Bound state influence on long-time non-exponential decay in open quantum systems

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1 Bound state influence on long-time non-exponential decay in open quantum systems Savannah Garmon JSPS Research Fellow, University of Tokyo Collaborators: Dvira Segal, University of Toronto Tomio Petrosky, University of Texas Naomichi Hatano, University of Tokyo Lena Simine, University of Toronto Support: Center for Quantum Information and Quantum Control Max Planck Institute for the Physics of Complex Systems Japan Society for the Promotion of Science

2 Outline: Orientation: description of open quantum systems and prototype model Deviations from exponential decay in quantum mechanics: Short time scales quantum Zeno and anti-zeno effects Long time scales connection with continuum threshold Survival Probability Formalism and Physical Motivation: Relevant studies in the literature Prototype model linear case: Bound state transition to anti-bound state (virtual state) Long time dynamics: long-time near zone P(t) ~ t - Long-time far zone P(t) ~ t -3 Timescale set by Δ Q gap between bound state and threshold Side-coupled impurity model: Bound state trapping below threshold

3 Open Quantum Systems Open quantum system consists of: Discrete system H D Embedded in a larger system (continuum) H C Coupled via H DC Prototype Model: semi-infinite chain with end-point impurity discrete component: H D ε d H DC H D = ε d d + d H C = 2 continuum: H C (c + i c i+ + c + i+ c i ) i=

4 Open Quantum Systems: Prototype model Prototype: /2 2 ε d H = H D + H C + H DC H = ε d d + d 2 (c + i c i+ + c + i+ c i ) 2 (c + d + d + c ) i=

5 Short-time deviations from exponential decay For decades it has been known that deviations from exponential decay exist in quantum systems at least on very short and very long time scales. C. B. Chiu, B. Misra, and E. C. G. Sudarshan, Phys. Rev. D 6, 520 (977). J. Martorell, J. G. Muga, and D. W. L. Spring, Lect. Notes. Phys. 789, 239 (2009). Short time scales typically give rise to parabolic decay: P(t) ~ t 2 Quantum Zeno effect repeated measurements result in decelerated decay quantum anti-zeno effect accelerated decay Experimental confirmation ultra-cold sodium atoms initially trapped in accelerating optical potential: S. R. Wilkinson, C. F. Bharucha, M. C. Fischer, K. W. Madison, P. R. Morrow, Q. Niu, B. Sunduram, and M. G. Raizen, Nature (London) 387, 575 (997). M. C. Fischer, B. Gutiérrez-Medina, and M. G. Raizen, Phys. Rev. Lett. 87, (200).

6 Long-time deviations from exponential decay Long time deviations intimately connected with the continuum threshold. Mathematically proven for quantum systems: L. A. Khalfin, Soc. Phys. JETP 6, 053 (958). M. N. Hack, Phys. Lett. A 90, 220 (982). Typically gives rise to inverse power law decay Typical asymptotic decay law: P(t) ~ t -3 Recent experimental verification: luminescence decay properties of dissolved organic materials following laser excitation: C. Rothe, S. I. Hintschich and A. P. Monkman, Phys. Rev. Lett. 96, 6360 (2006).

7 Formalism: survival probability for an initially prepared state Survival probability: P(t) = A(t) 2 A(t) = ψ 0 e iht ψ 0 Im z 2 = 2πi Γ dz e izt ψ 0 z H ψ 0 Im z 2 Γ Re z Re z 2 background integral 2

8 Physical motivations: bound state at threshold question: what happens as bound state approaches continuum threshold? Im z Re z 2 Answer: the long-time non-exponential decay effects will be amplified as bound state approaches the threshold. Note that bound state transitions to anti-bound state (2 nd sheet) after reaching threshold

9 Relevant studies in the literature T. Jittoh, S. Matsumoto, J. Sato, Y. Sako, and K. Takeda, Phys. Rev. A 7, 0209 (2005). Radial potential: for s-wave component, as energy of initially prepared state approaches threshold, exponential decay suppressed completely. (However, they do not consider bound states). Victor Dinu, Arne Jensen, and Gheorge Nenciu, J. Math. Phys. 50, 0356 (2009). From mathematical physics perspective, authors study a bound state near threshold. Unfortunately, they make several unphysical assumptions. (e.g. Discrete unperturbed energy appearing at threshold cannot form a bound state below threshold??) Other relevant studies TBA

10 Prototype model: continuum and discrete spectra Return to prototype: /2 2 ε d H = H D + H C + H DC H = ε d d + d 2 (c + i c i+ + c + i+ c i ) 2 (c + d + d + c ) i=

11 Prototype model: continuum spectrum take continuum limit and introduce half-chain Fourier series: H = ε d d d + Continuum: k [ 0,π ] on π ε 0 k c π k c k + V 0 k ε k = cosk ε k (c k d + d c k ) ( N ) p k threshold -

12 Prototype model: discrete spectrum Obtain the discrete spectrum from: d z H d = z ε d Σ(z) yields linear polynomial Linear dispersion: z ε d Σ(z) = z ε d 2 (z z2 ) yields solution: z L (ε d ) = ε d + 4ε d however, let s examine the dispersion more carefully

13 Prototype model: Bound state absorption into continuum Solution given from slightly re-written dispersion: 2 z ε d z2 = 0 Note that the root vanishes for z = ±, which occurs at: ε d = ± 2 These are the points where solution crosses one Riemann sheet into the other (localization/delocalization).

