Noise assisted transport in the Wannier-Stark system
Stephan Burkhardt, Matthias Kraft, Riccardo Mannella, Sandro Wimberger
aa r X i v : . [ qu a n t - ph ] F e b Noise assisted transport in the Wannier-Starksystem
Stephan Burkhardt , Matthias Kraft , , Riccardo Mannella and Sandro Wimberger Institut f¨ur theoretische Physik, Universit¨at Heidelberg, Philosophenweg 19, 69120Heidelberg, Germany Blackett Laboratory, Imperial College London, South Kensington Campus,SW7 2AZ London, UK Dipartimento di Fisica ‘E. Fermi’, Universit`a di Pisa, Largo Pontecorvo 3, 56127Pisa, Italy
Abstract.
We investigated how the presence of an additional lattice potential, drivenby a harmonic noise process, changes the transition rate from the ground band to thefirst excited band in a Wannier-Stark system. Alongside numerical simulations, wepresent two models that capture the essential features of the dynamics. The firstmodel uses a noise-driven Landau-Zener approximation and describes the short timeevolution of the full system very well. The second model assumes that the noiseprocess’ correlation time is much larger than the internal timescale of the system,yet it allows for good estimates of the observed transition rates and gives a simpleinterpretation of the dynamics. One of the central results is that we obtain a way tocontrol the interband transitions with the help of the second lattice. This could readilybe realized in state-of-the-art experiments using either Bose-Einstein condensates oroptical pulses in engineered potentials.
1. Introduction
During the last years, there have been enormous advances in the experimental controlover Bose-Einstein condensates (BECs) in optical lattices. States can be prepared witha high degree of control [1] and the systems can be isolated from external perturbationsfor long timescales [2]. This fine control over the system allows to investigate quantummechanical effects that were previously very difficult to observe in experiments. Amongthese effects are Bloch oscillations, Wannier-Stark ladders, and Landau-Zener tunneling[3–5]. Dynamics similar to the ones observed in dilute BECs can nowadays also berealized using purely optical means [6, 7].These advances have also paved the way to studying transport properties of spatially[8–12] or temporally disordered systems [13]. Disorder is to some degree present inalmost all naturally occurring systems; in particular it plays a large role in solid statesystems. New effects caused by disorder, such as Anderson localization have thereforepotential ramifications in a wide range of fields. oise assisted transport in the Wannier-Stark system et al. [29] reports results based on numericalsimulations of our system. In this paper, we will investigate the effects of the parametersof the noise process more thoroughly and introduce two different models that indeedexplain many facets of the behaviour of the noise-driven problem.In section 2 we will introduce the system studied in this paper and discuss it in thecontext of non-interacting BECs in optical lattices. We then introduce two models thatallow for an interpretation of the behaviour of the system in section 3. An extensivenumerical study is then presented in section 4; these results are compared to the modelspresented in the previous section and we give an outlook on weakly interacting BECs.We close with a short conclusion of the results in section 5.
2. Noisy Wannier-Stark system
In dimensionless units the Hamiltonian studied in this work readsˆ H = − ∂ x + V [cos( x ) + cos( α [ x − φ ( t )])] − F x, (1)where V gives the potential depth of the lattice, F is the Stark force, α is a realnumber ‡ and φ ( t ) is a so-called harmonic noise process (introduced below). For aconversion between our dimensionless units, SI-units and common experimental units(applied e.g. in [5, 29, 30]), see appendix Appendix B. ‡ If this number α is irrational, the resulting system is quasi-periodic, similar to those used to studyAnderson localization by Roati et al. [12]. oise assisted transport in the Wannier-Stark system | ψ i , which is sufficiently localized in momentum space (i.e. much smaller than theBrillouin zone) and prepared in the ground band, undergoes Bloch oscillations with theperiod T B = 1 /F and partially tunnels into the first excited band everytime it reachesthe edge of the Brillouin zone (corresponding to an avoided crossing of the respectiveenergy