Critical slowdown of non-equilibrium polaron dynamics

We study the quantum dynamics of a single impurity following its sudden immersion into a Bose-Einstein condensate. The ensuing formation of the Bose polaron in this general setting can be seen as impurity decoherence driven by the condensate, which we describe within a master equation approach. We derive rigorous analytical results for this decoherence dynamics, and thereby reveal distinct stages of its evolution from a universal stretched exponential initial relaxation to the final approach to equilibrium. The associated polaron formation time exhibits a strong dependence on the impurity speed and is found to undergo a critical slowdown around the speed of sound of the condensate. This rich non-equilibrium behaviour of quantum impurities is of direct relevance to recent cold atom experiments, in which Bose polarons are created by a sudden quench of the impurity-bath interaction.


I. INTRODUCTION
Understanding the non-equilibrium dynamics of many-body systems remains one of the most outstanding challenges in physics.
Cold atomic gases have emerged as an excellent platform to explore this question [1], since they can be well isolated from their environment and offer an extraordinary level of control. In particular, the precise tunability of interactions through Feshbach resonances [2][3][4] has opened the door to studies of interaction effects in quantum many-body systems with unprecedented control. While this offers unique perspectives for studying paradigmatic models in condensed matter physics [5,6], the scope of cold-atom research has extended well beyond such initial ideas. Exciting new research directions include the dynamical emergence of thermal equilibrium in isolated quantum systems [1], and the observation of many-body localization [7], linked to the breakdown of ergodicity [8,9]. Another example is the polaron quasiparticle, which was originally introduced by Landau [10] to describe the interaction of electrons with the atomic crystal of a solid, and has since been employed to understand a broad range of problems in condensed matter physics [11]. Experiments on ultracold Fermionic atoms provide an ideal quantum simulation platform for this problem [12][13][14][15][16] enabling precise studies of the Fermi-polaron [17][18][19]. At the same time, the possibility to realize so-called Bose polarons in atomic Bose-Einstein condensates (BECs) [20,21] has raised further questions and ushered in new theoretical investigations [22][23][24][25][26][27][28][29][30][31][32][33], expanding our understanding of quantum impurity physics. While much of previous efforts have been directed towards the equilibrium properties of the Bose polaron, its dynamics has spawned theoretical work only recently [34][35][36][37][38][39], predicting the formation of phonon-impurity bound states for strongly interacting systems [34] and studying trajectories and momentum relaxation of moving impurities [35][36][37][38], as well as the dynamics of phonon dressing in spinor condensates [39]. Here we investigate the non-equilibrium dynamics of a mobile impurity in a BEC and reveal distinct evolution stages with associated timescales that determine the formation dynamics of the Bose polaron from an initial interaction quench to its final equilibrium steady state. Treating the BEC as a bath, we use a master equation approach for such a systematic characterization and derive rigorous results in the limit of weak impurity-boson interactions. Within this picture, the Bose polaron emerges as a consequence of impurity-bath decoherence, whereby all many-body states apart from the polaron ground state dephase as the phonon-dressed impurity approaches the known equilibrium state of the Bose polaron. As illustrated in Fig. 1, we identify three distinct stages and corresponding timescales of the non-equilibrium impurity dynamics. The initial coherence of the impurity decays in a universal non-exponential fashion on a timescale that solely depends on the impurity-boson coupling. The subsequent relaxation however is strongly influenced by the boson-boson interaction, initiating the phonon dressing of the impurity. While the impurity eventually reaches its equilibrium polaron state [34], we find that the final polaron formation undergoes a critical slowdown as the impurity speed approaches the Landau critical velocity of the BEC. The determined characteristic timescales for polaron formation are within the resolution of current experiments, and should thus be observable in future measurements.

