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arXiv:1403.0543

Quantum tunneling and evolution speed in an exactly solvable coupled double-well system

Hideo Hasegawa

Published Mar 3, 2014• 0 citations
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Abstract

Exact analytical calculations of eigenvalues and eigenstates are presented for quantum coupled double-well (DW) systems with Razavy's hyperbolic potential. With the use of four kinds of initial wavepackets, we have calculated the tunneling period $T$ and the orthogonality time $τ$ which signifies a time interval for an initial state to evolve to its orthogonal state. We discuss the coupling dependence of $T$ and $τ$, and the relation between $τ$ and the concurrence $C$ which is a typical measure of the entanglement in two qubits. Our calculations have shown that it is not clear whether the speed of quantum evolution may be measured by $T$ or $τ$ and that the evolution speed measured by $τ$ (or $T$) is not necessarily increased with increasing $C$. This is in contrast with the earlier study [V. Giovannetti, S. Lloyd and L. Maccone, Europhys. Lett. {\bf 62} (2003) 615] which pointed out that the evolution speed measured by $τ$ is enhanced by the entanglement in the two-level model.

DOI
10.48550/arxiv.1403.0543

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Exact analytical calculations of eigenvalues and eigenstates are presented for quantum coupled double-well (DW) systems with Razavy's hyperbolic potential. With the use of four kinds of initial wavepackets, we have calculated the tunneling period TTT and the orthogonality time τττ which signifies a time interval for an initial state to evolve to its orthogonal state. We discuss the coupling dependence of TTT and τττ, and the relation between τττ and the concurrence CCC which is a typical measure of the entanglement in two qubits. Our calculations have shown that it is not clear whether the speed of quantum evolution may be measured by TTT or τττ and that the evolution speed measured by τττ (or TTT) is not necessarily increased with increasing CCC. This is in contrast with the earlier study [V. Giovannetti, S. Lloyd and L. Maccone, Europhys. Lett. {\bf 62} (2003) 615] which pointed out that the evolution speed measured by τττ is enhanced by the entanglement in the two-level model.
We have studied wavepacket dynamics in the Razavy hyperbolic double-well (DW) potential which is coupled to a harmonic oscillator (HO) by linear and quadratic interactions. Taking into account the lowest two states of DW and (N+1)(N+1)(N+1) states of HO (N=1N=1N=1 to 10), we evaluate eigenvalues and eigenfunctions of the composite system. An analytical calculation is made for N=1N=1N=1 and numerical calculations are performed for 1<N≤101 < N \leq 101<N≤10. Quantum tunneling of wavepackets is realized between two bottoms of composite potential U(x,y)U(x,y)U(x,y) where xxx and yyy denote coordinates in DW and HO potentials, respectively. It has been shown that with increasing NNN and/or the coupling strength, the tunneling period is considerably increased. Phase space plots of ⟨x⟩\langle x \rangle⟨x⟩ vs. ⟨px⟩\langle p_x \rangle⟨px​⟩ and ⟨y⟩\langle y \rangle⟨y⟩ vs. ⟨py⟩\langle p_y \rangle⟨py​⟩ are elliptic, where ⟨⋅⟩\langle \cdot \rangle⟨⋅⟩ denotes an expectation value for the two-term wavepacket. This result is quite different from the relevant one previously obtained for the quartic DW potential with the use of the quantum phase space representation [Babyuk, arXiv:0208070]. Similarity and difference between results calculated for linear and quadratic couplings, and the uncertainty relation in the model are discussed.
Effects of time-dependent applied fields on quantum coupled double-well (DW) systems with Razavy's hyperbolic potential have been studied. By solving the Schrödinger equation for the DW system, we have obtained time-dependent occupation probabilities of the eigenstates, from which expectation values of positions x1x_1x1​ and x2x_2x2​ of particles ($$), the correlation ($Γ(t)$) and the concurrence ($C(t)$) expressing a degree of the entanglement of the coupled DW system, are obtained. Analytical expressions for $$, Γ(t)Γ(t)Γ(t) and C(t)C(t)C(t) are derived with the use of the rotating-wave approximation (RWA) for sinusoidal fields. Model calculations have indicated that $$, Γ(t)Γ(t)Γ(t) and C(t)C(t)C(t) show very complicated time dependences. Results of the RWA are in good agreement with exact ones evaluated by numerical methods for cases of weak couplings and small applied fields in the near-resonant condition. Applications of our method to step fields are also studied.
Eects of time-dependent applied elds on quantum coupled double-well (DW) systems with Razavy's hyperbolic potential have been studied. By solving the Schrodinger equation for the DW system, we have obtained time-dependent occupation probabilities of the eigenstates, from which expectation values of positions x1 and x2 of particles (hx1 + x2i), the correlation (( t)) and the concurrence (C(t)) expressing a degree of the entanglement of the coupled DW system, are obtained. Analytical expressions forhx1 +x2i, ( t) and C(t) are derived with the use of the rotating-wave approximation (RWA) for sinusoidal elds. Model calculations have indicated that hx1 +x2i, ( t) and C(t) show very complicated time dependences. Results of the RWA are in good agreement with exact ones evaluated by numerical methods for cases of weak couplings and small applied elds in the near-resonant condition. Applications of our method to step elds are also studied.
It has been recently pointed out [V. Giovannetti, S. Lloyd, and L. Maccone, Europhys. Lett. 62, 615 (2003)] that, for certain classes of states, quantum entanglement enhances the ``speed'' of evolution of composite quantum systems, as measured by the time a given initial state requires to evolve to an orthogonal state. We provide here a systematic study of this effect for pure states of bipartite systems of low dimensionality, considering both distinguishable (two-qubits) subsystems, and systems constituted of two indistinguishable particles.
The concept of quantum speed limit-time (QSL) was initially introduced as a lower bound to the time interval that a given initial state ${\ensuremath{\psi}}_{I}$ may need so as to evolve into a state orthogonal to itself. Recently [V. Giovanetti, S. Lloyd, and L. Maccone, Phys. Rev. A 67, 052109 (2003)] this bound has been generalized to the case where ${\ensuremath{\psi}}_{I}$ does not necessarily evolve into an orthogonal state, but into any other ${\ensuremath{\psi}}_{F}$. It was pointed out that, for certain classes of states, quantum entanglement enhances the evolution ``speed'' of composite quantum systems. In this work we provide an exhaustive and systematic QSL study for pure and mixed states belonging to the whole 15-dimensional space of two qubits, with ${\ensuremath{\psi}}_{F}$ a not necessarily orthogonal state to ${\ensuremath{\psi}}_{I}$. We display convincing evidence for a clear correlation between concurrence, on the one hand, and the speed of quantum evolution determined by the action of a rather general local Hamiltonian, on the other one.
We study a wavepacket tunneling in one-dimensional periodically driven double-well system using entangled trajectory molecular dynamics method. The tunneling dynamics dependents on the amplitude and frequency of the driven force are present. Both resonant and nonresonant tunneling process are enhanced by the driven force when the system is chaotic under classical dynamics. We give entangled trajectory in phase space compared to corresponding classical trajectory with same initial state to visually show quantum tunneling process. The average values of quantum tunneling probability after long time evolution have been shown in the parameter spaces, the effect of resonance and chaos on the tunneling dynamics are present. The relation between chaos and the uncertainly product is discussed in the end. © 2016 Wiley Periodicals, Inc.