The quantum brachistochrone problem for non-Hermitian Hamiltonians

Assis, Paulo Eduardo Gonçalves de; Fring, Andreas

doi:10.1088/1751-8113/41/24/244002

Cited by 39 publications

(71 citation statements)

References 26 publications

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“…The arbitrarily short alternative complex pathway from an up state to a down state, as illustrated by this thought experiment, is reminiscent of the short alternative distance between two widely separated space-time points as measured through a wormhole in general relativity. This comparison is of course controversial, and it has subsequently motivated much research and it has generated a lively debate in the literature [2,15,33,35,36,49,52,53,56]. We emphasise that the entire package of flipping the spin is not realised by a unitary operation.…”

Section: Extension Of Non-hermitian Hamiltonians To Higher-dimensmentioning

confidence: 96%

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Bender¹,

Brody²

2009

Lecture Notes in Physics

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The shortest path between two truths in the real domain passes through the complex domain. -Jacques Hadamard, The Mathematical Intelligencer 13 (1991) Consider the set of all Hamiltonians whose largest and smallest energy eigenvalues, E max and E min , differ by a fixed energy ω. Given two quantum states, an initial state |ψ I and a final state |ψ F , there exist many Hamiltonians H belonging to this set under which |ψ I evolves in time into |ψ F . Which Hamiltonian transforms the initial state to the final state in the least possible time τ ? For Hermitian Hamiltonians, τ has a nonzero lower bound. However, among complex non-Hermitian P T -symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of τ can be made arbitrarily small because for P T -symmetric Hamiltonians the evolution path from the vector |ψ I to the vector |ψ F , as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here resembles the effect in general relativity in which two space-time points can be made arbitrarily close if they are connected by a wormhole. This result may have applications in quantum computing.

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Section: Extension Of Non-hermitian Hamiltonians To Higher-dimensmentioning

confidence: 96%

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Bender¹,

Brody²

2009

Lecture Notes in Physics

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“…However, Bender et al [2] have recently shown that for non-Hermitian PT -symmetric quantum systems, the answer is quite different and that the time evolution τ can be made arbitrarily small, despite the fact that the eigenvalue constraint is held fixed and identical to that for the corresponding Hermitian system. The mechanism described in [2] resembles the wormhole effect in general relativity and has generated discussion in the literature [6,7,8,9,10,11,12,13,14].…”

Section: Introductionmentioning

confidence: 99%

“…The generic non-Hermitian quantum brachistochrone problem poses the same question, with the exception that now the evolution of the system is described by the nonHermitian Hamiltonian, which is not necessarily PT -symmetric [2,6,12].…”

Section: Introductionmentioning

confidence: 99%

“…For the Hamiltonian (2.20), the straightforward computation yields 6) and, using equations (2.18) and (4.3), we obtain…”

mentioning

confidence: 99%

“…To compare our results with the PT -symmetric model widely discussed in the literature (see, e.g., [6,7,8,9,10,11,12,13,14]), we choose θ = π/2 + iη and assume λ 0 = ϕ = 0 to write the effective Hamiltonian of equation (2.20) as…”

mentioning

confidence: 99%

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Non-Hermitian Quantum Systems and Time-Optimal Quantum Evolution

Nesterov¹

2009

SIGMA

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Abstract. Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian PT -symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state |ψ i into a specific final state |ψ f can be made arbitrarily small. Additionally, it has been shown that finding the shortest possible time requires only the solution of the two-dimensional problem for the quantum system governed by the effective Hamiltonian acting in the subspace spanned by |ψ i and |ψ f . In this paper, we study a similar problem for the generic non-Hermitian Hamiltonian, focusing our attention on the geometric aspects of the problem.

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Trends in Supersymmetric Quantum Mechanics

David²

2019

Integrability, Supersymmetry and Coherent States

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Along the years, supersymmetric quantum mechanics (SUSY QM) has been used for studying solvable quantum potentials. It is the simplest method to build Hamiltonians with prescribed spectra in the spectral design. The key is to pair two Hamiltonians through a finite order differential operator. Some related subjects can be simply analyzed, as the algebras ruling both Hamiltonians and the associated coherent states. The technique has been applied also to periodic potentials, where the spectra consist of allowed and forbidden energy bands. In addition, a link with nonlinear second-order differential equations, and the possibility of generating some solutions, can be explored. Recent applications concern the study of Dirac electrons in graphene placed either in electric or magnetic fields, and the analysis of optical systems whose relevant equations are the same as those of SUSY QM. These issues will be reviewed briefly in this paper, trying to identify the most important subjects explored currently in the literature. KeywordsSupersymmetric quantum mechanics • Coherent states • Painlevé equations • Painlevé transcendents • Polynomial Heisenberg algebras • Factorization method • Exact solutions • Spectral design • Graphene Dedicated to my dear friend and colleague Véronique Hussin. D. J. Fernández C ( )

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The quantum brachistochrone problem for non-Hermitian Hamiltonians

Cited by 39 publications

References 26 publications

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Non-Hermitian Quantum Systems and Time-Optimal Quantum Evolution

Trends in Supersymmetric Quantum Mechanics

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