Anderson’s considerations on the flow of superfluid helium: Some offshoots

Nearly five decades have elapsed since the seminal 1966 paper of P. W. Anderson on the flow of superfluid helium, ${}^4\mathrm{He}$ at that time. Some of his “considerations”—the role of the quantum phase as a dynamical variable, the interplay between the motion of quantized vortices and potential superflow, its incidence on dissipation in the superfluid and the appearance of critical velocities, the quest for the hydrodynamic analogs of the Josephson effects in helium—and the way they have evolved over the past half century are recounted in this review. But it is due to key advances on the experimental front that phase slippage could be harnessed in the laboratory, leading to a deeper understanding of superflow, vortex nucleation, the various intrinsic and extrinsic dissipation mechanisms in superfluids, macroscopic quantum effects, and the superfluid analog of both ac and dc Josephson effects—pivotal concepts in superfluid physics—have been performed. Some of the experiments that have shed light on the more intimate effect of quantum mechanics on the hydrodynamics of the dense heliums are surveyed, including the nucleation of quantized vortices both by Arrhenius processes and by macroscopic quantum tunneling, the setting up of vortex mills, and superfluid interferometry.

Figure 1

Dispersion curve of the elementary excitations in superfluid ${}^4\mathrm{He}$ as devised by [169].

Figure 2

A 3D view of 2.6 Å diameter hard spheres randomly distributed in a $10\times 10\times 10\AA$ cube with a 3.6 Å mean spacing, as for liquid helium-4 at SVP. In helium, atoms are both strongly confined by hard-core repulsion and dynamically delocalized by the zero-point energy motion. Courtesy of Nelle Varoquaux.

Figure 3

Two different situations for the path of integration in Eq. (17): (a) $\mathrm{\Gamma }$ can be deformed continuously into ${\mathrm{\Gamma }}^{\prime }$; both paths give the same phase difference between point 1 and point 2. (b) Vortex $V$ stands between the two paths; the phase differences along $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\prime }$ differ by $2\pi$.

Figure 4

The geometrical representation of a vortex loop, an arbitrary surface $S$ spanning the loop, with element $d\mathbf{S}$, and the solid angle $\mathrm{\Omega }$ subtended by the loop from point $\mathbf{r}$, the tangent $\widehat{\mathbf{t}}$, normal $\widehat{\mathbf{n}}$, and binormal $\widehat{\mathbf{b}}$ at point $\mathbf{s}$. The flow lines of the vortex velocity field (dashed line) thread surface $S$ where the phase changes determination by $2\pi$. Solid angle $\mathrm{\Omega }$ offers a geometrical representation of the integrand in Eq. (21).

Figure 5

Stream of vortices ${\mathrm{v}}_1,{\mathrm{v}}_2,{\mathrm{v}}_3,\dots$ crossing a path between points 1 and 2, moving from bottom to top. The vortices are represented by dots and are assumed perpendicular to the figure and nearly straight in the vicinity of 1-2. Each contributes angle ${\theta }_i$ to the phase difference recorded between 1 and 2. In its travel from far below to far above line 1-2, each vortex contributes $2\pi$ to the phase difference. According to the ac Josephson effect, a pressure difference develops in the superfluid due to the stream of vortices [see Eq. (38)].

Figure 6

Virtual displacement by $\delta x$ of a small line element $\mathrm{\Delta }l$.

Figure 7

Potential streamlines past the cylindrical vortex core flowing from left to right in thin plain lines. The flow from the vortex, in dashed lines, subtracts from the potential flow in the upper half and adds to it in the lower half, inducing a Bernoulli pressure difference between top and bottom.

Figure 8

The Richards-Anderson cell (1965).

Figure 9

Schematic of the flexible-diaphragm double-hole hydromechanical resonator. The dashed line shows a closed loop threading the two holes (microaperture and long channel) located entirely in the superfluid.

Figure 10

Time chart of the peak resonance amplitude of the resonator membrane, in nm, for positive and for negative excursions. Each dot represents a measurement. The time interval between individual measurements is half a period, 177 ms here. The ticks labeled “in” and “out” indicate whether the jumps occur when the liquid flows in or out of the cell. The drive power applied to drive the resonance is very small ($2.4\times {10}^{-18}\mathrm{W}$). When the peak amplitude is subcritical, its value builds up linearly with time as seen at the left of the chart. The critical events are sharply defined and quite reproducible but occur at a threshold that varies slightly from event to event. The superfluid ${}^4\mathrm{He}$ contained 100 to 300 ppb of ${}^3\mathrm{He}$ and was cooled to about 10 mK under a very small hydrostatic pressure head. From [33].

