Fluctuations and Stochastic Processes in One-Dimensional Many-Body Quantum Systems

We study the fluctuation properties of a one-dimensional many-body quantum system composed of interacting bosons and investigate the regimes where quantum noise or, respectively, thermal excitations are dominant. For the latter, we develop a semiclassical description of the fluctuation properties based on the Ornstein-Uhlenbeck stochastic process. As an illustration, we analyze the phase correlation functions and the full statistical distributions of the interference between two one-dimensional systems, either independent or tunnel-coupled, and compare with the Luttinger-liquid theory.

Figure 1

Interference contrast distribution $W(\alpha )$ of a ${}^{87}\mathrm{Rb}$ quasicondensate with $n_{1\mathrm{D}}=59\mu {\mathrm{m}}^{-1}$ and ${\omega }_{\perp }=2\pi \times 3\mathrm{kHz}$ as a function of the sampling length $L$ for (a) $T=31\mathrm{nK}$, $J=0$, (b) $T=60\mathrm{nK}$, $J=0$, (c) $T=60\mathrm{nK}$, $J=2\pi \times 1\mathrm{Hz}$, and (d) $T=60\mathrm{nK}$, $J=2\pi \times 3\mathrm{Hz}$. Lines represent the results of the full Bogoliubov modeling for $L=10$ (red), $24$ (green), $37$ (blue), and $51\mu \mathrm{m}$ (black). Symbols (circles, squares, diamonds, and up triangles) of the respective colors show the results of the Luttinger-liquid approach [9] for the same values of $L$ and $J=0$.

Figure 2

Interference contrast distribution $W(\alpha )$ for (a) $k_T{l}_J\to \infty$ and (b) $k_T{l}_J=1.0$. Solid lines display the results of the Ornstein-Uhlenbeck stochastic process modeling for $k_{T}L=1.85$ (red) and 4.43 (black). Results of the full Bogoliubov [Eq. (3)] simulations and Bogoliubov simulations without quantum noise [Eq. (4)] are shown by triangles and circles, respectively.

Figure 3

(a) Interference contrast distribution $W(\alpha )$ for ${}^{87}\mathrm{Rb}$ atoms, $n_{1\mathrm{D}}=60\mu {\mathrm{m}}^{-1}$, $J=0$, and $L=10\mu \mathrm{m}$ calculated for both thermal and quantum fluctuations (solid line) and thermal fluctuations only (dashed line) for two different parameter sets: $T=10\mathrm{nK}$, ${\omega }_{\perp }=2\pi \times 3\mathrm{kHz}$ (I, red) and $T=40\mathrm{nK}$, ${\omega }_{\perp }=2\pi \times 60\mathrm{kHz}$ (II, blue). The inset shows the temperature dependence of $|C{|}^2\equiv ⟨|A(L){|}^2⟩/L^2$ (the average square of the dimensionless coherence factor) for the respective cases. (b) Ratio of the centered $s$th order momenta $R_s$ for $s=2$ (solid line), $s=3$ (dashed line; hardly distinguishable from $s=2$ in case I), and $s=4$ (dot-dashed line) as a function of temperature for two parameter sets at $L=10\mu \mathrm{m}$. Quantum fluctuations manifest themselves in $R_s<1$. (c) The same as in (b) but for $L=20\mu \mathrm{m}$. The effects of atomic shot noise on $⟨|A(L){|}^2⟩$ and $W(\alpha )$ are not taken into account.

Figure 4

The universal (dimensionless) functions $K{\mathcal{A}}_{\mathrm{th}}$ and $K{\mathcal{A}}_q$ as a function of distance (in units of healing length) for uncoupled quasicondensates. Thin solid line: Contribution of the quantum noise; dashed line: its logarithmic asymptotics. Thick lines: Contributions of the thermal noise; the curves are labeled by the respective values of $k_{B}T/(g{n}_{1\mathrm{D}})=1$, 0.3, 0.1, and 0.05.