Josephson tunnelling of a phase-imprinted Bose-Einstein condensate in a time-dependent double-well potential

This paper discusses the feasibility of experimental control of the flow direction of atomic Bose-Einstein condensates in a double-well potential using phase-imprinting. The flow is induced by the application of a time-dependent potential gradient, providing a clear signature of macroscopic quantum tunneling in atomic condensates. By studying both initial state preparation and subsequent tunneling dynamics we find the parameters to optimise the phase induced Josephson current. We find that the effect is largest for condensates up to a few thousand atoms, and is only weakly-dependent on trap geometry.


Introduction
The creation of superconducting [1] and superfluid [2] weak links has led to the experimental observation of Josephson effects [3], arising as a result of macroscopic quantum phase coherence. Josephson weak links are typically created by connecting two initially independent superconducting or superfluid systems via a barrier with dimensions of the order of the system healing length. Such junctions lead to a variety of interesting phenomena [4], including dc-and ac-Josephson effects. Observations in superconductors preceded those in superfluids, due to the much larger healing lengths, thus enabling easier fabrication of weak links. Evidence for Josephson-like effects has been observed in 4 He weak links [5], and unequivocally demonstrated for weaklycoupled 3 He systems [6]. The recent achievement of dilute trapped atomic Bose-Einstein condensation (BEC) [7] gives rise to a new system for studying Josephson effects.
In particular, such systems enable the investigation of dynamical regimes not easily accessible with other superconducting or superfluid systems. Remarkable experimental progress has led to the creation of atomic BEC Josephson junction arrays, in which the harmonically trapped atoms are additionally confined by an optical lattice potential, generated by far-detuned laser beams. Phase coherence in different wells was observed by interference experiments of condensates released from the lattice [8]. In addition, Josephson effects [9] and the control of tunnelling rate have been demonstrated [10,11]. Alternative insight into the diverse range of Josephson phenomena can be obtained by looking at a single Jospehson junction arising in a double-well system. This system has already received considerable theoretical attention, with treatments based on a twostate approximation [12][13][14][15][16][17][18][19][20][21][22][23], zero temperature mean field theory [24][25][26][27][28][29][30][31][32], quantum phase models [33,34] and instanton methods [35]. Experimentally, a double well potential may be produced by adding a blue-detuned laser beam which induces a repulsive gaussian barrier to a harmonic trap [36]. Atomic interferometry based on such a set-up was recently reported [37,38]. Alternatively, a condensate can be created directly in a magnetic double-well structure [39,40].
In this paper, we investigate the Josephson dynamics for a phase-imprinted atomic condensate in a double-well potential under the influence of a time-dependent potential gradient. We focus on the sensitivity of the Josephson flow direction to the initial state preparation. In particular, preparation in the odd parity energy eigenstate combined with the application of the potential gradient leads to a flow towards a region of higher potential energy providing a clear signature of Josephson tunnelling. Flow reversal in context of the Josephson effect is well known. For example in a superconducting π-junction [41], the addition of a macroscopic phase difference φ = π across the superconducting weak link leads to reversal of the sign of the current [42][43][44]. Also a related effect has been predicted for condensates in optical lattices as a result of the renormalization of the mass in the lattice, based on Bloch wave analysis [45].
The superfluid analogue of a supeconducting π-junction is a metastable π-state, recently observed in 3 He weak links [46]. Atomic BEC junctions behave similarly to those of 3 He-B. Thus, although superconducting Josephson junctions can be mapped onto a rigid pendulum, atomic condensate tunnel junctions map onto a non-rigid pendulum [14][15][16], thus exhibiting richer oscillation modes. For example, π oscillations arise in such systems [14][15][16][17]34]. For an atomic condensate in a double-well potential, π oscillation modes can be produced by imprinting a phase shift of π between the two wells. We study how these modes behave under the action of an external potential difference.