Figure 11

Critical velocity data vs channel width (⋄) [345]; (∘) and (▵) temperature-dependent and temperature-independent data as compiled by [325]; (◂) [273]; (⋆) [99]; (□) [290] as reanalyzed by [312]. For the temperature-dependent data, the highest value, i.e., that at the lowest temperature, has been retained. The dashed line is obtained from Eq. (45) for the Feynman critical velocity.

Figure 12

Critical velocity, normalized to the zero-temperature linear extrapolation value $v_0$, vs $T$, in kelvin: (⋄) [29], for ultrapure ${}^4\mathrm{He}$; (▵) [359]. The plain curves are computed from the half-ring model (see Sec. 5e) for $a_0=2.2$, 3.2, 4.5, and 6.0 Å and are normalized to match the experimental value at 0.5 K. The inset shows the influence of ${}^3\mathrm{He}$ impurities on $v_{\mathrm{c}}$: (∘) 3 ppb ${}^3\mathrm{He}$ in ${}^4\mathrm{He}$; (▵) 45 ppb, from [322]. Adapted from [321].

Figure 13

Statistical width of the critical velocity transition, normalized to the linear extrapolation limit at $T=0$, $v_0$, in terms of temperature: (⋄) [29]; (▵) [359]; ($\times$) [290]. The plain curves are computed as in Fig. 12 for four values for $a_0$. Adapted from [321].

Figure 14

Probability $p$ vs phase-slip velocity in winding number. The plain curve is a nonlinear least squares fit to the analytic form Eq. (47), which contains two adjustable parameters $v_{\mathrm{c}}$ and $\mathrm{\Delta }{v}_{\mathrm{c}}$. The critical velocity resulting from this distribution of the measured values is defined as the fit value for $p=1/2$. The critical velocity distribution width is given by the slope at $p=1/2$. From [314].

Figure 15

Nucleation rate $\mathit{\Gamma }$ expressed in ${\mathrm{s}}^{-1}$ on a semilogarithmic scale vs slip velocity in winding number in ultrapure ${}^4\mathrm{He}$ at 17.70 mK and saturated vapor pressure. The line is a linear fit to the data. From [314].

Figure 16

Potential well trapping a particle in one dimension. The particle can escape to the continuum of states to the right. The lowest mode at the bottom of the well has angular frequency ${\omega }_0$; ${\omega }_{\mathrm{b}}$ would be the corresponding quantity if the potential was inverted bottom over top. For the simple case of Eq. (51), ${\omega }_0={\omega }_{\mathrm{b}}$. There can exist intermediate energy levels within the well, which are populated according to the Boltzmann factor. Particle escape can take place by quantum tunneling “under” the barrier or by thermal activation “over” the barrier. The intermediate energy levels can be used as ladder rungs by the particle attempting to escape. These processes are embedded in Eq. (55).

Figure 17

Logarithm of the escape rate normalized to the attempt frequency in terms of inverse temperature, also normalized to ${\omega }_0$ for various values of the damping parameter $\alpha$ ([110]). The dotted line is a hand sketch of the situation where $\alpha$ increases with temperature, starting from zero at $T=0$ (see Sec. 5c). From [314].

Figure 18

The barrier energy $E_{\mathrm{b}}$ in kelvin vs $v$ the mean velocity in the aperture expressed in phase winding numbers and obtained from the nucleation rate data of [314]: (▪) low-temperature data transformed using the numerical tables of [110]; (⋄) high-temperature data. From [314].

Figure 19

Arrhenius plot $\ln \mathrm{\Gamma }(v)$ vs $1/T$, $\mathrm{\Gamma }$ being expressed in ${\mathrm{s}}^{-1}$ and $T$ in kelvin, as measured at varying $T$ and $v_{\mathrm{c}}$ for two ultrapure ${}^4\mathrm{He}$ samples (⋄) and (▵); (*) and (▴) after correction for the change of the velocity with $T$. The solid and the dotted curves are calculated from the experimental barrier energy $E_{\mathrm{b}}$ represented in Fig. 18. In the inset, $\ln \mathrm{\Gamma }(v_{\mathrm{q}})$ (▪) has been obtained with smoothed values of $v_{\mathrm{c}}$. The curves represent the calculated values of $\ln \mathrm{\Gamma }(v_{\mathrm{q}})$ with $\alpha =0$ (dashed) or varying with $T$ (solid). The latter gives a better representation of the data and illustrates the influence of damping depicted by the dotted line in Fig. 17. From [314].

