Observation of coherent quench dynamics in a metallic many-body state of fermionic atoms

Quantum simulation with ultracold atoms has become a powerful technique to gain insight into interacting many-body systems. In particular, the possibility to study nonequilibrium dynamics offers a unique pathway to understand correlations and excitations in strongly interacting quantum matter. So far, coherent nonequilibrium dynamics has exclusively been observed in ultracold many-body systems of bosonic atoms. Here we report on the observation of coherent quench dynamics of fermionic atoms. A metallic state of ultracold spin-polarised fermions is prepared along with a Bose-Einstein condensate in a shallow three-dimensional optical lattice. After a quench that suppresses tunnelling between lattice sites for both the fermions and the bosons, we observe long-lived coherent oscillations in the fermionic momentum distribution, with a period that is determined solely by the Fermi-Bose interaction energy. Our results show that coherent quench dynamics can serve as a sensitive probe for correlations in delocalised fermionic quantum states and for quantum metrology.

The investigation of nonequilibrium dynamics in interacting quantum many-body systems has emerged as a key approach to characterize the nature of quantum states, to study excitation spectra [1][2][3], and to shed light on thermalization processes [4,5]. So far, research on nonequilibrium dynamics has focused on manybody quantum states of bosonic particles, leading to the observation of coherent quench dynamics [6][7][8][9] and the exploration of relaxation and thermalization in isolated quantum systems [10][11][12].
Here we report on the observation of coherent quench dynamics in a many-body quantum state of fermionic particles. In the experiment, we prepare a metallic state of ultracold spin-polarized fermionic atoms in a shallow three-dimensional (3D) optical lattice. The delocalized fermions are in contact with a Bose-Einstein condensate (BEC) that is simultaneously loaded into the lattice. With a rapid increase of lattice depth, we take the system out of equilibrium and induce quench dynamics that is driven by the interactions between fermions and bosons. We observe the time evolution of the fermionic momentum distribution, which shows long-lived coherent oscillations for up to ten periods, both for attractive and repulsive Fermi-Bose interactions. A theoretical model reveals that the dynamics arises as a consequence of the delocalized nature of the initial fermionic state and the on-site number fluctuations of the BEC. Our work demonstrates that coherent quench dynamics constitutes a powerful technique to gain insight into the nature of fermionic quantum many-body states and to accurately determine Hamiltonian parameters used in their microscopic description.
When temperatures approach absolute zero, quantum statistics becomes increasingly relevant and gives rise to distinctly different many-body ground states for bosonic and fermionic systems [13]. Noninteracting bosons collectively condense into the single-particle state of lowest energy, forming a BEC. Fermions, on the other hand, obey the Pauli exclusion principle, which limits the occupation of single-particle states to a maximum of one fermion. Therefore, fermions fill the lowest energy single-particle states from bottom up and form a Fermi sea. When placed in a periodic lattice potential with M sites, the wavefunction of a weakly interacting BEC can be approximated by a product |Ψ BEC = M j=1 |α j of identical coherent states |α j = e −|α| 2 /2 e αâ † j |0 , where |α| 2 is the mean occupation per lattice site, andâ † j the bosonic creation operator at site j [14]. On the other hand, the wavefunction of a weakly interacting Fermi sea of N identical fermions can be expressed by the product |Ψ FS = E k ≤EF |k of the N quasi-momentum eigen- with energy eigenvalues E k smaller than the Fermi energy E F . Here,ĉ † j denotes the fermionic creation operator, and r j the position of site j. As long as the fermions do not completely fill up a lattice band, |Ψ FS represents a metallic state.
