The strongly driven Fermi polaron (2025)

The low-energy physics of quantum many-body systems can often be understood in terms of quasiparticle excitations. This description has been successful in explaining the thermodynamic and near-equilibrium transport properties of a wide range of materials¹. Methods to modify, and even controllably destroy, these quasiparticles have long been sought after^2,3,4. Such tuning capability could allow modifying the thermodynamics of a system, and potentially offer a route towards realizing strongly correlated systems without well-defined quasiparticles^5,6.

Fermi polarons—impurities that interact with a non-interacting Fermi gas—have attracted considerable interest because they constitute one of the simplest quantum many-body settings for studying both in-equilibrium and out-of-equilibrium properties in correlated systems^{7,8,9,10,11,12,13}. Both attractive and repulsive Fermi polarons have been experimentally realized with ultracold atoms^{9,10,14,15,16,17,18,19} and with semiconductor heterostructures²⁰. Furthermore, Rabi oscillations of impurities with internal states that interact differently with the bath have been used to probe the properties of these polarons, though the interpretation of these experiments remains challenging^{10,17,21,22,23,24}. Harnessing the non-equilibrium dynamics of driven Fermi polarons to manipulate their quasiparticle properties has, thus, remained a major challenge to both experiment and theory.

Here we investigate Fermi impurities embedded in a homogeneous atomic Fermi gas and driven with an external radio-frequency (rf) field as a platform for realizing quasiparticles with tunable properties (Fig. 1a). The impurities have two internal states: one of them essentially does not interact with the Fermi gas, whereas the other interacts unitarily with it. We drive the impurity by coupling the two internal states with the rf field and probe the impurities dressed by both their interactions with the bath and their coupling to the rf field; we measure their quasiparticle properties—energy, decay rate and residue—as a function of the drive strength.

Our experiment begins with a spatially uniform quantum gas of ⁶Li atoms confined in an optical box^25,26 (Fig. 1a). The gas is in a highly imbalanced mixture of the internal states |↓〉 and |B〉, where |↓〉 is one of the two internal states of the impurity and |B〉 is the bath state that forms the Fermi gas. The bath has a Fermi energy E_F/ℏ ≈ 2π × 6 kHz and a temperature T = 0.25(2)T_F, where T_F is the Fermi temperature of the bath and ℏ is the reduced Planck constant (Methods and Supplementary Sections II and III). The magnetic field is set to the Feshbach resonance B₀ of the |↑〉–|B〉 mixture, whereas |↓〉 and |B〉 are weakly interacting; that is, 1/k_Fa_↑B = 0 and k_Fa_↓B ≈ 0.16; |↑〉 is the second internal state of the impurity, k_F is the Fermi wavevector of the bath and a_↑B (a_↓B) is the s-wave scattering length between states |↑〉 (|↓〉) and |B〉. An in situ optical density image of the impurity species |↓〉 is shown in Fig. 1a, along with two central cuts.

After initialization, an rf field of Rabi frequency Ω₀ and detuning Δ (relative to the bare |↓〉–|↑〉 transition) is turned on for a duration t. We then measure the magnetization of the impurities M ≡ (N_↑ – N_↓)/(N_↑ + N_↓) as a function of t, where N_↑ (N_↓) is the population of the impurities in state |↑〉 (|↓〉). In Fig. 1b, we show typical measurements of the dynamics of the magnetization for different detunings. For all the detunings, we observe damped Rabi oscillations, which reach a detuning-dependent steady-state value at long times. The dependence of the magnetization on the detuning is visualized in Fig. 1c, where we show the magnetization as a function of detuning for various evolution times. At short times, we see a sharp peak with a broad shoulder reminiscent of the linear-response spectrum at unitarity^10,12,17(our measurement would be equivalent to the linear-response injection spectroscopy if performed in the regime of weak drive (ℏΩ₀/E_F ≪ 1) and short times (for which \({{M}}\approx -1\))). However, at longer times, we find a strong deviation from linear-response-type behaviour; in fact, \({{M}}\) converges to a monotonously increasing function of Δ in the steady state (Fig. 1c, black dashed lines). The steady-state magnetization spectrum \({{M}}(\Delta )\) vanishes at detuning Δ₀, the zero crossing.

