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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Generation of superpositions of displaced Fock states via the driven Jaynes - Cummings model

Shi-Biao Zheng, Guang‐Can Guo

Published Aug 1, 1996• 12 citations
Publisher

Abstract

We propose a method to produce superpositions of displaced Fock states via the driven Jaynes - Cummings model. We introduce the dressed-atom representation of the driven Jaynes - Cummings model to investigate the evolution of the state of the system. When the cavity field is in an initial coherent state ( is the amplitude of the classical driving field and g is the coupling constant between the atom and the cavity field), the evolution of the state of the system is similar to the vacuum Rabi oscillation of the normal Jaynes - Cummings model. After a period of interaction, the superposition of displaced Fock states of the cavity field can be generated by a state-selective measurement on the atom.

DOI
10.1088/1355-5111/8/4/016

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We propose a method to produce superpositions of displaced Fock states via the driven Jaynes - Cummings model. We introduce the dressed-atom representation of the driven Jaynes - Cummings model to investigate the evolution of the state of the system. When the cavity field is in an initial coherent state ( is the amplitude of the classical driving field and g is the coupling constant between the atom and the cavity field), the evolution of the state of the system is similar to the vacuum Rabi oscillation of the normal Jaynes - Cummings model. After a period of interaction, the superposition of displaced Fock states of the cavity field can be generated by a state-selective measurement on the atom.
We study the dynamical features of the Jaynes–Cummings model with the atom driven by an external classical field in the case that the cavity field is initially in a Fock state.We find the dynamical variables by using dressed states in the bare atom representation. We show that the dynamical behavior of the cavity field can be strongly modified by the external classical field. We also find that the cavity field is super-Poissonian after some interaction time if the external field is strong enough.
The eigenstates of the Jaynes-Cummings Hamiltonian (the dressed atomic states) are widely used to describe the interaction between a single two-level atom and a single mode of the electromagnetic field. They provide a particularly useful basis when the secular approximation is justified. For an atom interacting fog with n photons in a cavity, the secular approximation requires that the n-photon Rabi frequency 2 n g be much larger than the atomic and cavity linewidths. Normally this condition is met when n is large and the coupling constant g is much smaller than the atomic and cavity linewidths. In this paper we allow g to be larger than the linewidths, and we allow the photon number to be small. We show that when the cavity is driven by a coherent field, the usual secular approximation cannot be used. The natural basis states become are now the eigenstates of the Jaynes-Cummings Hamiltonian plus the external field interaction. We solve this eigenvalue problem by separating the field and atomic parts of the Hamiltonian using a displacement and squeeze in the Hilbert-space of the field. We find that the Rabi frequencies are renormalized downwards as the strength of the driving field is increased. They eventually vanish at a critical field strength. At this critical field strength we observe a spontaneous symmetry breaking in the steady-state solution of the master equation for a single atom interacting with a driven cavity mode.
We analyze the time-dependent physical spectrum of a driven Jaynes-Cummings model in which both the two-level system and the quantized cavity mode are subject to coherent classical driving. The time-dependent Hamiltonian is mapped, via well-defined unitary transformations, onto an effective stationary Jaynes-Cummings form. Within this framework, we derive closed-form expressions for the two-time correlation functions of both the atomic and field operators. These correlation functions are subsequently used to evaluate the time-dependent physical spectrum according to the Eberly-Wódkiewicz definition, which properly accounts for finite spectral resolution and transient emission dynamics. We show that the external driving leads to substantial modifications of the atomic spectral response, including controllable frequency shifts and asymmetric line shapes. Importantly, we identify a regime in which the driving parameters are chosen such that the coherent displacement induced in the cavity field exactly cancels out the initial coherent amplitude. In this limit, the system dynamics reduce to that of an effectively vacuum-initialized Jaynes-Cummings model, and the standard vacuum Rabi splitting is recovered. This behavior provides a clear and physically transparent interpretation of the spectral features as arising from coherent field displacement rather than from modifications of the underlying atom-cavity coupling.
We consider the resonant interaction of a cavity mode with a two-level atom that is driven by a coherent field while it crosses the cavity. Starting from the cavity field in a coherent state, we show that the state of the system can reach the maximum entanglement after a unitary evolution for long enough interaction times. Also we illustrate how the generation of cavity field superposition states can allow, in the open system dynamics, the observation of their decoherence in atomic correlation measurements, for any initial cavity field state, and even under the combined effects of dissipation, thermal noise, and atomic pumping.
There have recently been several proposals for generation of optical superposition states in the resonant atom-field interaction and more practically in microwave cavities. In the present paper we study the influence of the vacuum reservoir on properties of the near-superposition state of the cavity field which is described by the Jaynes-Cummings model at one-half of the revival time. Instead of introducing the cavity loss from the first instance of the atom-field interaction we consider the cavity loss only after the near-superposition state is produced and after the atom leaves the cavity. We solve the corresponding master equation with the initial condition being the Jaynes-Cummings field at one-half of the revival time. We find that under the influence of the vacuum reservoir the photon number distribution of the superposition state we study exhibits certain asymmetry around the mean photon induced by the decay process. We show that an analogous effect can be seen when the Jaynes-Cummings superposition state is amplified. For a basic test of our approach we study the dissipation and amplification of Fock states.