Non-Hermitian noncommutative quantum mechanics

Santos, Jonas F. G.; Luiz, F. S.; Duarte, O. S.; Moussa, M. H. Y.

doi:10.1140/epjp/i2019-12738-3

Cited by 15 publications

(10 citation statements)

References 44 publications

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“…In general, non-commutative Hamiltonians should be addressed by employing the Seiberg-Witten transformation, as discussed in [47]. However, the gauge in Eq.…”

Section: The κ-Jc and The κ-Ajc Modelsmentioning

confidence: 99%

Effects of quantum deformation on the Jaynes-Cummings and anti-Jaynes-Cummings models

Uhdre,

Cius,

Andrade

2021

Preprint

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The theory of non-Hermitian systems and the theory of quantum deformations have attracted a great deal of attention in the last decades. In general, non-Hermitian Hamiltonians are constructed by a ad hoc manner. Here, we study the (2+1) Dirac oscillator and show that in the context of the κ-deformed Poincaré-Hopf algebra its Hamiltonian is non-Hermitian but having real eigenvalues. The non-Hermiticity steams from the κ-deformed algebra. From the mapping in [Bermudez et al., Phys. Rev. A 76, 041801(R) 2007], we propose the κ-JC and κ-AJC models, which describe an interaction between a two-level system with a quantized mode of an optical cavity in the κ-deformed context. We find that the κ-deformation modifies the Zitterbewegung frequencies and the collapse and revival of quantum oscillations. In particular, the total angular momentum in the z-direction is not conserved anymore, as a direct consequence of the deformation.

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“…In general, non-commutative Hamiltonians should be addressed by employing the Seiberg-Witten transformation, as discussed in [47]. However, the gauge in Eq.…”

Section: The κ-Jc and The κ-Ajc Modelsmentioning

confidence: 99%

Effects of quantum deformation on the Jaynes-Cummings and anti-Jaynes-Cummings models

Uhdre,

Cius,

Andrade

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…And much has been done recently, such as the experimental realizations of Floquet PT -symmetric systems [5] and PTsymmetric flat bands [6], besides enhanced sensing based on PT -symmetric electronic circuits [7] and PT -symmetric topological edge-gain effect [8]. The linear response theory for a pseudo-Hermitian system-reservoir interaction was developed [9], as well as a protocol to approach non-Hermitian non-commutative quantum mechanics [10].…”

Section: Introductionmentioning

confidence: 99%

Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians

2022

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In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t)I(t) for a TD non-Hermitian Hamiltonian H(t)H(t). Assuming H(t)H(t) to be simultaneously \rho(t)ρ(t)-pseudo-Hermitian and \Xi(t)Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t)I(t)=Ξ−1(t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J. Math, Phys. 43, 3944 (2002)] for the time-independent (TI) scenario. We apply our method for the derivation of the symmetries associated with TD non-Hermitian linear and quadratic Hamiltonians. The computed TD symmetry operators for both cases are then particularized for their equivalent TI Hamiltonians and PT -symmetric restrictions. In the TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1I2k=1 with kk odd [J. Phys. A 35, L467 (2002)]. The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians. From this relation the TD metric is automatically derived from the TD symmetry of the problem. Then, when placed in perspective with the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t)I(t)=Ξ−1(t)ρ(t), the symmetry-metric relation finally allow us to derive the \Xi(t)Ξ(t)-anti-pseudo-Hermitian operator. Our results reinforce the prospects of going beyond \mathcal{PT}𝒫𝒯-symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.

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“…This new condition to guarantee real spectra is known as PT -symmetry, and it gave rise to the non-Hermitian quantum mechanics [16], once PTsymmetric Hamiltonians are not Hermitian in general. Indeed, the range of interest in effects by considering systems described by PT -symmetric Hamiltonians is vast and includes quantum optics and photonics [17,18], time-dependent Hamiltonians [19][20][21], applications to non-commutative geometries [22][23][24], as well as to fluctuation relations [25][26][27]. Since the conditions for an operator to be a viable candidate to represent a physical observable are that eigenvalues are real and that eingestates are complete, the condition of orthogonality for a non-Hermitian PT -symmetric Hamiltonian H can be relaxed and substituted by a biorthogonality [28], and the conection with the biorthogonal system with the ortogonal is made by a nontrivial metric operator [29,30].…”

Section: Introductionmentioning

confidence: 99%

Quantum thermodynamics aspects with a thermal reservoir based on $\mathcal{PT}$-symmetric Hamiltonians

Santos,

Luiz

2021

Preprint

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Non-Hermitian noncommutative quantum mechanics

Cited by 15 publications

References 44 publications

Effects of quantum deformation on the Jaynes-Cummings and anti-Jaynes-Cummings models

Effects of quantum deformation on the Jaynes-Cummings and anti-Jaynes-Cummings models

Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians

Quantum thermodynamics aspects with a thermal reservoir based on $\mathcal{PT}$-symmetric Hamiltonians

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