The paper is structured as follows. Sec. 2 introduces our main formalism, and briefly reviews the low-lying states of a condensate in a double-well, while Sec. 3 discusses the dynamics associated with particular initial states, including the π oscillation modes, under the addition of a time-dependent potential gradient. The experimental observation of controlling the Josephson flow direction by phase imprinting in current BEC set-ups is analysed in Sec. 4, with a short conclusion in Sec. 5.

Time-independent Properties of a BEC in a double-well potential
At low temperatures, the behaviour of a Bose-Einstein condensate is accurately described by a nonlinear Schrödinger equation known as the Gross-Pitaevskii (GP) equation. Throughout this paper we work in dimensionless (harmonic oscillator) units, by applying the following scalings:space coordinates transform according to and energy E ′ = (hω ⊥ ) −1 E. Here a ⊥ = h/mω ⊥ is the harmonic oscillator length in the transverse direction(s), where ω ⊥ the corresponding trapping frequency. We thus obtain the following dimensionless GP equation (primes henceforth neglected for convenience) describing the evolution of the condensate wavefunction (normalised to unity) The atom-atom interaction is parametrized byg = g/(a 3 ⊥h ω ⊥ ), where g = N (4πh 2 a/m) is the usual three-dimensional scattering amplitude, defined in terms of the s-wave scattering length a, and N is the total number of atoms (mass m). The total confining potential (see Fig. 1(a)) is given by The first term describes a cylindrically symmetric harmonic trapping potential, with a trap aspect ratio λ = ω z /ω ⊥ . The trap is spherical for λ = 1, 'cigar-shaped' for λ < 1 and 'pancake-like' for λ > 1. The second term describes a gaussian potential of height h generated by a blue detuned light sheet of beam waist w in the z direction, located at z = 0. In Eq. (2), the contribution δz corresponds to an additional linear potential of gradient δ pivoted at the centre of the trap. For δ > 0, the right well obtains higher potential energy and the trap centre is additionally shifted into the z > 0 region; however, this shift is negligible for the parameters studied throughout this work, and will be henceforth ignored. The eigenestates of the double-well condensate are calculated by substituting ψ (r, t) = e −iµt φ (r) and solving the resulting time-independent equation as discussed in [32]. As is well-known, sufficiently large interactions lead to the appearance of a loop structure, see e.g. [20,47]. The loop structure for the first excited state is shown in Fig.  1(b). Corresponding wavefunctions for ground and first excited state are shown in Fig.  1(c) for δ = 0. The three eigenstates are (i) asymmetric ground state Ψ g with equal population in both wells, (ii) an anti-symmetric state with equal population in both wells and a phase difference of π across the trap centre, which we shall henceforth refer to as Ψ e and (iii) two higher energy 'self-trapped' states with most of the population in either the left (dotted) or the right (dashed) well [13][14][15]32]. This paper is mainly concerned with states Ψ g and Ψ e and superpositions thereof. In particular, we will show that the dynamics of excited states in the presence of a time-dependent potential δz are remarkably different to that of the ground state, and offer a clear demonstration of Josephson tunnelling.

Tunnelling Dynamics under a time-dependent magnetic field gradient
The main theme of the present study is to consider the tunnelling of states with initial phase φ = 0 and π, whose symmetry is broken by the addition of a time-dependent potential gradient which increases linearly. The potential gradient is applied at t = 0, i.e., δ = Rt for t > 0, (dashed line in Fig.1(a) showing the right well with higher potential energy than the left).
Before considering the effect of asymmetry, we review the behaviour of the symmetric double-well. The solution of the GP equation in a double-well potential can be mapped onto a two-state model  by writing the wavefunction as a superposition of states localised in the left and right wells, ψ ℓ (r) and ψ r (r), i.e., The Hamiltonian for this two-state system is, where N = (N ℓ − N r )/N is the fractional population difference between the left and right well, ∆ is the potential energy difference between the left and right well (∆ = αδ, where α is a numerical factor determined numerically from the GP solution), E C =g ψ ℓ,r ||ψ ℓ,r | 2 |ψ ℓ,r is the self-interaction energy, E J = −2 ψ ℓ | − 1 2 ∇ 2 + V δ=0 |ψ r is the Josephson coupling energy. The energy splittings E C and E J are indicated in Fig. 1(b). By writing c i = √ N i exp(iφ i ) and defining a relative phase φ = φ ℓ − φ r the two-state Schrödinger equation can be re-written in terms of the coupled equationṡ To find the dynamics of N and φ one needs to know the value of E C , E J and ∆ for any particular barrier height, asymmetry and nonlinearity. Below (Fig. 3) we confirm that the two-state model is an excellent approximation to the full solution of the Schrödinger equation as long as |δ| is not too large.
