Revealing Quantum Statistics with a Pair of Distant Atoms

Quantum statistics have a profound impact on the properties of systems composed of identical particles. At the most elementary level, Bose and Fermi quantum statistics differ in the exchange phase, either 0 or $\pi$, which the wave function acquires when two identical particles are exchanged. In this Letter, we demonstrate that the exchange phase can be directly probed with a pair of massive particles by physically exchanging their positions. We present two protocols where the particles always remain spatially well separated, thus ensuring that the exchange contribution to their interaction energy is negligible and that the detected signal can only be attributed to the exchange symmetry of the wave function. We discuss possible implementations with a pair of trapped atoms or ions.

Figure 1

Detection of the wave function symmetry in a two-particle interference experiment. (a) Exchanging two identical particles multiplies the wave function by a global phase factor $e^{i{\phi }_{\text{ex}}}=\pm 1$, which—without a reference state—is not observable. Dynamical and geometrical phases are assumed to vanish. (b) By splitting the wave function into a reference path and another path for which the particles’ positions are switched, ${\phi }_{\text{ex}}$ can be detected by correlation measurements after recombining the two paths. The interference signal is controlled by an additional phase $\phi$, induced by a potential or by the geometry.

Figure 2

Two-atom Ramsey interferometer sequence probing quantum statistics with two distant neutral atoms. A spin parity measurement produces a two-atom Ramsey-like fringe, whose phase depends on ${\phi }_{\text{ex}}$. To recombine the atoms, a position-dependent $\pi$ pulse [41, 42, 43] is applied to the outermost sites, $L_3$, $R_3$, see Ref. [34]. The arrows indicate the spin state for the different paths, $n$ denotes the initial separation, and time is expressed in units of shift operations. A two-dimensional variant is presented in the Supplemental Material [34], which ensures that the two atoms always stay far apart.

Figure 3

Trapped-ion protocol. (a) The two-ion wave function is treated as a one-dimensional quantum rotor (shown for fermions in the same internal state). An adiabatic transformation splits a single-well (1) into a double well potential (3) that is subsequently merged again into a single well, but at a different position (5). In the case of fermions (bosons), the final state (5) has opposite (same) parity as compared to that of the initial state (1). (b) In the radial rf-quadrupolar potential of a linear trap, a two-ion rotor (yellow spheres) can be aligned with the $x$ axis by dc voltages reducing the confinement along the $x$ axis (top). For zero dc voltage, the radial symmetry of the confining potential is broken by the orientation of the micromotion (red arrows) with respect to the rotor axis [34]. The rotor will align under an angle $\theta =\pm \pi /4$ with respect to the $x$ axis (bottom). (c) Contour lines of the time-averaged Coulomb and trapping potential as a function of the relative position vector $\mathit{r}$ for three different potentials.