This book introduces the theoretical description and properties of quantum fluids. The focus is on gaseous atomic Bose-Einstein condensates and, to a minor extent, superfluid helium, but the underlying concepts are relevant to other forms of quantum fluids such as polariton and photonic condensates. The book is pitched at the level of advanced undergraduates and early postgraduate students, aiming to provide the reader with the knowledge and skills to develop their own research project on quantum fluids. Indeed, the content for this book grew from introductory notes provided to our own research students. It is assumed that the reader has prior knowledge of undergraduate mathematics and/or physics; otherwise, the concepts are introduced from scratch, often with references for directed further reading.
All content in this area was uploaded by Nicholas Parker on Sep 07, 2016
Content may be subject to copyright.
A preview of the PDF is not available
... They found that, at temperatures below a critical value , the de Broglie wavelength becomes larger then the average distance between the particles. Essentially, the particles behave collectively like a giant wave: the entire gas is governed by a macroscopic wavefunction Ψ, creating a new state of matter which is now called a Bose-Einstein condensate [2,3]. As the temperature is reduced below , a larger and larger fraction of the particles condenses into the ground state; at = 0, all particles are in the ground state. ...
... In 1972 Douglas Osheroff, David Lee and Robert Richardson discovered that the rare lighter isotope of helium, 3 He, if cooled to temperatures of few mK, also becomes superfluid. The effect is notable because 3 He is a fermion (the 3 He nucleus contains two protons and only one neutron, not two), so Bose-Einstein condensation occurs via the formation of Cooper pairs like in superconductors. ...
... In the following years other quantum fluids were discovered and investigated. Besides 4 He, 3 He and a large number of atomic condensates (lithium, sodium, potassium, rubidium, caesium, hydrogen, etc), the realm of quantum fluids now comprises polaritons, magnons, various two-component condensates, quantum ferrofluids, spinor condensates, the interior of neutron stars and probably even dark matter axions. ...
Near absolute zero, superfluid liquid helium displays quantum properties at macroscopic length scales. One property, superfluidity, means flow with zero viscosity. Another property, the existence of a complex wavefunction, constrains the rotation to thin, discrete vortex lines carrying one quantum of circulation each. Therefore, if liquid helium is stirred, it becomes turbulent, but in a peculiar way: it is a state of turbulence consisting of a tangle of quantised vortex lines in a fluid without viscosity. Surprisingly, this disordered state, which I called quantum turbulence years ago, shares many properties with ordinary turbulence as it represents its essential skeleton. These lectures attempt to relate the dynamics, the geometry and the topology of quantum turbulence. Although recently much progress has been made, there are still many open questions. Some of the methods which have been used and are described here (e.g. the smoothed vorticity, the use of crossing numbers) have a scope which clearly goes beyond the fascinating problem of turbulence near absolute zero.
... Distinctively different from classical fluids, quantum fluids exhibit several exotic symptoms such as Bose-Einstein condensation (BEC) and superconductivity that appear macroscopically, with roots in by quantum mechanics and quantum statistics. [3] Figure 1b is a snapshot of the bizarre fountain effect in liquid helium-4 as it undergoes temperature gradients and flows through small pores, revealing the properties of superfluids (SFs). [4][5][6] Ultracold atomic gases become condensed at the system ground state below a transition temperature. ...
... Spatial and temporal coherence properties are captured in interferograms, with which we can discuss spontaneous coherence build-up, BEC, and Berezinskii-Kosterlitz-Thouless (BKT) phase transition. [35][36][37][38][39] The second-order and higher-order correlations are quantified by coincidence photon counting of g (2) (0), [15,[40][41][42] g (3) (0), [40][41][42] and g (4) (0). [42] Dynamical processes are also kept an eye on via standard pump-probe techniques, [43] and lifetime measurements are routinely done with streak-cameras. ...
... The existence of vortices is closely connected to the order parameter of quantum phases. [3] A quantum vortex means the quantization of a circular flow as a topological defect in SFs or superconductors. In 1949, L. Onsager conjectured a vortex in SFs, whose order parameter is a complex function written as (⃗ r) = | √ n|e i (⃗ r) , where n is the polariton density and the phase (⃗ r) can be chosen for a single-valued wavefunction. ...
