Superfluid properties of bright solitons in a ring

Superfluidity is one of the most significant macroscopic quantum phenomena. From Kapitza’s experiment with liquid helium in 1938 kapitza , properties of superfluidity have been extensively investigated both theoretically and experimentally pethicksmith ; pitaevskiistringari ; svistunov . For a spatial dimension lesser than three, Mermin-Wagner’s theorem rules out the emergence of an off-diagonal long-range order merminwagner . However, in two-dimension, we can have a quasi-long-range order and find a Berezinskii-Kosterlitz-Thouless (BKT) transition at a BKT transition temperature, above which a proliferation of free vortices occurs berezinskii ; kosterlitz1 ; kosterlitz2 ; nelson . This BKT transition can be observed through abrupt changes of thermodynamic quantities such as sound velocities or the superfluid fraction stringarisound ; ozawa ; ota ; hadz ; furutani , and this is a crucial nature specific to two-dimensional systems different from three-dimensional ones.

On the other hand, in one dimension, we can observe distinct phenomena either from two-dimensional or three-dimensional systems. A one-dimensional Gross-Pitaevskii (GP) equation or nonpolynomial Schr$\ddot{\text{o}}$dinger equation involves a soliton, which is a solitary wave that keeps its inhomogeneous shape during the propagation, for a sufficiently attractive interparticle interaction carr ; ueda ; minguzzi ; polo ; salasnich ; crosta ; amico ; naldesi ; naldesi20 ; polo22 . This is peculiar to effective one-dimensional systems because an extra transverse dimension can destabilize solitons in higher spatial dimensions. Solitons show up in various context of physics from condensed-matter physics kishine ; togawa and biological systems yomosa ; scott to high-energy physics nitta ; brauner . In ultracold atomic systems, the dynamics of soliton has been studied under several conditions such as the quench dynamics cidrim ; muru , the dynamics in the presence of a spin-orbit coupling tononisoc ; muru , and solitons in an annular geometry nis ; polo ; minguzzi . Based on these developments, it is significant to clarify and characterize the property of superfluid that involves modulational changes.

In this paper, starting from the one-dimensional GP equation, we illustrate superfluid properties of the solitons in an annular geometry as depicted in Fig. 1 by applying Leggett’s formula leggett . First, we see the emergence of solitons in an annular one-dimensional superfluid with a boost. Secondly, we derive Leggett’s formula from the one-dimensional GP equation including a boost velocity. While Leggett’s formula is widely used mainly in the context of supersolid which involves a crystalline order associated with the nonclassical translational inertia leggett ; francesca ; tanzi , we can straightforwardly obtain it also from the one-dimensional GP equation (see also martone ; rica ; rica94 ; rica07 ). The superfluid fraction given by Leggett’s formula is determined by the modulus of the macroscopic wavefunction and detects modulational instabilities in the one-dimensional superfluid. We see this behavior of the superfluid fraction given by Leggett’s formula with varying parameters. As one increases the attractive interaction strength, the uniform configuration of the steady-state changes to a bright soliton at a critical interaction strength, which reduces the superfluid fraction. In particular, at a critical boost velocity, the soliton has a node leading to the vanishing superfluid fraction. Finally, we also mention the effects of the transverse dimensions and potential barriers along the ring on the superfluid fraction. We can see that the superfluid fraction decreases by increasing the transverse width and it undergoes dynamical instability at a certain width. A potential barrier makes a uniform configuration into an inhomogeneous one. Increasing the attractive interaction strength, we see that the inhomogeneous density develops a soliton at one of the potential minima.

