arXiv:1502.07333

Effects of applied fields on quantum coupled double-well systems

Hasegawa, Hideo

Published Feb 25, 2015• 0 citations

Abstract

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 $x_1$ and $x_2$ of particles ($$), the correlation ($Γ(t)$) and the concurrence ($C(t)$) expressing a degree of the entanglement of the coupled DW system, are obtained. Analytical expressions for $$, $Γ(t)$ and $C(t)$ are derived with the use of the rotating-wave approximation (RWA) for sinusoidal fields. Model calculations have indicated that $$, $Γ(t)$ 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 fields in the near-resonant condition. Applications of our method to step fields are also studied.

DOI: 10.48550/arxiv.1502.07333

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arXiv:1502.07333

Effects of applied fields on quantum coupled double-well systems

Hasegawa, Hideo

Published Feb 25, 2015• 0 citations

arXiv HTML PDF

Abstract

x_1

and

x_2

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)

and

C(t)

are derived with the use of the rotating-wave approximation (RWA) for sinusoidal fields. Model calculations have indicated that $$,

Γ(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 fields in the near-resonant condition. Applications of our method to step fields are also studied.

DOI: 10.48550/arxiv.1502.07333

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of applied elds on quantum coupled double-well systems

Hideo Hasegawa

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.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.

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Physical ReviewJan 20, 196411 citations

x_1

and

x_2

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)

and

C(t)

are derived with the use of the rotating-wave approximation (RWA) for sinusoidal fields. Model calculations have indicated that $$,

Γ(t)

and

C(t)

(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.

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

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.

z

is the axis of quantization, the projection of the wave function on an eigenstate of

{F}_{x}

oscillates at twice the resonance frequency at exact double quantum resonance. An approximate form is derived for the transition probability near double quantum resonance, showing that our solution is consistent with Salwen's in the limit of well-separated single and double quantum resonances. The theory is applied to the uppermost three substates of the upper hyperfine multiplet of an alkali metal of arbitrary nuclear spin, a case of interest to the analysis of magnetic resonance in optically pumped vapors.