## Sluggish turmoil in the Fermi sea

The nonequilibrium dynamics of many-body quantum systems are tricky to study experimentally or theoretically. As an experimental setting, dilute atomic gases offer an advantage over electrons in metals. In this environment, the heavier atoms make collective processes that involve the entire Fermi sea occur at the sluggish time scale of microseconds. Cetina *et al.* studied these dynamics by using a small cloud of ^{40}K atoms that was positioned at the center of a far larger ^{6}Li cloud. Controlling the interactions between K and Li atoms enabled a detailed look into the formation of quasiparticles associated with K “impurity” atoms.

*Science*, this issue p. 96

## Abstract

The fastest possible collective response of a quantum many-body system is related to its excitations at the highest possible energy. In condensed matter systems, the time scale for such “ultrafast” processes is typically set by the Fermi energy. Taking advantage of fast and precise control of interactions between ultracold atoms, we observed nonequilibrium dynamics of impurities coupled to an atomic Fermi sea. Our interferometric measurements track the nonperturbative quantum evolution of a fermionic many-body system, revealing in real time the formation dynamics of quasi-particles and the quantum interference between attractive and repulsive states throughout the full depth of the Fermi sea. Ultrafast time-domain methods applied to strongly interacting quantum gases enable the study of the dynamics of quantum matter under extreme nonequilibrium conditions.

The nonequilibrium dynamics of fermionic systems is at the heart of many problems in science and technology. The wide range of energy scales, spanning the low energies of excitations near the Fermi surface up to high energies of excitations from deep within the Fermi sea, challenges our understanding of the quantum dynamics in such fundamental systems. The Fermi energy *E*_{F} sets the shortest response time for the collective response of a fermionic many-body system through the Fermi time τ_{F} = *ħ*/*E*_{F}, where *ħ* is the Planck constant divided by 2π. In a metal (i.e., a Fermi sea of electrons), *E*_{F} is in the range of a few electron volts, which corresponds to τ_{F} on the order of 100 attoseconds. Dynamics in condensed matter systems on this time scale can be recorded by attosecond streaking techniques (*1*), and the initial applications were demonstrated by probing photoelectron emission from a surface (*2*). However, despite these advances, the direct observation of the coherent evolution of a fermionic many-body system on the Fermi time scale has remained beyond reach.

In atomic quantum gases, the fermions are much heavier and the densities far lower, which brings τ_{F} into the experimentally accessible range of typically a few microseconds. Furthermore, the powerful techniques of atom interferometry (*3*) now offer an opportunity to probe and manipulate the real-time coherent evolution of a fermionic quantum many-body system. Such techniques have been successfully used to measure bosonic Hanbury-Brown-Twiss correlations (*4*), to demonstrate topological bands (*5*), to probe quantum and thermal fluctuations in low-dimensional condensates (*6*, *7*), and to measure demagnetization dynamics of a fermionic gas (*8*, *9*). Impurities coupled to a quantum gas provide a unique probe of the many-body state (*10*–*16*). Strikingly, they allow direct access to the system’s wave function when the internal states of the impurities are manipulated using a Ramsey atom-interferometric technique (*17*, *18*).

We used dilute ^{40}K atoms in a ^{6}Li Fermi sea to measure the response of the sea to a suddenly introduced impurity. For near-resonant interactions, we observed coherent quantum many-body dynamics involving the entire ^{6}Li Fermi sea. We also observed in real time the formation dynamics of the repulsive and attractive impurity quasiparticles. In the limit of low impurity concentration, our experiments confirm that an elementary Ramsey sequence is equivalent to linear-response frequency-domain spectroscopy. We demonstrate that our time-domain approaches allow us to prepare, control, and measure many-body interacting states.

