National Cheng Kung University, Tainan Talk at AMO seminar, NTHU 2011/11/17
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1 Physics behind Quantum Photonic Dissipative Transport Theory Wei-Min Zhang National Cheng Kung University, Tainan wzhang@mail.ncku.edu.tw Talk at AMO seminar, NTHU 2011/11/17
2 Outline Ø Introduction of dissipative transport dynamics Ø General theory for electronic quantum transport Ø Development of photonic quantum transport theory Ø Applications to various nanophotonic devices Ø Prospective and further development
3 Outline Ø Introduction of dissipative transport dynamics Ø General theory for electronic quantum transport Ø Development of photonic quantum transport theory Ø Applications to various nanophotonic devices Ø Prospective and further development
4 Typical phenomena for dissipative transport dynamics Ø Bose-Einstein Condensation (BEC): Ø Molecular Dynamics: Ø Spintronics: Dissipative Dynamics Ø Early Universe: Open Quantum Systems
5 Dissipative photonic transport in photonic networks: D. Englund et al., Nature, 450, 857 (2007) The nanophotonic networks we concerned consist of all-optical circuits incorporating photonic bandgap waveguides and driven resonators embedded in nanostructured photonic crystals.
6 Nanostructured photonic crystals: a lossless material Photonic Crystals : Molding the Flow of Light, By Johns D. Joannopoulos Photonic crystals are artificial materials with periodic refractive index, its photonic band gap (PBG) structure together with its characteristic dispersion properties make the light manipulation and transmission much more efficient through the nanocavities and waveguides.
7 Photonic Crystals Ø Photonic crystals are artificial materials with periodic refractive index. Ø Photonic Band Gap M. Notomi, Rep. Prog. Phys. 73 (2010)
8 Defects ü Ultra high Q cavity Strong light confinement ü Waveguide Slow light Ø Well-defined defects incorporated in photonic crystal can become functional devices! M. Notomi, Rep. Prog. Phys. 73 (2010)
9 Devices ü All Optical EIT ü All Optical Switch K. Nozaki, et al. Nat. Photon. 4, 477 (2010) ü Single Photon Source X. Yang et al. PRL 102, (2009) W. H. Chang et al. PRL 96, (2006)
10 Controllability ü Adiabatic Wavelength Conversion ü Dynamical Tuning the Coupling T. Tanabe et al. PRL 102, (2009) Ø High Controllability X. Chew et al. Opt. Lett. 15, 2517(2010)
11 Modeling photonic circuits: Ø open optical systems Lei & WMZ, arxiv: (2010)
12 Ø Quantum theory for open systems: a long-standing problem Master Equation: Openness Quantum Mechanics Non-Markovian Memory However, it has been attempted for many years without a very satisfactory answer to find the exact master equation for an arbitrary open quantum system since Pauli first proposed the phenomenological master equation in 1928!
13 Outline Ø Introduction of dissipative transport dynamics Ø General theory for electronic quantum transport Ø Development of photonic quantum transport theory Ø Applications to various nanophotonic devices Ø Prospective and further development
14 Three basic approaches for electronic quantum transport: Ø Scattering theory approach Büttiker, Phys. Rev. B 46, (1992) using single particle scattering states to build up the multiparticle states with the proper symmetry Ø NE Green function approach Wingreen, Jauho & Meir, Phys. Rev. B 48,8487 (1993) based on Schwinger-Keldysh s Non-Equilibrium GF J. Schwinger, J. Math. Phys. 2, 407 (1961) L. V. Keldysh, Sov. Phys. JETP, 20, 1018 (1965) Ø Master equation approach Jin, Tu, WMZ & Yan, New J. Phys. 12, (2010) based on Feynman-Vernon influence functional Feynman and Vernon, Ann. Phys. 24, 118 (1963)
15 Quantum transport based on scattering theory: " Single-particle scattering matrix: Büttiker, Phys. Rev. B 46, (1992) a 1 a 2 b 1 b 2 " Transport current of each reservoir: " Landauer-Büttiker formula: with Limited to the simple steady transport phenomena with a black box.
16 Nonequilibrium GF approach: " closed-time Dyson equation: L. V. Keldysh, JETP 20, 1018 (1965) closed-path where retarded GF lesser GF quantum kinetic equation that can systematically explore all the nonequilibrium dynamics e.g. see WMZ & Wilets, PRC45, 1900 (1992)
17 Quantum transport based on nonequilibrium GF In terms of nonequilibrium Green functions, one has established the quantum transport theory of mesoscopic systems: u Transient current: Wingreen, Jauho & Meir, PRB48,8487 (1993) u tà, one can easily obtained the famous famous: Landauer-Buttiker formula: u can be applied to mesoscopic systems but still not convenient for studying transient dynamics and quantum decoherence.
