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Research in Optical Sciences

Study of the Temporal-Evolution of a Star-like Quantum State of Light Through the Wigner Function

Juan Camilo López Carreño, J. P. Restrepo Cuartas, Herbert Vinck-Posada

Published Jan 1, 2012• 0 citations
Publisher

Abstract

In this work the time evolution of a star-like state was studied. The state gets his name after the shape of it’s corresponding Wigner function. The physical situation considered is a confined atom inside a microcavity, whose boundaries are made of Bragg reflectors. After the atom is confined, a beam made of a Star state of light incides against the microcavity. The Bragg reflectors are made to let the radiation go inside the microvacity an to keep it inside the microcavity, in order to let it interact with the atom. The matter-radiation interaction was described using the Jaynes-Cummings model.The time evolution was made into two different parts. In the first one, the evolution we studied was purely hamiltonian; that is, there was no disipation terms. Once we did such evolution we saw that the Wigner function rotated and that furthermore, the rotation had a given frecuency. In order to fully characterize these behavior, an analitical expression for the rotation period was found. It was also seen that, while the Wigner function was rotating, it also varied between the star state and the vacuum state corresponding Wigner functions.When the dissipative terms were considered into the time evolution of the state, it was clear that the Wigner function does not rotate in the same way as when there were no dissipative terms. Moreover it was seen, that there were several time intervals in which the Wigner function did not rotate at all and that these behavior usually took place after sudden and fast oscilations. The latest oscilations meant that there were happening some phase swifts, in the sense that there were different physical situations taking place. In order to determine which states were involved in such phase swifts, a tomography of the density matrix corresponding to the time evolution was made. At last it was studied, if there were a set of dissipative parameters such that the stationary state reached by the time evolution coincided with the initial star state. To verify how similar those two states were, the fidelity parameter was used.In both cases entanglement measurements were used to characterized the states in which the system went throught during the time evolution.

