Preparing Quantum Backflow States by Large Momentum Transfer

TL;DR

Quantum backflow—negative probability flux despite a strictly positive momentum spectrum—has remained unobserved experimentally. This work proposes using large momentum transfer (LMT) pulses within an atom interferometry sequence to flexibly prepare highly backflowing states in a non-interacting Bose-Einstein condensate, with one arm receiving consecutive $\pi$ pulses to accumulate large momentum separation. A theoretical framework combines free-propagation dynamics in a moving frame with laser-induced transitions to derive the final backflow state and express the observable flux $J(x,t)$ and critical density $\rho_{\rm crit}$; simulations under realistic $^{88}$Sr parameters show tunable backflow flux and a substantial reduction of classical backflow, controlled by the beam-splitter phase and pulse sequence. The results extend the parameter space for backflow engineering and point toward experimental observation, while highlighting practical imaging challenges due to rapid density modulation and signaling a need for alternative density-extraction approaches.

Abstract

A quantum backflow state refers to a quantum state exhibiting negative probability density flux albeit a completely positive momentum spectrum. Extending earlier work that uses single laser pulse to prepare quantum backflow state in an ultracold atomic BEC [1], we theoretical investigated flexible quantum backflow state preparation via large momentum transfer technique, which to our knowledge, has not been studied before. By combining atom interferometry theory and non-interacting BEC wave function, we solve for the evolution of a BEC wavepacket under atom interferometry sequence. Simulation results show a highly tunable backflow flux and critical density under our scheme, and can be manipulated to go beyond existing numbers.

Figures (9)

• Figure 1: Flexible quantum backflow state preparation procedure. (a) Initial cloud launching upward with momentum $\hbar k_0$ (b) Beam splitter pulse (not necessarily $\pi/2$. (c),(d),(e) $\pi$ pulse on one arm to tailor momentum (f) Wavepackets encounter to form a backflow state.
• Figure 2: Qualitative illustration of LMT setup. (a): overall setup of LMT array. The left most arrow represents splitting pulse. The subsequent $\pi-$laser pulses only address on one arm, causing it to alternate between ground and excitation state as well as transferring momentum boost. (b): Zoom in view of two consecutive pulses. Duration of each pulse is fixed as $\tau$, while pulse interval $T=t_{n+1}-t_n$ may vary. States labeled as $\ket{\Psi_n}$ and $\ket{\Psi_{n+1}}$ are defined as the quantum state right after each pulse.
• Figure 3: Simulation setup for preparing backflow states. Pulses only address $\ket{\Psi_{b}}$ (blue curve), hence it undergoes large momentum transfer while $\ket{\Psi_f}$ (red curve) evolves freely. At $t=0$, a splitting pulse (shown as purple dashed line) stimulates a proportion of the condensate into excited state and creates initial velocity difference. $\ket{\Psi_b}$ is then accelerated by consecutive $\pi-$pulse.
• Figure 4: Momentum spectrum of final combined state $\braket{k|\Psi}$. The two peaks correspond to the two arms' momentum at encounter. No negative momentum is observed.
• Figure 5: Example probability flux $J$ for initial splitting parameter $\Omega \tau=0.6\pi$ plotted against relative position to COM $x-x_c$. (a) Overall view; a considerable proportion of flux is below zero, demonstrating successful backflow. (b) Zoom-in view of positions $0.3\;\rm\mu m$ near COM; dashed regions correspond to backflow occurring.