Shape of a sound wave in a weakly-perturbed Bose gas

The Gross-Pitaevskii equation is used to study acoustic emission generated in a uniform one-dimensional Bose gas by a static impurity. It is shown that the impurity creates and shapes a sound-wave packet, which propagates through the gas. Detection of this wave packet can be used to extract properties of the impurity as illustrated here for a Bose gas with a trapped impurity atom -- an example of a relevant experimental setup. Presented results are general for all one-dimensional systems described by the nonlinear Schr\"odinger equation and can also be used in nonatomic systems, e.g., to analyze light propagation in nonlinear optical media.

Time evolution of weakly perturbed quantum gases and liquids is often visualized as the dynamics of collective excitations, e.g., phonons. For example, the response of superfluid helium-4 to various weak perturbations is interpreted as generation of elementary excitations in the Landau's theory of superfluidity [1][2][3]. Similar approaches are used to understand cold atoms [4], polaritons [5,6], and other quantum many-body systems. The general idea is that low-energy perturbations lead to certain occupancies of collective modes, whose dynamics determines the later state of the system. Looking at the problem from another angle, the population of collective modes after excitation carries information about perturbation. This information could be used to study the source of perturbation, as done in acoustic emission testing in classical solids [7].
We use a weakly-perturbed Bose gas to illustrate an idea of reconstructing perturbing potential from sound waves. We choose to model the problem using the Gross-Pitaevskii equation (GPE) -the standard tool for studying degenerate Bose gases [8]. Our work focuses on the linear regime of the GPE, which has sound waves as elementary excitations. Nonlinear phenomena supported by the GPE (e.g., solitons, shock waves [9]) are not important for our study, and will be a subject of our future work. For simplicity, we focus on a quasi-one-dimensional Bose gas that can be modelled by a one-dimensional GPE [10][11][12].
Our work is summarized in Fig. 1. A static impurity inserted in a homogeneous Bose gas creates a defect in the Bose gas and two sound waves, which contain information about the spatial profile of the impurity. One can learn later properties of the impurity by analyzing the emitted sound. This could allow one to extract properties of the impurity even if its exact location is not known. The Letter illustrates this idea by studying time evolution of the system upon an introduction of a single weaklyinteracting impurity of a general kind. The problem is motivated, in particular, by Bose gases with a localized 1. An illustration of the system. A Bose gas which is homogeneous at t < 0 is perturbed at t = 0. At t > 0, the impurity (perturbation) creates a defect in the density of the Bose gas, which resembles the shape of the impurity. Moreover, the impurity generates sound waves, which carry away information about the spatial profile of the impurity. defect [4] or with a massive moving impurity [13].
Our findings are applicable to all systems that are described by the nonlinear Schrödinger equation (NLSE), e.g., to optical pulses propagating inside lossless optical fiber [14], because the Gross-Pitaevskii equation is mathematically equivalent to the NLSE. Furthermore, since linear excitations of the GPE are phonons, our results for weak couplings can be applied to other onedimensional Hamiltonians that describe an impurity in a bosonic bath, e.g., to the Fröhlich model with a static impurity (cf. Ref. [15]).
Formulation. -Our system consists of N repulsively interacting bosons that can be described via the onedimensional GPE. The system is confined to a ring of length L, otherwise it does not experience any external potential at t < 0. We focus on the thermodynamic limit N (L) → ∞, assuming homogeneous density ρ = N/L at t < 0. At t = 0 the system is weakly perturbed, and we study the time evolution at t > 0 using the equation where φ(x, t) is the order parameter, m is the mass of a boson, g determines the strength of the interaction between the atoms in the gas, V (resp. γ) defines the geom-etry (resp. strength) of the perturbation potential. For simplicity, we focus on parity-symmetric potentials, i.e., V (−x) = V (x), that are real and decay exponentially fast at infinity. Otherwise, there are no assumptions on the form of V , moreover, the generalization for nonsymmetric potentials is straightforward. Note that an important Gaussian perturbation has been extensively studied in Refs. [16][17][18][19][20] -these works provide reference points for our study. The function φ obeys the initial condi- , and normalized |φ(x, t)| 2 dx = 1. For later convenience, we associate a length scale, l, with the potential V , and define the healing length of the gas as ξ = √ 2mgρ . Linearization. We assume that the strength of the perturbation is a small parameter, γ → 0, that allows us to expand φ as We are interested in the evolution of the function f (x, t).
Inserting the expansion of Eq. (2) in the GPE (1) and taking into account the initial condition f (t = 0) = 0, we find that f (x, t) is a sum of running waves, phonons, plus a special time-independent solution, f sp (x), [21] f ( where f k (t) determines the population of phononic modes -the quantity of our interest. Both, f k and f sp depend on the Fourier transform of the external potential, where k = 2 k 2 2m , and ω k = |k| 2 k 2 4m 2 + gρ m defines Bogoliubov's phononic spectrum whose relevance for excitations of 1D Bose gases is confirmed by the Bethe ansatz results [22,23].
