Nonlinear Landau–Zener tunneling under biharmonic driving

Physica B 406 (2011) 1795–1798

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Nonlinear Landau–Zener tunneling under biharmonic driving Xiaobing Luo a,, Senping Luo b, Jun Xu c, Jinwang Chen a a b c

Department of Physics, Jinggangshan University, Ji’an 343009, China Department of Mathematics, Jinggangshan University, Ji’an 343009, China Center of Experimental Teaching for Common Basic Courses, South China Agriculture University, Guangzhou 510642, China

a r t i c l e i n f o

abstract

Article history: Received 13 June 2010 Received in revised form 23 January 2011 Accepted 10 February 2011 Available online 16 February 2011

We address the response of a two-mode Bose–Einstein condensate to a biharmonic high-frequency driving field that directly couples the two mode. By use of a high-frequency approximation we find that asymmetric biharmonic high-frequency driving yields new effects on Nonlinear Landau–Zener tunneling as compared to a symmetric off-diagonal modulation. In particular, we detect an unclosed loop structure and the disappearance of the energy level below the loop structure. & 2011 Elsevier B.V. All rights reserved.

Keywords: Bose–Einstein condensate Landau–Zener process Off-diagonal driving

1. Introduction Significant research efforts have been devoted to the nonlinear dynamics of a Bose–Einstein condensate (BEC) driven by an external periodic driving field in recent years. One main motivation is to understand the combined effects of periodic driving force and the self-interaction of cold atoms on the tunneling dynamics such as chaotic atomic tunneling [1–8], photon-assisted tunneling [9–11], coherent control of the BEC self-trapping [12–18], controlled Mott-insulator transitions associated with a BEC in an optical lattice [19–22], to name a few. It is well known that a high-frequency diagonal modulation (e.g., high-frequency tilting of a double-well potential) can only rescale effectively the tunneling parameter by a factor of the Bessel function [14,23,24]. In Refs. [25,26], Zhang et al. have studied how an adiabatic Landau–Zener(LZ) tunneling process of a two-mode BEC may be manipulated by a sinusoidal/cosine single-frequency field that directly couples the two modes. Using a high-frequency approximation, Zhang et al. have showed such an off-diagonal sinusoidal/cosine modulation can realize the complete suppression of nonlinear Landau–Zener (NLZ) tunneling. The topic concerning a quantum particle or BEC subjected to biharmonic modulation enjoys current interest [27–32]. It is well known that there exist two types of biharmonic driving field. One is symmetric driving, the other has an asymmetric sawtooth form, which breaks the time-reversal symmetry and is generally  Corresponding author. Tel.: + 867968100489.

E-mail address: [email protected] (X. Luo). 0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.02.030

known as harmonic mixing driving (for more details see Ref. [33]). The purpose of this paper is to consider the effects of a periodic biharmonic driving field on NLZ processes, and compare the difference between the symmetric and asymmetric case. In this work, we consider a two-mode BEC under a symmetric or asymmetric biharmonic driving field that directly couples the two modes (hence called an off-diagonal driving hereafter), rather than modulating their energy bias. Like the single-frequency case, the new additional level below the loop induced by the symmetric biharmonic driving field also offers a means to circumvent the loops structure which is observed in non-driven NLZ models, and hence totally suppresses the NLZ transition. In contrast, an effective stationary Hamiltonian approach reveals that the loop structure emerging in non-driven NLZ models can no longer be closed due to asymmetric high-frequency modulation. Therefore, the adiabatically following of the energy level necessarily breaks down at the position where the spectrum of loop structure is also breaking down.