Figure 20

Energy barrier $E_{\mathrm{v}}$ normalized by ${\rho }_{\mathrm{s}}{\kappa }_4^2{a}_0$ and expressed in kelvin vs the vortex radius $R/a_0$ for the vortex half-ring model. Illustration of various potential forms given by Eq. (59) at various superfluid flow velocities taken from the data in Fig. 12 at, from top down, $T\to T_{\lambda }$, 1 K, 0.5 K, and at the quantum crossover $T_{\mathrm{q}}$. The lowest curve illustrates the situation in which the potential barrier vanishes and the trapped particle runs away, the so-called lability point. Adapted from [29].

Figure 21

Schematic views in 2D (left) and 3D (right) of the vortex half-ring trajectory over a pointlike pinhole punched in an infinite horizontal plane. The dashed lines on the 2D plot are the potential flow streamlines that emerge from the pinhole. The 3D view shows the vortex half ring being first pushed by the potential flow to the right and then flying back over the pinhole to finally drift away to the upper left. Adapted from [310].

Figure 22

Operation of a helical vortex mill: (a) Spiral-helix configuration of a streamwise vortex filament pinned at the center of the bottom left section of the channel. The helical vortex self-velocity opposes the superflow while the vortex filament grows in size and spirals on itself. (b) End-on view of subsequent reconnections. The vortex is pinned near the upper left of the channel cut view. The outwardly growing spiral sporadically reconnects to the wall and releases a new line segment, which then moves to the lower right. These numerical simulations are carried out in the presence of a sizable mutual friction with the normal component in order to stabilize the numerical algorithms, which is why the generated half rings rapidly decrease in size. From [267].

Figure 23

Absolute peak amplitude $A_{\mathrm{p}}$ normalized to the amplitude jump of a single slip $\mathrm{\Delta }{A}_1$ during successive half cycles of the resonance plotted as a function of half-cycle index (time runs from left to right) at 100.7 mK (top) and 201.5 mK (bottom) in nominal purity helium at SVP. The expanded traces at the very top and bottom of the graph show the slip sizes in signed winding numbers (according to flow direction, in and out of the resonator chamber). The numerous multiple slips seen on these charts occur at or close to the critical velocity for single slips. From [314].

Figure 24

Absolute peak amplitudes at successive half cycles of the resonator motion, normalized as in Fig. 23, vs half-cycle index in a ${}^4\mathrm{He}$ sample containing 100 ppb of ${}^3\mathrm{He}$ impurity, at 24.0 bars and 12.5 mK. The resonance half period is 31.8 ms. The top trace shows a succession of amplitude drops which correspond, for its main part, to phase slips by $2\pi$ of opposite sign, with occasional larger slips—the multiple slips of Fig. 23. The large feature around the 1000th half cycle is a “singular” collapse, as defined in the text. The inset shows the details of this collapse, (•) being for positive peaks, (∘) for negative peaks. It is preceded by a slip by $-2(\times 2\pi )$ and followed by a slow recovery of the peak amplitude caused by the applied drive, punctuated by single and double slips. The time-resolved evolution of this collapse has actually been tracked. From [26].

Figure 25

Mean size of (positive) multiple slips vs velocity in phase winding number in nominal purity ${}^4\mathrm{He}$ (100 ppb ${}^3\mathrm{He}$): (▵) pressure sweep from 0.4 to 24 bars at 81.5 mK (for all even values of the pressure $P$, and 0.4, 1, 3, 5, and 7 bars); (⋄) temperature sweep at 16 bars; (∘) temperature sweep at 24 bars; (*) drive level sweep at 24 bars, 81.5 mK; (▪) temperature sweep at 0 bar. Lines connect successive data points in the temperature and pressure sweeps. For the temperature sweeps, from 14 to 200 mK approximately, $v$ first increases when the ${}^3\mathrm{He}$ impurities evaporate from the vortex core, reaches the quantum plateau, and then decreases, following the same pattern as shown in the inset of Fig. 12 for $v_{\mathrm{c}}$. Adapted from [321].

Figure 26

Schematic representation of a two-dimensional weak link: (top) cut view of the elongated slit in the partitioning wall; (bottom) amplitude of the macroscopic wave function $\mathrm{\Phi }$ in the region in which it is depleted by the energy barrier of height $U$ and extent $l_{\mathrm{b}}$ due to the constricting walls.

Figure 27

Staircase patterns (peak resonance amplitude vs applied drive level, both in arbitrary units) in ${}^3\mathrm{He}\text{−}\mathit{B}$ at a pressure of about 0.2 bar. The curves at various temperatures are shifted vertically for readability. Drive frequencies at the various temperatures are shifted from resonance so as to set comparable resonance conditions. From [37].

