Non-Markovian finite-temperature two-time correlation functions of system operators: beyond the quantum regression theorem.

An extremely useful evolution equation that allows systematically calculating the two-time correlation functions (CF's) of system operators for non-Markovian open (dissipative) quantum systems is derived. The derivation is based on perturbative quantum master equation approach, so non-Markovian open quantum system models that are not exactly solvable can use our derived evolution equation to easily obtain their two-time CF's of system operators, valid to second order in the system-environment interaction. Since the form and nature of the Hamiltonian are not specified in our derived evolution equation, our evolution equation is applicable for bosonic and/or fermionic environments and can be applied to a wide range of system-environment models with any factorized (separable) system-environment initial states (pure or mixed). When applied to a general model of a system coupled to a finite-temperature bosonic environment with a system coupling operator L in the system-environment interaction Hamiltonian, the resultant evolution equation is valid for both L = L(†) and L ≠ L(†) cases, in contrast to those evolution equations valid only for L = L(†) case in the literature. The derived equation that generalizes the quantum regression theorem (QRT) to the non-Markovian case will have broad applications in many different branches of physics. We then give conditions on which the QRT holds in the weak system-environment coupling case and apply the derived evolution equation to a problem of a two-level system (atom) coupled to the finite-temperature bosonic environment (electromagnetic fields) with L ≠ L(†).

[1]  A. Manolescu,et al.  Geometrical effects and signal delay in time-dependent transport at the nanoscale , 2008, 0807.4015.

[2]  Christoph Meier,et al.  Non-Markovian evolution of the density operator in the presence of strong laser fields , 1999 .

[3]  Yoshinori Takahashi,et al.  A generalized stochastic liouville equation. Non-Markovian versus memoryless master equations , 1977 .

[4]  Gert-Ludwig Ingold,et al.  Quantum Brownian motion: The functional integral approach , 1988 .

[5]  M. Esposito,et al.  Quantum master equation for electron transport through quantum dots and single molecules , 2006, cond-mat/0610004.

[6]  Stephen M. Barnett,et al.  Methods in Theoretical Quantum Optics , 1997 .

[7]  Ting Yu.  Non-markovian quantum trajectories versus master equations: Finite-temperature heat bath , 2004 .

[8]  G. Milburn,et al.  Charge Transport in a Quantum Electromechanical System , 2004, cond-mat/0406102.

[9]  Yijing Yan.  Quantum Fokker-Planck theory in a non-Gaussian-Markovian medium , 1998 .

[10]  F. Petruccione,et al.  Stochastic wave function method for non-markovian quantum master equations , 1999 .

[11]  U. Weiss.  Quantum Dissipative Systems , 1993 .

[12]  Yijing Yan,et al.  Dynamical semigroup Fokker–Planck equation approach to transient absorption and fluorescence upconversion spectroscopies , 2001 .

[13]  I. D. Vega,et al.  Non-Markovian reduced propagator, multiple-time correlation functions, and master equations with general initial conditions in the weak-coupling limit , 2006 .

[14]  S. B. Cavalcanti,et al.  Non-Markovian damping of Rabi oscillations in semiconductor quantum dots , 2009, Journal of physics. Condensed matter : an Institute of Physics journal.

[15]  R. F. O'Connell,et al.  Comment on “The Lax-Onsager Regression “Theorem” revisited” , 2000 .

[16]  Yijing Yan,et al.  Complex non-Markovian effect on time-dependent quantum transport. , 2009, The Journal of chemical physics.

[17]  I. Sinayskiy,et al.  Non-Markovian dynamics of an interacting qubit pair coupled to two independent bosonic baths , 2009, Journal of Physics A: Mathematical and Theoretical.

[18]  H. Goan,et al.  Non-Markovian finite-temperature two-time correlation functions of system operators of a pure-dephasing model , 2010, 1007.3791.

[19]  B. Muzykantskii,et al.  ON QUANTUM NOISE , 1995 .

[20]  P. Haikka.  Non-Markovian master equation for a damped driven two-state system , 2009, 0911.4600.

[21]  G. W. Ford,et al.  Calculation of Correlation Functions in the Weak Coupling Approximation , 1999 .

[22]  C. Emary,et al.  Weak-coupling approximations in non-Markovian transport , 2009, 0902.2118.

[23]  G. W. Ford,et al.  Driven systems and the Lax formula , 2000 .

[24]  Francesco Petruccione,et al.  The Theory of Open Quantum Systems , 2002 .