Our system consists of a small sample of typically 1.5 × 10^{4} ^{40}K impurity atoms immersed in a Fermi sea of 3 × 10^{5} ^{6}Li atoms (*19*, *20*). The mixture is held in an optical dipole trap (Fig. 1A) at a temperature of *T* = 430 nK after forced evaporative cooling. Because of the Li Fermi pressure, and because our optical potential for K has more than twice the strength of that for Li, the K impurities are concentrated in the central region of the large Li cloud. Here they experience a nearly homogeneous environment with an effective Fermi energy of ε_{F} = *k*_{B} × 2.6 μK (*20*), where *k*_{B} is the Boltzmann constant. The corresponding Fermi time, τ_{F} = 2.9 μs, sets the natural time scale for our experiments. The degeneracy of the Fermi sea is characterized by *k*_{B}*T*/ε_{F} ≈ 0.17. The concentration of K in the Li sea remains low, with ≈ 0.2, where is the average Li number density and is the average K number density sampled by the K atoms (*20*).

The interaction between the impurity atoms in the internal state K|3〉 (third-to-lowest Zeeman sublevel) and the Li atoms (always kept in the lowest Zeeman sublevel) is controlled using a rather narrow (*20*) interspecies Feshbach resonance near a magnetic field of 154.7 G (*19*, *21*). We quantify the interaction with the Fermi sea by the dimensionless parameter *X* ≡ –1/κ_{F}*a*, where is the Li Fermi wave number (with *m*_{Li} the Li mass) and *a* is the *s*-wave interspecies scattering length. Slow control of *X* is realized in a standard way by variations of the magnetic field, whereas fast control is achieved using an optical resonance shifting technique (*19*). The latter permits sudden changes of *X* by up to ±5 within τ_{F}/15 ≈ 200 ns.

Our interferometric probing method is based on a two-pulse Ramsey scheme (Fig. 1B), following the suggestions of (*17*, *18*). The sequence starts with the impurity atoms prepared in the spin state K|2〉 (second-to-lowest Zeeman sublevel), for which the background interaction with the Fermi sea can be neglected. An initial radio-frequency (rf) π/2 pulse, of duration 10 μs, drives the K atoms into a coherent superposition between this noninteracting initial state and the state K|3〉 under weakly interacting conditions (interaction parameter *X*_{1} with |*X*_{1}| ≈ 5). Using the optical resonance shifting technique (*19*), the system is then rapidly quenched into the strongly interacting regime (|*X*| < 1). After an evolution time *t*, the system is quenched back into the regime of weak interactions and a second π/2 pulse is applied. The population difference* N*_{3} – *N*_{2} in the two impurity states is measured as a function of the phase ϕ of the rf pulse. By fitting a sine curve to the resulting signal (*N*_{3} – *N*_{2})/(*N*_{3} + *N*_{2}), we obtain the contrast |*S*(*t*)| and the phase ϕ(*t*) (*20*), which yields the complex-valued Ramsey signal *S*(*t*) = |*S*(*t*)| exp[–*i*ϕ(*t*)]. In the limit of low impurity concentration and rapid quenching, describes the sensitivity of the time evolution to perturbations of the system. Here, the angle brackets denote the quantum statistical average, the Hamiltonian describes the noninteracting Fermi gas, and the interacting Hamiltonian differs from by the additional scattering between the Fermi sea atoms and the impurity atoms. The function *S*(*t*), which for pure initial states is often referred to as the Loschmidt amplitude (*22*), was introduced in the context of nuclear magnetic resonance experiments (*23*) and was also applied in the analysis of the orthogonality catastrophe (*24*) as well as in the study of quantum chaos (*25*).

