Generalized Josephson effect in an asymmetric double-well potential at finite temperatures

We investigate a noninteracting many-particle bosonic system, placed in an asymmetric double-well potential. We first consider the dynamics of a single particle and determine its time-dependent probabilities to be in the left or the right well of the potential. These probabilities obey the standard Josephson equations, which in their many-particle interpretation also describe a globally coherent system, such as a Bose-Einstein condensate. This system exhibits the widely studied Josephson oscillations of the population imbalance between the wells. In our paper we go beyond the regime of global coherence by developing a formalism based on an effective density matrix. This formalism gives rise to a generalization of Josephson equations, which differ from the standard ones by an additional parameter, that has the meaning of the degree of fragmentation. We first consider the solution of the generalized Josephson equations in the particular case of thermal equilibrium at finite temperatures, and extend our discussion to the nonequilibrium regime afterwards. Our model leads to a constraint on the maximum amplitude of Josephson oscillations for a given temperature and the total number of particles. A detailed analysis of this constraint for typical experimental scenarios is given.

Figure 1

Example geometries of (a) a symmetric double-well potential $V\left(x\right)$ with the two lowest-energy eigenstates ${\varphi }_{0/1}\left(x\right)$ of a contained quantum particle and (b) an asymmetric double-well potential $V\left(x\right)+\delta V\left(x\right)$ with corresponding states ${\psi }_{0/1}\left(x\right)$.

Figure 2

Left and right well states in (a) a symmetric and (b) an asymmetric double-well potential. The functional forms of both coincide, regardless of the value of $V_0\ll E$.

Figure 3

Bottom: Degree of condensation $f=\delta N_{01}/N$ as a function of temperature $T$ in thermal equilibrium. Top: Populations of the ground ${\tilde{E}}_0$ and excited states ${\tilde{E}}_1$ shown schematically.

Figure 4

Oscillations of the population imbalance $Z\left(t\right)$ as a function of dimensionless time $\mathrm{\Delta }Et/\hslash$. The initial potential step is chosen as $V_0^i=\mathrm{\Delta }E$ and the final potential step $V_0^f=0$. The number of particles is fixed to $N={10}^3$, while the temperature is $T=2.5\times {10}^{-4}$ K (red dotted), $T={10}^{-6}$ K (blue dashed), and $T={10}^{-8}$ K (blue solid).