Variational Approach for Impurity Dynamics at Finite Temperature

We present a general variational principle for the dynamics of impurity particles immersed in a quantum-mechanical medium. By working within the Heisenberg picture and constructing approximate time-dependent impurity operators, we can take the medium to be in any mixed state, such as a thermal state. Our variational method is consistent with all conservation laws and, in certain cases, it is equivalent to a finite-temperature Green’s function approach. As a demonstration of our method, we consider the dynamics of heavy impurities that have suddenly been introduced into a Fermi gas at finite temperature. Using approximate time-dependent impurity operators involving only one particle-hole excitation of the Fermi sea, we find that we can successfully model the results of recent Ramsey interference experiments on ${}^{40}\mathrm{K}$ atoms in a ${}^6\mathrm{Li}$ Fermi gas. We also show that our approximation agrees well with the exact solution for the Ramsey response of a fixed impurity at finite temperature. Our approach paves the way for the investigation of impurities with dynamical degrees of freedom in arbitrary quantum-mechanical mediums.

Figure 1

Ramsey response of a fixed impurity from the variational approach (blue) and exact solution (green) for range parameter $k_F{R}^{*}=1$. Dashed and solid lines correspond to temperatures $T=0$ and $T=0.2T_F$, respectively. The interactions during the time evolution are (a),(d) repulsive with $1/(k_{F}a)=1$; (b),(e) attractive, $1/(k_{F}a)=-1$; and (c),(f) at unitarity $1/a=0$.

Figure 2

(a)–(f) Ramsey response and (g)–(i) radio-frequency spectra of ${}^{40}\mathrm{K}$ impurities in a ${}^6\mathrm{Li}$ Fermi gas. We show the experimental results from Ref. [15] (circles) together with results from our variational approach (solid lines), thermally averaged over impurity momentum. In (a)–(f), the shading corresponds to the experimental uncertainty of the scattering length [15]. From left to right, the experimental parameters are $1/(k_{F}a)=\{0.23,-0.86,-0.08\}$, $T/T_F=\{0.17,0.16,0.18\}$, and $1/(k_F{a}_1)=\{3.9,-5.8,-4.8\}$. In all panels, $k_F{R}^{*}=1.1$ and $t_1=4.0{\tau }_F$.