Quantum dynamics featuring a dark state recently gained a lot of attraction since their implementation in the context of driven-open quantum systems represents a viable possibility to engineer unique, pure quantum states without the problem of extensive cooling. In this work, we analyze a driven many-body spin system, which undergoes a transition from a dark steady state to a mixed steady state as a function of the driving strength. Since this transition connects a zero entropy dark state with a finite entropy mixed state, it goes beyond the realm of equilibrium statistical mechanics and becomes a genuine non-equilibrium transition.
We focus on a parameter regime, where the transition is predicted to be discontinuous and analyze the relevant long-wavelength fluctuations driving the transition in the framework of the renormalization group. This allows us to approach the non-equilibrium dark state transition and identify similarities and clear differences to common, equilibrium phase transitions and to derive the phenomenology for a first order dark state phase transition. Understanding and probing phase transitions in non-equilibrium systems is an ongoing challenge in physics.
A particular instance are phase transitions that occur between a non-fluctuating absorbing phase, e. I will present recent results on the observation of such a non-equilibrium phase transition in an open driven quantum system. In our experiment, rubidium atoms in a cold disordered gas are laser-excited to Rydberg states under so-called facilitation conditions. This conditional excitation process competes with spontaneous decay and leads to a crossover between a stationary state with no excitations absorbing state and one with a finite number of Rydberg excitations active state.
I will discuss the implications of our findings for future investigations into non-equilibrium phase transitions in open many-body quantum systems. A long-lived prethermal state may emerge upon a sudden quench of a quantum system. In this talk, I consider a quantum quench of an initial critical state, and show that the resulting prethermal state exhibits a genuinely quantum and dynamical universal behavior. I also briefly discuss how the system approaches the prethermal state in a universal way described by a new exponent that characterizes a kind of quantum aging.
Ultracold quantum gases are usually well isolated from the environment. This allows for the study of ground state properties and non-equilibrium dynamics of many-body quantum systems under almost ideal conditions. Such an approach provides new opportunities to study fundamental quantum phenomena and to engineer robust many-body quantum states. I will present an experimental platform [1,2] that allows for the controlled engineering of dissipation in ultracold quantum gases by means of localized particle losses.
We also investigate the steady-states in a driven-dissipative Josephson array . We find that for small dissipation strength, the steady-states are a direct manifestation of coherent perfect absorption. This phenomenon is known from linear optics and is usually challenging to observe. Surprisingly, due to the nonlinearity of the condensate, coherent perfect absorption can be observed much even easier and the phenomenon is very robust.
References  T. Gericke et al. Barontini et al. Labouvie et al. Ultracold Rydberg atoms provide an ideal testbed for study the interplay between strong coherent interactions and dissipative processes, a subject that has recently seen great attention following the discovery of dissipative state engineering for tailored many-body quantum states. However, in contrast to their fully coherent counterparts, our insights into dissipative many-body dynamics are still in its infancy.
I will present the first steps towards a deeper understanding of driven-dissipative Rydberg gases based on a recently developed variational principle  as well as numerical simulations based on tensor network operators . Specifically, I will investigate phase transitions of the steady state, including the presence of a multicritical point that is triggered by the dissipation within the system .
References H. Weimer, Phys.
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Kshetrimayum, H. Weimer, R. Orus, Nature Commun. Overbeck, M. Maghrebi, A. Gorshkov, H. A 95, Critical phenomena in the nonequilibrium steady state of driven-dissipative quantum systems are emerging as a novel class of phase transitions that may be studied in modern hybrid quantum platforms like ultracold atoms or superconducting circuits.
Arrays of coupled, nonlinear optical resonators can in particular simulate the physics of the driven-dissipative Bose-Hubbard model. This symmetry can be spontaneously broken giving rise to a second order phase transition in analogy with a lattice spin model. We present the results of modeling this phase transition in the framework of the Gutzwiller mean-field ansatz and study its dependence on the system parameters.
