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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. 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. Anderson introduced the eponymous model of disordered lattices which exhibits localization, the confinement of the electrons' probability distributions within some small volume.

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.^[1] 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.^[2]

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.^[3] 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

Consider a wavepacket $Ψ (x, t) \propto \exp [- x^{2} / (2 σ)^{2} + i k_{0} x]$ with initial momentum $ℏ k_{0}$ which evolves in the general Hamiltonian of a Gaussian, uncorrelated, disordered medium:

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

where $\overline{V (x)} = 0$ and $\overline{V (x) V (x^{'})} = γ δ (x - x^{'})$ , 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 $| Ψ^{2} |$ 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.^[3]

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:

\partial_{t} ⟨ {\hat{x}}^{2} ⟩ = \frac{1}{2 i ℏ m} ⟨ [{\hat{x}}^{2}, {\hat{p}}^{2}] ⟩ = - \frac{1}{2 m} (\hat{x} \hat{p} + \hat{p} \hat{x}) \approx 2 v_{0} ⟨ \hat{x} ⟩_{+} - 2 v_{0} ⟨ \hat{x} ⟩_{-},

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

⟨ x ⟩_{\pm} \equiv \int_{- \infty}^{\infty} x n_{\pm} (x, t) d x .

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

\partial_{t} ⟨ {\hat{x}}^{2} ⟩ = 2 v_{0} ⟨ \hat{x} ⟩ .

This yields long-time behavior

⟨ \hat{x} (t) ⟩ = 64 ℓ {(\frac{τ}{t})}^{2} \log (1 + \frac{t}{τ}),

where $ℓ$ and $τ$ 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 $R_{[n / n]}$ extracted from a series expansion derived with the Berezinskii diagrammatic technique.^[3]

