Composite particles with minimum uncertainty in spacetime

Composite particles—atoms, molecules, or microspheres—are unique tools for testing joint quantum and general relativistic effects, macroscopic limits of quantum mechanics, and searching for new physics. However, all studies of the free propagation of these particles find that they delocalize into separate internal energy components, destroying their spatial coherence. This renders them unsuitable for experimental applications, as well as theoretical studies where they are used as idealized test masses or clocks. Here we solve this problem by introducing a class of states with minimal uncertainty in spacetime that fully overcome the delocalization. The relevant physics comes from minimizing the uncertainty between position and velocity, rather than position and momentum, while directly accounting for mass as an operator. Our results clarify the nature of composite particles, providing a currently missing theoretical tool with direct relevance for studies of joint foundations of quantum and relativistic phenomena, which removes a roadblock that could limit near-future quantum tests using composite particles.

Figure 1

Propagation of a generic Gaussian state in an equal superposition (${\alpha }_i\equiv 1/\sqrt{3}$) of mass-energies (top) and the propagation of our position-and-velocity minimum uncertainty state (bottom). Initial states at $t=0$ (dashed gray lines), final state at $t=5$ time steps (solid colored lines for each mass component, natural units). In the Gaussian state, the mass components become separated and spread out at different rates. In our minimum uncertainty state, the components propagate together for all times and spread at the same rate.

Figure 2

Wigner functions of time-evolved Gaussian state (left column) and our minimum uncertainty state (right column) in phase space (top row) and configuration space (bottom row). All states are initially (at $t=0$) centered at the origins of the plots, and the plots show the state after $t=3$ time steps (in natural units). Each state is comprised of three masses, $m=\{0.5,1,2\}$. The mass-energy components in the generic Gaussian state delocalize in both position and velocity due to different propagation speeds. In our minimum uncertainty state, the mass-energy components remain localized in position and velocity, and the full state follows a semiclassical trajectory. In phase space, our minimum uncertainty state shows correlations between mass-energies and peak wave-packet momenta as expected from their common velocity.

Figure 3

Double-slit interference of generic Gaussian superposition of two masses (top), and minimum uncertainty state (bottom). The mass-energy components of each (dotted and solid colored lines) are explicitly shown along with the overall superpositions (thick, black line). Note that the interference pattern for the MUS is more pronounced, while the Gaussian interference washes out quickly as we move out from the center. Insets: In the many-mass limit, both interference patterns approach a smooth classical distribution. The bright fringe in the center is an artefact of our idealized case. We assume $N$ masses with a gap $\sim 1/N$, keeping mean mass and variance the same for all plots.

Figure 4

Position-space probability amplitudes for propagating states over four time slices. Both generic Gaussian (top row) and our MUS (bottom row, shaded) are in a superposition of three masses, the smallest mass being the blue, thin line, the largest the green, dashed line. Axes are the same scale for all plots. While in the Gaussian states different mass components exhibit the position-focusing at different times, in the MUS all internal states reach the the minimum position variance simultaneously.

Figure 5

Position-space probability amplitude of the generic Gaussian (left) and our MUS (right) for different time slices, chosen such that maximal contraction occurs for a specific mass component (orange, thick line). The smallest mass is represented by the blue, solid line, the largest mass by the green, dashed line. Axes are adjusted from Fig. 1, but with the same scale for both plots. Apart from the different times of maximal contraction for each mass component in the Gaussian state, which is absent in our MUS, the minimal widths for the same mass also differ between these two cases. This difference arises from the difference in the initial state of the mass components in the Gaussian state and in the MUS.