Dynamics of a two-level system coupled to a quantum oscillator: transformed rotating-wave approximation

For studying the dynamics of a two-level system coupled to a quantum oscillator we have presented an analytical approach, the transformed rotating-wave approximation, which takes into account the effect of the counter-rotating terms but still keeps the simple mathematical structure of the ordinary rotating-wave approximation. We have calculated the energy levels of ground and lower-lying excited states, as well as the time-dependent quantum dynamics. It is obvious that the approach is quite simple and can be easily extended to more complicated situation. Besides, the results are compared with the numerically exact ones to show that for weak to intermediate coupling and moderate detuning our analytic calculations are quantitatively in good agreement with the exact ones.     

Effective Hamiltonian of the Jaynes–Cummings model beyond rotating-wave approximation

The Jaynes–Cummings model with or without rotating-wave approximation plays a major role to study the interaction between atom and light. We investigate the Jaynes–Cummings model beyond the rotating-wave approximation. Treating the counter-rotating terms as periodic drivings, we solve the model in the extended Floquet space. It is found that the full energy spectrum folded in the quasi-energy bands can be described by an effective Hamiltonian derived in the highfrequency regime. In contrast to the Z_2 symmetry of the original model, the effective Hamiltonian bears an enlarged U(1)symmetry with a unique photon-dependent atom-light detuning and coupling strength. We further analyze the energy spectrum, eigenstate fidelity and mean photon number of the resultant polaritons, which are shown to be in accordance with the numerical simulations in the extended Floquet space up to an ultra-strong coupling regime and are not altered significantly for a finite atom-light detuning. Our results suggest that the effective model provides a good starting point to investigate the rich physics brought by counter-rotating terms in the frame of Floquet theory.     

Geometric phases in qubit-oscillator system beyond conventional rotating-wave approximation

In this work we investigated the geometric phases of a qubit-oscillator system beyond the conventional rotating- wave approximation. We find that in the limiting of weak coupling the results coincide with that obtained under rotating-wave approximation while there exists an increasing difference with the increase of coupling constant. It was shown that the geometric phase is symmetric with respect to the sign of the detuning of the quantized field from the one-photon resonance under the conventional rotating-wave approximation while a red-blue detuning asymmetry occurs beyond the conventional rotating-wave approximation.     

Spontaneous emission of an excited two-level atom without both rotating-wave and Markovian approximation

The spontaneous emission of an excited atom is analyzed by quantum stochastic trajectory approach without both rotating-wave approximation and Markovian approximation. The atom finite size effect is also taken into account. We show by an example that the correction due to the counter-rotating wave term is rather small, even for the largest atomic number of real nuclei. Received 10 July 2002 / Received in final form 12 November 2002 Published online 4 February 2003     

Entanglement evolution of a two-qubit system with decay beyond the rotating-wave approximation

The entanglement property of two identical atoms, initially entangled in Bell states, coupled to a single-mode cavity is considered. Based on the reduced non-perturbative quantum master equation method, the entanglement evolution of the two atoms with decay is investigated beyond the conventional rotating-wave approximation. We show that the counter-rotating wave terms, usually neglected, have a great influence on the disentanglement behaviour of the system. The phenomena of entanglement sudden death and entanglement sudden birth will occur. In addition, we show that the entanglement can be strengthened by introducing the dipole--dipole interaction of the two atoms.     

Rabi-oscillations without rotating-wave approximation

We consider the interaction of an ensemble of two level atoms with linearly polarized light. This problem is solved in terms of the well-known Bloch equations for various amplitudes of the incident light field without rotating-wave approximation (RWA). Closed-form solutions are obtained for small and very high electric field amplitudes. In the latter case severe deviations from the results obtained with RWA are observed.     

Interactions of a two-level atom and a field with a time-varying frequency without rotating-wave approximation

The interactions between a two-level atom and a field with a time-varying frequency without rotating-wave approximation have been investigated. The frequency of the field varies in the form of rectangle. The dynamic behaviors of the atomic population inversion, photon anti-bunching and atomic dipole squeezing are investigated numerically as function of time. A number of novel phenomena are discovered and discussed.     

Spontaneous emission of a two-level system and the influence of the rotating-wave approximation on the final state. I

By using a modified Robertson projection technique an exact equation of motion for the expectation value of the population inversion operatorS ^z of a single two-level atom in the case of spontaneous emission is derived. Afterwards, by making the Markov approximation, it is shown that the ground state expectation value〈S ^z 〉 _t =? 1/2 fort→∞ will be reached only if the rotating-wave approximation or the Born approximation is made additionally.     

Generalized rotating-wave approximation for arbitrarily large coupling

A generalized version of the rotating-wave approximation for the single-mode spin-boson Hamiltonian is presented. It is shown that performing a simple change of basis prior to eliminating the off-resonant terms results in a significantly more accurate expression for the energy levels of the system. The generalized approximation works for all values of the coupling strength and for a wide range of detuning values, and may find applications in solid-state experiments.     

Dynamics of electromagnetic field in two-level doped systems

The dynamics of solitons of self-inducted transparency is theoretically and numerically investigated in the case when atoms described in the two-level approximation fit into a medium with ferroelectric properties. The interaction between electromagnetic pulses in the medium and two-level dopant atoms is considered.     

