W HY BALLISTIC TRANSPORT IS DIFFERENT ? Dephasing Quantum Disorder - - PowerPoint PPT Presentation

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W HY BALLISTIC TRANSPORT IS DIFFERENT ? Dephasing Quantum Disorder - - PowerPoint PPT Presentation

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D EPHASING IN QUANTUM CHAOTIC TRANSPORT A SEMICLASSICAL APPROACH Dephasing & Quantum Transport Cyril Petitjean 1 , 2 Cyril Petitjean Transport & Chaos 1 Institut fr Theoretische Physik, Universitt Regensburg Dephasing 2

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

DEPHASING IN QUANTUM CHAOTIC

TRANSPORT

A SEMICLASSICAL APPROACH Cyril Petitjean1,2

1Institut für Theoretische Physik, Universität Regensburg 2Department of Theoretical Physics, University of Geneva

Banff, February 2008

Collaborators : Philippe Jacquod (University of Arizona ) Robert S. Whitney (Institut Laue-Langevin)

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

OUTLINE

1

QUANTUM TRANSPORT AND CHAOS

2

QUANTUM TRANSPORT AND DEPHASING

Microscopic Model Phenomenological Model

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

OUTLINE

1

QUANTUM TRANSPORT AND CHAOS

2

QUANTUM TRANSPORT AND DEPHASING

Microscopic Model Phenomenological Model

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SLIDE 4

Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

QUANTUM TRANSPORT AND CHAOS

Quantum Mechanics : Probability is given by the square of an Amplitude.

Anderson, Abrahams and Ramarkrishnan (1979) Gorkov, Larkin and Khmelnitskii (1979)

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

QUANTUM TRANSPORT AND CHAOS

Quantum Mechanics : Probability is given by the square of an Amplitude.

Anderson, Abrahams and Ramarkrishnan (1979) Gorkov, Larkin and Khmelnitskii (1979)

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

QUANTUM TRANSPORT AND CHAOS

EXPERIMENTS : OPEN BALISTIC QUANTUM DOT

BALLISTIC CAVITY : ✓ Clean sample, very few impurity ✓ Randomness / chaoticity ⇒ sample boundaries

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

QUANTUM TRANSPORT AND CHAOS

EXPERIMENTS : OPEN BALISTIC QUANTUM DOT

BALLISTIC CAVITY : ✓ Clean sample, very few impurity ✓ Randomness / chaoticity ⇒ sample boundaries

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

WHY BALLISTIC TRANSPORT IS DIFFERENT ?

Quantum Disorder

✓ Huygens diffraction ✓ Universal average properties

Quantum Chaos

✓ Classical trajectories resolved ✓ Features of Classical dynamic ✓ ( ergodicity + mixing )

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

WHY BALLISTIC TRANSPORT IS DIFFERENT ?

Quantum Disorder

✓ Huygens diffraction ✓ Universal average properties

Quantum Chaos

!! Classical Chaotic Dynamics !! ✓ New "Ehrenfest " time scale ✓ Short time correction to the universality

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

THE EHRENFEST TIME SCALE

LARKIN AND OVCHINIKOV JETP 28, 1200(1969) & BERMAN AND ZASLAVSKY, PHYSICA A 91,450 (1978)

Classical Chaos Local exponential divergence (Lyapunov exponent) Quantum Chaos

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

THE EHRENFEST TIME SCALE

LARKIN AND OVCHINIKOV JETP 28, 1200(1969) & BERMAN AND ZASLAVSKY, PHYSICA A 91,450 (1978)

Classical Chaos Local exponential divergence (Lyapunov exponent) Quantum Chaos

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

THE EHRENFEST TIME SCALE

LARKIN AND OVCHINIKOV JETP 28, 1200(1969) & BERMAN AND ZASLAVSKY, PHYSICA A 91,450 (1978)

Classical Chaos Local exponential divergence (Lyapunov exponent) Quantum Chaos

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING APPROACH AND SEMICLASSICAL

FORMALISM

Scattering matrix

Büttiker, Phys. Rev. B 33, 3020 (1986).

ˆ S = r t t′ r′

  • Landauer- Büttiker

Formula :

g = Tr[t†t]

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING APPROACH AND SEMICLASSICAL

FORMALISM

Scattering matrix

Büttiker, Phys. Rev. B 33, 3020 (1986).

ˆ S = r t t′ r′

  • Landauer- Büttiker

Formula :

g = Tr[t†t]

Fisher,Lee, Phys. Rev. B 23, R6851 (1981) & Baranger et al. Phys. Rev. B 44, 10637 (1991).

