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arXiv:1805.00709 · SciPost Physics

Two distinguishable impurities in BEC: squeezing and entanglement of two Bose polarons

Christos Charalambous, Miguel Ángel García-March, Aniello Lampo, Mohammad Mehboudi, Maciej Lewenstein

Published Jan 22, 2019• 32 citations
arXivHTMLPDF

Abstract

We study entanglement and squeezing of two uncoupled impurities immersed in a Bose-Einstein condensate. We treat them as two quantum Brownian particles interacting with a bath composed of the Bogoliubov modes of the condensate. The Langevin-like quantum stochastic equations derived exhibit memory effects. We study two scenarios: (i) In the absence of an external potential, we observe sudden death of entanglement; (ii) In the presence of an external harmonic potential, entanglement survives even at the asymptotic time limit. Our study considers experimentally tunable parameters.

DOI
10.21468/scipostphys.6.1.010

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Quantum Brownian motion is one of the most prominent examples of an open quantum system, a system which cannot be treated in isolation from its environment. The simplest method to study the dynamics of a system undergoing such a type of motion, that satisfies Heisenberg Uncertainty principle is the approach of Quantum Generalized Langevin Equations (QGLE), which was used throughout this thesis. A Quantum Brownian motion approach is used in this work to study the Bose polaron problem. In this case, one transforms the original problem into one where the impurities are treated as quantum Brownian particles interacting with a bath composed of the Bogoliubov modes of the condensate. Then by deriving the relevant QGLE, it was shown that the dynamics of the Bose polaron exhibit memory effects. This was studied for both a free Bose-Einstein condensate (BEC) and a harmonically trapped one, in both cases for experimentally relevant parameters. Taking advantage of this recent theoretical development, we study a number of phenomena that can be examined under this prism and show how various microdevices can be constructed and controlled. In the first project, we study the creation of entanglement and squeezing of two uncoupled impurities that are immersed in a single common (BEC) bath. We treat these impurities as two quantum Brownian particles. We study two scenarios:(i) In the absence of an external potential, we observe sudden death of entanglement;(ii) In the presence of an external harmonic potential, where entanglement survives even at the asymptotic time limit. In our second work, we studied the diffusive behavior of a Bose Polaron immersed in a coherently coupled two-component BEC. The particle superdiffuses if it couples in the same manner to both components, i.e. if it couples either attractively or repulsively to both of them. This is the same behavior of an impurity immersed in a single BEC. Conversely, we find that it exhibits a transient nontrivial subdiffusive behavior if it couples attractively to one of the components and repulsively with the other. We show how the magnitude of the anomalous exponent reached and the duration of the subdiffusive interval can be controlled with the Rabi frequency of the coherent coupling between the two components and the coupling strength of the impurity to the BEC. Then we proceeded with the construction of two microdevices, a quantum sub-nk thermometer and a heat diode. In the first project, we introduced a novel minimally disturbing method for sub-nK thermometry in a BEC. In this case, the impurity acted as a thermometer, where one detects temperature fluctuations from measurements of the position and momentum of the impurity. Crucially, these cause minimal backaction on the BEC and hence, realize a nondemolition temperature measurement. Following the paradigm of the emerging field of quantum thermometry, we combine tools from quantum parameter estimation and the theory of open quantum systems to solve the problem in full generality. We thus avoid any simplification, such as demanding thermalization of the impurity atoms. In our final work, we investigated the heat transport and the control of heat current among two spatially separated trapped BECs, each of them at a different temperature. To allow for heat transport among the two independent BECs we consider a link made of two dipole-dipole interacting harmonically trapped impurities, each of them interacting with one of the BECs. We address the dependence of heat current and current-current correlations on the physical parameters of the system. Interestingly, we show that heat rectification, can occur in our system, when a periodic driving on the trapping frequencies of the impurities is considered. 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We study entanglement and squeezing of two uncoupled impurities immersed in a Bose-Einstein condensate. We treat them as two quantum Brownian particles interacting with a bath composed of the Bogoliubov modes of the condensate. The Langevin-like quantum stochastic equations derived exhibit memory effects. We study two scenarios: (i) In the absence of an external potential, we observe sudden death of entanglement; (ii) In the presence of an external harmonic potential, entanglement survives even at the asymptotic time limit. Our study considers experimentally tunable parameters.