Figure 28

Current-phase relations in ${}^3\mathrm{He}\text{−}\mathit{B}$ observed in an array weak link. CPR’s for the low current state (left panel) and the high current state (right panel) for temperatures ranging from $0.850T/T_{\mathrm{c}}$ down to $0.450T/T_{\mathrm{c}}$ in steps of approximately $0.05T/T_{\mathrm{c}}$. The mass current through the weak link increases as the temperature is lowered. At temperatures close to $T_{\mathrm{c}}$, the current-phase relations can be fitted well with the Deaver-Pierce model. As the temperature decreases (and the critical current increases), this model becomes inadequate as a $\pi$-periodic component gradually sets in. From [198].

Figure 29

Staircase patterns in the $A$ phase at 28.4 bars and $T=0.92T_{\mathrm{c}}$ ($T_{\mathrm{c}}=2.417\mathrm{mK}$). The left and right panels show the patterns slightly above and below the resonator resonance at ${\omega }_{\mathrm{m}}=1.783\mathrm{Hz}$, as observed (upper curves) and as computed (lower curve) with the Deaver-Pierce model. From [38].

Figure 30

Staircase patterns in the $A$ and $B$ phases at nearly identical frequencies and temperature. The horizontal ticks mark the periodicity of the staircase pattern (top) and of the low-level structure (bottom). This low-level structure has been attributed to solid friction of unknown origin somewhere in the resonator (but may also arise from subgap energy levels). From [38].

Figure 31

Staircase patterns in ${}^4\mathrm{He}$ close to $T_{\lambda }$ (amplitude in picometers vs drive, in arbitrary units for (i) $T_{\lambda }-T=61\mu \mathrm{K}$, (ii) $154\mu \mathrm{K}$, and (iii) 3.72 mK. The plots are shifted vertically by 2 pm with respect to one another for clarity. The data for (iii) has been divided by 20 along the $x$ axis and by 10 along the $y$ -axis. As in the case of ${}^3\mathrm{He}\mathit{\text{-B}}$, shown in Fig. 27, the height of the first step corresponds to the critical current $J_{\mathrm{c}}$. Each subsequent step corresponds to an additional phase difference of $2\pi$. From [294].

Figure 32

Superfluid interferences in a two-aperture resonator. (a) The hydromechanical resonator. The unshaded regions are filled with superfluid ${}^4\mathrm{He}$. The flexible metallized diaphragm at the top of the upper chamber serves both as a microphone to detect the resonant oscillations and as a pressure pump to drive the flow across the two aperture arrays indicated by crosses. These arrays interrupt the superfluid channel enclosing the sense area. (b) The resonator equivalent circuit showing the analogy with the electrodynamic dc SQUID. There are two superposed currents flowing through the weak links, one corresponding to the rotation flux picked up by the sensing loop, Eq. (76), the other being the common-mode readout current from the flexible diaphragm. Bottom panel: Peak amplitude of the diaphragm displacement on resonance as a function of the rotation flux $2\mathbf{\Omega }·\mathbf{A}$ picked up by the superfluid loop enclosing the sense area; see Sec. 8a for a description of these interferometer operations based on the Sagnac effect. The measured data are shown by the symbols; the solid lines are fits to the data of the equation modeling the resonator motion as described by [138]. The modulation curves were taken at temperatures $T_{\lambda }-T=12$, 7.0, 4.0, 3.0, 2.0, 1.5, 0.9, 0.6, 0.4, and 0.3 mK from top to bottom. This temperature span covers the coherent Josephson regime in the array discussed in Sec. 7f. From [138].

Figure 33

Schematics of the optical Sagnac interferometer. A collimated beam from the source enters the interferometer through a beam splitter and is separated into two beams traveling along the optical path defined by the three mirrors in opposite directions, as indicated by the arrows. Reentering the beam splitter, they recombine and produce a fringe pattern on the detector plate.

Figure 34

Photograph and schematic view of the cell of [32], approximately to scale for the inner parts, except for the loop, which is made of two turns of 0.4 mm internal diameter capillary (only one turn is shown) with total area $5.90\pm 0.10{\mathrm{cm}}^2$. The lower chamber of the resonator is a cylindrical duct, 1 mm in diameter, and connects the weak link to the flexible diaphragm (at the bottom) and to one end of the pickup loop (on the side); the upper chamber, a squat cylinder, is connected to the other end of the loop and to an inlet toward the main superfluid bath in which the resonator is immersed. Two pairs of coils produce fields parallel and perpendicular to the flow through the weak link to locally control the order parameter texture in ${}^3\mathrm{He}$. The cryostat is rotated about the vertical axis $z$ by angle $\beta$ from the north, shown by a compass needle; $\lambda$ is the latitude, $48°{43}^{\prime }$ at Saclay. This cell has been used to detect the rotation of the Earth from a “blind” laboratory with a sensitivity of $5\times {10}^{-3}{\mathrm{\Omega }}_{\oplus }{\text{Hz}}^{-1/2}$.