II. THE SYSTEM
We consider an impurity of mass m in a gas of bosons with a mass m B and a density n B . The boson-boson and boson-impurity interactions are both of short range nature and characterised by the scattering lengths a B and a, respectively. For weak interactions and close to zero temperature, the bosons form a BEC that is accurately described by Bogoliubov theory [40]. The Hamiltonian, of this system can be separated into three terms, where describes the BEC in terms of Bogoliubov modes with momenta k and associated energies E k = [ε B k (ε B k + 2n B T B )] 1/2 that are created by the operators β † k . Here, T B = 4πa B /m B is the scattering matrix for the bosonboson interaction, and ε B k = k 2 /2m B is the bare boson kinetic energy at momentum k. We work in units where = 1. The impurity Hamiltonian is given by where c † k creates an impurity with momentum k and energy which includes the mean-field shift ε MF = n B U 0 of the impurity energy due to its interaction with the BEC.
Here, U k is the impurity-boson interaction momentum space. Accordingly the term describes the impurity-boson interaction. Eqs. (2)- (5) correspond to the so-called Fröhlich model [41], originally put forth to describe the electrons coupled to optical phonons of a dielectric crystal. In the present case of an impurity in a BEC, there are interaction terms not included in Eq. (5), which describe the scattering of the impurity on bosons already excited out of the BEC. For small impurity-boson interactions, their contribution is, however, suppressed by a factor (n B a 3 B ) 1/2 [28], which we assume to be small. Focussing our analysis to the regime of validity of the Fröhlich Hamiltonian, we consider all observables to second order in the impurity scattering length a. Explicitly, we express the interaction as U k = U 0 g k , with g k a rescaled coupling and g 0 = 1. We then solve the Lippmann-Schwinger equation to express U 0 in terms of a to second order. This yields Here T = 2πa/m r is the zero energy impurity-boson scattering matrix and m r = m B m/(m + m B ) denotes the reduced mass. This allows to re-express our results in terms of the scattering length a, which yields well-defined result in the limit of a zero range interaction, i.e. a momentum independent impurity-boson interaction with g k = 1. We however retain a momentum dependent g k in the formalism in order to analyse how a small but finite interaction range can affect the initial impurity dynamics at early times.
In order to study decoherence, we consider a quench in which an impurity with momentum p is suddenly immersed into the condensate at time t = 0, creating the initial state where θ is the mixing angle that determines the probability, sin 2 θ, for initial impurity creation. The coherence between the vacuum and the single-impurity state can then be obtained from where c p (t) = exp(iHt)c p exp(−iHt) is the timedependent impurity operator in the Heisenberg picture. The evolution of C p (t) closely traces the dynamical formation of the polaron, as we shall discuss in the rest of the paper. Substituting the initial state Eq. (6) shows that the coherence is related to the time-dependent impurity Green's function, G p , for t > 0, and which also corresponds to the dynamical overlap considered in [34].
For long times, it should thus approach the asymptotic Greens function of the polaron and thereby define the polaron energy, E p , the polaron lifetime τ p , and the quasiparticle residue Z p . For an infinite polaron lifetime, |C p (t)| therefore converges to the quasiparticle weight Z p , as also found in [34].
In cold-atom experiments, the initial state Eq. (6) can be prepared by driving a transition between internal atomic states of the impurity which feature different interactions. Indeed recent experimental demonstrations of the Bose polaron were based on RF spectroscopy on a hyperfine transition between a weakly and a strongly interacting state of a single-component 39 K BEC [20] or of 40 K impurity atoms immersed in a 87 Rb condensate [21]. In both cases, a direct measurement of the coherence dynamics is possible via Ramsey spectroscopy [42][43][44].