If a system is initially prepared in one of its eigenstates, Ψ g or Ψ e , it will remain in that same state and there is no tunnelling current. This is shown in Fig. 2(a), where we plot the fractional relative population, N as a function of time with δ = 0 for t < 0. However, if the system is prepared in a superposition of Ψ g or Ψ e with a π phase difference, i.e., the population tunnels back and forth (see Fig. 2(b) and (c)) and the relative phase between the two wells oscillates around a mean value of π (π-oscillations [14-17, 34]).
The amplitude of the π-oscillations depends on the ratio Λ = E C /E J [15-17, 26, 28, 29]. By solving the two-state coupled equations (5) with initial conditions N(0) = 0.994 (determined from the GP solution for ψ π− ) and φ(0) = π, we find a critical ratio, Λ c ∼ 1.80. For Λ < Λ c , the population oscillates between ±N(0), as in Fig. 2(b)(t < 0), whereas for Λ > Λ c the oscillations in N are suppressed Fig. 2(c)(t < 0). Note that in Fig. 2(b) we have shifted the time origin by a quarter of an oscillation period such that N = 0 at t = 0. The effect of introducing an asymmetry depends sensitively on the initial state. For Ψ g , the potential gradient induces a Josephson current to the left (lower potential energy region), whereas for Ψ e flow occurs to the right (higher potential energy) (Fig. 2(a)). The situation is more complex for superposition states, such as Ψ π± . For Λ < Λ c , if the potential gradient is turned on rapidly (compared to the period of the π-oscillations), the oscillations in N are suppressed tending towards a mean N close to its initial value, Fig. 2(b). For Λ > Λ c , the induced flow is very different for Ψ π+ and Ψ π− . For Ψ π+ most of the population remains self-trapped in the higher energy well, whereas for Ψ π− a large fraction of the population flows from the lower to the upper well.
The parameters used throughout the rest of the paper,g = π and a spherical trap geometry (λ = 1), give E C = 0.220hω ⊥ and E J = 0.102hω ⊥ , corresponding to the regime, Λ > Λ c . In this regime, the experimental preparation of Ψ π± from the ground state is more difficult as it requires a transfer of population in addition to imprinting a phase diference of π. However, as we will see in Sec. 4, straightforward phase-imprinting can generate a superposition that contains a large fraction of Ψ π± .
The tunnelling behaviour of different initial states such as Ψ g and Ψ e under a potential gradient can be explained in terms of the time-independent solutions [32]. These can be plotted as a function of time via their dependence on the time-dependent potential gradient δ = Rt. The time evolution and the corresponding time-independent population difference for state Ψ e is shown in Fig. 3(a), from which it is found that, for slow velocities, the system follows the eigenstate almost adiabatically. The initial dynamics discussed above is also well described by the two-state model [12][13][14][15][16][17][18][19][20][21][22][23] (grey line in Fig. 3(a)). However, for larger gradients the full potential GP calculation predicts that the atoms return to the lower (or left) potential well, as illustrated by the density snapshots in Fig. 3(b), whereas the two-state model suggests they remain in the upper (right) well. This breakdown of the two-state model occurs because it does not take higher lying modes into consideration [32]. This is an important consideration for any experimental demonstration of macroscopic flow to the higher well.
