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: “1” for detection) RY, KZ, FT, EB 20/01/2020 Marseille 2020: QS & QW 2 / 15 ˆ U ( τ ) = exp( − iH τ ) ( ℏ = 1), ˆ D = | ψ d ⟩⟨ ψ d | Outcome: a string of length n: “0 , 0 , 0 , . . . , 1” (“0” for non-detection, and
If w Marseille 2020: QS & QW 20/01/2020 RY, KZ, FT, EB p k w w j j 3 / 15 Known results: (Grünbaum, Velázquez et al. 2013) • First-detected “return” time (FDRt): | ψ d ⟩ = | ψ in ⟩ • The system’s energy spectrum is discrete • The FDR amplitude, the generating function(Z Transform): ] n − 1 ˆ n = 1 z n ˆ ⟨ ψ in | ∑ ∞ [ ˆ U ( τ )( 1 − ˆ U ( τ ) | ψ in ⟩ , ˜ U ( n τ ) | ψ in ⟩ φ n = ⟨ ψ in | D ) φ ( z ) = n = 1 z n ˆ 1 + ⟨ ψ in | ∑ ∞ U ( n τ ) | ψ in ⟩ • The system is recurrent n = 1 n | φ n | 2 = w ∈ Z , = the number of zeros { z i } of • The mean: ⟨ n ⟩ = ∑ ∞ φ ( z ) [or the phases exp( iE k τ ) ] =winding number of ˜ ˜ φ ( e i θ ) • The variance: 2z i z ∗ − ⟨ n ⟩ 2 = ⟨ n 2 ⟩ ∑ Var ( n ) = 1 − z i z ∗ i , j = 1 • Charge Theory: { z i } (excluding z 0 = 0) p k ln | e iE k τ − z | ⇒ stationary points ∑ V ( z ) = k = 1 ∑ F ( z ) = e iE k τ − z ⇒ equilibrium points k = 1 l = 1 | ⟨ ψ in | E kl ⟩| 2 = where p k = ∑ g k ⇒ charges on the unit circle.
Example 7 RY, KZ, FT, EB 20/01/2020 Marseille 2020: QS & QW 4 / 15 ∑ H = − γ [ | x ⟩⟨ x + 1 | + | x + 1 ⟩⟨ x | − 2 | x ⟩⟨ x | ] . x = 0 E k /γ = 2 − 2 cos( π k / 4 )
Schematics RY, KZ, FT, EB 20/01/2020 Marseille 2020: QS & QW 5 / 15 Figure: Eight-site ring. τ = 1.
6 / 15 Some zero(s) approach the unit circle: Marseille 2020: QS & QW 20/01/2020 RY, KZ, FT, EB neighboring stars From basic knowledge of electrostatics: “Signature” to jumps of the topological number Mechanism of Jumps of ⟨ n ⟩ and Diverging Var ( n ) • 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 − z i z ∗ j ) = ⇒ diverging var ( n ) • some “weak” charge(s) (“Shrinking”, Grünbaum, Velázquez et al. 2013): L 1 point in sun-earth system • charges merging (“Fusion”, Grünbaum, Velázquez et al. 2013):
Perturbation Method w p k RY, KZ, FT, EB 20/01/2020 Marseille 2020: QS & QW 7 / 15 ∑ F ( z ) = e iE k τ − z = 0 ⇒ critical zero z c k = 1 2 | z c | 2 Var ( n ) ∼ 1 − | z c | 2
Perturbation Method Triple-Charge Theory: Symmetric Scenario Marseille 2020: QS & QW 20/01/2020 RY, KZ, FT, EB 1 p 8 / 15 ( ) d ( z − 2z + d ) ∗ 2 | z σ d | 2 ∑ Var ( n ) ∼ d | 2 + d ) ∗ + c . c . , 1 − z + d ( z − 1 − | z σ σ = ± � �� � M Var ( n ) ∼ 16 ( p 0 + 2p ) 2 ) 2 , the mixing term M is fjnite. τ 2 ( ¯ E + − ¯ E −
Perturbation Method Zeno Regime–Lower Bound RY, KZ, FT, EB 20/01/2020 Marseille 2020: QS & QW 9 / 15 [ 2 cot 2 (∆ E m τ/ 2 ) − w + 2 ] Var ( n ) ≥ ( w − 1 )
Time-Energy Uncertainty Principle RY, KZ, FT, EB Marseille 2020: QS & QW 20/01/2020 10 / 15 Assume ∆ E m τ ≪ ℏ (∆ E m ) 2 (∆ t det ) 2 ≳ 8 ( ⟨ n ⟩ − 1 ) ℏ 2 , where ∆ E m = E max − E min , √ ⟨ ( n τ ) 2 ⟩ − ⟨ n τ ⟩ 2 = τ √ ∆ t det = Var ( n ) In the mathematical limit ⟨ n ⟩ = w = 1, the particle is detected at the fjrst attempt for any sampling time τ . ∆ t det = 0 The presence of the factor ⟨ n ⟩ − 1 is physically reasonable.
Example Eight-Site Ring Marseille 2020: QS & QW 20/01/2020 RY, KZ, FT, EB 11 / 15 B B B (A) (B) C D C D C D A (C) (D) D D D B A C B C B C
Another Example n 2 Marseille 2020: QS & QW 20/01/2020 RY, KZ, FT, EB Interacting-Boson Model n 2 12 / 15 a † a † a † H = − J 2 (ˆ l ˆ a r + ˆ r ˆ a l ) + U (ˆ l + ˆ r ) , ˆ n l , r = ˆ l , r ˆ a l , r √ U 2 + J 2 , E 1 = 4U , E 2 = 3U + √ U 2 + J 2 . E 0 = 3U − 0.5 0.4 Overlaps 0.3 0.2 0.1 0.0 0 2 4 6 8 10 U Figure: | ψ in ⟩ = | 2 , 0 ⟩ , and J is set as 1. The green curve represents p 0 , and the blue/red is p 1 / p 2 . U is large, the ground state is almost | 1 , 1 ⟩ .
Another Example RY, KZ, FT, EB Marseille 2020: QS & QW 20/01/2020 13 / 15 10 4 Var( n ) 10 3 10 2 10 1 0 2 4 6 8 10 U Figure: J = 1, τ = 3
Summary return time based on four scenarios: single-weak charge, two-charge merging, three-charge merging, Zeno regime RY, KZ, FT, EB 20/01/2020 Marseille 2020: QS & QW 14 / 15 • Diverging Var ( n ) accompany isolated jumps of ⟨ n ⟩ • We quantify the magnitude of large fmuctuations of fjrst detected • Topology-dependent time-energy uncertainty relation
Thank You! R. Yin, K. Ziegler, F. Thiel and E. Barkai, Phys. Rev. Research 1, 033086(Editors’ Suggestion) RY, KZ, FT, EB 20/01/2020 Marseille 2020: QS & QW 15 / 15
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