bands). The characteristic time scale and frequency of the system are thus theBloch period T B and Bloch frequency ω B = 2 π/T B [31]. The second potential termrenders the system stochastic (via the phase φ ) and spatially disordered in the staticcase ( φ = const) if α / ∈ Q .Possible experimental realizations of such systems are for example BECs in opticallattices [5, 12]. Similar Hamiltonians are for example used for precision measurementsof forces acting on the atoms of the BEC [33]. If the BEC is either dilute enoughand/or interactions are suppressed by tuning with the help of a Feshbach resonance,their behavior can be well described by a single particle Hamiltonian as the one in(1) (see, e.g., [5, 30, 34]). In order to better understand how the observed effects wouldmanifest in BECs with non-vanishing interactions, we will also look at simulations usingthe Gross-Pitaevskii equation in section 4.2. There are also recent experiments usingoptical pulses in engineered potentials that realize similar Hamiltonians [6, 7].Let us briefly summarize the most important properties of harmonic noise.Harmonic noise can be imagined as a damped harmonic oscillator driven by gaussianwhite noise. It is defined via the two coupled stochastic differential equations [21, 22]˙ φ = µ (2)˙ µ = − µ − ω φ + √ T ξ ( t ) , (3)where Γ is the damping coefficient, ω sets the characteristic frequency of the noiseand T determines the noise strength. ξ ( t ) represents gaussian white noise [35, 36]. Theequilibrium distributions of the noise variables are of bivariate gaussian type with firstand second moments given by [22, 35] h φ i = 0 , h µ i = 0 , h φ i = Tω and h µ i = T . (4)Just as the damped harmonic oscillator, harmonic noise can be classified into twodifferent regimes, ω > √
2Γ and ω < √ ω = p ω − ≈ ω for ω ≫ Γ [22, 29, 35].While the work of Tayebirad et al. provided a first numerical investigation of thissystem and showed that the noise properties have an influence on the observed transport,it could not answer the question after the mechanism of this influence [29]. In orderto provide an answer to this problem, we present two simple models that explain howthe noise process acts on the wavefunction in different regimes. We then compare thepredictions of the two models with a more systematic numerical study and find thatthey give a good agreement with, and provide an intuitive understanding of the relevant oise assisted transport in the Wannier-Stark system
3. Simplified Models
It turned out that the incommensurability of the two lattices does not have a greateffect on the survival probability [37] due to the time-dependent phase φ ( t ). We willthus set α = 1 and present a two state model that allows us to approximate the system’sdynamics around the band edge. The Hamiltonian of (1) now reads,ˆ H = − ∂ x + V [cos( x ) + cos( x − φ ( t ))] − F x. (5)The translational invariance (at fixed time) of this Hamiltonian can be recovered byapplying the gauge transformation ψ = e iF xt ˜ ψ (changing the frame of reference fromthe lab system to the accelerated lattice frame) [5, 31]. This yields,ˆ H = 12 (ˆ p + F t ) + V [cos( x ) + cos( x − φ ( t ))] , (6)where we identified − i∂ x as the momentum operator ˆ p . It is instructive to express thisHamiltonian in the momentum basis, this givesˆ H = Z p dp (cid:2) ( p + F t ) | p ih p | + V (cid:0) (1 + e iφ ) | p ih p | + (1 + e − iφ ) | p ih p | (cid:1)(cid:3) , (7)from which it is clear that it only couples states with ∆ p = p − p ′ ∈ Z (due tothe time evolution). The momentum states can thus be written as | p i = | n + k i ,with n ∈ Z and k ∈ [ − . , . ⊂ R being the quasimomentum of the system. Dueto the conservation of quasimomentum, the Hamiltonian in (7) can be decomposedinto independent Hamiltonians each acting on a subsystem of constant quasimomentum k [5, 38]. A system initially in state | p i = | k + n i thus evolves according to the(tridiagonal) matrix Hamiltonian,ˆ H k = 12 . . . V (1 + e iφ ) ( k − + F t ) V ( + e i φ ) V ( + e − i φ ) ( k + F t ) V (1 + e iφ ) V (1 + e − iφ ) ( k + 1 + F t ) V (1 + e − iφ ) . . . , (8)where the V (1 + e ± iφ ) terms stem from the optical lattices and φ enters as a complexphase. The dynamics of the full system around an avoided crossing of the ground andfirst excited energy band (at the edge of the Brillouin zone) can be approximated bythe highlighted part of the above matrix § with k = 0 .