III. MASTER EQUATION DESCRIPTION
We determine the dynamics of the coherence using the impurity density operator ρ I . First, we consider the density operator ρ(t) of the entire system, i.e. the impurity and the BEC. In the interaction picture, it obeys the von Neumann equation One can formally solve this equation by integrating both sides and resubstituting the result into the right hand side of (10) to obtain We can now trace out the bosonic degrees of freedom on both sides of Eq. (11) to obtain an evolution equation for the reduced density operator, ρ I = Tr B ρ, of the impurity. For our chosen initial state Eq. (6) the initial density operator of the entire system factorizes according to ρ(0) = ρ I (0) ⊗ ρ B (0), where ρ B (0) = |BEC BEC| is the density operator for the BEC. Taking the trace of the first term in Eq. (11) only yields terms proportional to β = β † = 0, such that we obtain the following equation of motion for the impurity density operator Here, we have made the Born approximation, ρ(s) = ρ I (s) ⊗ ρ B (0) assuming that the density matrix of the BEC is unaffected by the impurity, which is justified for small impurity interactions. In the same limit we can also replace ρ I (s) by ρ I (t) to obtain a time-local equation that contains all relevant contributions up to second order in the interaction strength [45,46]. Altogether this makes it possible to evaluate the trace over the commutator in Eq. (12). Using further that BEC| β k β † k |BEC = δ k,k and β k (t) = e −iE k t β k (0) in the interaction picture, we finally obtain Further details of this derivation are given in App. A. Equation (13) constitutes an effective von Neumann equation for the impurity density operator, and will now be used to determine the dynamics of the impurity coherence.

IV. IMPURITY COHERENCE
Knowing the time dependent density operator of the impurity we can calculate its coherence from where we now trace over the impurity degrees of freedom, and have used that only the vacuum |0 contributes to the trace for the initial state Eq. (6). It turns out to be convenient to introduce the coherenceC p (t) = e iεpt C p (t), stripped of the single particle and mean field phase rotation. Using Eq. (14) and c p (t) = e −iεpt c p (0) we can write for its time evolution ∂ tCp (t) = 0| c p (0)∂ t ρ I (t) |0 , which, upon substituting Eq. (13) and after some algebra, yields with the time dependent rate coefficient Equation (15) implies a pure decay of the impurity coherence driven by its interaction with the surrounding BEC. Indeed, population redistribution to other impurity momenta, as studied in [35][36][37][38] does not affect the evolution ofC p (t), since any momentum change due to the scattering or generation of Bogoliubov excitations acts as a projective measurement of the internal impurity state and can therefore not generate coherence. The solution of Eq. (15) is readily obtained, and upon reintroducing the phase factor e iεpt gives with Here, we have used which follows from the Lippmann-Schwinger equation, relating U 0 to the zero-energy impurity-boson scattering matrix T , as discussed in Sec. II. Equation (17) explicitly shows how the phase factor corresponding to the polaron energy E p = p 2 /2m + n B T + Σ p naturally emerges from our formalism. Indeed, Eq. (18) coincides with the second order contribution to the ground state energy of the Bose polaron obtained in [25,28] in the limit of a zero range potential with g k = 1. Comparing Eq. (17) with the t → ∞ limit given by Eq. (9), we see that the integral t 0 dsΓ p (s) determines the dynamical formation of the polaron state. Upon evaluating the integral in Eq. (16), the expression for Γ p (t) can be written as where is a rate constant, and γ(α, pξ/ √ 2, t/t B ) is a time dependent function that only depends on the mass ratio α = m/m B and the dimensionless impurity momentum pξ/ √ 2 in units of the inverse of the BEC coher- Moreover, Eq. (21) contains the characteristic timescale t B = ξ/ √ 2c for the impurity dynamics, determined by the the speed of sound c = √ 4πn B a B /m B of the condensate. This timescale reflects how fast Bogoliubov modes can build up distortions on a length scale ξ, which corresponds to the typical size of the formed screening cloud surrounding the impurity.
In Fig. 2, we display the time dependent rate coefficient Γ p (t) for different impurity momenta and fixed mass ratio α = m/m B = 1. As can be seen, the real part of Γ p (t) vanishes asymptotically for t/t B 1 so that the phase factor of the coherence Γ p (t) approaches exp(−iE p · t) in the long time limit, in agreement with Eq. (9). We also see from Fig. 2 that the imaginary part of Γ p (t) vanishes for t/t B 1 for impurity momenta below the Landau critical momentum p c = mc of the BEC. This reflects the infinite polaron lifetime τ p → ∞, due to the absence of damping as it moves through the superfluid with p < p c . However, when the impurity momentum exceeds p c , Fig. 2 shows that the imaginary part of Γ p (t) approaches a finite value and yields a finite polaron lifetime τ p due to the continual emission of Cherenkov radiation.