We first consider the interaction conditions for which earlier experiments have demonstrated that the spectral response is dominated by polaronic quasi-particles (*15*). Figure 2, A to D, shows the evolution of the contrast and the phase measured in the repulsive and the attractive polaron regimes, where *X* = –0.23 ± 0.06 and *X* = 0.86 ± 0.06, respectively. For short evolution times up to ~4 τ_{F}, we observed that both contrast signals exhibit a similar initial parabolic transient, which is typical of a Loschmidt echo (*25*). For longer times, this connects to an exponential decay of the contrast and a linear evolution of the phase. In (*19*), we showed that the exponential decay of the contrast in this regime can be interpreted in terms of quasiparticle scattering. Here, the linear phase evolution corresponds to the energy shift of the quasiparticle state, for which we obtain (+0.29 ± 0.01) ε_{F} for the repulsive case in Fig. 2C and (–0.27 ± 0.01) ε_{F} for the attractive case in Fig. 2D. The longer-time behavior reflects the quasiparticle properties, whereas the observed initial parabolic transient reveals the ultrafast real-time dynamics of the quasiparticle formation.

On resonance, for the strongest possible interactions, a description of the dynamics in terms of a single dominant quasiparticle excitation breaks down. In this regime, our measurements—displayed in Fig. 2, E and F, for *X* = 0.08 ± 0.05—reveal the striking quantum dynamics of an interacting fermionic system forced into a state far out of equilibrium. The contrast |*S*(*t*)| shows pronounced oscillations reaching almost zero, which indicates that the time-evolved state can become almost orthogonal to the initial state. Meanwhile, the phase ϕ(*t*) exhibits plateaus with jumps of π near the contrast minima.

To further interpret our measurements, we used two different theoretical approaches: the truncated basis method (TBM) (*20*) and the functional determinant approach (FDA) (*18*). The TBM models our full experimental procedure assuming zero temperature and considering only single particle-hole excitations. This approximation, first introduced in (*26*) to model the attractive polaron, was later applied to predict repulsive quasi-particles in cold gases (*27*). The predictions of the TBM are represented by the blue lines in Fig. 2. This method accurately describes the initial transient as well as the period of the oscillations of *S*(*t*) on resonance. Although the zero-temperature TBM calculation naturally overestimates the contrast in the thermally dominated regime (*t* > 6 τ_{F}), it accurately reproduces the observed linear phase evolution and thus the quasiparticle energy. The FDA is an exact solution for a fixed impurity at arbitrary temperatures, taking into account the nonperturbative creation of infinitely many particle-hole pairs. The FDA calculation is represented by the solid red lines in Fig. 2. We see excellent agreement with our experimental results, which indicates that the effects of impurity motion remain small in our system. This observation can be explained by the fact that our impurity is sufficiently heavy so that the effects of its recoil with energies of ~0.25 ε_{F} (*20*) are masked by thermal fluctuations. To identify the effect of temperature, we performed a corresponding FDA calculation for *T* = 0; the results are shown as dashed lines in Fig. 2. Here, we see a slower decay of |*S*(*t*)|, which follows a power law at long times (*20*) under the idealizing assumption of infinitely heavy impurities.

Time-domain and frequency-domain methods are closely related, as is well known in spectroscopy. In the limit of low impurity density, where the interactions between the impurities can be neglected, *S*(*t*) is predicted to be proportional to the inverse Fourier transform of the linear excitation spectrum *A*(ω) of the impurity (*24*). To benchmark our interferometric method, we measured *A*(ω) using rf spectroscopy, similar to our earlier work (*12*) but with great care taken to ensure a linear response (*20*). The measured excitation spectra are shown in Fig. 3, A to C, together with a schematic energy diagram of the quasiparticle branches (Fig. 3D). In the repulsive and attractive polaron regimes, we observed the characteristic structure of a peak on top of a broad pedestal (*15*). The peak corresponds to the long-time evolution of the quasiparticle, whereas the pedestal is associated with the rapid dynamics related to the emergence of many-body correlations. For resonant interactions, the rf response is broad and nearly symmetric. The latter implies that the imaginary part of *S*(*t*) remains small. Consequently, as seen in Fig. 2, E and F, the phase ϕ(*t*) essentially takes values near 0 and π, and each phase jump is accompanied by a pronounced minimum of |*S*(*t*)|.