DOI
10.1364/hilas.2012.jt2a.16

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In this work the time evolution of a star-like state was studied. The state gets his name after the shape of it’s corresponding Wigner function. The physical situation considered is a confined atom inside a microcavity, whose boundaries are made of Bragg reflectors. After the atom is confined, a beam made of a Star state of light incides against the microcavity. The Bragg reflectors are made to let the radiation go inside the microvacity an to keep it inside the microcavity, in order to let it interact with the atom. The matter-radiation interaction was described using the Jaynes-Cummings model.The time evolution was made into two different parts. In the first one, the evolution we studied was purely hamiltonian; that is, there was no disipation terms. Once we did such evolution we saw that the Wigner function rotated and that furthermore, the rotation had a given frecuency. In order to fully characterize these behavior, an analitical expression for the rotation period was found. It was also seen that, while the Wigner function was rotating, it also varied between the star state and the vacuum state corresponding Wigner functions.When the dissipative terms were considered into the time evolution of the state, it was clear that the Wigner function does not rotate in the same way as when there were no dissipative terms. Moreover it was seen, that there were several time intervals in which the Wigner function did not rotate at all and that these behavior usually took place after sudden and fast oscilations. The latest oscilations meant that there were happening some phase swifts, in the sense that there were different physical situations taking place. In order to determine which states were involved in such phase swifts, a tomography of the density matrix corresponding to the time evolution was made. At last it was studied, if there were a set of dissipative parameters such that the stationary state reached by the time evolution coincided with the initial star state. To verify how similar those two states were, the fidelity parameter was used.In both cases entanglement measurements were used to characterized the states in which the system went throught during the time evolution.
We found analytically the exact joint Wigner function (WF) of an atom-field system described by the Jaynes-Cummings model in the dispersive limit for a given initial state. Using the solution of a master equation with a dissipative term calculated by Peixoto de Faria and Nemes (Peixoto de Faria J G and Nemes M C 1999 Phys. Rev. A 59 3918) and a general expression for the joint WF proposed by Czirják and Benedict (Czirják A and Benedict M G 1996 Quantum Semiclass. Opt. 8 975), we obtained the exact reduced atom and field WFs and analysed their time evolution. Our results show that, at the maximum entanglement times, the atom tends to a statistical mixture state represented by a spherical polar plot. Also, the atomic WF shows how fast the negative-valued part disappears as a function of dissipation constant and the strength of the field coherent state.
ABSTRACT: The processes of quantum decoherence in QED cavities have been extensively studied in the last decade, both experimentally by Haroche, Raimond et. al., as theoretically by Nemes, Davidovich et. to the. We present here the dynamics of the Wigner function for a system consisting of a two-level atom and a cavity in a dissipative environment, using a numerical integration scheme that allows us to consider initial Squeezed and Schrodinger's Cat States. The interaction between the atom and the single-mode field is considered by a dispersive model, that is, the interaction in the Hamiltonian of Jaynes-Cummings appears as a small disturbance. The interest is focused both on the quantum characteristics of the dispersive approach (preparation of mesoscopic states) and on the aspects that characterize dynamics and dissipation. In particular, the gradual loss of quantum coherence is observed through the use of the Wigner function, where the disappearance of the interference regions of the initial quantum states is evidenced.
Based on the Weyl expansion representation of Wigner operator and its invariant property under similar transformation, we derived the relationship between input state and output state after a unitary transformation including Wigner function and density operator. It is shown that they can be related by a transformation matrix corresponding to the unitary evolution. In addition, for any density operator going through a dissipative channel, the evolution formula of the Wigner function is also derived. As applications, we considered further the two-mode squeezed vacuum as inputs, and obtained the resulted Wigner function and density operator within normal ordering form. Our method is clear and concise, and can be easily extended to deal with other problems involved in quantum metrology, steering, and quantum information with continuous variable.
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Cavity quantum electrodynamics (QED) has become one of themost successful tools for quantum information processing, butalso for the study of other pure quantum effects. In thisthesis we investigate one such system, the micromaser; an atomtraverse a cavity and interacts with the quantized modes of thecavity field. Today, the relaxation times of the atom and thecavity field can be made orders of magnitude longer than singleinteraction times, which makes it possible to carry out severaloperations on the system before decoherence becomessignificant. One can often, to a good approximation, treat themicromaser problem by considering the atom as a two-levelsystem interacting with just one of the modes, and losses ofthe combined system can be neglected during the interaction.Any rotations of the atomic Bloch vector during the interactionare, in principle, achievable with modern laser techniques. Forexample, the atom can be Stark shifted in a controlled fashionwith external lasers while it passes through the cavity.The micromaser is theoretically well described by the famousJaynes-Cummings model. Here we look into the problem arisingwhen the parameters of the Jaynes-Cummings model, which are theatom-field coupling and detuning, are allowed to vary in time.What possibilities may occur by letting the parameters dependon time? This question is fairly unanalyzed in the literatureand we are especially interested in preparing variousnon-classical states of the field. The states of the atoms aremeasured after the interactions with thefield and we considerboth the cases when the results of the detections are collectedor not. It is shown that, both in the non-selective andselective measurement procedure, any Fock state can begenerated. We also discuss how one can heat up or cool down thefield and how to prepare entangled states including more thanone mode. The photon field statistics are investigated when theparameters becomes time dependent and field losses areconsidered.
We consider the time evolution of the radiation field (R) and a two-level atom (A) in a resonant microcavity in terms of the Jaynes-Cummings model with an initial general pure quantum state for the radiation field. It is then shown, using the Cauchy-Schwarz inequality and also a Poisson resummation technique, that perfect coherence of the atom can in general never be achieved. The atom and the radiation field are, however, to a good approximation in a pure state ∥ψ〉A⊗R=∥ψ〉A×∥ψ〉R in the middle of what has been traditionally called the ‘collapse region’, independent of the initial state of the atoms, provided that the initial pure state of the radiation field has a photon number probability distribution which is sufficiently peaked and phase differences that do not vary significantly around this peak. An approximate analytic expression for the quantity Tr[ρ2 A(t)], where ρA (t) is the reduced density matrix for the atom, is derived. We also show that under quite general circumstances an initial entangled pure state will be disentangled to the pure state ∥ψ〉PSA⊗R.