Non-linear Relaxation of Interacting Bosons
Coherently Driven on a Narrow Optical Transition
We study the dynamics of a two-component Bose-Einstein condensate (BEC) of Yb atoms coherently driven on a narrow optical transition. The excitation transfers the BEC to a superposition of states with different internal and momentum quantum numbers. We observe a crossover with decreasing driving strength between a regime of damped oscillations, where coherent driving prevails, and a decay regime, where relaxation takes over. We investigate the role of several relaxation mechanisms to explain the experimental results: inelastic losses involving two excited atoms, leading to a non-exponential decay of populations; Doppler broadening due to the finite momentum width of the BEC, leading to a damping of the oscillations; and inhomogeneous elastic interactions leading to dephasing. We compare our observations to a two-component Gross-Pitaevskii (GP) model that fully includes these effects. For small or moderate densities, the damping of the oscillations is mostly due to Doppler broadening. In this regime, we find excellent agreement between the model and the experimental results. For higher densities, the role of interactions increases and so does the damping rate of the oscillations. The GP model underestimates these effects, possibly a hint for many-body effects not captured by the mean-field description.
In the recent years, ultranarrow optical “clock” transitions interrogated by lasers with sub-Hertz frequency stability have enabled dramatic progress in time-frequency metrology Ludlow et al. (2015). The very small radiative linewidth (low spontaneous emission rate) characterizing such transitions opens many unprecedented opportunities, e.g. for quantum information processing Stock et al. (2008); Gorshkov et al. (2010); Daley (2011); Shibata et al. (2014), to reach new regimes in quantum optics Bohnet et al. (2012); Norcia et al. (2016), or to simulate complex many-body systems such as high-spin magnetism or impurity problems Gorshkov et al. (2009); Martin et al. (2013); Riegger et al. (2018). Moreover, the recoil effect – the increase of the atomic momentum upon absorbing a laser photon – couples the motional state of the atoms to their internal state. This feature distinguishes single-photon transitions in the optical domain from hyperfine transitions in the radio-frequency or microwave domain, where the recoil is negligible. This enables in principle a fully coherent manipulation of the internal and external atomic state, with applications in atom interferometry Hu et al. (2017), or in the realization of artificial gauge potentials Gerbier and Dalibard (2010); Livi et al. (2016); Kolkowitz et al. (2017).
In many of these applications, interatomic interactions play an essential role. In atomic clocks, interactions limit the clock accuracy and their role has been studied extensively Ludlow et al. (2015). Even for fermions, where one would a priori expect vanishing clock shifts at low temperatures, interactions lead to tiny clock shifts because of inhomogeneous excitation Campbell et al. (2009). While atomic clocks usually operate far from quantum degeneracy, a host of new phenomena appears in quantum degenerate gases due to the interplay between quantum statistics, the quantized motion of atoms and intra- and inter-state interactions. Optical spectroscopy has been instrumental to reveal Bose-Einstein condensation of spin-polarized atomic hydrogen Fried et al. (1998); Killian et al. (1998); Killian (2000). These experiments were performed in a weak coupling, irreversible regime suitable for spectroscopy. Still, the experimental results have not been fully understood Fried et al. (1998); Gardiner and Bradley (2001); Oktel et al. (2002); Landhuis et al. (2003). More recently, one-photon spectroscopy on ultra-narrow optical transitions with spontaneous linewidth Hz has been reported and used to probe interaction shifts in an Yb Bose-Einstein condensate (BEC) Yamaguchi et al. (2010); Notermans et al. (2016) or a Sr degenerate Fermi gas Campbell et al. (2017); Marti et al. (2018), to measure scattering properties of bosonic Yb atoms Bouganne et al. (2017); Franchi et al. (2017), to study the superfluid-Mott insulator transition in an optical lattice Kato et al. (2012), or to reveal the change in the density of states in spin-orbit coupled Fermi gases Livi et al. (2016); Kolkowitz et al. (2017).
