Polariton dynamics in strongly interacting quantum many-body systems

We develop a non-perturbative theory describing light propagating in an atomic Bose-Einstein condensate (BEC) in the presence of strong interactions. We show that the resulting correlations in the system can have profound effects onto its optical properties. For weak atom-light coupling, there is a well-defined quasiparticle, the polaron-polariton, supporting the formation of dark-state polaritons whose spectral features differ significantly from the non-interacting case. Its damping depends non-monotonically on the light-matter coupling strength, initially increasing and then decreasing. For strong interactions, there is a cross-over regime where the light is carried by a lossy quasiparticle giving rise to a substantial optical depth. Eventually, interaction effects are suppressed and EIT mediated by an ideal gas dark state polariton is recovered for strong light coupling. We furthermore show that the presence of a continuum of many-body states leads to an increased light transmission away from the EIT window.

The ability to prepare, control, and probe cold matter systems via external light fields is at the heart of modern developments in atomic physics, quantum optics, many-body physics, and quantum technologies. Here, EIT presents a particularly powerful approach to achieve strong light-matter coupling at greatly reduced losses. This effect opens up numerous applications, from cooling [1] and trapping [2] techniques, to the realization of quantum memories [3] and ultraslow propagation of light in the form of so-called dark-state polaritons [4]. EIT has been observed in a wide variety of media including hot atomic vapors [5], cold atomic gases [6][7][8], Rydberg gases [9], and solids [10]. When light propagates through an interacting system its polaritonic spin wave component will typically interact and develop correlations with its surrounding. These correlations can give rise to the formation of quasiparticles, such as polarons. While the formation of polarons have been explored in great detail in cold atomic gases [11][12][13][14][15][16][17], polaron-polaritons in interacting optical media are expected to yield new insights into light-matter interactions [18][19][20].
Here, we address this problem and develop a diagrammatic theory for light propagation under EIT conditions in the presence of strong many-body correlations between the photons and a surrounding Bose-Einstein condensate (BEC). Despite the fact that the underlying dark-state is almost entirely composed of the impurity excitation [4], that forms a polaron in the absence of light, we find that its propagation dynamics cannot in general be described in terms of an effective polaron-polariton theory. Yet, we identify a parameter regime in which light propagation can be described in terms of polaron-polaritons, where pronounced absorption minima persists and feature large shifts that are indicative of the underlying polaron. In fact, the emerging combined quasiparticle is found to have a narrowed EIT linewidth and reduced group velocity compared to the bare slow-light polariton, whereas the many-body continuum in the polaron spectrum causes enhanced transmission away from the EIT resonance. (c) Λ-scheme, here g denotes the single-photon coupling, Ω the Rabi frequency of the classical control field, Γ ee the decay rate of the excited state, ∆ is the one-photon detuning, and δ is the two-photon detuning.
System.-We consider atoms of mass m with three atomic states b⟩, e⟩, and c⟩. A quantised light-field couples the b⟩ and e⟩ states with a single-photon coupling g, whereas a classical control field couples the e⟩ and c⟩ states with Rabi frequency Ω, forming a so-called Λ-scheme as illustrated in Fig. 1c. Within the rotating wave approximation, the Hamiltonian can be written as where the operators b † p , c † p , and e † p create an atom with momentum p and kinetic energy p = p 2 2m in the atomic state b⟩, c⟩, and e⟩ respectively. The atomic states are such that  p includes the Lamb shift due to the coupling g to the b⟩⊗ γ⟩ continuum. The operator γ † p creates a photon with momentum p and kinetic energy cp with c the speed of light in a vacuum. The second line of Eq. (1) describes the coupling between the atoms and the probe photons as well as the classical control field. Note that the classical field with wave vector k cl (ω cl = c k cl ) decreases the momentum of the c⟩ atoms by k cl compared to the b⟩ and e⟩ atoms. The interaction V B (q) describes the interaction between two atoms in state b⟩, and V (q) denotes the interaction between a b⟩-and a c⟩-state atom. Both interactions are short range and accurately characterised by the scattering lengths a B and a respectively. We use units where the system volume and ̵ h are both one. The b⟩-atoms form a weakly interacting BEC with density n = k 3 n 6π 2 and 0 < k n a B ≪ 1. Its excitation spectrum is given by Bogoliubov theory, i.e. E p = p ( p + 2µ B ) with µ B = 4πa B n m the chemical potential of the BEC. The densities of the e⟩-and c⟩-atoms created via the Λ-scheme are small compared to n, such that we can regard the c-atoms as impurities immersed in the BEC, which acts as a particle reservoir. Typically, k n a is tuneable with k n a ≳ 1 characterising the regime of strong b⟩-c⟩ interaction.