We have shown that the knowledge of f k grants access toṼ (k), hence, by measuring the occupation of the excitation spectrum one learns properties of the perturbing impurity. Next, we analyze f k for perturbations with long wavelengths, i.e., we focus on the limit ξ l where the GPE works best. In the energy domain, this limit reads where k pert = 1/l determines the range ofṼ (k). Equation (6) allows us to simplify f k and to write the real and imaginary parts of f as where c = ρg m is the speed of sound. Using the convolution theorem for inverse Fourier transform, we obtain These equations show that two counterpropagating sound waves are formed upon excitation of the Bose gas. For t ml 2 / , the waves support a phase difference between parts of the Bose gas, visualized by One can extract information about the perturbing potential by observing the density of the Bose gas. Indeed, the density of the gas n(x, t) = N |φ| 2 is written as n(x, t) = ρ(1 + 2γRe(f )) or (11) Note that Ref dx = 0 to ensure correct normalization of |φ(x, t)| 2 . In the linear regime, the density of the gas at t > 0 is fully defined by ρ and the shape of the perturbation potential. The perturbation creates a stationary defect in the density of the gas given by V (x). It also leads to two waves propagating in the opposite directions with the speed of sound, V (x ± ct). One can learn about the shape of the impurity by observing the propagation of these sound waves. Furthermore, one can speculate that the running waves can potentially be useful for short-distance communication between different points of the Bose gas provided that V is tailored to the needs of information transfer.
Exact solution versus linearization. -To demonstrate the applicability of the results above, we compare them to a numerical solution of the GPE. For the sake of discussion, we consider the perturbation potential, inspired by the second excited eigenstate of a harmonic oscillator. This choice is made to ensure that the numerical solution of the GPE would be well-behaved and would clearly show the shape of the running waves. In Fig. 2, we compare the normalized density of the gas, n(x, t)/ρ − 1, to 2γRe(f ) from Eq. (11). The numerically 'exact' density is given by L|φ (num) | 2 − 1 where φ (num) is a solution of the GPE. This solution is obtained using the pseudospectral method (equipped with the fast Fourier transform routine) for space discretization, in combination with the Runge-Kutta time-stepping scheme [24]. after a rather long evolution time, when the running waves are clearly separated. For strong interactions, the nonlinear effects of the GPE muddle the quantitative comparison; still a qualitative agreement is observed. Note that the shape of the sound waves resembles the shape of the impurity even for rather large values of the perturbation parameter, γ, for which the density variation can be of the order of 10%. The effect of this size is within reach of current experimental setups, allowing one to extract properties of the impurity from the generated sounds waves. The presence of the left-and rightmoving wave packets is useful for averaging out random noise. Note also that one can study the impurity from the static defect created by the impurity in the density of the Bose gas, provided that the position of the impurity is known.
A mobile impurity in a Bose gas. -Finally, we discuss systems where observation of the sound waves examined above may provide a valuable tool for a weakly destructive measurement of the perturbation's properties. Such systems can be Bose gases with localized or slowly moving defects [4,13], or systems with mobile impurities.
We choose to consider a homogeneous Bose gas with a mobile impurity atom trapped by an external potential. A mobile impurity could be used to study properties of the Bose gas [25,26], to simulate Bose polarons [27], and to store and process quantum information [28,29]. These applications have already motivated a number of works to study the quench dynamics of impurities in similar setups [27,[30][31][32][33][34]. In this Letter, we investigate the corresponding time dynamics of the Bose gas.
We assume that the impurity occupies a certain state of the harmonic trap, and argue that information about this state can be extracted in a weakly destructive manner by observing the density of the Bose gas following a sudden change of the boson-impurity interaction. For simplicity, we focus on an impurity that is either in the ground, |g , or in the first excited, |e , states. A more challenging measurement of a general quantum state: |g + A|e , where A is some complex number, requires knowledge of both the density and the phase of the Bose gas. A corresponding discussion is left for future studies.
To study the dynamics of the impurity we employ a strong coupling approach [35][36][37], which is based on the Hartree approximation to the wave function. The system (Bose gas plus impurity) is described within this approach by the system of coupled equations where φ is the order parameter of the Bose gas, and ψ describes the impurity; g ib is the strength of the bosonimpurity interaction. The impurity is trapped by an external harmonic oscillator, 2 x 2 2ml 4 . For simplicity, we assume that the impurity and a boson are of equal masses. We again consider ξ l, which means that the density distortion created by the impurity has a range, which is smaller than the oscillator length. Interestingly, this condition is also required to define a one-dimensional Bose polaron [34,38,39].
In the limit of small values of g ib , Eq. (13) maps onto Eq. (1) with γV (x) = g ib e −x 2 /l 2 /( √ πl) for the ground state and γV (x) = 4g ib x 2 e −x 2 /l 2 /( √ πl 3 ) for the excited state. The Bose gas is homogeneous at t < 0, which means that in the leading order in g ib , only the phase of the impurity atom is affected. By changing g ib one creates a sound wave in the Bose gas with amplitude proportional to g ib , whereas the corresponding change of the state of the impurity is of the order g 2 ib . Let us show that the sound wave indeed contains information about the state of the impurity. To this end, we solve Eqs. (13) and (14) numerically and via linearization. The comparison is presented in Fig. 3. We see that numerical results agree well with linearization. As expected, the shape of the sound wave is given by the state of the impurity. If the impurity is used to store information in the form of either |g or |e , then the read-out of this information can be achieved from the sound waves emitted upon the change of g ib . The measurement process is weakly destructive, since the state of the impurity is perturbed as ∼ g 2 ib upon the change of g ib , and the observation of the density of the Bose gas may (in principle) have no effect on the impurity.
To summarize, population of collective modes provides information on the properties of their source. To illustrate this, we have calculated excitations generated in a Bose gas by a static impurity. In the linear regime, these excitations have been expressed through the potential imposed by the impurity. As a relevant example, we have considered a Bose gas with a trapped impurity atom. The impurity has been initialized in either the ground or the first excited states of a harmonic oscillator, although, it is not difficult to argue that our analysis can be extended to more complicated cases. For example, one could measure not only motional but also internal states of an impurity, provided that the impurity-gas interaction strength, g ib , depends on the pseudospin of the impurity. The next step is to extend our discussion to higher-dimensional systems, and to assess beyond-meanfield effects which are particularly important for onedimensional quantum gases.