Quantum Brownian motion is one of the most prominent examples of an open quantum system, a system which cannot be treated in isolation from its environment. The simplest method to study the dynamics of a system undergoing such a type of motion, that satisfies Heisenberg Uncertainty principle is the approach of Quantum Generalized Langevin Equations (QGLE), which was used throughout this thesis. A Quantum Brownian motion approach is used in this work to study the Bose polaron problem. In this case, one transforms the original problem into one where the impurities are treated as quantum Brownian particles interacting with a bath composed of the Bogoliubov modes of the condensate. Then by deriving the relevant QGLE, it was shown that the dynamics of the Bose polaron exhibit memory effects. This was studied for both a free Bose-Einstein condensate (BEC) and a harmonically trapped one, in both cases for experimentally relevant parameters. Taking advantage of this recent theoretical development, we study a number of phenomena that can be examined under this prism and show how various microdevices can be constructed and controlled. In the first project, we study the creation of entanglement and squeezing of two uncoupled impurities that are immersed in a single common (BEC) bath. We treat these impurities as two quantum Brownian particles. We study two scenarios:(i) In the absence of an external potential, we observe sudden death of entanglement;(ii) In the presence of an external harmonic potential, where entanglement survives even at the asymptotic time limit. In our second work, we studied the diffusive behavior of a Bose Polaron immersed in a coherently coupled two-component BEC. The particle superdiffuses if it couples in the same manner to both components, i.e. if it couples either attractively or repulsively to both of them. This is the same behavior of an impurity immersed in a single BEC. Conversely, we find that it exhibits a transient nontrivial subdiffusive behavior if it couples attractively to one of the components and repulsively with the other. We show how the magnitude of the anomalous exponent reached and the duration of the subdiffusive interval can be controlled with the Rabi frequency of the coherent coupling between the two components and the coupling strength of the impurity to the BEC. Then we proceeded with the construction of two microdevices, a quantum sub-nk thermometer and a heat diode. In the first project, we introduced a novel minimally disturbing method for sub-nK thermometry in a BEC. In this case, the impurity acted as a thermometer, where one detects temperature fluctuations from measurements of the position and momentum of the impurity. Crucially, these cause minimal backaction on the BEC and hence, realize a nondemolition temperature measurement. Following the paradigm of the emerging field of quantum thermometry, we combine tools from quantum parameter estimation and the theory of open quantum systems to solve the problem in full generality. We thus avoid any simplification, such as demanding thermalization of the impurity atoms. In our final work, we investigated the heat transport and the control of heat current among two spatially separated trapped BECs, each of them at a different temperature. To allow for heat transport among the two independent BECs we consider a link made of two dipole-dipole interacting harmonically trapped impurities, each of them interacting with one of the BECs. We address the dependence of heat current and current-current correlations on the physical parameters of the system. Interestingly, we show that heat rectification, can occur in our system, when a periodic driving on the trapping frequencies of the impurities is considered. Therefore, our system is a possible setup for the implementation of a phononic circuit, and hence contributes in the general framework of using BECs as platforms for quantum information processing. El movimiento Browniano, es un ejemplo de un sistema abierto, es decir un sistema que no se puede tratar en aislamiento. El método más simple para estudiar la dinámica de dicho sistema, que cumple el principio de la incertidumbre de Heisenberg es el de Quantum Generalized Langevin Equations (QGLE), que es el método que se usa en esta tesis. La perspectiva de Quantum Brownian motion se ha usado para estudiar muchos sistemas, entre ellos el problema de Bose polaron. En este caso, uno pasa el problema original a uno donde las impurezas se tratan como partículas Brownianas quanticas interactuando con un baño compuesto de modos de Bogoliubov del condensado. Después