The distinct ground state properties of bosons and fermions have direct implications for their respective many-body quantum dynamics. For the case of bosons, coherent quench dynamics was experimentally studied by preparing an atomic BEC in a shallow optical lattice and taking it out of equilibrium by a sudden quench to a deep lattice [6][7][8]. The rapid suppression of tunneling and the enhanced interactions between the atoms gave rise to characteristic collapses and revivals of the bosonic matter wave interference pattern, whose periodicity is determined by the strength of the on-site interaction U BB . In homogeneous lattice potentials, this phenomenon can be understood from the dynamics of a single lattice site: The time evolution of the many-body state is governed by the operator e −iĤt/ = M j=1 e −iĤj t/ witĥ H j = U BBn j (n j − 1)/2 being the on-site interaction term of the Bose-Hubbard Hamiltonian, wheren j =â † jâ j counts the number of bosons at site j. Consequently, the dynamics of the entire system, e −iĤt/ |Ψ BEC = M j=1 e −iĤj t/ |α j , is comprised of a product of identical dynamics at each lattice site.
In this work, we are concerned with the dynamics of a delocalized many-body state of fermions, for which, even in a homogeneous lattice, an effective single-site description is not possible. Specifically, we consider a metallic state of spin-polarized fermionic atoms in a shallow lattice, immersed in a BEC as schematically shown in Fig. 1a. Initially, the interactions between fermions Emergence of coherent quench dynamics in a metallic many-body state of fermionic atoms. a, A Fermi-Bose quantum gas mixture is loaded into a shallow optical lattice. The spin-polarized fermions form a metallic many-body quantum state, delocalized across the lattice (green spheres). The bosons form a BEC (blue background). b, A rapid increase of lattice depth quenches the system by suppressing the tunneling between lattice sites and increasing the on-site interaction between fermions and bosons, U BF . The resulting coherent quench dynamics in the fermionic momentum distribution can be understood from a two-site system, illustrated by three sample cases. If the system features a localized fermion (left) or bosonic Fock states with an equal number of bosons on each site (middle), no dynamics occurs. Dynamics occurs if the fermion is delocalized, ĉ † 1ĉ2 = 0, and the number of bosons is different on each site (right). c, Fermionic singleparticle correlations in a 1D lattice at times t = 0, h/(8U FB ), and h/(2U FB ) after the quench. The fermionic filling is chosen to bem = 0.3 and stays constant as a function of time (black point at j − l = 0). and bosons are weak. Therefore, the quantum state of this hybrid Fermi-Bose system can be approximated by the direct product |Ψ FS ⊗ |Ψ BEC . When the system is quenched by a rapid increase of lattice depth, tunneling between lattice sites is suppressed and interparticle interactions dominate. Interactions among the bosonic component give rise to typical collapse and revival dynamics that have been analyzed previously [15]. It is the key finding of the present work that the fermionic component also undergoes coherent dynamics. Although the fermions do not interact among themselves, the interaction with the bosons drives the dynamics of the quenched metallic state. Similar to the purely bosonic case, the time evolution operator e −iĤt/ factorizes into M j=1 e −iĤ FB j t/ withĤ FB j = U FBn jmj being the on-site interaction term of the Fermi-Bose Hubbard Hamiltonian [16], wherem j =ĉ † jĉj counts the number of fermions at site j and U FB is the on-site Fermi-Bose interaction energy. However, due to its delocalized nature, the initial metallic state |Ψ FS does not factorize into a product of on-site wavefunctions, which is crucial for fermionic dynamics to occur.
In order to illustrate the emergence of dynamics, we consider an elementary setup with two lattice sites (labeled 1 and 2, spaced by distance a), occupied by a single fermion and multiple bosons (see Fig. 1b). For finite tunneling and vanishing interactions, U FB = 0, the fermionic ground state has the form (ĉ † 1 +ĉ † 2 ) |0 / √ 2, corresponding to the fermion being in the k = 0 momentum eigenstate. After the quench, the site occupations ĉ † 1ĉ1 = ĉ † 2ĉ2 = 1/2 remain constant due to the absence of tunneling. The off-diagonal correlations of the single-particle density matrix, however, evolve in time: is the number of bosons on site 1 (2). Consequently, the momentum distribution, n(k) ≡ M −1 j,l e ika(j−l) ĉ † jĉ l , undergoes dynamics. The evolution for the two allowed momentum eigenstates k = 0 and k = π/a reads (1 ± cos[U FB (n 1 − n 2 )t/ ])/2, respectively, and indicates that the fermion oscillates between the k = 0 and k = π/a states with a period T ∝ h/U FB . In contrast, no dynamics occurs if the fermion occupies a localized state,ĉ † 1 |0 orĉ † 2 |0 , since the off-diagonal correlations vanish, or if the number of bosons is identical on both sites (n 1 = n 2 ). For n(k) to evolve with time, delocalized fermions and spatially varying bosonic occupancies are required. In the experiment, the latter is provided by the quantum fluctuations of the on-site occupation that are characteristic to a BEC.
