D EPHASING IN QUANTUM CHAOTIC TRANSPORT A SEMICLASSICAL APPROACH Dephasing & Quantum Transport Cyril Petitjean 1 , 2 Cyril Petitjean Transport & Chaos 1 Institut für Theoretische Physik, Universität Regensburg Dephasing 2 Department of Theoretical Physics, University of Geneva Microscopic Model Phenomenological Model Banff, February 2008 Summary Collaborators : Philippe Jacquod ( University of Arizona ) Robert S. Whitney ( Institut Laue-Langevin )
O UTLINE Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Q UANTUM T RANSPORT AND C HAOS 1 Dephasing Microscopic Model Phenomenological Model Summary Q UANTUM T RANSPORT AND D EPHASING 2 Microscopic Model Phenomenological Model
O UTLINE Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Q UANTUM T RANSPORT AND C HAOS 1 Dephasing Microscopic Model Phenomenological Model Summary Q UANTUM T RANSPORT AND D EPHASING 2 Microscopic Model Phenomenological Model
Q UANTUM T RANSPORT AND C HAOS Dephasing & Quantum Transport Quantum Mechanics : Cyril Petitjean Probability is given by the square of an Amplitude. Transport & Chaos Dephasing Microscopic Model Phenomenological Model Summary Anderson, Abrahams and Ramarkrishnan (1979) Gorkov, Larkin and Khmelnitskii (1979)
Q UANTUM T RANSPORT AND C HAOS Dephasing & Quantum Transport Quantum Mechanics : Cyril Petitjean Probability is given by the square of an Amplitude. Transport & Chaos Dephasing Microscopic Model Phenomenological Model Summary Anderson, Abrahams and Ramarkrishnan (1979) Gorkov, Larkin and Khmelnitskii (1979)
Q UANTUM T RANSPORT AND C HAOS E XPERIMENTS : O PEN B ALISTIC Q UANTUM D OT Dephasing B ALLISTIC CAVITY : & Quantum ✓ Clean sample, very few impurity Transport ✓ Randomness / chaoticity ⇒ sample boundaries Cyril Petitjean Transport & Chaos Dephasing Microscopic Model Phenomenological Model Summary
Q UANTUM T RANSPORT AND C HAOS E XPERIMENTS : O PEN B ALISTIC Q UANTUM D OT Dephasing B ALLISTIC CAVITY : & Quantum ✓ Clean sample, very few impurity Transport ✓ Randomness / chaoticity ⇒ sample boundaries Cyril Petitjean Transport & Chaos Dephasing Microscopic Model Phenomenological Model Summary
W HY BALLISTIC TRANSPORT IS DIFFERENT ? Dephasing Quantum Disorder & Quantum Transport Cyril Petitjean Transport & Chaos ✓ Huygens diffraction Dephasing ✓ Universal average properties Microscopic Model Phenomenological Model Summary Quantum Chaos ✓ Classical trajectories resolved ✓ Features of Classical dynamic ✓ ( ergodicity + mixing )
W HY BALLISTIC TRANSPORT IS DIFFERENT ? Dephasing Quantum Disorder & Quantum Transport Cyril Petitjean Transport & Chaos ✓ Huygens diffraction Dephasing ✓ Universal average properties Microscopic Model Phenomenological Model Summary Quantum Chaos !! Classical Chaotic Dynamics !! ✓ New "Ehrenfest " time scale ✓ Short time correction to the universality
T HE E HRENFEST TIME SCALE L ARKIN AND O VCHINIKOV JETP 28 , 1200(1969) & B ERMAN AND Z ASLAVSKY , P HYSICA A 91 ,450 (1978) Dephasing Classical Chaos & Quantum Local exponential divergence (Lyapunov exponent) Transport Quantum Chaos Cyril Petitjean Transport & Chaos Dephasing Microscopic Model Phenomenological Model Summary
T HE E HRENFEST TIME SCALE L ARKIN AND O VCHINIKOV JETP 28 , 1200(1969) & B ERMAN AND Z ASLAVSKY , P HYSICA A 91 ,450 (1978) Dephasing Classical Chaos & Quantum Local exponential divergence (Lyapunov exponent) Transport Quantum Chaos Cyril Petitjean Transport & Chaos Dephasing Microscopic Model Phenomenological Model Summary
T HE E HRENFEST TIME SCALE L ARKIN AND O VCHINIKOV JETP 28 , 1200(1969) & B ERMAN AND Z ASLAVSKY , P HYSICA A 91 ,450 (1978) Dephasing Classical Chaos & Quantum Local exponential divergence (Lyapunov exponent) Transport Quantum Chaos Cyril Petitjean Transport & Chaos Dephasing Microscopic Model Phenomenological Model Summary
S CATTERING A PPROACH AND SEMICLASSICAL FORMALISM Scattering matrix Dephasing � r & Büttiker, Phys. Rev. B 33 , 3020 (1986). � Quantum t ˆ Transport S = t ′ r ′ Cyril Petitjean Landauer- Büttiker Transport & Chaos Formula : Dephasing g = Tr[ t † t ] Microscopic Model Phenomenological Model Summary