Diagrammatic theory.-In order to develop a nonperturbative theory that can simultaneously account for strong-light matter coupling as well as atomic interactions, we introduce the imaginary time Green's function Here T τ denotes time ordering and Ψ p = [c p−k cl , γ p , e p ] T . Due to the coupling between light and atoms, the Green's function G is a 3 × 3 matrix obeying a Dyson equation, which in frequency space reads G(p, z) −1 = G (0) (p, z) −1 − FIG. 2: (color online) The diagrams describing EIT and polaron formation. A dashed line is a b⟩-atom emitted from or absorbed into the BEC, a red line is the c⟩-propagator, a green line is the e⟩-propagator, a wavy blue line is the photon propagator, and a black line is a b⟩-propagator. The classical field Ω is indicated by a * , a • is the dipole matrix element g between the photons and the atoms, and a wavy black line is the b⟩-c⟩ interaction. The double red line is the impurity propagator including the light coupling given by Eq. (4).
of the c⟩-atoms, the photons, and the e⟩-atoms. The many-body problem has now been reduced to calculating the 3 × 3 self-energy matrix Σ. We focus on an incident photon propagating with momentum k. We now present the diagrammatic scheme illustrated in Fig. 2, which shows the key set of diagrams. The offdiagonal self-energies are Here, Σ ce describes the excitation of an atom from the c⟩-to the e⟩-state by the classical field, and Σ γe gives the absorption of a photon by a b⟩-atom in the condensate exciting it into the e⟩-state, where we ignore quantum fluctuations of the BEC. The self-energies Σ γγ = Σ cγ = Σ γc = 0, while Σ ee gives the decay of the e⟩-atom due to the coupling to the γ⟩ ⊗ b⟩ continuum described within Weisskopf-Wigner theory [21][22][23]. Finally, Σ cc = nT describes the scattering of a b⟩-atom out of the condensate by a c⟩-atom, which is the dominant process leading to the formation of the Bose polaron [24][25][26]. Since the light coupling is crucial for EIT physics, one needs to include the self-energies given by Eq. (4) in the scattering matrix T , see Supplemental Material [27]. This goes beyond the usual ladder approximation based on bare c-propagators [24] or the equivalent variational Chevy ansatz [25]. As we shall see below, this simultaneous treatment of strong atom-atom and strong light-atom coupling has important consequences on the resulting photon propagation. Light propagation in the BEC can be described by the retarded photon Green's function G γγ (k, ω), which is obtained from G γγ (k, z) by the analytical continuation z → ω + i0 + , where ω is a real frequency and 0 + is an infinitesimal positive number. Solving the Dyson equation with the self-energies given in Fig. 2 yields the photon is the optical susceptibility. The decay of the excited state is denoted by Γ ee = −ImΣ ee , and we have defined Expressions for the excited state G ee , impurity propagators G cc , and the scattering matrix can be obtained analogously and are given in [27].
Polaron-polaritons.-To gain some intuition and to simplify the discussion, we first ignore the effects of the light coupling on the atom-atom scattering, which amounts to assuming bare c-propagators inside the scattering matrix T .
This approximation is valid for Ω 2 Γ ee ≪ E n as we shall see below. Then, G P in Eq. (6) is identical to the known Bose polaron propagator in the ladder approximation [24], and it has a real undamped pole at the polaron energy c + ω cl + E (P ) k−k cl for attractive interactions k n a < 0. It follows from Eq. (5) that the on-shell susceptibility χ(k, ck) vanishes at this pole, i.e. when G −1 P (k−k cl , ck) = 0. Physically, this means that the photon can propagate undamped under EIT conditions when its energy ck matches that of a polaron with momentum k − k cl , i.e. when ck = E (P ) k−k cl + c + ω cl . Defining the two-photon detuning as δ = c + ω cl − ck (see Fig. 1a), one thus finds that the EIT resonance is shifted from its non-interacting value δ = − k−k cl , indicating the underlying mechanism is the formation of an undamped quasiparticle, the dark state polaron-polariton.
