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

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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

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Macroscopic Zeno effect and stationary flows in nonlinear waveguides with localized dissipation

  • D. A. Zezyulin1, V. V. Konotop1, G. Barontini2, and H. Ott2

1Center for Theoretical and Computational Physics, University of Lisbon, Portugal 2Department 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

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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

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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

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Model of a nonlinear waveguide

iΨt = −Ψxx + g|Ψ|2Ψ − iγ(x)Ψ localized dissipation: γ(x) = Γ0f (x/ℓ)

|f (x)| → 0 as x → ±∞ max|f (x)| = f (0) ∼ 1, |fx(x)| ∼ 1 Γ0 — amplitude ℓ — characteristic width ∞

−∞ |γ(x)| dx ∝ Γ0ℓ

  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 4 / 17

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Stationary flows

“hydrodynamic” form: Ψ(t, x) = ρ(x) exp

  • i

x

0 v(s)ds − iµt

  • µ — chemical potential

n(x) = ρ2(x) — density v(x) — superfluid velocity j(x) = v(x)n(x) — superfluid current

stationary equations: ρxx + µρ − gρ3 − j2ρ−3 = 0, jx + γ(x)ρ2 = 0 boundary conditions for stationary flows:

density is a fixed constant: |ρ(x)| → ρ∞ as x → ±∞ (ρ∞ = 1 for all

  • ur results)

current is some constant: j(x) → ∓j∞ as x → ±∞

The main question: how j∞ depends on Γ0 and ℓ?

  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 5 / 17

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  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 6 / 17

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  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 7 / 17

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Example: An exact solution

ρxx + µρ − gρ3 − j2ρ−3 = 0, jx + γ(x)ρ2 = 0 γ(x) = 3Γ0 sech2(x/ℓ) A branch of stationary flows:

ρ(x) = tanh(x/ℓ) j(x) = −Γ0ℓ tanh3(x/ℓ) additional constraint: ℓ2(g + Γ2

0ℓ2) = 2

j∞ = Γ0ℓ j∞ grows monotonously with Γ0 and ℓ. No macroscopic Zeno effect.

  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 8 / 17

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Another example: Dissipation with finite support

ρxx + µρ − gρ3 − j2ρ−3 = 0, jx + γ(x)ρ2 = 0 γ(x) =

  • Γ0
  • 1 − x2/ℓ22

if |x| < ℓ,

  • therwise

no exact solution; let’s start from Γ0 = 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 Γ0 = 0

  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 9 / 17

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  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 10 / 17

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Dependence on the width

  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 11 / 17

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Two different situations:

γ(x) = 3Γ0sech2(x/ℓ)

ρ(x) = tanh(x/ℓ) j(x) = −Γ0ℓ tanh3(x/ℓ) j∞ = Γ0ℓ 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

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Difference in asymptotic behavior

γ(x) = 3Γ0sech2(x/ℓ)

ρ(x) = tanh(x/ℓ) ρ(x) ∼ 1 − 2e−2x/ℓ as x → ∞ asymptotic behavior is determined by ℓ

dissipation with finite support

ρ∞ − ρ(x) ∝ e−

√ Λx, where Λ = 2g − 4j2 ∞

asymptotic behavior is determined by j∞ Λ > 0: j∞ < jmax

∞

=

  • g/2 — maximal possible value of j∞
  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 13 / 17

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Making a bridge

γ(x) = 3Γ0sech2(x/ℓ)

  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 14 / 17

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Making a bridge

γ(x) = 3Γ0sech2(x/ℓ) ρ1(x) = 1 − ρ(x), ρ1(x) → 0 as x → ±∞ ρ1,xx − Λρ1 = 12j∞Γ0ℓe−2x/ℓ, Λ = 2g − 4j2

∞

  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 14 / 17

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Making a bridge

γ(x) = 3Γ0sech2(x/ℓ) ρ1(x) = 1 − ρ(x), ρ1(x) → 0 as x → ±∞ ρ1,xx − Λρ1 = 12j∞Γ0ℓe−2x/ℓ, Λ = 2g − 4j2

∞

behavior of ρ1(x) is determined by the Λ-term or by r.h.s. In order to observe MZE we want ρ1(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

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Making a bridge

γ(x) = 3Γ0sech2(x/ℓ) ρ1(x) = 1 − ρ(x), ρ1(x) → 0 as x → ±∞ ρ1,xx − Λρ1 = 12j∞Γ0ℓe−2x/ℓ, Λ = 2g − 4j2

∞

behavior of ρ1(x) is determined by the Λ-term or by r.h.s. In order to observe MZE we want ρ1(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∞ < jmax

∞

=

  • g/2

√ Λ < 2/ℓ – gives the minimal current: j∞ > jmin

∞ =

  • g/2 − 1/ℓ2
  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 14 / 17

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Making a bridge

γ(x) = 3Γ0sech2(x/ℓ) ρ1(x) = 1 − ρ(x), ρ1(x) → 0 as x → ±∞ ρ1,xx − Λρ1 = 12j∞Γ0ℓe−2x/ℓ, Λ = 2g − 4j2

∞

behavior of ρ1(x) is determined by the Λ-term or by r.h.s. In order to observe MZE we want ρ1(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∞ < jmax

∞

=

  • g/2

√ Λ < 2/ℓ – gives the minimal current: j∞ > jmin

∞ =

  • g/2 − 1/ℓ2

As ℓ grows, jmin

∞

asymptotically approaches jmax

∞ .

The range of currents allowing for MZE decreases.

  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 14 / 17

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Making a bridge

γ(x) = 3Γ0sech2(x/ℓ) ρ1(x) = 1 − ρ(x), ρ1(x) → 0 as x → ±∞ ρ1,xx − Λρ1 = 12j∞Γ0ℓe−2x/ℓ, Λ = 2g − 4j2

∞

behavior of ρ1(x) is determined by the Λ-term or by r.h.s. In order to observe MZE we want ρ1(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∞ < jmax

∞

=

  • g/2

√ Λ < 2/ℓ – gives the minimal current: j∞ > jmin

∞ =

  • g/2 − 1/ℓ2

As ℓ grows, jmin

∞

asymptotically approaches jmax

∞ .

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

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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

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  • D. A. Zezyulin (CFTC, University of Lisbon)

Macroscopic Zeno effect Lencos’12 16 / 17

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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

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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