Nonlinearity-assisted quantum tunneling in a matter-wave interferometer
aa r X i v : . [ c ond - m a t . o t h e r] O c t Nonlinearity-assisted quantum tunneling in amatter-wave interferometer
Chaohong Lee, Elena A. Ostrovskaya and Yuri S. Kivshar
Nonlinear Physics Centre and ARC Centre of Excellence for Quantum-Atom Optics,Research School of Physical Sciences and Engineering, Australian NationalUniversity, Canberra ACT 0200, Australia
Abstract.
We investigate the nonlinearity-assisted quantum tunneling and formationof nonlinear collective excitations in a matter-wave interferometer, which is realised bythe adiabatic transformation of a double-well potential into a single-well harmonictrap. In contrast to the linear quantum tunneling induced by the crossing (or avoidedcrossing) of neighbouring energy levels, the quantum tunneling between differentnonlinear eigenstates is assisted by the nonlinear mean-field interaction. When thebarrier between the wells decreases, the mean-field interaction aids quantum tunnelingbetween the ground and excited nonlinear eigenstates. The resulting non-adiabaticevolution depends on the input states. The tunneling process leads to the generationof dark solitons, and the number of the generated dark solitons is highly sensitive tothe matter-wave nonlinearity. The results of the numerical simulations of the matter-wave dynamics are successfully interpreted with a coupled-mode theory for multiplenonlinear eigenstates.PACS numbers: 03.75.Lm, 39.20.+q, 03.75.Kk
Submitted to:
J. Phys. B: At. Mol. Opt. Phys. onlinearity-assisted quantum tunneling in a matter-wave interferometer
1. Introduction
Matter-wave interferometry involves coherent manipulation of the external or internaldegrees of freedom of massive particles [1, 2]. Utilizing the well-developed techniques oftrapping and cooling, the matter-wave interferometers have been realised with atomicBose-Einstein condensates (BECs) [3, 4, 5, 6]. Almost all BEC interferometers based onspatial interference measure the phase coherence by merging two initially separatedcondensates [3, 4, 5, 6]. To recombine two condensates confined in a double-wellpotential, one has to transform the double-well potential into a single-well potentialby decreasing the barrier height.Intrinsic interparticle interactions in atomic condensates have stimulated variousstudies of the nonlinear behaviour of condensed atoms [7]. A balance between matter-wave dispersion and nonlinear interaction supports a number of nontrivial collectiveexcitations, including bright solitons in condensates with attractive interactions [8, 9]and dark solitons in condensates with repulsive interactions [10, 11]. In a harmonicallytrapped condensate with repulsive interparticle interactions, the nodes of excitednonlinear eigenstates correspond to dark solitons [16], so that the formation of darksolitons can be associated with populating excited states [16, 17]. Several methodsof condensate excitation have been suggested, the most experimentally appealing onesinvolving time-dependent modifications of trapping potentials [18]. The operation ofBEC interferometers and splitters based on spatiotemporal Y- and X-junctions [12, 14]is greatly affected by the possibility of nonlinear excitations. The nonlinear excitationsin BEC interferometers with repulsive interparticle interactions lead to the generation ofdark solitons [13, 14], and can be utilised to enhance the phase sensitivity of the devices[14, 15].The extensively explored mechanisms for population transfer between differenteigenstates of a trapped BEC include non-adiabatic processes [12], Josephson tunneling[19, 20], and Landau-Zener tunneling [21, 22, 23], which are also responsible forpopulation transfer in linear systems. However, in a sharp contrast to linear systems,the quantum tunneling between different nonlinear eigenstates can be assisted by thenonlinear mean-field interaction even in the absence of crossing (and avoided crossing)of the energy levels. Up to now, this peculiar type of quantum tunneling remains poorlyexplored.In this paper, we explore the intrinsic mechanism for the quantum tunnelingassisted by repulsive nonlinear mean-field interactions in a matter-wave interferometer.We consider the dynamical recombination process of a BEC interferometer, in whichan initially deep one-dimensional (1D) double-well potential is slowly transformedinto a single-well harmonic trap. Our numerical simulations, employing a time-dependent 1D mean-field Gross-Pitaevskii (GP) equation, show that multiple movingdark solitons are generated as a result of the nonlinearity-assisted quantum tunnelingbetween the ground and excited nonlinear eigenstates of the system, and the qualitativemechanism is independent on the particular shape of the symmetric double-well onlinearity-assisted quantum tunneling in a matter-wave interferometer
Figure 1.
