Exploring the anisotropic Kondo model in and out of equilibrium with alkaline-earth atoms

 We propose a scheme to realize the Kondo model with tunable anisotropy using alkaline-earth atoms in an optical lattice. The new feature of our setup is Floquet engineering of interactions using time-dependent Zeeman shifts, that can be realized either using state-dependent optical Stark shifts or magnetic fields. The properties of the resulting Kondo model strongly depend on the anisotropy of the ferromagnetic interactions. In particular, easy-plane couplings give rise to Kondo singlet formation even though microscopic interactions are all ferromagnetic. We discuss both equilibrium and dynamical properties of the system that can be measured with ultracold atoms, including the impurity spin susceptibility, the impurity spin relaxation rate, as well as the equilibrium and dynamical spin correlations between the impurity and the ferromagnetic bath atoms. We analyze the nonequilibrium time evolution of the system using a variational non-Gaussian approach, which allows us to explore coherent dynamics over both short and long timescales, as set by the bandwidth and the Kondo singlet formation, respectively. In the quench-type experiments, when the Kondo interaction is suddenly switched on, we find that real-time dynamics shows crossovers reminiscent of poor man's renormalization group flow used to describe equilibrium systems. For bare easy-plane ferromagnetic couplings, this allows us to follow the formation of the Kondo screening cloud as the dynamics crosses over from ferromagnetic to antiferromagnetic behavior. On the other side of the phase diagram, our scheme makes it possible to measure quantum corrections to the well-known Korringa law describing the temperature dependence of the impurity spin relaxation rate. Theoretical results discussed in our paper can be measured using currently available experimental techniques.

Figure 1

Experimental realization. (a) Blue (red) atoms denote the $\left|g\right.〉$ ($\left|e\right.〉$) states of alkaline-earth atoms. Only two of the $2I+1$ nuclear spin states of $\left|g\right.〉$ atoms are populated initially. A dim laser pulse excites a small fraction of $\left|g\uparrow \right.〉$ atoms into the $\left|e\Uparrow \right.〉$ state, whereas the $\left|g\downarrow \right.〉$ atoms are left unaltered. The $\left|e\right.〉$ atoms are anchored by a deep optical lattice, acting as impurities that interact with the itinerant $\left|g\right.〉$ atoms through strong on-site interaction. (b) In the quench experiment discussed in Sec. 5b, the $\left|e\right.〉$ atoms are excited at time $\tau =0$ into the $|\Uparrow \rangle$ state, during a time that can be considered instantaneous on the timescales of the Kondo dynamics. They gradually lose their spin orientation due to the spin exchange with the $\left|g\right.〉$ atoms. The magnetization of the impurity $\langle S_e^z\left(\tau \right)\rangle$ can be measured after an evolution time $\tau$.

Figure 2

Anisotropic Kondo phase diagram in the absence of an effective magnetic field. $j_z$ is the dimensionless coupling between the $z$ components of the impurity and the bath atoms, whereas $j_{\perp }$ is associated with spin-flip processes. Gray arrows indicate the poor man's scaling renormalization group flows, arising from the second-order perturbation theory [31]. Couplings in the white region flow to the line of ferromagnetic fixed points $(j_{\perp }=0,j_z<0)$, where the Kondo impurity remains unscreened. In contrast, all points in the dark (green) region flow into the antiferromagnetic fixed point, where the Kondo impurity forms a singlet with the surrounding cloud of atoms.

Figure 3

Low-energy spin dynamics in the Hubbard-Anderson model. (a) Whereas the impurity atom $|e\rangle$ is localized by a strong optical potential, the bath of $|g\rangle$ atoms is itinerant, with a hopping energy $t$. Energy scales of the system are shown in (b), with ${\epsilon }_F$ denoting the Fermi energy. The impurity interacts with the bath through on-site interactions $U_{eg}^{-}$ and $U_{eg}^{+}$, corresponding to the triplet and singlet spin channels, respectively. The on-site interactions are much larger than the tunneling matrix element to the impurity site. Interactions with the impurity, therefore, happen only virtually through second-order processes. Since $U_{eg}^{-}\ll U_{eg}^{+}$, the virtual state is dominated by the spin triplet channel, which leads to FM Kondo couplings $J_z=J_{\perp }<0$ in the low-energy effective Hamiltonian. An external Zeeman field creates ${\mathrm{\Delta }}_e$ and ${\mathrm{\Delta }}_g$ Zeeman splittings, acting on the $|e\rangle$ and $|g\rangle$ atoms, respectively. The Zeeman splitting $\mathrm{\Delta }={\mathrm{\Delta }}_e-{\mathrm{\Delta }}_g$ leads to level repulsion and mixing between the singlet and the triplet channels. As a result, the Kondo parameters become anisotropic and a finite magnetic term appears in the Kondo Hamiltonian, see Eqs. (2) and (3).