V. SHORT-TIME DYNAMICS AND THE ROLE OF THE INTERACTION RANGE
We begin by discussing the impurity dynamics due to decoherence at short times t t B prior to emergence of the polaron. To model a finite range impurity-boson interaction, we consider a simple step function in momentum space, g k = Θ(κ − k) where κ is a high momentum cut-off. This choice provides a simplified model for the actual interaction between the atoms, whose typical range, r 0 , relates to the characteristic momentum κ ∼ r −1 0 . In Fig. 3, we plot the absolute value of the coherence for different values of the cut-off κ. As can be seen, the initial decoherence is characterised by a Gaussian damping ln |C p (t)/C p (0)| ∼ −(t/t i ) 2 . The characteristic time for this initial decoherence is determined by the cut-off as t i ∼ m r /( √ n B aκ 3/2 ). This simple initial behavior applies for short times t < ∼ m r /κ 2 . Beyond this time, the decoherence proceeds in a way that is largely independent Times and momenta are scaled by tB = ξ/ √ 2c and pc = mc, respectively, while Γp(t) is shown in units of the rate constant Γ given in Eq. (21). The mass ratio is α = m/mB = 1.
of κ. In fact, the characteristic extent of interactions between alkaline atoms is below 1nm and, thus, about two orders of magnitudes smaller than the typical coherence length ξ ∼ 0.1µm of atomic BECs [40]. It follows that the initial Gaussian decoherence determined by the finite range of the interaction is only present at very short times. Under relevant experimental conditions, we can therefore neglect this initial Gaussian dynamics and take the zero-range limit κ → ∞. In this limit, the rate Γ p (t) diverges at the initial time t → 0. As shown in Fig. 3, this leads to fast initial decoherence but yields a well behaved coherence dynamics close to the results for finite values of κξ 1. To investigate this further, we plot in Fig. 4(a) the subsequent dynamics for different impurity momenta, p, and different values of the scattering length a. The initial decoherence appears to proceed independently of the impurity momentum, which can be readily understood from Eq. (16). At early times t, it takes large momenta k for the phase to rotate significantly in the exponential factor in the integrand for Γ p (t). The dominant contributions to the integral therefore stems from large momenta that eventually exceed the impurity momentum and render the initial decoherence of the impurity independent of p. In this regime, which is dominated by coupling to free-particle excitations, the initial impurity dynamics is also independent of the interaction among the condensate atoms, and only depends on the strength of their coupling to the impurity as parametrised by the scattering length a.
By expanding the integral for Γ p (t) in Eq. (16) for short times and evaluating the resulting integrals analytically, one can show that The appearing timescale is given by which indeed only depends on a, the reduced mass m r and the density whereas it is independent of p and a B . Details of this derivation are given in App. F. In Fig. 4(b), we plot the decoherence |C p (t)/C p (0)| as a function of t/t 0 for the same momenta and scattering lengths as in Fig. 4(a). Indeed all data points collapse to a single curve given by Eq. (22) for times t < ∼ t B /2. In the limit of an ideal Bose gas, the non-exponential decoherence given by Eq. (22) persists for all times, such that a polaron can consequently not be defined at any stage of the impurity dynamics. Thus, the decoherence initially follows a universal non-exponential relaxation |C p (t)| = |C p (0)|e − √ t/t0 , while the polaron properties,  such as the energy shift Σ p and residue Z p , have yet to emerge as we will discuss in the next section.