Dynamics of pulsed saturation in inhomogeneously broadened two-level systems

On the basis of Bloch's equations an analytical expression is obtained for the pulsed saturation signal of a two-level quantum system with allowance for inhomogeneous line broadening. It is shown that in this case the signal drop is exponential over the initial portion and nonexponential for long times. It is found that starting from a certain time, which depends on the field magnitude, relaxation parameters, and inhomogeneous width of the line, a through appears in the line contour, the depth and width of which increases with an increase in pulse duration. At smaller durations of the pulse, when the line is bell-shaped, an exponential decay of the signal indicates saturation of homogeneously broadened line. Comparison of the results obtained in an experiment on pulsed saturation in ruby shows that at large amplitudes of the field and considerable inhomogeneous broadening, one observes a deviation from the Bloch theory manifesting itself as a biexponential drop in the pulsed saturation signal. A conclusion is drawn that the Bloch theory satisfactorily describes the experiment until the effective time of energy transfer to the lattice is smaller than the time of the spectral propagation of excitation. Institute of Solid-State and Semiconductor Physics, Academy of Sciences of Belarus, 17, P. Brovka St., Minsk, 220072, Belarus. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 64, No. 2, pp. 197–203, March–April, 1997.     

Dynamics of a coupled pair of two-level tunnelling systems

An exact solution for the dynamics of a coupled pair of symmetric two-level-systems is given by calculating the resolvent of the Liouvillian and the statistical operator of the problem. At low temperatures interference effects between the two systems turn out to be of major importance. Depending on the value and the sign of the interaction parameter the susceptibility of the pair increases or decreases compared to the situation of isolated systems. For high temperatures the interference contributions disappear.     

Dynamics of strongly driven two-level systems: analytical solutions

The dynamics of a two-level system, strongly driven by an external sinusoidal field, is studied using a systematic iteration procedure. Analytical solutions are presented for the quasi-eigenenergies and Floquet states. They are valid for arbitrary strengths of the driving field and thus encompass the existing low- and high-field solutions. The higher-order corrections to the Floquet states are essential for properly describing the dynamics on the time scale of the driving field (as opposed to the time scale associated with the tunnel splitting), which is relevant for harmonic-frequency generation. The convergence of the procedure is good for driving frequencies larger than half the tunnel splitting.     

Analytical approach to dynamics of transformed rotating-wave approximation with dephasing

We study the dynamics of the Jaynes-Cummings model within transformed rotating-wave approximation （TRWA）. We analyze this model coupled to a dephasing reservoir, through the Lindblad formalism in the master equation. Then, we examine the expectation value of the number operator. Finally, we investigate the validity of this model under dephasing using the Mandel parameter and the total number of quanta.     

Energy spectrum of the Hamiltonian of the Jaynes-Cummings model without rotating-wave approximation

The energy spectrum of the Hamiltonian of the Jaynes-Cummings model without a rotating-wave approximation is studied. The trajectories of eigenvalues as functions of the dimensionless coupling constant are constructed. It is shown that the spectrum approaches the equidistant spectrum, i.e., the oscillator spectrum, as the coupling constant increases. As a result, each energy level becomes doubly degenerate.     

Chaotic dynamics in a nonautonomous Dicke model without the rotating-wave approximation

The properties of a system consisting of two-level atoms interacting with a mode of the electromagnetic field in a cavity are considered for the case when the cavity detuning or the coefficient of the atom-field interaction is modulated. The consideration is performed with account taken of the Hamiltonian terms that are neglected in the rotating-wave approximation. It is shown that in the semiclassical equations for such a model, the effect of extension (compared to the autonomous system) of the range of variation of the quantity characterizing the number of photons can manifest itself; in this case, the energy oscillations have a chaotic character. The dependence of this phenomenon on parameters characterizing the model is studied. It is numerically demonstrated that with account taken for the relaxation, the system studied can have attractors different from the equilibrium positions, i.e., the number of photons in the mode does not tend to a constant value. The limits of validity of the rotating-wave approximation in the parametrically perturbed Dicke model are discussed.     

A modern review of the two-level approximation

The paradigm of the two-level atom is revisited and its perturbative analysis is discussed in view of the principle of duality in perturbation theory. The models we consider are a two-level atom and an ensemble of two-level atoms both interacting with a single radiation mode. The aim is to see how the latter can be actually used as an amplifier of quantum fluctuations to the classical level through the thermodynamic limit of a very large ensemble of two-level atoms [M. Frasca, Phys. Lett. A 283 (2001) 271] and how can remove Schrödinger cat states. The thermodynamic limit can be very effective for producing both classical states and decoherence on a quantum system that evolves without dissipation. Decoherence without dissipation is indeed an effect of a single two-level atom interacting with an ensemble of two-level atoms, a situation that proves to be useful to understand recent experiments on nanoscale devices showing unexpected disappearance of quantum coherence at very low temperatures.     

Squeezing and rabi oscillations in the Dicke model within and without rotating-wave approximation

The Dicke model is investigated within and without rotating-wave approximation in the case of a lossless resonant cavity. Numerical results for the time evolution of the atomic population inversion, dipole moment and atomic and field squeezing parameters for an initial field are presented for different numbers of atoms. The significant influence of the counter-rotating terms on the squeezing and collapse (revival) phenomena is pointed out.