Macroscopic Zeno effect and stationary flows in nonlinear waveguides - PowerPoint PPT Presentation

Macroscopic Zeno effect and stationary flows in nonlinear waveguides with localized dissipation D. A. Zezyulin 1 , V. V. Konotop 1 , G. Barontini 2 , and H. Ott 2 1 Center for Theoretical and Computational Physics, University of Lisbon, Portugal 2 Department of physics and OPTIMAS research center, University of Kaiserslautern, Germany Preprint available: arXiv:1112.4324; to appear in Phys. Rev. Lett. 2nd Conference on Localized Excitations in Nonlinear Complex Systems D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 1 / 17

Quantum Zeno effect QZE — slowing down the dynamics of a quantum system subjected to frequent measurements or to a strong coupling to another quantum system. theory: B. Misra and E. C. G. Sudarshan, J. Math. Phys. Sci. 18 , 756 (1977) experiment: single ions: W. M. Itano et. al., Phys. Rev. A 41 , 2295 (1990) ultracold atoms in accelerated optical lattices: M. C. Fischer et al., Phys. Rev. Lett. 87 , 040402 (2001) photons in a cavity: J. Bernu et al. , Phys. Rev. Lett. 101 , 180402 (2008) cold molecular gases in an optical lattice: N. Syassen et al., Science 320 , 1329 (2008) More generally, ZE can be understood as the effect of changing a decay law depending on the frequency of measurements. L. A. Khalfin, Usp. Fiz. Nauk 160 , 185 (1990); [Sov. Phys. Usp. 33 , 868 (1990)] D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 2 / 17

Zeno effect in a macroscopic system gas of condensed bosonic atoms in the macroscopic dynamics the frequency of measurement can be interpreted as the strength of the induced dissipation V. S. Shchesnovich and V. V. Konotop, Phys. Rev. A 81 , 053611 (2010) Macroscopic ZE — the effect of the dissipation on the macroscopic characteristics of the system. D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 3 / 17

Model of a nonlinear waveguide i Ψ t = − Ψ xx + g | Ψ | 2 Ψ − i γ ( x )Ψ localized dissipation: γ ( x ) = Γ 0 f ( x /ℓ ) | f ( x ) | → 0 as x → ±∞ max | f ( x ) | = f (0) ∼ 1, | f x ( x ) | ∼ 1 Γ 0 — amplitude ℓ — characteristic width � ∞ −∞ | γ ( x ) | dx ∝ Γ 0 ℓ D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 4 / 17

Stationary flows � x � � “hydrodynamic” form: Ψ( t , x ) = ρ ( x ) exp i 0 v ( s ) ds − i µ t µ — chemical potential n ( x ) = ρ 2 ( x ) — density v ( x ) — superfluid velocity j ( x ) = v ( x ) n ( x ) — superfluid current stationary equations: ρ xx + µρ − g ρ 3 − j 2 ρ − 3 = 0 , j x + γ ( x ) ρ 2 = 0 boundary conditions for stationary flows : density is a fixed constant: | ρ ( x ) | → ρ ∞ as x → ±∞ ( ρ ∞ = 1 for all our results) current is some constant: j ( x ) → ∓ j ∞ as x → ±∞ The main question: how j ∞ depends on Γ 0 and ℓ ? D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 5 / 17

D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 6 / 17

D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 7 / 17

Example: An exact solution ρ xx + µρ − g ρ 3 − j 2 ρ − 3 = 0 , j x + γ ( x ) ρ 2 = 0 γ ( x ) = 3Γ 0 sech 2 ( x /ℓ ) A branch of stationary flows: ρ ( x ) = tanh( x /ℓ ) j ( x ) = − Γ 0 ℓ tanh 3 ( x /ℓ ) additional constraint: ℓ 2 ( g + Γ 2 0 ℓ 2 ) = 2 j ∞ = Γ 0 ℓ j ∞ grows monotonously with Γ 0 and ℓ . No macroscopic Zeno effect. D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 8 / 17

Another example: Dissipation with finite support ρ xx + µρ − g ρ 3 − j 2 ρ − 3 = 0 , j x + γ ( x ) ρ 2 = 0 1 − x 2 /ℓ 2 � 2 � � Γ 0 if | x | < ℓ, γ ( x ) = 0 otherwise no exact solution; let’s start from Γ 0 = 0 symmetric mode: ρ ( x ) = 1, j ( x ) = 0 � antisymmetric mode: ρ ( x ) = tanh( g / 2 x ), j ( x ) = 0 we numerically construct “symmetric” and “antisymmetric” branches starting from Γ 0 = 0 D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 9 / 17

D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 10 / 17

Dependence on the width D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 11 / 17

Two different situations: γ ( x ) = 3Γ 0 sech 2 ( x /ℓ ) ρ ( x ) = tanh( x /ℓ ) j ( x ) = − Γ 0 ℓ tanh 3 ( x /ℓ ) j ∞ = Γ 0 ℓ no MZE dissipation with finite support: different manifestations of MZE What is the difference? Let’s look at the asymptotic behavior of the flows at x → ±∞ D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 12 / 17