The apparent double-hump structure of the spectral response in the resonance regime suggests an interpretation of the observed oscillations of *S*(*t*) (Fig. 2E) in terms of a quantum beat between the repulsive and attractive branches of our many-body system. The two branches are strongly broadened and overlap (Fig. 3D), which results in a strong damping of the oscillations.

A detailed comparison of our time- and frequency-domain measurements reveals the potential of our approach to prepare and control many-body states. This is illustrated in Fig. 3, where we show the Fourier transform of the *S*(*t*) data from Fig. 2 as gray curves. We observed that time-domain measurements where the rf pulses are applied in the presence of weakly repulsive interactions (Fig. 3A) emphasize the upper branch of the many-body system, whereas in the attractive case (Fig. 3, B and C), the lower branch is emphasized relative to the rf spectra. We interpret this as a consequence of the fact that the residual interactions during the rf pulse already bring the system into a weakly interacting polaron state before it is quenched to resonance (*20*). Relative to the noninteracting initial state used in frequency-domain spectroscopy, these polarons have an increased wave function overlap with the corresponding strongly interacting repulsive and attractive branches, leading to the observed shift in the spectral weight. Our measurements show that the control over the initial state of many particles can be used to manipulate quantum dynamics in the strongly interacting regime. This unique capability of time-domain techniques can potentially be exploited in a wide range of applications, including the study of the dynamical behavior near the phase transition from a polaronic to a molecular system (*15*) and the creation of specific excitations of a Fermi sea down to individual atoms (*28*).

Our interpretation of the results in Figs. 2 and 3 relies on the assumption that our fermionic impurities are sufficiently dilute so that any interactions between them can be neglected. By increasing the impurity concentration, we can extend our experiments into a complex many-body regime where the impurities interact both with the Fermi sea and with each other (*20*). Figure 4 shows the time-dependent contrast measured for *k*_{B}*T* = 0.24 ± 0.02 ε_{F} and = 0.20, 0.33, and 0.53. An extrapolation of the *S*(*t*) data to zero concentration (open red circles) lies close to the data points for = 0.20, which is the typical concentration in our measurements and agrees with the FDA calculation. This confirms that the physics that we access in the measurements with a small sample of fermionic impurities is close to that of a single impurity, which we posit to be a consequence of the fermionic nature of the impurities. When the impurity concentration is increased, we find that the contrast for* t* > 5 τ_{F} is decreased and the period of the revivals of |*S*(*t*)| is prolonged. We interpret this as arising from effective interactions between the impurities induced by the Fermi sea (*29*, *30*). Such interactions between fermionic impurities are predicted to lead to interesting quantum phases (*31*).

Our results demonstrate the power of many-body interferometry to study ultrafast processes in strongly interacting Fermi gases in real time, including the formation dynamics of quasi-particles and the nonequilibrium dynamics arising from quantum interference between different many-body branches. Of particular interest is the prospect of observing Anderson’s orthogonality catastrophe (*18*, *20*) by further cooling the Li Fermi sea (*32*) while pinning the K atoms in a deep species-selective optical lattice (*33*).

## Supplementary Materials

www.sciencemag.org/content/354/6308/96/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S11

Table S1

## References and Notes

**Acknowledgments:**We thank M. Baranov, F. Schreck, G. Bruun, N. Davidson, and R. Folman for stimulating discussions. Supported by NSF through a grant for ITAMP at Harvard University and the Smithsonian Astrophysical Observatory (R.S.); the Technical University of Munich-Institute for Advanced Study, funded by the German Excellence Initiative and the European Union FP7 under grant agreement 291763 (M.K.); the Harvard-MIT Center for Ultracold Atoms, NSF grant DMR-1308435, the Air Force Office of Scientific Research Quantum Simulation Multidisciplinary University Research Initiative (MURI), the Army Research Office MURI on Atomtronics, M. Rössler, the Walter Haefner Foundation, the ETH Foundation, and the Simons Foundation (E.D.); and the Austrian Science Fund (FWF) within the SFB FoQuS (F4004-N23) and within the DK ALM (W1259-N27).