Bose-Einstein Condensation Measurements and Superflow in Condensed Helium

We review the formulation and measurement of Bose-Einstein condensation (BEC) in liquid and solid helium. BEC is defined for a Bose gas and subsequently for interacting systems via the one-body density matrix (OBDM) valid for both uniform and non-uniform systems. The connection between the phase coherence created by BEC and superflow is made. Recent measurements show that the condensate fraction in liquid ⁴He drops from 7.25±0.75 % at saturated vapor pressure (p≈0) to 2.8±0.2 % at pressure p=24 bars near the solidification pressure (p=25.3 bar). Extrapolation to solid densities suggests a condensate fraction in the solid of 1 % or less, assuming a frozen liquid structure such as an amorphous solid. Measurements in the crystalline solid have not been able to detect a condensate with an upper limit set at n ₀≤0.3 %. Opportunities to observe BEC directly in liquid ⁴He confined in porous media, where BEC is localized to patches by disorder, and in amorphous solid helium is discussed.     

Fluctuations and BEC at the Free Surface of 4He

There is some theoretical evidence that full Bose–Einstein condensation (BEC) is achieved at the free surface of superfluid ⁴ He where the density is small. We present new variational results of the BEC and of the density fluctuation spectrum in the surface region of a slab of ⁴ He. We use both a standard wave function (wf) with bulk correlations and one body shape terms and a novel shadow wave function with a glue term (G-SWF). This last one describes the self- binding of ⁴ He only via interparticle correlations. In both cases we find that BEC increases from the bulk like value well inside the slab to a much larger value in the surface region but a striking different behavior is found in the low density region. With a standard wf the BEC reaches essentially 100% in the surface region. With the G-SWF the condensate only increases up to about 51%. Further out of the surface the condensate decreases and correspondingly there is an enhanced population of small momentum states. This different behavior is correlated with the presence in the G-SWF of enhanced density fluctuations in the surface region due to the zero point motion of ripplons. This enhancement is absent in the case of a standard wf.     

Thermodynamics of a Trapped Bose Condensate with Negative Scattering Length

We study the Bose–Einstein condensation (BEC) for a system of ⁷Li atoms, which have negative scattering length (attractive interaction), confined in a harmonic potential. Within the Bogoliubov and Popov approximations, we numerically calculate the density profile for both condensate and non-condensate fractions and the spectrum of elementary excitations. In particular, we analyze the temperature and number-of-boson dependence of these quantities and evaluate the BEC transition temperature T _BEC. We calculate the loss rate for inelastic two- and three-body collisions. We find that the total loss rate is strongly dependent on the density profile of the condensate, but this density profile does not appreciably change by increasing the thermal fraction. Moreover, we study, using the quasi-classical Popov approximation, the temperature dependence of the critical number N _c of condensed bosons, for which there is the collapse of the condensate. There are different regimes as a function of the total number N of atoms. For N<N _c the condensate is always metastable but for N>N _c the condensate is metastable only for temperatures that exceed a critical value T _c.     

The Quest for Bose-Einstein Condensation in Solid 4He

Ever since the seminal torsional oscillator (TO) measurements of Kim and Chan which suggested the existence of a phase transition in solid ⁴He, from normal to a ??supersolid?? state below a critical temperature T _c = 200 mK, there has been an unprecedented amount of excitement and research activity aimed at better understanding this phase. Despite much work, this remarkable phase has yet to be independently confirmed by conventional scattering techniques, such as neutron scattering. We have carried out a series of neutron scattering measurements, which we here review, aimed at observing Bose-Einstein condensation (BEC) in solid ⁴He at temperatures below T _c . In bulk liquid ⁴He, the appearance of BEC below T _?? signals the onset of superfluidity. The observation of a condensate fraction in the solid would provide an unambiguous confirmation for ??supersolidity??. Although, our measurements have not yet revealed a non-zero condensate fraction or algebraic off diagonal long-range order n ₀ in solid ⁴He down to 65 mK, i.e. n ₀=(0±0.3)%, our search for BEC and its corollaries continues with improved instrumentation.     

