Essay: Exploring Ratchet Effects with Two-Component Bose-Einstein Condensate



SOLITON RATCHETS IN TWO COMPONENT BOSE EINSTEIN CONDENSATE

Abstract

We study dynamics of two component Bose Einstein condensate trapped by an ratchet typed potential. Using variational method we obtain the equations of motion and get the trajectories of solitons depending upon initial conditions. We verify our findings by making direct numerical solutions of coupled Gross-Pitaevskii equations.

1.Introduction

Ratchet effect can be defined as direct transport in the absence of net external force [1]. Direct transport in periodical systems arise from symmetry broken such as asymmetric lattice or dissipation. Ratchet system assigned originally to Brownian particles is a system consisting of particles moving in a asymmetric potential in the presence of damping [1,2]. The system is under the influence of ac forces of zero average. This effect can be seen generalized for dynamical systems such as solitons. The space-time symmetry in driving potential should be broken for ratchet effect  [3].  Therefore, direct transport is an attractive topic for the studies of ultracold atoms and Bose Einstein condensate (BEC) trapped by asymmetric  potentials [4- 7]. The dynamics of BEC trapped by a toroidal trap has been studied and found that atom-atom interactions leads ratchet effect by breaking quantum symmetry [6]. In Reference [7], quantum ratchets were proposed for quantum information transport between cold atoms trapped by optical superlattices and found that the transport was depend on the lattice site number. The relationship between symetry breaking and directed transport has been experimentally achieve for cold caesium atoms [8]. Ratchet effect has several applications in physics and biology, such as biological transports, moleculer motors, kuantum dots, optical rathets, ve josephson junction [6-12]. Ratchets have been classically investigated in dissipative conditions and also studied for quantum mechanical systems in Hamiltonian regimes [13-17]. Ratchets in cold atom systems are examples to quantum Hamiltonian ratchets.

 Soliton ratchets were introduced in literature [18,19], and there are several studies about this kind of solitons [20-24]. Ratchet effect induces one directional transport of soliton and this is called soliton ratchets. Soliton ratchets behaves like a point particle ratchets in the absence of damping [20] and collective coordinates are responsible for the motion of the center of mass. The soliton ratchets in[18-20] are topological solitons and were examined in symmetric potentials for overdamped situations. The dynamics of topological soliton, taking asymmetric double sine-Gordon equation as a model, was studied in the absence and presence of nose [18]. In this reference, it has been found that soliton transport arises from phase locking between external force and internal mode induced by ratchet potential. The ratchet effect for non-topological solitons in non-dissipative medium was fistly studied in [21] and showed that this effect could exist in non-dissipative systems too which is in contrast to early studies [23]. The authors also found atom number dependent direct transport. Soliton velocity and current depend on atom number. The multisoliton analytical solutions to nonautonomous nonlinear Schrodinger equation have been done in Fourier synthesized optical lattice, and the  interactions between solitons have been investigated in ratchet potential [25].

There are several papers analyzing brigth and dark soliton dynamics under the influence of time modulated potentials for BEC with both one component and two component s [26-33].  For the latter case, BEC is defined by two coupled condensate which means that the system composed of two different kinds of atoms. In this study, we theoretically investigate the dynamics of coupled BEC trapped by ratchet potential. We use both variational and numerical methods. Altough this kind of investigation has been done for one component BEC in [21,22], there is not any study related with two component BEC in literature so far. The paper is organized in this way: In section 2, we introduces the theoretical model. In section 3 and 4, we give the results found from variational and numerical methods, respectively. In tha last section, we sum up the results.

2.Theoretical Model

BEC is a dilute system consisted of weakly interacting ultracold atoms. Two component BEC defines the condensate with two different kinds of atoms. The dynamics of this sytem is investigated with coupled Gross Pitaevskii equations (CGPEs)  which contain two partial diffrential eqautions. The nonlinear term appearing in the equation system ensures the coupling between the equations. CGPEs for non-dissipative system and in the mean field approach are given by [32, 33]

i (∂u_j)/∂t=-1/2  ã€–∂^2 u〗_j/(∂x^2 )+(g_j |u_j |^2+g_12 |u_(3-j) |^2 ) u_j+V(x,t)u_j (1)

