Quantum limits to center-of-mass measurements

We discuss the issue of measuring the mean position (center of mass) of a group of bosonic or fermionic quantum particles, including particle number fluctuations. We introduce a standard quantum limit for these measurements at ultralow temperatures, and discuss this limit in the context of both photons and ultracold atoms. In the case of non-interacting harmonically trapped fermions, we present evidence that the Pauli exclusion principle has a strongly beneficial effect, giving rise to a $1∕N$ scaling in the position standard deviation—as opposed to a $1∕\sqrt{N}$ scaling for bosons. The difference between the actual mean-position fluctuation and this limit is evidence for quantum wave-packet spreading in the center of mass. This macroscopic quantum effect cannot be readily observed for noninteracting particles, due to classical pulse broadening. For this reason, we also study the evolution of photonic and matter-wave solitons, where classical dispersion is suppressed. In the photonic case, we show that the intrinsic quantum diffusion of the mean position can contribute significantly to uncertainties in soliton pulse arrival times. We also discuss ways in which the relatively long lifetimes of attractive bosons in matter-wave solitons may be used to demonstrate quantum interference between massive objects composed of thousands of particles.

Figure 1

Deviation of the variance (relative to the wave-packet variance, ${\sigma }^2$) of the quasi-intensive c.m. coordinate (dashed) from the true intensive c.m. coordinate (solid) for a heralded coherent state.

Figure 2

Mean position variances for 1D coherent pulses of $\overline{n}={10}^4$ particles propagating in a dispersive medium. The three figures correspond to three different wave-packet shapes: (a) the soliton solution to the 1D nonlinear Schrödinger equation $⟨\widehat{\varphi }\left(\zeta \right)⟩=\frac{1}{2}\mathrm{sech}\left(\frac{1}{2}\zeta \right)$, (b) a wider pulse of the form $⟨\widehat{\varphi }\left(\zeta \right)⟩\propto \mathrm{sech}\left(\frac{1}{4}\zeta \right)$, and (c) a narrower pulse $⟨\widehat{\varphi }\left(\zeta \right)⟩\propto \mathrm{sech}\left(\zeta \right)$. The solid lines represent the analytically determined total variance $⟨\mathrm{\Delta }{\widehat{\mathfrak{X}}}^2⟩∕x_0^2$, while the dashed and dash-dotted lines represent the numerical results for the classical $({\sigma }_{SQL}^2∕x_0^2)$ and quantum $({\sigma }_{QM}^2∕x_0^2)$ contributions, respectively.

Figure 3

Possible soliton double slit interference experiment. The soliton wave packet (I) broadens over time due to the linear increase in the quantum uncertainty in the mean position. Lasers then eliminate solitons outside of two “slits” (II). Any remaining soliton is left to propagate until it undergoes absorption imaging (III).