Nonlocal pair correlations in the one-dimensional Bose gas at finite temperature

The behavior of the spatial two-particle correlation function is surveyed in detail for a uniform one-dimensional Bose gas with repulsive contact interactions at finite temperatures. Long-, medium-, and short-range effects are investigated. The results span the entire range of physical regimes from ideal gas to strongly interacting and from zero temperature to high temperature (Gross-Pitaevskii) and strongly interacting (Tonks-Girardeau) gases. We present perturbative analytic methods, available at strong and weak couplings, and first-principles numerical results using imaginary time simulations with the gauge-$P$ representation in regimes where perturbative methods are invalid. Nontrivial effects are observed from the interplay of thermally induced bunching behavior versus interaction induced antibunching.

Figure 1

Nonlocal pair correlation $g^{\left(2\right)}\left(r\right)$ in the nearly ideal-gas regime: (a) decoherent classical regime, $\tau ⪢\text{max}\left\{1,{\gamma }^2\right\}$ [Eq. (39)], with $r$ in units of the thermal de Broglie wavelength ${\Lambda }_T=\sqrt{4\pi /\left(\tau n^2\right)}$ and (b) decoherent quantum regime, $\sqrt{\gamma }⪡\tau ⪡1$ [Eq. (45)], with $r$ in units of the phase coherence length $l_{\varphi }=2/n\tau$.

Figure 2

Approach of the pair correlation function to the ideal-gas solution (shown dashed) in the decoherent quantum regime at $\tau =0.1$, with $r$ in units of the thermal de Broglie wavelength, ${\Lambda }_T=\sqrt{4\pi /\tau n^2}$. The thickness of the solid lines (numerical results) comes from the superimposed $1\sigma$ error bars which are below resolution.

Figure 3

Exact behavior of $g^{\left(2\right)}\left(r\right)$, with $r$ in units of ${\Lambda }_T$, in the nearly ideal-gas regime with $\gamma =0.001$ and varying $\tau$ around the quantum/classical crossover. In panel (b), the derivative $f=\partial \left[\text{ln}\mathbf{\left(}{g}^{\left(2\right)}\left(r\right)-1\mathbf{\right)}\right]/\partial r$ shows a clear distinction between exponential decay (when $f$ is constant) and Gaussian thermal-like behavior when $f$ is linear. The triple lines indicate the numerical curves together with $1\sigma$ error bars which are mostly below resolution.

Figure 4

Approach to the classical decoherent gas solution (shown dashed) [Eq. (39)] for finite but small interaction with $\gamma /\sqrt{\tau }=0.03$, which corresponds to a variation in density while keeping the coupling $g$ and $T$ constant. Here $g^{\left(2\right)}\left(0\right)\to 1.925$ in the $\tau \to \infty$ or equivalently $n\to 0$ limit. Triple solid lines are the numerical results, with $1\sigma$ error bars below resolution.

Figure 5

Behavior of $g^{\left(2\right)}\left(r\right)$ in the crossover region between decoherent classical and quantum gas at $\tau =1$. Values of $\gamma$ shown are 0.001, 0.003, 0.01, 0.03, 0.06, 0.1, and 0.2 as the curves for $g^{\left(2\right)}\left(r\right)$ descend.

Figure 6

Nonlocal pair correlation $g^{\left(2\right)}\left(r\right)$ in the weakly interacting regime, with $r$ in units of the healing length $\xi =1/\sqrt{\gamma }n$: (a) low-temperature weekly interacting gas at $\tau ⪡\gamma ⪡1$ [Eq. (59)] and (b) weakly interacting gas at $\gamma ⪡\tau ⪡\sqrt{\gamma }$ [Eq. (62)].

Figure 7

Nonlocal pair correlation $g^{\left(2\right)}\left(r\right)$ as a function of the relative distance $r$ in the strongly interacting regime, $\gamma ⪢1$: (a) regime of high-temperature fermionization, $1⪡\tau ⪡{\gamma }^2$ [Eq. (70)], with $r$ in units of the thermal de Broglie wavelength ${\Lambda }_T=\sqrt{4\pi /\left(\tau n^2\right)}$ and (b) low-temperature Tonks-Girardeau regime [Eq. (71)] for $\tau =0.01$, with $r$ in units of mean interparticle separation $1/n$.

Figure 8

Behavior on the verge of the high-$T$ fermionization regime for ${\gamma }^2/\tau =4$. The dashed line is Eq. (70).

Figure 9

Crossover from decoherent classical to high-temperature fermionization regimes at high temperature.

Figure 10

Situation when $T>T_d$ and the local second-order coherence is apparently unity. All curves plotted correspond to parameter values for which $g^{\left(2\right)}\left(0\right)=1$ in the crossover region between the classical decoherent and high-temperature fermionized gas. The dots (rather than triple lines here, for clarity) indicate $1\sigma$ error bars.

Figure 11

Heights of the anomalous peak of $g^{\left(2\right)}\left(r\right)$ that occurs at nonzero $r_{\text{max}}$, for different values of $\tau$, as functions of (a) $g^{\left(2\right)}\left(0\right)$ and (b) ${\gamma }^2/\tau$. The height is taken to be $h\equiv g^{\left(2\right)}\left(r_{\text{max}}\right)-g^{\left(2\right)}\left(0\right)$ at high temperatures when $g^{\left(2\right)}\left(0\right)>1$ and $h\equiv g^{\left(2\right)}\left(r_{\text{max}}\right)-1$ when $g^{\left(2\right)}\left(0\right)<1$. The two regimes are separated by the dotted-dashed vertical line in (a). Analytic results from Eq. (39) in the decoherent quantum regime are shown as a dashed line. Dots (rather than triple lines here, for clarity) indicate $1\sigma$ error bars on the numerical results.

Figure 12

Regimes and their numerical accessibility: the asterisks indicate the lowest $\tau$ and highest $\gamma$ reachable using the gauge-$P$ method as described in Appendix . The dark dashed line indicates the point at which $g^{\left(2\right)}\left(0\right)=1$.

Figure 13

Deterministic phase space for Stratonovich for of the $d{N}_j$ equation when ${\mu }_e=0$: (a) ungauged and (b) using gauge (A6). The moving singularity in (a) is shown with a large arrow and the attractor in (b) at $\left|N_j\right|=N_a$ with a thick dashed line.