where the indice j=1,2 indictaes the components. Since t and x are in units of oscillation period, 1/ω_⊥, harmonic oscillator length, l=√(ℏ/mω_⊥ ), respectively, Equation (1) is dimensionless. uj is the wave function of the component j, V(x,t)  is time and space dependent trapping potential, gj  is the interaction strength between the same kind of atoms, g12 is the interaction strength between diffrent kinds of atoms. In this paper, we assume that interactions between all atoms are equal, g_1=g_2=g_12=g. If g>0, the interactions between atoms are repulsive and Equation (1) has dark soliton solutions. For g<0 which corresponds to attractive interactions between atoms, brigth soliton solutions of Equation (1) exist. The potential V(x,t) is a flashing optical potential, and the time average of its amplitude vanishes. This biharmonic driving potential is given by [21]

V(x,t)=V_0 f(t)[cos⁡(x)+cos⁡(2x+φ)] (2)

where f(t)=sin⁡(ωt)+sin⁡(2ωt).  is oscillation frequency***, V0 is the amplitude** depending on the laser forming lattice. This potential changes sinusoidally by time. Since the sapece dependent part of the potential contains two terms, it is a biharmonic driver. The potential is asymmetric provided φ≠0,π. The maximum simetry occurs at φ=π/2 and with this value space and time symmetry of the potential are broken. This symmetry breaking leads to soliton to have a non-zero time average velocity (cumulative velocity) and ratchet effect. It has been showed in [28] that this kind of potential can be formed experimentally.

In this study, we focus on* brigt soliton (g<0) solutions of CGPEs. Since the system is defined by a coupled equation system, two soliton solutions exist. In the absence of an external potential,  the brigth soliton solutions of  CGPEs has the following form  [34, 35]:

u_j (x,t)=A_j/2 sech(A_j/2(x-x_0j))exp[ix ̇_0j  ((x-x_0j))/2+iθ_j ] (3)

Here xoj is the position of the soliton center of mass, A is the amplitude of the wave packet, θ is soliton phase. The amplitude A is a measure of mass as well [35]. This parameters are time dependent. We investigate the dependence of the dynamic of the solitons on these parameters by variational and numerical methods below.

 3. Variational Approach

Variational approach is a method which is extensively used for studying the dynamics of  BEC trapped by time dependent potentials. Using variational approach, the average Lagrange of CGPEs is given by [33, 34].

L(t)=∫_(-∞)^∞▒[∑_(j=1)^2▒〖{i/2 (u_j^* u ̇_j-u_j u ̇_j^* )-1/2 |(∂u_j)/∂x|^2-1/2 g|u_j |^4-V|u_j |^2 }-g〗 |u_j |^2 |u_(3-j) |^2 ]  dx (4)

With Equation (3), we find for tha average Lagrangian

L(t)=〖3/8 A〗_j x ̇_0j^2-A_j θ ̇_j-1/24 〖A_j〗^3 (1+2g)-V_j-L' (5)

where

L^'=g∫_(-∞)^∞▒〖|u_j |^2 |u_(3-j) |^2 dx〗 (6)

and

V_j=πV_0 f(t)[(cos⁡(x_0j))/(sinh⁡(π/A_j))+2 (cos⁡(x_0j+φ))/(sinh⁡(2π/A_j))] (7)

Time dependent parameteres are  Î²=(θ_j,A_j,x_0j  ), and we find equations of motion by means Euler-Lagrange equation

(∂L(t))/∂β-d/dt (∂L(t)/(∂β ̇ ))=0 (8)

Using Equation (5) and (8), we can write the equations of motion of the system:

(dA_j)/dt=0 (9)

(dx_oj^2)/(dt^2 )=-4/(3A_j )  âˆ‚/(∂x_0j )(V_j+L^') (10)

3/8 x ̇_0j^2-θ ̇_j-(A_j^2)/8 (1+2g)-(∂V_j)/(∂A_j )-(∂L^')/(∂A_j )=0 (11)

According to Equation (9), mass is conserved. Equation (10) is the ordinary diffrential equation which determines the dynamics of BEC. Equation (11)  which we don’t concern in this paper gives soliton phase.

