Regimes

The predominant contribution toQresults from thedX₂variation⁶, therefore we refer from now on toG₂=Gonly. An overview of the numerical results for the conductance is shown

6If we have an rectangular pumping cycle inX-space, it can be closed atX₂=±∞where the influence of the change inX₁can be safely neglected since the monopoles are located on theX₁=0 axis aroundε≈ −1.

Figure 6.5.:Conductance during the first half of the pumping cycle. (a) the conductanceG₂ as a function of the on-site potentialε, for various values ofU. In the numerical simulations (see also Fig. 6.6) we are using exact diagonalization of the trimer Hamiltonian (6.1). For the evaluation of Gwe use Eq. (6.5) while for Qwe use Eq. (6.3). The other parameters areN =16 particles, ¯k= 0.0003/√

2 and∆k=0.0001/√

2. As the interactionU becomes larger one observes the crossover from a single to individual peaks in the conductance. (b), theU-dependence of the integrated charge Q^∗, calculated for wide rectangular cycles for whichX₂ is varied within[−∞,ε^∗]. The values ofε^∗ are indicated by arrows of the same color in the main panel.

in Fig. 6.5, where we plot G as a function of X₂ for various interaction strengthsU. In Fig. 6.6 more details are presented: BesidesGwe also plot theX₂-dependence of the energy levels and of the site population. Four representative values ofU are considered including also the case of weak attractive interactions,U<0.

Let us try to understand the observed results. ForU =0 the analytical calculation is just

6.3. Controlled atom current in the Bose-Hubbard trimer 101

Figure 6.6.: Evolution of the energy levels, the site occupation and the conductance. Further details relating to the data of Fig. 6.5. We refer to four representative values ofU, which are indicated on top of each set of panels. Upper panels: the lowest N+1 energy levels En which dominate the conductanceG₂ are plotted as a function ofX₂=ε. The insets represent magnifications of the indicated areas. Middle panels: the site occupationsn₀(blue∆),n₁(black◦),n₂(red ). Note the steps of size 1 for the dot-occupation and 1/2 in the wire-sites occupation in subfigure d. Lower panels: the corresponding conductanceG₂as a function ofε. Numerical results are represented by solid black lines while the dotted red line corresponds to the analytical result (6.31) in (b) and to (6.36) in (c), (d).

Ntimes the single particle result (6.14):

G=−N∆_k 2¯k

2¯k²

[(ε−ε−)²+ (2¯k)²]^3/2, (6.31) which can be expressed as a function of the control parameters(X₁,X₂). Integrating over a full cycle one obtains

Q=N[1+ (kR)¯ ²]^1/2−1

kR¯ , (6.32)

whereRis the radius of the pumping cycle (see Fig. 6.2a). For small cycles we get

Q≈NkR/2¯ , (6.33)

while for large cycles we get the limiting value

Q≈N. (6.34)

ForU =0 and also for small values ofU all the particle cross “together” from the shuttle orbital to theε₋canal orbital. We call this type of dynamics “mega crossing”.

-120

Figure 6.7.: Evolution of the energy levels, the site occupation and the conductance for strong attractive interactionU=−1 (see Fig. 6.6 for legend). 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 occupationsn₀(blue

∆),n₁(black◦),n₂(red) are plotted.

For very repulsive interaction (U >0) we get G=−∆_k We overplot this formula in the lower panel of Fig. 6.6d) where an excellent agreement is observed. For intermediate values ofU (weak repulsive interaction), namely in the range k¯U Nk, we find neither the sequential crossing of Eq. (6.35), nor the mega-crossing¯ of Eq. (6.31), but rather a gradual crossing. Namely, in this regime, over a range∆X₂= 3/2(N−1)U we get a constant geometric conductance:

G ≈ −∆_k k¯

1

3U (6.36)

which reflects in a simple way the strength of the interaction. This formula has been de-duced by extrapolating Eq. (6.35), and then was validated numerically (see lower panel of Fig. 6.6c).

The above scenarios can be also identified by studying the evolution of the energy levels E_n as a function of the on-site potential ε. As discussed previously for large positive U the N-fold “degeneracy” of theU =0 level crossing is lifted, and we get a sequence of Navoided crossings (for schematic illustration see Fig. 6.2c, and compare with the numer-ical results in the upper panels of Fig. 6.6). Also forU <0 this N-fold “degeneracy” is lifted, but in a different way: The levels separate in the “vertical” (energy) direction rather than “horizontally” (see upper panels of Fig. 6.6). In the latter regime all the particles ex-ecute a single two-level transition from the shuttle to the canal (see Fig. 6.6a). In fact, for sufficiently strong attractive interactionsNU −1 all the particles are glued together and behave like a classical ball that rolls from the shuttle tooneof the canal sites (see Fig. 6.7).

When the sign ofX₁is reversed the ball rolls from one end of the canal to the other end (not

6.4. Conclusions 103 shown). This should be clearly distinguished from theN-fold degenerated transition to the lower canallevelwhich is observed in theU=0 case.

6.4. Conclusions

The theoretical [142, 147, 1, 216, 122, 73, 196] and experimental [50, 74] study of driven dynamics in single and double site systems is state of the art. The study of three-site sys-tems adds the exciting topological aspect: Controlled atomic current can be induced using optical lattice technology [3]. Our “driven vortex” should be distinguished though from the ignited stirring of [175, 145]. The actual measurement of induced neutral currents poses a challenge to experimentalists. In fact, there are a variety of techniques that have been proposed for this purpose. For example one can exploit the Doppler effect at the perpen-dicular direction, which is known as the rotational frequency shift [29]. The analysis of the prototype trimer system reveals the crucial importance of interactions. The interactions are not merely a perturbation but determine the nature of the transport process. We expect the induced circulating atomic current to be extremely accurate, which would open the way to various applications, either as a new metrological standard, or as a component of a new type of quantum information or processing device.