Suppression and control of prethermalization in multicomponent Fermi gases following a quantum quench

We investigate the mechanisms of control and suppression of prethermalization in $N$-component alkaline-earth-metal gases. To this end, we compute the short-time dynamics of the instantaneous momentum distribution and the relative population for different initial conditions after an interaction quench, accounting for the effect of initial interactions. We find that switching on an interaction that breaks the $\mathrm{SU}\left(N\right)$ symmetry of the initial Hamiltonian, thus allowing for the occurrence of spin-changing collisions, does not necessarily lead to a suppression of prethermalization. However, the suppression will be most effective in the presence of $\mathrm{SU}\left(N\right)$-breaking interactions provided the number of components $N\ge 4$ and the initial state contains a population imbalance that breaks the $\mathrm{SU}\left(N\right)$ symmetry. We also find the conditions on the imbalance initial state that allow for a prethermal state to be stabilized for a certain time. Our study highlights the important role played by the initial state in the prethermalization dynamics of multicomponent Fermi gases. It also demonstrates that alkaline-earth-metal Fermi gases provide an interesting playground for the study and control of prethermalization.

Figure 1

Two possible scattering processes in an $\mathrm{SU}\left(4\right)$ Fermi gas following a quench of the interaction that breaks the emergent $\mathrm{SU}\left(4\right)$ symmetry, starting from an initial state with population imbalance. Right panel: Spin-changing collision (as understood in this work) defines a two-atom collision that modifies the relative spin populations. Left panel: Other scattering processes that conserve total spin and do not change the relative spin-population.

Figure 2

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 of at which observable in which we are interested is evaluated. $|\mathrm{GS},\nu \rangle$ is the initial state, which we take to be the ground state of a noninteracting $N$-component Fermi gas characterized by a certain population ratio $\nu$ (see the discussion below).

Figure 3

Schematic representation of the only scattering process allowed in a two-component gas with Dirac-delta interactions. Notice that $(\uparrow +\downarrow )\to (\uparrow +\downarrow )$, thus preserving the spin populations for each spin component.

Figure 4

Left panel: Dynamics of discontinuity of the momentum distribution at Fermi momentum $Z_{\sigma }\left(t\right)$ after a sudden interaction quench in a two-component Fermi gas. After a short transient, a plateau is observed indicating the existence of a prethermalized regime. Right panel: Dynamics of the change in momentum distribution ${(k/{\overline{k}}_F)}^2\delta n_{k\sigma }$ show stationary behaviors for both components. The ${(k/{\overline{k}}_F)}^2$ is chosen since ${\sum }_k(\cdots )=\int (\cdots )k^{2}dkd\mathrm{\Omega }/{\left(2\pi \right)}^3$ with the isotropic angular integral $\int d\mathrm{\Omega }=4\pi$. Time is measured in units of the inverse of mean Fermi energy, ${\overline{E}}_F=({\epsilon }_F^{\uparrow }+{\epsilon }_F^{\downarrow })/2={\overline{k}}_F^2/2m$. The interaction strength is ${\overline{k}}_F{a}_s^0=0.0158$ for the quenched interaction and we take ${\overline{k}}_F{a}_s=0.0097$ for the initial interaction.

Figure 5

Upper panel: $(1-Z_{\sigma }^{st})/(1-Z_{\sigma }^{eq})$ as a function of the ratio of the scattering lengths. $Z_{\sigma }^{st}$ and $Z_{\sigma }^{eq}$ are the stationary and equilibrium values of the discontinuity at the Fermi momentum of the momentum distribution for the spin $\sigma =\uparrow$ in a two-component Fermi gas following a sudden interaction quench. $a_s^0$ is the scattering length of the quenched interaction and $a_s$ is the scattering length of the initial interaction. The post-quench total scattering length is $a_s+a_s^0$. As discussed in the main text, the ratio $(1-Z_{\uparrow }^{st})/(1-Z_{\uparrow }^{eq})$ ranges between 1 and 2 and depends on the ratio of the initial to the final interaction strengths. Lower panel: The evolution of discontinuity at Fermi momentum for different values of the ratio $a_s/a_s^0$ shows a plateau after a short-time transient. The interaction strength for the quenched interaction is ${\overline{k}}_F{a}_s^0=0.0158$. Time is measured in units of the inverse of the mean Fermi energy ${\overline{E}}_F=({\epsilon }_F^{\uparrow }+{\epsilon }_F^{\downarrow })/2={\overline{k}}_F^2/2m$. The scaling ${\left({\overline{k}}_F{a}_s^0\right)}^{-2}$ is used because the results are obtained using second-order perturbation theory.

