Controlled quantum stirring of Bose-Einstein condensates

By cyclic adiabatic change of two control parameters of an optical trap one can induce a circulating current of condensed bosons. The amount of particles that are transported per period depends on the “radius” of the cycle, and this dependence can be utilized in order to probe the interatomic interactions. For strong repulsive interaction the current can be regarded as arising from a sequence of Landau-Zener crossings. For weaker interaction one observes either gradual or coherent mega crossings, while for attractive interaction the particles are glued together and behave like a classical ball. For the analysis we use the Kubo approach to quantum pumping with the associated Dirac monopoles picture of parameter space.

Figure 1

(Color online) Illustration of the model system. Initially, all particles are located on the upper site $\left(i=0\right)$, which represents the shuttle. In the first half of the cycle (a) the on-site potential $v_0=\epsilon$ is raised adiabatically slow from a very negative initial value and the particles are mainly transported via the $k_1$ bond to the canal, which is represented by the strongly coupled canal sites $\left(i=1,2\right)$. In the second half of the cycle (b) the bias in the coupling is inverted and the particles are mainly transported back from the canal to the shuttle via the $k_2$ bond.

Figure 2

(Color online) Scheme of the avoided crossings for the one particle problem. Left panels: The two-site system (36) is prepared with one particle in the ground state, which initially corresponds to having the particle occupying the left site. As the potential $\epsilon$ is raised the particle encounters an avoided crossing. Due to the slowness of the driving it stays in the ground state, which implies an adiabatic passage to the right site. Right panels: The three-site trimer system is prepared with one particle in the ground state, which initially corresponds to having the particle occupying the shuttle site. As the potential $\epsilon$ is varied the particle encounters avoided crossings with the lower canal orbital. A full stirring cycle consists of an adiabatic passage through $k_1$ in the first half of the cycle, and another adiabatic passage through $k_2$ in the second half of the cycle. See the text for further explanations.

Figure 3

Analysis of the pumping cycle for $N$ particles. See the text for further details. For a large cycle that encircles the whole shaded region we have $Q\approx N$. The position of the monopoles is depicted by black dots: (a) no interactions (all monopoles are “piled up” at the same position) (b) with interactions. In panel (c) we schematically plot the energy levels along the $X_1=0$ axis for a system corresponding to $N=3$ bosons. Note that energy levels that correspond to nonparticipating states (those with nonzero occupation of the upper canal orbital) are not plotted.

Figure 4

(Color online) Evolution of the energy levels, the site occupation, and the conductance for $N=16$ particles and $k_1=2\times {10}^{-4},k_2=1\times {10}^{-4}$. We refer to three representative values of $U$, which are indicated on top of each set of panels. Upper panels: the lowest $N+1$ energy levels $E_n$, which dominate the conductance $G_2$, are plotted as a function of $X_2=\epsilon$. The insets represent magnifications of the indicated areas. Middle panels: the site occupations $n_0$ (blue △), $n_1$ (black ◯), $n_2$ (red ◻). Note the steps of size 1 for the dot occupation and 1/2 in the wire-sites occupation in (c). Lower panels: the corresponding conductance $G_2$ as a function of $\epsilon$. Numerical results are represented by solid black lines, while the dotted red line corresponds to the analytical result (47) in (a) and to Eq. (55) in (b) and (c).

Figure 5

(Color online) Evolution of the energy levels, the site occupation, and the conductance for strong attractive interaction $U=-1$ (see Fig. 4 for the legend and parameters). The particles are “glued together” and roll like a classical ball from the shuttle to the left wire site as can be seen from the middle panel where the site occupations $n_0$ (blue △), $n_1$ (black ◯), $n_2$ (red ◻) are plotted. The width of the transition is exponentially small in $N$, as explained in the text, and therefore cannot be resolved numerically.

Figure 6

(Color online) Integrated density (IDoS) of avoided crossings (“magnetic monopoles”) for various values of the parameters $U$, $k_1$, $k_2$, and the boson number $N$ as a function of the rescaled on-site potential $\widehat{\epsilon }$. The support of the IDoS is predicted by Eq. (28) to be $\widehat{\epsilon }=\left[-1,0.5\right]$, which is nicely confirmed.

Figure 7

(Color online) Scaling behavior of the conductance $G$ in the gradual crossing regime $\left(U/\kappa \approx 4.7\right)$. We have $N=16$ particles, and the curves correspond to various values of $\kappa$ and $\lambda$. The $x$ axis is the rescaled control variable $\widehat{\epsilon }$ (28) and the $y$ axis is scaled accordingly to preserve the net charge. Additionally, the ordinate was scaled by the expected charge $Q$ for a half-cycle according to Eq. (48) and indeed, the area under the curve is $Q\approx 0.5$. One observes that the curves fall on top of one another, which confirms the dependence of $G$ on the ratio $U/\kappa$. Additionally, we overplotted the theoretical expression (solid orange line) for the conductance (55). Although this expression is not expected to be valid in the intermediate regime, the agreement is pretty good. Also, the agreement with the estimation from Eq. (56) corresponding to a constant value of $-G\approx 0.315$ is apparent.

Figure 8

(Color online) Dependence of $Q$ on the $X_2$ radius $R$ of the pumping cycle for the system presented in Fig. 4. The curves correspond to different values of the interaction strength $U$ and represent the net amount of particles $\left(Q\right)$ transported during the first half of a symmetric pumping cycle centered around $X_2=\epsilon =0$. For vanishing interactions $U$ (dotted line) all the particles are transported once the cycle encloses the value $X_2=-1$ (mega crossing). As $U$ is being increased the transport gradually changes leading eventually to a steplike behavior (solid line) for large repulsive interactions (sequential crossing).