Microcavity exciton‐polaritons are attractive quantum quasi‐particles resulting from strong light–matter coupling in a quantum‐well‐cavity structure. They have become one of the most stimulating solid‐state material platforms to explore beautiful collective quantum phenomena originating from macroscopic coherence in condensation and superfluidity, Berezinskii–Kosterlitz–Thouless transition, and various topological excitations in the form of solitons, vortices, and skyrmions. They can also provide opportunities for the development of pioneering photonic devices by exploiting bistability and parametric scatterings due to strong nonlinearity that possess remarkable performance advantages of power‐efficient operation, ultrafast response time, and scalable planar geometries. This story becomes profound and fascinating when the spins of excitons are taken into account, that can be directly accessed through light polarization states. The purpose of this review is to give central principles of microcavity exciton‐polariton spins and their anisotropic interactions, which can couple with the effective magnetic fields from mode‐splitting of microcavity photons and spin‐dependent relaxation processes of quantum‐well excitons. Furthermore, notable theoretical and experimental research activities are summarized to reveal extraordinary quantum phenomena of spin‐resolved topological states and exotic spin textures and to devise novel spin‐based photonic devices based on microcavity exciton‐polaritons.
... In any real gas, particles interacting with each other deviate from the ideal gas assumptions [73]. These particle interactions will amplify quantum fluctuations in a BEC and will lead to a depletion of the condensate [73]. ...
... In any real gas, particles interacting with each other deviate from the ideal gas assumptions [73]. These particle interactions will amplify quantum fluctuations in a BEC and will lead to a depletion of the condensate [73]. Modeling a condensate with interacting quantum particles needs the manybody Schrödinger equation. ...
... Due to the complexity of this approach, it will limit modelling for only a few atoms. But typically a BEC contains thousands or millions of atoms [73]. ...
This thesis consists of two main sections. In the first section we present main steps that were taken in the process of constructing a Bose-Einstein condensation (BEC) apparatus. A detailed description of the laser-optical system and vacuum system is provided. We also present the assembly and characterization of the first three-dimensional (3D) magneto-optical trap (MOT) in this doubleMOT system. In the second part we present numerical simulations on manipulating BECs at a fast rate while maintaining the coherence properties of its initial quantum state. Two-dimensional (2D) simulations of BEC transport are performed by numerically solving the Gross-Pitaevskii equation (GPE). In our simulations, we use trapping potentials in the form of painted potentials because it is possible to achieve arbitrary, dynamic traps with this method. To achieve high quantum fidelity, we use shortcuts-to-adiabaticity (STA) for high-speed BEC transport. With these simulations, we
compared different time intervals for a particular spatial displacement that a BEC can travel while keeping high quantum fidelity using experimentally feasible parameters.
... Equation (31) can also be written as a quantum Bernoulli equation as follows [13]: ...
... The non-local constitutive Equation (13) implies that density changes in r' determine an instantaneous variation of the free energy at r, with an infinite propagation velocity. To correct this obviously unrealistic assumption, Equation (13) is replaced with the following expression: ...
Assuming that the energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm’s quantum potential and the Madelung equation are obtained, showing explicitly that some of the hypotheses that led to the formulation of quantum mechanics do admit a classical interpretation based on non-locality. Here, we generalize this approach imposing a finite speed of propagation of any perturbation, thus determining a covariant formulation of the Madelung equation.
... We first obtain the stationary solution of the GPE in the presence of the rough boundary, with bulk density n 0 , by solving the GPE in imaginary time [50]. The GPE is then transformed into a frame moving at speed u in the x-direction (corresponding to the imposed flow) by the addition of a Galilean boost term ihu∂Ψ/∂x to the right-hand side of the GPE. ...
We model the superfluid flow of liquid helium over the rough surface of a wire (used to experimentally generate turbulence) profiled by atomic force microscopy. Numerical simulations of the Gross-Pitaevskii equation reveal that the sharpest features in the surface induce vortex nucleation both intrinsically (due to the raised local fluid velocity) and extrinsically (providing pinning sites to vortex lines aligned with the flow). Vortex interactions and reconnections contribute to form a dense turbulent layer of vortices with a non-classical average velocity profile which continually sheds small vortex rings into the bulk. We characterise this layer for various imposed flows. As boundary layers conventionally arise from viscous forces, this result opens up new insight into the nature of superflows.
... The experimental discovery [1,2] and theoretical explanation [3] of this macroscopic quantum phenomenon were among the most important achievements of 20th-century physics. While initial studies were restricted to liquid helium, superfluid properties were recently observed and explored in a dilute gas of bosons, called Bose-Einstein condensates (BECs) [4,5]. The advantage of ultracold gases in studies of superfluidity over liquid helium is that they are analytically tractable, weakly interacting systems, and free of surface tension effects. ...