In order to quantify the fermionic quench dynamics in a setup with many lattice sites, it is convenient to study the visibility of the fermionic momentum distribution. We define it as the ratio between the number of fermions with quasi-momenta in the interval [−k 0 , k 0 ] (see Fig. 2a) and the total fermion number N F , V F ≡ 1 NF k0 −k0 dk x n(k x ). For the case of a 1D lattice and an initial state |Ψ FS ⊗|Ψ BEC , the time evolution of the visibility after the quench can be calculated analytically in the thermodynamic limit (see Methods Summary). Assuming k 0 ≤ k F , we obtain where k F is the Fermi momentum, k lat = π/a is the quasi-momentum corresponding to the edge of the Brillouin zone and a is the lattice spacing. Analogous to the two-site case, the periodicity of the oscillation is determined by T = h/U FB , and the coherent dynamics originates from the presence of off-diagonal single particle correlations and on-site occupancy fluctuations of the BEC. Figure 1c illustrates the time evolution of the off-diagonal correlations ĉ † jĉ l (t) =m sin[πm(j − l)] exp[2|α| 2 {cos(U FB t/ ) − 1)}]/(j − l), wherem = k F /k lat is the fermionic filling. The theoretical analysis can be extended to 3D lattices, where n(k x , t) is taken to be the projection of the full fermionic momentum distribution n(k x , k y , k z , t) onto one dimension, n(k x , t) = k lat −k lat dk y k lat −k lat dk z n(k x , k y , k z , t). The results are qualitatively similar to Eq. (1) (see Methods Summary).
The experiment begins with the preparation of a quantum degenerate mixture of 2.1(4)×10 5 fermionic 40 K and 1.7(3) × 10 5 bosonic 87 Rb atoms in their absolute hyperfine ground states |9/2, −9/2 and |1, +1 , respectively. The temperature of the spin-polarized Fermi gas is typically T /T F = 0.20 (2). Its interaction with the bosons is tuned by means of a Feshbach resonance at 546.75(6) G [17], addressing interspecies scattering lengths a FB in a range between −161.2(1) a 0 and +59(10) a 0 . Subsequently, a 3D optical lattice operating at a wavelength of λ lat = 738 nm is adiabatically ramped up within 50 ms to a depth of V L = 3.5(2)E F rec , where E F rec = 2 k 2 lat /(2m F ) denotes the recoil energy, k lat = 2π/λ lat and m F is the atomic mass of 40 K. For these parameters, the fermions form a metallic many-body state within the first lattice band (see Methods Summary) [18] and the bosons form a BEC. Then, we quench the system by rapidly increasing the lattice depth to V H = 18(1)E F rec , suppressing the tunneling coupling between lattice sites and initiating coherent nonequilibrium dynamics of the Fermi-Bose many-body state. After letting the system evolve for variable hold times t, all trapping potentials are suddenly switched off and an absorption image of the momentum distribution of 40 K is recorded after 9 ms time-of-flight expansion (see Fig. 2a, inset).