We then present an analysis of the phase transition beyond mean field, obtained using the corner-space renormalization method , and discuss the dependence on the system dimensionality. Savona, Physical Review A 96, Finazzi, et al. Driven dissipative systems opened up a novel research area in the field of quantum criticality. Phase transitions in these systems lie beyond the standard classification of classical dynamical or equilibrium phase transitions and define completely new universality classes.
In an open quantum system, the critical behavior appears in the state formed by the dynamical equilibrium of the external driving and dissipation processes.
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The correlation functions at the critical point are determined by nonequilibrium noise rather than thermal or ground-state quantum fluctuations. We study the open-system realization of the Dicke model, where a bosonic cavity mode couples to a large spin formed by two motional modes of an atomic Bose-Einstein condensate. The cavity mode is driven by a high-frequency laser and it decays to a Markovian bath, while the atomic mode interacts with a colored bath.
We demonstrate that the critical exponent of the superradiant phase transition is determined by the low-frequency behavior of the spectral density function of the colored bath. We show that a finite temperature in the colored bath leads to qualitatively similar dependence on the spectral density function.
Thursday, 29 March Chair: Jonathan Simon. This is one. Read it and you will see. Want to read more? Register to unlock all the content on the site. E-mail Address. Hamish Johnston is the general-physics editor of Physics World. Read next Everyday science Blog What steps have you taken to pursue your career in physics? Discover more from Physics World. Everyday science Blog Eel delivers record-breaking voltage, pricey helium grounds Boris Johnson blimp, did top journals ban quantum foundations?
Everyday science Blog Science flourishes when leading researchers die, Hippocratic oath for scientists, walking on the Moon. Related jobs. To this end we consider both the effective 1D Hamiltonian with contact interactions, see equation 1 , and a full 3D Hamiltonian. The latter takes into account the geometry of the optical lattice potential as well as the three-dimensional scattering length. Even though the experimental geometry is rather complicated, with different optical lattice potentials felt by the Rb and K atoms, the DMC method can still be applied.
We focus on two densities, first corresponding to strong interactions in the bath, or , as in the Florence experiment [ 7 ]. The second considered value, , corresponds to deep in the Bogoliubov regime. For the first case we consider impurities Rb and bath particles K of different mass. In the second case we will also consider the limit of an infinitely repulsive pinned impurity, in which Bethe ansatz can be used in order to find the ground state energy of the Bose polaron.
This permits us to verify the consistency of the DMC energy with the Bethe ansatz result in this exactly integrable limit. In addition we obtain the correlation functions and the density profile of bosons around the impurity from DMC calculations.
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Here and are the boson—boson and impurity—boson interaction potentials, and , denote the Laplace operators with respect to the impurity and the boson labeled by i , respectively. The external potentials, and , are felt by bosons and the impurity respectively. In the experiment [ 7 ] they have been created by two-dimensional lattices forming an array of 1D tubes. We consider the case when a single tube in the array is populated. We have checked that similar result can be obtained by considering a simple harmonic external potential, , with the transverse oscillator frequency kHz for Rb K atoms.
We ignore the residual shallow trapping along the longitudinal direction in DMC calculations. The relation of the three-dimensional s -wave scattering length, , to the 1D one, , for the tight transverse confinement with oscillator length , is given by Olshanii's formula [ 4 ]. In an optical lattice geometry no simple analytical result is known and the corresponding relation is obtained following [ 65 ].
In the Florence experiment [ 7 ] the boson—boson s -wave scattering length was fixed to nm and was not changed. The boson—impurity s -wave scattering length, in contrast, is tunable over a wide range by changing the strength of the applied magnetic field. In our quasi-1D simulations we model the three-dimensional interaction potential by hard-spheres, when and otherwise. The diameter of the hard-sphere potential coincides with its s -wave scattering length and is set to reproduce Rb—Rb Rb—K value for the boson—boson boson—impurity scattering amplitude. In our simulations we consider a single impurity and impose periodic boundary conditions along the longitudinal direction of the tube.