VI. LONG-TIME DYNAMICS
At later times, t > t B /2, coupling to phonon modes starts to become significant and the coherence dynamics becomes dependent on the boson-boson interaction and the impurity momentum p. Asymptotically, the coherence approaches a finite steady state for p < p c , while it continues to decay above the Landau critical momentum p c = mc. This is illustrated in Fig. 5, showing how |C p (t)| relaxes to a constant value for p < p c but continues to decay exponentially for higher momenta. In this regime the dynamics is driven by phonon dressing leading to the eventual formation of the polaron, as we now demonstrate explicitly.
By evaluating the time dependent rate coefficient, Eq. (16), we can determine the asymptotic decay time τ p = 1/[2ImΓ p (∞)] (see App. C) the spectral width of the polaron in equilibrium [25]. In Fig. 6 we show the instantaneous damping rate ImΓ p (t). For p < p c the decay rate steadily decreases and vanishes at long times, reflecting the superfluidity of the condensate which leaves the eventually formed polaron state unaffected. For higher momenta, p > p c , the impurity moves faster than the speed of sound and therefore emits Cherenkov phonon radiation, causing decoherence and exponential decay on a timescale of τ p . We can separate this long time behavior in Eq. (17) such that C p (t) = e −iEpt−t/(2τp) e t 0 ds(iΓp(s)+1/(2τp)) .
Since the integral over the rate coefficient becomes purely imaginary at long times, we can use this expression to define the asymptotic quasiparticle residue Z p = e − ∞ 0 ds(ImΓp(s)−1/(2τp)) .
As discussed further in App. D, this indeed coincides with the equilibrium result obtained in Ref. [28] from a diagrammatic expansion for zero momentum and a 2 /(a B ξ) 1, and under more general conditions in Ref. [35].
Finally, we turn to the phase factor of the coherence. In Fig. 2 we saw that the real part of Γ p (t) vanishes asymptotically such that the phase factor of C p (t) approaches exp(−iE p t). For p = 0 and α = 1, the long-time phase rotation takes on a particularly simple form where Σ 0 = 32 √ 2n B a 2 /3mξ for α = 1 as obtained from Eq. (18) once more coincides with the expected equilibrium result, obtained for p = 0 and a/ξ 1 [28]. While a more general expression for arbitrary momenta and mass ratios is derived in App. E, this simple result shows that one can define an asymptotic energy of the immersed impurity, which approaches the polaron ground state energy at long times. Equation (26) shows explicitly that the timescale for this approach is on the order of t B for α = 1 and zero momentum. As shown in Figs. 7 and 5, however, the corresponding characteristic timescale, t f , for polaron formation can have a strong dependence of the impurity momentum, as we will discuss in the next section.

VII. POLARON FORMATION TIME
Below p c , we find that the coherence always overshoots its eventual steady state. One can thus identify an intermediate relaxation stage (see Fig. 1), at the end of which the coherence, |C p (t)|, goes through a minimum and the impurity starts its final approach to the polaron state. The location of this minimum can therefore be used to define the polaron formation time, t f , which marks the onset of the final stage of the impurity dynamics. Since ∂ t |C p (t)| = −ImΓ p (t)|C p (t)|, this is equivalent to finding the first zero of ImΓ p , as illustrated in Fig. 8a. We can capture the asymptotic coherence dynamics by expanding Eq. (16) in orders of 1/t. As outlined in detail in App. E, this gives the following asymptotic behavior where ln Z p = −a 2 /(a B ξ)A p , as shown in App. D. Both coefficients, A p and B p , depend only on the mass ratio, α, and the momentum ratio p/p c . Regardless of these parameters, however, one finds that B p > 0 is positive such that the coherence always approaches its final value from below. Since the coherence initially decreases, this implies that |C p (t)| indeed goes through a minimum for any p < p c . Since, the expansion is only valid for t t B B p /A p beyond t f , we determine the formation time numerically. Interestingly, however, we find that the coefficient B p diverges when the impurity momentum approaches p c , indicating a logarithmic relaxation dynamics with a diverging formation time.