Close to the polaron energy, we have where Z P is the polaron residue. Using this in Eq. (5), we obtain for the photon propagator around the EIT condition δ = −E k−k cl , and q, which is taken to be parallel to k. We have neglected terms involving ∇ k E corresponding to the residue Z of the EIT pole in the photon propagator, the group velocity v g of light, the width σ of the EIT window, and the Rabi frequency Ω P renormalised by many-body correlations. From Eq. (8), we see in addition to moving the condition for EIT away from δ = 0, the formation of the polaron decreases both the group velocity of light in the BEC and the width of the EIT window through its residue Z P < 1.
We show in Fig. 1 the light transmission through a BEC of length L as a function of the b⟩-c⟩ scattering length a and the detuning δ. The transmission is described by the optical depth OD = Γ γ L v g = ImχL where Γ γ = ZcImχ is the damping rate of the photons. The optical depth OD 0 = ng 2 L Γ ee c in the absence of the classical control field serves as a reference. For concreteness, we take the 4 2 S 1 2 to 4 2 P 1 2 transition in 39 K, as employed in recent EIT and polaron experiments [15,28]. For this atomic transition, Γ ee = π × 2.978MHz corresponding to a wavelength λ = 2π k ≃ 700.1nm. Taking a BEC density of n = 2 × 10 14 cm −3 , this gives the typical many-body energy E n = k 2 n 2m ≃ 420kHz, and using g 2 = 3π cΓ ee k 2 from Weisskopf-Wigner theory yields √ ng ≃ 6.1×10 5 E n . In order to resolve many-body physics in the EIT spectrum, we choose a classical light coupling Ω so that the width σ ≃ Ω 2 Γ ee = 518kHz is comparable to E n . Finally, the impurity momentum k − k cl , the temperature and the one-photon detuning ∆ = ε (e) k − ck are all zero. We also plot as dashed lines in Fig. 1a the attractive and repulsive polaron energies in the absence of light, determined by the pole of G P in Eq. (6) with Ω = 0. Figure 1a clearly demonstrates that the optical depth essentially vanishes when the photon energy is resonant with the polaron energy, −δ = E (P ) k−k cl , for weak attractive coupling 1 k n a ≲ −1. This corresponds to the formation of a dark state polaron-polariton leading to EIT with the shifted resonance condition as described above.
We however also see that for stronger interactions, the optical depth increases at the minimum, which moreover is shifted away from the polaron energy. To illustrate this further, we show in Fig. 1b vertical cuts for several values of the interaction strength. The vertical lines correspond to the polaron energy in absence of any light. EIT with very small damping is achieved when the photon energy matches that of the attractive polaron for 1 k n a = −5. As the attraction increases, the EIT minimum is displaced away from the polaron energy and the optical depth increases becoming substantial at unitarity 1 k n a = 0. This is due to the interplay between the scattering and the light coupling leading to decay of the polaron, even when it is the ground state for a < 0 in the absence of light. Note also that the optical depth at the EIT resonance is in general larger for k n a > 0 compared to the attractive side, reflecting that the repulsive polaron is not the ground state so that it can decay even in the absence of light. In addition, we see an interesting double dip structure in the optical depth for strong interactions 0 ≲ 1 k n a ≲ 1. This is caused by a continuum of many-body states involving Bogoliubov excitations of the BEC, which increases the transparency of the BEC for detunings away from the polaron energies. Figure 1a clearly shows that the transmission spectrum is very different from the non-interacting case where it would simply exhibit a single horizontal minimum of zero damping for δ = 0.
Light induced damping.-Let us analyse the light induced energy shift and additional damping of the polaron giving rise to the strong coupling effects shown in Fig. 1.
