Dissipation in mesoscale superfluids

TL;DR

This paper analyzes the maximum flow speed of superfluids through narrow channels, focusing on dissipation mechanisms like vortex nucleation, and presents a universal framework for different superfluid systems.

Contribution

It introduces a universal approach to understanding critical velocities in superfluids considering vortex nucleation and phase slips across various conditions.

Findings

Critical velocity decreases sharply in ultra-narrow channels.

Universal scaling laws for superfluid flow across different systems.

Thermal vortex nucleation limits superfluid flow at small scales.

Abstract

We investigate the maximum speed at which a driven superfluid can flow through a narrow constriction with a size on the order of the healing length. Considering dissipation via the thermal nucleation of quantized vortices, we calculate the critical velocity for superfluid $^4$He and ultracold atomtronic circuits, identify fundamental length and velocity scales, and are thus able to present results obtained in widely different temperature and density ranges in a universal framework. For ultra-narrow channels we predict a drastic reduction in the critical velocity as the energy barrier for flow reducing thermally activated phase slip fluctuations is suppressed.

Tables

Table 1. Table 1: Critical velocity and related velocity, length and temperature scales extracted from four neutral bosonic mass flow experiments in ultracold atomic and molecular condensates.

Reference	System	$v_c$ ($\mathrm{mm}\text{/}s$)	$v_0$ ($\mathrm{mm}\text{/}s$)	$a$ ($\mu m\text{/}$)	${\xi }_0$ ($\mu m\text{/}$)	$T/T_c$
Neely et al. [Neely et al., 2010]	87Rb	0.2	1.2	47	0.3	0.6
Ramanathan et al. [Ramanathan et al., 2011]	23Na	0.9	0.7	7	0.51	0.2
Raman et al. [Raman et al., 1999]	23Na	1.6	4.6	15	0.3	0.8
Weimer et al. [Weimer et al., 2015]	6Li (molecule)	1.7	3.1	10	0.85	0.5

Equations

$$E_{\mathrm{tension}}=\frac{\mathcal{L}}{4\pi }{\rho }_s{\kappa }^2\left[\ln \frac{\ell }{\xi }+\alpha \right]$$

$$E_{\mathrm{flow}}=\kappa {\rho }_s\iint {\mathbf{v}}_s\cdot d\mathbf{S},$$

$${\beta }_c{E}_{\mathrm{line}}(x)$$

$$-\frac{v_J}{v_0}{\left(\frac{a}{\xi }\right)}^2\frac{{\pi }^2\xi }{2{\xi }_0}\left(1-\frac{x}{a}\right).$$

$${\beta }_c{E}_{\mathrm{ring}}(R)$$

$$-\frac{v_J}{v_0}{\left(\frac{a}{\xi }\right)}^2\frac{{\pi }^2\xi }{{\xi }_0}\left[1-\sqrt{1-{\left(\frac{R}{a}\right)}^2}\right]$$

$$\Gamma ={\Gamma }_0\left[{\mathrm{e}}^{-\beta E_{\max }(+v_J)}-{\mathrm{e}}^{-\beta E_{\max }(-v_J)}\right],$$

$$\Gamma_{0}=\frac{1}{\tau_{GL}}\frac{La}{\xi^{2}}\left\{\begin{array}[]{cl}\frac{\pi a}{\xi}&{\rm vortex\ ring}\\ 2\pi&{\rm vortex\ line}\end{array}\right.,$$

$$\frac{R^{\ast }}{\xi }$$

$$\frac{v_c}{v_0}$$

$${\beta }_c{E}_{\mathrm{line}}(x=0)=2\pi \frac{a}{\xi }\left(\ln \frac{a}{\xi }+\alpha \right)-\frac{{\pi }^2\xi }{2{\xi }_0}\frac{v_s}{v_0}{\left(\frac{a}{\xi }\right)}^2,$$

$$\frac{v_c}{v_0}$$

$$\times \exp \left[\frac{2\pi a}{\xi }\frac{T_c}{T}\left(\ln \frac{a}{\xi }+\alpha \right)\right]\}.$$

$${\beta }_c{E}_{1d}=\pi {\left(\frac{a}{\xi }\right)}^2\left[\frac{4}{3\sqrt{2}}-\pi \frac{\xi }{{\xi }_0}\frac{v_s}{v_0}\right].$$

$${\mathbf{v}}_s=v_J\hat{z}=\frac{J}{\pi a^2{\rho }_s}\hat{z}$$

$$x$$

$$y$$

$$z$$

$${\mathbf{v}}_s=\frac{v_J}{2}\frac{1}{\sqrt{({\zeta }^2+{\eta }^2)({\zeta }^2+1)}}\hat{\zeta }$$