Figure 6

Left panel: Dynamics of the discontinuity, $Z_{\sigma }\left(t\right)$, at Fermi momentum of the momentum distribution for a $N=4$ Fermi gas following a sudden quench of the interaction that breaks the $\mathrm{SU}\left(4\right)$ symmetry. The population ratio in the initial state $\nu =N_{\pm 1/2}/N_{\pm 3/2}=1.83$ is indicated in the inset. Right panel: Evolutions of the change in momentum distribution. They show an increase (decrease) in the populations of the minority (majority) component. Time is measured in units of the inverse of the mean Fermi energy ${\overline{E}}_F=({\epsilon }_F^{3/2}+{\epsilon }_F^{1/2})/2={\overline{k}}_F^2/2m$, and $\overline{a_s}=(a_s^0+a_s^2)/2$ is the mean value of the scattering lengths of the quenched interaction. The interaction strengths are determined by the parameters ${\overline{k}}_F{a}_s^0=0.0158$ and ${\overline{k}}_F{a}_s^2=0.0032$, for the quenched interaction and ${\overline{k}}_F{a}_s=0.0097$ for the initial interaction. The scaling ${\left({\overline{k}}_F{\overline{a}}_s\right)}^{-2}$ is used because the results are obtained from second-order perturbation theory in the scattering length of the quenched interaction.

Figure 7

Evolution of the population ratio $\delta N_{\sigma }/N_{\mathrm{tot}}{\left(\overline{k_F}{\overline{a}}_s\right)}^{-2}$ for $\sigma =3/2$, and different initial conditions with symmetry ($N_{1/2}=N_{-1/2}$, $N_{3/2}=N_{-3/2}$). The scaling ${\left({\overline{k}}_F{\overline{a}}_s\right)}^{-2}$ is used because the results are obtained using second-order perturbation theory in the scattering length of the quenched interaction. Time is measured in units of the inverse of the mean Fermi energy ${\overline{E}}_F=({\epsilon }_F^{3/2}+{\epsilon }_F^{1/2})/2={\overline{k}}_F^2/2m$. The interaction strengths are determined by the parameters ${\overline{k}}_F{a}_s^0=0.0158$ and ${\overline{k}}_F{a}_s^2=0.0032$ for the quenched interaction and ${\overline{k}}_F{a}_s=0.0097$ for the initial interaction. ${\overline{a}}_s=(a_s^0+a_s^2)/2$ is the mean value of the scattering lengths of the quenched interactions.

Figure 8

Left panel: Discontinuity at the Fermi momentum starting from an imbalanced initial state. The plateau indicates the emergence of prethermalization in the presence of SCCs. The inset shows the initial condition schematically where the imbalanced initial state is ${\epsilon }_F^{-3/2}=0$ and $2{\epsilon }_F^{\pm 1/2}={\epsilon }_F^{3/2}$. Right panel: Time evolution of the relative population. After an initial exchange of particles, the spin populations reach stationary values. Time is measured in units of the inverse of the mean Fermi energy ${\overline{E}}_F=\frac{1}{4}{\sum }_{\sigma }{\epsilon }_F^{\sigma }={\overline{k}}_F^2/2m$. The interaction strengths satisfy ${\overline{k}}_F{a}_s^0=0.0158$ and ${\overline{k}}_F{a}_s^2=0.0032$ for the quenched interaction and ${\overline{k}}_F{a}_s=0.0097$ for the initial interaction. The scaling ${\left({\overline{k}}_F{\overline{a}}_s\right)}^{-2}$ is for second-order perturbation calculation.

Figure 9

First- and second-order diagram for momentum distribution. $V\left(t\right)=U_0\left(t\right)+U_1\left(t\right)$ is the sum of the total (initial plus quenched) interaction.