The statistical fluctuations in Monte Carlo simulation can be greatly reduced by using importance sampling. This is done on the basis of a distribution function which we derive from a trial guiding wave function. Motivated by our physical insights, it is chosen as a product of one- and two-body terms,. The Gaussian one-body terms and localize the particles inside the central tube. The two-body Jastrow terms are taken in the following form.
Here corresponds to boson—boson and to the case of impurity—boson scattering. We also perform calculations for the 1D Hamiltonian from equation 1. In this case the guiding wave function can be conveniently written in a pair-product form. Instead the 'phononic' long-range part in equation 49 , for , is obtained from the hydrodynamic approach [ 66 ]. The variational parameter corresponds to the crossover distance between the two-body and phononic regimes.
It is optimized by minimizing the variational energy, which leads to the VMC results presented earlier in this paper. The variation parameter for large system size coincides with the Luttinger parameter of the bath K. Its dependence on the gas parameter, , is known from the Bethe ansatz solution to the Lieb—Liniger model [ 67 ]. We use the thermodynamic value of parameter for the bath and optimize parameter by minimizing the variational energy. VMC method. The VMC method evaluates averages over the trial wave function. The Metropolis algorithm [ 68 ] is used to sample its square, by generating a Markov chain with corresponding probability distribution.
The average of the Hamiltonian, , provides an upper bound to the ground state energy E 0. It is interesting to compare the value of with prediction of MF theory. DMC method. For large times, the contribution to the energy from the excited states is exponentially suppressed, permitting to obtain the exact ground state energy E 0.
The density profile of the polaron can be calculated using the technique of pure estimators [ 69 , 70 ]. The variational value for is obtained from DMC algorithm without branching, which is an alternative method to the Metropolis algorithm to generate the probability distribution according to. A physically important limit is that of an impurity with an infinite mass,. It corresponds to two realistic situations: i a pinned impurity, ii a static potential created by a focused laser beam. Furthermore this limit is interesting as it is allows to obtain physical insights to the effects of phonon—phonon interactions on the polaron cloud.
When the Bogoliubov approximation is justified—which is the case for or —we can study this situation using the Gross—Pitaevskii mean-field equation GPE [ 47 ]:. Here describes the boson field in the 1D system, and denotes the total energy of the combined impurity—boson system. The GPE is valid even for strong impurity—boson interactions and goes beyond perturbative expansions in orders of. This is different from perturbative theories which linearize boson—impurity interactions, which is justified when is small.
Because the boson—boson interactions are treated on a MF level, the validity of GPE is limited to the Bogoliubov regime as we discuss below. Dark soliton solution. The repulsive nonlinear GPE possesses a famous class of solutions known as gray solitons. These correspond to a depletion in the boson density which maintains its shape while propagating with a constant speed. In the case of zero velocity the density completely vanishes in a single point, and the solution is referred to a a dark soliton. Thus we expect that the effect of a massive impenetrable impurity, with , is to localize a dark soliton in the Bose gas.
The wave function of a dark soliton is given by with the corresponding energy. If we take into account that bosons are repelled from the homogeneous Bose gas, we find that the energy E 0 of the impurity in a system with fixed total particle number N and density n 0 is. This energy has the same physical meaning as other polaron energies E 0 which we calculated before. On the other hand, we can calculate the polaron energy starting from an impurity and using MF theory as described in section 5. In that case we obtain. The reason is that in our MF theory we ignore the nonlinearity in equation By making the Bogoliubov approximation we effectively linearize.
Comparison to DMC. To benchmark our theoretical methods we calculated the energy of an impenetrable, localized impurity for different parameters of the Bose gas. Our comparison in figure 14 shows that DMC is in perfect agreement with the exact result in the limit where Bogoliubov theory is valid, or. The energy shift E 0 caused by an impenetrable static impurity and is calculated as a function of the interaction parameter of the surrounding Bose gas.