$${\mathbf{v}}_s(r,z=0)=\frac{v_J}{2\sqrt{1-r^2/a^2}}\hat{z}$$

$$E_{\mathrm{line}}(x)$$

$$-\kappa {\rho }_s{v}_J\left[a^2{\cos }^{-1}\left(\frac{x}{a}\right)-x\sqrt{a^2-x^2}\right].$$

$$\xi (T)=\frac{4{\pi }^2{k}_{\mathrm{B}}{T}_c}{{\kappa }^2{\rho }_s(T)}$$

$${\beta }_c{E}_{\mathrm{line}}(x)$$

$$-\frac{v_J}{v_0}{\left(\frac{a}{\xi }\right)}^2\frac{\pi \xi }{{\xi }_0}\left[{\cos }^{-1}\left(\frac{x}{a}\right)-\frac{x}{a}\sqrt{1-{\left(\frac{x}{a}\right)}^2}\right]$$

$$E_{\mathrm{ring}}(R)=\frac{1}{2}{\kappa }^2{\rho }_{s}R\left(\ln \frac{R}{\xi }+\alpha \right)-\kappa {\rho }_s\pi R^2{v}_J$$

$${\beta }_c{E}_{\mathrm{ring}}(R)=2{\pi }^2\frac{R}{\xi }\left(\ln \frac{R}{\xi }+\alpha \right)-\frac{v_J}{v_0}{\left(\frac{R}{\xi }\right)}^2\frac{{\pi }^2\xi }{{\xi }_0}.$$

$$E_{\mathrm{flow},\mathrm{line}}(x)$$

$${\beta }_c{E}_{\mathrm{line}}(x)$$

$$-\frac{v_J}{v_0}{\left(\frac{a}{\xi }\right)}^2\frac{{\pi }^2\xi }{2{\xi }_0}\left(1-\frac{x}{a}\right).$$

$$E_{\mathrm{flow},\mathrm{ring}}(R)=\kappa {\rho }_s{v}_J\pi a^2\left[1-\sqrt{1-{\left(\frac{R}{a}\right)}^2}\right]$$

$${\beta }_c{E}_{\mathrm{ring}}(R)$$

$$-\frac{v_J}{v_0}{\left(\frac{a}{\xi }\right)}^2\frac{{\pi }^2\xi }{{\xi }_0}\left[1-\sqrt{1-{\left(\frac{R}{a}\right)}^2}\right].$$

$${\rho }_s(t)=\frac{16{\pi }^{3}mn}{{\left[\zeta \left(\frac{3}{2}\right)\right]}^{4/3}}(n{a}^3{)}^{1/3}{\left(1-t\right)}^2{f}_s\left(t,n{a}^3\right)$$

$$\frac{{\xi }_0}{{\xi }_h}=\mathcal{A}{\left(n{a}^3\right)}^{(2\nu -1)/6}$$

Full text

The flow of dissipationless atomic supercurrents in neutral superfluids is one of the most dramatic manifestations of macroscopic quantum coherence Anderson (1966); Varoquaux (2015); Chien et al. (2015), with applications to matter wave interferometry Sato and Packard (2012); Schumm et al. (2005); Chiow et al. (2011). Recently, there has been increased interest in dimensionally confined superfluids, due to progress in manufacturing nanoscale channels and fountain effect devices for studying the flow of superfluid helium Hoskinson et al. (2005a, b); Cheng et al. (2015); Cheng and Beamish (2016); Vekhov et al. (2014); Vekhov and Hallock (2012); Ray and Hallock (2010, 2008); Boninsegni et al. (2007); Savard et al. (2011, 2009); Duc et al. (2015); Velasco et al. (2012, 2014); Botimer and Taborek (2016); Fil and Shevchenko (2009); Del Maestro et al. (2011); Pollet and Kuklov (2014) and the availability of trapped non-equilibrium atomic Bose-Einstein condensates Raman et al. (1999); Watanabe et al. (2009); Neely et al. (2010); Ramanathan et al. (2011); Lee et al. (2013); Wright et al. (2013a); Murray et al. (2013); Wright et al. (2013b); Eckel et al. (2014); Jendrzejewski et al. (2014); Mathey et al. (2014); Weimer et al. (2015); Husmann et al. (2015); Singh et al. (2016); Li et al. (2016). Common to these experiments in vastly different density and interaction regimes is an observed increase in dissipation for highly confined systems.

