Quantum boomerang effect

The quantum boomerang effect is a quantum mechanical phenomenon whereby wavepackets launched through disordered media return, on average, to their starting points, as a consequence of Anderson localization and the inherent symmetries of the system. At early times, the initial parity asymmetry of the nonzero momentum leads to asymmetric behavior: nonzero displacement of the wavepackets from their origin. At long times, inherent time-reversal symmetry and the confining effects of Anderson localization lead to correspondingly symmetric behavior: both zero final velocity and zero final displacement.^[1]

History

In 1958, Philip W. Anderson introduced the eponymous model of disordered lattices which exhibits localization, the confinement of the electrons' probability distributions within some small volume.^[2] In other words, if a wavepacket were dropped into a disordered medium, it would spread out initially but then approach some maximum range. On the macroscopic scale, the transport properties of the lattice are reduced as a result of localization, turning what might have been a conductor into an insulator. Modern condensed matter models continue to study disorder as an important feature of real, imperfect materials.^[3]

In 2019, theorists considered the behavior of a wavepacket not merely dropped, but actively launched through a disordered medium with some initial nonzero momentum, predicting that the wavepacket's center of mass would asymptotically return to the origin at long times — the quantum boomerang effect.^[1] Shortly after, quantum simulation experiments in cold atom settings confirmed this prediction^[4][5][6] by simulating the quantum kicked rotor, a model that maps to the Anderson model of disordered lattices.^[7]

Description

 

The center-of-mass trajectory of a quantum wavepacket in a disordered medium with some initial momentum. A classical wavepacket obeying the Boltzmann equation localizes at the terminus of its mean free path ${\displaystyle \ell }$ , but a quantum wavepacket returns to and localizes at the origin.^[1] The infinite limit of the Padé approximant factor is approximated with ${\displaystyle R_{[5/5]}}$  for the purposes of this figure.

Consider a wavepacket ${\displaystyle \Psi (x,t)\propto \exp \left[-x^{2}/(2\sigma )^{2}+ik_{0}x\right]}$  with initial momentum ${\displaystyle \hbar k_{0}}$  which evolves in the general Hamiltonian of a Gaussian, uncorrelated, disordered medium:

${\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V({\hat {x}}),}$ 

where ${\displaystyle {\overline {V(x)}}=0}$  and ${\displaystyle {\overline {V(x)V(x')}}=\gamma \delta (x-x')}$ , and the overbar notation indicates an average over all possible realizations of the disorder.

The classical Boltzmann equation predicts that this wavepacket should slow down and localize at some new point — namely, the terminus of its mean free path. However, when accounting for the quantum mechanical effects of localization and time-reversal symmetry (or some other unitary or antiunitary symmetry^[8]), the probability density distribution ${\displaystyle |\Psi ^{2}|}$  exhibits off-diagonal, oscillatory elements in its eigenbasis expansion that decay at long times, leaving behind only diagonal elements independent of the sign of the initial momentum. Since the direction of the launch does not matter at long times, the wavepacket must return to the origin.^[1]

The same destructive interference argument used to justify Anderson localization applies to the quantum boomerang. The Ehrenfest theorem states that the variance (i.e. the spread) of the wavepacket evolves thus:

${\displaystyle \partial _{t}\langle {\hat {x}}^{2}\rangle ={\frac {1}{2i\hbar m}}\left\langle \left[{\hat {x}}^{2},{\hat {p}}^{2}\right]\right\rangle =-{\frac {1}{2m}}\left({\hat {x}}{\hat {p}}+{\hat {p}}{\hat {x}}\right)\approx 2v_{0}\langle {\hat {x}}\rangle _{+}-2v_{0}\langle {\hat {x}}\rangle _{-},}$ 

where the use of the Wigner function allows the final approximation of the particle distribution into two populations ${\displaystyle n_{\pm }}$  of positive and negative velocities, with centers of mass denoted

${\displaystyle \langle x\rangle _{\pm }\equiv \int \limits _{-\infty }^{\infty }xn_{\pm }(x,t)\mathrm {d} x.}$ 

