Maximum Energy Growth Rate in Dilute Quantum Gases

In this Letter we study how fast the energy density of a quantum gas can increase in time, when the interatomic interaction characterized by the $s$-wave scattering length $a_s$ is increased from zero with arbitrary time dependence. We show that, at short time, the energy density can at most increase as $\sqrt{t}$, which can be achieved when the time dependence of $a_s$ is also proportional to $\sqrt{t}$, and especially, a universal maximum energy growth rate can be reached when $a_s$ varies as $2\sqrt{\hslash t/(\pi m)}$. If $a_s$ varies faster or slower than $\sqrt{t}$, it is, respectively, proximate to the quench process and the adiabatic process, and both result in a slower energy growth rate. These results are obtained by analyzing the short time dynamics of the short-range behavior of the many-body wave function characterized by the contact, and are also confirmed by numerically solving an example of interacting bosons with time-dependent Bogoliubov theory. These results can also be verified experimentally in ultracold atomic gases.

Figure 1

(a1)–(a2) The time dependence of the scattering length $a_s(t)$ (in units of $l_0$) with different power-law functions of Eq. (6). (a1) $\alpha =1/4,1/2,3/4$, and $\beta$ is fixed at $\beta =1$. (a2) $\alpha$ is fixed at $\alpha =1/2$ and $\beta =1/2,1.128,1.596$, and 4. (b1)–(b2) The short time behavior of the contact $\mathcal{C}$ [in units of $g_2(0)l_0^2$] with $a_s(t)$ plotted in (a1) and (a2), respectively. (c1)–(c2) The time dependence of the energy density change $\delta \mathcal{E}$ [in units of $g_2(0)l_0{\hslash }^2/m$] with $a_s(t)$ plotted in (a1) and (a2), respectively.

Figure 2

Initial growth rate for contact (a) and for energy (b) as a function of $\beta$ for $a_s(t)=\beta \sqrt{\hslash t/m}$. Arrows mark ${\beta }_{c1}$ and ${\beta }_{c2}$ where the maximum contact growth rate and the maximum energy growth rate are reached. ${\upsilon }_C$ and ${\upsilon }_E$ are plotted in units of $g_2(0)\hslash /m$ and $g_2(0)\sqrt{{\hslash }^3/m}$, respectively. The inset in (b) highlights the peak at the negative $\beta$ side where $a_s$ changes to negative values.

Figure 3

Dynamics of the total energy density of Bose gas for $a_s(t)$ with different power-law functions of Eq. (6). (a) $\beta =1$ and $\alpha =1/4,1/2,3/4$. (b) $\alpha =1/2$ and $\beta =4$, 0.5, 1.128. $\delta \mathcal{E}$ is plotted in units of $n^2{l}_0{\hslash }^2/m$ and we have set $t_0=t_n$ and thus $l_0=1/k_n$ in the numerical calculation.