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American Journal of Physics

The forced harmonic oscillator: Coherent states and the RWA

Luis Octavio Castaños-Cervantes, Arturo Zúñiga-Segundo

Published Sep 18, 2019• 15 citations
Publisher

Abstract

We consider a forced harmonic oscillator in one-dimension. Using coherent states, we show that the treatment of the system is simplified, that the relationship between the classical and quantum solutions becomes transparent, and that the evolution operator of the system can be calculated easily as the free evolution operator of the harmonic oscillator followed by a displacement operator that depends on the classical solution. In addition, we consider the system in the rotating-wave-approximation (RWA), an application of the Averaging Theorem. We determine the relationship between the exact solution and the one in the RWA, test the accuracy of the RWA, and explain why the RWA gives accurate results in the realm of cavity quantum electrodynamics. Finally, we apply the results to a charged particle interacting with an electromagnetic field.

DOI
10.1119/1.5115395

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We consider a forced harmonic oscillator in one-dimension. Using coherent states, we show that the treatment of the system is simplified, that the relationship between the classical and quantum solutions becomes transparent, and that the evolution operator of the system can be calculated easily as the free evolution operator of the harmonic oscillator followed by a displacement operator that depends on the classical solution. In addition, we consider the system in the rotating-wave-approximation (RWA), an application of the Averaging Theorem. We determine the relationship between the exact solution and the one in the RWA, test the accuracy of the RWA, and explain why the RWA gives accurate results in the realm of cavity quantum electrodynamics. Finally, we apply the results to a charged particle interacting with an electromagnetic field.
The interaction of an atomic system with a single mode of a quantized electromagnetic field is investigated in the dipole approximation. Two approximations that are almost always made in this analysis are the neglect of the diamagnetic term in the Hamiltonian and the dropping of the rotating-wave approximation (RWA) terms that correspond to processes that do not conserve energy. Both approximations involve the neglect of terms that are of the same order of magnitude so that consistency demands that they be considered together. It is demonstrated that, in the dipole approximation, a minor modification of the usual application of quantum electrodynamics to any atomic system can exactly incorporate the effects of the diamagnetic term into a new frequency of the electromagnetic cavity mode; this frequency is slightly different from that of the empty cavity. This result is then applied to the specific case of a charged harmonic oscillator interacting with a single mode. For this simple model an analytic solution can then be found that includes the effects of the diamagnetic term and does not use the RWA. It is found that, for off resonance, the normal modes of the coupled system oscillate at shifted frequencies which are in agreement with the second-order perturbation-theory calculation of the Lamb shift. When the cavity mode is tuned to the atomic oscillator frequency, the perturbation-theory Lamb-shift calculation is no longer valid, and the exact solution derived in this paper predicts a resonant Lamb shift that is significantly larger than the off-resonant single-mode Lamb shift. Furthermore, the frequency spectrum is qualitatively changed from the single frequency of the free atomic system to a doublet spectrum. An analytic solution of this harmonic-oscillator problem can also be found when the RWA terms are dropped from the Hamiltonian. Comparison of the two solutions reveals that the RWA terms contribute a frequency shift of magnitude independent of the field intensity in the simple case of a charged harmonic oscillator.When trajectories of the exact and RWA solutions are compared in phase space, it is found that one effect of the RWA terms is to lift a twofold degeneracy of the phase-space orbits that occurs when the RWA terms are neglected. The harmonic-oscillator problem has several features in common with the interaction of a two-level atom with a single quantized field mode. Both Hamiltonians exhibit inversion symmetry and hence have parity as a good quantum number. Both Hamiltonians contain RWA terms. When these RWA terms are neglected, both Hamiltonians have a new constant of motion that can be identified with the total number of excitations in the combined system.
Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. We have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution $P(\ensuremath{\alpha}, t)$ of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition amplitudes between number states.
The aim of this paper is to investigate dynamical properties of a harmonic oscillator coupled to a non-equilibrium bath without the rotating wave approximation. At first, we investigate the theory of this model and obtain the Heisenberg equations of motion for the oscillator and bath annihilation operators in the absence and in the presence of a rotating wave approximation. Then, we obtain the fluctuation and fluctuation-dissipation relations in the absence and in the presence of a rotating wave approximation for the non-equilibrium bath annihilation operator, respectively. After that, we study the decay of system modes using the Wigner-Weisskopf approximation. Finally, we calculate the time-dependent spectrum of a cavity field mode coupled to a non-equilibrium reservoir. We show that even under the condition in which the RWA is considered to be valid (weak interaction), there are significant effects of virtual-photon field on the diffusion coefficient, the energy density, line width and the shift frequency.
The system of one harmonic oscillator interacting with a continuum is studied. The problem is solved for two cases: the exact (full) Hamiltonian and the truncated Hamiltonian [the rotating-wave approximation (RWA)]. Comparing these two solutions, we formulate the conditions of the applicability of the RWA. The case in which the solution can be treated as Markovian is also discussed. The conditions for the existence of discrete levels are obtained. It was shown that under certain conditions, nonoscillating terms in the solution, leading to the instability of the whole system, can exist. Such terms cannot be obtained within either the Markovian or the rotating-wave approximation. Two particular cases of coupling are proposed and the results of numerical simulations are discussed.
One of the cornerstones of many areas of physics (such as quantum optics, laser theory, and cavity QED) is the rotating-wave approximation (RWA) to the Hamiltonian for a dipole-oscillator coupled to the radiation field. Here it is shown that within the RWA one can obtain explicitly an exact expression for the autocorrelation of the force exerted by the field on the oscillator. This exact result differs from that appearing in much of the standard literature in that the spectrum is not that of white noise and there is no nontrivial dependence on the oscillator frequency.
The interaction of qubits with quantized modes of electromagnetic fields has been largely addressed in the quantum optics literature under the rotating wave approximation (RWA), where rapid oscillating terms in the qubit-mode interaction picture Hamiltonian can be neglected. At the same time, it is generally accepted that, provided the interaction is sufficiently strong or for long times, the RWA tends to describe physical phenomena incorrectly. In this work, we extend the investigation of the validity of the RWA to a more involved setup where two qubit-mode subsystems are brought to interaction through their harmonic coordinates. Our treatment is all analytic thanks to a sequence of carefully chosen unitary transformations, which allows us to diagonalize the Hamiltonian within and without the RWA. By also considering qubit dephasing, we find that the purity of the two-qubit state presents non-Markovian features which become more pronounced as the coupling between the modes gets stronger and the RWA loses its validity. In the same regime, there occurs fast generation of entanglement between the qubits, which is also not correctly described under the RWA. The setup and results presented here clearly show the limitations of the RWA in a scenario amenable to exact description and free from numerical uncertainties. Consequently, it may be of interest for the community working with cavity or circuit quantum electrodynamic systems in the strong coupling regime.