Time-Dependent Impurity in Ultracold Fermions: Orthogonality Catastrophe and Beyond

The recent experimental realization of strongly imbalanced mixtures of ultracold atoms opens new possibilities for studying impurity dynamics in a controlled setting. In this paper, we discuss how the techniques of atomic physics can be used to explore new regimes and manifestations of Anderson’s orthogonality catastrophe (OC), which could not be accessed in solid-state systems. Specifically, we consider a system of impurity atoms, localized by a strong optical-lattice potential, immersed in a sea of itinerant Fermi atoms. We point out that the Ramsey-interference-type experiments with the impurity atoms allow one to study the OC in the time domain, while radio-frequency (RF) spectroscopy probes the OC in the frequency domain. The OC in such systems is universal, not only in the long-time limit, but also for all times and is determined fully by the impurity-scattering length and the Fermi wave vector of the itinerant fermions. We calculate the universal Ramsey response and RF-absorption spectra. In addition to the standard power-law contributions, which correspond to the excitation of multiple particle-hole pairs near the Fermi surface, we identify a novel, important contribution to the OC that comes from exciting one extra particle from the bottom of the itinerant band. This contribution gives rise to a nonanalytic feature in the RF-absorption spectra, which shows a nontrivial dependence on the scattering length, and evolves into a true power-law singularity with the universal exponent $1/4$ at the unitarity. We extend our discussion to spin-echo-type experiments, and show that they probe more complicated nonequilibirum dynamics of the Fermi gas in processes in which an impurity switches between states with different interaction strength several times; such processes play an important role in the Kondo problem, but remained out of reach in the solid-state systems. We show that, alternatively, the OC can be seen in the energy-counting statistics of the Fermi gas following a sudden quench of the impurity state. The energy distribution function, which can be measured in time-of-flight experiments, exhibits characteristic power-law singularities at low energies. Finally, systems in which the itinerant fermions have two or more hyperfine states provide an even richer playground for studying nonequilibrium impurity physics, allowing one to explore the nonequilibrium OC and even to simulate quantum transport through nanostructures. This provides a previously missing connection between cold atomic systems and mesoscopic quantum transport.

Popular Summary

When an impurity particle is suddenly introduced into a gas of a large number of fermions (for example, electrons) that can be viewed as noninteracting with each other, what happens to the perturbed gas? Naively, one might expect that the changes caused by the single impurity may be little. It turns out, however, that the ground state of the perturbed gas essentially has very little resemblance to the ground state of the original gas: In the technical language of quantum mechanics, the two ground states are “orthogonal” to each other, as first realized and named “orthogonality catastrophe” by P. W. Anderson in 1967.

Since then, many nontrivial consequences, such as certain universal features of the dynamics of the impurity, have been derived from studies of orthogonality catastrophe in the context of condensed matter physics, where the problem becomes frequently relevant. But, experimental observations of impurity dynamics have been few, made challenging by the fact that noninteracting-fermion (electron) gases in condensed matter systems are an idealization often difficult to achieve. In this paper, we propose a shift of focus from condensed matter systems to gases of ultracold atoms and present a comprehensive program of how to probe orthogonality catastrophe, and more broadly, quantum impurity problems, on the new platform.

The extraordinary development in the field of physics of ultracold atoms during the past two decades has put a number of powerful experimental observational techniques at our disposal: Ramsey interferometry, spin-echo spectroscopy, and radio-frequency spectroscopy. Tuning the interaction between the impurity and the “native” fermions at will is also relatively straightforward. We show that these techniques can be used to demonstrate the universality of the dynamics of orthogonality catastrophe in complementary time and frequency domains. In particular, the proposed applications of these techniques have led to us to the discovery of a new type of universal behavior characterized by its own power law, which is inaccessible in the condensed matter setting. We hope that our work will help to establish a new connection between atomic and condensed matter physics, and will stimulate experiments on quantum impurities in ultracold atoms.