Semiclassic :

  • ver classical paths from mode m to n
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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING APPROACH AND SEMICLASSICAL

FORMALISM

Scattering matrix

Büttiker, Phys. Rev. B 33, 3020 (1986).

ˆ S = r t t′ r′

  • Landauer- Büttiker

Formula :

g = Tr[t†t]

Fisher,Lee, Phys. Rev. B 23, R6851 (1981) & Baranger et al. Phys. Rev. B 44, 10637 (1991).

Semiclassic :

  • ver classical paths from mode m to n
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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING APPROACH AND SEMICLASSICAL

FORMALISM

Scattering matrix

Büttiker, Phys. Rev. B 33, 3020 (1986).

ˆ S = r t t′ r′

  • Landauer- Büttiker

Formula :

g = Tr[t†t]

Fisher,Lee, Phys. Rev. B 23, R6851 (1981) & Baranger et al. Phys. Rev. B 44, 10637 (1991).

Semiclassic :

  • ver classical paths from mode m to n
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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING APPROACH AND SEMICLASSICAL

FORMALISM

Scattering matrix

Büttiker, Phys. Rev. B 33, 3020 (1986).

ˆ S = r t t′ r′

  • Landauer- Büttiker

Formula :

g = Tr[t†t]

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING APPROACH AND SEMICLASSICAL

FORMALISM

Scattering matrix

Büttiker, Phys. Rev. B 33, 3020 (1986).

ˆ S = r t t′ r′

  • Landauer- Büttiker

Formula :

g = Tr[t†t]

Baranger et al., Phys. Rev. Lett. 70, 3876 (1993)

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING APPROACH AND SEMICLASSICAL

FORMALISM

Scattering matrix

Büttiker, Phys. Rev. B 33, 3020 (1986).

ˆ S = r t t′ r′

  • Landauer- Büttiker

Formula :

g = Tr[t†t]

Aleiner, Larkin, (1996) Sieber, Richter, (2001) Adagideli, (2003) Rahav, Brouwer, (2005) Jacquod, Whitney,(2006)

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

QUANTUM TRANSPORT AND DEPHASING

WHAT IS LEFT OUT IN THE STANDARD SCATTERING APPROACH ?

✓ All dissipative processes occur in the leads ✓ Apart from this lead connection System is isolated Effects due to the coupling with an external environment cannot be described

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

QUANTUM TRANSPORT AND DEPHASING

WHAT IS LEFT OUT IN THE STANDARD SCATTERING APPROACH ?

✓ All dissipative processes occur in the leads ✓ Apart from this lead connection System is isolated Effects due to the coupling with an external environment cannot be described

Webb, Washburn, Umbach, and Laibowitz, Phys. Rev. Lett. 54, 2696 (1985)

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

QUANTUM TRANSPORT AND DEPHASING

WHAT IS LEFT OUT IN THE STANDARD SCATTERING APPROACH ?

Treatement of Decoherence = Ways : MICROSCOPIC MODEL ⇒ New scattering formalism PHENOMENOLOGICAL MODEL ⇒ Dephasing Model

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

STANDARD TREATMENT OF DECOHERENCE

JOOS, ZEH, KIEFER, GIULINI, KUPSCH, AND STAMATESCU, SPRINGER, (2003). ZUREK, REV. MOD. PHYS. 75, 715 (2003).

Reduced Density Matrix

ηSys = TrEnv[ηTot]

Application to : ✓ Coupled chaotic system

  • Jacquod, Phys. Rev. Lett. 92, 150403 (2004)
  • C.P

., Jacquod, Phys. Rev. Lett. 97, 194103 (2006) ✓ Fidelity ( Boltzmann echo)

  • C.P

., Jacquod, Phys. Rev. Lett. 97, 124103 (2006) ✓ Quantum Transport, · · · Now

  • C.P

., Jacquod, Whitney, JETP Lett. 86, 736 (2007)

  • Whitney, Jacquod, C.P

., Phys. Rev. B 77, 045315 (2008)

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Coupled Dot Model

H = HSys + HEnv + U(U, ξ) U = Strength ξ = Correlation length

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Coupled Dot Model

H = HSys + HEnv + U(U, ξ) U = Strength ξ = Correlation length

⇒ New Scattering matrix

:

1 Include external non- current carrying

degrees of freedom

2 in a Time-resolved manner .

