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AIP conference proceedings

Quantum behaviour mirrored by semi-classical states

Eduardo Nahmad-Achar, Octavio Castaños, Ramón López–Peña, Jorge G. Hirsch

Published Jan 1, 2012• 1 citation
Publisher

Abstract

We show that the ground state of the Tavis-Cummings (TC) model of N 2-level systems in interaction with a 1-mode radiation field can be mirrored by a semi-classical state to an excellent approximation. The latter is obtained through a projection of a variational tensorial product of coherent states onto the value of the constant of motion of the TC Hamiltonian which minimises the (classical) energy of the ground state. This projection restores the Hamiltonian symmetry, and is amiable to analytical calculations. The observed super-radiant phase-transitions are reflected by this projected state, and an excellent agreement with the exact quantum solution for the ground state is obtained as regards to the entanglement entropy between the matter and field, the squeezing parameter of the state, the probability distributions of photons and atoms, and the expectation value of all field and matter observables of interest, including the photon number dispersion. All these are given in analytic form. The fitness between our approximation and the exact quantum solution is measured through the evaluation of the fidelity parameter; this is very close to 1, except in the region of classical phase transitions, as is to be expected. The presented approach may enable novel investigations of super-radiant and sub-radiant states of artificial atoms, where the relevant control parameters can be manipulated to cross the separatrix.

DOI
10.1063/1.3688977

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We show that the ground state of the Tavis-Cummings (TC) model of N 2-level systems in interaction with a 1-mode radiation field can be mirrored by a semi-classical state to an excellent approximation. The latter is obtained through a projection of a variational tensorial product of coherent states onto the value of the constant of motion of the TC Hamiltonian which minimises the (classical) energy of the ground state. This projection restores the Hamiltonian symmetry, and is amiable to analytical calculations. The observed super-radiant phase-transitions are reflected by this projected state, and an excellent agreement with the exact quantum solution for the ground state is obtained as regards to the entanglement entropy between the matter and field, the squeezing parameter of the state, the probability distributions of photons and atoms, and the expectation value of all field and matter observables of interest, including the photon number dispersion. All these are given in analytic form. The fitness between our approximation and the exact quantum solution is measured through the evaluation of the fidelity parameter; this is very close to 1, except in the region of classical phase transitions, as is to be expected. The presented approach may enable novel investigations of super-radiant and sub-radiant states of artificial atoms, where the relevant control parameters can be manipulated to cross the separatrix.
We show that an excellent approximation to the exact quantum solution of the ground state of the Tavis–Cummings model is obtained by means of a semi-classical projected state. This state has an analytical form in terms of model parameters and, in contrast to the exact quantum state, allows for an analytical calculation of the expectation values of field and matter observables, entanglement entropy between field and matter, squeezing parameter and population probability distributions. The fidelity between this projected state and the exact quantum ground state is very close to 1, except for the region of classical phase transitions. We compare the analytical results with those of the exact solution obtained through direct Hamiltonian diagonalization as a function of atomic separation energy and matter–field coupling.
We show that semiclassical states adapted to the symmetry of the Hamiltonian are an excellent approximation to the exact quantum solution of the ground and first excited states of the Dicke model. Their overlap with the exact quantum states is very close to 1 except in a close vicinity of the quantum phase transition. Furthermore, they have analytic forms in terms of the model parameters and allow us to calculate analytically the expectation values of field and matter observables. Some of these differ considerably from results obtained via the standard coherent states and by means of Holstein-Primakoff series expansion of the Dicke Hamiltonian. Comparison with exact solutions obtained numerically supports our results. In particular, it is shown that the expectation values of the number of photons and of the number of excited atoms have no singularities at the phase transition. We comment on why other authors have previously found otherwise.
We introduce a combination of coherent states as variational test functions for the atomic and radiation sectors to describe a system of N a three-level atoms interacting with a one-mode quantised electromagnetic field, with and without the rotating wave approximation, which preserves the symmetry presented by the Hamiltonian. These provide us with the possibility of finding analytical solutions for the ground and first excited states. We study the properties of these solutions for the V -configuration in the double resonance condition, and calculate the expectation values of the number of photons, the atomic populations, the total number of excitations, and their corresponding fluctuations. We also calculate the photon number distribution and the linear entropy of the reduced density matrix to estimate the entanglement between matter and radiation. For the first time, we exhibit analytical expressions for all of these quantities, as well as an analytical description for the phase diagram in parameter space, which distinguishes the normal and collective regions, and which gives us all the quantum phase transitions of the ground state from one region to the other as we vary the interaction parameters (the matter-field coupling constants) of the model, in functional form.
The transition to the super-radiant regime in the Dicke model of two-level atoms interacting with a one-mode radiation field has been argued to be impossible for many years. Recently, however, experimental results show that this transition is possible in an open system [1, 2]. Here we exhibit the power of the use of coherent states with restored symmetry to describe the superradiant regime of the Dicke Hamiltonian. These symmetry-adapted states allow for an analytic description of both the ground and first excited states. From these, analytical expressions for all expectation values of interest are obtained, valid for any number of atoms. All these results are exact in the thermodynamic limit.
It is shown, by conventional mean-field technics, that su(2)⊗su(1, 1) coherent states are a good choice for describing the two-photon Jaynes-Cummings model in the sense that results based on mean-field methods for those states are in good agreement with exact results. We also find that this agreement is much improved if the constant of motion of the model is implemented exactly instead of being implemented only in the average. This is done by projecting the su(2) ⊗ su(1, 1) coherent states onto the physical subspaces. It is seen that the Hamiltonian of the model is unbounded from below, i.e., it lacks a ground-state. We also present and study a stabilized version of the model such that the relevant physical features are preserved. Along the article, special attention is given to the description of the super-radiant phase transition exhibited by the model.
We consider N a three-level atoms (or systems) interacting with a one-mode electromagnetic field in the dipolar and rotating wave approximations. The order of the quantum phase transitions is determined explicitly for each of the configurations Ξ, Λ and V , with and without detuning. The semi-classical and exact quantum calculations for both the expectation values of the total number of excitations and photon number have excellent correspondence as functions of the control parameters. We prove that the ground state of the collective regime obeys sub-Poissonian statistics for the and n distribution functions. Therefore, their corresponding fluctuations are not well described by the semi-classical approximation. We show that this can be corrected by projecting the variational state to a definite value of .