Method for the construction of the Lewis-Riesenfeld time-dependent invariants and their eigenvalue equations

Ponte, M. A. de; Cônsoli, Pedro M.; Moussa, M. H. Y.

doi:10.1103/physreva.98.032102

Cited by 18 publications

(20 citation statements)

References 25 publications

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“…The Eq. ( 10) is exactly that defining the Lewis & Riesenfeld linear dynamical invariant for a Hermitian Hamiltonian H(t) [43][44][45][46]. For a TI symmetry I, the Eq.…”

Section: A Methods For the Construction Of A General Td Symmetry Oper...mentioning

confidence: 97%

“…Following the reasonings in Ref. [44], where a method for the construction of nonlinear Lewis & Riesenfeld TD invariants is presented, we define the general symmetry operator as the product I(t) = Λ(t)U (t), with Λ(t) being either a unitary or nonunitary operator. Regarding U(t), from now on we assume it to be antilinear in accordance with the condition imposed in references [4,29], whose results we want to rescue in the scenario of TI Hamiltonian, symmetry and metric operators.…”

Section: A Methods For the Construction Of A General Td Symmetry Oper...mentioning

confidence: 99%

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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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“…The Eq. ( 10) is exactly that defining the Lewis & Riesenfeld linear dynamical invariant for a Hermitian Hamiltonian H(t) [43][44][45][46]. For a TI symmetry I, the Eq.…”

Section: A Methods For the Construction Of A General Td Symmetry Oper...mentioning

confidence: 97%

Section: A Methods For the Construction Of A General Td Symmetry Oper...mentioning

confidence: 99%

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

2022

View full text Add to dashboard Cite

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“…The construction of the invariant operator and its associated eigenvalue equation can be a difficult process, and various methods have been introduced to overcome this [37,[57][58][59]. However, since we have the form of the Hamiltonian and its eigenvectors, we need only to parameterize Î(t) and |φ n (t) in the same functional forms.…”

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confidence: 99%

Speeding Up Particle Slowing using Shortcuts to Adiabaticity

Bartolotta,

Reilly,

Holland

2020

Preprint

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We propose a method for slowing particles by laser fields that potentially has the ability to generate large forces without the associated momentum diffusion that results from the random directions of spontaneously scattered photons. In this method, time-resolved laser pulses with periodically modified detunings address an ultranarrow electronic transition to reduce the particle momentum through repeated absorption and stimulated emission cycles. We implement a shortcut to adiabaticity approach that is based on Lewis-Riesenfeld invariant theory. This affords our scheme the advantages of adiabatic transfer, where there can be an intrinsic insensitivity to the precise strength and detuning characteristics of the applied field, with the advantages of rapid transfer that is necessary for obtaining a short slowing distance. For typical parameters of a thermal oven source that generates a particle beam with a central velocity on the order of meters per second, this could result in slowing the particles to near stationary in less than a millimeter. We compare the slowing scheme to widely-implemented slowing techniques that rely on radiation pressure forces and show the advantages that potentially arise when the excited state decay rate is small. Thus, this scheme is a particularly promising candidate to slow narrow-linewidth systems that lack closed cycling transitions, such as occurs in certain molecules.

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“…As stated in Ref. (33), the Lewis and Riesenfeld theorem states that a particular solution can be found for the Schrödinger equation…”

Section: The Superradiance and Superabsorption Hamiltonian And Its Solutionmentioning

confidence: 93%

“…Although a general method for the construction of Lewis and Riesenfeld TD invariants was proposed in Ref. (33), we are going to limit ourselves to the invariant proposed (7, 8), as our Hamiltonian is similar to the one used in those works.…”

Section: The Superradiance and Superabsorption Hamiltonian And Its Solutionmentioning

confidence: 99%

The superradiance-superabsorption interplay and a PT-symmetric harmonic oscillator with complex frequency

Dourado¹

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A superradiant pulse emitted by a sample of two-level atoms can be combined with superabsorption in order to increase the speed of the excitation interplay. We showcase this with two different experiments, first by subjecting the atomic sample to a competition between dissipation and amplification, the latter being a time-dependent multi-modal linear pumping; second, by coupling the atomic sample with a single mode of a high finesse cavity. The mean-field approximation was used in both cases. In the first, to solve the Schrödinger equation for the resulting time-dependent nonlinear Hamiltonian exactly we use the Lewis and Riesenfeld method. In the second case, we obtain a quantum master equation in the Lindblad form and solve it numerically. We observe that, in particular, the higher the number of atoms on the sample the higher will be the frequency of the interplay in comparison with that coming from the Jaynes-Cummings model. For a certain regime of parameters this system also leads to a superradiant laser, whose characteristics we discuss. These results show that many-particle effects can be further explored for the implementation of quantum information processes as well as for constructing of a superradiant laser. At last, we study non-Hermitian PT -symmetric bosonic nonautonomous Hamiltonians under linear and parametric amplifications. We present a generic approach to solve these Hamiltonians through the use of non-unitary similarity transformations. The choice for the metric associated with these transformations has a dramatic impact on the pseudo-Hermitian operators of the system.

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Method for the construction of the Lewis-Riesenfeld time-dependent invariants and their eigenvalue equations

Cited by 18 publications

References 25 publications

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

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

Speeding Up Particle Slowing using Shortcuts to Adiabaticity

The superradiance-superabsorption interplay and a PT-symmetric harmonic oscillator with complex frequency

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