Wave-Particle Duality of Many-Body Quantum States

We formulate a general theory of wave-particle duality for many-body quantum states, which quantifies how wavelike and particlelike properties balance each other. Much as in the well-understood single-particle case, which-way information—here, on the level of many-particle paths—lends particle character, while interference—here, due to coherent superpositions of many-particle amplitudes—indicates wavelike properties. We analyze how many-particle which-way information, continuously tunable by the level of distinguishability of fermionic or bosonic, identical and possibly interacting particles, constrains interference contributions to many-particle observables and thus controls the quantum-to-classical transition in many-particle quantum systems. The versatility of our theoretical framework is illustrated for Hong-Ou-Mandel-like and Bose-Hubbard-like exemplary settings.

Popular Summary

The duality of wave- and particlelike behavior played a key role in the development of quantum theory. For single quantum objects, it manifests in a complementarity between the interference of different paths the particle can take (wavelike behavior) and information about which path the particle took (particlelike behavior). This was expressed quantitatively by modern uncertainty relations, which were repeatedly confirmed in experiments. In ensembles of many identical particles, however, particle indistinguishability can additionally give rise to interference between different many-particle paths—in the sense of which particle took which way—on top of single-particle interference. Whether this intricate behavior on the many-particle level can likewise be understood in terms of modern wave-particle-duality relations remained an open question. In our work, we formulate such many-body wave-particle-duality relations.

We develop a general theoretical framework for the description of many partially distinguishable particles. This framework introduces—on the many-body level—a measure for the particle character, which quantifies the maximum available information about “which particle took which way,” as well as a measure of the wave character, which quantifies how well different many-particle paths can interfere with each other. We find that these measures are complementary, in the sense that the extent of one of these measures limits the extent of the other measure.

Our analysis provides a general framework for many-particle interference phenomena in terms of a many-body complementarity principle and opens new insights into the quantum-to-classical transition on the many-particle level.

Figure 1

Three-particle transition amplitudes of (a) indistinguishable (indicated by identical coloring), (b) partially (two different colors), and (c) fully (three different colors) distinguishable particles. In the fully indistinguishable case, six indistinguishable three-particle transition amplitudes add coherently to determine the output event’s probability [32]. In the partially distinguishable case, the six transition amplitudes from diagram (a) fall apart into three mutually distinct pairs of interfering two-particle amplitudes. In the fully distinguishable case, no—now perfectly distinguishable—transition amplitudes superimpose coherently anymore.

Figure 2

Single-particle double-slit experiment in the presence of a which-path detector (transparent box). A single particle (blue ball) passes through a double slit and is detected on a screen. The amount of information on the particle’s path, obtained by the which-path detector, determines the visibility of the interference pattern on the screen.

Figure 3

Example of a state of $N=3$ partially distinguishable particles with mode occupation list $\overrightarrow{R}=(2,1,0,\dots ,0)$ and, correspondingly, mode assignment list $\overrightarrow{E}=(1,1,2)$. External states are depicted by black arrows, particles by colored balls, and internal states by the balls’ coloring, with the yellow envelope illustrating correlations between the particles. Particle distinguishability is determined by the listed coefficients $C_{\overrightarrow{\mathcal{I}}}$.

Figure 4

Complementarity relations for 300 randomly generated states of three partially distinguishable particles, with $l$ referring to the number of mixed internal states (see main text for details). Panels (a)–(d) show the wave character measures ${\mathcal{W}}_P^2$ (blue circles) and ${\mathcal{W}}_C^2$ (red triangles) plotted against the particle character quantifier ${\mathcal{P}}_F^2$, while in panels (e)–(h), they are plotted against ${\mathcal{P}}_T^2$. In all panels, the solid black line corresponds to the upper bound according to Eq. (43).

Figure 5

General setting of a many-particle interference experiment with a state $\rho$ of $N$ partially distinguishable particles in $n$ modes (colored balls covered by a yellow envelope illustrating correlations in the internal degrees of freedom). The external state ${\rho }_E$ evolves according to a many-particle, possibly interacting, unitary $\mathcal{U}\in \mathrm{U}(n^N)$ (illustrated in blue) and is measured by a POVM $\mathcal{M}$ with outcomes $1,\dots ,k$ (illustrated in red). As a consequence of wave-particle duality, the visibility of many-particle interference is bounded by the particles’ degree of distinguishability.

Figure 6

Experimental settings to illustrate the complementarity relations (67) and (68), for systems of partially distinguishable, possibly interacting particles. (a) Hong-Ou-Mandel experiment: $N=2$ noninteracting partially distinguishable particles (colored balls), which can be correlated in their internal degrees of freedom (yellow envelope), evolve according to $\mathcal{U}=u^{\otimes N}$, with $u$ the single-particle unitary matrix of the balanced beam splitter. The output mode occupation $\overrightarrow{S}$ is measured according to the measurement ${\mathcal{M}}_O$ (see main text) with an interference contrast controlled by Eq. (67). (b) Double-well Bose-Hubbard model: Each site (or mode) initially contains two particles, with particles on distinct sites prepared in distinct internal states with mutual overlap $\gamma$ (see main text). The difference in the potential wells’ on-site energies is controlled by the tilt $F$. Particles can tunnel with rate $J$ and interact with strength $U$. After an evolution for some time $t$, different observables exhibit an interference contrast, which is bounded by the complementarity relation (67); see Fig. 8.

Figure 7

Many-particle interference visibility ${𝒱}_T$ in the double-well Bose-Hubbard model (78) for the particle occupation measurement ${\mathcal{M}}_O$ and tilt $F=0$. Panels (a)–(d) show the visibility ${𝒱}_T$ as a function of the evolution time $t$ for different levels of particle distinguishability in the internal state (79), as controlled by $\gamma =1/4$, $1/2$, $3/4$, and 1, respectively. Independently of the interaction strength $U/J$, and as predicted by Eq. (65), the visibility ${𝒱}_T$ is bounded for all evolution times $t$ by the particle indistinguishability $1-{\mathcal{D}}_T$ (red dashed line), which is a monotonously increasing function of $\gamma$, as illustrated in the inset of panel (a).

Figure 8

Many-particle interference visibility ${𝒱}_T$ in the double-well Bose-Hubbard model for the evolution time $t=4\hslash /J$ and tilt $F=J$. For the particle occupation measurement ${\mathcal{M}}_O$, as well as the correlation measurements ${\mathcal{M}}_{1P}$, ${\mathcal{M}}_{2P}$, ${\mathcal{M}}_{3P}$, and ${\mathcal{M}}_{4P}$, panels (a)–(d) show the visibility ${𝒱}_T$ as a function of the interaction strength $U/J$ for $\gamma =1/4$, $1/2$, $3/4$, and 1, respectively. Just like in Fig. 7, for all measurements and all interaction strengths $U/J$, the visibility ${𝒱}_T$ is bounded by the particle indistinguishability $1-{\mathcal{D}}_T$ (red dashed line). See the inset in Fig. 7 for the relation between $1-{\mathcal{D}}_T$ and $\gamma$.