Loschmidt echo in one-dimensional interacting Bose gases

We explore Loschmidt echo in two regimes of one-dimensional interacting Bose gases: the strongly interacting Tonks-Girardeau (TG) regime, and the weakly interacting mean-field regime. We find that the Loschmidt echo of a TG gas decays as a Gaussian when small (random and time independent) perturbations are added to the Hamiltonian. The exponent is proportional to the number of particles and the magnitude of a small perturbation squared. In the mean-field regime the Loschmidt echo shows richer behavior: it decays faster for larger nonlinearity, and the decay becomes more abrupt as the nonlinearity increases; it can be very sensitive to the particular realization of the noise potential, especially for relatively small nonlinearities.

Figure 1

Decay of the Loschmidt echo (fidelity) with time for $\varepsilon =0.05$. (a) The averaged values ${\langle F\left(t\right)\rangle }_{\text{noise}}$ for $N=10$ (crosses), $N=20$ (asterisks), and $N=50$ (circles). Solid black lines represent the Gaussian functions fitted to the numerically obtained values. Error bars depict the standard deviation for 50 different realizations of the noise potential $V_{\varepsilon }\left(x\right)$. (b) The fidelity ${\langle F\left(t\right)\rangle }_{\text{noise}}$ (circles), fidelity product ${\langle F_P\left(t\right)\rangle }_{\text{noise}}$ (black dot-dashed lines), the values obtained via $\det \left({\mathbf{PP}}^{†}\right)$, where $P_{ij}={\sum }_n{a}_n^{i*}{a}_n^j\exp \left(i{\omega }_{n}t\right)$ is obtained via approximation presented by Peres [3] [solid (blue) line], and the fidelity obtained via trace-log formula Eq. (14) [dotted (red) line].

Figure 2

The Gaussian exponent of the fidelity as a function of $\varepsilon$ and $N$. (a) The quantities $\langle \lambda \rangle /{\varepsilon }^2$ are plotted for different particle numbers, and they are ordered just as in the legend (higher lines are for larger values of $N$); obviously $\langle \lambda \rangle \propto {\varepsilon }^2$. (b) The quantities $\langle {\lambda }_P\rangle /N$ (red circles, upper line) and $\langle \lambda \rangle /N$ (blue asterisks, lower line) are plotted as a function of $N$ for $\varepsilon =0.05$. For larger $N$ the lines become horizontal, indicating that $\langle \lambda \rangle \propto N\propto \langle {\lambda }_P\rangle$.

Figure 3

(a) Fidelities of evolving BECs for $N=50$ and $\varepsilon =0.05$, averaged over 50 realizations of the noise potential, and standard deviations from the noise average. The averaged fidelity for a noninteracting BEC is shown with the black dot-dashed line, and its standard deviation with open black circles. The averaged fidelity for a weakly interacting BEC is shown with the solid (blue) line, and its standard deviation with solid (blue) circles. (b) Fidelities of the evolving BECs for two different realizations of the noise potential.

Figure 4

Comparison of (a) the averaged fidelities and (b) their standard deviations for the TG gas (${\langle F\left(t\right)\rangle }_{\text{noise}}$, red dotted line), the weakly interacting BEC (${\langle F_{\text{GP}}^N\left(t\right)\rangle }_{\text{noise}}$, solid blue line), and the noninteracting BEC (black dot-dashed line) for the same number of particles. The parameters are $N=50$ and $\varepsilon =0.05$.

Figure 5

The fidelity decays in the mean-field theory obtained for large linear densities. (a) The values ${\langle F_{\text{GP}}\left(t\right)\rangle }_{\text{noise}}$ and ${\langle F_{\text{GP}}^N\left(t\right)\rangle }_{\text{noise}}$ averaged over 30 realizations of the noise potential obtained with the GP equation (16) and with the NPSE (19). Error bars correspond to the standard deviations which are smaller than in the first set of simulations [compare with Fig. 3a]. (b) The values of $F_{\text{GP}}\left(t\right)$ for a single realization of the noise potential for $N=500$, 750, and 1500; the inset shows the dependence of $T_{1/2}$ (blue circles) and $T_{1/2}^N$ (red asterisks) on $N$. See text for details.