The flow towards the right (higher) potential well shown in Fig. 3(b) provides a clear macroscopic demonstration of quantum tunnelling. To consider whether this effect is observable in current experimental set-ups, we have studied the effect of varying the nonlinearity, trap geometry, and the time dependence of the ramp. Note that, the effect of interactions has also been considered in [27], with the effective interaction also modified by atom losses [22]. Our studies reveal that increasing the nonlinearity causes a reduction in the amount of initial flow to the upper well, and thus tends  Figure 2. Evolution of fractional population difference N as a function of time (calculated using the GP equation) without (t < 0) and with (t > 0) a potential gradient δ = Rt (shown at the top of each figure) for different initial states: (a) a pure ground Ψ g or first excited Ψ e state with equal populations in both wells. In this case, tunnelling only arises due to the additional external potential; (b) the superposition states Ψ π± forg = π/2, corresponding to the regime Λ < Λ c (E C = 0.123hω ⊥ and E J = 0.095hω ⊥ ), showing maximum amplitude π-oscillations for t < 0: and (c) the superposition states Ψ π± forg = π, corresponding to Λ > Λ c (E C = 0.220hω ⊥ and E J = 0.102hω ⊥ ). The other parameters used here are λ = 1, h = 4hω ⊥ , R = 2 × 10 −3 (hω 2 ⊥ /a ⊥ ).
to inhibit the experimental observation (see next section for experimental estimates). For example, using the parameters of Fig. 3 with a nonlinearity ten times larger (i.e. g = 10π), leads to a reduction of the average population imbalance induced by the applied potential gradient of slightly more than a factor of 2. It is thus natural to ask if other factors (e.g. modifying initial trap aspect ratio, or changing barrier height h) will have the opposite effect. Enhanced tunnelling has been predicted for 'pancake' traps (λ > 1) [28]. Furthermore, such geometries feature a larger energy splitting between the ground and first excited state, making them more robust to external perturbations (e.g. thermal [18,21,23,48]). However, the suppression of tunnelling induced by increased nonlinearities cannot be compensated by changing the geometry. We should further comment on the extent to which the above findings depend on the rate R with which the linear potential gradient δ = Rt is applied. Fig. 3(a) shows the dependence for R = 10 −3 (hω 2 ⊥ /a ⊥ ). If R is increased by a factor of 10 then the maximum flow to the right well is reduced by roughly a factor of 2. Also flow to higher potential region can only be observed for approximately one tenth of the time, unless the gradient is ramped up to a particular value and subsequently kept constant. If the gradient is kept constant at the point of maximum population difference, the population remains trapped in the Here h = 4hω ⊥ , λ = 1,g = π and the potential gradient δ = Rt increases at constant rate R = 10 −3 (hω 2 ⊥ /a ⊥ ). (b) Snapshots of the evolution of the density distribution for case (a) when (i) t = 0, (ii) 100, (iii) 300 and (iv) 800 in units of (ω −1 ⊥ ). The population of both wells is initially equal (t = 0). As the gradient is increased in (ii), (iii), population starts being transferred towards the right (z > 0), upper well. Increasing the asymmetry beyond a threshold value leads the population to be once again transferred to the left (z < 0), lower well. Eventually, (iv), a transition to the second excited state occurs (d) (see movie). right upper well, i.e., macroscopic quantum self-trapping [13][14][15]49] occurs to a state with higher potential energy. In this regime, where the gradient does not exceed the value at which the flow is reversed, the two-state model correctly predicts the behaviour.

Experimental Considerations
We now discuss the feasibility of observing flow to the upper well using phase imprinting [50,51]. Starting from the condensate ground state in a symmetric double-well trap, population can be transferred to the excited states by applying a light-induced potential of the form V r (z, t) = α sin (πt/τ 0 ) tanh (z) (7) for (t < τ 0 ), where α and τ 0 are constants which we vary. At t = τ 0 , the potential V r is suddenly switched off such that there is a π phase shift between the two wells. This simple phase-imprinting method does not distinguish between states with similar density and phase profiles such as Ψ e or Ψ π± . Other more sophisticated methods of preparing the initial state such as 2-photon adiabatic passage [52] could also be explored.