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Coupled Dot Model

H = HSys + HEnv + U(U, ξ) U = Strength ξ = Correlation length

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Coupled Dot Model

H = HSys + HEnv + U(U, ξ) U = Strength ξ = Correlation length

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Coupled Dot Model

H = HSys + HEnv + U(U, ξ) U = Strength ξ = Correlation length

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SLIDE 29

Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Coupled Dot Model

H = HSys + HEnv + U(U, ξ) U = Strength ξ = Correlation length

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SLIDE 30

Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Coupled Dot Model

H = HSys + HEnv + U(U, ξ) U = Strength ξ = Correlation length

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SLIDE 31

Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Coupled Dot Model

H = HSys + HEnv + U(U, ξ) U = Strength ξ = Correlation length y0 y q q0 ¯ Γ = Γ ¯ γ ≈ γ

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Coupled Dot Model

H = HSys + HEnv + U(U, ξ) U = Strength ξ = Correlation length y0 y q q0 ¯ Γ = Γ ¯ γ ≈ γ

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SLIDE 33

Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Coupled Dot Model

H = HSys + HEnv + U(U, ξ) U = Strength ξ = Correlation length y0 y q q0 ¯ Γ = Γ ¯ γ ≈ γ

Diagonal Approximation for Environment Noise

∆ΦU ⇒ ∆ΦNoise

δg = NLNR exp[−τE/τD] (NL + NR)2 exp[−τξ/τΦ] 1 + τξ/τΦ

ξ ∝ √ D

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Voltage1-Dephasing2 Model

Add Fictitious probe Net current is Null

  • 1. Büttiker, Phys. Rev. B 33, 3020 (1986)
  • 2. De Jong, Beenakker, Physica A 230, 219 (1996)
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SLIDE 35

Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Voltage1-Dephasing2 Model

Add Fictitious probe Net current is Null

  • 1. Büttiker, Phys. Rev. B 33, 3020 (1986)
  • 2. De Jong, Beenakker, Physica A 230, 219 (1996)

Two terminal conductance : Büttiker, Phys. Rev. B 33, 3020 (1986)

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SLIDE 36

Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Voltage1-Dephasing2 Model

Add Fictitious probe Net current is Null

  • 1. Büttiker, Phys. Rev. B 33, 3020 (1986)
  • 2. De Jong, Beenakker, Physica A 230, 219 (1996)

Tunnel barriere semiclassic : Whitney, Phys. Rev. B 75, 235404 (2007)

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SCATTERING FORMALISM IN PRESENCE

OF AN ENVIRONMENT

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

Voltage1-Dephasing2 Model

Add Fictitious probe Net current is Null

  • 1. Büttiker, Phys. Rev. B 33, 3020 (1986)
  • 2. De Jong, Beenakker, Physica A 230, 219 (1996)

δg = NLNR exp[−τE/τD] (NL + NR)2 exp[−2τE/τφ] 1 + τD/τφ

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

DEPHASING MODEL VERSUS MICROSCOPIC MODEL

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

δg = NLNR exp[−τE/τD] (NL + NR)2 exp[−τ ∗/τφ] 1 + τD/τφ

Microscopic Model ✓ Dephasing is model-dependent

τ ∗ ∝ λ−1 ln [L/ξ]

Dephasing Model ✓ No independent parameter ξ

τ ∗ ∝ λ−1 ln [L/λF]

✓ ξ is replaced by λF

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

DEPHASING MODEL VERSUS MICROSCOPIC MODEL

C.P., JACQUOD, WHITNEY, JETP LETT. 86, 736 (2007) WHITNEY, JACQUOD, C.P., PHYS. REV. B 77, 045315 (2008)

δg = NLNR exp[−τE/τD] (NL + NR)2 exp[−τ ∗/τφ] 1 + τD/τφ

Microscopic Model ✓ Dephasing is model-dependent

τ ∗ ∝ λ−1 ln [L/ξ]

Dephasing Model ✓ No independent parameter ξ

τ ∗ ∝ λ−1 ln [L/λF]

✓ ξ is replaced by λF Dephasing Model cannot mimic a system-environment model with ξ ∼ L

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

NUMERICAL SIMULATIONS

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

NUMERICAL SIMULATIONS

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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

SUMMARY

Dephasing First trajectory based treatment of an ✓ environment ("New scattering formalism") ✓ of the dephasing model.

Dephasing is model-dependent τ ∗ = {τξ, τE}

For more see (Shot noise, UCF , ...): Whitney, Jacquod, C.P .,

  • Phys. Rev. B 77, 045315 (2008)

See also Altland, Brouwer, Tian,

  • Phys. Rev. Lett. 99, 036804 (2007)
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Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing

Microscopic Model Phenomenological Model

Summary

ACKNOWLEDGMENTS for useful and stimulating discussions to : Piet Brouwer Markus Büttiker Philippe Jacquod Mikhail Polianski Robert S. Whitney

Thank for your attention