The dashed—dotted line corresponds to the dark-soliton solution of the Gross—Pitaevskii equation GPE. To ensure proper finite-size scaling, DMC results are compared to exact analytical calculations using Bethe ansatz and in the Tonks—Girardeau limit. In section 5. This leads to the condition that or. Although in figure 14 we never obtain quantitative agreement of the Bogoliubov approximation with the numerically exact DMC results, we find that the qualitative behavior of the impurity energy E 0 is correctly described in this framework for or small.
In the opposite limit or in contrast, the Bogoliubov description completely fails. In figure 15 we compare the polaron energy for repulsive interactions and parameters as in the experiment of [ 7 ]. For stronger couplings we observe sizable quantitative differences. The corrections of the RG to the MF results is pronounced at large couplings, but DMC predicts even smaller polaron energies in this regime. Comparison of the polaron energy E 0 computed by different methods and using Hamiltonians as indicated in the legend.
We have chosen parameters as in the experiment by Catani et al [ 7 ]. Note that the RG predicts a larger energy than MF theory for weak interactions because for the latter we ignored the logarithmically divergent term equation It is included and properly regularized in the RG. For DMC also the extrapolated value expected in the thermodynamic limit is shown, see appendix C for details of our analysis. In figure 3 we compared predictions for the effective polaron mass to the experimental results from analyzing polaron oscillations.
There we have found large deviations for. Interestingly this is exactly where the strong coupling regime starts and DMC predicts different energies than our RG approach. In this regime the density modulations of the Bose gas around the impurity are expected to become large. As a result, our analysis of polaron oscillations showed that large deviations from the adiabatic result can be expected.
We speculate that this may be related to the large difference between theory and experiment at strong couplings in figure 3. In [ 7 ] it was moreover suggested that could also be the point where higher transverse modes are important. Population of such modes would imply that the system can no longer be treated as strictly 1D.
However, the argument of [ 7 ] was based on a comparison of the bare energy with the transverse trapping frequency. As can be seen from figure 15 , the relevant polaron energies are well below for all interaction strengths. Therefore we conclude that a cross-over into a higher-dimensional regime cannot explain the experimental observations. This supplements our analysis of higher transverse modes in figure 6 , where we arrived at the same conclusion.
We conclude by noting that phonon—phonon interactions play an important role for understanding polarons at strong couplings in the experiment by Catani et al [ 7 ]. On the attractive side, where , we expect their influence to be even more dramatic, because larger deformations of the Bose gas around the impurity are possible. In this paper we studied theoretically mobile impurities interacting with a 1D Bose gas. We provided general theoretical analysis of such problems and considered a specific experimental system realized in experiments by Catani et al [ 7 ].
We extended our analysis to include two-phonon scattering terms, which become important for stronger impurity—boson interactions. Finally we also discussed the effects of boson—boson interactions on the polaron cloud. Main new results. The main new theoretical insights of our work are related to how two-phonon terms affect Bose polarons at strong coupling. Simple MF does not work in one dimension and needs to be corrected by RG calculations. For sufficiently weak boson—boson interactions we find that qualitative features of the polaron phase diagram remain the same as obtained from MF description of polarons in three dimensions [ 24 , 40 ].
In particular we find a repulsive polaron branch for repulsive couplings, , and an attractive polaron branch for sufficiently weak attractive interactions,.
On the other hand, for sufficiently strong attractive interactions, , we expect multi-particle bound states at low energies and a meta-stable repulsive polaron branch at high energies. The latter is adiabatically connected to the polaron at infinitely repulsive microscopic interactions. The enhanced role of quantum fluctuations in 1D manifests itself in the logartihmic divergence with system size of the MF polaron energy. In appendix B we generalized the RG approach from [ 40 ] to one dimension. We showed that the resulting polaron energy is regularized in the RG approach and converges to finite value when the system size is increased.