Generation of Vortices and Observation of Quantum Turbulence in an Oscillating Bose-Einstein Condensate

We report on the experimental observation of vortex formation and production of tangled vortex distribution in an atomic BEC of ⁸⁷Rb atoms submitted to an external oscillatory perturbation. The oscillatory perturbations start by exciting quadrupolar and scissors modes of the condensate. Then regular vortices are observed finally evolving to a vortex tangle configuration. The vortex tangle is a signature of the presence of a turbulent regime in the cloud. We also show that this turbulent cloud has suppression of the aspect ratio inversion typically observed in quantum degenerate bosonic gases during free expansion.     

On the Structure of the Superfluid State and Quasiparticle Spectrum in a Bose Liquid with a Suppressed Bose–Einstein Condensate

A self-consistent model of the superfluid (SF) state of a Bose liquid with strong interaction between bosons and a weak single-particle Bose–Einstein condensate (BEC) is considered. The ratio of the BEC density n ₀ to the total particle density n of the Bose liquid is used as a small parameter of the model, n ₀/n?1, unlike in the Bogolyubov theory of a quasi-ideal Bose gas, in which the small parameter is the ratio of the number of supracondensate excitations to the number of particles in an intensive BEC, (n?n ₀)/n ₀?1. A closed system of nonlinear integral equations for the normal ~Σ₁₁(p, ω) and anomalous ~Σ₁₂(p, ω) self-energy parts is obtained with account for terms of first order in the BEC density. A renormalized perturbation theory is used, which is built on combined hydrodynamic (at p→0) and field (at p≠0) variables with analytic functions ~Σ _ij (p, ε) at pε0 and ε→0 and a nonzero SF order parameter ~Σ₁₂(0, 0)≠0, proportional to the density ρ _s of the SF component. Various pair interaction potentials U(r) with inflection points in the radial dependence and with an oscillating sign-changing momentum dependence of the Fourier component V(p) are considered. Collective many-body effects of renormalization (“screening”) of the initial interaction, which are described by the bosonic polarization operator Π(p, ω), lead to a suppression of the repulsion [V(p)<0] and an enhancement of the effective attraction [V(p)<0] in the respective domains of nonzero momentum transfer, due to the negative sign of the real part of Π(p, ω) on the “mass shell” ω=E(p). In the framework of the “soft spheres” model with the single fitting parameter—the value of the repulsion potential at r=0—the quasiparticle spectrum E(p) is calculated, which is in good accordance with the experimental spectrum E _exp(p) of elementary excitations in superfluid ⁴He. It is shown that the roton minimum in the quasiparticle spectrum is directly associated with the first negative minimum of the Fourier component of the renormalized (“screened”) potential of pair interaction between bosons.     

Bose-Einstein Condensation in Liquid 4He in Vycor

We present neutron scattering measurements of Bose-Einstein condensation (BEC) in liquid ⁴He confined in Vycor. The data show clear evidence of a condensate in Vycor with a condensate fraction comparable to that of bulk superfluid ⁴He, approximately 7.5% at low temperature and SVP. The temperature dependence of n₀(T) is also similar to that in the bulk with critical temperature for BEC, T_BEC, in the range 1.80BEC<2.05 K. The data are not accurate enough to show whether T_BEC for Vycor is the same or greater than the depressed critical temperature for superfluidity, T_c=1.95 K.     

Bose-Einstein Condensation and Superfluidity of Strongly Correlated Bose Fluid in a Random Potential

No Heading It was recently observed that liquid ⁴He confined in porous glass lost its superfluidity at high pressures and very low temperatures, causing the quantum phase transition by the strong correlation. Motivated by this experiment, we study the behavior of a strongly correlated Bose fluid in a disordered environment, such as the Bose-Einstein condensate (BEC) and superfluid critical temperatures, by using a model of 3-dimensional Bose fluid in a random potential. For the perturbation of the repulsive interaction between particles, we introduced two-loop renormalized self energy by the self-consistent calculation, which is effective near the BEC critical temperature. Calculating the second order perturbation with respect to the random potential, we found that the BEC disappears at high densities, which is qualitatively consistent with the experimental results.PACS numbers: 67.40 –w, 05.30 Jp, 64.60 Cn     

Vacancies in Solid 4He and Bose Einstein Condensation

We find that a finite concentration of vacancies induces Bose Einstein condensation (BEC) of the atoms in solid ⁴ He at density close to the T=0 K melting where vacancies are delocalized. No BEC is present in the perfect crystal and in the defected solid at higher densities. These results are based on variational Monte Carlo method with shadow wave function which allows for relaxation and delocalization of vacancies. Extension of the theory at finite temperature leads to several possibilities. Depending on presence or not of ground state vacancies and on parameters like energy of formation and effective mass of a vacancy one can have a normal transition with BEC at low temperature or an inverted transition with BEC at high T. ⁴ He on graphite appears to be a favorable candidate for a state with BEC and spatial order.     

