Effect of Non-Commutative Space on Quantum Correlations in Two Bilinearly Coupled Harmonic Oscillators Interacting with its Environment

Koumetio, Armel Azangue; Deuto, Germain Yiande; Tene, Alain Giresse; Tchoffo, Martin

doi:10.1007/s10773-022-05065-2

Cited by 3 publications

(1 citation statement)

References 52 publications

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“…Since the connection (Darboux transformation) between the commutative space and NCspace is not symplectic, the direct use of generalized RSUP ( Vnc + i 2 e Ĵ ≥ 0) is not immediate to predict the separability of the bipartite Gaussian state in NC-space [39,40]. Nonetheless, we can transform the NC-space system into an equivalent system with commutative space operators through Bopp's shift (Darboux transformation) and use the formalism of usual quantum mechanics, including RSUP [41][42][43][44].…”

Section: Introductionmentioning

confidence: 99%

Squeezed coherent states for gravitational well in noncommutative space

Patra¹,

Saha

Biswas

2021

Indian J Phys

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Gravitational well is a widely used system for the verification of the quantum weak equivalence principle (WEP). We have studied the quantum gravitational well (GW) under the shed of non-commutative (NC) space so that the results can be utilized for testing the validity of WEP in NC-space. To keep our study widely usable, we have considered both position-position and momentum-momentum non-commutativity. Since coherent state (CS) structure provides a natural bridge between the classical and quantum domain descriptions, the quantum domain validity of purely classical phenomena like free-fall under gravity might be verified with the help of CS. We have constructed CS with the aid of a Lewis-Riesenfeld phase space invariant operator. We deduced the uncertainty relations from the expectation values of the observables and shown that the solutions of the time-dependent Schrödinger equation are squeezed-coherent states.

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Section: Introductionmentioning

confidence: 99%

Squeezed coherent states for gravitational well in noncommutative space

Patra¹,

Saha

Biswas

2021

Indian J Phys

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Tuning the separability in noncommutative space

Patra

2024

Journal of Mathematical Physics

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With the help of the generalized Peres–Horodecki separability criterion (Simon’s condition) for a bipartite Gaussian state, we have studied the separability of the noncommutative (NC) space coordinate degrees of freedom. Non-symplectic nature of the transformation between the usual commutative space and NC space restricts the straightforward use of Simon’s condition in NCS. We have transformed the NCS system to an equivalent Hamiltonian in commutative space through the Bopp shift, which enables the utilization of the separability criterion. To make our study fairly general and to analyze the effect of parameters on the separability of bipartite state in NC-space, we have considered a bilinear Hamiltonian with time-dependent (TD) parameters, along with a TD external interaction, which is linear in field modes. The system is transformed (Sp(4,R)) into canonical form keeping the intrinsic symplectic structure intact. The solution of the TD-Schrödinger equation is obtained with the help of the Lewis–Riesenfeld invariant method (LRIM). Expectation values of the observables (thus the covariance matrix) are constructed from the states obtained from LRIM. It turns out that the existence of the NC parameters in the oscillator determines the separability of the states. In particular, for isotropic oscillators, the separability condition for the bipartite Gaussian states depends on specific values of NC parameters. Moreover, particular anisotropic parameter values for the oscillator may cease the separability. In other words, both the deformation parameters (θ, η) and parameter values of the oscillator (mass, frequency) are important characteristics for the separability of bipartite Gaussian states. Thus tuning the parameter values, one can destroy or recreate the separability of states. With the help of a toy model, we have demonstrated how the tuning of a TD-NC space parameter affects the separability.

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On the entanglement of co-ordinate and momentum degrees of freedom in noncommutative space

Nandi,

Rahaman,

Patra

2024

Mod. Phys. Lett. A

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In this paper, we investigate the quantum entanglement induced by phase-space noncommutativity. Both the position–position and momentum–momentum noncommutativity are incorporated to study the entanglement properties of coordinate and momentum degrees of freedom under the shade of oscillators in noncommutative space. Exact solutions for the systems are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp’s shift to the noncommuting degrees of freedom. It is shown that the bipartite Gaussian state for an isotropic oscillator is always separable. To extend our study for the time-dependent system, we allow arbitrary time dependency on parameters. The time-dependent isotropic oscillator is solved with the Lewis–Riesenfeld invariant method. It turns out that even for arbitrary time-dependent scenarios, the separability property does not alter. We extend our study to the anisotropic oscillator, which provides an entangled state even for time-independent parameters. The Wigner quasi-probability distribution is constructed for a bipartite Gaussian state. The noise matrix (covariance matrix) is explicitly studied with the help of Wigner distribution. Simon’s separability criterion (generalized Peres–Horodecki criterion) has been employed to find the unique function of the (mass and frequency) parameters, for which the bipartite states are separable. In particular, we show that the mere inclusion of non-commutativity of phase-space is not sufficient to generate the entanglement, rather anisotropy is important at the same footing. We explore the experimental viability of our result through the computation of extractable work for the current situation.

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Effect of Non-Commutative Space on Quantum Correlations in Two Bilinearly Coupled Harmonic Oscillators Interacting with its Environment

Cited by 3 publications

References 52 publications

Squeezed coherent states for gravitational well in noncommutative space

Squeezed coherent states for gravitational well in noncommutative space

Tuning the separability in noncommutative space

On the entanglement of co-ordinate and momentum degrees of freedom in noncommutative space

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