We now study how \({{M}}(\Delta )\) varies with the drive strength ℏΩ₀/E_F. In Fig. 2a, we show the steady-state spectra for ℏΩ₀/E_F = 1.1 and 9.2. Although the zero crossing and the typical width of the spectra depend on ℏΩ₀/E_F, Fig. 2a (inset) shows that the spectra collapse onto a universal curve when the detuning is rescaled to (Δ – Δ₀)/Ω₀, where Δ₀ depends on Ω₀.

A first description of this behaviour is provided by the ground state of an effective spin 1/2 coupled to a field of Rabi frequency \(\tilde{\varOmega }\) and detuning \({\tilde{\Delta}}\) from its resonance (Supplementary Section I):

Our measured magnetization is indeed well described by this model when setting \({\tilde{\Delta}}=\Delta -{\Delta }_{0}\) and \(\tilde{\varOmega }={\varOmega }_{0}\) (Fig. 2a, black dashed line). This model captures the rescaling of the spectrum, with \({{{M}}}_{0}=f({\tilde{\Delta}}/{\varOmega}_{0})\) and \(f(x)=x/\sqrt{1+{x}^{2}}\).

This universality suggests that our protocol realizes a novel steady-state spectroscopy, that is, the response of the many-body system is captured by a small number of (drive-dependent) parameters (in this case, Δ₀ and the response’s typical width). Interestingly, Δ₀ shifts closer to zero with increasing drive strength (Fig. 2a, vertical colour bands). To quantify this effect, we show the extracted Δ₀ value over nearly two orders of magnitude of the drive strength in Fig. 2b. We also extract the typical width of the spectrum (Fig. 2b, inset), which we characterize by the on-resonance susceptibility \(\chi \equiv (\partial {{M}}/\partial \Delta ){| }_{\Delta = {\Delta }_{0}}\). The behaviour of χ(Ω₀) is well explained with the finite-temperature generalization of the model equation (1) (Supplementary Section I): for such a system, the T > 0 magnetization is given by \({{M}}={{{M}}}_{0}\tanh\left(\sqrt{{(\hslash {\varOmega }_{0})}^{2}+{(\hslash {\tilde{\Delta}})}^{2}}/(2{k}_{{\rm{B}}}{T}_{{\rm{spin}}})\right)\). A fit of the measured χ to this model yields a spin (that is, the internal-state) temperature for the impurity of T_spin = 0.24(1)T_F (Fig. 2b (inset), black dashed line). Because the finite-temperature correction becomes negligible when ℏΩ₀ ≳ 2k_BT_spin, the impurity is effectively in its internal ground state for ℏΩ₀ ≳ E_F (Fig. 2b (inset), black dashed line and red solid line).

The fact that the spin temperature T_spin matches the bath temperature T indicates that the internal degrees of freedom of the driven polaron have thermalized with the fermionic bath. Two additional observations support the fact that our system consisting of the impurities plus the bath obeys closed-system dynamics. First, the coherence time of the Rabi oscillations in the absence of the bath exceeds—by about two orders of magnitude—the typical duration of our experiments in the presence of the bath. Thus, the decay is not due to dephasing caused by, for instance, magnetic-field noise or inhomogeneity (Supplementary Section IV). Second, the impurity and bath atom numbers are constant during the dynamics (Fig. 1b, pink symbols).

A non-trivial shift in the zero crossing, Δ₀ ≠ 0, is a prime indication of interactions between the impurity and the Fermi gas. To gain insight into Δ₀, we measure the rf linear-response injection spectrum, that is, the fraction of impurities transferred from |↓〉 into |↑〉 using a weak rf pulse (ℏΩ₀ ≪ E_F). A typical measurement is shown in Fig. 2c (green diamonds). We find that Δ₀ matches the location of the peak E_p of the linear-response spectrum (that is, the energy of the attractive polaron¹⁸), even for a steady-state spectrum taken at a moderate rf power ℏΩ₀ ≈ E_F (Fig. 2c, red points). We thus conclude that the plateau at low ℏΩ₀/E_F (Fig. 2b) matches the linear-response peak position. However, with increasing drive strength ℏΩ₀/E_F ≳ 1, the zero crossing smoothly departs from its low-Ω₀ plateau, towards zero.

