Intertwining relations of non-stationary Schrödinger operators

Cannata, F.; Ioffe, M. V.; Junker, Georg; Nishnianidze, D. N.

doi:10.1088/0305-4470/32/19/309

Cited by 34 publications

(71 citation statements)

References 24 publications

(67 reference statements)

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“…We have demonstrated that metric representations lead to consistent descriptions equivalent to the operator representation by providing further solutions to the time-dependent quasi-Hermiticity relation (4) and the time-dependent Dyson relation (5). For the spin models we considered here we observed that the determining relation for the metric operator (4) converts into as many equations as unknown functions.…”

Section: Discussionmentioning

confidence: 70%

“…Assuming the Dyson operator η(t) to be Hermitian, it can in principle be computed from ρ(t) by taking its square root. Subsequently one may compute the Hermitian counterpart h(t) by direct evaluation of the right-hand side of the time-dependent Dyson relation (5). As we will demonstrate in more detail below, taking the square root in this case can be rather awkward and to avoid this step we pursue here a different approach by solving the time-dependent Dyson relation first.…”

Section: H(t)φ(t) = I ∂ T φ(T) and Hψmentioning

confidence: 99%

“…Instead we solve the time-dependent Dyson equation (5). We assume a similar form for our Hermitian target Hamiltonian as in (9) and take S z to be a spin-1 matrix, denote χ = X and take η(t) to be of the Hermitian form…”

Section: Solutions Of the Time-dependent Dyson Relationmentioning

confidence: 99%

“…These physically equivalent representations are made possible in this setting as it always involves nontrivial metric operators on the non-Hermitian side. For Hermitian systems one can employ time-dependent Darboux transformation to map time-dependent Hamiltonian systems to time-independent ones [5,6].…”

Section: Introductionmentioning

confidence: 99%

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Metric versus observable operator representation, higher spin models

Fring

Frith

2018

Eur. Phys. J. Plus

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Abstract. We elaborate further on the metric representation that is obtained by transferring the timedependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.

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Section: Discussionmentioning

confidence: 70%

Section: H(t)φ(t) = I ∂ T φ(T) and Hψmentioning

confidence: 99%

Section: Solutions Of the Time-dependent Dyson Relationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Metric versus observable operator representation, higher spin models

Fring

Frith

2018

Eur. Phys. J. Plus

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“…For the one-dimensional case intertwining relations (30) were investigated in [19] for differential operators U(y, t) of first and second order in derivatives. While in the onedimensional problem a wide class of solutions was found, a straightforward extension to the two-dimensional case does not appear to be obvious.…”

Section: Time-dependent Exactly Solvable 3-body Matrix Problemsmentioning

confidence: 99%

Exactly solvable three-body systems with internal degrees of freedom

Cannata

Ioffe

2001

J. Phys. A: Math. Gen.

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The approach of multi-dimensional SUSY Quantum Mechanics is used in an explicit construction of exactly solvable 3-body ( and quasi-exactly-solvable N -body ) matrix problems on a line. From intertwining relations with time-dependent operators, we build exactly solvable non-stationary scalar and 2 × 2 matrix 3-body models which are time-dependent extensions of the Calogero model. Finally, we investigate the invariant operators associated to these systems.

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Experimental Realization of Parrondo's Paradox in 1D Quantum Walks

Jan

Wang

et al. 2020

Adv Quantum Tech

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The Parrondo effect is a well‐known apparent paradox where a combination of biased random walks displays a counterintuitive reversal in direction. These random walks can be expressed in terms of classical coin tossing games, leading to the surprising result that a combination of losing games can result in a winning game. There is now a large body of literature on quantum walks theoretically analyzing the quantum version of this effect, but to date, there have been no experimental observations of quantum Parrondo walks. Here, the first experimental verification of a quantum Parrondo walk within a quantum optics scenario is demonstrated. Based on the compact large‐scale experimental quantum‐walk platform, two rotation operators are implemented to realize the quantum Parrondo effect. The effect of quantum coherence in a quantum Parrondo walk is also investigated based on a delayed‐choice scheme that cannot be realized with classical light. It is demonstrated that the Parrondo effect vanishes when the quantum walk has a completely decoherent initial state in a delayed‐choice setting. Quantum walks are fundamental to multiple quantum algorithms, and this research provides motivation to expand the results to further explore quantum Parrondo walks.

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Intertwining relations of non-stationary Schrödinger operators

Cited by 34 publications

References 24 publications

Metric versus observable operator representation, higher spin models

Metric versus observable operator representation, higher spin models

Exactly solvable three-body systems with internal degrees of freedom

Experimental Realization of Parrondo's Paradox in 1D Quantum Walks

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