Total energy dynamics and asymptotics of the momentum distribution following an interaction quench in a two-component Fermi gas

The absence of a characteristic momentum scale in the pseudopotential description of atomic interactions in ultracold (two-component Fermi) gases is known to lead to divergences in perturbation theory. Here we show that they also plague the calculation of the dynamics of the total energy following a quantum quench. A procedure to remove the divergences is devised, which provides finite answers for the time evolution of the total energy after a quench in which the interaction strength is ramped up according to an arbitrary protocol. An important result of this analysis is the time evolution of the asymptotic tail of the momentum distribution (related to Tan's contact) to leading order in the scattering length. Explicit expressions for the dynamics of the total energy and the contact for a linear interaction ramp are obtained, as a function of the interaction ramp time in the crossover from the sudden quench to the adiabatic limit. In the sudden quench limit, the contact, following a rapid oscillation, reaches a stationary value which is different from the equilibrium one. In the adiabatic limit, the contact grows quadratically in time and later saturates to the equilibrium value for the final value of the scattering length.

Figure 1

Closed-time contour $C$. Times $\tau$ and $\overline{\tau }$ lie on the time-ordered and anti-time-ordered branches, respectively. $\tau$ is earlier than $\overline{\tau }$ in contour ordering. The turning point $t$ is the time at which the observable in which we are interested is evaluated. $|{\mathrm{\Phi }}_0\rangle$ is the initial state.

Figure 2

First- and second-order diagram for momentum distribution. $\tilde{V}\left(t\right)$ is the quench interaction and its time dependence can be chosen to be on the two time branches. Therefore, we need to consider four types of Green function from the contour ordering.

Figure 3

Feynman diagrams for the first- and second-order corrections to the interaction energy.

Figure 4

(Left panel) Dynamics of the second-order correction to the total energy density (after renormalization), $\delta {\tilde{E}}_{\mathrm{tot}}^{\left(2\right)}(t,T_r)/\mathrm{\Omega }$, for different interaction ramp times $T_r$. Note that the second-order correction vanishes as the sudden-quench limit (i.e., $T_r\to 0$) is approached; see discussion in Sec. 4. (Right panel) Dynamics of the momentum distribution after a linear ramp of the interaction strength. We observe the development of a stationary distribution for $k\sim k_F$, which is characteristic of a prethermalized regime. The dashed line corresponds to the largest time of the series, i.e., $t/T_r=20$, and we have chosen $E_F{T}_r=6$. The interaction strength is $a_s=0.02k_{\mathrm{F}}^{-1}$ ($k_F$ is the Fermi momentum). The initial gas temperature $T=0.1T_F$, where $T_F$ is the Fermi temperature.

Figure 5

Log-log plot of the stationary value of the second-order correction to the momentum distribution $\delta n_{st}(k,T_r)$ derived from the renormalization procedure explained in Sec. 3, for $E_{\mathrm{F}}t=20$. The high-momentum $\sim k^{-4}$ tail is displayed (dashed line). Note that the curves for the sudden quench limit $T_r\to 0$ and $T_r{E}_F=6$ exhibit the same asymptote for $k/k_F\gg 1$. Closer inspection reveals a crossover of the $T_r{E}_F=6$ curve between two different asymptotes. See explanation in Sec. 5c. The interaction strength is chosen as $k_{\mathrm{F}}{a}_s=0.02$ and $T/T_F=0.1$.

Figure 6

(Left panel) Coefficient controlling the asymptote of the $\sim k^{-4}$ tail of the instantaneous momentum distribution, i.e., $k^4\delta n_k(T_r,t)$, divided by the equilibrium value of Tan's contact, $C_{\mathrm{eq}}$. The calculation is carried out to second order in the scattering length $a_s$ for $k=5k_{\mathrm{F}}$ for a linear ramp of the interaction strength with characteristic time $T_r$. The time is measured in units of ${\epsilon }_k^{-1}$. For small ${\epsilon }_k{T}_r$, after some oscillatory behavior, the asymptote saturates to a stationary value which differs from $C_{\mathrm{eq}}$. As ${\epsilon }_k{T}_r$ increases, the short-time behavior exhibits a growth that is quadratic in time for $t<T_r$ and reaches the equilibrium value of Tan's contact. (Right panel) Stationary value of the asymptote $k^4\delta n_k(T_r,t)$ as a function of the ramp time $T_r$. The dependence on $T_r$ shows a crossover between the sudden and adiabatic limits. The temperature and interaction strength are chosen as in Fig. 5.