OFFPRINT Breathing oscillations of a trapped impurity in a Bose gas

Motivated by a recent experiment (Catani J. et al., Phys. Rev. A, 85 (2012) 023623) we study breathing oscillations in the width of a harmonically trapped impurity interacting with a separately trapped Bose gas. We provide an intuitive physical picture of such dynamics at zero temperature, using a time-dependent variational approach. The amplitudes of breathing oscillations are suppressed by self-trapping, due to interactions with the Bose gas. Further, exciting phonons in the Bose gas leads to damped oscillations and non-Markovian dynamics of the width of the impurity, the degree of which can be engineered through controllable parameters. Our results, supported by simulations, reproduce the main features of the dynamics observed by Catani et al. despite the temperature of that experiment. Moreover, we predict novel effects at lower temperatures due to self-trapping and the inhomogeneity of the trapped Bose gas. Copyright c © EPLA, 2012 The ability to trap and cool atoms of different species to ultra-low temperatures has led to the realisation of various theoretical models in which the intriguing physics of binary mixtures of spin hyperfine states or different elements can be studied [1–5]. In particular, highly imbalanced mixtures have made it possible to investigate the dynamics, interactions and decoherence of single atoms, generally referred to as impurities, immersed in a background atomic gas [6–11]. As a prominent example, signatures of polaron effects, caused by impurity-induced density fluctuations of the background gas, have been investigated theoretically [12,13] and observed in experiments [14,15]. More recently, Catani et al. created a harmonically trapped impurity suspended in a separately trapped Bose gas [16]. They studied the dynamics of the system following a sudden lowering of the trap frequency of the impurity. Primarily, they observed breathing oscillations of the width σ of the impurity density distribution for various impurity-Bose gas interaction strengths. Several features of the experiment were amenable to interpretation in terms of a quantum Langevin equation in conjunction with a polaronic mass shift; however, we show that at even lower (a)E-mail: [email protected] temperatures a different model is required to fully describe the dynamics of the system. In this letter, we develop a versatile analytical model to describe the breathing oscillations of an impurity in a Bose gas at zero temperature. At first, our analysis is based on a variational approach in the Gross-Pitaevskii (GP) regime. We show that the impurity density distribution, which we describe by a Gaussian, has a width obeying a Newtonian equation of motion for a fictitious particle with position σ. The potential governing this motion accounts for the quantum pressure of the impurity, the inhomogeneity of the trapped Bose gas and the localised distortion of this background gas induced by the impurity. The latter leads to a strong confinement of the impurity, known as self-trapping [17–20]. Subsequently, we extend our model by including excitations of the Bose gas in the form of Bogoliubov phonons. A variational ansatz, which describes the bosons as a product of coherent phonon states, allows us to track the evolution of the system, including the exchange of energy between the impurity and the phonon bath. The dynamics turns out to be non-Markovian because of the back-action of phonons created by the impurity, and we show that the timescale of memory effects can be varied by adjusting the trapping parameters, allowing for a comprehensive study