Squeezed-field path-integral description of second sound in Bose-Einstein condensates

We propose a generalization of the Feynman path integral using squeezed coherent states. We apply this approach to the dynamics of Bose-Einstein condensates, which gives an effective low-energy description that contains both a coherent field and a squeezing field. We derive the classical trajectory of this action, which constitutes a generalization of the Gross-Pitaevskii equation, at linear order. We derive the low-energy excitations, which provides a description of second sound in weakly interacting condensates as a squeezing oscillation of the order parameter. This interpretation is also supported by a comparison to a numerical c-field method.

Figure 1

(a) Illustration of the ordered state of a weakly interacting condensate, described by ${\left|\varphi \right|}^4$ theory. The bosons condense in the minimum of the Mexican hat potential $V\left(\psi \right)=-{\mu \left|\psi \right|}^4+g{\left|\psi \right|}^4/2$, with $\mu ,g>0$. The distribution, shown in red, is squeezed. Panel (b) shows a sketch of a single path in the path integral in which both the expectation value and the squeezing of the distribution vary in time. Panel (c) shows schematically the corresponding fields $\psi \left(t\right)$ and $\eta \left(t\right)$.

Figure 2

(a) Illustration of the ground state, cf. Fig. 1, and its representation as an equilibrium state of the coherent field ${\psi }_0$ and the squeezing field ${\eta }_0$. Panel (b) shows the coherent field eigenmodes, which correspond to the Bogolyubov modes, and panel (c) shows the squeezing field eigenmodes, which are breathing modes around the equilibrium state. In panel (d), we show the dispersion of the squeezing field $\hslash {\omega }_{\mathbf{k},\eta }$ as a red continuous line, its low-energy approximation $\hslash c_2\left|\mathbf{k}\right|$ as a red, dashed line, the Bogolyubov mode $\hslash {\omega }_{\mathbf{k}}$ as a blue line, and its low-energy approximation $\hslash c_1\left|\mathbf{k}\right|$ as a blue, dashed line. These are shown in units of the mean-field energy $g{n}_0$ and as a function of $k\xi$, where $\xi$ is the healing length $\xi =\hslash /\sqrt{2mg{n}_0}$.

Figure 3

Histogram of the field ${\psi }_i\left(t\right)$ at times $t_j$, following a momentum kick at time $t=0$, as given in Eq. (12). For the full time evolution see Ref. [21]. In the upper row, we depict the full distribution, in the lower row, we remove the rotational motion. The white circle has the radius $\sqrt{n_0}$, where $n_0$ is the numerically determined condensate fraction. The dynamics consist of a rotation of the cloud around the origin, as well as a breathing motion, which we capture within the squeezed-field approach.

Figure 4

Single-particle correlation function $c_1(\mathbf{k},\omega )$ as a function of frequency $\omega$ and momentum $k_x$, for the interaction $U/J=0.05$. The two excitation branches are compared to the Bogolyubov dispersion ${\omega }_{\mathbf{k}}$, blue dashed line, and the side peak at ${\omega }_{\mathbf{k}}+{\omega }_{\mathbf{k},\eta }$, red dashed line, motivated by the pole structure of Eq. (10). The inset shows half of the Brillouin zone, the large figure the low-energy regime, where the approximations ${\omega }_{\mathbf{k}}\approx c_1\left|\mathbf{k}\right|$ and ${\omega }_{\mathbf{k},\eta }\approx c_2\left|\mathbf{k}\right|$ are used.

Figure 5

(a) Illustration of Eq. (A41) via Feynman diagrams, where the forward propagators correspond to particle and the backward to hole excitations. The inclusion of the squeezing field complements the “coherent propagator” (continuous lines) with the “squeezing propagator” (dashed lines). (b) $\left|g_1(k\xi ,\omega )\right|$ as a function of $\omega$ and dimensionless $k\xi$. The four branches correspond to the poles of $g_1(\mathbf{k},\omega )$. For increasing $\mathbf{k}$, the branches that correspond to the poles $\pm ({\omega }_{\mathbf{k}}+{\omega }_{\mathbf{k},\eta })$ vanish first, followed by the pole $-{\omega }_{\mathbf{k}}$. The branch that corresponds to the pole $+{\omega }_{\mathbf{k}}$ remains. (c) Weights of the branches in (b) as a function of $k\xi$. The weights $g_{\mathbf{k}}^{(\pm )}$ that emerged from the inclusion of the squeezing field decay very quickly, followed by $v_{\mathbf{k},0}^2$. The only surviving weight, i.e., $u_{\mathbf{k},0}^2\to 1$, gives rise to the single-particle Green's function of a free particle.

Figure 6

Single-particle correlation function $c_1(\mathbf{k},\omega )$ as a function of frequency $\omega$ and momentum $k_x$, for the interactions $U/J=0.05$, 0.09, and 0.13. The two excitation branches are compared to the Bogolyubov dispersion ${\omega }_{\mathbf{k}}$, lower (blue) dashed line, and the side peak at ${\omega }_{\mathbf{k}}+{\omega }_{\mathbf{k},\eta }$, upper (red) dashed line, motivated by the pole structure of Eq. (A41). The vertical dashed lines denote the inverse healing length ${\xi }^{-1}$.