de derivar la QGLE relevante, se puede demostrar que la dinámica del Bose polaron muestra efectos de memoria. Esto se ha estudiado tanto en un Bose Einstein Condensate (BEC) libre como en uno atrapado en un trapo harmónico, para parámetros relevantes en experimentos. Aprovechando de este reciente desarrollo, estudiamos muchos fenómenos que se pueden investigar bajo este prisma y mostramos cómo se pueden construir y controlar varios microdispositivos. En el primer proyecto, estudiamos la creación de enlazamiento y squeezing de dos impurezas no acopladas, inmersas en un único BEC baño común. Estudiamos dos senarios: (i) en la ausencia de un potencial externo, donde observamos la muerte repentina del enlazamiento (ii) en la presencia de un trapo externo harmónico, donde el enlazamiento sobrevive incluso en el límite asintótico de largos tiempos. En nuestro segundo trabajo, estudiamos el comportamiento difusivo de un Bose polaron inmerso en un BEC de dos componentes que están acopladas coherentemente. La partícula es superdffusa si se acopla en la misma manera a los dos componentes, i.e. atractivamente o repulsivamente, como en el caso de un unico BEC. En el caso contrario, encontramos que la partícula muestra un comportamiento transitorio non-trivial. Mostramos como la magnitud del exponente anómalo y la duración del periodo transitorio se pueden controlar a través de la frecuencia Rabi del acoplamiento coherente entre los dos componentes y la fuerza del acoplamiento de la impureza a los dos componentes del BEC. En seguida, procedemos con la construcción de dos microdispositivos, un termómetro quántico y un diodo térmico. En el primer proyecto, hemos introducido un nuevo método de mínimo disturbio, que sirve para termometría en temperaturas sub-nK en un BEC. Nuestra técnica está basada otra vez en el modelo de Bose polaron, donde esta vez la impureza inmersa en un BEC sirve como un termómetro. La propuesta es detectar fluctuaciones de la temperatura de las medidas de la posición y el impulso de la impureza. Crucialmente, estas causan una reacción mínima en el BEC y, por lo tanto, realizan una medida de la temperatura no demoledora. En nuestro trabajo, evitamos cualquiera simplificación, como la imposición de la termalización de la impureza, o del acoplamiento débil de la impureza con el BEC. En el último trabajo, investigamos el transporte de calor y el control de corrientes de calor entre dos BECs espacialmente separados y atrapados harmónicamente, en temperaturas distintas. El flujo de calor entre los dos BECs, esta facilitado a través de dos impurezas harmónicamente atrapadas, cada una interactuando con su propio BEC. Las impurezas estan acopladas a traves de interacciones de dipolo-dipolo. Examinamos la dependencia del corriente de calor y sus correlaciones en los parámetros físicos del sistema. Mostramos que la rectificación del corriente del calor, i.e. el flujo de calor unidireccional puede ocurrir en el sistema, cuando aplicamos una conducción periódica en las frecuencias de los trapos de las impurezas. Por lo tanto, nuestro sistema es una posible configuración para la implementación de un circuito fononico.
We study the dynamics of a quantum impurity immersed in a Bose-Einstein condensate as an open quantum system in the framework of the quantum Brownian motion model. We derive a generalized Langevin equation for the position of the impurity. The Langevin equation is an integrodifferential equation that contains a memory kernel and is driven by a colored noise. These result from considering the environment as given by the degrees of freedom of the quantum gas, and thus depend on its parameters, e.g. interaction strength between the bosons, temperature, etc. We study the role of the memory on the dynamics of the impurity. When the impurity is untrapped, we find that it exhibits a super-diffusive behavior at long times. We find that back-flow in energy between the environment and the impurity occurs during evolution. When the particle is trapped, we calculate the variance of the position and momentum to determine how they compare with the Heisenberg limit. One important result of this paper is that we find position squeezing for the trapped impurity at long times. We determine the regime of validity of our model and the parameters in which these effects can be observed in realistic experiments.