In general, superflow is possible at speeds less than a superfluid critical velocity set by the Landau criterion $v_{c}\leq\min\varepsilon(p)/p$ below which there are no accessible excitations $\varepsilon(p)$ with momentum $p$ Landau (1941). Among the different types of excitations in superfluids, quantized vortices Anderson (1966); Iordanskii (1965); Feynman (1955); Langer and Fisher (1967); Volovik (1972) give rise to the smallest $v_{c}$. For flow through a cylindrical channel, if the total kinetic energy is converted into vortex rings with the size $a$ of the constriction, the Landau criterion predicts a critical velocity $v_{c,F}\sim(\kappa/a)\ln(a/\xi_{0})$ Feynman (1955) where $\kappa=h/m$ is the quantum of circulation for condensed bosons of mass $m$ and $\xi_{0}$ is a characteristic length scale of the superfluid. This prediction (due to Feynman) has been born out by nearly a half-century worth of superfluid massflow observations with temperature independent critical velocities Varoquaux (2015); Harrison and Mendelssohn (1974). However, it must ultimately break down as the constriction radius approaches $\xi_{0}$. Moreover, any observed temperature dependence of $v_{c}$ can only be described by the existence of an energy activation barrier for the creation of vortices.

In this letter, we consider confined mesoscale superflow through quasi-one-dimensional (1d) constrictions with a characteristic size $a$ approaching the temperature $(T)$ dependent correlation (healing) length $\xi(T)$, and find a strong increase in dissipation when $a/\xi(T)$ approaches one. Going beyond previous studies Varoquaux (2015); Weimer et al. (2015), we (i) quantitatively predict the temperature, size, and drive dependence of the critical velocity without adjustable parameters, (ii) use a paradigmatic orifice geometry to model the enhancement of vortex creation in spatially inhomogeneous flow near a sharp boundary, which significantly lowers critical velocities, (iii) point out the universality between high density 4He Hoskinson et al. (2005b); Duc et al. (2015) and low density atomic condensates Neely et al. (2010); Ramanathan et al. (2011); Raman et al. (1999); Weimer et al. (2015), by characterizing constrictions via the dimensionless length $a/\xi$ and measuring velocities in units of $v_{0}={\kappa}/\left({4\pi\xi_{0}}\right)$, and (iv) describe the crossover to the purely 1d limit, a Luttinger liquid in the thermal regime. Predictions are expected to be logarithmically accurate in the critical regime while corrections of order unity may arise when extrapolating to lower $T$.

We begin by considering superflow between reservoirs with a chemical potential difference $\Delta\mu$ (pressure difference $\Delta P$) connected by prototypical geometric constrictions with either a “channel” or “orifice” shape as seen in Fig. 1. Channel flow is spatially homogeneous with a constant superfluid velocity $v_{s}$ that is representative of flow through long narrow cylindrical pores. Flow through an orifice can be studied by considering a hyperbolic surface of revolution connecting two bulk reservoirs where the potential flow problem can be solved analytically sup . The solution is characterized by divergent flow near the boundaries as seen in Fig.1c, where the creation of line vortices (Fig. 2 left) is facilitated. Ring vortices in the center of the orifice (Fig. 2 right) are not strongly affected by the spatial dependence of flow near the boundaries.

The energy cost for creating a quantized vortex is due to (i) the kinetic energy of circular superflow around the core region, and (ii) the loss of condensation energy within the core Iordanskii (1965). For a vortex ring of radius $R$ (length $\mathcal{L}=2\pi R$) or a line vortex (length $\mathcal{L}$) in a constriction of radius $a$, the combination of these effects yields

[TABLE]

where $\ell=a$ for line vortices, $\ell=R$ for ring vortices, and the constant $\alpha$ depends of the vortex type and model of the core. Solving the Gross-Pitaevski (GP) equation, one finds that $\alpha=0.385$ for line vortices Pitaevskii (1961) and $\alpha=\ln 8-2+0.385\simeq 0.464$ for ring vortices Roberts and Grant (1971). Obtaining a more accurate model of the vortex core is possible via numerical simulations in low Winiecki et al. (1999) and high density superfluids Ortiz and Ceperley (1995); Galli et al. (2014). The results are consistent with the GP value of $\alpha$ and show only only weak density dependence. The reduction of the kinetic energy of superflow due to interaction with the vortex is

[TABLE]

where the integral is over the area bounded by the vortex ring, or between the vortex line and boundary. The total energy is $E=E_{\rm tension}+E_{\rm flow}$. To unify the description of driven quantum fluids, we employ the Josephson relation in 3d Josephson (1966) $\kappa^{2}\rho_{s}(T)\xi(T)=4\pi^{2}k_{\rm B}T_{c}$ where $\rho_{s}(T)$ is the superfluid mass density for a transition temperature $T_{c}$ and $\xi(T)=\xi_{0}(1-T/T_{c})^{-\nu}$ and $\nu$ is the correlation length critical exponent. We numerically checked that the Josephson relation is valid to within $20\%$ down to $T/T_{c}\approx 0.7$, for details see sup . For flow through an orifice with speed $v_{J}$, we obtain the energy barrier for a line vortex located a distance $x$ from its center:

[TABLE]

and that for a centered ring vortex with radius $R$:

[TABLE]

where $\beta_{c}=1/(k_{\rm B}T_{c})$. From these expressions (and those for channel flow derived in the supplementary material sup ) we observe the emergence of natural length ($\xi_{0}$) and velocity $(v_{0}=\kappa/(4\pi\xi_{0}))$ scales that are essential for constructing a universal theory of dissipative superfluids.