[25]  Francesco Petruccione,et al.  The Time-Convolutionless Projection Operator Technique in the Quantum Theory of Dissipation and Decoherence , 2001 .

[26]  Ford,et al.  There is No Quantum Regression Theorem. , 1996, Physical review letters.

[27]  G. Milburn,et al.  Dynamics of a mesoscopic charge quantum bit under continuous quantum measurement , 2001, cond-mat/0103005.

[28]  M. S. Zubairy,et al.  Quantum optics: Dedication , 1997 .

[29]  The influence of ultrafast laser pulses on electron transfer in molecular wires studied by a non-Markovian density-matrix approach. , 2005, The Journal of chemical physics.

[30]  I. D. Vega,et al.  Emission spectra of atoms with non-Markovian interaction: Fluorescence in a photonic crystal , 2007, 0711.1499.

[31]  Yijing Yan,et al.  Quantum master equation scheme of time-dependent density functional theory to time-dependent transport in nanoelectronic devices , 2006, cond-mat/0606788.

[32]  H. Grabert.  Nonlinear relaxation and fluctuations of damped quantum systems , 1982 .

[33]  M. S. Zubairy,et al.  Quantum optics: Frontmatter , 1997 .

[34]  Kuan-Liang Liu,et al.  Non-markovian entanglement dynamics of quantum continuous variable systems in thermal environments , 2007, 0706.0996.

[35]  Ping Cui,et al.  Spontaneous relaxation of a charge qubit under electrical measurement. , 2005, Physical review letters.

[36]  J. Shao,et al.  Quantum measurement of a solid-state qubit: A unified quantum master equation approach , 2003, cond-mat/0309574.

[37]  Michael Schreiber,et al.  Calculation of absorption spectra for light-harvesting systems using non-Markovian approaches as well as modified Redfield theory. , 2006, The Journal of chemical physics.

[38]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[39]  Peter Talkner,et al.  The failure of the quantum regression hypothesis , 1986 .

[40]  Melvin Lax,et al.  The Lax–Onsager regression `theorem' revisited , 2000 .

[41]  S. Girvin,et al.  Introduction to quantum noise, measurement, and amplification , 2008, 0810.4729.

[42]  H. Umezawa,et al.  Functional methods in thermofield dynamics: A real-time perturbation theory for quantum statistical mechanics , 1983 .

[43]  Yijing Yan,et al.  Non-equilibrium quantum theory for nanodevices based on the Feynman–Vernon influence functional , 2010 .

[44]  Nathan,et al.  Continuous quantum measurement of two coupled quantum dots using a point contact: A quantum trajectory approach , 2000, cond-mat/0006333.

[45]  Daniel Alonso,et al.  Multiple-time correlation functions for non-markovian interaction: beyond the quantum regression theorem. , 2005, Physical review letters.

[46]  H. Goan,et al.  Non-Markovian dynamics of a nanomechanical resonator measured by a quantum point contact , 2010, 1101.2393.

[47]  Y. Blanter,et al.  Shot noise in mesoscopic conductors , 1999, cond-mat/9910158.

[48]  F. Shibata,et al.  Time-convolutionless projection operator formalism for elimination of fast variables. Applications to Brownian motion , 1979 .

[49]  Sabrina Maniscalco,et al.  Non-Markovian Dynamics of a Damped Driven Two-State System , 2010, 1001.3564.

[50]  H. Carmichael.  Statistical Methods in Quantum Optics 1 , 1999 .

[51]  G. J. Milburn,et al.  Spin-detection in a quantum electromechanical shuttle system , 2006, cond-mat/0601448.

[52]  I. D. Vega,et al.  Hierarchy of equations of multiple-time correlation functions , 2007 .

[53]  Po-Wen Chen,et al.  Decoherence-free subspace and disentanglement dynamics for two qubits in a common non-Markovian squeezed reservoir , 2010, 1007.3792.

[54]  A. Manolescu,et al.  Time-dependent transport via the generalized master equation through a finite quantum wire with an embedded subsystem , 2009, 0903.3491.

[55]  N. Gisin,et al.  Non-Markovian quantum state diffusion , 1998, quant-ph/9803062.

[56]  Quantum noise in the electromechanical shuttle: Quantum master equation treatment , 2005, cond-mat/0509748.

[57]  Wei-Min Zhang,et al.  Non-Markovian decoherence theory for a double-dot charge qubit , 2008, 0809.3490.

[58]  Ulrich Kleinekathöfer,et al.  Non-Markovian theories based on a decomposition of the spectral density. , 2004, The Journal of chemical physics.