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.
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 x1 and x2 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.
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.
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≤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 ⟨x⟩ vs. ⟨px⟩ and ⟨y⟩ vs. ⟨py⟩ are elliptic, where ⟨⋅⟩ 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.
The solution of the time-dependent wave equation describing the interaction of two coupled spins with a rotating magnetic field in the presence of a steady field, as in the usual magnetic resonance situation, has been reexamined. By extending the method previously developed by Salwen, an exact expression is derived for all components of the wave function describing a three-state system under magnetic-resonance conditions. This solution describes in analytical form all the single and multiple resonances of which the system is capable for fields of any intensity. It is shown that when z is the axis of quantization, the projection of the wave function on an eigenstate of Fx 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.
We propose an analytical approach for computing the eigenspectrum and corresponding eigenstates of a hyperbolic double well potential of arbitrary height or width, which goes beyond the usual techniques applied to quasi-exactly solvable models. We map the time-independent Schrödinger equation onto the Heun confluent differential equation, which is solved by using an infinite power series. The coefficients of this series are polynomials in the quantisation parameter, whose roots correspond to the system's eigenenergies. This leads to a quantisation condition that allows us to determine a whole spectrum, instead of individual eigenenergies. This method is then employed to perform an in depth analysis of electronic wave-packet dynamics, with emphasis on intra-well tunneling and the interference-induced quantum bridges reported in a previous publication Chomet et al (2019 New J. Phys. 21 123004). Considering initial wave packets of different widths and peak locations, we compute autocorrelation functions and Wigner quasiprobability distributions. Our results exhibit an excellent agreement with numerical computations, and allow us to disentangle the different eigenfrequencies that govern the phase-space dynamics.
We study the influence of nodal structures in two-dimensional quantum mechanical densities on wave packet entanglement. This is motivated by our recent study [Entropy, 25, 970 (2023)], which showed that the mutual information derived from the momentum-space probability density of a coupled two-particle system exhibits an unusual time dependence, which is not encountered if the position-space density is employed in the calculation. In studying a model density, here, we identify cases where the mutual information increases with the number of nodes in the wave function and approaches a finite value, whereas in this limit, the linear correlation vanishes. The results of the analytical model are then applied to interpret the correlation measures for coupled electron-nuclear dynamics, which are treated by numerically solving the time-dependent Schrödinger equation.