A path contributing to ${\displaystyle \langle {\hat {x}}\rangle _{-}}$  at some time must have negative momentum ${\displaystyle -\hbar k_{0}}$  by definition; since every part of the wavepacket originated at the same positive momentum ${\displaystyle \hbar k_{0}}$  behavior, this path from the origin to ${\displaystyle x}$  and from initial ${\displaystyle \hbar k_{0}}$  momentum to final ${\displaystyle -\hbar k_{0}}$  momentum can be time-reversed and translated to create another path from ${\displaystyle x}$  back to the origin with the same initial and final momenta. This second, time-reversed path is equally weighted in the calculation of ${\displaystyle n_{-}(x,t)}$  and ultimately results in ${\displaystyle \langle {\hat {x}}\rangle _{-}=0}$ . The same logic does not apply to ${\displaystyle \langle {\hat {x}}\rangle _{+}}$  because there is no initial population in the momentum state ${\displaystyle -\hbar k_{0}}$ . Thus, the wavepacket variance only has the first term:

${\displaystyle \partial _{t}\langle {\hat {x}}^{2}\rangle =2v_{0}\langle {\hat {x}}\rangle .}$ 

This yields long-time behavior

${\displaystyle \langle {\hat {x}}(t)\rangle =64\ell \left({\frac {\tau }{t}}\right)^{2}\log \left(1+{\frac {t}{\tau }}\right),}$ 

where ${\displaystyle \ell }$  and ${\displaystyle \tau }$  are the scattering mean free path and scattering mean free time, respectively. The exact form of the boomerang can be approximated using the diagonal Padé approximants ${\displaystyle R_{[n/n]}}$  extracted from a series expansion derived with the Berezinskii diagrammatic technique.^[1]

References

1. Prat, Tony; Delande, Dominique; Cherroret, Nicolas (27 February 2019). "Quantum boomeranglike effect of wave packets in random media". Physical Review A. 99 (2): 023629. arXiv:1704.05241. Bibcode:2019PhRvA..99b3629P. doi:10.1103/PhysRevA.99.023629. S2CID 126938499. Retrieved 3 February 2022.
2. Anderson, P. W. (1 March 1958). "Absence of Diffusion in Certain Random Lattices". Physical Review. 109 (5): 1492–1505. Bibcode:1958PhRv..109.1492A. doi:10.1103/PhysRev.109.1492. Retrieved 11 February 2022.
3. Abanin, Dmitry A.; Altman, Ehud; Bloch, Immanuel; Serbyn, Maksym (22 May 2019). "Colloquium: Many-body localization, thermalization, and entanglement". Reviews of Modern Physics. 91 (2): 021001. arXiv:1804.11065. Bibcode:2019RvMP...91b1001A. doi:10.1103/RevModPhys.91.021001. S2CID 119270223. Retrieved 1 July 2022.
4. Sajjad, Roshan; Tanlimco, Jeremy L.; Mas, Hector; Cao, Alec; Nolasco-Martinez, Eber; Simmons, Ethan Q.; Santos, Flávio L.N.; Vignolo, Patrizia; Macrì, Tommaso; Weld, David M. (23 February 2022). "Observation of the Quantum Boomerang Effect". Physical Review X. 12 (1): 011035. arXiv:2109.00696. Bibcode:2022PhRvX..12a1035S. doi:10.1103/PhysRevX.12.011035. S2CID 237385885. Retrieved 23 February 2022.
5. Chen, Sophia (23 February 2022). "A Bose-Einstein-Condensate Boomerang". Physics. 15: s24. Bibcode:2022PhyOJ..15..s24C. doi:10.1103/Physics.15.s24. S2CID 247113461. Retrieved 1 July 2022.
6. Emily Conover (8 February 2022). "The quantum 'boomerang' effect has been seen for the first time". ScienceNews. Retrieved 20 June 2022.
7. Fishman, Shmuel; Grempel, D. R.; Prange, R. E. (23 August 1982). "Chaos, Quantum Recurrences, and Anderson Localization". Physical Review Letters. 49 (8): 509–512. Bibcode:1982PhRvL..49..509F. doi:10.1103/PhysRevLett.49.509. Retrieved 11 February 2022.
8. Janarek, Jakub; Grémaud, Benoît; Zakrzewski, Jakub; Delande, Dominique (26 May 2022). "Quantum boomerang effect in systems without time-reversal symmetry". Physical Review B. 105 (18): L180202. arXiv:2203.11019. Bibcode:2022PhRvB.105r0202J. doi:10.1103/PhysRevB.105.L180202. S2CID 247593916. Retrieved 1 July 2022.