Effects of applied fields on quantum coupled double-well systems

Effects of applied fields on quantum coupled double-well systems

Hideo Hasegawa hideohasegawa@goo.jp Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan
July 19, 2019

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 and of particles (), the correlation () and the concurrence () expressing a degree of the entanglement of the coupled DW system, are obtained. Analytical expressions for , and are derived with the use of the rotating-wave approximation (RWA) for sinusoidal fields. Model calculations have indicated that , and 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.

Keywords: coupled double-well potential, Razavy’s potential, rotating-wave approximation, entanglement

03.65.-w, 03.67.Mn

I Introduction

Extensive studies have been made for quantum double-well (DW) systems in physics and chemistry where a tunneling is one of intrigue quantum phenomena Tannor07 (). Effects of applied fields on DW systems have been studied (for review see Grifoni98 ()). Various phenomena such as a coherent destruction of tunneling by applied fields were pointed out Grossmann91 (). The two-level (TL) system which is a simplified model of a DW system, has been employed for a study on qubits which play important roles in quantum information and quantum computation. Many theoretical studies on effects of fields applied to single and coupled qubits have been reported with the use of the TL model Storcz01 (); Satanin12 (); Bina14 (); Pal14 (). In contrast to the simplified TL model, studies on coupled DW systems which are commonly described by the quartic potentials are scanty Gupta06 (), because a calculation of such a system is much tedious than that of the coupled TL model, even for the absence of applied fields. One of difficulties in studying coupled DW systems is that one cannot obtain exact eigenvalues and eigenfunctions of the Schrödinger equation for quartic DW potential. Then one has to apply various approximate approaches such as perturbation and spectral methods to quartic DW models. Razavy Razavy80 () proposed quasi-exactly solvable hyperbolic DW potential for which one may exactly determine a part of whole eigenvalues and eigenfunctions. A family of quasi-exactly solvable potentials has been investigated Finkel99 (); Bagchi03 ().

Recently the present author Hasegawa15 () has investigated the relation between the entanglement and the speed of evolution in coupled DW system described by Razavy’s potential. It would be interesting to study effects of applied fields on coupled DW systems with Razavy’s potential, which is the purpose of the present study. Some sophisticated methods like the Floquet approach have been developed in solving Schrödinger equation for time-dependent periodic fields. In order to treat the periodic as well as non-periodic dynamical fields, we solve in this study the time-dependent Schrödinger equation by a straightforward method. An advantage of our approach is that we may exactly determine eigenvalues and eigenfunctions of driven coupled DW systems. We calculate expectation values of various quantities such as positions of particles, the correlation and the concurrence which expresses a measure of the entanglement of a coupled DW system. Effects of applied fields are analytically studied with the use of the rotating-wave approximation (RWA) which has been widely adopted for sinusoidal periodic field, in particular for the TL model. The validity of the RWA may be examined by a comparison between results of the RWA and exact ones evaluated by numerical methods.

The paper is organized as follows. In Sec. II, the calculation method employed in our study is explained with a brief review on Razavy’s hyperbolic potential Razavy80 (). Equations of motion for populations of four energy levels are obtained from the time-dependent Schrödinger equation of driven coupled DW systems. Expressions for expectation values of particle positions, the correlation and the concurrence are calculated. For sinusoidal fields, we present their analytical expressions by using the RWA. In Sec. III, we report model calculations with the use of the RWA and numerical methods when the sinusoidal fields are applied to the initial ground state. In Sec. IV, calculations are made for sinusoidal fields applied to the initially wavepacket state. Our method is applied also to the case of applied step fields. Sec. V is devoted to our conclusion.

Ii Coupled double-well system with Razavy’s potential

ii.1 Calculation method

We consider coupled two DW systems whose Hamiltonian is given by




Here and stand for coordinates of two distinguishable particles of mass , signifies a DW system with Razavy’s potential Razavy80 (), means the coupling term with an interaction , and includes the time-dependent applied field whose explicit form will be given shortly [Eq. (40) or (88)]. The case of which is more general than Eq. (4) will be studied in the Appendix. The potential with adopted in this study is plotted in Fig. 1(a). Minima of locate at with and its maximum is at .

Firstly we consider only in Eq. (2), whose eigenvalues are given Razavy80 ()


and whose eigenfunctions are given by


() denoting normalization factors. Eigenvalues for the adopted parameters are , , and . Both and locate below as shown by dashed curves in Fig. 1(a), and and are far above . In this study, we take into account only the lowest two states with and , which is justified because of () (). Figure 1(b) shows eigenfunctions of and , which are symmetric and anti-symmetric, respectively, with respect to the origin.