5, i.e. the value at the edge of § This corresponds to reducing the system to the two lowest energy states. At the edge of the Brillouinzone, this approximation is very accurate as long as the off-diagonal coupling terms are not too large,since only the lowest energy states contribute to the first excited as well as the ground band. oise assisted transport in the Wannier-Stark system H N,LZ = 12 − F t V (1 + e iφ ) V (1 + e − iφ ) F t ! , (9)where we additionally subtracted + ( F t ) , i.e. we shifted the absolute value of theenergy scale. This 2-by-2 matrix can be seen as an effective ‘noisy’ Landau-Zener (LZ)Hamiltonian; its instantaneous eigenvalues are given by, E N ;1 , = ∓ q ( F t ) + 2 V (cos( φ ( t )) + 1) . (10)We can thus introduce an effective band gap as a function of time by averaging over theprobability distribution of the noise process φ . At t = 0, this leads to ∆ E eff = V h h p φ (0)) + 1) i ± Std (cid:16)p φ (0)) + 1)) (cid:17)i , (11)where h f ( φ ) i denotes the average over the noise process and ‘Std’ means standarddeviation (see figure 6 for a schematic representation of ∆ E eff ).In the case of a single (noiseless) optical lattice only the constant term in theanti-diagonal of the Hamiltonian in (9) is present and one recovers the standard LZmodel [39–42]. Starting in the ground state at time t = −∞ , the probability to remainin that state until time t = ∞ is given by the Landau-Zener formula [39–42], P sur ( t = ∞ ) = 1 − exp (cid:18) − πV F (cid:19) . (12)This allows to give estimates for the survival probability in the ‘noisy’ LZ model byreplacing V with ∆ E in (12).The two state approximation around an avoided crossing is only applicable if theinitial momentum distribution is sufficiently localized in momentum space (i.e. muchsmaller than the Brillouin zone) [30] and if the characteristic time of the system is largerthan the transition time in the two state model [43, 44], i.e. the transition time shouldbe smaller than one Bloch period . Since the noisy LZ model is limited to short timescales and can only be applied if bothlattices in (1) have the same wavelength, we will present another model which is ableto describe the behaviour of the system in cases where those conditions are not met.This model is based on a quasistatic approximation of the noise process; φ ( t )is replaced by a term βt linear in time. The resulting system is analyzed and itsobservables are averaged according to the properties of the noise process. In the followingparagraphs, we will provide a more detailed description of this model.For short timescales, the dynamics of the system can be approximated by a Taylorexpansion of the noise term φ ( t ): φ ( t + ǫ ) = φ ( t ) + ǫ dd t φ ( t ) + O( ǫ ) = φ ( t ) + ǫµ + O( ǫ ) . (13) oise assisted transport in the Wannier-Stark system P s u r v ( β ) β Figure 1: The fluctuating red curve shows the survival probability for the noiselessbichromatic lattice for a relative lattice velocity of β . The parameters are α = 0 . , F =0 . , Γ = 0 . V = 0 . β which leadto an intertwined position of the barriers (see figure 2). The blue gaussian curve ofvariance T gives the probability distribution of µ given in (4). In the quasistatic model,we set β = µ and P sur ( β ) is therefore averaged over this gaussian distribution.A rigorous mathematical treatment of the ǫ term in (13) is far from trivial k .Nevertheless, it is clear that it depends on d d t φ ( t ) = dd t µ ( t ) as given in (3) and isthus negligible for ω , Γ ≪
1. In this case, (1) can therefore be approximated as H = − ∂ x + V cos( x ) + V cos ( α ( x − φ − βt )) + F x, (14)where the noise variable µ ( t ) has been replaced by a constant term β . Equation(14) describes a tilted bichromatic optical lattice system where the two lattices havea constant relative velocity β . The dynamics of such a system can be seen in figure 1.The fluctuating red curve shows the survival probability of the state in the ground bandafter t = T B . It can be seen that this survival probability drops sharply for intermediatevalues of β ( | β | ≈ . − .
