Length scales involved in decoherence of trapped bosons by buffer-gas scattering

We ask and answer a basic question about the length scales involved in quantum decoherence: how far apart in space do two parts of a quantum system have to be before a common quantum environment decoheres them as if they were entirely separate? We frame this question specifically in a cold atom context. How far apart do two populations of bosons have to be before an environment of thermal atoms of a different species (“buffer gas”) responds to their two particle numbers separately? An initial guess for this length scale is the thermal coherence length of the buffer gas; we show that a standard Born-Markov treatment partially supports this guess, but predicts only inverse-square saturation of decoherence rates with distance, and not the much more abrupt Gaussian behavior of the buffer gas's first-order coherence. We confirm this Born-Markov result with a more rigorous theory, based on an exact solution of a two-scatterer scattering problem, which also extends the result beyond weak scattering. Finally, however, we show that when interactions within the buffer-gas reservoir are taken into account, an abrupt saturation of the decoherence rate does occur, exponentially on the length scale of the buffer gas's mean free path.

Figure 1

Decoherence resolution length scale. Bosons of one type are coherently distributed between two tight single-mode traps. Surrounding reservoir particles interact with them by scattering, causing decoherence between different distribution states. At what separation length scale does the decoherence rate saturate?

Figure 2

Rate function $R\left(x\right)$ vs separation between the two scattering centers in units of reservoir coherence length, $x=s/\lambda$. Blue solid line: Born-Markov decoherence rate coefficient $R\left(x\right)$. At large $x$, $R\left(x\right)$ falls off only as $1/\left(2x^2\right)$ (black dotted line), instead of the much more abrupt $e^{-x^2}$ decay (red dashed line) of spatial coherence in the thermal gas between any two points separated by a distance $x\lambda$.

Figure 3

Contour of integration for the $k$ integral in (A2), for some arbitrary fixed $r_{\pm }$, at a time before (black dashed line) and after (red solid line) the time-dependent saddle point $k_{\mathrm{sp}}=\frac{m}{\hslash t}{r}_{\pm }$ reaches the pole in $+k_0$. The points $k_{\mathrm{sp}}^{0,1}$ here denote the saddle points at the two different times.

Figure 4

Comparison between the outgoing spherical wave from one of the scatterers obtained by steepest descent (black dashed line) and by numerical integration in (A2) (red solid line) for $a_{\pm }=3/2,1/2$, $k_0=10$ and incidence angle $\pi /6$ at time $t=10$ (the separation $s$ has been taken as length unit).

Figure 5

Examples of crossing points of curves $e^{-2q_{n}s}$ (blue line) and $(q_{n}s-\frac{s}{a_{-}})(q_{n}s-\frac{s}{a_{+}})$ (black lines) for different values of $\frac{a_{\pm }}{s}$. The crossing points (red) determine the roots of $\Lambda \left(k\right)$, i.e., energies of bound states when there are two (dotted line, with $\frac{s}{a_{\pm }}=2,3$), one (dashed line, $\frac{s}{a_{\pm }}=1,\frac{1}{2}$), or no such bound states (solid line, $\frac{s}{a_{\pm }}=-3,-\frac{1}{2}$).