In this article, we report on a study of the dynamics of a BEC of Yb atoms coherently driven on such a narrow transition. The excitation transfers the BEC coherently in a superposition of states with different internal and momentum quantum numbers. The coherent excitation competes against a number of relaxation processes, including linear dephasing due to the finite initial momentum width and non-linear interactions, in particular inelastic processes involving two excited atoms. We observe a crossover with decreasing driving strength from a regime of damped oscillations, where coherent driving prevails, to a decay regime, where relaxation takes over. Throughout the crossover (except for very small driving strength), the populations relax in time with a non-exponential law. We compare our observations to a two-component Gross-Pitaevskii (GP) model that fully includes elastic and inelastic interactions and atomic motion. We find excellent agreement between the model and the experiments for moderate values of the interactions. For larger interactions, we find that the oscillations are damped more strongly in the experiment than in the model. This could point to additional effects beyond the GP description at play in the experiments.
We produce nearly pure BECs of Yb atoms in an optical crossed dipole trap (CDT) Dareau et al. (2015). The CDT operates at the so-called magic wavelength nm, where the light shifts of the electronic ground state and of the metastable excited state (denoted respectively by and in the following) are almost equal. The trapping potential of the CDT is then almost independent of the internal state. A laser near-resonant with the - transition induces a coupling between the two internal states (see Dareau et al. (2015) for more details on the optical setup and on the frequency stabilization). After preparing a BEC in the state, we illuminate the sample with a pulse of duration , with a coupling strength and detuning from the bare atomic resonance frequency , with the laser frequency. The coupling laser propagates in the horizontal plane, with a wavevector making an angle with the weak axis (-axis) of the trap (see Fig. 1a). We switch off the CDT immediately after the pulse, let the cloud expand for a time of flight of ms and record an absorption image of the atoms in using the dipole-allowed transition. In the following, we focus on the normalized population in , noted , that is the atom number deduced from absorption images normalized to the initial one.
Fig. 1b-f show the time evolution of after the coupling laser is turned on, for various Rabi frequencies. We observe damped, Rabi-like oscillations with a contrast that decreases when increasing the Rabi frequency. For all data shown in Fig. 1, the initial condensate contains typically atoms for a chemical potential kHz. The laser detuning is fixed to the value where we observe maximum transfer after a given pulse time (see Supplemental Material for more details). The observation of oscillations shows that the condensate is coherently transfered in a quantum superposition of and . For a uniform condensate, the transfer would couple two single quantum states and with momenta and , respectively. For our sample of finite size, the two states correspond to two wavepackets centered around the same momenta and with a width , where is a typical condensate size.
Our observations are reminiscent of the behavior of two-state quantum systems, both coherently driven and incoherently coupled to a bath, such as the paradigmatic two-level atom of quantum optics or a driven qubit undergoing relaxation. In these examples, under the assumption of short memory of the bath, one expects for weak driving an exponential decay analogous to the Wigner-Weisskopf (W-W) desintegration of a discrete level into a continuum of states Cohen-Tannoudji et al. (1997). The transition from W-W desintegration to underdamped oscillations with increasing driving strength can be estimated by comparing the spectral width of the bath (the inverse of its memory time) to the coupling strength : Underdamped oscillations take place in the strong driving regime and W-W desintegration in the weak coupling regime , with a continuous change from one regime to the other. The same conclusions hold for an ensemble of independent two-level systems, where inhomogeneities in the coupling strength or detuning also lead to dephasing between the different members of the ensemble. This induces an additional decay of the - coherence when considering ensemble-averaged quantities, translating to a reduced contrast of the oscillations.
In our experiment, the detuning , with the atomic mass, depends on the atomic momentum due to the Doppler effect. The finite momentum width of the BEC then translates into a Doppler broadening of the resonance by Hz, with the recoil velocity and the size of the condensate in the most confined direction (see Supplemental Material for a more detailed discussion). The Doppler width plays the role of the spectral width, and oscillations in Fig. 1b-f are indeed observed when . The Doppler effect couples together the internal and external dynamics of the atoms. If we describe the atoms by an internal density matrix , with external degrees of freedom integrated out, Doppler broadening leads to a decay of the off-diagonal elements on a time . For weak driving strength, assuming the damping can be accounted for by and performing adiabatic elimination of the off-diagonal elements Cohen-Tannoudji et al. (1997), one finds that the slowly-evolving population decays exponentially at a rate .