31, region between the black vertical lines). This correspondsto strongly enhanced interband transport for these parameters.As in section 3.1, the key to understanding lies in considering the effect of opticallattices on the wavefunction in momentum space. Let us first look at a single tiltedoptical lattice V cos( αx ). We assume that the timescale of the Landau-Zener transitionfrom the ground band to the first excited band is short compared to a Bloch periodand is thus accurately described by the Landau-Zener model. Due to the static force F,a momentum-eigenstate scans through momentum space; once it reaches a momentum p = ± α , it undergoes a reflection on the lattice with the probability of P sur given in (12).This process is schematically displayed in figure 2 (left). For the ground band, the effectof the lattice can therefore be represented as a pair of barriers at ± α (blue barriers inleft of figure 2), which trap the state in between them. This momentum range between k As the second time derivative of the noise term φ ( t ) is not well-defined, it is necessary to look at avalue h φ ( t ) i which is averaged over a timescale, which is much shorter than the characteristic one ofthe system. oise assisted transport in the Wannier-Stark system β ,the position of these barriers in momentum space is shifted to ( ± α + β ). Adding upthe effects of both, static and moving lattice thus results in the presence of two pairsof barriers as seen in the right part of figure 2. If the two pairs of barriers are in anintertwined position (as shown in the right of figure 2), reflections on the moving latticecan help the state escape from the ground band. This can be understood from theright part of figure 2. If the state ‘hits’ the barrier at k = − α + β from the left (greenbarriers in figure 2), it can either be transmitted or reflected. While the transmittedfraction of the state is still trapped in-between the barriers at k = ± α , the reflectedfraction appears at the right side of the barrier at k = α + β and has thus escaped theground band of the non-moving lattice. This can be interpreted as absorbing momentumfrom the moving lattice by means of a Bragg reflection and thus being ‘boosted’ outof the ground band. It should be noted that the same process can also take place fornegative beta (transporting the state to the left of ground band in figure 2, thereforealso out of the ground band). Transport to higher energy bands is therefore enhancedfor intertwined barriers. This is supported by the data visible in figure 1, where thevalues of β which lead to an intertwined configuration are the area in-between the blackvertical lines. These values of β do indeed correspond to a heavily suppressed survivalprobability in the ground band and thus to strongly enhanced transport.Keeping this picture in mind, we can understand how the noise process acts uponthe momentum states of the system. The constant β is responsible for the relativeposition of the barriers in momentum space. In the real system, β is given by the noisevariable µ ( t ) and is thus a function of the time. The positions of the barriers of thenoise driven lattice thus move in momentum space.In our quasistatic model we assume µ ( t ) to be a constant β , which is a realisticapproximation as long as the noise process is slow, i.e. µ ( t ) changes slowly with time.For µ ( t ) = β the survival probability in the ground band is given by figure 1. To obtainan approximation for the behavior of the full, noise-driven system, we thus average thissurvival probability P ( β ) over the equilibrium distribution of µ ( t ), which is a gaussianof width T (see (4)) as pictured in figure 1.While we would expect this model to be accurate for slow noise processes, it ignores k − α α k −
12 12 − α + β α + β Figure 2: Left: Effect of a single lattice on momentum eigenstates. Right: Combinedeffect of a static and moving lattice. oise assisted transport in the Wannier-Stark system µ ( t ) has on the Landau-Zener transitions and doesalso not account for time-correlations of the noise-process. Especially for a fast noiseprocess where µ ( t ) changes significantly during one Bloch period, we therefore expectdiscrepancies between the quasistatic model and the full system [45].
4. Numerical Results
We compare now the predictions made by the two models introduced in the previoussection with data from the full system described by the Hamiltonian in (1).The most important property of the noise process φ is its variance . In the following,we will analyze the survival probability of the system in the ground band for the twocases: Var( φ ) = constant (section 4.1) and Var( φ ) = changing (section 4.2). The latteris realized by keeping ω constant and varying T .Figure 3 shows the survival probability in the ground band after one Bloch period.The survival probability strongly depends on the noise parameters T and ω .10 − ω /ω B − T / ω B t = T B , i.e.after one transition, versus the rescalednoise frequency ω /ω B and the noisestrength T /ω B . The vertical solid linerepresents a line of constant ω . Theinclined dashed line represents a line ofconstant variance. Data from numericalsimulations of the Hamiltonian in (5),for details see appendix Appendix A.The parameters are F = 0 . . V = 0 . T /ω = const . Here, we show the survival probability in the ground band after one Bloch period forconstant variance (cut along the dashed line in figure 3) versus the noise frequency. Wefocus on the region of the initial decrease and minimum of the survival probability, i.e. ω /ω B .