Figure 1: (Color online) (a) Razavy’s DW potential (solid curve), dashed and chain curves expressing eigenvalues of and , respectively, for [Eq.(5)]. (b) Eigenfunctions of (solid curve) and (dashed curve).

Secondly we include the coupling term in Eq. (3). With basis states of , , and , the energy matrix of the Hamiltonian of is expressed by




Eigenvalues of the energy matrix of are given by




Corresponding eigenfunctions are given by




We hereafter assume Hasegawa15 (). The dependence of () is shown in Fig. 2 of Ref. Hasegawa15 ().

Thirdly we take into account for an applied field in Eq. (4). The energy matrix of the time-dependent total Hamiltonian () with basis states of , , and is expressed by


Alternatively, the energy matrix of may be expressed with basis states of , , and in Eqs. (22)-(25) by



In our following analysis, we adopt the energy matrix given by Eq. (LABEL:eq:A12) because it has more transparent physical meaning than Eq. (27). We expand the eigenstate of in terms of () with the time-dependent expansion coefficients as


where expansion coefficients satisfy the relation


The Schrödinger equation: becomes


Multiplying from the left side of Eq. (33) and integrating it over and , we obtain equations of motion for




With the use of the energy matrix in Eq. (LABEL:eq:A12), equations of motion for become [the argument in is hereafter suppressed]


When we apply the sinusoidal field given by


Eqs. (36)-(39) become


where and denote magnitude and frequency, respectively, of the applied field.

Rotating-wave approximation (RWA)

In the rotating-wave approximation (RWA) where only terms with are taken into account in Eqs. (41)-(44), we obtain


For a given initial condition of at (), we obtain the solution of Eqs. (45)-(47)




where stands for Rabi’s frequency given by


For a later purpose, we may rewrite and as




where , , and are time independent.

ii.2 Various physical quantities

Once time-dependent are obtained from Eqs. (36)-(39) or from Eqs. (55) and (56), we may evaluate various physical quantities such as expectation values, the correlation and concurrence.

(1) Expectation values

Time-dependent expectation values of and are expressed by


Substituting Eqs. (55) and (56) to Eq. (63), the expectation value in the RWA with is given by


which includes time-dependent components with frequencies of besides of the applied field.

(2) Correlation

The correlation is defined by Hasegawa15 ()


which is unity at .

In the RWA, the correlation is given by


With the use of Eqs. (55) and (56), the correlation in the RWA is expressed by


which consists of components with frequencies of , and . When we take the average of over a long period, oscillating terms vanish and its average becomes


(3) Concurrence

Substituting Eqs. (22)-(25) into Eq. (31), we obtain




where with . The concurrence of the state given by Eq. (71) is defined by Wootters01 ()


The state given by Eq. (71) becomes factorizable if and only if the relation: holds. Substituting Eqs. (72)-(75) into Eq. (76), we obtain the concurrence Hasegawa15 ()


In the RWA with , the concurrence is given by


With the use of Eqs. (55) and (56), Eq. (78) becomes




which include contributions from multiple components with frequencies of , , , , , and . The concurrence averaged over a long period is given by

Figure 2: (Color online) Time developments of (solid curves), (dashed curves) and (bold solid curve) for (a) , (b) and (c) in exact calculations, and those for (d) , (e) and (f) in the RWA ( and ), bottom curves in (a)-(f) expressing applied fields.
Figure 3: (Color online) (a)-(c) for (a) , (b) and (c) ; (d)-(f) for (d) , (e) and (f) ; (g)-(i) for (g) , (h) and (i) , solid and dashed curves expressing results of exact and RWA calculations, respectively ( and ). Bottom curves in (a)-(i) denote applied fields.

Iii Model calculations

Assuming the initial ground state given by


we have made numerical calculations by changing model parameters of , and .

Figure 4: (Color online) Time developments of (solid curves), (dashed curves) and (bold solid curve) for (a) , (b) , and (c) in exact calculations, and those for (d) , (e) , and (f) in the RWA ( and ), bottom curves in (a)-(f) expressing applied fields.

iii.1 dependence

Figures 2(a), (b) and (c) show time developments of populations of in levels () for , 0.01 and 0.02, respectively with and (=0.07431) obtained by numerically solving Eqs. (41)-(44) which is hereafter referred to as an exact calculation: note that because of and . For comparison, relevant results obtained in the RWA are plotted in Figs. 2(d)-(f). Exact calculations in Fig. 2(a) show that for a field with , magnitude of is decreased while that of is increased at with small