The velocity of superflow at finite $T$ is limited by the thermal nucleation of quantized vortices which traverse the flow lines leading to a change of phase of the superfluid order parameter by $\pm 2\pi$, (see e.g. Ref. [Langer, 1968]). The decay of a persistent current is then governed by vortex energetics via:

[TABLE]

where $h\Gamma=\Delta\mu=m\Delta P/\rho_{s}$ drives total mass flow $J=\rho_{s}\iint\boldsymbol{v}_{s}\cdot d\boldsymbol{S}\equiv\pi a^{2}\rho_{s}v_{J}$ and $E_{\rm max}\equiv\max_{\mathcal{L}}E$ is the saddle point of the vortex excitation energy over the domain of the channel or orifice. The difference of rates in Eq. (5) corresponds to the contributions from vortices which reduce and increase the superflow, respectively. The attempt rate $\Gamma_{0}$ is related to the phase space available for vortex excitations and is geometry dependent:

[TABLE]

with $\tau_{\rm GL}^{-1}=16k_{\rm B}(T_{c}-T)/h$ the Ginzburg-Landau relaxation rate. The decay rate in Eq. (5) contains the main contribution from integration over zero modes, corresponding to a translation of the vortex with action $S_{v}$ in both time and space. In addition, there are Jacobians due to the change of coordinates from the superfluid phase $\Phi$ to the radius $R$ of the vortex ring or the location $x$ of the vortex line, and a contribution from integration over the negative eigenvalue mode. As previously discussed Coleman (1977); *coleman2; Schulman (1981), the Jacobian is $\sqrt{S_{v}/2\pi}$ for the zero modes, and the negative eigenvalue mode contributes a factor of similar magnitude. We have verified that the combination of all these factors is of order unity and thus neglect them. Other modifications to the pre-factor in Eq. (5) could result when considering the microscopic details of dynamics and vortex evolution inside the constriction Bulgac et al. (2011); Yefsah et al. (2013); Ku et al. (2014); Donadello et al. (2014); Serafini et al. (2015); Ku et al. (2016) and would introduce quantitative logarithmic corrections to the nucleation theory.

Evaluation of the critical velocity from Eq. (5) proceeds as follows. For a given flow profile and vortex type, we maximize the vortex energy as a function of $\pm v_{J}$ over the spatial domain of possible configurations. This leads to critical vortex positions $x^{\ast}(\pm v_{J})\in(-a,a)$ for line vortices and critical radii $R^{\ast}(\pm v_{J})\in(0,a)$ for ring vortices. Vortices with a length smaller than the critical one will tend to collapse, or anihilate at boundaries, while those larger can proliferate, leading to dissipation. For a fixed constriction radius $a/\xi_{0}$, temperature $T/T_{c}$ and external drive potential $\Gamma_{0}/\Gamma$, Eq. (5) can be numerically solved self-consistently giving the critical velocity when $v_{J}=v_{c}$.

When $a/\xi_{0}\gg 1$, the boundaries of the constriction no longer play an important role and only uniform channel flow is relevant. Due to the resulting large critical velocities, energy increasing fluctuations can be neglected and the saddle point energies can be found analytically. The resulting pair of transcendental equations

[TABLE]

can be solved numerically for $R^{\ast}$ and $v_{c}$ sup .

Our main results for the critical velocity in both the channel and orifice flow profiles are shown as lines in Fig. 3. When employing the scales $v_{0}$ and $\xi_{0}$, mass flow measurements in drastically different density, interaction, and temperature regimes are well bounded by the vortex nucleation theory, and experiments on confined superfluid 4He and low-dimensional Bose-Einstein condensates can be directly compared.