14. Figure 4 compares numerical data from the ‘noisy’ LZ model in (9), withdata from the full Hamiltonian in (5) for two values of the potential depth V .There is an excellent qualitative and a good quantitative agreement between the‘noisy’ LZ predictions and the simulations of the full system. The ‘noisy’ LZ accuratelyreproduces the initial decay of the survival probability (as obtained for the stochasticWannier-Stark system) for increasing noise frequency and gives a good approximation oise assisted transport in the Wannier-Stark system P s u r ( t = T B ) ω /ω B ω /ω B Figure 4: Survival probability in the ground band at t = T B versus the rescaled noisefrequency ω /ω B . Shown are data for the real system (blue dashed lines), the ‘noisy’ LZmodel (red solid line) and the quasistatic model (black dotted line). The vertical linesindicate the position of the first minimum in the survival probability for their respectivedata sets. The parameters are F = 0 . . h φ i = 0 . V = 0 . V = 0 .
125 (right).for the position of the minimum. Yet, it systematically overestimates the survivalprobability. This overestimation is less pronounced for higher potential depths V (compare figure 4 (left) and figure 4 (right)). Most importantly, the position of theminimum changes with varying potential depth V . The quasi-static model also showsa decrease in the survival probability with increasing frequency, but cannot reproduceany of the finer features visible in the stochastic Wannier-Stark system and the ‘noisy’LZ-model.The varying position of the minimum leads us to the next figure, figure 5, inwhich the position of the first minimum in the survival probability is plotted versus thepotential depth V . Here, the predictions of the ‘noisy’ LZ model and the quasistaticmodel are compared to the full system; also the effective band gap ∆ E eff is plottedas a function of V . The data for the full system shows that there exists a linearrelationship between the position of the minimum in (4) and the potential depth V .This relationship is accurately reproduced by the ‘noisy’ LZ model, but it can not beseen in the quasistatic model. The fact that the quasistatic model can not account forthe position of the minimum is easily understood by looking at figure 4. Since the modeldoes not predict the local minima visible in P sur of both the full system as well as thenoisy LZ model, it is clear that the position of the first minimum will in general be atmuch larger values of ω .Another observation that can be made in figure 5 is that the slope of the linearcurve (for the full system and the LZ model) is approximately given by the one for theeffective band gap. In fact, linear fits to the data give a slope of 1 .
93 for the real systemand 1 .
76 for the LZ model, compared to 1 . ± .
15 for the effective band gap (11). oise assisted transport in the Wannier-Stark system F [37].051015 0 0.1 0.2 0.3 ω m i n / ω B V Figure 5: Frequency position ofminimum versus potential depth V . Shown are the data forthe real system (blue lines), the‘noisy’ LZ prediction (red squares),the quasistatic model prediction(green crosses) and the effectiveband gap ∆ E eff (dashed-dotted line).The parameters for the numericalsimulations are F = 0 . . h φ i = 0 . V can be interpreted as follows. Asmentioned in section 2, the harmonic noise ‘feeds’ an energy of ≈ ω into the system.Once this energy is high enough to overcome the band gap between the ground and firstexcited band, given by ≈ ∆ E eff , the system can be excited into the upper band via a‘phononic’ excitation process ¶ (red wiggly lines in figure 6).The overestimation of the survival probability by the ‘noisy’ LZ model occursbecause the ‘LZ’ model overestimates the separation of the bands in the real systemfar away from an avoided crossing (the model only works in the blue shaded regionof figure 6); hence transitions to higher bands are suppressed (the effective bandgapincreases linearly with time t for large times, see figure 6). For large V , only transitionsclose to an avoided crossing can occur, but for smaller V transitions across the fullBrillouin zone happen in the real system and this can not be ignored. t/T Bloch E ∆ E eff −
12 12 ω ′ ω ′′′ ω ′′ Figure 6: Instantaneous eigenvalues(solid black lines) of the ‘noisy’ LZmodel for a typical noise realization.Schematically represented are ∆ E eff andtransitions between energy states (redwiggly lines). The blue shaded areaextends over one Bloch period andapproximates the energy band structureof the ‘noisy’ WSS around an avoidedcrossing. The crossed out transition isnot allowed. ¶ ‘Phononic’ is not to be taken literally, but indicates that the energy is transferred to the atoms viaan optical lattice shaking/vibrating with a more or less well-defined frequency of ≈ ω . oise assisted transport in the Wannier-Stark system ω because it is not just a twoband system and transitions to higher bands are actually possible. ω and varying T . In figure 3 we observe the following: once ω assumes a small enough value it stopshaving a discernible influence on the survival probability P sur in the ground band. Inthis section we will therefore take a more detailed look at how P sur in this regime dependson T while keeping ω = 1 constant.On the left side of figure 7, we see the survival probability of a system initiallyprepared in the ground state in comparison to the predictions from the two models.The observable P sur ( t = T Bloch ) is the same as previously looked at in figure 4.Good qualitative agreement between the full system and both models is observed.Furthermore, within the numerical errors, the minimum of the survival probability iscorrectly predicted by both models. There are, however, quantitative discrepancies forsmall (noisy LZ model) as well as large values of T (quasistatic model).On the right side of figure 7, a system with two different lattice constants is studied( α = 0 . P sur ( t + T Bloch ) /P sur ( t ) for t >
5. This quantityis similar to the previously examined survival probability P sur ( t = T Bloch ), but it is0.40.60.81 10 P s u r T /ω T /ω
Figure 7: Survival probability in the ground band for fixed noise parameter ω . Plottedare the results from the full system (blue stars), the predictions by the noisy LZ model(red line) as well as the predictions from the quasistatic model (black dashed line).Parameters are F = 0 . , Γ = 0 . , V = 0 .
