Work fluctuations in bosonic Josephson junctions

We calculate the first two moments and full probability distribution of the work performed on a system of bosonic particles in a two-mode Bose-Hubbard Hamiltonian when the self-interaction term is varied instantaneously or with a finite-time ramp. In the instantaneous case, we show how the irreversible work scales differently depending on whether the system is driven to the Josephson or Fock regime of the bosonic Josephson junction. In the finite-time case, we use optimal control techniques to substantially decrease the irreversible work to negligible values. Our analysis can be implemented in present-day experiments with ultracold atoms and we show how to relate the work statistics to that of the population imbalance of the two modes.

Figure 1

Numerical (blue dots) and analytical (solid line) log-log plots of the quantum part of the average work (top, main), its variance (inset) and the irreversible work (bottom) against $(U_f-U_i)/J$, obtained by changing $U$ instantaneously from $U_i=0$, where the system is in the Rabi regime, to different values of $U_f$ in the range ${10}^{-3}J\text{–}{10}^{4}J$, including all the regimes for $N=100$. In the bottom panel, the dotted line fitting the numerical points corresponds to the QHO analytical prediction, the green (dashed) and the orange (solid) lines correspond to the limits $(U_f-U_i)/J$ much smaller and much larger than 1, respectively [see Eqs. (22) and (25)].

Figure 2

Analytical (green dotted lines) and numerical (blue dots) log-log plots of the quantum part of the average work (top), its variance (middle) and the irreversible work (bottom) vs $U_i$, obtained by changing $U$ instantaneously from different values of $U_i$ in the range ${10}^{-3}J\text{–}{10}^{3}J$, including all the regimes, to a value $U_f=10J$, where the system is in the Josephson regime. We plot the absolute value of the work since it is negative for $U_i\ge U_f$. We set $N=100$. The blue (dashed) and orange (solid) lines represent respectively the analytical formulas obtained for the limiting cases of $\frac{U_{i}N}{2J}\ll 1$ and $\frac{U_{i}N}{2J}\gg 1$.

Figure 3

Semilog plot of the probability distribution of the work calculated analytically by using (26) and numerically, for a sudden quench from $U_i=0$ to $U_f=0.1J$ and $N=100$.

Figure 4

Analytical (dashed) and numerical (solid) plots of variance of the work (top) and the irreversible work (bottom) vs $\tau$, obtained by changing $U$ from $U_i=0$ to $U_f=0.2J$ with $N=200$.

Figure 5

Irreversible work vs the parameters $A_1$ and $B_2$ calculated for the ansatz $U_{\mathrm{LCS}}\left(t\right)$ (Linear + Cos + Sin ramp) for the values $U_i=0.2J,U_f=0.8J$, and $\tau =0.1/J$.

Figure 6

Semilog plot of the irreversible work vs $\tau$ evaluated with the optimal parameters of every ansatz, for values $U_i=0.2J$ and $U_f=0.8J$. The blue line corresponds to the linear ramp, the orange crosses to the linear + Sin + Cos ramp, the triangles to the ramp $U_{2\mathrm{LCS}}\left(t\right)$, the green points and the diamonds respectively to the cubic and quintic ones.

Figure 7

Plot of the irreversible work vs the duration of the quench $\tau$, evaluated for the exact optimized parameters (orange crosses) and as an average for random fluctuations of these parameters up to the $15\%$ and $5\%$ of the optimal values for $A_1$ and $B_1$ respectively (blue dots). These results are obtained for $U_i=0.2J$ and $U_f=0.8J$.

Figure 8

Comparison of the plots of the variance in units of $J/{\omega }_i$ (where ${\omega }_i$ is the plasma frequency at $t=0$) vs time obtained for the cases of the numerical Bose-Hubbard model, the semianalytical approach to the QHO and the asymptotic limit. The parameters used are $N=200,U_i=0,U_f=0.2J$.