The key point is that while the coupling Ω of the c⟩-state to the lossy e⟩-state by the classical field is suppressed for the EIT resonant momentum k − k cl , it can be significant for other momenta where the photon is off resonant. The remaining light coupling to the e⟩-state is controlled by the ratio Ω Γ ee and leads to damping of the impurity.
When Ω Γ ee ≪ 1, the damping of impurities with offresonant momenta is proportional to Ω 2 Γ ee [27]. This is of course irrelevant for EIT physics in the absence of interactions where the impurity momentum is fixed to k − k cl by the incoming light. In the presence of interactions, atom-atom scattering can however change the impurity momentum to values where the mixing with the lossy e⟩-state is significant. This scattering is described by the self-energy Σ cc = nT , and it gives rise to a non-zero damping Γ P ∝ (1 − Z P )Ω 2 Γ ee of the polaritonic impurity with resonant momentum k − k cl = 0 for Ω Γ ee ≪ 1 [27]. This in turn results in a damping of the polaron-polariton and a corresponding non-zero optical depth at −δ = E (P ) k−k cl given respectively by for Ω Γ ee ≪ 1, Ω 2 Γ ee ≪ E n and √ ng ≫ Ω P [27]. Equation (9) provides a close link between the physics of polarons and polaritons, and shows how the damping of the polaron-polariton initially increases with increasing light coupling Ω. The coupling to resonant photons can furthermore be shown to result in an additional source of decay of the polaron-polariton due to Cherenkov radiation [27].
All these effects caused by the simultaneous presence of light and interactions are described by our theoretical scheme. Most crucially it includes the light coupling inside the scattering matrix as explained above. Indeed, it is the underlying competition between atomatom and atom-light interactions that causes the shift of the EIT minimum away from the polaron energy and its significant non-zero optical depth visible in Fig. 1b for strong interactions. Here, the light carrying quasiparticle is a complex mixture of photons and matter with an energy substantially different from the polaron-polariton and with significant damping.
For large Ω, the impurity states with momenta different from k − k cl eventually become so strongly damped that scattering into them is suppressed. As a result, both the energy shift and the damping of the impurity with resonant momentum decreases, and the EIT spectrum approaches that of a ideal gas. In other words, interaction effects are suppressed for a strong control field giving rise to a non-monotonic dependence of the damping and the eventual re-emergence of the non-interacting polariton for large Ω. We note this surprising effect can only be described using a non-perturbative theory taking into account the repeated scattering of impurities on the BEC [27].
Regimes of light propagation.-Our findings for the subtle interplay between interactions and light coupling are summarized in Fig. 3a. It shows the damping rate of the impurity for the detuning δ giving the minimal optical depth as a function of the interaction strength 1 k n a and the classical light coupling Ω Γ ee , keeping Γ ee fixed. All other parameters are as in Fig. 1. In agreement with the discussion above, we see that the damping of the impurity depends non-monotonically on Ω Γ ee for fixed coupling strength 1 k n a. For Ω Γ ee ≪ 1, the damping is small and light propagates in the form of a well-defined polaron-polariton giving rise to EIT with a transparency window that is shifted by the polaron energy and shows a small but finite residual absorption. The damping in- creases with increasing Ω Γ ee and it becomes substantial for strong coupling 1 k n a ≫ 1 and intermediate classical atom-light coupling 0.3 ≲ Ω Γ ee ≲ 1. Finally, both the decay and the energy shift of the impurity start to decrease for even stronger light coupling and the ideal gas EIT spectrum governed by the noninteracting polariton emerges.
To illustrate this subtle non-monotonic behaviour in more detail, we plot in Fig. 3b the polaron energy E P = E (P ) 0 and its decay rate Γ P at unitarity 1 k n a = 0 as a function of Ω Γ ee in units of the polaron energy E P (Ω = 0) in the absence of light. The damping initially increases as Ω 2 in agreement with the analysis above, and it is much smaller than the energy shift for Ω Γ ee ≪ 1 so that the polaron-polariton is well-defined. For intermediate values 0.3 ≲ Ω Γ ee ≲ 1, the damping is substantial and the energy significantly shifted away from the polaron energy in the absence of light. In this region, light propagation is carried by a damped quasiparticle consisting of a complicated mixture of light and atoms. Finally, the non-monotonic behaviour of the damping together with the steady decrease in the energy shift eventually makes both small for large Ω Γ ee , so that the non-interacting dark state polariton emerges.