This exponential behavior is observed for the weakest coupling used in our experiment (Fig. 1f), but not for larger driving strengths where we find instead a much slower algebraic decay at long times (insets of Fig. 1b-e). Moreover, the normalized populations in Fig. 1 do not settle to the value that would be expected from ensemble averaging of different momentum classes. Hence, the simple picture of the driven two-component BEC as a collection of Doppler-broadened, independent two-level systems is not sufficient to fully explain our experimental observations.
The algebraic decay can be ascribed to inelastic two-body losses due to principal quantum number changing collisions between two excited atoms (the rate for inelastic processes involving one ground and one excited atom is negligible Bouganne et al. (2017); Franchi et al. (2017)). Due to inelastic losses, the spatial density in state decays according to
with a two-body inelastic rate constant. In cases where (for instance, a uniform system prepared in and in the absence of driving), the total atom number obeys a similar equation and decays according to
with a relaxation time . We find that this decay law is compatible with our observations in Fig. 1b-e.
To describe the crossover more quantitatively we have fitted an empirical function of the form
to the data. We chose as in eq. (2) for the amplitude damping function following the preceding discussion, and an exponential damping of coherences for simplicity111Other choices return a similar behavior for the fit parameters versus Rabi frequency.. The parameters , and are the angular frequency, contrast and damping time of the oscillations. The best fit parameters are shown in Fig. 2 versus the expected222We compute from the formula given in Taichenachev et al. (2006); Barber et al. (2006). The applied magnetic field enabling the coupling on the otherwise “doubly forbidden” transition is G. The laser waist m is calculated from Gaussian beam propagation and the laser power is measured for each experiment. Rabi frequency . We find that the measured oscillation frequencies agree well with the expected ones (Fig. 2a). Fig. 2b shows how the contrast of the oscillations decreases with decreasing Rabi frequency, terminating below Hz. The inverse population and coherence damping times are also shown in Fig. 2c and d, respectively. The threshold in Fig. 2b coincides with , as expected from the picture of an ensemble of Doppler-broadened two-level systems previously discussed. The same picture explains the trend observed for weak coupling, where the effective amplitude damping rate scales as (dashed line in Fig. 2c).
To go beyond this empirical analysis, we analyze the experimental data with a set of two GP equations describing two coherently-coupled interacting Bose gases with non-Hermitian evolution,
Interactions between two ultracold atoms occupying states and are modeled by contact potentials Pitaevskii and Stringari (2003) with coupling constants related to the wave scattering length by . For Yb, nm is accurately known from photoassociation spectroscopy Kitagawa et al. (2008), and other elastic and inelastic scattering parameters have been measured recently using isolated atom pairs or triples in deep optical lattices Bouganne et al. (2017); Franchi et al. (2017). In this work, we use the most accurate measurements, namely , and cm/s Bouganne et al. (2017); Franchi et al. (2017). Inelastic losses are taken into account by the imaginary term . The spatial densities in are given by , and we have defined the single-particle Hamiltonian , with the momentum operator, the harmonic trapping potential, and , with the recoil frequency. The lossy GP equations (4,5) can be derived from a master equation treated in the mean-field approximation (see Supplementary Material), and take into account all effects discussed so far – coherent driving, intra- and inter-state interactions, coupling between internal state dynamics, atomic motion by the Doppler term and inelastic losses. Interactions, losses and internal-motional coupling are of the same order of magnitude (a few hundred Hz) for our experimental parameters.
We solve eqs. (4,5) numerically (see Supplemental Material) and fit the numerical solution to the experimental data with the initial atom number, coupling strength and detuning as free parameters. For all data shown in Fig. 1, we find a good agreement between the predicted evolution of the coherently-coupled lossy GP model with the observed dynamics. The fitted Rabi frequencies are close to the expected ones (less than % difference), and the fitted detunings are compatible with our accuracy in finding the resonance (see Supplementary Material).