Figure 4

(a) Dependence of the dimensionless Kondo parameters $j_z$ and $j_{\perp }$ in Eqs. (20) and (21) on a static Zeeman splitting $\mathrm{\Delta }$ in the anisotropic Kondo phase diagram. Red line shows the effect of increasing $\mathrm{\Delta }$ on the isotropic system $j_z=-j_{\perp }=0.2$ (black dot). Gray lines denote the directions of the poor man's scaling flow [31] in the absence of magnetic terms. [(b) and (c)] Dependence of Kondo model parameters of Eqs. (2) and (3) on $\mathrm{\Delta }$. Here, $k=K\varrho$ denotes the dimensionless potential scattering term and $\tilde{m}=m_e\varrho =-m_g\varrho$ corresponds to the dimensionless magnetic couplings. These parameters become resonant at the Zeeman field ${\mathrm{\Delta }}_{*}$ at the edges of the plots. The Schrieffer-Wolff transformation and the Kondo description is not valid anymore in the vicinity of ${\mathrm{\Delta }}_{*}$, as indicated by the dotted line in (a). (Parameters of the plot: $U_{eg}^{+}=15U_{eg}^{-},t=0.35U_{eg}^{-}$, and ${\epsilon }_F=0$.)

Figure 5

Kondo model parameters in an oscillating Zeeman field $\mathrm{\Delta }\left(\tau \right)={\mathrm{\Delta }}_{0}cos\left(\omega \tau \right)$. The dimensionless couplings $j_z$ and $j_{\perp }$ as well as the potential scattering term $k=K\varrho \left(0\right)$ are determined at driving frequencies [(a) and (b)] $\omega =0.8U_{eg}^{-}$ and [(c) and (d)] $\omega =0.95U_{eg}^{+}$ as the Zeeman energy ${\mathrm{\Delta }}_0$ increases. The panels on the right show these parameters as functions of ${\mathrm{\Delta }}_0$. As the field has no static component, the effective magnetic couplings $m_e$ and $m_g$ vanish. [(e) and (f)] Resonant behavior of the Kondo parameters at frequencies around (e) $U_{eg}^{-}$, with a driving amplitude ${\mathrm{\Delta }}_0=0.6U_{eg}^{-}$, and (f) near $U_{eg}^{+}$, with an amplitude ${\mathrm{\Delta }}_0=1.5U_{eg}^{-}$. Close to the resonance (dotted part of the curve) the Schrieffer-Wolff transformation becomes unreliable. [Parameters of the plot: $t=0.35U_{eg}^{-}\text{and}{U}_{eg}^{+}=15U_{eg}^{-}$.]

Figure 6

(a) The evolution of the Kondo temperature $T_K$ in the AF regime and the characteristic temperature $E_0$ in the FM domain. The parameters ($j_z,j_{\perp }$) are continuously tuned along the red (dotted) line in the inset from ($-0.5$, 0) towards (0, 0.5). When $-0.5<j_z<-0.25$, the system displays the FM behavior and when $-0.25<j_z<0$ the system is in the AFM state. (b)–(e) Zero- and finite-temperature equilibrium magnetization of the impurity $\langle S_e^z\rangle$ across the phase transition. Different line colors corresponds to different magnetic fields, as indicated in (b).

Figure 7

Magnetic susceptibility of the anisotropic Kondo impurity across the FM to AFM phase transition for $T=0$. Figures on the left (right) show results at zero (finite) magnetic fields. At zero magnetic field in the AF regime [(a) and (b)], the susceptibility depends linearly on the driving frequency ${\chi }_{zz}^{\prime \prime }\left(\mathrm{\Omega }\right)\sim \mathrm{\Omega }$, indicating AFM Kondo screening. This behavior changes on the other side of the phase transition, where the FM ground state exhibits ${\chi }_{zz}^{\prime \prime }\left(\mathrm{\Omega }\right)\sim 1/\mathrm{\Omega }$ scaling. In case of finite magnetic fields, the low-frequency behavior of the imaginary part of the susceptibility always shows $\sim \mathrm{\Omega }$ scaling at frequencies $\mathrm{\Omega }\lesssim B$. The symbols in the inset in (a) indicate the points ($j_z,j_{\perp }$) in the phase diagram where the susceptibility has been computed in each panel.