2. Two-mode system under symmetric or asymmetric offdiagonal modulation The nonlinear two-mode system under off-modulation is described by i‘

d dt

  a b

¼ HðtÞ

  a b

,

ð1Þ

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X. Luo et al. / Physica B 406 (2011) 1795–1798

H012 ¼ ½Dc igsinðyÞ þ csin2 ðyÞða0 b0 a0 b0 Þ þicsinðyÞcosðyÞðja0 j2 jb0 j2 Þ=2,

where 1 2

HðtÞ ¼

g þ cðjbjjaj2 Þ

Dc þ f ðtÞ

Dc þf ðtÞ

gcðjbjjaj2 Þ

! :

ð2Þ

Here g ¼ at denotes an energy bias between two modes and is being varied at rate a. jaj2 and jbj2 represent occupation probabilities on the two modes, with the normalization condition taken as jaj2 þ jbj2 ¼ 1. c characterizes the strength of the selfinteraction of the BEC, proportional to the number of bosons and the s-wave scattering length. Dc represents the static coupling between the two modes. f ðtÞ ¼ AcosðotÞ þBcosð2ot þ fÞ is a zerobias biharmonic driving field with period T ¼ 2p=o, and f is the relative phase between the o and 2o fields. For f ¼ 0, the driving is symmetric under t!t inversion. The time-reversal symmetry is maximally broken if f ¼ p=2, modeling an asymmetric sawtooth driving field. Throughout we set ‘ ¼ 1 and hence all parameters are scaled dimensionless variables. Within the BEC context, the above Hamiltonian may be experimentally realized in several ways. For example, one may consider a BEC in a doublewell potential, with the barrier height periodically modulated, or a BEC in an optical lattice occupying two bands, with the welldepth of the optical lattice periodically modulated. One may also consider a BEC occupying two hyperfine levels, where the offdiagonal modulation may be realized by modulating the intensity of the coupling field. It should be noted that our system may be realized in nonlinear optics by using two nonlinear optical waveguides [34–37]. For f(t) ¼0, the above Hamiltonian reduces to the standard model of non-driven NLZ processes. Therein the energy loop structure is discovered within the context of NLZ tunneling [38,39]. Such a loop structure, absent in linear systems, directly leads to break down of adiabatic following of the nonlinear system. As shown below, we will consider the effects of a periodic biharmonic driving field on NLZ processes, and compare the difference between the symmetric and asymmetric case. Without loss of generality, we will restrict ourselves to the c 4 0 case. Our approach and results will be very similar if one considers instead the c o 0 case. When the driving field is turned on, the NZL dynamics under the condition ocg,c, Dc may be drastically modified. To expose Heff

1 ¼ 2

A

In the high-frequency limit ocfg,c, Dc g such that the oscillation in y is much faster than the natural time scale of the system as characterized by g,c, and Dc , the amount of change in a0 and b0 during a period, T ¼ 2p=o, can be regarded as being infinitesimal, at most of order 1=o. Thus Eq. (5) can be integrated approximately over a period 2p=o by assuming that a0 ðtÞ and b0 ðtÞ are constants, since the rapidly terms are separated out by the transformation (3). Note that experimentally the driving frequency of high-frequency driving filed should not conflict with the two-mode approximation. By use of the following expressions     Z T 1 X 1 A B cosðyÞ dt ¼ J2m J cosðmfÞ T 0 o m 2o m ¼ 1 1 T 1 T 1 T 1 T

Z

sinðyÞ dt ¼

0

Z

sinðotÞ þ

B sinð2ot þ fÞ: 2o

ð3Þ

sinð2yÞ dt ¼

0

Z

H ðtÞ ¼

H012

H021

H022

1 X m ¼ 1

T

sin2 ðyÞ dt ¼

0

Z

J2m

m ¼ 1 T

T

0

    A B J sinðmfÞ o m 2o

J2m

  2A

o

Jm

  B

o

sinðmfÞ

" #     1 X 1 2A B 1 J2m Jm cosðmfÞ 2 o o m ¼ 1

" #     1 X 1 2A B 1þ cos ðyÞ dt ¼ J2m J cosðmfÞ , 2 o m o m ¼ 1 2

ð8Þ

where Jm is the m th-order Bessel function of the first kind, and upon a time averaging of Eq. (5), we find the effective equations of motion     a d a i ¼ Heff , ð9Þ dt b b where

g1 ¼ g

! ð10Þ

1 X

1 X m ¼ 1

cX ¼ ð5Þ

with H011 ¼ ½gcosðyÞ þ ccos2 ðyÞðjb0 j2 ja0 j2 Þ þ icsinðyÞcosðyÞða0 b0 a0 b0 Þ=2,