For both flow profiles, line vortices have lower activation energies than ring vortices, giving smaller velocities and a lower bound on $v_{c}$. An absolute upper bound is provided by ring vortices in the orifice flow profile. In the limit $a\gg\xi_{0}$, where the geometry approaches that of bulk flow, we recover the intrinsic superfluid velocity due to the nucleation of vortex rings analyzed by Langer and Fisher Langer and Fisher (1967). For 4He, we have used a vortex core size determined from specific heat measurements Singasaas and Ahlers (1984); sup , and have thus been able to correct a long-standing inconsistency of 27 orders of magnitude in the applied pressure difference employed in Ref. [Langer and Fisher, 1967] to obtain agreement with experiments. For tight constrictions, both the lower and upper bound turn towards smaller velocities indicative of enhanced dissipation.

While details of additional experiments are discussed in the supplemental material sup , Fig. 3 includes data from two 4He studies whose critical velocities display a clear temperature dependence – a signature of activated behavior. Harrison et al. Harrison and Mendelssohn (1974) measured flow in networks of imperfect pores of varying radii which should provide a lower bound on flow speeds, behavior consistent with our results. Recent measurements on single nanopores by Duc et al. Duc et al. (2015) exhibit drastically different behavior: large critical velocities that decrease as the radius approaches the correlation length. While the microscopic details of the flow profiles are not known, $v_{c}$ for the narrowest pore is bounded by the channel prediction, consistent with the reported aspect ratio of 10:1 and suggesting a crossover to strongly dissipative quasi-1d flow.

Fig. 3 also includes results from analogous neutral atomtronic circuits using utracold bosonic and fermionic condensates Chien et al. (2015). Here, the “orifice” can be replaced with a quantum point contact between two resonantly coupled Fermi gases Husmann et al. (2015) or channel-like flow can be driven by the discharge of an bosonic atom capacitor Lee et al. (2013) or by dragging an optical potential through a simply Neely et al. (2010); Weimer et al. (2015) or multiply connected Ramanathan et al. (2011); Wright et al. (2013a); Eckel et al. (2014) Bose-Einstein condensate. For the latter, our nucleation theory yields $v_{c}\approx 100\leavevmode\nobreak\ {\rm\mu m/s}$ for the drag and $v_{c}\approx 1\leavevmode\nobreak\ \text{mm/s}$ for the toroidal flow in remarkable agreement with measurements and more microscopic theoretical analysis Crescimanno et al. (2000); Eckel et al. (2014); Mathey et al. (2014).

Intuition for the dissipation in narrow pores with radius $a\approx\xi$ can be gained by considering the unoptimized energy of line vortices with position $x=0$ at the center of the channel. This approximation is qualitatively correct since narrow pores can be expected to be in the channel flow regime with line vortex activation energies comparable to temperature. The vortex energy is found from the sum of Eqs. (1) and (2) with $x=0$:

[TABLE]

with resulting critical velocity

[TABLE]

As $a/\xi\to 1^{+}$, the activation energy in the exponent of Eq. (10) decreases rapidly, and the small multiplier of the exponential is no longer compensated. As a consequence, the critical velocity drops by several orders of magnitude.

In the quasi-1d $(a\lesssim\xi)$ limit there are no transverse degrees of freedom, and the system can be described in analogy to fluctuating superconducting wires by computing the resistance due to thermally activated phase slips Langer and Ambegaokar (1967); McCumber and Halperin (1970). Translated into the language of 1d superfluidity, the phase slip energy is (see supplemental material sup ):

[TABLE]

For line vortices in the channel flow profile, the tension scales with $(a/\xi)^{2}$ in contrast to the linear dependence in Eq. (9). This difference is unimportant at the crossover $a/\xi\approx 1$ and the critical velocity from the nucleation of line vortices and the 1d theory should be in agreement as demonstrated in Fig. 4a.

The lower bound on the critical velocity due to line vortices drops three orders of magnitude, and crosses over to the 1d result for single mode channels. An analogous crossover from the linear to non-linear Josephson junction regime has been observed in superflow through an array of orifices near $T_{c}$ Hoskinson et al. (2005b).

Fig. 4b shows the $T$-dependence of $v_{c}$ for constrictions with $a/\xi_{0}=10$ and we observe a reduction as $T\to T_{c}$. Qualitatively, this is due to the fact that $\xi(T)$ increases as $T\to T_{c}$, thus reducing the ratio $a/\xi(T)$ which determines the effective constriction radius. Experiments on 4He have reported an apparent temperature power-law scaling of $v_{c}$ Duc et al. (2015) which we have confirmed is the spurious result of an interplay between the thermal activation energy and vortex attempt rate. As $T\to 0$, vortices will no longer be thermally activated and dissipation will be dominated by the nucleation of quantum phase slips Khlebnikov (2004); Danshita (2013).

