction. Hence, making the RWA modifies the stochastic terms in the Hamiltonian. Modification of the decay terms is also required in a master equation approach (i.e., the Liouville--von Neumann density matrix equation) for describing the dynamics. For isotropic Gaussian white noise, the RWA dynamics remains Markovian, although the Lindblad terms in the master equation for the density matrix become time-dependent when the non-commutation of the RWA transformation and the noise Hamiltonian is properly accounted for. We also treat Ornstein--Uhlenbeck noise, and find, in contra-distinction to the white noise case, a significant difference in the dynamics calculated with the RWA when the non-commutation of the RWA transformation and the noise Hamiltonian is taken into account. These findings are applicable to the modeling of any open quantum system coupled to an electromagentic field.
It is our aim to study the dynamics of a two-level atom immersed in the modified radiation field of a photonic band-gap material using non-Markovian stochastic Schr\"odinger equations. Up to now, such methodology has only been applied to toy models and not to physically realistic systems as the one presented here. In order to check its validity, we shall study several of the physical phenomena already described in the literature within non-Markovian master equations, such as the long-time-limit residual population in the excited level of the atom and the population inversion which occurs in the atomic system when applying an external laser field. In addition to the stochastic equation, we propose a non-Markovian master equation derived from the stochastic formalism, which in contrast to the current models of master equation preserves positivity. We propose a correlation function for the radiation field (environment) that captures many of the physically relevant aspects of the problem and describes the short-time behavior in a more accurate way than previously proposed ones. This characteristic permits a correct description of the fluctuations of the electromagnetic field, which in the stochastic formalism are represented by the noise, and a better description of the non-Markovian effects in the atomic dynamics. The methodology presented in this paper to apply stochastic Schr\"odinger equations can be followed to study more complex systems, like many-level atoms embedded in more complicated photonic band-gap structures.