We choose the phase imprinting parameters such that the amplitude of the subsequent number oscillations between the wells in a symmetric double-well potential (i.e. in the absence of a potential gradient) are minimized. The time dynamics for this case are shown by the grey lines in Fig. 4, and essentially correspond to π-oscillations with N(t) = 0, as discussed in [14][15][16][17]34] and shown in Fig. 2(c). For an imprinted phase of π, the population oscillates with most of the condensate in the left well (grey line in Fig. 4(a)), while for an imprinted phase of −π, the population oscillations are contained within the right well (grey line in Fig. 4(b)). In both cases, the addition of the potential gradient at a time indicated by the open circle in Fig. 4, induces a flow to the right or upper potential well (solid lines in Fig. 4). Even at the time in the π-oscillation cycle when most of the population is already on the right well and would subsequently flow back to the left, the addition of the gradient induces more flow to the right, as shown in Fig. 4(b). Note that, in this case, the population remains trapped in the right well until the influence of the second excited state becomes important. If the correct initial state parameters are obtained from the full GP calculation, then the results shown in Fig. 4 can be reproduced using the two-state model, except for the transition to the second excited state. However, a full potential calculation is required to correctly predict the initial state and the dynamics for larger potential gradients, when the two-state model breaks down.  . Evolution of fractional population difference N as a function of time for initial states prepared by phase imprinting (|α| =hω ⊥ , τ 0 = 3ω −1 ⊥ ). The grey and black curves correspond respectively to the absence of a potential gradient (i.e. symmetric double well), and the addition of a potential gradient δ = R(t − τ 1 ) increased linearly at rate R = 2 × 10 −3 (hω 2 ⊥ /a ⊥ ) from time τ 1 , with this time indicated by the open circles. (a) α =hω ⊥ , τ 1 = 10ω −1 ⊥ and (b) α = −hω ⊥ τ 1 = 85ω −1 ⊥ . Other parameters as in Fig. 2(c).
Finally, we discuss typical experimental parameters required for the demonstration of Josephson flow to the upper potential well. The number of atoms is given by The total atom number is independent of the trap aspect ratio, therefore, for given dimensionless nonlinearityg, large atom numbers can be obtained for light, weaklyinteracting, transversally weakly-confined systems. For a large number of atoms, one should preferably choose species with a small value of a √ m. For example, takingg = 4π and ω ⊥ = 2π ×5 Hz, we find: N = 3300 ( 23 Na) and 1000 ( 87 Rb). An enhancement of the atom number by a factor of 10 may be possible by tuning around a Feshbach resonance (e.g. 23 Na, 85 Rb, 133 Cs) [53]. In the case of 7 Li and 85 Rb, the number of atoms needed to observe such Josephson flow is not likely to exceed the critical value [54] for collapse. Note that for fixed, reasonably small, nonlinearity (g < 10π), such that the effect can be clearly observed, one needs weak transverse confinement ω ⊥ in order to obtain a reasonable number of atoms which can be imaged easily. However, small ω ⊥ imply long timescales, such that the observation of this effect becomes limited by other factors (e.g. thermal damping [18,21,23,48], atom losses [22], etc.). If we choose ω ⊥ = 2π × (5 − 100)Hz, then the preparation of the Ψ e state requires a time τ 0 ∼ (300 − 150)ms. and an applied field gradient R = (10 −3 − 10 −2 )(hω ⊥ /a ⊥ ) (which translates into a Zeeman shift of (1−100 MHz)/cm) ramped up over a time t exp ∼ (1.5 s−75 ms).

Conclusions
We have studied the Josephson dynamics of phase-imprinted condensates in a doublewell potential in the presence of a time-dependent potential gradient. We show that phase imprinting can lead to a significant change in the flow direction producing a clear signature of macroscopic quantum tunnelling. We have discussed the range of parameters for optimum experimental demonstration of this effect, and find that it is only weakly-dependent on the aspect ratio of the trap. However, suppression of the flow for large nonlinearities, restricts the condensate size to a few thousand atoms. An attractive candidate for the observation of the flow reversal is the recently realized atom chips [55] with a blue detuned laser beam to create the weak link. The observation of the flow of phase imprinted states would provide a clear experimental demonstration of the Josephson effect.