In summary, we considered two characteristic confined flow geometries and the thermal activation of representative low energy excitations, ring and line vortices, inside them. The resulting bounds they place on the critical velocity of neutral superflow through narrow constrictions agree with a large body of measurements on confined superfluid 4He and low-dimensional ultracold gases. As the confinement radius approaches the healing length, we find an exponential suppression of the critical velocity of three orders of magnitude. The experimental observation of this dramatic reduction would be a clear signal of entering the strongly fluctuating mesoscale regime.

Acknowledgements – We thank B.I. Halperin, G. Gervais, P.F. Duc, P. Taborek and K. Wright for helpful discussions. A.D. appreciates the University of Leipzig’s hospitality and his participation in a grand challenges workshop on quantum fluids and solids supported by NSF DMR-1523582. B.R. acknowledges support by DFG grant No. RO 2247/8-1.

I Velocity flow profiles

As illustrated in Fig. 1a-b of the main text, we consider two different superfluid flow profiles through long narrow channels, and hole-like orifices.

I.1 Channel Flow

For a long cylindrical channel of radius $a$ and length $L\gg a$ oriented with its axis along the $z$-direction we neglect any acceleration of the fluid at the entry and exit to obtain a spatially independent velocity field

[TABLE]

where $J$ is the total mass flow rate and $\rho_{s}$ is the superfluid mass density.

I.2 Orifice Flow

The geometry of an orifice of radius $a$ oriented in the $x-y$ plane centered at the origin can be conveniently described in oblate spheroidal coordinates $(\zeta,\eta,\phi)$ by the surface $\eta=0$ where

[TABLE]

In general, $\phi\in[0,2\pi)$, $\zeta\in[0,\infty)$ and $\eta\in[-1,1)$ with the surface $|\eta|=\eta_{0}$ corresponding to a hyperboloid of revolution about the $y$-axis (see Fig. 1b). Steady incompressible flow through the orifice with total rate $J$ can be studied by defining a potential field $\Phi(\zeta,\eta,\phi)$ such that $\boldsymbol{v}_{s}=(\hbar/m)\boldsymbol{\nabla}\Phi$ and solving $\nabla^{2}\Phi=0$ by requiring that $\Phi$ is continuous at $\zeta=0$, and subject to the boundary condition that the velocity component perpendicular to the surface defined by $\eta=0$ vanishes outside the orifice. This yields the velocity field: Morse and Feshbach (1953); Blount and Varma (1976); Schwarz (1993)

[TABLE]

which is plotted in the $y-z$ plane in Fig. 1c. Inside the orifice at $\zeta=0$ this expression simplifies to:

[TABLE]

which diverges near the boundary as $r\to a$.

II Vortex Energies

Employing Eqs. (1)–(2) in the main text in combination with the spatial dependence of $v_{s}$ we may now obtain the energy cost of nucleating line and ring vortices.

II.1 Channel Flow

For channels, we use the velocity profile in Eq. (S1).

II.1.1 Line Vortices

For a line vortex attached to the walls of a channel of radius $a$ offset a distance $x$ from the center with length $\mathcal{L}(x)=2\sqrt{a^{2}-x^{2}}$ as seen in Fig. 2 the total energy is the sum of $E_{\rm tension}+E_{\rm flow}$ and given by:

[TABLE]

Now, the superfluid density and correlation length $\xi$ can be related via the Josephson scaling relation in three dimensions Josephson (1966)

[TABLE]

where $\xi(T)=\xi_{0}(1-T/T_{c})^{-\nu}$. Fig. S1 shows the accuracy of this relation for superfluid 4He Donnelly and Barenghi (1998) and a weakly interacting Bose gas Prokof’ev et al. (2004) down to $T/T_{c}\approx 0.7$.

All temperature dependence now enters expressions through the correlation length and the resulting dimensionless vortex energy is given by:

[TABLE]

with $\beta_{c}\equiv 1/(k_{\rm B}T_{c})$ and we have identified the fundamental velocity scale $v_{0}=\kappa/(4\pi\xi_{0})$. This expression simplifies to Eq. (9) in the main text when the vortex line is located at the center of the channel $x=0$.

II.1.2 Ring Vortices

For a ring vortex of radius $R$, with length $\mathcal{L}=2\pi R$ the energy can be written as:

[TABLE]

yielding

[TABLE]

II.2 Orifice Flow

For the orifice flow profile, the velocity field now has the spatial dependence seen in Eq. (S4) and Fig. 1c with a divergence at the boundary. While the effective line tension $E_{\rm tension}$ (first term) in Eqs. (S5) and (S8) are unchanged, a modified spatial integral in $E_{\rm flow}$ needs to be computed.

II.2.1 Line Vortices

The energy cost for flow captured by a line vortex at position $x$ is given by

[TABLE]

leading to the total dimensionless vortex energy:

[TABLE]

II.2.2 Ring Vortices

For ring vortices, the flow integral is given by

[TABLE]

which leads to

[TABLE]

III Application to Mass Flow Experiments

In this section we provide a more complete analysis of neutral bosonic mass flow experiments that were discussed in Fig. 3 of the main text.