Figure 8

Time evolution after the creation of the impurity. (a–c) show the time-resolved impurity-bath spin correlations $C_i\left(\tau \right)$ [shown in Eq. (31)]. The excess polarization is emitted ballistically after the Kondo cloud forms around the impurity. The time evolution of the correlations $C_{i=0}\left(\tau \right)$ at the impurity site and that of the impurity magnetization are shown in (d) and (e), respectively. The dynamics of these observables follow our expectations based on the RG flow shown in (f). The bare couplings corresponding to figures (a) $j_z=-0.35,j_{\perp }=0.15$; (b) $j_z=-0.15,j_{\perp }=0.35$; and (c) $j_z=0.15,j_{\perp }=0.35$ are denoted as FM (rectangle), $\mathrm{FM}\to \mathrm{AFM}$ (star), and AFM (circle). The easy-axis couplings in (a) flow to the ferromagnetic line of fixed points ($j_{\perp }=0$), the dynamics remains ferromagnetic, as the impurity becomes ferromagnetically correlated with the surrounding bath of atoms. In contrast, the bare couplings, shown in (b) and (c), flow into the antiferromagnetic fixed point. After the formation of the Kondo screening cloud, the impurity magnetization decays and the impurity becomes antiferromagnetically aligned with the surrounding bath atoms. The bare ferromagnetic couplings in (b) determine the initially ferromagnetic dynamics. However, this quickly crosses over to antiferromagnetic behavior [see also (d)]. [The calculation was done for a 1D open chain of range $[-L,L]$ with $L=200$ sites, and with the impurity at the origin. $\tilde{t}$ denotes the tunneling matrix element along the chain.]

Figure 9

Nonequilibrium and equilibrium impurity-bath spin correlations. The correlation $C_i\left(\tau \right)$ [defined in Eq. (31)] in the steady-state regime $\tau =50/\tilde{t}$ agrees with the equilibrium values of the corresponding ground state obtained by the imaginary-time evolution. At the intermediate time $\tau =5/\tilde{t}$, the emitted spin polarization forms an effective light cone in which the AFM correlations are partially developed. All the parameters are set to be the same as in Fig. 8.

Figure 10

Density of states $\varrho \left({\epsilon }_F\right)$ in a (a) three (b) two and (c) one-dimensional optical lattice (solid line), as a function of the Fermi energy ${\epsilon }_F$. The ${\rho }_{\mathrm{bath}}$ density of states of the bath (dashed line) is also shown. In three dimensions, $\varrho \left({\epsilon }_F\right)$ is slightly suppressed near half-filling as compared to ${\rho }_{\mathrm{bath}}$, whereas it is enhanced towards the band edges. In contrast, the density of states is always smaller than ${\rho }_{\mathrm{bath}}$ in lower dimensions.

Figure 11

Schematics of the laser configurations realizing effective Zeeman shifts for the $|e\rangle$ atoms, using circularly polarized ${\sigma }^{+}$ laser fields. Only the couplings to the $m_I=\pm 5/2$ fields are shown. The laser frequencies are detuned relative to the $|e\rangle \to \left|{}^3{D}_1\right.〉$-transition with a detuning comparable to the hyperfine splitting. Due to their different matrix elements, the ${\sigma }^{+}$ lasers couple differently to the two nuclear spin states. Thick arrows denote large matrix elements whereas narrow ones correspond to weak couplings. By choosing appropriate detunings of the lasers, a Zeeman shift (a) ${\mathrm{\Delta }}_e<0$ and (b) ${\mathrm{\Delta }}_e>0$ can be realized. Modulated Zeeman fields can be realized by modulating the amplitudes of the laser configurations in (a) and (b).

Figure 12

Dimensionless isotropic Kondo coupling $j=j_z=j_{\perp }$ in the presence of periodically modulated optical lattice potentials. The driving frequency $\omega =14U_{eg,0}^{-}$ is red detuned. As the amplitude $\delta u$ in Eq. (E1) grows, the FM coupling $j<0$ initially decreases in amplitude, then it crosses over to the AFM regime $j>0$. [Parameters of the plot: $U_{eg,0}^{+}=15U_{eg,0}^{-},t=0.35U_{eg}^{-}$, and ${\epsilon }_F=0$.]