Localization of Bose–Einstein Condensation by Disorder

Recent experiments suggest that Bose–Einstein condensation (BEC) in liquid⁴ can be localized when the liquid is confined in porous media. We demonstrate in a simple model of hard core bosons using Monte Carlo that the condensate can be separated into two parts. The two regions of condensate are separated by a region of uncondensed fluid that forms in response to a local attractive external potential. The aim is to illustrate that separated condensates, and therefore localized BEC, can be created in porous media.     

Freely decaying Turbulence and Bose–Einstein Condensation in Gross–Pitaevski Model

We study turbulence and Bose–Einstein condensation (BEC) within the two-dimensional Gross–Pitaevski (GP) model. In the present work, we compute decaying GP turbulence in order to establish whether BEC can occur without forcing and if there is an intensity threshold for this process. We use the wavenumber–frequency plots which allow us to clearly separate the condensate and the wave components and, therefore, to conclude if BEC is present. We observe that BEC in such a system happens even for very weakly nonlinear initial conditions without any visible threshold. BEC arises via a growing phase coherence due to anihilation of phase defects/vortices. We study this process by tracking of propagating vortex pairs. The pairs loose momentum by scattering the background sound, which results in gradual decrease of the distance between the vortices. Occasionally, vortex pairs collide with a third vortex thereby emitting sound, which can lead to more sudden shrinking of the pairs. After the vortex anihilation the pulse propagates further as a dark soliton, and it eventually bursts creating a shock.     

Moment of Inertia Just Above λ Point

The moment of inertia of a quantum liquid of a bulk sample is examined just above λ point. Although the macroscopic condensate does not yet appear above  T _λ , under the strong influence of Bose statistics, the coherent many-body wave function grows to an intermediate size between a macroscopic and a microscopic one as a thermal equilibrium state, which must affect rotational properties of a liquid. Beginning with the bosons without the condensate, we make a perturbation calculation of its susceptibility with respect to the repulsive interaction. By taking peculiar diagrams obeying Bose statistics, we examine how, with decreasing temperature, the growth of the coherent wave function gradually changes the rotational behavior of a liquid, with a result that the moment of inertia slightly decreases prior to T _λ . This means that at the vicinity of T _λ the superfluid density defined in the mechanical phenomena does not always agree with that defined in thermodynamics. Its implication to the recent discovery of nonclassical moment of inertia in a solid helium 4 is discussed.     

Ground-State Properties and Collective Excitations in a 2D Bose-Einstein Condensate with Gravity-Like Interatomic Attraction

We study the ground-state properties of a Bose-Einstein condensate (BEC) with the short-range repulsion and gravitylike 1/r interatomic attraction in two-dimensions (2D). Using the variational approach, we obtain the ground-state energy and show that the condensate is stable for all interaction strenghts in 2D. We also determine the collective excitations at zero temperature using the time-dependent variational method. We analyze the properties of the Thomas-Fermi-gravity (TF-G) and gravity (G) regimes.      

Splitting of a Quadruply Quantized Vortex in the Rb Bose-Einstein Condensate

We observed the sudden deformation of a quadruply quantized vortex into a linear shape in a ⁸⁷Rb (F=2,m _F =2) Bose-Einstein condensate (BEC). Multiply quantized vortex is predicted to split into singly quantized vortices and the observed deformation is considered to be the onset of the splitting. The displacement of the vortex with respect to the center of the BEC would induce the observed splitting. Our theoretical simulation qualitatively supports these arguments.     

Self-similar Expansion of the Density Profile in a Turbulent Bose-Einstein Condensate

In a recent study we demonstrated the emergence of turbulence in a trapped Bose-Einstein condensate of ⁸⁷Rb atoms. An intriguing observation in such a system is the behavior of the turbulent cloud during free expansion. The aspect ratio of the cloud size does not change in the way one would expect for an ordinary non-rotating (vortex-free) condensate. Here we show that the anomalous expansion can be understood, at least qualitatively, in terms of the presence of vorticity distributed throughout the cloud, effectively counteracting the usual reversal of the aspect ratio seen in free time-of-flight expansion of non-rotating condensates.     

