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arXiv:1106.3394 · Journal of Physics A Mathematical and Theoretical

Variance squeezing and entanglement of theXXcentral spin model

Faisal A A El-Orany, M Sebawe Abdalla

Published Dec 13, 2010• 24 citations
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Abstract

In this paper, we study the quantum properties for a system that consists of a central atom interacting with surrounding spins through the Heisenberg XX couplings of equal strength. Employing the Heisenberg equations of motion we manage to derive an exact solution for the dynamical operators. We consider that the central atom and its surroundings are initially prepared in the excited state and in the coherent spin state, respectively. For this system, we investigate the evolution of variance squeezing and entanglement. The nonclassical effects have been remarked in the behavior of all components of the system. The atomic variance can exhibit revival–collapse phenomenon based on the value of the detuning parameter.

DOI
10.1088/1751-8113/44/3/035302

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In this paper, we study the quantum properties for a system that consists of a central atom interacting with surrounding spins through the Heisenberg XX couplings of equal strength. Employing the Heisenberg equations of motion we manage to derive an exact solution for the dynamical operators. We consider that the central atom and its surroundings are initially prepared in the excited state and in the coherent spin state, respectively. For this system, we investigate the evolution of variance squeezing and entanglement. The nonclassical effects have been remarked in the behavior of all components of the system. The atomic variance can exhibit revival–collapse phenomenon based on the value of the detuning parameter.
In this paper we introduce a Hamiltonian model which represents the interaction between SU (1, 1) and SU (2) quantum systems. The Heisenberg equations of motion are invoked to derive the exact time-dependent dynamical operators. The phenomena of the collapse and revival are observed for the intermediate states, however, for a large value of the excitation number m . It is also shown that the phenomenon of squeezing is observed in the intermediate state. Moreover, it is sensitive to the variation in the coupling parameter λ, the detuning parameter Δ, as well as in the Bargmann index k . The examination of the correlation function shows that the system displays anti-bunching for all periods of time except for a large value of the excitation number when k = 3/4. Our discussion for the variance squeezing also shows that the phenomenon of squeezing is pronounced in the quadrature variances for the odd parity case when the detuning parameter is large.
A model of a two-coupled two-level atom under the action of an external magnetic field is introduced. The system is assumed to be isotropic and the coupling between the atoms is taken to be explicitly time dependent. Direct solution for the equations of motion in the Heisenberg picture is used to obtain the explicit expression for the dynamical operators when we assumed the coupling parameter is constant. In the meantime we have employed the rotating wave approximation for the case in which the coupling parameter is time dependent. The variance and entropy squeezing for both cases are discussed. It has been shown that the variance squeezing and entropy squeezing are sensitive to any change in the atomic states, so that there is no variance squeezing reported for the case in which one or both the atoms are in the pure states (e.g., lower or excited states). Further, the phenomenon of squeezing is also affected by the value of the coupling parameter, where we have noted that for a strong coupling parameter the squeezing is pronounced for a long period of time.
We introduce the problem of three types of interaction between an N-level quantum system and a two-level atom where three coupling parameters are involved. The system can be deduced from the Heisenberg chain. The canonical transformation is used to remove two coupling parameters from the system and consequently it is reduced to atom–atom interaction. The wave function is calculated using the evolution operator and hence we have managed to obtain the expectation value of some dynamical operators. During our study of the atomic inversion we noted that the collapses period is shifted up when we take the effect of λ2 into consideration. While it is shifted up and down in the presence of λ3. The atomic angle plays a crucial role for controlling the degree of entanglement. For the variance squeezing we noted that the coupling parameter λ2 shows amounts of squeezing more than the case of λ3. Similar behavior is noted for the entropy squeezing.
We discuss two central spin models coupled, respectively, to a quantum Heisenberg XY spin star environment [1] and to an antiferromagnetic environment [2]. In the first model [1], the exact quantum dynamics of the reduced density matrix of two coupled spin qubits in a quantum Heisenberg XY spin star environment in the thermodynamic limit at arbitrarily finite temperatures is obtained using a novel operator technique. In this approach, the transformed Hamiltonian becomes effectively Jaynes-Cumming like and thus the analysis is also relevant to cavity quantum electrodynamics. This special operator technique is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. To study their entanglement evolution, the concurrence of the reduced density matrix of the two coupled central spins is also obtained exactly. It is shown that the dynamics of the entanglement depends on the initial state of the system and the coupling strength between the two coupled central spins, the thermal temperature of the spin environment and the interaction between the constituents of the spin environment. We also investigate the effect of detuning which in our model can be controlled by the strength of a locally applied external magnetic field. It is found that the detuning has a significant effect on the entanglement generation between the two spin qubits.
We obtain exact dynamics of a two-qubit central spin model (CSM) consisting of two interacting qubits homogeneously coupled to a spin bath via the XXZXXZXXZ-type coupling, with the bath initially prepared in linear superpositions of the symmetric Dicke states. Using the interaction picture Hamiltonian with respect to the non-spin-flipping part of the model, we derive a sequence of equations of motion within each magnetization sector satisfied by the probability amplitudes of the time-evolved state. These equations of motion admit analytical solutions for the single-qubit CSM in which one of the two central qubits decouples from the rest of the system. Based on this, we provide a quantitative interpretation to the observed collapse-revival phenomena in the single-qubit Rabi oscillations when the bath is prepared in the spin coherent state. We then study the disentanglement and coherence dynamics of two initially entangled noninteracting qubits when the two qubits interact with individual baths or with a common bath. For individual baths, the coherent dynamics is found to be positively correlated to the single-qubit purity dynamics, and entanglement sudden disappearance and revivals are observed in both cases. The entanglement creation of two initially separable qubits coupled to a common bath is also studied and collapse and revival behaviors in the entanglement dynamics are observed. Choosing the equally weighted state and the WWW-class states as the bath initial states, we finally study the dynamics of entanglement between two individual bath spins and demonstrate the entanglement sharing mechanism in such a system.