    A B J cosðmfÞ o m 2o

J2m

    A B J sinðmfÞ o m 2o

    1 c X 2A B J2m J sinðmfÞ 2 m ¼ 1 o m o

" #     1 X c 2A B 1 J cosðmfÞ J2m cY ¼ 2 o m o m ¼ 1

! ð6Þ

J2m

m ¼ 1

ð4Þ

where H011

1 X

T

g2 ¼ g

Then one finds the equations of motion for ða0 ,b0 Þ    0 a d a0 i ¼ H0 ðtÞ 0 , 0 dt b b

0

ð7Þ

with

where

o

H022 ¼ ½gcosðyÞccos2 ðyÞðjb0 j2 ja0 j2 ÞicsinðyÞcosðyÞða0 b0 a0 b0 Þ=2:

g1 þcZ ðjbj2 jaj2 Þ þ icX ða bab Þ Dc ig2 þ cY ða bab Þ þ icX ðjaj2 jbj2 Þ   2 2  Dc þig2 cY ða bab ÞicX ðjaj jbj Þ g1 cZ ðjbj2 jaj2 ÞicX ða bab Þ

the new physics, we take advantage of the transformation    !   isin 2y  a0  cos 2y a  y y , ¼ isin 2 cos 2 b0 b

y¼

H021 ¼ ½Dc þ igsinðyÞcsin2 ðyÞða0 b0 a0 b0 ÞicsinðyÞcosðyÞðja0 j2 jb0 j2 Þ=2,

cZ ¼

" #     1 X c 2A B 1þ J2m Jm cosðmfÞ : 2 o o m ¼ 1

ð11Þ

Note that we have replaced a0 and b0 by a and b in effective Eqs. (9) and (10), respectively. Comparing the effective Hamiltonian (10) with the original one in Eq. (2) for f(t)¼0, we see that

X. Luo et al. / Physica B 406 (2011) 1795–1798

the nonlinear parameter cZ can be regarded as a re-scaled parameter c, and some other new terms containing cX ,cY , g2 arise as a surprise. Apparently, these newly defined parameters reflect the action of the high-frequency off-diagonal modulation. As listed in Eq. (11), there are five effective parameters with some obvious relations. The ratios g1 =g2 , cX/cY, cY/cZ are easily adjustable by tuning the parameters of the driving field, such as the driving amplitudes A and B, the driving frequency o, and the phase f. For the case of symmetric driving (f ¼ 0), the terms containing cX and g2 in the effective Hamiltonian (10) vanish. Such an effective Hamiltonian (10) is analogous to the previously studied one for the case of sinusoidal/cosine single-frequency driving, except that the factor of g is different [25,26].

3. Detailed results For simplicity, all the effective parameters listed in Eq. (11) are P scaled by the sum 1 m ¼ 1 J2m ðA=oÞJm ðB=2oÞcosðmfÞ such that g1 is replaced by g. We present the eigenvalues of Heff as a function of g, which are shown by the discrete squares in Fig. 1, by solving     a a Heff ðgÞ ¼ meff ðgÞ : ð12Þ b b Here the eigenvalues meff ðgÞ are also called adiabatic energy levels. As expected in Fig. 1(a) for the case of symmetric two-frequency driving (f ¼ 0), the eigenvalues meff ðgÞ display the same structure 1