Superfluid and dissipative regimes in the dynamics of a two-component quasi-one-dimensional Bose–Einstein condensate (BEC) with unequal atom numbers in the two components have been explored. The system supports localized waves of the symbiotic type owing to the same-species repulsion and cross-species attraction. The minority BEC component moves through the majority component and creates excitations. To quantify the emerging excitations, we introduce a time-dependent function called disturbance. Through numerical simulations of the coupled Gross–Pitaevskii equations with periodic boundary conditions, we have identified a critical velocity of the localized wave, above which a transition from the superfluid to dissipative regime occurs, as evidenced by a sharp increase in the disturbance function. The factors responsible for the discrepancy between the actual critical velocity and the speed of sound, expected from theoretical arguments, have been discussed.
... To obtain the ground state, we perform an imaginary time propagation [99], that is, we use imaginary time steps −i dτ . Furthermore, we solve the differential equation numerically on a spatial grid with a discrete time split step method (see [99] and Appendix A of [100]). ...
Ensembles of ultra-cold atoms have been proven to be versatile tools for high precision sensing applications. Here, we present a method for manipulation and readout of the state of trapped clouds of ultra-cold bosonic atoms. In particular, we discuss the creation of coherent and squeezed states of quasiparticles and the coupling of quasiparticle modes through an external cavity field. This enables operations like state swapping and beam splitting which can be applied to realize a Mach-Zehnder interferometer (MZI) in frequency space. We present two explicit example applications in sensing: the measurement of the healing length of the condensate with the MZI scheme, and the measurement of an oscillating force gradient with a pulsed optomechanical readout scheme. Furthermore, we calculate fundamental limitations based on parameters of state-of-the-art technology.
... Bose-Einstein condensates well below their transition temperatures, i.e., with negligible thermal fraction, are well described by the Gross-Pitaevskii equation (GPE). Solitons are exact solutions of the onedimensional GPE [17,18], ...
Established techniques for deterministically creating dark solitons in repulsively interacting atomic Bose-Einstein condensates (BECs) can only access a narrow range of soliton velocities. Because velocity affects the stability of individual solitons and the properties of soliton-soliton interactions, this technical limitation has hindered experimental progress. Here we create dark solitons in highly anisotropic cigar-shaped BECs with arbitrary position and velocity by simultaneously engineering the amplitude and phase of the condensate wavefunction, improving upon previous techniques which only explicitly manipulated the condensate phase. The single dark soliton solution present in true 1D systems corresponds to the kink soliton in anisotropic 3D systems and is joined by a host of additional dark solitons including vortex ring and solitonic vortex solutions. We readily create dark solitons with speeds from zero to half the sound speed. The observed soliton oscillation frequency suggests that we imprinted solitonic vortices, which for our cigar-shaped system are the only stable solitons expected for these velocities. Our numerical simulations of 1D BECs show this technique to be equally effective for creating kink solitons when they are stable. We demonstrate the utility of this technique by deterministically colliding dark solitons with domain walls in two-component spinor BECs.
We collect and describe the observed geometrical and dynamical properties of turbulence in quantum fluids, particularly superfluid helium and atomic condensates for which more information about turbulence is available. Considering the spectral features, the temporal decay, and the comparison with relevant turbulent classical flows, we identify three main limiting types of quantum turbulence: Kolmogorov quantum turbulence, Vinen quantum turbulence, and strong quantum turbulence. This classification will be useful to analyse and interpret new results in these and other quantum fluids.
We study the properties of a spin-polarized Fermi gas in a harmonic trap, using the semiclassical Thomas-Fermi approximation. Universal forms for the spatial and momentum distributions are calculated, and the results compared with the corresponding properties of a dilute Bose gas. S1050-29479705306-7 PACS numbers: 03.75.Fi, 05.30.Fk
Low-temperature decaying superfluid turbulence is studied using the nonlinear Schrödinger equation in the geometry of the Taylor-Green (TG) vortex flow with resolutions up to 5123. The rate of (irreversible) kinetic energy transfer in the superfluid TG vortex is found to be comparable to that of the viscous TG vortex. At the moment of maximum dissipation, the energy spectrum of the superflow has an inertial range compatible with Kolmogorov's scaling. Physical-space visualizations show that the vorticity dynamics of the superflow is similar to that of the viscous flow, including vortex reconnection. The implications to experiments in low-temperature helium are discussed.