Above p c , the formation time is also determined as the time when the rate first crosses its steady state value, which in this case corresponds to ImΓ p (t f ) = 1 2τp . The final coherence dynamics exhibits damped oscillations which are well described by the asymptotic damping rate as derived in App. E. The coefficient D p and the frequency ω p only depend on α and the scaled impurity momentum p/p c . As shown in Fig. 8b, ω p determines both the frequency as well as the amplitude damping of the oscillation, since scaling time by ω −1 p renders the asymptotic coherence dynamics nearly independent of p. It is given by the simple relation which corresponds to the energy cost ∆E p,k0 associated with phonon generation, whereby the corresponding phonon momentum k 0 lies at the stationary point that minimizes ∆E p,k . This thus corresponds to energetic off-shell processes with phonon momenta around k 0 where the phases in Γ p add up constructively (see Eq. (16)), and which naturally cease to occur at large times t → ∞ when off-shell scattering is suppressed by energy conservation.
In Fig. 9 we show our numerical results for t f for equal masses α = 1. In general we find that the formation decreases with α and eventually diverges as α → 0. This reflects the increasing effects of phonon recoil which tends to slow down the relaxation to a steady state. The found timescale for small momenta around p = 0 is consistent with the developed picture of polaron formation in terms of a phonon-induced decoherence process. It implies that the formation time corresponds to the time it takes to propagate the presence of the impurity. At zero momentum the size of the screening phonon cloud is on the order of the coherence length, ξ. Thus the expected formation time can be estimated by the time for a phonon to propagate this distance, t f ∼ ξ/c = √ 2t B , which is consistent with the results shown in Fig. 9.
Most strikingly, we find that the formation time diverges when the impurity momentum approaches p c from below as well as from above. At p = p c the coherence dynamics undergoes a critical slowdown and never reaches the polaron steady state as illustrated in p c the formation time can be described by the following critical behavior where the proportionality constant b depends on the mass ratio. For α = 1, the critical exponent is found to be η = 2 above and below p c . This behavior can be understood as follows. As discussed above, the relevant timescale for the asymptotic approach to the polaron state above p c is given by 1/ω p . From Eqs. (29) and (30) one can show that ω p t B = α(p/p c − 1) 2 /4 close to p c (see App. E). Hence, we have a universal critical exponent of η = 2 for p > p c . For p < p c , on the other hand, the exponent turns out to depend on the mass ratio α. Explicitly, we find that for a light impurity with α ≤ 1 the critical exponent remains to be η = 2, while it acquires a mass dependence η = 1 + (1 + 1/α)/2 for α ≥ 1.
Eq. (29) together with Eq. (30) shows that the oscillation and damping timescale above p c is related to forward scattering of phonons with respect to the moving impurity. In fact, the same mechanism causes the divergence of B p in Eq. (27), for p < p c . When the impurity moves precisely at the speed of sound p c /m, forward emitted phonons cannot dissipate away from the impurity as they propagate at the same velocity, rendering decoherence ineffective. Hence, forward scattering of phonons turns out to be the general bottleneck for the impurity relaxation and ultimately leads to the increasing polaron formation time when approaching the Landau critical momentum p c .

VIII. CONCLUSION
In summary, we have studied the non-equilibrium dynamics of a quantum impurity immersed into a Bose-Einstein condensate. By tracing the dynamics of the impurity coherence via a master equation approach we could identify different dynamical regimes from an initially universal non-exponential coherence decay and a subsequent phonon-driven relaxation to the final formation of the polaron steady state. Considering different impurity momenta, we have found that the dynamical polaron formation undergoes a critical slowdown when approaching the Landau critical velocity of the condensate.
The thereby established link between the critical equilibrium properties of the condensate and the dynamical critical behavior of the non-equilibrium impurity state raises a number of open questions. For example it suggests that a similar slowdown might occur in condensates at finite temperature [47] and cause very long polaron formation times close to the critical temperature for Bose-Einstein condensation. Clarifying this open question should indeed be important for future experiments to study the Bose polaron at finite temperatures and across the BEC phase transition.