125 and α = 1 (left) and α = 0 . P sur ( t = T Bloch )(left) and the average survival rate for long timescales P sur ( t + T Bloch ) /P sur ( t ) (right). oise assisted transport in the Wannier-Stark system α = 1. For the quasistatic model, good quantitative agreement with the full system isobserved for most values of T . Nevertheless, significant differences are visible aroundthe predicted minimum. These differences, however, result from the limitations of thequasistatic model and vanish if ω as well as Γ are chosen small enough. In this case,where α = 1, the quasistatic model is very successful in quantitatively predicting thelong-term survival probabilities, especially if Γ as well as ω are small [45].0.20.40.60.81 10 P s u r ( t = T B l o c h ) T /ω
Figure 8: Survival probability in theground band after one Bloch periodusing the Gross-Pitaevskii equation (sameparameters as left of figure 7). Resultsfrom the (linear) Schr¨odinger equation areshown in solid blue, while green crossesshow results for N = 5 · and redstars for N = 10 atoms in a cigar-shapedcondensate. The simulated setup was theone used in the experiments reported in [46]and is described in Appendix A. The usedatom numbers correspond to dimensionlessnonlinearity parameters of C ≈ .
08 (greencrosses) and C ≈ .
13 (blue stars), c.f.ref. [32, 46].In order to understand whether the described effects would be visible in a realisticexperimental setup using BECs, it is important to know how they are affected byinteractions between the atoms in a condensate. Using the 3D Gross-Pitaevskii equation,we performed numerical calculations for an experimental setup similar to the one usedin [46]: Rb atoms that are placed in a cigar-shaped optical dipole trap with 250Hzradial and 20Hz longitudinal frequency. The results can be seen in figure 8 and showthat, while clear differences in the survival probability exist, the enhanced tunneling ratefor intermediate noise frequencies persists and is hardly influenced by the nonlinearity.It should especially be noted that for a nonlinearity parameter (as defined in [46]) upto C ≈ .
13, the position of the minimum is hardly changed from the single particledynamics + . + Nonlinearity parameters are only given approximately, as they depend on the maximal density of thecondensate. Since our setup is governed by a stochastic equation, the peak density of the condensateis not only time-dependent due to the interband tunneling [47], but also fluctuates for different noiserealizations. oise assisted transport in the Wannier-Stark system
5. Conclusion
We have investigated the transport properties of a bichromatic Wannier Stark problemunder the influence of harmonic noise. Our main interest was understanding thecircumstances under which the noise process can enhance or suppress the transportin momentum space.An extensive numerical study of the influence of the noise parameters revealed thatthe system’s behaviour can be divided into two different regimes, as seen in figure 3. Inthe regime of low variance of the noise process, the noise only has very little influence,while in the regime of large variance, the properties of the system almost exclusivelydepend on the noise strength T . In this regime of large variance, the transport inmomentum space has a clear maximum for a certain value of T .We presented two different models that both offer an interpretation of these results.The noisy LZ model approaches the system as a noise-driven Landau-Zener transitionand argues that transport should be maximized if the noise frequency matches theband gap between the ground and the first excited band. While this model can onlypredict results for the case where both lattices have the same wavelength, it reproducesthe numerical results of the full system in this case very well. Furthermore, it givesan accurate prediction of the noise parameters necessary to maximize transport (seefigure 5). The quasistatic model assumes the noise process to be slow compared to thetimescales of the noise-free system. Linking the relative velocity of the optical latticesto their action upon the wavefunction in momentum space, this model predicts thatthe transport rate only depends on the noise parameter T and achieves a maximumfor intermediate values of T . Within the low-variance regime seen in figure 3, thisprediction matches very well with the results of the full system, even if full quantitativeagreement is only reached for very slow noise