We compare our measurements with the predictions of a diagrammatic theory based on the non-self-consistent T matrix (Methods and Supplementary Section I). We directly include the drive in the T matrix²³ to describe the drive-induced changes in the scattering properties of the impurities. We also include corrections due to the non-zero a_↓B (Methods). In Fig. 2b, the black line shows the theoretical prediction for Δ₀(Ω₀), calculated from the spectral functions (Methods and Supplementary Section I); the theoretical Δ₀ is normalized to its weak-drive limit \({\Delta }_{0}^{{\rm{ref}}}\equiv {\Delta }_{0}({\varOmega }_{0}\to 0)\). We find a behaviour that qualitatively reproduces the experimental data. Furthermore, we find that the polaron energy, defined as the solution of E_p = ReΣ(E_p) (where Σ is the zero-momentum self-energy of the impurity evaluated at the Rabi frequency Ω₀ and detuning Δ₀), matches the theoretical ℏΔ₀ in the limit ℏΩ₀/E_F→0 (ref. ²⁷), reproducing the experimental finding shown in Fig. 2c. Interestingly, E_p is in remarkable agreement with the experimental data, when normalized to the weak-drive limit \({\Delta }_{0}^{{\rm{ref}}}\) (Fig. 2b, green solid line). This observation suggests that Δ₀ may be the dressed polaron energy at all drive strengths, and motivates the development of more refined methods to calculate Δ₀.

We now study the pre-steady-state dynamics, focusing on the resonant case Δ = Δ₀. The magnetization undergoes damped oscillations towards \({{M}}=0\) (Fig. 1b, middle). We find that the data are well fitted by \({{M}}(t)=-\cos ({\varOmega }_{{\rm{R}}}t)\exp (-\Gamma t/2)\), from which we extract a renormalized Rabi frequency Ω_R and a decay rate Γ (Methods).

These quantities have been connected to the equilibrium properties of the polaron^10,17,22. By assuming that the bath is not perturbed by the impurities (and that ℏΩ₀ ≪ E_F, such that the Rabi oscillations are underdamped), the dynamics of the Rabi oscillations at early times can be approximately captured by an equilibrium correlation function that is given at low temperature by the impurity’s zero-momentum spectral function in the presence of the drive (Supplementary Section I). This approach leads to the following expressions for the zero-momentum quasiparticle decay rate Γ and the renormalized Rabi frequency Ω_R in terms of the impurity self-energy Σ (refs. ^22,23): ℏΓ = –2Z ImΣ(E = E_p) and \({\varOmega }_{{\rm{R}}}=\sqrt{Z}{\varOmega }_{0}\) (ref. ¹⁰), where \(Z=1/\left(1-{\frac{\partial ({\rm{Re}}\Sigma )}{\partial E}\vert }_{E = {E}_{{\rm{p}}}}\right)\) is the quasiparticle residue, which is the overlap between the Fermi polaron wavefunction and the non-interacting impurity wavefunction.

In Fig. 3a, we show Γ as a function of the drive strength, along with the T-matrix predictions. At our weakest drives, the decay rate appears to slowly level off. Note that a previous measurement at a given Ω₀ is in good agreement with our data (Fig. 3, red square¹⁷). For stronger drives, that is, ℏΩ₀ ≳ E_F, Γ shows an unexpected non-monotonic behaviour. First, it increases by over an order of magnitude between our lowest Ω₀ and a maximum located at around ℏΩ₀/E_F ≈ 3; it then decreases as a power law of the drive strength.