Furthermore we showed that RG analysis can be used to study other properties of 1D Bose polarons, incudling the effective mass and impurity boson correlations. We compared these predictions to our numerically exact DMC calculations and found good agreement for a weakly interacting Bose gas in the Bogoliubov regime. We concluded that for a full quantitative description of Bose polarons at strong couplings, phonon—phonon interactions always need to be included.
In addition we identified regimes in the phase diagram where the full microscopic Hamiltonian is required for reaching even a qualitative understanding of the polaron properties figure 2. Analysis of the experiment by Catani et al . Original analysis of the experiments showed a disagreement between theoretical results and experimentally measured effective mass already for weak impurity—boson interaction. We explained that this disagreement results from the high energy modes with that were not included properly in earlier analysis.
We showed that both analytical RG and numerical DMC methods give results in good agreement with experiments when the high energy modes are included more accurately. Theoretical methods based on the Bogoliubov approximation predict a saturation of the polaron mass at a finite value of the impurity—boson interaction strength.
Although qualitatively this behavior has been observed in the experiment, large quantitative deviations from our theoretical calculations are found in this regime, where effects beyond the Bogoliubov approximation are expected to play a role. We performed full numerical simulations of the polaron trajectories in a harmonic trapping potential and argued that the disagreement between theory and experiment could be related to the inhomogeneity of the Bose gas.
By performing DMC simulations for two different Hamiltonians: i strictly 1D, see equation 1 , ii three-dimensional, see equation 46 , with strong transverse confinement, we found no significant differences for the dynamic and static properties, even in the regime of strong interactions. This means that the use of a strictly 1D Hamiltonian is justified. Closer inspection of different theoretical models revealed that in the strong coupling regime phonon—phonon interactions need to be included if one wants to do accurate comparison to experiments.
In contrast to Bose polarons in three dimensions [ 8 , 9 ], the relative size of quantum fluctuation corrections to the ground state energy of the Bose gas was sizable in [ 7 ]. We showed here for 1D systems that this is a suitable indicator for the applicability of the Bogoliubov approximation for describing Bose polarons. We conclude that a detailed quantitative analysis of the experiment by Catani et al [ 7 ] at strong couplings is extremely challenging. Moreover, from full dynamical simulations of this problem we found indicators that the inhomogeneity of the Bose gas needs to be taken into account as well.
Our analysis moreover ignored effects of finite temperatures, which can also contribute to the observed differences between theory and experiment. Possible future experiments. Here we performed full dynamical simulations of polaron trajectories inside a shallow trapping potential. We showed by using a time-dependent MF ansatz combined with the LDA that polaron oscillations inside a homogeneous trap provide a powerful means for measuring the effective polaron mass. When the Bose gas can be assumed to be homogeneous, the frequency renormalization provides accurate results.
We suggest to use species-selective optical traps in the future to perform such measurements, in a regime where the Bose gas is as large as possible to avoid effects of the inhomogeneous density profile. The energy provides another important quantity to characterize Bose polarons. It can be obtained directly from the impurity's radio-frequency spectrum, which has been measured in 3D [ 8 , 9 ]. We suggest to repeat these experiments in 1D systems, possibly even in the time-domain [ 73 ]. It would be particularly interesting to study quenches from strong repulsive to strong attractive interactions and show the existence of a repulsive polaron branch for attractive microscopic interactions in one dimension.
FG acknowledges support from the Gordon and Betty Moore foundation. Authors thankfully acknowledge the computer resources at MareNostrum and the technical support provided by Barcelona Supercomputing Center FI The authors gratefully acknowledge the Gauss Centre for Supercomputing e. In this appendix we briefly derive the MF theory for Bose polarons in 1D. To solve the full Hamiltonian 6 , 8 we start by applying the unitary LLP transformation [ 55 ]. Following [ 10 , 11 , 24 ] this gives rise to the Hamiltonian. Note that we normal-ordered the two-phonon scattering terms denoted by , which gives rise to the constant energy shift in the first term of the second line.
