Large fmuctuations of the fjrst detected quantum return time Ruoyu - - PowerPoint PPT Presentation

• R. Yin, K. Ziegler, F. Thiel and E. Barkai,
• Phys. Rev. Research 1, 033086

Large fmuctuations of the fjrst detected quantum return time

Ruoyu Yin Bar Ilan University 20th Jan. 2020

RY, KZ, FT, EB 20/01/2020 Marseille 2020: QS & QW 1 / 15

Measurement Protocol

Repeated strong measurements until the fjrst success: ˆ U(τ) = exp(−iHτ) (ℏ = 1), ˆ D = |ψd⟩⟨ψd|

Outcome: a string of length n: “0, 0, 0, . . . , 1” (“0” for non-detection, and “1” for detection)

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If

• First-detected “return” time (FDRt): |ψd⟩ = |ψin⟩
• The system’s energy spectrum is discrete

Known results: (Grünbaum, Velázquez et al. 2013)

• The FDR amplitude, the generating function(Z Transform):

φn = ⟨ψin| [ˆ U(τ)(1 − ˆ D) ]n−1 ˆ U(τ) |ψin⟩ , ˜ φ(z) =

⟨ψin| ∑∞

n=1 zn ˆ

U(nτ)|ψin⟩ 1+⟨ψin| ∑∞

n=1 zn ˆ

U(nτ)|ψin⟩

• The system is recurrent
• The mean: ⟨n⟩ = ∑∞

n=1 n|φn|2 = w ∈ Z, = the number of zeros {zi} of

˜ φ(z) [or the phases exp(iEkτ)] =winding number of ˜ φ(eiθ)

• The variance:

Var(n) = ⟨ n2⟩ − ⟨n⟩2 =

w

∑

i,j=1

2ziz∗

j

1 − ziz∗

j

• Charge Theory: {zi} (excluding z0 = 0)

V(z) =

w

∑

k=1

pk ln |eiEkτ − z| ⇒ stationary points F(z) =

w

∑

k=1

pk eiEkτ − z ⇒ equilibrium points where pk = ∑gk

l=1 | ⟨ψin| Ekl⟩|2 =

⇒ charges on the unit circle.

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Example

H = −γ

7

∑

x=0

[|x⟩⟨x + 1| + |x + 1⟩⟨x| − 2|x⟩⟨x|] . Ek/γ = 2 − 2 cos(πk/4)

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Schematics

Figure: Eight-site ring. τ = 1.

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Mechanism of Jumps of ⟨n⟩ and Diverging Var(n)

Some zero(s) approach the unit circle:

• Mean: when the zero(s) “reach” the unit circle, it(they) does(do) not

count into ⟨n⟩ (canceled out)

• Var(n): when the zero(s) is(are) approaching to the unit circle =

⇒ vanishing denominator (1 − ziz∗

j ) =

⇒ diverging var(n) “Signature” to jumps of the topological number From basic knowledge of electrostatics:

• some “weak” charge(s) (“Shrinking”, Grünbaum, Velázquez et al.

2013): L1 point in sun-earth system

• charges merging (“Fusion”, Grünbaum, Velázquez et al. 2013):

neighboring stars

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Perturbation Method

F(z) =

w

∑

k=1

pk eiEkτ − z = 0 ⇒ critical zero zc Var(n) ∼ 2|zc|2 1 − |zc|2

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Perturbation Method

Triple-Charge Theory: Symmetric Scenario

Var(n) ∼ ∑

σ=±

2|zσ

d|2

1 − |zσ

d|2 +

( 2z+

d (z− d )∗

1 − z+

d (z− d )∗ + c.c.

) ,

• M

Var(n) ∼ 16(p0 + 2p)2 p 1 τ 2 (¯ E+ − ¯ E− )2 , the mixing term M is fjnite.

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Perturbation Method

Zeno Regime–Lower Bound

Var(n) ≥ (w − 1) [ 2 cot2 (∆Emτ/2) − w + 2 ]

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Time-Energy Uncertainty Principle

Assume ∆Emτ ≪ ℏ (∆Em)2(∆tdet)2 ≳ 8(⟨n⟩ − 1)ℏ2, where ∆Em = Emax − Emin, ∆tdet = √ ⟨(nτ)2⟩ − ⟨nτ⟩2 = τ √ Var(n) In the mathematical limit ⟨n⟩ = w = 1, the particle is detected at the fjrst attempt for any sampling time τ. ∆tdet = 0 The presence of the factor ⟨n⟩ − 1 is physically reasonable.

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Example

Eight-Site Ring

(A) (C) (B) (D)

A B B C

D C D B C D

A B C B

D C D B C D

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Another Example

Interacting-Boson Model

H = −J 2(ˆ a†

l ˆ

ar + ˆ a†

rˆ

al) + U(ˆ n2

l + ˆ

n2

r), ˆ

nl,r = ˆ a†

l,rˆ

al,r E0 = 3U − √ U2 + J2, E1 = 4U, E2 = 3U + √ U2 + J2.

2 4 6 8 10 U 0.0 0.1 0.2 0.3 0.4 0.5 Overlaps

Figure: |ψin⟩ = |2, 0⟩, and J is set as 1. The green curve represents p0, and the blue/red is p1/p2. U is large, the ground state is almost |1, 1⟩.

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Another Example

2 4 6 8 10 U 101 102 103 104 Var(n)

Figure: J = 1, τ = 3

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Summary

• Diverging Var(n) accompany isolated jumps of ⟨n⟩
• We quantify the magnitude of large fmuctuations of fjrst detected

return time based on four scenarios: single-weak charge, two-charge merging, three-charge merging, Zeno regime

• Topology-dependent time-energy uncertainty relation

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Thank You!

• R. Yin, K. Ziegler, F. Thiel and E. Barkai,
• Phys. Rev. Research 1, 033086(Editors’ Suggestion)

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