18 Master equation approach Feynman & Vernon, Ann. Phys. (1963) " a truly nonperturbation way to fully trace over the environmental degrees of freedom, explicitly and completely: ρ(t 0 ) ρ(t) closed-path propagating function influence functional HPZ master equation for quantum Brownian motion. Hu, Paz & Zhang, PRD45, 2843 (1992)
19 Quantum transport with master equation. Tu & WMZ, PRB78, (2008) Jin, Tu, WMZ, Yan, NJP12, (2010) l The super-operators are exactly derived: Neither Born-Markov approx. nor Lindblad form l Transient current: where
20 Equations of Motion for u(t) and v(t) Non-perturbation equations retarded and lesser Green functions device energy levels Non-Markovian memory where system-reservoir couplings reservoir s spectra Dissipation-fluctuation theorem initial particle distribution in reservoirs Tu & WMZ, PRB78, (2008)
21 Reproduce NEGF: Jin, Tu, WMZ & Yan, NJP 12, (2010) Ø We reproduce and further generalize the transient current: where Wingreen, Jauho & Meir, PRB48, 8487 (1993)
22 As a result of the exact transport theory based on master equation u full nonequilibrium dynamics can be described with the exact master equation. u quantum decoherence in transport dynamics can be explicitly addressed from the time-evolution of the reduced density matrix. u the initial state dependence is included so that the non-markovian memory structure in various transport processes and quantum measurement can be explored explicitly. u the theory can be used to study various transport phenomena, including energy transfer and heat transfer, etc. Ø It may also be used to develop the theory for quantum feedback controlling???
23 Outline Ø Introduction of dissipative transport dynamics Ø General theory for electronic quantum transport Ø Development of photonic quantum transport theory Ø Applications to various nanophotonic devices Ø Prospective and further development
24 Nonequilibrium dynamics of nanophotonic devices: The System Tu & WMZ, Phys. Rev. B 78, (2008) In the coherent state representation WMZ et al., Rev. Mod. Phys. 62, 867 (1990) The Environment
25 Transport theory for photonic network Lei & WMZ, arxiv: (2010) Exact master equation: l The super-operators are exactly derived: l Transient photocurrent: where Photonic Crystals : Molding the Flow of Light, By Johns D. Joannopoulos
26 Non-Markovian dynamics Non-perturbation equations:
27 Generalize the NEGF theory: Ø Non-equilibrium GFs Jin, Tu, WMZ & Yan, NJP (2010) Ø Explicit and complete solution: Lei & WMZ, arxiv: (2010)
28 Photonic quantum transport theory Generalized quantum kinetic equation Lei & WMZ, arxiv: (2010) Transient photocurrent
29 Outline Ø Introduction of dissipative transport dynamics Ø General theory for electronic quantum transport Ø Development of photonic quantum transport theory Ø Applications to various nanophotonic devices Ø Prospective and further development
30 Quantum devices with micro/nano cavities High-Q photonic nanocavity A typical quantum device: with micro/nano-cavity build on photonic crystals coupled to waveguides, which has the potential application for light propagating and for storage. Let us start with such a cavity coupled to a general reservoir
31 Exact master equation: Xiong, WMZ, Wang, Wu, PRA 82, (2010) where and
32 Born-Markov approximation: Born-Markov approx.: taking the coefficients in the master equation up to the second order of the coupling between the cavity field and the thermal field Markov limit: t» τ ε (the character time of the thermal field), or equivalently, taking t : Carmichael s textbook κ with
33 Photon confinement in photonic crystals a nanocavity (defect) in photonic crystals: Nonmarkovian dynamics and Photon confinement! S. John, PRL58, 2486 (1990) Wu & WMZ, (2011) photonic crystals are lossless materials
34 Nanocavity coupled to a waveguide u Taking the waveguide as a reservoir u Spectral density: Wu, Lei, WMZ & Xiong, Opt. Express, 18, (2010)
35 Non-Markovian dynamics Wu, Lei, WMZ & Xiong, Opt. Express, 18, (2010) Non-Markovian A phase transition from Markovian to Non- Markovian dynamics Markovian
36 Analysis Strong Interaction Weak Strong Bound modes Critical Coupling Non-dissipative Steady state solution
37 Time Evolution of Wigner function Initial Coherent State η=0.15 T=0.5K η=2.0 T=0.05K
38 Time Evolution of Wigner function Initial Squeezed State η=0.15 T=0.5K η=2.0 T=0.05K
39 Time Evolution of Wigner function Initial Schrodinger-like Cat State η=0.15 T=0.5K η=2.0 T=0.05K Ø Quantum Coherence Protection via Strong Coupling!