As shown in [ 24 ], minimization of the variational energy with respect to leads to the MF solution. The two remaining parameters and are obtained by solving the following set of coupled self-consistency equations,. In this appendix we extend the MF analysis from section 5. We also study the polaron phase diagram of the Bogoliubov polaron model, where phonon—phonon interactions are neglected.
It was checked by showing excellent agreement with numerically exact diagrammatic Monte Carlo calculations [ 18 ] for the polaron energy. In [ 40 ] we extended this approach to Bose polarons, including two-phonon terms. To benchmark the method also in this case, we compare to our DMC calculations in figure 1 b. In the considered regime of large Bose gas densities n 0 , corresponding to a small gas parameter , we find good quantitative agreement. Starting from the MF polaron solution one can rewrite the Hamiltonian by an effective Hamiltonian describing quantum fluctuations around the MF solution.
Using the same notation and following the derivation of [ 40 ] we obtain an effective Hamiltonian. A few explanations are in order. First of all, note that the impurity operators and have been eliminated by applying the LLP transformation [ 55 ] and considering a polaron with vanishing total momentum. The polaron energy , starting at for , decreases until it reaches the ground state energy of the polaron for. The effective phonon frequency in the frame co-moving with the impurity is given by. Note that can be used as an approximation for the effective polaron mass [ 39 ]. The last term in the first line of equation B1 describes phonon—phonon interactions induced by the mobile impurity, where the operators are defined as.
Note that the MF amplitude is flowing in the RG,. The second line in equation B1 is an alternative formulation of the two-phonon scattering terms in equation 8 , with coupling constants running in the RG. We introduced the following pairs of conjugate operators,. In one dimension the RG flow equations read [ 40 ],.
In three dimensions [ 40 ] all coupling constants converge when the cut-off. In that case the RG flows stop when the dispersion relation becomes linear for. Except for , this is also true in one dimension. Here the coupling constant always flows to the weak coupling fixed-point in the IR limit,. To see this, note that we obtain a divergent RG flow when.
The reason for this divergence is the unphysical assumption of MF theory that the coupling constant is unmodified by quantum fluctuations. Now we show that the RG flow of to the universal weak coupling fixed point , see equation B12 , leads to a regularized polaron energy. As discussed around equation 27 the MF energy has a contribution. From the terms in the second line of equation B11 we obtain a contribution. From the explicit solution of the RG flow of we obtain the exact expression. Because for , the last term in equation B15 is irrelevant in the IR limit.
Finally, combining equations B13 , B15 and using we obtain. Before moving on, a comment is in order about the number of phonons in the polaron cloud, which according to MF theory diverges logarithmically with the IR cut-off. Now we have proven that the coupling constant gives rise to two different couplings flowing in the RG, where the renormalization of to zero at low energies regularizes the polaron energy.
We emphasize that the number of phonons in the polaron cloud is still diverging as , as can be readily checked from analyzing the IR behavior of the renormalized MF amplitudes in equation B3. Therefore we conclude that the orthogonality catastrophe [ 60 ] also exists for mobile impurities interacting with 1D quantum gases.
As a direct way of detecting this effect for ultracold atoms, Ramsey interferometry can be used as suggested in [ 73 ]. Now we analyze the RG flows of the coupling constants more closely and derive the polaron phase diagram. We work in a regime where phonon—phonon interactions can be neglected. We will show that the phase diagram shares all qualitative features with the 3D case discussed in [ 40 ]. Static impurity in a non-interacting Bose gas.
Let us start by considering the exactly solvable case of an infinitely heavy impurity, , localized in the origin. Furthermore we assume that the bosons are non-interacting. For repulsive impurity—boson interactions, , the ground state corresponds to a wave function where all bosons populate the same single-particle state, forming a repulsive polaron.