Let us consider one-dimensional attractively interacting bosons in a ring with a boost velocity $u$ as in Fig. 1 described by

with

where $\Psi(x,t)$ is the complex macroscopic wavefunction satisfying $\int^{L}_{0}dx\absolutevalue{\Psi(x,t)}^{2}=N$ with system size $L$, number of particle $N$, and $V(x)$ the interparticle interaction potential. Equation (1) is, in the absence of the boost velocity $u=0$, identical to the usual GP equation. Now we assume a steady state

where $\psi(x)$ is a time-independent complex field, which describes the stationary configuration. The parameter $\mu$ is the chemical potential. This provides

With the contact interaction $V(x)=g\delta(x)$, we introduce the following dimensionless quantities

where $L=2\pi R$, $\bar{n}=N/L$ is the average density, $\gamma<0$ is the dimensionless attractive interaction strength, $\theta$ is the azimuthal angle, $\Omega$ is the angular frequency of rotation corresponding to the dimensionless boost velocity, and $\varphi(\theta)$ is the dimensionless macroscopic wave function normalized as $\int^{2\pi}_{0}d\theta\absolutevalue{\varphi(\theta)}^{2}=1$ and periodic $\varphi(0)=\varphi(2\pi)$. Then, Eq. (4) can be written as

The GP equation has a bright soliton solution as a stationary solution given by carr ; ueda

where $K(m)$ and $E(m)$ are the first and second kind complete elliptic integrals, respectively, under the condition $K(m)E(m)=\pi^{2}\gamma/2$ and $\mathrm{dn}(u|m)$ is a Jacobi elliptic function with an elliptic modulus $m$. Here $\gamma_{\mathrm{c}}=-1/2$ is the critical interaction strength. Due to the broken continuous translational symmetry, Eq. (7) includes a parameter $\theta_{0}$ that specifies the center of the bright soliton. This translational symmetry breaking results in the emergence of a Nambu-Goldstone (NG) mode in addition to the one associated with the $\mathrm{U}(1)$ symmetry breaking ueda ; kanamoto ; martone . This emergence of two distinct NG modes are analogous to supersolid, which has both a crystalline order and a superfluid order leggett ; rica ; rica07 ; martone ; stringari ; watanabe ; rica94 ; tanzi . In the following, we set $\theta_{0}=0$ for brevity. It is important to stress that the NG mode related to the formation of the bright soliton breaking the continuous translational symmetry is gapless, i.e. zero energy with respect to the ground state. The first excitation energy calculated under the Bogoliubov approximation is gapped at $\gamma=0$, while it closes at $\gamma=\gamma_{\mathrm{c}}$ ueda . The latter phenomenon is also usually called modulational instability. Indeed, the Bogoliubov spectrum of the uniform configuration reads $E_{l}^{(\mathrm{B})}=\sqrt{l^{2}(l^{2}+2\gamma)}$ with $l=\pm 1,\pm 2,\pm 3,...$ ueda ; kanamoto and for $l=\pm 1$ one gets $E^{(\mathrm{B})}=0$ precisely at $\gamma=\gamma_{\mathrm{c}}=-1/2$. Note that at $\gamma=0$ our bosonic system is noninteracting and uniform but also fully superfluid just because there is a finite energy gap due to the finite size.

With a boost velocity, we can define phase winding number $\nu$, which is an integer such that $\Omega-1/2<\nu\leq\Omega+1/2$, and the relative angular frequency

which varies within $-1/2\leq\omega<1/2$ kanamoto ; ueda08 ; ueda10 . Below the critical interaction strength $\gamma<\gamma_{\mathrm{c}}\equiv 2\omega^{2}-1/2$, it has a bright soliton. The number density $n(\theta)=\absolutevalue{\varphi(\theta)}^{2}$ is kanamoto

where $m^{\prime}=1-m$ and

Equation (9) indicates that, for $0\leq\absolutevalue{\omega}<1/2$, the number density is nodeless while it has a node for $\omega=-1/2$. The chemical potential $\bar{\mu}$ is determined by kanamoto

and the elliptic modulus $m$ is determined by

with $\varepsilon\equiv\mathrm{arcsin}\left(\sqrt{f_{\mathrm{c}}/(m^{\prime}f)}\right)$ and

Here, $F(u|m)$ and $\mathcal{E}(u|m)$ are the elliptic integrals of the first kind and second kind respectively.

