One-dimensional system. Bethe-ansatz theory

We saw in the previous subsection that for a weakly interacting impurity the drag force appears only when the impurity velocity $V$ is larger than the Landau critical velocity, which is equal to the velocity of sound $c$. The situation is, however, different in the Bethe-ansatz Lieb-Liniger theory of a 1D Bose gas [LL63]. According to this theory excitations in the system actually have a fermionic nature. Even a low frequency perturbation can create a particle-hole pair with a total momentum near $2p_{F}\equiv 2\hbar k_{F}=\hbar 2\pi n_{1D}$. To calculate the drag force for this case we will use the dynamic form factor of the system $\sigma \left(\omega, k\right)$ (we follow notation of [LP80], §87). The dissipated energy at $T=0$ can be calculated as

$$\dot{E}=-\int\limits_{-\infty}^{\infty}\frac{dk}{2\pi}\int\l... ...ega,k\right) \left\vert U\left(\omega, k\right) \right\vert^2,$$ (7.50)

where $U(\omega, k)=2\pi {g_{i}}\delta (\omega -kV)$ is the Fourier transform of the impurity potential $U(t,z)={g_{i}}\delta (z-Vt)$. One has $\left\vert U\left(\omega,k\right)\right\vert^2=2\pi{g_{i}}^2t\delta(\omega-kV)$, where $t$ is ''time of observation''. Thus the energy dissipation per unit of time is

$$\dot{E}=-F_{V}V=-\frac{{g_{i}}^2n_{1D}V}{\hbar}\int\limits_0^{\infty}%% \frac{dk}{2\pi}k\sigma\left(kV,k\right),$$ (7.51)

where $F_{V}$ is the drag force. We will try to estimate the velocity dependence of $F_{V}$.

For low frequency dissipation the important values of $\k$ are near $2k_{F}$. According to [NLCC94]

$$\sigma \left(\omega, 2k_{F}\right) \sim \omega^{\left(\eta -2\right) },\omega \rightarrow 0,$$ (7.52)

where $\eta =\frac{2\hbar k_{F}}{mc}=\frac{2\pi \hbar n_{1D}}{mc}\geq 2$ is the characteristic parameter of a 1D Bose gas. In the mean-field limit when $n_{1D}\rightarrow\infty$ the parameter $\eta\rightarrow \infty$. In the opposite case of a small density bosons behave as impenetrable particles (Tonks-Girardeau limit [Gir60]) and the dynamic form-factor coincides with the one of an ideal Fermi gas. In this limit $\eta = 2$.

In the general case one can calculate $\sigma \left(\omega, k\right)$ at small $\omega$ and $k\approx 2k_{F}$ generalizing the method of Haldane [Hal81] for the case of time-dependent correlation functions. Calculations give

$$\sigma \left(\omega, k\right) =\frac{n_{1D}c}{\omega^2}\left... ...t)^{\eta}f\left(\frac{c\Delta k}{\omega}\right), \omega>0,k>0,$$ (7.53)

where $k=2k_{F}+\Delta k$ and the function $f(x)$ is

$$f\left(x\right) = A\left(\eta\right)\left(1-x^2\right)^{\eta/2-1}$$ (7.54)

in the interval $\left\vert x\right\vert <1$ and is equal to zero at $\vert x\vert\geq 1$ (see also [KBI93]). The constant $A\left(\eta \right)$ can be calculated in two limiting cases: $A\left(\eta =2\right) =\pi/4$ (see [PS03] §17.3) and $A\left(\eta\right) \approx 4\pi^2/\left[\left(8C\right)^{\eta}\Gamma^2\left(\frac{\eta}2\right)\right]$, where $C=1.78$... is the Euler 's constant (see Eq. 1.186), for $\eta \gg 1$.

Substituting $(\ref{So})$ into $(\ref{deltaF})$ we finally find velocity dependence of the drag force:

$$F_{V}=\frac{\Gamma \left(\frac{\eta}2\right)}{2\sqrt{\pi}\Ga... ...i}}^2n_{1D}^2}{%% \hbar V}\left(\eta \frac{V}{c}\right)^{\eta}$$ (7.55)

Equation $\left(\ref{FV}\right)$ is valid for the condition $V\ll c$.

Thus in the Tonks-Girardeau strong-interaction limit $F_{V}\sim V$ and Bose gas behaves, from the point of view of friction, as a normal system, where the drag force is proportional to the velocity. On the contrary, in the mean-field limit the force is very small and the behavior of the system is analogous to a 3D superfluid. However, even in this limit the presence of the small force makes a great difference. Let us imagine that our system is twisted into a ring, and that the impurity rotates around the ring with a small angular velocity. If the system is superfluid in the usual sense of the word, the superfluid part must stay at rest. Presence of the drag force means that equilibrium will be reached only when the gas as a whole rotates with the same angular velocity. From this point of view the superfluid part of the 1D Bose gas is equal to zero even at $T=0$. Notice that in an earlier paper [Son71] the author concluded that $\rho_{s}=\rho$ at $T=0$ for arbitrary $\eta$. We believe that this difference results from different definitions of $\rho_{s}$ and reflects the non-standard nature of the system.

Equation $\left(\ref{FV}\right)$ is equivalent to a result which was obtained by a different method in [BGB01], with a model consisting of an impurity considered as a Josephson junction. Notice that the process of dissipation, which in the language of fermionic excitations can be described as creation of a particle-hole pair, corresponds in the mean-field limit to creation of a phonon and a small-energy soliton. It seems that such a process cannot be described in the mean-field approach in the linear approximation.

Experimental confirmation of these quite non-trivial predictions demands a true one-dimensional condensate, where non mean-field effects can be sufficiently large. Such condensates have been investigated for the first time in experiments [SKC+01,GVL+01]. In experiments [GVL+01,SMS+04] condensates have been created in the form of elongated independent ''needles'' in optical traps, consisting of two perpendicular standing laser waves. The role of an impurity in this case must be played by a light sheet, perpendicular to the axis of condensates and moving along them.

Notice also, that application of the additional light waves in this experiments of this type allows one to create a harmonic perturbation of the form

$$U(t,z)=U_0\cos \left(\omega t-kz\right),\qquad k=2k_{F}+\Delta k$$ (7.56)

with small $\omega$ and $\Delta k$. Such potential with was used in [GVL+01,SMS+04] for experiments with 1D condensate in a periodic lattice. However, for a small amplitude $U_0$, measurement of the dissipation energy $Q$ gives, according to $\left(\ref{Q}\right)$, the dynamic form-factor $S\left(\omega, k\right)$ directly.