III.1 Superfluid Helium-4

A recent review by Varoquaux Varoquaux (2015) includes a compilation of superfluid critical velocity results from a diverse set of experiments (both dependent and independent of temperature) that are reproduced in Fig. S2 along with theoretical upper and lower bounds computed within the vortex nucleation theory using parameters relevant for superfluid helium. The experimental data from Ref. [Varoquaux, 2015] can be grouped into two distinct regions. For larger channels, a temperature independent critical velocity increases as the channel size is reduced (grey circles).

This is consistent with the Feynman prediction $v_{F}\sim\kappa/a$, where vortex rings are dynamically ejected from the end of the channel.

A second group of temperature dependent critical velocities in pores with radii $a\sim$100\text{,}\mathrm{nm}\text{/}$-$10\text{,}\mathrm{\SIUnitSymbolMicro m}\text{/} have considerably larger velocities and are in a thermally activated dissipation regime that is well bounded by the vortex nucleation theory. Additionally, Fig. [S2](#S3.F2) includes a systematic set of experiments performed by Harrison and Mendelssohn Harrison and Mendelssohn ([1974](#biba.bib8)) (green circles) employing the fountain effect to drive superfluid mass flow through arrays of $10^{4}-10^{7}$ $L=$5\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/} pores etched in irradiated mica using HF acid with radii $a=20,50,80,120$ and $200\text{\,}\mathrm{nm}\text{/}$. These results, taken at $T=$1.5\text{,}\mathrm{K}\text{/}$$, should be considered as representing a lower bound on the critical velocity as all pores were assumed to be open in the analysis and both a variation in radii and a taper along the channel were observed.

Experiments by Duc et al. Duc et al. (2015) shown in Fig. S2 are in the interesting mesoscopic regime where $a/[\xi_{0}(1-T/T_{c})^{-\nu}]\sim\mathrm{O}(1)$. For superfluid helium, $\xi_{0}\simeq$3.45\text{,}\mathrm{\SIUnitSymbolAngstrom}\text{/}, $T_{c}=T_{\lambda}\simeq$2.1768\text{\,}\mathrm{K}\text{/} and $\nu\simeq 0.6717$. They observed a decrease in the critical velocity as the pore radius was reduced. In these experiments, 4He mass flow is studied through single pores nanofabricated using a transmission electron beam incident on a $L\simeq$30\text{,}\mathrm{nm}\text{/} thick silicon nitride wafer resulting in smooth constrictions with radii $a\simeq\text{$3.147.8120\text{\,}\mathrm{nm}\text{/}$}$. Critical velocities were reported for $T=$1.5\text{\,}\mathrm{K}\text{/}, and flow was driven via pressure differences between $\Delta P=$250830\text{,}\mathrm{mbar}\text{/}$$ which corresponds to an external drive frequency $\Gamma\simeq$27\text{,}\mathrm{GHz}\text{/}$\sim k_{\rm B}T_{c}/h$.

III.2 Ultracold Gases

In systems of neutral ultracold atoms, flow is driven by dynamically shaping an optical potential which causes a local imbalance in the chemical potential (see Ref. [Chien et al., 2015] for current experimental designs). For dilute gases, the prefactor $\xi_{0}$ which appears in the critical scaling relation for the correlation length can be related to the zero temperature healing length $\xi_{h}=1/\sqrt{8\pi a_{s}n}$ via known universal results for the weakly interacting Bose gas Prokof’ev et al. (2004). Here $n$ is the number density and $a_{s}$ is the scattering length and provided that $na^{3}\ll 10^{-5}$ the superfluid density can be written in terms of a universal scaling function $f_{s}$:

[TABLE]

where $m$ is the mass, $t=1-T/T_{c}$ is the reduced temperature and the Riemann zeta function, $\zeta(3/2)$, appears through the use of the critical temperature of the non-interacting Bose gas: $k_{\rm B}T_{c}=({2\pi\hbar^{2}}/{m})[{n}/{\zeta(3/2)}]^{2/3}$. The function $f_{s}$ is known from high precision Monte Carlo calculations Prokof’ev et al. (2004); Capogrosso-Sansone et al. (2010) and when $t\ll 1$, $f_{s}\propto(t/\sqrt[3]{na^{3}})^{\nu}$. Writing the temperature dependent correlation length in the critical region as $\xi(T)=(\xi_{0}/\xi_{h})\xi_{h}t^{-\nu}$, we can use the Josephson relation in Eq. (S6) to determine

[TABLE]

where $\mathcal{A}\simeq 3.8$ is a universal number. Thus, for experimentally accessible weakly interacting Bose gases $\xi_{0}\simeq\mathcal{B}\xi_{h}$ with $\mathcal{B}\sim 1-2$. Away from the critical region, the full scaling function can be used to test the accuracy of the Josephson relation with the result shown in Fig. S1.