14 Prototype model: linear dispersion plot z L (ε d ) = ε d + 4ε d Re z L 3 2 bound state Ε d transitions to anti-bound state 2 3 we have purely non-exponential decay for ε d < /2

15 Long-time dynamics for prototype model Focus on the purely non-exponential case: Im z 2 vicinity of lower threshold 2 2 Re z define z L + Δ Q A th (t) = 2πi i dz e izt dz e izt η I (z) i η II (z) 2 z 2 = i dz e izt 2πiε d z z L (Δ Q )

16 Long-time dynamics: near zone and far zone A th (t) = e it s 2 2its dz e s 2πiε d t 2 0 Δ Q + i s t Consider the timescale: << t << (Δ Q ) (Long-time near zone ) A th (t) 2 ~ t Asymptotic limit: (Δ Q ) << t (Long-time far zone ) A th (t) 2 ~ t 3 Note that for Δ Q = 0, the near zone becomes fully asymptotic

17 Long-time dynamics: numerical results for prototype model g=0.5, ε d = 0.5 g=0.5, ε d = 0.4 P(t) t S. Garmon, T. Petrosky, L. Simine, and D. Segal, Fortschritte der Physik, DOI: 0.002/prop ; arxiv:

18 Long-time dynamics for general open quantum systems Similar effect observed in the following works: S. Longhi, Phys. Rev. Lett. 97, 0402 (2006). S. Garmon, Ph.D. thesis, University of Texas at Austin (2007). Axel D. Dente, Raúl A. Bustos-Marún, and Horacio M. Pastawski, Phys. Rev. A 78, 0626 (2008). Straight-forward to demonstrate: time scale separating near and far zones should always be inversely related to Δ Q in OQS S. Garmon, T. Petrosky, L. Simine, and D. Segal, Fort. Physik, DOI: 0.002/prop ; arxiv: (202). The transition from bound to anti-bound states also leads to a maximum in the local density of states: Raúl A. Bustos-Marún, Eduardo A. Coronado, and Horacio M. Pastawski, Phys. Rev. B 82, (200).

Further demonstration: bound state trapped below threshold Side-coupled impurity model (or T-model): -N/2-2

19 Further demonstration: bound state trapped below threshold Side-coupled impurity model (or T-model): -N/ N/2! µ "/2! d "g H = ε d d d + π ε π k π c k c k + V k (c k d + d c k ) π integra(on over the full chain

20 Side-coupled impurity model: van Hove singularities H = ε d d d + π ε π k π c k c k + V k (c k d + d c k ) π E k = cosk E k DOS singularities at E k = ±B k ρ(e k ) = dk de k = B 2 E k 2

21 Side-coupled impurity: bound state trapped below threshold We again find (quartic) dispersion from resolvent method: z ε d = g 2 z 2 Ε S. Garmon, H. Nakamura, N. Hatano, and T. Petrosky, Phys. Rev. B 80, 538 (2009). van Hove DOS singularity prevents persistent bound state reaching threshold

22 Long-time dynamics for bound state trapped below threshold Bound state eigenvalue expansion: z = Δ Q + O(g 8 ) with Δ Q 2(+ε d ) 2 g4 In the near zone: A th (t) 2 = Δ 2 Q 2πg 2 t = g 4 2π(+ε d ) 4 t Hence closing the gap requires ε d goes to infinity, which kills the effect in any case.

23 Long-time dynamics: numerical results for side-coupled impurity model P(t) g=0., ε = 0.5 d g=0.2, ε d = 0.5 g=0.3, ε = 0.5 d t Due to bound states, resonance, difficult to see much

24 Conclusions Bound state influence on long time dynamics in OQS: Bound state transition to anti-bound state at continuum threshold Purely non-exponential dynamics when only anti-bound states are present Long time dynamics for prototype model: Long-time near zone: P(t) ~ t - Long-time far zone: P(t) ~ t -3 Amplification of non-markovian decay as bound state transitions to anti-bound state; near zone becomes asymptotic dynamics

Bound state influence on long-time power law decay in open quantum systems

Bound state influence on long-time power law decay in open quantum systems Savannah Garmon University of Toronto, Chemical Physics Theory Group Collaborators: Dvira Segal, University of Toronto Tomio Petrosky,