40 Driven nanocavity Ø Hamiltonian Lei &WMZ, arxiv: (2010).
41 Photonic coherence controlled by external driving field Th T 5K 5mK η = 2.0
42 Photonic transport controlled by external driving field I 1 I 2
43 " Entanglement generation between two spatially-separated nanocavities through a waveguide Tan, WMZ & Li, PRA83, (2011) Entangled squeezed state: Logarithmic negativity
44 Other interesting development " Exact master equation with initial system-reservoir correlations Tan & WMZ, Phys. Rev. A83, (2011)
45 Energy transfer in photosynthesis: 1. Recent experiments show that the energy transfer in photosynthesis may involve quantum coherence channel. H. Lee, Y.-C. Cheng and G. R. Fleming, Science, 316, 1462 (2007) Liang, WMZ & Zhuo, PRE81, (2010)
46 Nanoparticles make leaves glow: A new idea on Bio-LED Su, Tu, Tseng, Chang & WMZ, Nanoscale 2, 2639 (2010). Ø Highlight in Chemistry World Ø Interviews by NewScientist and Reuters Ø Selected as Cutting Edge Chemistry in 2010 Ø Reported by Discovery News and other over hundred medias over the world
47 Outline Ø Introduction of dissipative transport dynamics Ø General theory for electronic quantum transport Ø Development of photonic quantum transport theory Ø Applications to various nanophotonic devices Ø Prospective and further development
48 Bio-junctions: further development 1. Molecular electronics 3. Photonic electronics 2. DNA junctions
49 Theory for quantum feedback controls?? State evolution Measurement Input: controller output feedback estimator l How to make Feedback controls???
50 Summary: Ø An & WMZ, PRA76, (2007) An, Feng & WMZ, QIC 9, 317 (2009) Ø Tu & WMZ, PRB78, (2008) Tu, Lee & WMZ, QIP 8, 631 (2009) Ø Jin, Tu, WMZ & Yan, NJP 12, (2010) Ø Tan & WMZ, PRA83, (2011) Ø Lei & WMZ, arxiv: (2010)
51 Applications: Ø Tu, WMZ & Jin, PRB83, (2011) Tu, WMZ, Jin, Entin-Wohlman & Aharony (in preparation) Ø Ø Ø Ø Ø Jin, WMZ, Tu & Wang, arxiv: (2011) Xiong, WMZ, Wang & Wu, PRA 82, (2010) Wu, Lei, WMZ & Xiong, Opt. Express. 18, (2010) Tan & WMZ, PRA83, (2011) Lin & WMZ, APL 99, (2011) Jin et al., arxiv: (2011)
52 Conclusions: " It has been attempted for many decades without a very satisfactory answer to find the exact master equation for an arbitrary open quantum system since Pauli first proposed the phenomenological master equation in " We utilized the coherent state path integral approach to reformulate Feynman-Vernon influence functional approach and derived an exact master equation for a large class of nanoelectronic devices (electronic nanostructures coupled with multiple electrodes for control and measurement) and various nanophotonic devices. " We believe that the new master equation is a crucial step toward establishing the nonequilibrium quantum theory for arbitrary open systems, one of the most difficulty problems that has been struggled for many decades without a significant achievement during the 20th century. Ø Hopefully, such theory can also be extended to the study of biosystems and other more complex open systems in nature
53 Prospective: Physics in 21th Century Ø Closed systems: most of problems have been solved with well-developed perturbation theories except for some strongly correlated systems that need a nonperturbation treatment which has not been developed yet Ø Open systems: most of problems have been over-looked in 20th century and also no well-developed theory has be established to address many unsolved issues
54 Acknowledgement: Chun U Lei (NCKU, Caltech.) Matisse W. Y. Tu (NCKU) Jinshuang Jin (Hongzhou Normal Univ.) Jun Hong An (Lazhou University, NUS) Ming-Tsung Lee (RCAS, Academic Sinica) Menh-Hsiu Wu (NCKU) Hua-Tang Tan (Huachung Normal Univ.) Heng-Na Xiong (Zhejiang Univ.) Chuan-Yu Lin (NCKU) Nien-An Wang (NCKU)
55 Historical development of the master equation for open systems u Pauli Master equation (W. Pauli, Festschrift zum 60. Geburtstage A. Sommerfelds (Hirzel, Leipzig, p.30, 1928) u Generalized master equation (S. Nakajima, Prog. Theo. Phys. 20, 948, 1958; R. Zwanzig, J. Chem. Phys. 33, 1338, 1960): u Master equation under Born Approximation (e.g. F. Haake, Z. Phys. 223, 364, 1969): GME has been mainly used for investigating non-markovian processes due to the fact that the master equation involves an explicit time integration
56 Markovian processes Ø Since 1970 s, one made further approximation: taking the perturbation up to the second order è Born-Markov (or Redfield or Lindblad) master equation, for example:
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