Thus, to analyze non-equilbirium mass flow experiments employing ultracold Bose-Einstein condensates, we approximate $\xi_{0}\approx\xi_{h}\sim$1\text{,}\mathrm{\SIUnitSymbolMicro m}\text{/}, which yields the critical velocity scale $v_{0}=h/(4\pi m\xi_{0})\sim$1\text{\,}\mathrm{mm}\text{/}\mathrm{s}. In Fig. 3 of the main text, we have shown results for the critical velocity from three ultracold gas experiments using the parametrization shown in Table 1. We have not included the low temperature data point from Ref. [Ramanathan et al., 2011] where the errors in our critical theory are difficult to estimate. We find that the other experiments are in the same flow regime as tightly confined superfluid helium.

IV Intrinsic critical velocity of bulk superfluids

In the $a\gg\xi_{0}$ limit we can directly find the critical ring vortex radius $R^{\ast}$ that maximizes the energy barrier in the constant channel flow profile in Eq. (S8) as

[TABLE]

In this bulk regime, one only needs to consider the effects of vortices which reduce the total energy and Eq. (5) gives

[TABLE]

which can be solved for the critical velocity $v_{c}$. Combining Eq. (S17) with Eq. (S16) with $v_{J}=v_{c}$ gives the transcendental equation

[TABLE]

which can be simplified to yield Eq. (7). Replacing $\ln\Gamma_{0}/\Gamma$ in Eq. (S17) via Eq. (S18) we obtain the critical velocity in the large radius limit in Eq. (8).

Eq. (7) in the main text is equivalent to Eq. (13) in Langer and Fisher Langer and Fisher (1967) (LF), noting the change of variables $\eta_{c}\equiv\ln 8R^{\ast}/\xi$, their use of a rigid vortex core with $\alpha=\ln 8-7/4$ and setting $\ln\Gamma_{0}/\Gamma\simeq 83$. The un-physically large value of $\ln\Gamma/\Gamma_{0}$ employed by LF was required to ensure that their computed value of $v_{c}=v_{c0}(1-T/T_{c})^{\nu}$ in the scaling regime was below the absolute upper bound given by the Landau criterion. In order to understand the origin of this long-standing discrepancy, we have investigated $v_{c0}$ as a function of both the external drive $\Gamma_{0}/\Gamma$ and vortex core details $\xi_{0}$ with the results shown in Fig. S3.

V The one-dimensional limit

To derive Eq. (11) in the main text, we begin with a free energy functional near $T_{c}$:

[TABLE]

where $f_{0}$ is a condensate energy density and the wave function is given by: $\Psi(\boldsymbol{r})=\sqrt{n(\boldsymbol{r})}\mathrm{e}^{i\Phi(\boldsymbol{r})}$ with $n(\boldsymbol{r})$ the number density. In a spatially homogeneous superfluid, the free energy is minimized when $\lvert\Psi\rvert^{2}\equiv n_{0}=-A/B$. Substituting this value for the field into Eq. (S19):

[TABLE]

where the last equality is schematic and in analogy to superconductors where $H_{c}$ is the critical field. Identifying $-A/B$ with the density and performing the usual rescaling: $\tilde{\Psi}=\Psi/\sqrt{n_{0}}$ one introduces the correlation length $\xi(T)$ such that $\xi^{2}(T)=\hbar^{2}/2mA(T)$, and we can now write $f-f_{0}=\rho_{s}\kappa^{2}/16\pi^{2}\xi^{2}(T)$. According to Refs. [Langer and Ambegaokar, 1967-McCumber and Halperin, 1970], the free energy barrier of a phase slip excitation inside a quasi-one-dimensional channel of cross-sectional area $\pi a^{2}$ is given by

[TABLE]

The energy reduction due to a superflow $J$ is $\delta F=\kappa\rho_{s}v_{J}\pi a^{2}$, and adding the free energy barrier Eqs. (S21) one obtains the total phase slip energy in Eq. (11) of the main text. The total phase slip rate is $\Gamma=2\Gamma_{0}\mathrm{e}^{-\Delta F_{0}/k_{\rm B}T}\sinh(\delta F/k_{\rm B}T)$, and solving for $v_{c}$ we find

[TABLE]

where McCumber and Halperin (1970)

[TABLE]

Eq. (S22) is shown in Fig. 4 of the main text.