Higher-order interference between multiple quantum particles interacting nonlinearly

The double-slit experiment is the most direct demonstration of interference between individual quantum objects. Since similar experiments with single particles and more slits produce interference fringes reducible to a combination of double-slit patterns, it is usually argued that quantum interference occurs between pairs of trajectories, compactly denoted as second-order interference. Here we show that quantum mechanics in fact allows for interference of arbitrarily high order. This occurs naturally when one considers multiple quantum objects interacting in the presence of a nonlinearity, both of which are required to observe higher-order interference. We make this clear by treating a generalized multislit interferometer using second quantization. We then present explicit experimentally relevant examples both with photons interacting in nonlinear media and an interfering Bose-Einstein condensate with particle-particle interactions. These examples are all perfectly described by quantum theory, and yet exhibit higher-order interference based on multiple particles interacting nonlinearly.

Figure 1

Measuring interference. The simplest interference experiment involves two paths that can be individually blocked. This figure shows a configuration where the upper path is open and the lower path is blocked. Here $\rho$ denotes the input state, and U is some unitary interaction between the two modes. Interference is present if the intensity measured with both paths open is different from the sum of the intensities measured with the individual paths open.

Figure 2

Schematic for measurement of higher-order interference in quantum optics. The input coherent states propagate through blockers and then through a sequence of nonlinear phase shifters $U_{1M},\cdots ,U_{13}$. The final element, marked BS, is a 50-50 beam splitter between the first and the second path, and the intensity is monitored in the first path. This setup gives rise to the $M\mathrm{th}$-order interference, see Eq. (7).

Figure 3

Higher-order interference with Bose-Einstein condensates. The condensate is initially prepared in an even superposition of Gaussian wave functions each with position spread (standard deviation) of $1\mu \mathrm{m}$ centered at $\pm 5\mu \mathrm{m}$ and 0. The solid line presents the values of ${\mathcal{I}}_3$ for the repulsive condensate of ${}^{87}\mathrm{Rb}$ with parameters $N={10}^3$ atoms, $a=5.8$ nm, and $m=1.45\times {10}^{-25}$ kg after an evolution time of $\tau =1$ ms. The dashed line is for the attractive condensate of ${}^7\mathrm{Li}$ with parameters $N=500$ atoms, $a=-1.2$ nm, and $m=1.16\times {10}^{-26}$ kg. Both are experimentally feasible with present day technology [31, 32].