Let us now set

and

Inserting these formulas into Eq. (4) we get

Equation (18) implies that

where $J$ is a constant current density. Equation (19) is very interesting because it says that if $n(x)$ has spatial variations then also $v(x)$ must have spatial variations.

We now introduce the average value of the velocity $v(x)$ in a spatial region $[a,b]$ as

Then, from Eqs. (19) and (20) we obtain

where

We will see that the number density ${\bar{n}}_{\mathrm{s}}$ can be interpreted as the superfluid number density of the stationary state in the spatial region $[a,b]$. Indeed, Eq. (22) is the 1D version of the formula obtained by Leggett in 1970 leggett for a supersolid with spatial periodicity $b-a$, and recently discussed by many others rica ; chomaz ; martone . In other words, if the stationary state $\psi(x)$ moves with the average velocity ${\bar{v}}$, its current density reads

where ${\bar{v}}$ is the average velocity in the region $[a,b]$ and ${\bar{n}}_{s}$ the corresponding superfluid number density.

Let us now consider the linear canonical momentum in the region $[a,b]$. It is given by

Note that Eq. (24) is different from the physical momentum, which takes into account the boost velocity, $P=\mathrm{Re}\int^{b}_{a}dx\psi^{*}(x)\left[-i\hbar\partial_{x}-mu\right]\psi(x)$. With the help of Eqs. (15) and (19), we then find

where

is the average number density in the region $[a,b]$.

As usual (see for instance Refs. leggett ; rica ), the normal density ${\bar{n}}_{\mathrm{n}}$ in the spatial region $[a,b]$ can be defined as the response of the linear momentum ${\cal P}$ to the boost velocity $u$, namely

By using Eq. (25), we immediately find

and this result fully justifies that ${\bar{n}}_{\mathrm{s}}$ of Eq. (22) is indeed the superfluid number density in the region $[a,b]$. Similarly, a relation with the nonclassical translational inertia (NCTI), which is expected in the case of a supersolid system, is provided by the definition of superfluid fraction $f_{\mathrm{s}}$ associated with NCTI rica

with $N=\int_{a}^{b}n(x)\,dx=(b-a){\bar{n}}$. Using the above definition for $\cal{P}$, one finds

In particular, by setting $a=0$ and $b=L=2\pi R$, we can obtain

where $n(x)=\absolutevalue{\psi(x)}^{2}$ is the number density.

A realistic 1D system would have a finite width in the transverse direction as in Fig. 1. This effect can be captured by the 1D nonpolynomial Schr$\mathrm{\ddot{o}}$dinger (NPS) equation salasnicha ; salasnichb ; adhikari ; salasnich . It is given by

with

with $\tilde{g}\equiv g_{\mathrm{3D}}/(2\pi a_{\perp}^{2})$, $g_{\mathrm{3D}}=4\pi\hbar^{2}a_{s}/m$, $a_{\perp}=\sqrt{\hbar/(m\omega_{\perp})}$ the transverse width, and $\eta\equiv[1+2a_{s}\absolutevalue{\Psi(x,t)}^{2}]^{1/4}$ salasnicha ; salasnichb . The steady-state equation can be found under Eq. (3) as

Using the dimensionless quantities in Eq. (5), one obtains

with

Here we have introduced the dimensionless $s$-wave scattering length $\tilde{a}_{s}\equiv Na_{s}/R$ and the dimensionless transverse width $\tilde{a}_{\perp}\equiv a_{\perp}/R$.

In the thin transverse length limit $a_{\perp}\to 0$ with a fixed interaction strength $\tilde{\gamma}$, one finds $\eta(\theta)\to 1$ and the 1D NPS equation in Eq. (47) recovers the 1D GP equation as

with $\bar{\mu}_{\mathrm{eff}}\equiv\bar{\mu}-2\tilde{a}_{\perp}^{-2}$ the dimensionless effective chemical potential.