In a weak ratchet potential, the shape of the soliton does not change and the motion of the center of mass specifies the dynamics. Within this condition, the soliton behaves like a classical paricle moving in an effective potential with an effective mass A [20-22]. This approach is known as effective particle approach (EPA). According to EPA, equation  (d^2 x)/(dt^2 )=-(∂V_eff)/∂x  gives ** . As can be seen from Equation (10), the effective potential of coupled BEC  V_(eff,j) is given by

V_(eff,j)=4/3A(V_j+L^') (12)

The coupled system consists of two soliton each of which behaves like a particle moving in own effective potential. For symmetric solitons, taking A_1=A_2=A and x_01=y/2 , x_02=-y/2 , and writing these parameters in Equation (10), we can find

(dy^2)/(dt^2 )=-d/dy (16/3A){V_0 f(t)Ï€(cos⁡(y/2)/sinh⁡(2Ï€/A) +2  cos⁡(φ±y)/sinh⁡(2Ï€/A) )-〖gA〗^3/(2〖sinh⁡(Ay/2)〗^2 )(1-Aycoth(Ay/2)/2)} (13)

y is the distence between two solitons.  Therefore, Equation (13) explains how effective potentials changes with the distence between the solitons. This variaiton is plotted in Figure 1. Figure 1-a, and 1-b shows effective potentials for the first and second component of condensate. We take φ≠π/2 and f(t)=1. The difference between two potentials arises from the term  cos⁡(φ±y) in Equation (13). In the absence of the first term, assuming V0=0 , the effective potential of the system is like a potential well. The deep of the potential well increases with A, and the distance at which soliton interact each other decreases. Therefore, the second term determines the deep of the potential. For the case g=0, that is only the first term is considered, the effective potentials become periodic. With these two terms  we find the plots given in Figure 1. As the amplitude of the potential increases with increasing V0, its deep increases with A. Since the symmetry of the potential decreses with decreasing A, direct transport cannot exist for small A values.

Figure 1: The variation of effective potentials with the distance between solitons. We take A=3, V0=0.3, g =-1,  Ï†â‰ Ï€/2 and f(t)=1.

The studies in which one component BEC has been studied shows that the existence of soliton ratchets depends on the initial position of center of mass,  x0j(0), and effective mass A [21, 22]. The similar results are achieved for coupled BEC in this paper. Depending on these parameters,  soliton makes direct transport with a non-zero time avege velocity or oscillates around its initial position with zero velocity (or with a velocity which is very close to zero). Time average velocity is called cumulative velocity. In EPA, cumulative velocity is given by (v(t)=(dx_0)/dt)

v ̅=(1/T)∫_0^T▒〖v(t)dt〗 (14)

( T≈〖10〗^3×2π/ω ) [21,22]. Cumulative velocity can be calculated numerically using Equation (10) and Equation (14). Altough analytical results cannot be achieved, graphics plotted using numerical datas give informtion about the dynamics of the system. In Figure 2, we plot the variation of the cumulative velocity versus initial condition x0j(0) for a constant value A. Figure 2-a shows the cumulative velocity of the first soliton for A=10, Şekil 2-b shows the cumulative velocity of ths second soliton for A=10. According to these plots, we can say that the dependence of cumulative velocity on x0j(0) is not regular. Instead, it fluctuates. The soliton makes direct transport with initial conditions at which its cumulative velocity is non-zero**. Otherwise, it oscillates with zero velocity. We plot some examples Figure 2-c and 2-d to verify these results. Figure 2-c plots two soliton trajectories changing with time for x0j(0)=0.9. Solid line is relating to the first soliton, and its cumulative velocity at this initial condition (x01(0)=0.9) is found to be 0.112. Soliton makes diredt tranport under this condition. The dashed line belongs to the second soliton, and its cumulative velocity with initial condition x02(0)=-0.9 is 0.0035. This velocity is very small compared with the velocity of the first soliton. It can be understood from the figure that soliton makes oscillation instead direct transport. Figure 2-d shows the variaton of the first soliton with time for x0j(0)=1.2. As can be also seen from Figure 2-a, the velocity of the first soliton is zero. It oscillates, but the amplitude of the oscillation is very small. We calculated numerically the cumulative velocity of the second soliton as 0.0014. It also makes oscillation, but amplitude of its oscillation is higher than that of the first soliton.