For arbitrarily weak attraction, , a bound state of bosons to the impurity always exists in one dimension.
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In this regime the spectrum is unbounded, because can be occupied by any integer number of bosons. Note that the MF and RG theories provide a description of the polaron state at finite energy, where no bosons are bound to the impurity [ 40 ]. The polaron is meta-stable on the attractive side because it can decay and form a molecule. In the RG theory the existence of a bound state is indicated by a divergence of the effective interaction strength during the RG flow,. For the case without boson—boson interactions described above it holds. As shown in equation B12 , always flows to the repulsive weak coupling fixed point.
Because the RG flow starts at , there always exists a divergence at some intermediate on the attractive side. As explained in detail in [ 40 ] this is a direct manifestation for the bound state existing at low energies in this regime. When the mass of the impurity is slowly decreased, we expect the bound states to remain stable because their energy spacings are sizable.
Note however that the finite mass of the impurity introduces correlations between the bosons and requires us to solve a full many-body problem. Stable repulsive polarons. Now we extend our discussion to finite mass and non-vanishing boson—boson interactions. The RG flows of , which differ in this case, are shown in figure On the repulsive side, , the only qualitative change is that saturates at a finite value in the IR limit.
In this regime the ground state is a repulsive polaron. In figure 7 we show the density profile of the Bose gas around the impurity, and indeed the impurity repels bosons in this regime. For very strong repulsive interactions, the quasi-1D Bose gas is completely depleted around the impurity, reminiscent of the bubble polarons predicted in this regime in [ 74 ] or, equivalently, a dark soliton as described in section 7. Figure B1. The RG flows of blue, yellow are shown, which start form indicated by the dashed line.
The IR values are shown by thick lines; note that and the RG flows have divergencies on the attractive side. The ground states in the different parameter regimes are indicated in the bottom row. In the regime between the RG breaks down. We used parameters as in the experiment by Catani et al [ 7 ]. Attractive polarons. As discussed above, diverges during the RG flow for arbitrary attractive interactions.
On the other hand, the flow of the coupling constant stops in the IR limit due to finite. For sufficiently weak attraction,. This corresponds to attractive interactions of the impurity with density fluctuations in the Bose gas and gives rise to an attractive polaron. Compared to the MF result for the critical interaction strength , see equation 26 , the RG predicts a transition already somewhat earlier,. In figure B1 the attractive polaron regime, where , can be identified. Comparison with figure 6 shows that, indeed, this is the regime where the polaron energy is negative. The density profile of the Bose gas around the impurity also shows a pronounced peak for this set of parameters, see figure 7.
It is worth emphasizing that the existence of attractive polarons in one dimension is due solely to non-vanishing boson—boson interactions. As pointed out before, when the interactions are always repulsive, , in the long-wavelength limit. The effect manifests in the relation for. In view of this conclusion, the agreement between attractive polaron energies in figure 8 a is remarkable, because it suggests that indeed a meta-stable polaronic eigenstate exists which has negative energy.
The divergence of during the RG in the attractive polaron regime suggests that there exists a mode bound to the impurity at energies below the polaron. In [ 40 ] this effect is discussed in detail and it is shown that the polaron becomes dynamically unstable in this case. Because only is negative, while remains positive, the spectrum of is continuous and unbounded in the attractive polaron regime [ 40 ]. In our DMC calculations we fully included phonon—phonon interactions, which are expected to stabilize the attractive polaron [ 40 ].
Indeed we find a nodeless state at energies corresponding to the attractive polaron, see figure 8 a. Break-down of the RG. We find that the number of phonons increases dramatically when approaches. In the regime. This is a result of quantum fluctuations of the mobile impurity, because for it holds. As discussion further in [ 40 ] phonon—phonon interactions are required to stop the divergence of the MF amplitude.
Metastable repulsive polarons. When both coupling constants diverge during the RG flow, see figure B1. While they both start out attractive at high energies, they become repulsive in the long-wavelength limit. Therefore the polaron energy is positive in this regime, corresponding to a repulsive polaron, see figure 6.