We illustrated the density configuration in the left panel of Fig. 5 for $\tilde{\gamma}=-1$. As one increases the transverse width $\tilde{a}_{\perp}$, we have a sharper bright soliton. The right panel in Fig. 5 illustrates the superfluid fraction without a boost calculated by Leggett’s formula in Eq. (31) solving Eq. (47) numerically. We can check that the superfluid fraction approaches the results by the 1D GP equation in $a_{\perp}\to 0$, which is consistent with Eq. (49). As one increases the transverse width $a_{\perp}/R$, for $\tilde{\gamma}=-0.5$ and $\tilde{\gamma}=-0.7$, the superfluid fraction decays up to a certain value of $a_{\perp,\mathrm{c}}/R\equiv\tilde{a}_{\mathrm{c}}$. Beyond $\tilde{a}_{\mathrm{c}}$, $\eta^{4}(\theta)$ in Eq. (48) is negative and therefore $\eta^{2}(\theta)$ changes to purely imaginary. In this case, one can see $\mathrm{Im}[\mu]>0$, which indicates $\Psi(x,t)\propto e^{\mathrm{Im}[\mu]t/\hbar}$ and the steady-state solution in Eq. (3) is dynamically unstable. This is the collapse of the condensate implicitly induced by the transverse dynamics encoded into the NPS equation (44). The stability condition of the bright soliton is given by salasnichb

which gives the critical transverse width $\tilde{a}_{\mathrm{c}}\equiv 4/\left(3\pi\absolutevalue{\tilde{\gamma}}\right)$. On the other hand, in the case of $\tilde{\gamma}=-0.3$, the superfluid fraction does not decay at all by increasing the transverse width. This is because the density configuration is uniform instead of a bright soliton as in the case of a 1D GP equation below the critical interaction strength $\absolutevalue{\gamma}<1/2$. The phase diagram for the density configuration is summarized in Fig. 6.

In this section, we examine the effects of potential barriers in the ring on the density configuration and superfluid fraction.

We elucidated superfluid properties of a one-dimensional annular superfluid, which exhibits solitons with sufficiently attractive interaction, subject to a rotational boost, a transverse dimension, or a potential barrier by using the superfluid fraction given by Leggett’s formula. We derived Leggett’s formula for the superfluid fraction, which does not contradict Landau’s criterion and is valid even in the noninteracting case with a finite excitation gap in a finite-size system. The superfluid fraction is $f_{\mathrm{s}}=1$ for a uniform configuration and decreases for an inhomogeneous density. Therefore, Leggett’s formula characterizes modulational instabilities and we have seen the formation of soliton abruptly reduces the superfluid fraction. In particular, at $\omega=-1/2$, we have a black soliton and the superfluid fraction vanishes. We have included effects of transverse width through the one-dimensional nonpolynomial Schr$\ddot{\text{o}}$dinger equation. The superfluid fraction monotonically decreases as one increases the transverse width up to the critical width, above which the superfluid is unstable. We have also investigated the superfluid fraction in the presence of a potential barrier. The potential barrier alters a uniform configuration without a barrier into an inhomogeneous one, which reduces the superfluid fraction. As one increases the attractive interaction strength, the system selects one of the potential minima to develop a soliton. In a dipolar supersolid, the superfluid fraction associated with the nonclassical rotational inertia has been experimentally observed through the measurement of the scissors mode frequency tanzi . We expect that our results of the superfluid fraction in an annular superfluid are also experimentally accessible in a similar manner. A one-dimensional superfluid is a simple and useful platform to analyze various macroscopic quantum phenomena such as Josephson dynamics bouchoule ; tononi20 ; binanti ; vinh . Moreover, the formation of solitons that we have focused on appears ubiquitously not only in condensed-matter physics togawa ; kishine but in high-energy physics nitta ; brauner . Our work on the superfluid properties of the one-dimensional annular superfluid would have a wide range of applications and contributions to further theoretical investigations in diverse physical systems.