We use the T-matrix approach to calculate Γ (Fig. 3a, black solid line). It is in qualitative agreement with the data but is quantitatively imprecise for all but the largest drives. The power law at large drive strengths originates from the properties of the two-body scattering: scattered particles have an energy ∝ℏΩ₀ and the T matrix is dominated by the two-body contribution, so that it is proportional to the inverse square root of Ω₀ (ref. ²⁸). Our data are in good agreement with the corresponding analytical prediction \(\hslash \Gamma /{E}_{{\rm{F}}}\approx 16/\left(3\uppi \sqrt{\hslash {\varOmega }_{0}/{E}_{{\rm{F}}}}\right)\) (Fig. 3a, grey line). Motivated by this observation, we introduce a two-body T-matrix approximation that neglects in-medium scattering effects but still takes into account the presence of the Fermi sea in the self-energy (Fig. 3a, black dotted line). Surprisingly, this approximation captures the data remarkably well for all drive strengths.

The linear-response spectrum provides another measure for the quasiparticle decay rate. From our injection spectroscopy measurements (Fig. 2c, green points), we extract a width Γ_lin = 0.13(2)E_F/ℏ, which is close to the width measured by ejection spectroscopy¹⁸. This width is a little larger than our weakest-drive Γ = 0.072(5)E_F/ℏ (Fig. 3a). It is possible that a finite impurity concentration could have a different effect on the Rabi dynamics and the linear-response spectroscopy²³. Their respective relationship to the linear-response properties in the impurity limit is not straightforward and remains to be clarified^10,22.

The normalized (dynamical) Rabi frequency Ω_R/Ω₀ extracted from the damped oscillations also depends on the bare Rabi frequency Ω₀. In Fig. 3b, we show Ω_R/Ω₀ versus ℏΩ₀/E_F. At ℏΩ₀/E_F ≈ 2, Ω_R/Ω₀ abruptly increases, reaches a maximum exceeding the one at around ℏΩ₀/E_F ≈ 5 and then decreases towards one. In the regime of strong drives, we find Ω_R > Ω₀. An explanation for this effect could have been that the minimum of Ω_R is not reached at the detuning Δ = Δ₀. However, we measure Ω_R as a function of the detuning (Fig. 3b, top inset) and find that Ω_R is consistently larger than Ω₀; its minimum is reached at Δ = 1.03(5)Δ₀.

To settle the range of ℏΩ₀/E_F over which our Rabi oscillation spectroscopy is valid, we show the quality factor of the Rabi oscillations Ω_R/Γ in Fig. 3b (bottom inset). At weak drives, Ω_R/Γ roughly plateaus; at even weaker drives (ℏΩ₀ ≲ 0.4E_F), the ratio decreases further and oscillations ultimately become overdamped (Supplementary Section V shows a set of measurements across the critical damping threshold).

The observation that Ω_R can exceed Ω₀ is at odds with the expectation that (Ω_R/Ω₀)² is the quasiparticle residue. Indeed, the residue Z, defined as the area under the quasiparticle peak in the spectral function, must always be ≤1. We can formally define a pseudo-residue\(\bar{Z}\equiv 1/\left(1-{\frac{\partial ({\rm{Re}}\Sigma )}{\partial E}\vert }_{E = {E}_{{\rm{p}}}}\right)\), which coincides with Z for textbook quasiparticles²⁹ (and is often used as a definition for the residue³⁰). We find that both two-body and in-medium T-matrix approximations predict that \(\bar{Z}\) follows a similar behaviour to (Ω_R/Ω₀)² (Fig. 3b, dotted and solid black lines, respectively), suggesting the intriguing generalization \({({\varOmega }_{{\rm{R}}}/{\varOmega }_{0})}^{2}\approx \bar{Z}\). In particular, we analytically calculate \(\bar{Z}\approx 1+16/(6\uppi {(\hslash {\varOmega }_{0}/{E}_{{\rm{F}}})}^{3/2}) > 1\) in the limit ℏΩ₀/E_F ≫ 1 (Supplementary Section I). This prediction (Fig. 3b, grey line) is in very good agreement with our measurements. The fact that \(\bar{Z} > 1\) indicates that the standard description of a (Lorentzian-shaped) quasiparticle breaks down. This is another interesting illustration of a textbook quasiparticle breakdown that is not signalled by the usual criterion \(\bar{Z}\to 0\) (as in the case of immobile Fermi impurity^31,32); see, for example, a recent condensed-matter experiment in which a quasiparticle disappearance, accompanied by an increase in Z, was caused by a widening of the quasiparticle into an incoherent background³⁰.