In Figure 3, we investigate the variation of the cumulative velocity with effective mass A taking x0j(0) constant. We choose x01(0)=2 for the first soliton and x02(0)=-2 for the other one. It is seen from Figure 3-a that the cumulative velocity of the first soliton is non-zero between A=2 and A=6 while it is zero for other regions. The first soliton velocity does not change with A regularly, and we find similar results at different x0j(0) conditions. The cumulative velocity of the second soliton is zero for the region A<2. For A>2, it increases first, and then it gives a constant value. It is interesting that this result obtained for the second soliton is almost the same with that found for one component BEC [21].  As it can be seen from Figure 3-a and 3-b, for A=2 and x0j(0)=2, both soliton have zero velocity. The trajectories of solitons under these conditions are plotted in Figure Åžekil 3-c. Both soliton oscillates. We plot Figure 3-d for A=6 and x0j(0)=2. The trajectories we plot are consistent with Figure 3-a and 3-b.

Şekil 2: a) The cumulative velocity of the first soliton versus x01(0). b) The cumulative velocity of the second soliton versus x02(0) c) The trajectory of center of mass of both soliton versus time for x0j(0)=0.9. d) The trajectory of center of mass of both soliton versus time for x0j(0)=1.2. Solid line belongs to the first soliton and dashed line belongs to the second one. We take A=10.

Åžekil 3: a) The cumulative velocity of the first soliton versus A for x01(0)=2. b) The cumulative velocity of the second soliton versus A for x02(0) =-2 c) The trajectory of the first soliton versus time for A=2 ve x0j(0) =2. d) The trajectory of the second soliton versus time for A=6 ve x0j(0) =2. Solid line belongs to the first soliton and dashed line belongs to the second one.

4. Numerical Approach

In this section, we investigate soliton dynamics by numerical methods. We solve directly Equation (1) by using Split Step Fourier Method (SSFM). We choose time step dt=0.0012 and space step dx=0.04. We take bright soliton solution Equation (3) as the initial wave function with x ̇_0j (0)=0, θ_j (0)=0. Figures 4(a-d) plot the trajectories of solitons versus time***. We use the same parameters employed in the previous section for comparision. The upper trajectory belongs to the first soliton, while the lower trajectory belongs to the second soliton. Figure 4-a is plotted for A=10 and x0j(0)=0.9 , Figure 4-b is plotted for A=10 and x0j(0)=1.2. The plots are compatible with Figures 2-c  and Figure 2-d which are drawn with the same parameters. Figure 4-c shows the trajectories for A=2 and x0j(0)=2, Figure 4-d shows the trajectories for A=6 and x0j(0)=2. The soliton trajectory in Figure 4-c is similar to the one Figure 3-c, but there seems to be some differences between them. At these initial positions, the center of mass of the solitons are very close the each other. Since the interactions are attractive, solitons which velocities are nearly zero interacts. This may lead to the differences between the results obtained by two methods. In Figure 4-d, the second soliton moves away from the first one. Since interactions between them is less, Figure 4-d  is compatible with Figure 3-d. The dynamics of the system can be analysied better with long time, however, it needs more computational efforts.

Åžekil 4: The trajectory of both solitons versus time. The upper trajectory belongs to the first soliton, while the lower trajectory belongs to the second soliton. The parameters are a) A=10, x0j(0)=0.9 b) A=10,  x0j(0)=1.2 c) A=2, x0j(0)=2, d) A=6, x0j(0)=2. These results are compatible with the results of variational methods.  

5. Conclusions

In this paper, we study theoretically mass dependent direct transport for two component BEC. The dynamics of two component BEC trapped by a weak ratchet potantial is investigated by variational and numerical methods. As in the case of one component BEC, soliton dynamics is dependent on mass and initial positions. The cumulative velocity of the solitons changes according to these conditions. One or two solitons can make direct transport at the same time with non-zero cumulative velocities. If cumulative velocity is zero, ratchet effect does not exist and solitons oscillates around their initial positions. If cumulative velocity is close to zero, solitons again make oscillation, but the oscillation amplitude is higher than the former case. The results we get from variaitonal and numerical methods are compatible with each other.

This study can be extended by taking into account soliton scattering and multisoliton solutions. Since the results of this paper can be confirmed by experimental studies, it can motivate new studies about soliton dynamics.

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