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Non-Hermitian Quantum Fermi Accelerator

Andreas Fring, Takano Taira

Abstract

We exactly solve a quantum Fermi accelerator model consisting of a time-independent non-Hermitian Hamiltonian with time-dependent Dirichlet boundary conditions. A Hilbert space for such systems can be defined in two equivalent ways, either by first constructing a time-independent Dyson map and subsequently unitarily mapping to fixed boundary conditions or by first unitarily mapping to fixed boundary conditions followed by the construction of a time-dependent Dyson map. In turn this allows to construct time-dependent metric operators from a time-independent metric and two time-dependent unitary maps that freeze the moving boundaries. From the time-dependent energy spectrum, we find the known possibility of oscillatory behavior in the average energy in the PT-regime, whereas in the spontaneously broken PT-regime we observe the new feature of a one-time depletion of the energy. We show that the PT broken regime is mended with moving boundary, equivalently to mending it with a time-dependent Dyson map.

Non-Hermitian Quantum Fermi Accelerator

Abstract

We exactly solve a quantum Fermi accelerator model consisting of a time-independent non-Hermitian Hamiltonian with time-dependent Dirichlet boundary conditions. A Hilbert space for such systems can be defined in two equivalent ways, either by first constructing a time-independent Dyson map and subsequently unitarily mapping to fixed boundary conditions or by first unitarily mapping to fixed boundary conditions followed by the construction of a time-dependent Dyson map. In turn this allows to construct time-dependent metric operators from a time-independent metric and two time-dependent unitary maps that freeze the moving boundaries. From the time-dependent energy spectrum, we find the known possibility of oscillatory behavior in the average energy in the PT-regime, whereas in the spontaneously broken PT-regime we observe the new feature of a one-time depletion of the energy. We show that the PT broken regime is mended with moving boundary, equivalently to mending it with a time-dependent Dyson map.
Paper Structure (6 sections, 26 equations, 3 figures, 1 table)

This paper contains 6 sections, 26 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Equivalence of time-dependent boundary and Dyson map
  3. Swanson model: Mending $\mathcal{PT}$-broken regime via moving boundary
  4. Average energy
  5. Swanson model: Equivalence of time-dependent boundary and Dyson map
  6. Conclusion

Figures (3)

  • Figure 1: Commutative scheme, showing the relations between two time-independent Schrödinger equations with time-dependent boundary conditions on the top row, and two time-dependent Schrödinger equations with time-independent boundary conditions on the bottom row.
  • Figure 2: Average energy for the $\mathcal{PT}$-symmetric/broken regimes over time for the first metric (\ref{['Eq. Dyson map 1']}) in panels (a) and (c), respectively. The results computed with the second metric (\ref{['Eq. Dyson map 2']}) for $\mathcal{PT}$-symmetric/broken regimes are shown in panels (b) and (d), respectively. The case involving the third metric (\ref{['Eq. Dyson map 3']}) is omitted as it is almost identical to the first metric with a slight scale difference.
  • Figure 3: Showing the infinite spreading of the probability density $\Tilde{\phi}^\dagger \Tilde{\phi}$ of the wave function (\ref{['Eq. Metric-unitary wave function tilde']}) with time in $\mathcal{PT}$-broken regime. The vertical line is the boundary $\ell(t)$, which is found by solving the Ermakov-Pinney equation. In the $\mathcal{PT}$-symmetric regime, the boundary moves periodically with a similar spreading of the probability density.