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arXiv:1403.2368

Wavepacket dynamics in a quantum double-well system with Razavy's potential coupled to a harmonic oscillator

Hideo Hasegawa

Published Mar 10, 2014• 0 citations
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Abstract

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 $(N+1)$ states of HO ($N=1$ to 10), we evaluate eigenvalues and eigenfunctions of the composite system. An analytical calculation is made for $N=1$ and numerical calculations are performed for $1 < N \leq 10$. Quantum tunneling of wavepackets is realized between two bottoms of composite potential $U(x,y)$ where $x$ and $y$ denote coordinates in DW and HO potentials, respectively. It has been shown that with increasing $N$ and/or the coupling strength, the tunneling period is considerably increased. Phase space plots of $\langle x \rangle$ vs. $\langle p_x \rangle$ and $\langle y \rangle$ vs. $\langle p_y \rangle$ 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.

DOI
10.48550/arxiv.1403.2368

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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.
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.
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.
A behavior of quantum states (superposition of two lowest eigenstates, Gaussian wave packet) in phase space is studied for one and two dimensional double well potential. Two dimensional potential is constructed from double well potential coupled linearly and quadratically to harmonic potential. Quantum trajectories are compared with classical ones. Preferable tunneling path in phase space is found. An influence of energy of initial Gaussian wave packet and trajectory initial condition on tunneling process is analyzed.. Coupling effect on trajectories and preferable tunneling path for two dimensional potential is 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.
The familiar Rabi model, comprising a two-level system coupled to a quantum harmonic oscillator, continues to produce rich and surprising physics when the coupling strength becomes comparable to the individual subsystem frequencies. We construct approximate solutions for the eigenstates and energies in the regime in which the oscillator frequency is small compared to that of the two-level system and the coupling strength matches or exceeds the oscillator frequency. The resulting oscillator dynamics closely resembles that of a particle tunneling in a classical double-well potential. Relating our calculation to an earlier semiclassical approximation in which coupling to the two-level system creates an effective potential for the oscillator, we examine the extent to which this picture is valid. We find that, for certain parameter regimes, the interpretation of the oscillator dynamics in terms of tunneling holds to a good approximation despite the fundamentally entangled nature of the joint system. We assess the prospects for observation of oscillator tunneling in the context of nano- or micromechanical experiments and find that it should be possible if suitably high coupling strengths can be engineered.
Abstract We study the tunneling dynamics of N = 10 one-dimensional interacting bosons confined in a temporally driven double well potential that imitates a quantum seesaw and how we can manipulate these dynamics by changing the drive of the seesaw potential. We emulate the seesaw with a driven double well potential and consider two driving protocols: an harmonic constant-frequency drive and a chirped drive with linearly increasing frequency. We consider the time-dependent many-body Schrödinger equation of a repulsively interacting quasi-one-dimensional few-boson system. We solve it by using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB) as implemented in the MCTDH-X software. For an harmonic drive and at small values of the driving amplitude, the dynamics of the particles become very slow rendering a stationary-like state. In a phase-space picture the population imbalance between the wells follows a trajectory which is restricted to a comparatively small region of space. For an harmonic drive at intermediate amplitudes, the dynamics become periodic in nature, implying that the bosons populate each of wells periodically. At comparatively large amplitudes of the harmonic drive, the dynamics show features of chaos in phase-space representation. For the chirped drive with a driving frequency increasing linearly in time, the imbalance of the atoms in the seesaw, however, has a temporal evolution that is faster for certain frequency ranges. The tunneling dynamics in such cases, for small amplitudes, show the appearance of quasi-periodicity with simultaneously present slow and fast oscillations. Increasing the amplitude of the chirped drive, we observe that the dynamics, although being periodic, become severely damped in their amplitude. Our study establishes that by tuning the temporal evolution of the quantum seesaw, a precise control of tunneling dynamics of the correlated bosons can be achieved. Since harmonic driving and chirp frequency modulation of the seesaw are experimentally achievable, our simulations can be experimentally realized in laboratories dealing with cold atomic gases.