processes. Our quasistatic model, wherethe particle is effectively kicked by the moving second lattice, see figure 2, may turn outto be relevant as well for the understanding of damping effects of Bloch oscillations in‘noisy’ solid-state devices [48, 49].Since experiments that study the expansion of the wavefunction in noise-drivenlattices are underway [13], an experimental realization of our system and a comparisonbetween experimental and numerical data would be interesting. As shown at the end ofsection 4.2, the observed effects are fairly stable against interactions between the atomsof a BEC and should therefore be easily observable. Besides possible realizations withBECs in optical lattices [12], purely optical techniques such as the one of [6] could also beused. While control over the BEC using a noise process is subject to large fluctuations,a deterministic bichromatic lattice, as presented in the beginning of section 3.2, couldbe used to control the transport between different bands with a high degree of precision.In figure 1 we, e.g., can see that with small changes of the relative lattice velocity β ,the tunneling rates can be easily tuned by various orders of magnitude. oise assisted transport in the Wannier-Stark system Acknowledgments
We thank Niels L¨orch for help and discussions at an early stage of this work.Furthermore, we are grateful for the support of the Excellence Initiative through theHeidelberg Graduate School of Fundamental Physics (Grant No. GSC 129/1), the DFGFOR760, the Heidelberg Center for Quantum Dynamics and the Alliance Program ofthe Helmholtz Association (HA216/EMMI).
AppendixAppendix A. Numerical methods
The data for the ‘noisy’ LZ model is obtained by numerically integrating the Schr¨odingerequation corresponding to Hamiltonian (9). The survival probability has been obtainedby starting in the diabatic ground state (i.e., in the momentum eigenbasis of theuncoupled problem [34]) at t = − . T B and then propagate it to t = 0 . T B . Thesurvival probability at each time step is obtained by projecting onto the diabatic firstexcited state [5, 37]. Then the survival probability is averaged over the last 10% of thetotal integration time to estimate the asymptotic value of the survival probability. Inthe end, we average over 100 noise realisations and plot the standard deviation of theaverage survival probability as error bars.The data for the noise-driven Wannier-Stark problem are generated in a similar way,by integrating the time-dependent Schr¨odinger equation for a 1D or 3D wavefunction.At t = 0, the system is prepared in the ground state of the bichromatic latticepotential superposed by a small harmonic trap potential (to keep the size of theinitial wavefunction finite). At t = 0, the harmonic trap potential is disabled andreplaced by the static force F . The survival probability in the ground band ismeasured by integration over the momentum distribution using appropriate boundaries(see [5, 34, 45, 50] for more details). While simulations for single-particle wave functionwere performed in one dimension, realistic results for the Gross-Pitaevskii equationcan only be achieved using simulations in three dimensions (for the used algorithm,see [45]). For realistic values of the nonlinearity parameter, we used the values of thesetup described in [46]: Rb atoms in an optical lattice ( d l = 426nm) that are initiallyloaded into a cigar-shaped, quasi-one-dimensional optical trap, with 250Hz radial and20Hz longitudinal confinement. oise assisted transport in the Wannier-Stark system Appendix B. Unit conversion
Table B1: Summary of the new unit system. The table gives relations between our setof dimensionless units, SI-units and a set of common experimental units [5,29,30]. Here, M is the mass of the atoms in the BEC and k L is the wavelength of the laser light usedto generate the optical lattice. E rec sets the characteristic energy scale of the system. Energy Momentum1 photon exchange E rec = ~ k L M p rec = ~ k L SI-units dimensionless experimentalEnergy E SI E = E SI E rec E exp = E SI E rec Time t SI t = t SI 8 E rec ~ t exp = t SI Space x SI x = 2 x SI k L x exp = 2 x SI k L Force F SI F = F SI E rec k L F exp = F SI πE rec k L Potential V SI V = V SI E rec V exp = V SI E rec References [1] Bason M G, Viteau M, Malossi N, Huillery P, Arimondo E, Ciampini D, Fazio R, Giovannetti V,Mannella R and Morsch O 2012/02
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