Schematic diagram. Top: time-dependence of the barrier height B ( t ).Bottom left: initial BEC density distribution (shaded) at t = 0 in the double-wellpotential (solid line). Bottom right: density distribution (shaded) at t = 80 in thesingle-well potential (solid line).
2. Model and numerical results
We consider a condensate under strong transverse confinement, mω ρ ( y + z ) /
2, so thatthe 3D mean-field model can be reduced to the following 1D model [24]: i ¯ h ∂∂t Ψ( x, t ) = H Ψ( x, t ) + λ | Ψ( x, t ) | Ψ( x, t ) , (1)where H = − (¯ h / m )( ∂ /∂x ) + V ( x, t ), m is the atomic mass, λ > repulsive , and V ( x, t ) is an externalpotential. If the condensate order parameter Ψ( x, t ) is normalised to one, the effectivenonlinearity λ = 2 N a s ω ρ ¯ h is determined by the total number of atoms N , the s-wavescattering length a s and the transverse trapping frequency ω ρ [25]. In what follows weuse the dimensionless version of the model equation obtained by choosing the naturalunits of m = ¯ h = 1.We assume the time-dependent potential V ( x, t ) as a spatiotemporal Y-shapepotential generated by the superposition of a 1D time-independent harmonic potential onlinearity-assisted quantum tunneling in a matter-wave interferometer Figure 2.
Evolution of the condensate density (left) and phase (right) of the groundand first-excited states for the system of the effective nonlinearity λ = 20. The firstand second rows correspond to the evolution of the ground and first-excited states,respectively. and a time-dependent Gaussian barrier (see Fig. 1): V ( x, t ) = 12 ω x + B ( t ) · exp − x d ! , (2)where ω is the trapping frequency, d is the barrier width, and the barrier height dependson time as follows: B ( t ) = ( B − αt, for t < B /α, , for t ≥ B /α, (3)where α is the rate at which the barrier between the wells is ramped down. When thebarrier height B ( t ) > ω d , the time-dependent potential is a double-well potential withtwo minima at x = ± d q B / ( d ω )]. Thus, the 1D description is valid for weaklongitudinal confinements satisfying ω db = ω q B / ( d ω )] ≪ ω ρ and ω ≪ ω ρ . Toensure the adiabatic evolutions of the symmetric and antisymmetric initial eigenstates,the rate α must be sufficiently small. In Fig. 2, we show the evolutions of the ground andfirst-excited eigenstates of the system of the effective nonlinearity λ = 20 and rampingrate α = 1 /
4. For such a small value of α , both the ground and first excited eigenstatesof the initial double-well potential adiabatically evolve into the corresponding groundand first excited eigenstates of the final single-well potential. This means that the non-adiabatic effects are negligible for such a slowly varying process. onlinearity-assisted quantum tunneling in a matter-wave interferometer no significant overlap between theWannier states of the two wells. The fully localised initial state can be viewed asthe equal-probability superposition of the ground and first-excited eigenstates, so thatit can be used to observe the interference of these two eigenstates [26]. For BECstrapped in such a deep potential, the mean-field ground and first excited states aredegenerate or quasi-degenerate. Even for low barriers, if the tight-binding conditionis still satisfied, the two-mode approximation will give the picture of a classical Bose-Josephson junction. In the framework of second quantisation, the system obeys a two-site Bose-Hubbard Hamiltonian. In this fully quantum picture, a completely localisedinitial state corresponds to the highest excited state for repulsive interactions. Thisstate exhibits degeneracy with the sub-highest excited state [27], which corresponds tothe bistability in a classical Bose-Josephson junction [28]. Figure 3.
Evolution of the condensate density (left) and phase (right) for differentvalues of the effective nonlinearity λ . Cases (a-d) correspond to λ = 0, 5, 10 and 15,respectively. onlinearity-assisted quantum tunneling in a matter-wave interferometer ω = 0 . π , the initial barrier height B = 20 .
0, the barrier width d = √ /
2, theramping rate α = 1 /
4, and different values of the effective nonlinearity λ . For the chosensmall ramping down rate, all symmetric (antisymmetric) states of the deep double-wellpotential will adiabatically evolve into the corresponding ground (or excited) states ofthe single-well potential. Figure 4.
