Dissipative Dynamics of a Driven Quantum Spin Coupled to a Bath of Ultracold Fermions

We explore the dynamics and the steady state of a driven quantum spin coupled to a bath of fermions, which can be realized with a strongly imbalanced mixture of ultracold atoms using currently available experimental tools. Radio-frequency driving can be used to induce tunneling between the spin states. The Rabi oscillations are modified due to the coupling of the quantum spin to the environment, which causes frequency renormalization and damping. The spin-bath coupling can be widely tuned by adjusting the scattering length through a Feshbach resonance. When the scattering potential creates a bound state, by tuning the driving frequency it is possible to populate either the ground state, in which the bound state is filled, or a metastable state in which the bound state is empty. In the latter case, we predict an emergent inversion of the steady-state magnetization. Our work shows that different regimes of dissipative dynamics can be explored with a quantum spin coupled to a bath of ultracold fermions.

Figure 1

(a) A driven quantum spin, which is dissipatively coupled to a bath of fermions, can be experimentally realized with localized impurity atoms (sphere with arrow) that are immersed in a Fermi gas (small spheres). (b) Two hyperfine states of the impurities are driven by rf fields of strength ${\mathrm{\Omega }}_0$ detuned from the bare transition by $\mathrm{\Delta }$. (c) The sign of the steady-state magnetization $m_z$ of the driven quantum spin is shown as a function of inverse scattering length $1/k_{F}a$ and detuning $\mathrm{\Delta }/E_F$. Along the solid lines $m_z$ vanishes and nontrivial power-law frequency renormalizations and damping of the Rabi oscillations are found. A transition from normal $m_z>0$ to inverted $m_z<0$ magnetization emerges below the metastable state, red shaded area.

Figure 2

Time dependent occupation $n_{\downarrow }$ of the state $|\downarrow ⟩$ at resonance $\epsilon =0$ for driving strength ${\mathrm{\Omega }}_0=0.1E_F$. The Rabi oscillations are strongly damped and their frequencies renormalized. (a) $n_{\downarrow }$ for negative scattering length $k_{F}a=\{-0.5,-5.0\}$. (b) $n_{\downarrow }$ for the positive scattering length $k_{F}a=2$ and rf field tuned to $\mathrm{\Delta }=\mathrm{\Delta }E$ and to $\mathrm{\Delta }=\mathrm{\Delta }E+E_b-E_F$, respectively. In all cases the steady-state magnetization is zero. Solid lines are numerical results from the simulation of (3) and dashed lines are obtained from perturbation theory.

Figure 3

Power-law renormalization of the dressed Rabi frequency ${\mathrm{\Omega }}_0$ at resonance $\epsilon =0$ due to coupling to the bath. (a) Dressed Rabi frequency $\mathrm{\Omega }/{\mathrm{\Omega }}_0$ as a function of the inverse scattering length $1/k_{F}a$ for two different driving field strengths ${\mathrm{\Omega }}_0=0.05E_F$ and ${\mathrm{\Omega }}_0=0.1E_F$. (b) Scaling of $\mathrm{\Omega }/{\mathrm{\Omega }}_0$ as a function of ${\mathrm{\Omega }}_0$ for different values of the interaction strength $k_{F}a$. The data are shown on a double logarithmic scale. The numerical results (symbols) are well described by analytic formula (6) (solid lines).

Figure 4

Time dependent occupation $n_{\downarrow }$ off resonance $\epsilon \ne 0$ for a driving strength ${\mathrm{\Omega }}_0=0.1E_F$ and scattering length (a) $k_{F}a=-2$ and (b) $k_{F}a=2$. For the latter, the detuning is chosen with respect to the metastable branch. Solid lines are numerical and dashed lines perturbative results.