To obtain more insight on the dynamics described by the dissipative GP equations, we simplify the experimental situation and consider a uniform system of linear size and density . Neglecting elastic interactions and the Doppler term, we are interested in the competition between the coherent driving and the inelastic losses in the limit . We then expect Rabi oscillations to develop, with the spatial densities in and given by and . The envelope slowly decays because of the inelastic losses according to eq. (1). After averaging over one Rabi cycle and integrating the resulting equation, we find that the cycle-averaged population obeys eq. (2) with . The expected dynamics for is thus underdamped Rabi oscillations around an average value decaying algebraically, as observed experimentally for strong driving.
Both the experiments and the GP calculations show that the algebraic decay persists well beyond the regime of validity of the calculation.This is demonstrated in Fig. 3, where we plot versus , with the total atom number (normalized to the initial one) and the initial peak density calculated from the GP model. The quantity depends linearly on for the algebraic decay law in eq. (2), and grows exponentially with for an exponential decay. We find that the decay remains algebraic unless the driving becomes very small, . In this last regime, the excited state population remains small and we recover the model introduced earlier, where the BEC is treated as an ensemble of Doppler-broadened two-level systems experiencing exponential damping. This is observed in the simulation [Fig. 3d] and also in the experiment for the smallest Rabi frequency [Fig. 1f]. We note that even for large driving strengths, the GP calculation deviates from the algebraic law after sufficiently many Rabi cycles. For such long times, elastic interactions, strong depletion due to inelastic losses and the motion in the trap potential can no longer be neglected. It is then not surprising that the simple law in eq. (2) fails to reproduce, at long times, the complete dynamics captured by the GP equations.
In the experiments discussed so far, relaxation of coherence or populations are mostly determined by Doppler broadening or inelastic losses, respectively. We present in Fig. 4a another set of experiments for stronger interactions (kHz), where elastic collisions contribute substantially to the relaxation dynamics. Fig. 4b compares the contrast of the oscillations, determined by the empirical fit in eq. (3) as before. We find that the contrast is reduced as interactions become stronger. Consistent with this observation, the fit to the two-component GP model still reproduces well the long-time decay of the population, but underestimates the damping of coherences that we observe experimentally. Fig. 4c shows that the effect is systematic. We quantify the agreement between the GP model and the observations by a reduced , i.e.,the sum of the fit residuals weighted by the standard deviation and normalized to the number of data points. We find that the reduced increases systematically with the initial chemical potential (see Fig. 4c). This indicates that effects beyond the GP description, e.g. due to thermal population of quasi-particles in the initial state or to additional fluctuations of the fields and , due to the stochastic nature of the losses, become increasingly important.
In conclusion, we have studied the coherent dynamics of a two-component, laser-driven BEC. Whereas spontaneous emission is negligible, a number of other dephasing and relaxation processes take place. We identify three effects leading to relaxation: Doppler broadening due to the finite momentum width of the trapped BEC, inelastic losses between excited atoms, and elastic interactions. We compare our observations to a two-component GP model that includes all these effects in a mean-field approach. We find excellent agreement between the model and the experiments for moderate values of the interactions, but also that the oscillations are damped more strongly in the experiment than predicted by the model for larger interactions. The discrepancy for large interactions could point to additional effects beyond the GP description, for instance the role of quasiparticles present in the initial state due to quantum or thermal fluctuations. In the context of hydrogen spectroscopy experiments Fried et al. (1998), it has been pointed out that taking quasiparticles into account was probably necessary to explain certain features in the spectra and to resolve apparent paradoxes in the interpretation of the data Gardiner and Bradley (2001). Although the theory is more involved for strong driving than in the weak-driving, spectroscopic regime, theoretical tools, e.g. classical field methods Blakie et al. (2008); Polkovnikov (2010), are in principle available. Comparing such a calculation with our experimental results could provide an experimental test for such time-dependent classical field simulations in three dimensions.
Acknowledgements.We acknowledge stimulating discussions with M. Höfer, S. Fölling, members of the BEC group at LKB, R. Le Targat, J. Lodewyck, Y. Le Coq and with C. Kollath. This work was supported by the ERC (Starting Grant 258521–MANYBO). LKB is a member of the DIM SIRTEQ of Région Ile-de-France.
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