This finding motivates us to examine the quasiparticle quality factor Q, defined as the ratio of the quasiparticle energy and its decay rate. Using the zero crossing as a proxy for the drive-dressed polaron energy, we define Q ≡ Δ₀/Γ. In Q versus ℏΩ₀/E_F (Fig. 4a), we find that Q ≫ 1 for low Ω₀, indicating a well-defined polaron. However, the quality factor decreases by over an order of magnitude to a minimum of \(\approx 0.9\) at ℏΩ₀/E_F ≈ 4, indicating that the polaron is no longer well defined, specifically, it is close to the critical damping threshold Q = 1/2.

The fact that Q ≈ 1 in conjunction with (Ω_R/Ω₀)² > 1 at ℏΩ₀/E_F ≈ 4 further substantiates the failure of the textbook quasiparticle picture in that regime. We have seen earlier that the pseudo-residue \(\bar{Z}\) defined through the Taylor expansion of the self-energy around the polaron energy E_p yields non-physical values for the quasiparticle residue. We indeed find that within the T-matrix calculations, \(\bar{Z}\) depends on the energy at which it is evaluated, even within the spectral width Γ; this goes beyond the usual quasiparticle paradigm (Supplementary Section I). Nevertheless, it is intriguing that even though \(\bar{Z}\) (>1) is no longer the quasiparticle residue in the strong-drive regime, its formal expression is in good agreement with the measured (Ω_R/Ω₀)².

Putting all this together, we can now interpret the different regimes of the driven polaron. As shown in the energy diagram (Fig. 4b; built from the theoretical spectral function), there are three distinct regimes. In the first weak-drive regime ℏΩ₀/E_F ≪ 1 (labelled A), there are three spectral features. The attractive polaron is split into two rf-dressed states \(\left\vert \pm \right\rangle =\) (|↑〉±|↓〉) \(/\sqrt{2}\) (red and blue boxes), whose energy separation is roughly ∝ℏΩ₀. The third feature is the remnant of the unstable repulsive polaron, seen as the broad continuum at high energies; it is basically unaffected by the rf. In the strong-drive regime 1 ≲ ℏΩ₀/E_F ≲ 10 (labelled B), the dressed state |+〉 is pushed into the continuum and hybridization leads to the disintegration of the quasiparticle for ℏΩ₀/E_F ≈ 3. For ultrastrong drives ℏΩ₀/E_F ≳ 10 (labelled C), which we do not reach experimentally, many-body effects (dominated by two-body scattering) are a perturbation to the large rf splitting and the driven impurity essentially behaves as a dressed two-level system. In the experiment, we are probing the transient (Rabi) dynamics of the system towards thermalization after a rapid quench into a superposition of those dressed states; the hybridization manifests itself as the non-monotonic evolution of Γ and Ω_R around ℏΩ₀/E_F ≈ 3 (Supplementary Section I). This interpretation, although based on the standard but unsettled connection between the Rabi oscillations and the equilibrium spectral function, is supported by the good agreement between our experimental observation and our calculations.

In conclusion, we have shown that the driven impurity in a Fermi gas is a powerful platform for studying non-equilibrium quantum dynamics. In the future, our many-body steady-state spectroscopy could be used to study many-body systems without well-defined quasiparticles, such as the spin-balanced unitary Fermi gas in the normal phase³³, or the heavy impurity system^34,35. It would also be interesting to study how polaron–polaron interactions could be modified by the drive, as well as observe the dressed-state polarons with a secondary (probing) field. Furthermore, in the 1/k_Fa_↑B > 0 regime, one could observe richer steady-state magnetization spectra for pre-thermal states containing both stable attractive and repulsive polarons³⁴. Theoretically, it is imperative to establish a systematic theoretical framework to elucidate the connection between the impurity’s equilibrium spectral properties and the Rabi oscillations, since the reasonable agreement of the measurements of both Ω_R and Γ with our (state-of-the-art yet uncontrolled) T-matrix calculations is rather surprising. This could be done, for example, by using the Keldysh approach²⁴.