Condensate Fraction in Liquid 4He at Zero Temperature

We present results of the one-body density matrix ρ ₁(r) and the condensate fraction n ₀ of liquid ⁴He calculated at zero temperature by means of the Path Integral Ground State Monte Carlo method. This technique allows to generate a highly accurate approximation for the ground state wave function Ψ ₀ in a totally model-independent way, that depends only on the Hamiltonian of the system and on the symmetry properties of Ψ ₀. With this unbiased estimation of ρ ₁(r), we obtain precise results for the condensate fraction n ₀ and the kinetic energy K of the system. The dependence of n ₀ with the pressure shows an excellent agreement of our results with recent experimental measurements. Above the melting pressure, overpressurized liquid ⁴He shows a small condensate fraction that has dropped to 0.8% at the highest pressure of p=87?bar.     

Zero-Point Vacancies in Quantum Solids

A Jastrow wave function (JWF) and a shadow wave function (SWF) describe a quantum solid with Bose–Einstein condensate; i.e. a supersolid. It is known that both JWF and SWF describe a quantum solid with also a finite equilibrium concentration of vacancies x _v . We outline a route for estimating x _v by exploiting the existing formal equivalence between the absolute square of the ground state wave function and the Boltzmann weight of a classical solid. We compute x _v for the quantum solids described by JWF and SWF employing very accurate numerical techniques. For JWF we find a very small value for the zero point vacancy concentration, x _v =(1.4±0.1)×10^?6. For SWF, which presently gives the best variational energy of solid ⁴He, we find the significantly larger value x _v =(1.4±0.1)×10^?3 at a density close to melting. We also study two and three vacancies with SWF. We find that there is a strong short range attraction but the vacancies do not form a bound state, at variance with the exact finite temperature PIMC results.     

Experimental study on time evolution of Marangoni flow instability in molten silicon bridge

Temperature oscillations due to Marangoni flow instability in molten silicon half zone bridges with various aspect ratios of As = 0.5–2.0 were measured using six thermocouples set azimuthally 60^° apart in the liquid bridge close to the cold rod. The Marangoni number was estimated to range from 3000 to 14000, based on the measured axial temperature difference. Fourier spectra of the temperature oscillations were broad and continuous; each peak was not clearly distinguished but rather appeared as a frequency band. Thus, the convection was estimated to be turbulent-like. The time evolution of the azimuthal wave number was observed by analyzing the time-dependence of the phase relationship of the temperature oscillation detected by the six thermocouples. Analyzing the mode appearance coefficient MAC as a function of the aspect ratio, the relationship between the azimuthal mode number m and the aspect ratio As was observed to be m ? As ≈ 2.4; the basic structure of flow instability is sustained even under high Marangoni number. The temperature oscillation data was decomposed into that for each frequency band by using wavelet analysis. The frequencies for the m = 1 and m = 3 modes were estimated to be 0.08 to 0.2 Hz and 0.01 to 0.2 Hz, respectively.     

Separation characteristics of fluid flow inside two parallel plates with wavy surface

Separation characteristics of fluid flow inside two parallel wavy plates for steady-laminar flow is investigated numerically in the present study. Governing equations are discretized using control volume based finite-volume method with collocated variable arrangement. SIMPLE algorithm is used and SIP solver is applied for solution of system of equations. Effect of surface waviness (defined by amplitude to average interwall spacing ratio, a/H) and aspect ratio (defined by wavelength to average interwall spacing ratio, w/H) on separation characteristics of fluid flow is presented. The present work has been carried out for surface waviness a/H=0-0.3, aspect ratio w/H=1.5-2.25. A critical Reynolds number (Re_c) is used to identify the appearance of first separation of fluid flow in the channel. Critical Reynolds (Re_c) number is calculated for wide range of surface waviness and aspect ratio. The structure of separation bubble depends strongly on waviness of the surface and aspect ratio for a particular Reynolds number and changes little with wave number (n). Finally pressure drop characteristics is presented in terms of average friction factor as a function of Reynolds number.