1797

as the ones in the case of single-frequency driving. As shown in Fig. 1(a), level bifurcation takes place on the lowest branch and additional level can be below the loop structure. The results of asymmetric driving (f ¼ p=2) are shown in Fig. 1(b) where we see appearance of an unclosed loop structure in the lower energy level, and disappearance of the additional level below the loop. We stress that there is always such an unclosed loop structure in energy level for the asymmetric high-frequency modulation, as long as we choose parameters permitting the conditions cZ 4 Dc , cX a0 and g2 a 0. So, how does the different topological structures shown in Fig. 1 affect the adiabatic dynamics? To answer the question we compute the time-evolving expectation value of Heff in Fig. 1 (red lines) by numerically solving Eq. (9). Suppose we start with a state on the lower adiabatic level, and move it up along the branch by changing g so slowly such that little tunneling to the upper level is generated. For symmetric driving shown in Fig. 1(a), the system’s state is found to move along the lowest level up to the bifurcation point. When g increases beyond the bifurcation point where new level emerges below the loop, the new level (two-fold degenerate) becomes the lowest and the state follows the new level. As g increases further, the additional level induced by the driving field finally disappears and system reaches the nondegenerate lowest level again, thus completely suppressing the NLZ tunneling. For asymmetric driving (f ¼ p=2) instead shown in Fig. 1(b), since the loop structure can be no longer closed, the state remains in the course moving up in energy until hitting the terminal point, where it has no way to go any further except to jump to the upper and lower levels, thus leading to rapid oscillation of the expectation value of Heff.

0.5 1

μeff

0

0.8

–0.5 |a|2 (|b|2)

–1 –1.5 –2 –1.5

–1

–0.5

0 γ

0.5

1

0.6 0.4 0.2

1.5

0

1

0

500

1000

1500 t

2000

2500

3000

0

500

1000

1500 t

2000

2500

3000

μeff

0.5 0

1.4

–0.5

1.2

–1

1 |a|2 (|b|2)

–1.5 –2 –2.5 –3 –1.5

0.8 0.6 0.4

–1

–0.5

0 γ

0.5

1

1.5

Fig. 1. Eigenvalues of the effective Hamiltonian in Eq. (10) (squares) as a function of g for (a) symmetric driving (f ¼ 0, and hence g2 ¼ 0,cX ¼ 0:0), and (b) asymmetric driving (f ¼ p=2) with the choice of g2 ¼ 1:2g,cX ¼ 0:1. The effective Hamiltonian is obtained with a high-frequency approximation. The red lines represent the time-evolving expectation value of Heff when g increases adiabatically. The other parameters are Dc ¼ 1,cZ ¼ 2,cY ¼ 3, a ¼ 0:001. All variables are scaled and hence dimensionless. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

0.2 0

Fig. 2. Transition probability jaðtÞj2 (black solid lines) and jbðtÞj2 (red dashed lines) as a function of time for (a) symmetric driving (f ¼ 0), and (b) asymmetric driving (f ¼ p=2). The other parameters are the same as in Fig. 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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X. Luo et al. / Physica B 406 (2011) 1795–1798

In Fig. 2, we depict in detail the time-evolving occupation probability jaðtÞj2 (black solid lines) and jbðtÞj2 (red dashed lines) as a function of time with the initial state put on the lowest level. For the case of symmetric driving, if g changes adiabatically, a complete quantum population inversion is realized, thus completing LZ process, as shown in Fig. 2(a). Contrary to the case of symmetric driving, asymmetric driving field induces that the actual time-evolve state follows the adiabatic energy level until reaching the point where the adiabatic energy level breaks down, and then undergoes rapid transition, as shown in Fig. 2(b).

4. Conclusion In summary, we have examined some interesting features of driven NLZ processes, by considering symmetric and asymmetric two-frequency driving field. Using high-frequency approximation, we have showed that driven NLZ dynamics can be effectively described by a stationary Hamiltonian. For symmetric two-frequency driving, the field can induce a new energy level below the loop structure, analogous to the single-frequency case, thus suppressing the undesired NLZ tunneling. For asymmetric twofrequency driving, the spectrum of the effective Hamiltonian displays an unclosed loop structure as a surprise. Thus, the adiabatic following of the energy level breaks down at the point where the adiabatic energy level is also breaking down.

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Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant no. 10965001), the Natural Science Foundation of Jiangxi Province (Grant no. 2010GQW0033), and the Principal Foundation of South China Agriculture University. References [1] F.Kh. Abdullaev, R.A. Kraenkel, Phys. Rev. A 62 (2000) 023613-1. [2] C. Lee, W. Hai, L. Shi, X. Zhu, K. Gao, Phys. Rev. A 64 (2001) 053604-1. [3] B. Xia, W. Hai, G. Chong, Phys. Lett. A 351 (2006) 136.

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