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Physical Review A

Spin squeezing and entanglement in a dispersive cavity

Ram Narayan Deb, M. Sebawe Abdalla, S. S. Hassan, N. Nayak

Published May 17, 2006• 29 citations
Publisher

Abstract

We consider a system of $N$ two-level atoms (spins) interacting with the radiation field in a dispersive but high-$Q$ cavity. Under an adiabatic condition, the interaction Hamiltonian reduces to a function of spin operators which is capable of producing spin squeezing. For a bipartite system $(N=2)$, the expressions for spin squeezing get very simple, giving a clear indication of close to 100% noise reduction. We analyse this squeezing as a measure of bipartite entanglement.

DOI
10.1103/physreva.73.053817

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We consider a system of NNN two-level atoms (spins) interacting with the radiation field in a dispersive but high-QQQ cavity. Under an adiabatic condition, the interaction Hamiltonian reduces to a function of spin operators which is capable of producing spin squeezing. For a bipartite system (N=2)(N=2)(N=2), the expressions for spin squeezing get very simple, giving a clear indication of close to 100% noise reduction. We analyse this squeezing as a measure of bipartite entanglement.
We study the von Neumann entropy of the partial trace of a system of 2 two-level atoms (qubits) in a dispersive cavity where the atoms are interacting collectively with a single-mode electromagnetic field in the cavity. We make a contrast of this entanglement entropy with the spin squeezing property of the system. We find a close relationship between this von Neumann entropy and the spin squeezing of the system. We observe that this entanglement entropy attains its maximum value when the spin squeezing in the system goes to its maximum and is zero when the spin squeezing in the system vanishes.
Spin squeezing via atom-field interactions is considered within the context of the Tavis-Cummings model. An ensemble of N two-level atoms interacts with a quantized cavity field. For all the atoms initially in their ground states, it is shown that spin squeezing of both the atoms and the field can be achieved provided the initial state of the cavity field has coherence between number states differing by 2. Most of the discussion is restricted to the case of a cavity field initially in a coherent state, but initial squeezed states for the field are also discussed. Optimal conditions for obtaining squeezing are obtained. An analytic solution is found that is valid in the limit that the number of atoms is much greater than unity and is also much larger than the average number of photons, ${\ensuremath{\alpha}}^{2},$ initially in the coherent state of the cavity field. In this limit, the degree of spin squeezing increases with increasing $\ensuremath{\alpha},$ even though the field more closely resembles a classical field for which no spin squeezing could be achieved.
The generation of entanglement in atomic systems plays a central topic in the fields of quantum information storage and processing. Moreover, a special category of entangled states of multi-atom ensembles, spin squeezed states, have been proven to lead to considerable improvement in the sensitivity of precision measurements compared to systems involving uncorrelated atoms. A treatment of entanglement in open systems is, however, incomplete without a precise description of the process of decoherence which necessarily accompanies it. The theory of entanglement and decoherence are the two main topics of this thesis. Methods are described for the generation of strong correlations in large atomic ensembles using either cavity quantum electrodynamics or measurement outcome conditioned quantum dynamics. Moreover, the description of loss of entanglement resulting from the coupling to a noise reservoir (electromagnetic vacuum) is explored. A spin squeezing parameter is used throughout this thesis as both a measure of entanglement strength and as an indication of the sensitivity improvement above the so-called standard quantum limit (sensitivity obtained with uncorrelated particles) in metrology. The first scheme considered consists of a single mode cavity field interacting with a collection of atoms for which spin squeezing is produced in both resonant and off-resonant regimes. In the resonant case, transfer of squeezing from a field state to the atoms is analyzed, while in the off-resonant regime squeezing is produced via an effective nonlinear interaction (one-axis twisting Hamiltonian). A second, more experimentally realistic case, is one involving the interaction of free space atoms with laser pulses; a projective measurement of a source field originating from atomic fluctuations provides a means of preparing atomic collective states such as spin squeezed and Schrodinger cat states. A new unravelling is proposed, that employs the detection of photon number in a single output channel and is capable of producing mesoscopic Schrodinger cat states in a single step. As a first step to understanding the role of cooperative decoherence, of importance in the case of dense pencil-shaped atomic ensembles, the collective spin decoherence of a two multilevel atom system is derived. This calculation is also relevant to entanglement loss for two qubits manipulated using with reading/writing pulses. Finally, a scheme is proposed in which arbitrarily strong entanglement can be produced in a four-wave mixing setup, where the preparation of the atoms in a dark state limits the decoherence to negligible values.
We show that spin squeezing implies entanglement for quantum tripartite-state, where the subsystem of the bipartite-state is identical. We study the relation between spin squeezing parameters and entanglement through the quantum entropy of a system starts initially in a pure state when the cavity is binomial. We show that spin squeezing can be a convenient tool to give some insight into the subsystems entanglement dynamics when the bipartite subsystem interacts simultaneously with the cavity field subsystem, specially when the interaction occurs off-resonantly without and with a nonlinear medium contained in the cavity field subsystem. We illustrate that, in case of large off-resonance interaction, spin squeezing clarifies the properties of entanglement almost with full success. However, it is not a general rule when the cavity is assumed to be filled with a non-linear medium. In this case, we illustrate that the insight into entanglement dynamics becomes more clearly in case of a weak nonlinear medium than in strong nonlinear medium. In parallel, the role of the phase space distribution in quantifying entanglement is also studied. The numerical results of Husimi QQQ-function show that the integer strength of the nonlinear medium produces Schrödinger cat states which is necessary for quantum entanglement.
We show that spin-squeezing implies entanglement for the quantum tripartite state, where the subsystem of the bipartite state is identical. We study the relation between spin-squeezing parameters and entanglement through the quantum entropy of a system, starting initially in a pure state when the cavity is binomial. We show that spin-squeezing can be a convenient tool to give some insight into the subsystems entanglement dynamics when the bipartite subsystem interacts simultaneously with the cavity field subsystem, especially when the interaction occurs off-resonantly without and with a nonlinear medium contained in the cavity field subsystem. We illustrate that, in the case of large off-resonance interaction, spin-squeezing clarifies the properties of entanglement almost with full success. However, it is not a general rule when the cavity is assumed to be filled with a nonlinear medium. In this case, we illustrate that the insight into entanglement dynamics becomes more clear in the case of a weak nonlinear medium than in strong nonlinear medium. In parallel, the role of the phase-space distribution in quantifying entanglement is also studied. The numerical results of Husimi Q-function show that the integer strength of the nonlinear medium produces Schrödinger cat states, which is necessary for quantum entanglement.