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Purification of the atom-field interaction Hamiltonian

Jorge Andres Anaya-Contreras, Arturo Zúñiga-Segundo, H. M. Moya-Cessa

Published Nov 5, 2019• 2 citations
Publisher

Abstract

We study the Jaynes-Cummings model with initial mixed states, for the atom or the field, are considered. The evolved mixed field density matrix is purified to a wavefunction that describes the interaction between a quantised field and an artificial four-level atom. This allows us to use the Araki-Lieb inequality to calculate the field entropy from the atomic entropy. We then suggest Hamiltonians that reproduce the field entropy dynamics. We finally show that more realistic Hamiltonians may be considered by using two entangled two-level atoms.

DOI
10.1016/j.physo.2019.100007

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We study the Jaynes-Cummings model with initial mixed states, for the atom or the field, are considered. The evolved mixed field density matrix is purified to a wavefunction that describes the interaction between a quantised field and an artificial four-level atom. This allows us to use the Araki-Lieb inequality to calculate the field entropy from the atomic entropy. We then suggest Hamiltonians that reproduce the field entropy dynamics. We finally show that more realistic Hamiltonians may be considered by using two entangled two-level atoms.
We investigate some of the fundamental features of the interaction of a mixture of coherent states, namely, a Kaleidoscope states mixture, with two-level atoms in the Jaynes–Cummings model framework. We begin our analysis by calculating the von Neumann entropy of the field, which is determined with the help of the virtual atom method. The oscillations appearing in the entropy indicate a state of purity greater than the initial state, i.e., a purification of the initial state due to a transfer of coherence from the atom to the field. In this oscillatory region, we obtain a negative Wigner function that hints at a (noisy) multiple Schrödinger cat.
We study the dynamics of a two-level atom interacting with a single mode of a damped cavity at 0 K when the cavity is initially in the vacuum state and the atom enters it in an arbitrary (pure or mixed) state. A complete analytical solution of this simple model is presented. On the basis of this solution we firstly investigate the pseudo-spin dynamics of the atom and the cavity field, secondly give an illustration of the Araki-Lieb theorem concerning the von Neumann entropies of interacting quantum systems and thirdly demonstrate the generation of entangled states of the atom and cavity field that are of interest in connection with the Bell inequalities.
The Araki-Lieb inequality is commonly used to calculate the entropy of subsystems when they are initially in pure states, as this forces the entropy of the two subsystems to be equal after the complete system evolves. Then, it is easy to calculate the entropy of a large subsystem by finding the entropy of the small one. To the best of our knowledge, there does not exist a way of calculating the entropy when one of the subsystems is initially in a mixed state. For the case of a two-level atom interacting with a quantized field, we show that it is possible to use the Araki-Lieb inequality and find the von Neumann entropy for the large (infinite) system. We show this in the two-level atom-field interaction.
The interaction between an initially pure two-level atom and the mixed thermal quantized field mode in the framework of the Jaynes?Cummings model is considered with the employment of quantum causal analysis . At the high temperature of the field, a distinction between the resulting properties of initially excited and ground atom states smoothes over and the whole state turns out to be causally asymmetric, entangled and ?classical? in the entropic sense. The average asymptotic entanglement is found. It is revealed that the thermalization acting on the field corresponds to the stronger causality and entanglement decay, rather than the thermalization acting only on the atom.
We considered the anti-Jaynes-Cummings (AJC) interaction of a two-level atom in an initial ground state interacting with a field mode in an initial squeezed coherent state at arbitrary values of squeeze parameter r and provided the Jaynes-Cummings (JC) interaction as a comparison. We analysed the degree of entanglement (DEM) measured by the von Neumann entropy and the nature of the field quantified by the Mandel Q parameter in relation to the atomic population inversion during the AJC interaction and separately the corresponding JC interaction. We noted in our examples that at r>1.4, photon statistics evolved to super-Poissonian from sub-Poissonian during the respective AJC, JC interactions. Further, for high values of r, the form of the time evolution of atomic population inversion depicted enhanced ringing revivals at the collapse region in comparison to the case of an initial coherent state. What is more, at higher values of r the time evolution of DEM showed more rapid oscillations and recorded higher values, concurrently, an increase in the degree of mixedness.
The Jaynes-Cummings model (JCM) describes the interaction of a two-level atom with a single quantised field mode in an optical cavity. In this BSc project we derive the basic Rabi model Hamiltonian and show how it leads to the JCM Hamiltonian. The results are then used to analyse the general time evolution of a bipartite atom-field system and its properties, such as entanglement. We extend a scheme developed by Browne and Plenio [D. Browne and M. B. Plenio, Physical Review A, vol. 67, no. 1, Jan. 2003.] to entangle two cavities that interact with an atom. Our scheme allows to produce states that are arbitrarily close to a Bell state with a success probability greater than 0.75, whereas the success rate of the original method tends to 0 for perfect fidelity. Further, we extend both schemes by allowing the atom or the cavity to be in a thermal state and find that the fidelity of the produced states is reduced with increasing temperature. Finally, we apply our scheme to quantum teleportation, where we find that the fidelity of the teleported state compared to the original state also decreases as the temperature is increased.