Formation of dark solitons in the system with the effective nonlinearity λ = 15. Left: density distributions | Ψ( x, t ) | for different times. Right: phasedistributions φ ( x, t ) for the corresponding density distributions. onlinearity-assisted quantum tunneling in a matter-wave interferometer λ . In the linear case ( λ = 0), the fully localised initial state canbe viewed as an equal-probability superposition of the symmetric and antisymmetricstates, so that, according to the adiabatic theorem, the evolving state is always theequal-probability superposition of the ground and the first excited states of the system.Due to the nonlinear interactions, the superposition principle becomes invalid, andthe resulting behavior can be interpreted as the coupled dynamics of the ground andmultiple excited states of the nonlinear system. Akin to the linear systems, the quantumtunneling appears once the quasi-degeneracy between the ground and the first excitedstate is broken, and gradually becomes significant with decreasing barrier height. Thetime scale on which the quantum tunneling appears in the nonlinear system shortenswith the growth of nonlinear interaction strength λ . The excited nonlinear states of theBEC in a single-well potential can be thought as stationary configurations of single ormultiple dark solitons [16]. As a result of the population transfer to such excited modes,the condensate develops multi-peak distribution with significant phase gradients acrossdensity notches between neighboring peaks. These notches are dark or gray solitonswith well-defined phase gradients close to π (see Fig. 4).The number of dark solitons formed in this process varies with the effectivenonlinearity λ (see Fig. 5). This dependence exhibits multiple plateaus as the effectivenonlinearity λ changes, as shown in Fig. 5. Given the relationship between thenonlinear interaction strength and the key parameters of the system, λ = 2 N a s ω ρ ¯ h ,one can control the number of generated solitons by adjusting the s-wave scatteringlength with the Feshbach resonance, the total number of atoms in the condensate withinitial preparation, and/or the transverse trapping frequency by tuning the transversetrapping field strength. The number of solitons remains unchanged for a long periodof time before multiple collisions between solitons take place. The inelastic collisionslead to the radiation of small-amplitude waves, and after a large number of collisionsthe number of solitons oscillating in the trap changes. Figure 5.
Number of dark solitons generated in the condensate at t = 80 versus theeffective nonlinearity parameter λ . onlinearity-assisted quantum tunneling in a matter-wave interferometer
3. Modal decomposition
To obtain the quantitative picture of the population transfer, we decompose an arbitrarystate of our time-dependent system as [15, ? ],Ψ( x, t ) = N X j C j ( t ) φ j ( x, t ) , (4)where φ j ( x, t ) is the j -th stationary state for the nonlinear system with the potential V ( x, t ), which obeys the (dimensionless) equation: µ j ( t ) φ j ( x, t ) = " − d dx + V ( x, t ) φ j ( x, t ) + λφ j ( x, t ) . (5)Here µ j ( t ) is the chemical potential for the j -th stationary state. For each instantaneousform of the time-dependent potential, the nonlinear eigenstates { φ j ( x, t ) } form anorthogonal set, similarly to their linear counterparts [15, 16, 29]. Due to the nonlinearinterparticle interactions, there exist additional stationary states (e.g., self-trappedstates) which have no linear counterparts [29, 30]. Nevertheless, every stationarystate of the nonlinear system can be composed from the orthogonal basis { φ j ( x, t ) } .The nonlinear eigenstates of time-independent potentials are also time-independent.However, due to the violation of the superposition principle, population transfer betweendifferent nonlinear eigenstates also occurs in time-independent systems [29]. This typeof population transfer, which originates from the exchange collisions between atoms indifferent eigenstates, will be considered in detail below.The population dynamics for different nonlinear eigenstates can be described bythe evolution of the complex coefficients C j ( t ), which obey a series of coupled first-orderdifferential equations, i dC l ( t ) dt = N X j E l,j + X k,k ′ Q l,jk,k ′ C ∗ k ( t ) C k ′ ( t ) C j ( t ) . (6)Due to the conservation of the total number of particles, C j ( t ) satisfy the normalisationcondition P j | C j ( t ) | = 1. Here the linear coupling parameters are E l,j ( t ) = Z φ ∗ l ( x, t ) H φ j ( x, t ) dx, (7)and the nonlinear coupling parameters are Q l,jk,k ′ ( t ) = λ Z φ ∗ l ( x, t ) φ ∗ k ( x, t ) φ k ′ ( x, t ) φ j ( x, t ) dx. (8)For a spatially symmetric potential V ( x, t ) = V ( − x, t ), we have Q l,jk,k ′ ( t ) = 0, and then( k + k ′ + l + j ) are odd integer numbers.In our numerical simulations, we generalise the direct relaxation method for linearquantum systems [31] to calculate the eigenstates and their eigenvalues (chemicalpotentials) for our nonlinear system with different effective nonlinearities at any momentof time. Projecting the condensate wavefunction Ψ( x, t ) onto the nonlinear eigenstates φ j ( x, t ), we find the population probabilities P j ( t ) = | C j ( t ) | = | R φ ∗ j ( x, t )Ψ( x, t ) dx | which depend on the time and the effective nonlinearity λ . onlinearity-assisted quantum tunneling in a matter-wave interferometer Figure 6. (a) Time evolution of population probabilities in different eigenstates for λ = 2, where P and P j ( j = 1 , ,
3) are the population probabilities of the groundstate and the j − th excited state, respectively. The total probability of the first foureigenstates is denoted as P tot = P + P + P + P . (b) The corresponding populationevolution for t >
80 obtained from the coupled-mode equation (6) with first four lowesteigenstates. (c) Population probabilities for the first six lowest nonlinear eigenstatesat t = 80. In Fig. 6(a), we show the time evolution of the population probabilities P j ( t ) forthe effective nonlinearity λ = 2 .