Concluding remarks.-We developed a nonperturbative theory for light propagating in a BEC in the presence of strong interactions, and demonstrated that the interplay of many-body correlations with light coupling leads to several interesting effects. Our theory can easily be generalised to include other many-body effects such as quantum fluctuations of the BEC and interactions between the excited state and the BEC. In addition to opening up new approaches to optically probing and controlling nonequilibrium polaron dynamics, the developed framework is applicable to other physical systems [18,19,29] and will enable future explorations beyond the present scenario, such as strong few-photon nonlinearities, generated by polaron-polaron interactions [26].
We thank Jan Arlt and Luis Ardila for helpful discus- The T -matrix for the b⟩ − c⟩ scattering is given by where is the regularised propagator for a pair of b⟩-and c⟩-atoms in the presence of a BEC [24,30]. Here, the c⟩-state propagator G cc is given by which includes the coupling to the photon γ⟩ and excited state e⟩. The Dyson's equation yields for the excited state where Σ ee (p, ω) accounts for the Lamb shift and the decay of the excited state, while G P is given in the main text.

Light induced damping
Here, we provide more details concerning the light induced damping of the polaron discussed in the main text. This damping enters via the impurity states with momenta differing from k − k cl inside the scattering matrix in Eq. S1. Let the momentum of the impurity inside the scattering matrix be p − k cl and define q = p − k. When cq ≪ ng 2 Γ ee , it follows from Eq. (S3) that where we have used v g ≃ cΩ 2 gn 2 . Equation (S4) shows that the impurities with momenta close to the resonant momentum k − k cl are only weakly coupled to the excited state and thus have a long lifetime. The linear dispersion leads to an additional source of decay, Cherenkov radiation. For cq ≫ ng 2 Γ ee on the other hand, we get from Eq. (S3) That is, for momenta far away from the resonant momentum the impurity couples to the excited state, which results in decay.
To estimate how this decay of impurities with momenta different from k−k cl gives rise to a decay of the polaron with resonant momentum k − k cl via the interaction, we first consider the case Ω Γ ee ≪ 1 and Ω 2 Γ ee ≪ E n . Estimating the propagators inside the scattering matrix to be given by Eq. (S5) then yields In the opposite regime where Ω 2 Γ ee ≫ E P , the pair-propagator in Eq. (S1) can be approximated by Π(p, ω) ∝ −im 3 2 ω + iΩ 2 Γ ee . For Ω 2 Γ ee larger that the typical atomic energies, this suppresses the boson-impurity scattering matrix in in Eq. (S1) and thereby the impurity self-energy. Thus, one recovers the non-interacting dark state polariton for large Ω Γ ee .
To illustrate the imprints of the light on the atomic scattering, we neglect those in Fig. 4 (left), and compare to the physical case discussed in the main text. For illustration purposes, we show the latter in Fig. 4 (right), which fully includes the light-matter coupling. In Fig. 4 (left) the optical depth at resonance δ = −E P is strictly zero, illustrating that the ground-state polaron in absence of any light-matter coupling is undamped. In agreement with the theory, the width of the EIT reduces, as a consequence of the normalised Rabi frequency Ω 2 P = Z P Ω 2 that decreases with the residue of the polaron Z P . The idealised undamped polaron-polariton corresponds to ck n ≫ ng 2 Γ ee and Ω 2 Γ ee ≪ E n where the scattered c⟩-states are effectively decoupled from the resonant photons and the classical control field. In this limit, the atomic interactions can be described by the scattering matrix in absence of any light-coupling [24].

Damping of the polaron-polariton
The damping of the polaron in turn gives rise to a damping of the polaron-polariton given by is the modified residue of the EIT pole due to the light coupling. For Ω P 2 ≪ Γ P and taking ng 2 ≫ Ω P 2 , the decay of the photon is Γ γ = Γ P 1 + Γ P Γ ee Ω P 2 . The optical depth OD ∝ OD 0 (1 − Z P ) can be obtained by using Eq. (S6) in Eq. (S8).