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In this paper, we investigate into the numerical and analytical relationship between the dynamically generated quadrature squeezing and entanglement within a coupled harmonic oscillator system. The dynamical relation between these two quantum features is observed to vary monotically, such that an enhancement in entanglement is attained at a fixed squeezing for a larger coupling constant. Surprisingly, the maximum attainable values of these two quantum entities are found to consistently equal to the squeezing and entanglement of the system ground state. In addition, we demonstrate that the inclusion of a small anharmonic perturbation has the effect of modifying the squeezing

Entanglement is a fundamental resource for non-classical tasks in the field of quantum information [

Indeed, the relation between quantum squeezing and quantum entanglement has been actively pursued in recent years [

In this paper, we have extended our earlier work discussed above by investigating into the dynamical relation between quadrature squeezing and entanglement entropy of the coupled harmonic oscillator system. Coupled harmonic oscillator system has served as useful paradigm for many physical systems, such as the field modes of electromagnetic radiation [

The Hamiltonian of the coupled harmonic oscillator system is described as follow:
_{1} and _{2} are the position co-ordinates, while _{1} and _{2} are the momenta of the oscillators. The interaction potential between the two oscillators is assumed to depend quadratically on the distance between the oscillators, and is proportional to the coupling constant λ. For simplicity, we have set _{1} = _{2} = _{1} = _{2} =

Next, let us discuss on the relation between the squeezing and entanglement of the lowest energy eigenstate of this coupled harmonic oscillator system. Note that
_{0} being the lowest eigen-energy of the coupled oscillator system with Hamiltonian given by _{l}

Indeed, the position uncertainty squeezing and the entanglement entropy of the ground state of this oscillator have been solved analytically by previous studies [

In this paper, we have gone beyond the static relation between squeezing and entanglement based on the stationary ground state. In particular, we have explored numerically into the dynamical generation of squeezing and entanglement via the quantum time evolution, with the initial state being the tensor product of the vacuum states (|0, 0〉) of the oscillators. Note that the obtained results hold true for any initial coherent states (|_{1}, _{2}〉) since the entanglement dynamics of the coupled harmonic oscillator system is independent of initial states [_{1} = 1 and

In this section, we shall perform an analytical study on the dynamical relationship between quantum squeezing and the associated entanglement production. We first yield the second quantized form of the Hamiltonian of the coupled harmonic oscillator system as follow:
_{j}

With these results, we are now ready to determine the analytical expressions of both the quantum entanglement and squeezing against time. For entanglement, we shall employ the criterion developed by Duan _{1} + _{2} and _{1} − _{2} are two EPR-type operators, whereas Δ_{D}_{1}, _{2}|_{1}, _{2}), where _{1}, _{2}〉 represents a tensor product of arbitrary initial coherent states. Recall that the subsequent results are indepedent of the initial states as mentioned in the last section. After substituting _{D}_{vN}_{D}_{x}_{x}_{x}_{D}_{D}_{vN}_{D}_{vN}_{D}_{vN}_{D}_{vN}

When projected into the _{1} − _{2} or _{2} − _{1} plane, the initial coherent state can be represented by a circular distribution with equal uncertainty in both _{1} − _{2} or _{2} − _{1} plane with rotation of the ellipse's major axis away from the

Next, let us investigate the effect of including an anharmonic potential on the dynamical relation between squeezing and entanglement through the following Hamiltonian systems:

With a small anharmonic perturbation, the dynamically generated entanglement entropy is no longer a smooth monotonically increasing function of the quadrature squeezing as before (see

We have studied into the dynamical generation of quadrature squeezing and entanglement for both coupled harmonic and anharmonic oscillator systems. Our numerical and analytical results show that the quantitative relation that defines the dynamically generated squeezing and entanglement in coupled harmonic oscillator system is a monotonically increasing function. Such a monotonic relation vanishes, however, when a small anharmonic potential is added to the system. This result implies the possibility of characterizing the dynamically generated entanglement by means of squeezing in the case of coupled harmonic oscillator system. In addition, we have uncovered the unexpected result that the maximum attainable entanglement and squeezing obtained dynamically matches exactly the entanglement-squeezing relation of the system's ground state of the coupled harmonic oscillators. When an anharmonic potential is included, we found that the dynamically generated squeezing can be further enhanced. We percieve that this result may provide important insights to the construction of precision instruments that attempt to beat the quantum noise limit.

L. Y. Chew would like to thank Y. S. Kim for the helpful discussion on this work during the ICSSUR 2013 conference held in Nuremberg, Germany.

The authors declare no conflict of interest.

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A plot on the dynamical relation between entanglement and squeezing obtained numerically for coupled harmonic oscillator system with the coupling constant λ = 0.75 (squares), 2 (triangles), 3.75 (circles) and 6 (crosses). Note that the ground state entanglement-squeezing curve given by

This plot shows the monotonic relation between _{D}_{vN}_{D}_{vN}_{vN}_{x}

A plot on the dynamical relation between entanglement and squeezing given by

The effect of anharmonicity (

The effect of anharmonicity (