We study the quantum motion of an impurity atom immersed in a Bose-Einstein condensate in arbitrary dimensions. It was shown, for all dimensions, that the Bogoliubov excitations of the Bose-Einstein condensate act as a bosonic bath for the impurity, where linear coupling is possible for a certain regime of validity, which was assessed only in one dimension. Here we present the detailed derivation of the ddd-dimensional Langevin equations that describe the quantum dynamics of the system, and of the associated generalized tensor that describes the spectral density in the full generality, and assesses the linear assumption in all dimensions. As results, we obtain, when the impurity is not trapped, the mean square displacement in all dimensions, showing that the motion is superdiffusive. We obtain also explicit expressions for the superdiffusive coefficient in the small and large temperature limits. We find that, in the latter case, the maximal value of this coefficient is the same in all dimensions, but is only reachable in one dimension, within the validity of the assumptions. We study also the behavior of the average energy and compare the results for various dimensions. In the trapped case, we study squeezing and find that the stronger position squeezing can be obtained in lower dimensions. We quantify the non-Markovianity of the particle's motion and find that it increases with dimensionality.
We study the dynamics of an impurity embedded in a trapped Bose-Einstein condensate, i.e., the Bose polaron problem. This problem is treated by recalling open quantum systems techniques: the impurity corresponds to a particle performing quantum Brownian motion, while the excitation modes of the gas play the role of the environment. It is crucial that the model considers a parabolic trapping potential to resemble the experimental conditions. Thus, we detail here how the formal derivation changes due to the gas trap, in comparison to the homogeneous gas. More importantly, we elucidate all aspects in which the gas trap plays a relevant role, with an emphasis on the enhancement of the non-Markovian character of the dynamics. We first find that the presence of a gas trap leads to a new form of the bath-impurity coupling constant and a larger degree in the super-Ohmicity of the spectral density. We then solve the quantum Langevin equation to derive the position and momentum variances of the impurity, where the former is a measurable quantity. For the particular case of an untrapped impurity, the asymptotic behavior of this quantity is found to be motion superdiffusive. When the impurity is trapped, we find position squeezing, casting the system suitable for implementing quantum metrology and sensing protocols. We detail how both superdiffusion and squeezing can be enhanced or inhibited by tuning the Bose-Einstein condensate trap frequency. Compared to the homogeneous gas case, the form of the bath-impurity coupling constant changes, and this is manifested as a different dependence of the system dynamics on the past history. To quantify this, we introduce several techniques to compare the different amount of memory effects arising in the homogeneous and inhomogeneous gas. We find that it is higher in the second case. This analysis paves the way to the study of non-Markovianity in ultracold gases, and the possibility to exploit such a property in the realization of new quantum devices.
We consider two heavy impurities immersed in a Bose-Einstein condensate, and calculate the self-energy using the Wilsonian renormalization. The polaron energy, quasiparticle residue and damping rate are extracted from the self-energy. We demonstrate that various effective potentials emerge from the polaron energy under the specific conditions. In the limit of large separation between the impurities, the polaron spectrum converges to the results for a single impurity, exhibiting an attractive-repulsive crossover across the Feshbach resonance. The boundary of this crossover is identified through the analysis of the damping rate. We highlight that repulsive-dominant polarons can exist as long as the impurities are sufficiently close, even when the impurity-boson interactions are attractive. Additionally, we observe that the two impurities become asymptotically free in the repulsive polaron regime. These results are verifiable and offer a fresh perspective on the interaction dynamics between two polarons.
We consider impurity atoms embedded in a two-component Bose-Einstein condensate in a quasi-one dimensional regime. We study the effects of repulsive coupling between the impurities and Bose species on the equilibrium of the system for both miscible and immiscible mixtures by numerically solving the underlying coupled Gross-Pitaevskii equations. Our results reveal that the presence of impurities may lead to a miscible-immiscible phase transition due to the interaction of the impurities and the two condensates. Within the realm of the Bogoliubov-de Gennes equations we calculate the quantum fluctuations due to the different types of interactions. The breathing modes and the time evolution of harmonically trapped impurities in both homogeneous and inhomogeneous binary condensates are deeply discussed in the miscible case using variational and numerical means. We show in particular that the self-trapping, the miscibility and the inhomogeneity of the trapped Bose mixture may strongly modify the low-lying excitations and the dynamical properties of impurities. The presence of phonons in the homogeneous Bose mixture gives rise to the damping of breathing oscillations of impurities width.