Dynamical stability of composite quantum particles: When entanglement is enough and when interaction is needed

We consider an evolution of two elementary quantum particles and ask the question: under what conditions does such a system behaves as a single object? It is obvious that if the attraction between the particles is stronger than any other force acting on them the whole system behaves as one. However, recent insight from the quantum information theory suggests that in bipartite systems it is not attraction per se that is responsible for the composite nature, but the entanglement between the parts. Since entanglement can be present between the subsystems that interacted in the past, but do not interact anymore, it is natural to ask when such an entangled pair behaves as a single object. We show that there are situations when entanglement is enough to observe single-particle behavior. However, due to the no-signaling condition, in general an interaction, or a post-selective measurement, is necessary for a complex collective behavior.

Figure 1

Stability of a classical composite system (schematic representation). (a) Two noninteracting particles with correlated momenta stay close to each other after scattering from a flat wall. The wall reflects both particles the same way, so the momenta stay correlated. (b) Two noninteracting particles with correlated momenta drift away from each other after scattering from an irregular wall. Each particle is reflected at a different angle so the correlation between momenta is disturbed. (c) Two interacting particles stay close to each other after scattering from an irregular wall. Although each particle is reflected at a different angle, the attraction keeps them together and guarantees that momenta stay correlated.

Figure 2

Probability density showing example evolution of a double Gaussian wave packet for a separable state (top) and an entangled one (bottom). We assume natural units ($\hslash =1$ and $m=1$).

Figure 3

Probability density showing thermalization of a double Gaussian wave packet for various temperatures. We used natural units $k_B=1$ and $\hslash =1$.

Figure 4

Schematic representation of Mach-Zehnder interferometer.

Figure 5

Two-particle probability density plots showing the evolution in the MZI-like setup for $\phi =0$. The first row corresponds to entangled initial conditions, the second one to separable initial conditions, and the third one to the evolution with an interaction ($\gamma =-10$) between the particles. $T=11$ for the first two cases and $T=50$ for the last one.

Figure 6

Average values $\langle D_3\rangle$ as functions of $\phi$. The first row corresponds to entangled initial conditions, the second one to separable initial conditions, and the third one to the evolution with an interaction ($\gamma =-10$) between the particles. The graphs in each row represent different coarse graining of space which corresponds to different resolutions of the detector $D_3$. The coarse graining effect is especially important in the first two cases with no interaction.

Figure 7

Time evolution of the particle density plots $\langle a_x^{†}{a}_x\rangle +\langle b_x^{†}{b}_x\rangle$ for $\eta =0.4$, which gives $T_{\text{BO}}\approx 15.7$. Left: Noninteracting particles in the entangled state (9) with $\sigma =2,\mathrm{\Sigma }=0.01$ and initially centered around $x=20$. Center: Noninteracting particles in the separable state (9) with $\sigma =0.01,\mathrm{\Sigma }=0.01$ and initially centered around $x=20$. Right: Interacting particles with $\gamma =-2.5$ and initial state $a_{20}^{†}{b}_{20}^{†}|0\rangle$. Bloch oscillations with a fractional period are only visible when interaction is turned on.

Figure 8

Time evolution of the average value $\langle D_2\rangle$. The first row corresponds to entangled initial conditions, the second one to separable initial conditions, and the third one to the evolution with an interaction ($\gamma =-2.5$) between the particles. The graphs in each row represent different coarse graining of space which corresponds to different resolutions of the detector $D_2$. The oscillations with a fractional period can be observed only when interaction is turned on, provided the observed region is smaller than the amplitude of oscillations (amplitude is larger than the resolution of the detector). It is possible to observe more complex oscillations if one can detect exact positions of both particles (the third column).