Occupation Statistics of a Bose-Einstein Condensate for a Driven Landau-Zener Crossing

We consider an atomic Bose-Einstein condensate loaded in a biased double-well trap with tunneling rate $K$ and interatomic interaction $U$. The Bose-Einstein condensate is prepared such that all $N$ atoms are in the left well. We drive the system by sweeping the potential difference $\mathcal{E}$ between the two wells. Depending on the interaction $u=NU/K$ and the sweep rate $\dot{\mathcal{E}}$, we distinguish three dynamical regimes: adiabatic, diabatic, and sudden and consider the occupation statistics of the final state. The analysis goes beyond mean-field theory and is complemented by a semiclassical picture.

Figure 1

Phase space and corresponding energy levels (conceptual plot). Note that a ground state preparation of a $U<0$ system with increasing bias $\mathcal{E}(t)$ is equivalent to a preparation in the most excited state of a $U>0$ system with decreasing bias. The latter convention has been adopted in the right panel and in our simulations.

Figure 2

Dynamical evolution in a situation in which the mean-field (Gaussian) approximation does not hold: $N=100$ particles are prepared in one site of a symmetric ($\mathcal{E}=0$) double well and then evolved with a $u=2$ Hamiltonian. The initial wave packet is stretched along the separatrix. (a) (Semi)classical distribution in phase space. (b) Wigner function of the corresponding quantum state. (c) Quantum $P(n)$ compared with the (semi)classical and with the mean-field (binomial or Gaussian) predictions. The average occupation $⟨n⟩=47$ is associated here with a huge superbinomial variance $\mathrm{Var}(n)=1530$ instead of the binomial value $\mathrm{Var}(n)\sim 25$. In all of the simulations, we use a 4th order Runge-Kutta and made sure that during the integration the probability leakage is $\ll {10}^{-6}$.

Figure 3

(a) Parametric evolution of the adiabatic energy levels versus the decreasing ${\mathcal{E}}_1(t)$ for $N=10$ particles and $u=2.5$. We use units of time such that $K=1$. The dashed line corresponds to the level $n_c=4$. (b) Average occupation $⟨n⟩$ versus the sweep rate $\dot{\mathcal{E}}$. (c) PN and $\mathrm{Var}(n)$ versus $\dot{\mathcal{E}}$. The vertical lines indicate the various adiabatic and diabatic (dashed) thresholds.

Figure 4

$Y=\mathrm{Var}(n)/N$ versus $X=⟨n⟩/N$. Symbols correspond to numerical data, while lines indicate the sub-binomial scaling relation Eq. (6) with $u=2.5$ (dotted line) and $u=4.05$ (solid line). No fitting is involved. As a reference we plot the standard binomial scaling (dashed line) which would strictly apply in the absence of interaction ($u=0$) and the results of semiclassical (SC) simulations which are obtained by running ensemble of trajectories in phase space. There is a clear crossover from a binomial to the sub-binomial scaling, and the agreement becomes better for large $N$. Note that the superbinomial data point $(X,Y)=(0.47,15.3)$ that corresponds to the distribution in Fig. 2 is out-of-scale.

Figure 5

Diagram of the $(U,\dot{\mathcal{E}})$ regimes for a ground state preparation. In the Rabi regime $|U|<K/N$, we have a crossover from adiabatic to sudden behavior. For $K/N<U<NK$ (or $U>NK$), we have a broad crossover from adiabatic gradual (or sequential [22]) behavior to sudden behavior. For $U<-K/N$, we have two crossovers: the first from quantum adiabatic to diabatic behavior and the second from diabatic to sudden behavior. The diabatic behavior can be regarded as a classical nonlinear adiabatic behavior. For $U<-NK$, the distinction between the diabatic and the sudden regime is blurred because the final state is the same.