The dynamical evolution of the fermions is revealed via oscillations in the momentum distribution (see Fig. 2). The recorded absorption images are integrated along the direction of gravity to obtain 1D momentum profiles n(k x , t) at discrete hold times t, sampled in steps of 40 µs. Figure 2a compares two such profiles that are recorded at the approximate times of a half and a full oscillation cycle for a fixed value of Fermi-Bose interactions. The residual after subtracting the two profiles from each other (see Fig. 2b) illustrates how the interaction-driven dynamics leads to a redistribution of population from momenta at the center to momenta at the edge of the Brillouin zone. The coherent quench dynamics can be observed as a periodic modulation of the peak height at k x = 0 for hold times shorter than the time scale set by residual tunneling of the fermions. For longer times, the momentum profiles relax towards a state with a more uniform distri- bution across the Brillouin zone (Fig. 2c), as expected in thermal equilibrium. A quantitative understanding of the coherent fermionic dynamics can be gained from the time traces of the visibility V F . For each momentum profile n(k x , t) the fermionic visibility is evaluated as illustrated in Fig. 2a, where k 0 = 2k lat /3 is chosen for the best signal. Figure 3a shows exemplary time traces of the evolution with up to ten observable oscillation periods. Upon increasing the attraction between fermions and bosons, the period of the oscillations becomes shorter as expected from the theoretical analysis. This confirms that the quench dynamics is driven by the interspecies interaction U FB ∝ a FB . In general, we observe oscillation amplitudes that are significantly smaller than in the case of of bosonic collapse and revival dynamics [6,8]. A fundamental reason lies in the different occupation of the momentum distributions for fermions and bosons, as a direct consequence of quantum statistics. In addition, the overlap between 40 K and 87 Rb is limited, because the in-trap size of the Fermi gas is significantly larger than the BEC. Correlations between fermions on sites without bosons do not contribute to the dynamics. Furthermore, finite temperature and Fermi-Bose interactions in the inital state can lead to enhanced localization of fermions [19,20] and a reduction of visibility. Finally, residual tunneling after the quench is expected to enhance damping and to reduce the oscillation amplitude [21,22].
The spectral content of the fermionic quench dynamics is revealed via Fourier transform of the visibility time traces. As shown in Fig. 3b, the spectra are dominated by a single peak, in remarkable contrast to the rich spectra of the bosonic collapse and revival dynamics in the same experimental setting [15]. Nevertheless, the dominant peak displays a comb-like substructure with several frequencies of order U FB /h. We assign this substructure to the deformation of on-site orbitals as as a result of interactions [8,20,23], that effectively gives rise to an explicit dependence of the Fermi-Bose interaction energy on the bosonic on-site occupation n, U FB n (see Methods The dashed line shows a linear fit C|aFB| to the dominant spectral features, yielding C = 36.7(3) Hz/a0. The shaded area shows a numerical calculation assuming the prescription U FB = (2π 2 aFB/µ) dr 3 |wB(r)| 2 |wF(r)| 2 of a single-band Hubbard model, taking into account the experimental uncertainties in lattice depth and interspecies scattering length aFB. Here, wF(r) and wB(r) denote the fermionic and bosonic Wannier functions, respectively, and µ = mFmB/(mF + mB) is the reduced mass. Summary). The progression of U FB as a function of the interspecies scattering length is displayed in Fig. 4, both for attractive and repulsive Fermi-Bose interactions. We compare the data to a numerical calculation of U FB that uses Wannier functions as on-site orbitals. On the attractive side, the highest frequency components are compatible with the calculation, while on the repulsive side all frequency components are located within the bounds of the calculation.
Our measurements demonstrate that coherent quench dynamics of metallic states can be observed with ultracold fermionic atoms in an optical lattice. In the hybrid Fermi-Bose system investigated here, the time evolution arises from interspecies interactions and an initial bosonic state that exhibits site-to-site fluctuations of the atom number. Similar dynamics emerges in spin-1/2 interacting fermionic systems [24], and it is also expected to emerge in higher-spin fermionic systems [25], following a similar quench protocol. Since the dynamics relies on the delocalized nature of fermionic atoms, it can also serve as an interferometrically sensitive probe for delocalization. This offers a novel approach to identify delocalized quantum phases in fermionic systems, such as the Hubbard model, in one, two and three dimensions [26,27] and in chains of spin-polarized fermions with nearest and next-nearest neighbor interactions [28,29]. In addition, coherent quench dynamics can be used as a precise tool to measure on-site interactions and to reveal complex interaction effects in hybrid few-body systems.