The repulsive polaron branch is adiabatically connected to the polaronic states realized for repulsive microscopic interactions, similar to the physics of the super-Tonks—Girardeau metastable state [ 58 , 59 ]. The accumulation of bosons around the impurity is an indicator for molecule formation at.
Already for we find pronounced oscillations in the impurity—boson correlation function, which decay with the distance from the impurity. Indeed, when both coupling constants diverge during the RG, the appearance of a bound state with a discrete energy is expected [ 24 , 40 ]. This state is adiabatically connected to the molecular bound state discussed above at and. In the spectral function it is expected to give rise to a series of peaks separated by the bound state energy [ 24 ].
The possibility to decay into molecular states leads to a finite life-time of repulsive polarons when , a well-known phenomenon close to a Feshbach resonance in three dimensions [ 8 , 9 , 23 , 24 ]. This makes a direct calculation of the polaron energy using DMC difficult, because the polaron is no longer the ground state. To observe repulsive polarons at experimentally, we suggest to study quenches from the strongly repulsive side to where the microscopic interactions are attractive.
This approach has successfully been used to realize the super-Tonks—Girardeau regime of an interacting 1D gas, see [ 58 , 59 ]. We expect that the finite life-time of the repulsive polaron at should be observable, for example by using Ramsey interferometry between two spin states which interact differently with the bosons. Comparison to the experiment. In figure 3 we plotted the critical values and corresponding to the parameters in [ 7 ]. For weakly attractive interactions, , the measured values for the effective mass are in good agreement with predictions for an attractive polaron. Around the qualitative behavior of the data changes.
The range of parameters where the RG breaks down and we expect a polaron cloud with many phonons is too narrow to draw any conclusions from the comparison. The MF and RG theories discussed in the main text predict system properties in the thermodynamic limit of the bath. Instead, quantum Monte Carlo QMC simulations are carried out for a finite-size system in a box with periodic boundary conditions. Thus, for making a comparison between different theories it is preferable first to do the extrapolation of the QMC results to the thermodynamic limit.
Between two considered densities, corresponding to the Florence experiment [ 7 ] and to deep Bogoliubov regime, the latter is expected to have the strongest finite-size effects and we analyze it here. It is instructive first to study how the energy of the bath depends on the system size in the absence of the impurity. Figure C1 shows how the difference of the total energy of the bath and its thermodynamic value depends on the number of particles N.
Even if the convergence in the energy per particle has a fast dependence for large system sizes, in the total energy the asymptotic dependence is weaker, as can be seen from the fit in figure C1. It should be noted, that the polaron energy is obtained from the total energy E N , which diverges linearly with number of particles N. This imposes severe requirements for the numerical accuracy goal, especially when large system sizes are used.
Figure C1. Finite-size dependence of the ground-state energy of the bath deep in the Bogoliubov regime, , and in the absence of the impurity. The dependence on the number of atoms in the bath N is obtained from Bethe ansatz theory [ 48 ]. The asymptotic dependence is shown with a solid line obtained as a fit. For the same high density, , we now add a finite interaction with the impurity. We consider the extreme case of , corresponding to the strongest interaction. The resulting energy scaling obtained from QMC calculations is reported in figure C2.
By comparison with the non-interacting case of shown in figure C1 one can see that the effect is greatly enhanced by a finite interaction with the impurity, as can be perceived by contrasting the scales of the vertical axis. At the same time, the asymptotic convergence law is clearly seen. Figure C2. Finite-size dependence of the polaron energy deep in the Bogoliubov regime, , and for the strongest interaction with the impurity,.
The thermodynamic value is obtained from fits, shown with lines. The polaron energy which we report in the thermodynamic limit in the main part of the paper is obtained from DMC by adjusting a fit to system sizes. Google Scholar. Crossref Google Scholar.