Laser cooling is a relatively new technique that has led to insights into the behavior of atoms as well as confirming with striking detail some of the fundamental notions of quantum mechanics, such as the condensation predicted by S.N. Bose. This elegant technique, whereby atoms, molecules, and even microscopic beads of glass, are trapped in small regions of free space by beams of light and subsequently moved at will using other beams, provides a useful research tool for the study of individual atoms and clusters of atoms, for investigating the details of chemical reactions, and even for determining the physical properties of individual macromolecules such as synthetic polymers and DNA. Intended for advanced undergraduates and beginning graduate students who have some basic knowledge of optics and quantum mechanics, this text begins with a review of the relevant results of quantum mechanics, it then turns to the electromagnetic interactions involved in slowing and trapping atoms and ions, in both magnetic and optical traps. The concluding chapters discuss a broad range of applications, from atomic clocks and studies of collision processes to diffraction and interference of atomic beams at optical lattices and Bose-Einstein condensation.
The study of ultracold atomic Fermi gases is a rapidly exploding subject, which is defining new directions in condensed matter and atomic physics. Quite generally what makes these gases so important is their remarkable tunability and controllability. Using a Feshbach resonance, one can tune the attractive two-body interactions from weak to strong and thereby make a smooth crossover from a BCS superfluid of Cooper pairs to a Bose–Einstein condensed superfluid. Furthermore, one can tune the population of the two spin states, allowing observation of exotic spin-polarized superfluids, such as the Fulde Ferrell Larkin Ovchinnikov (FFLO) phase. A wide array of powerful characterization tools, which often have direct condensed matter analogs, are available to the experimenter. In this chapter, we present a general review of the status of these Fermi gases with the aim of communicating the excitement and great potential of the field.
Preface 1. The Kortewag-de Vries equation 2. Elementary solutions of the Korteweg-de Vries equation 3. The scattering and inverse scattering problems 4. The initial-value problem for the Korteweg-de Vries equation 5. Further properties of the Korteweg-de Vries equation 6. More general inverse methods 7. The Painleve property, perturbations and numerical methods 8. Epilogue Answers and hints Bibliography and author index Motion picture index Subject index.
We simulate the motion of a massive object through a dilute Bose-Einstein condensate
by numerical solution of the Gross-Pitaevskii equation coupled to an
equation of motion for the object. Under a constant applied force, the
object accelerates up to a maximum velocity where a vortex ring is
formed which slows the object down.
If the applied force is less than a critical value, the object becomes
trapped within the vortex core. We show that the motion follows
the time-independent solutions, and use these solutions
to predict the conditions required for vortex detachment.
Publisher Summary The aim of this chapter is to describe the physical ideas that have been suggested to explain the behavior of helium, which can most easily be related to the properties of the Schriidinger equation. Since the discovery of liquid helium, considerable progress has been made in understanding its behavior from first principles. Some of the properties are more easily understood than others. The most difficult of these concern the resistance to flow above critical velocity. Liquid helium exhibits quantum mechanical properties on a large scale in a manner somewhat differently than do other substances. No other substance remains liquid to a temperature low enough to exhibit the effects. Classically at absolute zero all motion stops, but quantum mechanically this is not so. In fact the most mobile substance known is one at absolute zero, where on the older concepts one should expect hard crystals. Helium stays liquid, as London has shown, because the inter-atomic forces are very weak and the quantum zero point motion is large enough, since the atomic mass is small, to keep it fluid even at absolute zero.
* Introduction * Experimental and Theoretical Background on He II. * Elementary Excitations * Elementary Excitations in He II * Superfulid Behavior: Response to a Transverse Probe. Qualitative Behavior of a Superfluid * Superfluid Flow: Macroscopic Limit * Basis for the Two-Fluid Model * First, Second, and Quasi-Particle sound * Vortex Lines * Microscopic Theory: Uniform Condensate * Microscopic Theory: Non-Uniform Condensate * Conclusion
This paper contains a proof that the description of the phenomenon of Bose-Einstein condensation is the same whether (1) an open system is contemplated and treated on the basis of the grand canonical ensemble, or (2) a closed system is contemplated and treated on the basis of the canonical ensemble without recourse to the method of steepest descents, or (3) a closed system is contemplated and treated on the basis of the canonical ensemble using the method of steepest descents. Contrary to what is usually believed, it is shown that the crucial factor governing the incidence of the condensation phenomenon of a system (open or closed) having an infinity of energy levels is the density of states N(E) En for high quantum numbers, a condition for condensation being n > 0. These results are obtained on the basis of the following assumptions: (i) For large volumes V (a) all energy levels behave like V−θ, and (b) there exists a finite integer M such that it is justifiable to put for the jth energy level Ej= c V−θand to use the continuous spectrum approximation, whenever j ≥ M c θ τ are positive constants, (ii) All results are evaluated in the limit in which the volume of the gas is allowed to tend to infinity, keeping the volume density of particles a finite and non-zero constant. The present paper also serves to coordinate much of previously published work, and corrects a current misconception regarding the method of steepest descents.