0. Here we only consider four lowest eigenstates (i.e., N = 4), so that P is the ground state population probability, P j ( j = 1 , ,
3) are thepopulation probabilities of the j − th excited state, and P tot = P + P + P + P is the totalprobability of the first four eigenstates. For t <
50, the population probabilities keepalmost unchanged. In the region of 50 < t <
80, we observe a fast population transferfrom the ground state to the first excited state. After the recombination of the two wells,i.e. for t >
80, the populations in different nonlinear eigenstates oscillate with time,even though the nonlinear system has a time-independent potential, time-independenteigenstates, and no energy degeneracy between neighbouring eigenstates. This behaviordiffers drastically from the linear dynamics where populations in different eigenstatesalways remain unchanged. We find that at this value of the effective nonlinearity thetotal population probability P tot ( t ) in the first four eigenstates is always close to one.The low-frequency population oscillations are dominated by the linear coupling betweendifferent modes and the high-frequency ones are due to the nonlinear cross-coupling ofthe nonlinear modes which corresponds to the exchange collision of atoms in different onlinearity-assisted quantum tunneling in a matter-wave interferometer λ , the dynamics of P j after the merging of the two wells can beapproximately captured by the projection of the BEC state at the moment of themerging (here t = 80) onto the set of N stationary nonlinear states φ j ( x ) of the single-well potential V ( x ) of B ( t ) = 0. This is confirmed in Fig. 6(b), where we employ thecoupled-mode theory (6) with N = 4 eigenstates of V ( x ) [cf. Fig. 6(a)]. The number ofeigenstates, N , that must be considered in the coupled-mode theory, increases with theeffective nonlinearity. The highest-order mode ( j = N ) of the harmonic potential V ( x )with significant (non-zero) excitation probability P N at the merging time will thereforedetermine the number N of dark solitons that are likely to be formed. Figure 6(c) showsthe excitation probabilities at t = 80 for the first six lowest eigenstates of V ( x ) anddifferent λ . By comparing the number of significantly excited states for different valuesof λ , one can see that the number of dark solitons formed is indeed approximatelydetermined by the highest-excited nonlinear mode of the harmonic trap that is stillsufficiently populated. For instance, N = 1 solitons are expected to form for λ = 2, and N = 2 for λ = 10 (cf. Fig. 5).
4. Conclusions
We have explored the nonlinearity-assisted quantum tunneling and formation ofnonlinear collective excitations in the matter-wave interferometer based on a time-dependent double-well potential, dynamically reconfigured to form a single-wellharmonic trap . In contrast to the Josephson tunneling and Landau-Zener tunneling, thenonlinearity-assisted quantum tunneling is brought about by the nonlinear inter-modepopulation exchange scattering. The excitations caused by this type of tunneling leadto the dark soliton generation in the process that differs dramatically from the phaseimprinting [10, 11] or condensates collisions [13]. The number of generated solitons canserve as a sensitive measure of the degree of the nonlinearity in the system. With thewell-developed techniques for preparing and manipulating condensed atoms in double-well potentials [3, 4, 5, 6], loading the condensed atoms in one well of a deep double-wellpotential and adjusting the barrier height, the experimental observation of this effectseems feasible.
Note added:
After this manuscript was prepared for submission, the group of PeterEngels from the Washington State University reported the experimental observation ofmatter-wave dark solitons due to quantum tunneling [32], in the process of sweeping ahigh potential barrier from one edge of the trap to the other.The authors thank R. Gati and M. Oberthaler for stimulating discussions. Thiswork was supported by the Australian Research Council (ARC).
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