S CATTERING A PPROACH AND SEMICLASSICAL FORMALISM Scattering matrix Dephasing � r & Büttiker, Phys. Rev. B 33 , 3020 (1986). � Quantum t ˆ Transport S = t ′ r ′ Cyril Petitjean Landauer- Büttiker Transport & Chaos Formula : Dephasing g = Tr[ t † t ] Microscopic Model Phenomenological Model Summary Fisher,Lee, Phys. Rev. B 23 , R6851 (1981) & Baranger et al. Phys. Rev. B 44 , 10637 (1991). Semiclassic : � over classical paths from mode m to n
S CATTERING A PPROACH AND SEMICLASSICAL FORMALISM Scattering matrix Dephasing � r & Büttiker, Phys. Rev. B 33 , 3020 (1986). � Quantum t ˆ Transport S = t ′ r ′ Cyril Petitjean Landauer- Büttiker Transport & Chaos Formula : Dephasing g = Tr[ t † t ] Microscopic Model Phenomenological Model Summary Fisher,Lee, Phys. Rev. B 23 , R6851 (1981) & Baranger et al. Phys. Rev. B 44 , 10637 (1991). Semiclassic : � over classical paths from mode m to n
S CATTERING A PPROACH AND SEMICLASSICAL FORMALISM Scattering matrix Dephasing � r & Büttiker, Phys. Rev. B 33 , 3020 (1986). � Quantum t ˆ Transport S = t ′ r ′ Cyril Petitjean Landauer- Büttiker Transport & Chaos Formula : Dephasing g = Tr[ t † t ] Microscopic Model Phenomenological Model Summary Fisher,Lee, Phys. Rev. B 23 , R6851 (1981) & Baranger et al. Phys. Rev. B 44 , 10637 (1991). Semiclassic : � over classical paths from mode m to n
S CATTERING A PPROACH AND SEMICLASSICAL FORMALISM Scattering matrix Dephasing � r & Büttiker, Phys. Rev. B 33 , 3020 (1986). � Quantum t ˆ Transport S = t ′ r ′ Cyril Petitjean Landauer- Büttiker Transport & Chaos Formula : Dephasing g = Tr[ t † t ] Microscopic Model Phenomenological Model Summary
S CATTERING A PPROACH AND SEMICLASSICAL FORMALISM Scattering matrix Dephasing � r & Büttiker, Phys. Rev. B 33 , 3020 (1986). � Quantum t ˆ Transport S = t ′ r ′ Cyril Petitjean Landauer- Büttiker Transport & Chaos Formula : Dephasing g = Tr[ t † t ] Microscopic Model Phenomenological Model Summary Baranger et al., Phys. Rev. Lett. 70 , 3876 (1993)
S CATTERING A PPROACH AND SEMICLASSICAL FORMALISM Scattering matrix Dephasing � r & Büttiker, Phys. Rev. B 33 , 3020 (1986). � Quantum t ˆ Transport S = t ′ r ′ Cyril Petitjean Landauer- Büttiker Transport & Chaos Formula : Dephasing g = Tr[ t † t ] Microscopic Model Phenomenological Model Summary Aleiner, Larkin, (1996) Sieber, Richter, (2001) Adagideli, (2003) Rahav, Brouwer, (2005) Jacquod, Whitney,(2006)
Q UANTUM T RANSPORT AND D EPHASING W HAT IS LEFT OUT IN THE STANDARD SCATTERING APPROACH ? ✓ All dissipative processes occur in the leads Dephasing ✓ Apart from this lead connection System is isolated & Quantum Transport Effects due to the coupling with an external environment Cyril Petitjean cannot be described Transport & Chaos Dephasing Microscopic Model Phenomenological Model Summary
Q UANTUM T RANSPORT AND D EPHASING W HAT IS LEFT OUT IN THE STANDARD SCATTERING APPROACH ? ✓ All dissipative processes occur in the leads Dephasing ✓ Apart from this lead connection System is isolated & Quantum Transport Effects due to the coupling with an external environment Cyril Petitjean cannot be described Transport & Chaos Dephasing Microscopic Model Phenomenological Model Summary Webb, Washburn, Umbach, and Laibowitz, Phys. Rev. Lett. 54 , 2696 (1985)
Q UANTUM T RANSPORT AND D EPHASING W HAT IS LEFT OUT IN THE STANDARD SCATTERING APPROACH ? Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Treatement of Decoherence � = Ways : Dephasing Microscopic Model Phenomenological M ICROSCOPIC M ODEL ⇒ New scattering formalism Model Summary P HENOMENOLOGICAL M ODEL ⇒ Dephasing Model
S TANDARD TREATMENT OF DECOHERENCE J OOS , Z EH , K IEFER , G IULINI , K UPSCH , AND S TAMATESCU , S PRINGER , (2003). Z UREK , R EV . M OD . P HYS . 75 , 715 (2003). Dephasing & Quantum Transport Cyril Petitjean Transport & Chaos Dephasing Reduced Density Matrix Microscopic Model Phenomenological η Sys = Tr Env [ η Tot ] Model Summary 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)
S CATTERING FORMALISM IN PRESENCE OF AN ENVIRONMENT C.P., J ACQUOD , W HITNEY , JETP L ETT . 86 , 736 (2007) W HITNEY , J ACQUOD , C.P., P HYS . R EV . B 77 , 045315 (2008) Dephasing & Quantum Transport Coupled Dot Model Cyril Petitjean Transport & Chaos = H Sys + H Env + U ( U , ξ ) H Dephasing U = Strength Microscopic Model Phenomenological Model = ξ Correlation length Summary
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