For the parameters of a recent experiment [20], in which Bose polarons have been created and studied by driving the (F = 1, M F = −1) → (F = 1, m F = 0) hyperfine transition of a 39 K BEC, we obtain for our relevant timescales t B ≈ 0.2ms, t 0 ≈ 14ms and t f ≈ 1ms for a = 20a B and p = 0. It turns out that the shortest timescale, t B = 0.2ms, is just about the duration of the microwave excitation pulse used in [20], such that the predictions of the present work should be observable with present technology. This would enable direct experimental access to the non-equilibrium properties of the Bose polaron.
The results of this work should also be relevant to other systems and similar problems, such as the decoherence of molecular rotational states in superfluid helium droplets [48], whose equilibrium properties have been linked to the quantum impurity physics in recent theoretical work [49][50][51]. Following these ideas, the theoretical framework described in the present work appears also applicable for the description of the short rotational dynamics of such molecules [48]. While the corresponding decoherence timescales are expected to be much shorter than for cold atom systems, they can be well resolved using fs-laser spectroscopy [48].
The formulated theory can also account for a finite range of the impurity interaction, which we used to demonstrate converged and well behaved dynamics in the limit of zero-range interactions in order to avoid ambiguities stemming from otherwise ad hoc momentum cutoffs.
While the effects of a finite interaction range were shown to be small in typical alkaline atom BECs, they are most likely relevant for systems with larger interaction ranges, such as recently studied Rydberg state impurities [52], where it is possible to observe the initial impurity dynamics on timescales t < t i well before the polaron formation time t f > t i .
The process of impurity decoherence is intrinsic to the formation of the polaron. Understanding its interplay with coherent driving of internal impurity states is important for the quality of light matter interfaces involving polaron physics. Examples include the fidelity of storage [53] and propagation [54] or quantum light coupled to atomic Rydberg excitations in the inevitable presence of interaction with their surrounding ground state atoms, Bose condensed atoms coupled to an optical cavity [55] for which the generation of dark-state polaritons has been suggested as a means to tune the effective polaron mass.
The derivation of rigorous analytical results and scaling relations has been made possible by focussing on the limit of weak impurity interactions. Yet, we expect several characteristics of the discussed impurity dynamics to survive under more general conditions. For example, the discovered critical slowdown of the polaron formation due to the discussed inhibition of phonon emission as well as the predicted universal short time behavior |C p (t)/C p (0)| ≈ 1 − t/t 0 should persist for stronger interactions and beyond the Fröhlich model for the Bose polaron.

ACKNOWLEDGMENTS
We thank Jan Arlt, Christopher Pethick, Matteo Zaccanti, and Richard Schmidt for fruitful discussions. This work has been supported by the DNRF through a Niels Bohr Professorship.
Appendix A: The effective von Neumann equation In this section we derive the effective von Neumann Eq. (13). We write the interaction Hamiltonian Eq. (1) in the interaction picture and the equation of motion for the impurity density operator after tracing over the bath and making the Born-Markov approximation (12) Here c p (t) = e −εpt c p (0) and β k (t) = e −iE k t β k (0) in the interaction picture. We then let Examining the commutator of the second line this yields terms like In the second equality we write U k = U 0 g k and replace U 0 by the impurity-boson scattering matrix at zero energy T = 2πa/m r , with a the impurity-boson scattering length. This is consistent to second order in a. We hereby derived Eq. (13). We find a closed form expression for Γ 0 . Using Φp(k) = −t B (ε p − ε p−k − E k ) =k k2 + 1 +k 2 − 2kp cos(θ), with k = kξ/ √ 2 and equal masses m = m B we write Eq. (16) of the main text on unitless form Here cos θ ∈ (−1, 1) andk ∈ (0, ∞). For zero impurity momentum, p = 0, Φ has the simple inverse:k = Φ √ 1+2Φ . Using this we get The problem has thus been reduced to calculating the above Fourier transform. A primitive to the integrand is found in Mathematica. With a bit of rewriting we then arrive at the result with erf the error function. This has also been checked numerically. It is also possible to achieve an analytical expression for the integral of Γ 0 (t). Integrating (B2) we get where the last expression is accurate asymptotically to ordert −2 = (t B /t) 2 .