@@ Line 1: / Line 1: @@
 {{Short description|Quantum mechanical effect in disordered media}}
 {{Quantum book backlink|Quantum dynamics and evolution}}
+{{Quantum article nav|previous=Physics:Quantum jump|previous label=Jump|next=Physics:Quantum chaos|next label=Chaos}}
 {{EngvarB|date=July 2022}}
+<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">
+<div style="width:280px;">
+__TOC__
+</div>
+<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
+'''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. 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. Anderson introduced the eponymous model of disordered lattices which exhibits localization, the confinement of the electrons' probability distributions within some small volume.
+</div>
+<div style="width:300px;">
+[[File:QBE Theory.svg|thumb|280px|Quantum boomerang effect.]]
+</div>
-The '''quantum boomerang effect''' is a [[Physics:Quantum mechanics|quantum mechanical]] phenomenon whereby [[Physics:Wave packet|wavepackets]] launched through [[Randomness|disordered]] media return, on average, to their starting points, as a consequence of [[Physics:Anderson localization|Anderson localization]] and the inherent symmetries of the system. At early times, the initial [[Physics:Parity|parity]] asymmetry of the nonzero momentum leads to asymmetric behavior: nonzero [[Displacement (geometry)|displacement]] of the wavepackets from their origin. At long times, inherent [[Physics:T-symmetry|time-reversal symmetry]] and the confining effects of Anderson localization lead to correspondingly symmetric behavior: both zero final [[Physics:Velocity|velocity]] and zero final displacement.<ref name="PratPRA">
+</div>
-{{Cite journal
-| doi = 10.1103/PhysRevA.99.023629
-| volume = 99
-| issue = 2
-| article-number = 023629
-| last1 = Prat
-| first1 = Tony
-| last2 = Delande
-| first2 = Dominique
-| last3 = Cherroret
-| first3 = Nicolas
-| title = Quantum boomeranglike effect of wave packets in random media
-| journal = Physical Review A
-| access-date = 2022-02-03
-| date = 2019-02-27
-| arxiv = 1704.05241
-| bibcode = 2019PhRvA..99b3629P
-| s2cid = 126938499
-| url = https://link.aps.org/doi/10.1103/PhysRevA.99.023629}}</ref>
 ==History==
-In 1958, [[Biography:Philip W. Anderson|Philip W. Anderson]] introduced the [[Physics:Anderson localization|eponymous model]] of disordered [[Bravais lattice|lattices]] which exhibits localization, the confinement of the electrons' probability distributions within some small volume.<ref name="Anderson">
+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.<ref name="Anderson">
 {{Cite journal
 | doi = 10.1103/PhysRev.109.1492
@@ Line 40: / Line 34: @@
 | bibcode = 1958PhRv..109.1492A
 | url = https://link.aps.org/doi/10.1103/PhysRev.109.1492| url-access = subscription
-}}</ref> 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 [[Physics:Transport phenomena|transport properties]] of the lattice are reduced as a result of localization, turning what might have been a [[Physics:Electrical conductor|conductor]] into an [[Physics:Insulator (electricity)|insulator]]. Modern [[Physics:Condensed matter physics|condensed matter]] models continue to study disorder as an important feature of real, imperfect materials.<ref name="Abanin">
+}}</ref> 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.<ref name="Abanin">
 {{Cite journal
 | doi = 10.1103/RevModPhys.91.021001
@@ Line 63: / Line 57: @@
 | url = https://link.aps.org/doi/10.1103/RevModPhys.91.021001}}</ref>
-In 2019, theorists considered the behavior of a wavepacket not merely dropped, but actively launched through a disordered medium with some initial nonzero [[Finance:Momentum|momentum]], predicting that the wavepacket's [[Physics:Center of mass|center of mass]] would [[Asymptotic analysis|asymptotically]] return to the origin at long times — the quantum boomerang effect.<ref name="PratPRA"/> Shortly after, [[Quantum simulator|quantum simulation]] experiments in [[Physics:Ultracold atom|cold atom]] settings confirmed this prediction<ref name="SajjadPRX">
+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.<ref name="PratPRA">
+{{Cite journal
+| doi = 10.1103/PhysRevA.99.023629
+| volume = 99
+| issue = 2
+| article-number = 023629
+| last1 = Prat
+| first1 = Tony
+| last2 = Delande
+| first2 = Dominique
+| last3 = Cherroret
+| first3 = Nicolas
+| title = Quantum boomeranglike effect of wave packets in random media
+| journal = Physical Review A
+| access-date = 2022-02-03
+| date = 2019-02-27
+| arxiv = 1704.05241
+| bibcode = 2019PhRvA..99b3629P
+| s2cid = 126938499
+| url = https://link.aps.org/doi/10.1103/PhysRevA.99.023629}}</ref> Shortly after, quantum simulation experiments in cold atom settings confirmed this prediction<ref name="SajjadPRX">
 {{Cite journal
 | doi = 10.1103/PhysRevX.12.011035
@@ Line 117: / Line 130: @@
 | date=8 February 2022
 | access-date=20 June 2022}}</ref>
-by simulating the [[Physics:Kicked rotator|quantum kicked rotor]], a model that maps to the Anderson model of disordered lattices.<ref name="FishmanPRL">
+by simulating the quantum kicked rotor, a model that maps to the Anderson model of disordered lattices.<ref name="FishmanPRL">
 {{Cite journal
 | doi = 10.1103/PhysRevLett.49.509
@@ Line 139: / Line 152: @@
 ==Description==
-[[File:QBE Theory.svg|thumb|600px|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 [[Physics:Mean free path|mean free path]] <math>\ell</math>, but a quantum wavepacket returns to and localizes at the origin.<ref name="PratPRA"/> The infinite limit of the [[Padé approximant]] factor is approximated with <math>R_{[5/5]}</math> for the purposes of this figure.]]
+[[File:QBE Theory.svg|thumb|600px|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 <math>\ell</math>, but a quantum wavepacket returns to and localizes at the origin.<ref name="PratPRA"/> The infinite limit of the Padé approximant factor is approximated with <math>R_{[5/5]}</math> for the purposes of this figure.]]
 Consider a wavepacket <math>\Psi(x,t)\propto\exp\left[-x^2/(2\sigma)^2+ik_0x\right]</math> with initial momentum <math>\hbar k_0</math> which evolves in the general Hamiltonian of a Gaussian, uncorrelated, disordered medium:
@@ Line 147: / Line 160: @@
 where <math>\overline{V(x)}=0</math> and <math>\overline{V(x)V(x')}=\gamma\delta(x-x')</math>, 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 [[Physics:T-symmetry|time-reversal symmetry]] (or some other [[Unitary operator|unitary]] or [[Antiunitary operator|antiunitary]] symmetry<ref name="JanarekPRB">
+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<ref name="JanarekPRB">
 {{Cite journal
 | doi = 10.1103/PhysRevB.105.L180202
@@ Line 168: / Line 181: @@
 | bibcode = 2022PhRvB.105r0202J
 | s2cid = 247593916
-| url = https://link.aps.org/doi/10.1103/PhysRevB.105.L180202}}</ref>), the [[Physics:Probability density function|probability density distribution]] <math>|\Psi^2|</math> exhibits off-diagonal, oscillatory elements in its [[Eigenvalues and eigenvectors#Eigenspaces, geometric multiplicity, and the eigenbasis|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.<ref name="PratPRA"/>
+| url = https://link.aps.org/doi/10.1103/PhysRevB.105.L180202}}</ref>), the probability density distribution <math>|\Psi^2|</math> 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.<ref name="PratPRA"/>
-The same [[Physics:Wave interference|destructive interference]] argument used to justify Anderson localization applies to the quantum boomerang. The [[Ehrenfest theorem]] states that the [[Variance|variance]] (''i.e.'' the spread) of the wavepacket evolves thus:
+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:
 ::<math>\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_-,</math>
-where the use of the [[Wigner quasiprobability distribution|Wigner function]] allows the final approximation of the particle distribution into two populations <math>n_\pm</math> of positive and negative velocities, with centers of mass denoted
+where the use of the Wigner function allows the final approximation of the particle distribution into two populations <math>n_\pm</math> of positive and negative velocities, with centers of mass denoted
 ::<math>\langle x\rangle_\pm\equiv\int\limits_{-\infty}^\infty x n_\pm(x,t)\mathrm{d}x.</math>
@@ Line 186: / Line 199: @@
 ::<math>\langle\hat{x}(t)\rangle=64\ell\left(\frac{\tau}{t}\right)^2\log\left(1+\frac{t}{\tau}\right),</math>
-where <math>\ell</math> and <math>\tau</math> are the scattering [[Physics:Mean free path|mean free path]] and scattering [[Physics:Mean free time|mean free time]], respectively. The exact form of the boomerang can be approximated using the diagonal [[Padé approximant|Padé approximants]] <math>R_{[n/n]}</math> extracted from a series expansion derived with the [[Biography:Vadim Berezinskii|Berezinskii]] diagrammatic technique.<ref name="PratPRA"/>
+where <math>\ell</math> and <math>\tau</math> 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 <math>R_{[n/n]}</math> extracted from a series expansion derived with the Berezinskii diagrammatic technique.<ref name="PratPRA"/>
-==References==
+== See also ==
+{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
+= References =
 {{reflist}}
@@ Line 194: / Line 210: @@
 [[Category:Quantum mechanics]]
-{{Sourceattribution|Quantum boomerang effect}}
+{{Author|Harold Foppele}}
+{{Sourceattribution|Physics:Quantum boomerang effect|1}}

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