Difference in asymptotic behavior γ ( x ) = 3Γ 0 sech 2 ( x /ℓ ) ρ ( x ) = tanh( x /ℓ ) ρ ( x ) ∼ 1 − 2 e − 2 x /ℓ as x → ∞ asymptotic behavior is determined by ℓ dissipation with finite support √ Λ x , where Λ = 2 g − 4 j 2 ρ ∞ − ρ ( x ) ∝ e − ∞ asymptotic behavior is determined by j ∞ Λ > 0: j ∞ < j max � = g / 2 — maximal possible value of j ∞ ∞ D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 13 / 17

Making a bridge γ ( x ) = 3Γ 0 sech 2 ( x /ℓ ) D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 14 / 17

Making a bridge γ ( x ) = 3Γ 0 sech 2 ( x /ℓ ) ρ 1 ( x ) = 1 − ρ ( x ), ρ 1 ( x ) → 0 as x → ±∞ ρ 1 , xx − Λ ρ 1 = 12 j ∞ Γ 0 ℓ e − 2 x /ℓ , Λ = 2 g − 4 j 2 ∞ D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 14 / 17

Making a bridge γ ( x ) = 3Γ 0 sech 2 ( x /ℓ ) ρ 1 ( x ) = 1 − ρ ( x ), ρ 1 ( x ) → 0 as x → ±∞ ρ 1 , xx − Λ ρ 1 = 12 j ∞ Γ 0 ℓ e − 2 x /ℓ , Λ = 2 g − 4 j 2 ∞ behavior of ρ 1 ( x ) is determined by the Λ-term or by r.h.s. In order to observe MZE we want ρ 1 ( x ) to be governed by the Λ-term (as it was in the case of the dissipation with finite support) D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 14 / 17

Making a bridge γ ( x ) = 3Γ 0 sech 2 ( x /ℓ ) ρ 1 ( x ) = 1 − ρ ( x ), ρ 1 ( x ) → 0 as x → ±∞ ρ 1 , xx − Λ ρ 1 = 12 j ∞ Γ 0 ℓ e − 2 x /ℓ , Λ = 2 g − 4 j 2 ∞ behavior of ρ 1 ( x ) is determined by the Λ-term or by r.h.s. In order to observe MZE we want ρ 1 ( x ) to be governed by the Λ-term (as it was in the case of the dissipation with finite support) two requirements: Λ > 0 – gives the maximal current: j ∞ < j max � = g / 2 ∞ √ Λ < 2 /ℓ – gives the minimal current: j ∞ > j min � g / 2 − 1 /ℓ 2 ∞ = D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 14 / 17

Making a bridge γ ( x ) = 3Γ 0 sech 2 ( x /ℓ ) ρ 1 ( x ) = 1 − ρ ( x ), ρ 1 ( x ) → 0 as x → ±∞ ρ 1 , xx − Λ ρ 1 = 12 j ∞ Γ 0 ℓ e − 2 x /ℓ , Λ = 2 g − 4 j 2 ∞ behavior of ρ 1 ( x ) is determined by the Λ-term or by r.h.s. In order to observe MZE we want ρ 1 ( x ) to be governed by the Λ-term (as it was in the case of the dissipation with finite support) two requirements: Λ > 0 – gives the maximal current: j ∞ < j max � = g / 2 ∞ √ Λ < 2 /ℓ – gives the minimal current: j ∞ > j min � g / 2 − 1 /ℓ 2 ∞ = As ℓ grows, j min asymptotically approaches j max ∞ . ∞ The range of currents allowing for MZE decreases. D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 14 / 17

Making a bridge γ ( x ) = 3Γ 0 sech 2 ( x /ℓ ) ρ 1 ( x ) = 1 − ρ ( x ), ρ 1 ( x ) → 0 as x → ±∞ ρ 1 , xx − Λ ρ 1 = 12 j ∞ Γ 0 ℓ e − 2 x /ℓ , Λ = 2 g − 4 j 2 ∞ behavior of ρ 1 ( x ) is determined by the Λ-term or by r.h.s. In order to observe MZE we want ρ 1 ( x ) to be governed by the Λ-term (as it was in the case of the dissipation with finite support) two requirements: Λ > 0 – gives the maximal current: j ∞ < j max � = g / 2 ∞ √ Λ < 2 /ℓ – gives the minimal current: j ∞ > j min � g / 2 − 1 /ℓ 2 ∞ = As ℓ grows, j min asymptotically approaches j max ∞ . ∞ The range of currents allowing for MZE decreases. Rapidly decaying dissipation is favorable for the observation of the MZE. D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 14 / 17

In summary, Conjecture: if the dissipation decays slower than exponentially, then there is no MZE. If the dissipation decays exponentially, then MZE can manifest itself but only for the stationary flows that obey a specific asymptotic behavior. If the dissipation decays faster than exponentially, then all the flows have a necessary asymptotics. This situation is favorable to encounter MZE. D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 15 / 17

D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 16 / 17

Conclusion Stationary flows for NLS MZE Role of the parameters of the defect Dynamics and possibilities of experimental observation D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 17 / 17

Appendix A: Preprint available D. A. Zezyulin, V. V. Konotop, G. Barontini, and H. Ott, arXiv:1112.4324; to appear in Phys. Rev. Lett. D. A. Zezyulin (CFTC, University of Lisbon) Macroscopic Zeno effect Lencos’12 18 / 17