Figure 1

Proposed experimental setup: Impurity atoms (blue dots with arrows, indicating internal states) are immersed in a sea of itinerant host fermions (red dots). The optical lattice, indicated by the black parabolas, localizes the impurity atoms without affecting the itinerant host fermions. We assume a low concentration of impurity atoms, so that we can consider scattering on a single impurity. The scattering of fermions can be controlled by applying magnetic fields and by manipulating the internal impurity states with RF pulses. The system’s parameters can be controlled quickly (compared to the intrinsic time scales of the many-body system of host fermions), which makes this setup ideal for studying nonequilibrium impurity physics, including the new regimes of the OC.

Figure 2

RF-absorption spectra for different values of scattering length $a$. (a) When $a<0$, no bound state is present and the spectrum exhibits one power-law edge singularity at $\hslash {\omega }_0=\Delta E$ due to the processes illustrated in Fig. 3b. In addition, a weak nonanalyticity at $\hslash {\omega }_1=\Delta E+E_F$ is found, attributed to the excitations from the bottom of the band, illustrated in Fig. 3c. This nonanalyticity is magnified and plotted for additional large values of $k_{F}a=-12,-18,-24$ in the inset in (a). At unitarity, the nonanalyticity transforms into a true divergence with exponent $1/4$, which is described by a universal result shown in Eq. (18). (b) When $a>0$, the impurity creates a bound state. The spectra in this case exhibit two threshold singularities, which are offset by an energy $|E_F-E_b|$, as well as an additional cusplike singularity, the origin of which is illustrated in Fig. 3f.

Figure 3

Schematic illustration of the intermediate excited sets of states giving power-law contributions to the overlap function, which result in the threshold singularities in the RF-absorption spectrum. (a) Undisturbed Fermi sea (impurity is in the $|\downarrow ⟩$ state). (b)–(c) Case $a<0$ (no bound state): (b) Multiple low-energy particle-hole pairs, giving rise to the leading power-law decay, Eq. (12), of the overlap; (c) an additional particle is promoted from the bottom of the band to the vicinity of the Fermi surface. This contribution leads to the oscillations with period $2\pi \hslash /E_F$ in the overlap function [see Fig. 4a], as well as to the new cusplike singularity in the absorption spectra [Fig. 2a]. (d)–(f) For the case $a>0$, when the impurity potential creates a bound state, there are three important sets of states: (d) The bound state is empty and multiple low-energy excitations are created, (e) the bound state is filled, and (f) an additional particle is excited from the bottom of the band to the Fermi surface. The processes, (d) and (e), lead to the behavior, Eq. (15), of the overlap function while (f) leads to a faster-decaying oscillating contribution. The three contributions result in the absorption spectra shown in Fig. 2b.

Figure 4

Universal overlap function $|S(t)|$, which can be measured in the Ramsey interferometry experiment, is shown for different values of the scattering length $a$; see legend. Panel (a) shows $|S(t)|$ for the case $a<0$, when the impurity potential does not create a bound state. The dashed lines are power-law fits, Eq. (12), at long times. At intermediate times, oscillations coming from exciting an extra hole near the bottom of the band are visible. Panel (b) shows $|S(t)|$ for the case $a>0$, when the impurity potential creates a bound state. The asymptotic behavior is described by Eq. (15) and exhibits strong oscillations.

Figure 5

The prefactors that appear in the asymptotic behavior of the overlap function, as well as in the RF response near the thresholds at ${\omega }_0,{\omega }_b$; see Eqs. (12, 15, 16, 17). The prefactors were obtained from the asymptotic form of the numerically evaluated overlap functions.

Figure 6

The spin-echo response, Eq. (19), of the Fermi gas. At long times, it is characterized by a power-law decay, Eq. (20), with an exponent that is 3 times larger than that of the standard OC.

Figure 7

The generalized spin-echo response, Eq. (21), of the Fermi gas for $n=1,2,3$ compared to the Ramsey interference. At long times, the generalized spin-echo response is characterized by a power-law decay, Eq. (22), with an exponent $(4n-1)$ times larger than that of the standard OC. The gray lines correspond to $T=0$, while the colored lines are for finite temperatures (a) $T=0.01T_F$, (b) $T=0.03T_F$, and (c) $T=0.05T_F$. The scattering length is $k_{F}a=-1.5$ in all cases. The power-law decay of the spin-echo response gives way to a faster exponential decay at times $t\gtrsim \hslash /T$.