Appendix C: Damping in the long time limit We calculate the polaron life-time, 1/2τ p = lim t→∞ ImΓ p (t). We can write the damping as For t → ∞ the temporal integral yields ∞ 0 ds e +ixs = Pr(−i/x)+πδ(x), where Pr stands for the principal value and x = ε p − ε p−k − E k . Taking the real part of this expression gives the damping rate at long times with τ p the polaron life-time. For momenta below the Landau critical momentum, p < p c = mc, the integral is 0. For p > p c the integral can be evaluated analytically to yield for p > p c , withp = pξ/ √ 2 and withk max =p − 1 4p in the equal mass case α = 1. We hereby have a closed form expression for the long time damping for all mass ratios.

Appendix D: Momentum dependent residue
In this section we calculate the quasiparticle residue as given in Eq. (25) Using that ∂Φp(k)/∂ cos(θ) = −2pk/α we can perform the angle integral in Eq. (B1) and get x 0 du sin(u)/u the sine integral, and Φ ± p (k) = k k2 + 1 + 1 α (k 2 ± 2kp). Then integrating the sine integrals we get t 0 ds ImΓ p (s) = √ We recognise the first two terms in the integral as ImΓ p (t)t which goes to t/2τ p for t t B . We therefore a 2 a B ξ Pr (D3) In the second equality we use that (cos(Φt) − 1)/Φ = − t 0 ds sin(Φt) = Im t 0 ds exp(−iΦt) → Pr(−1/Φ), for t → ∞, with Pr the principal value. The resulting integral above can readily be solved in the equal mass case α = 1. Here, using x = 1 +k 2 +k we get Re arctanh 1 2p .
The integral is performed using the transformation tanh(θ) = x/2p.
thus defining A p (α) for α = 1. At p = 0 and p = p c a limiting process must be performed. We have thus obtained an analytical result for the residue as a function of momentum in the equal mass case. Below we give the limiting values of A p (α) for general α calculated from Eq. (D3) where f (α) = α+1 α−1 arctan α−1 α+1 and √ −1 = i. Here A 0 (α = 1) = 2 √ 2/(3π) p. It turns out that there are two regimes in which this behaves quite differently: p < p c and p > p c , where p c = mc is the Landau critical momentum of the condensate.
p < p c : We calculate t 0 dsΓ p (s) to leading order in 1/t. We will see that where A p (α) is defined in Eq. (D4). This analysis is detailed below.
For reference we write Eq.
where Φp(k) =k k2 + 1 + (k 2 − 2kp cos θ)/α, and k = kξ/ √ 2. Here cos θ ∈ (−1, 1) andk ∈ (0, ∞). The second equality defines wp(k) =k 3 /( k2 + 1Φp(k)). We now wish to expand the integral asymptotically for t = t/t B 1. Specifically we investigate thek-integral and write Λ now takes on the role of an initial momentum cut-off. The initial expansion, Eq. (F1), only works for finite Λ, but the above expression is valid for t t B for any large value of Λ. Importantly, we recognize the integral as the second order polaron energy shift, Σ p , which is independent of Λ as long as it is sufficiently high. As for the integral of Γ p we get to linear order in t t 0 ds Γ p (s) (i − 1) where we define 1/t 0 = 16πn 2 B a 4 /m r . The coherence at short times is then C p (t) = C p (0)e −i(p 2 /2m+nBT +Σp)t e i t 0 dsΓp(s) which explicitly shows that the coherence does not depend on the polaron energy at short times, only the overall mean-field shift n B T is present. Also, the decay of |C p (t)| is independent of momentum. As noted in the main text when we approach the ideal Bose gas limit, a B → 0, the coherence collapses to this short-time behavior for all times.