Non-Markovian Open Quantum Systems Sabrina Maniscalco Quantum Optics Group, Department of Physics University of Turku

Contents Motivation Theoretical approaches to the study of open systems dynamics Markovian approximation and Lindblad Master Equation Non-Markovian systems Examples: Quantum Brownian Motion and two-level atom

Open Quantum Systems Quantum Mechanics The theory of open quantum systems describes the interaction of a quantum system with its environment Quantum Mechanics closed systems unitary dynamics reversible dynamics Schrödinger Equation Liouville – von Neumann Equation open quantum systems reduced density operator master equation non-unitary and irreversible dynamics

Entanglement between the degrees of freedom of the system and those Motivation Fundamentals of Quantum Theory Quantum Technologies Quantum Measurement Theory Border between quantum and classical descriptions Decoherence Entanglement between the degrees of freedom of the system and those of the enviornment PROBLEM: All new quantum technologies rely on quantum coherence Qubit quantum computation quantum cryptography quantum communication

Approaches to the dynamics of open quantum systems 1 Derivation of an equation of motion for the reduced density matrix of the system of interest: The master equation microscopic approach phenomenological approach Microscopic hamiltonian for the total closed system (model of environment and interaction) * Trace over the environmental degrees of freedom * Approximations * 2 Solution of the master equation analytical methods numerical methods damping basis method1 green function algebraic superoperatorial methods2 * 1 H.-J. Briegel, B.-G. Englert, PRA 47, 3311 (1993) 2 F. Intravaia, S.M., A. Messina, PRA 67, 042108 (2003) Monte Carlo wave function (and variants)3,4 Influence functional – path integral5 3 J. Dalibard, Y. Castin and K. Molmer, Phys. Rev.Lett. 68, 580 (1992) 4 J. Piilo, S.M., A. Messina, F. Petruccione, PRE 71, 056701 (2005) 5 U. Weiss, Quantum Dissipative Systems 2nd ed. (World Scientific, Singapore, 1999). *

Approximations weak coupling approximation weak coupling between system and environment perturbative approaches (expansions in the coupling constant) Markovian approximation assumes that the reservoir correlation time is much smaller than the relaxation time of the open system coarse-graining in time changes in the reservoir due to the interaction with the system do not feed back into the system

Lindblad master equation 1,2 Weak coupling + Markovian approximation + RWA or secular approx. Vj and Vj†transition or jump operators depending on the specific physical system  density matrix of the reduced system ADVANTAGES [ ] simulation techniques important properties dynamical map Semigroup property: * Markovian dynamical map Positivity: * 1 G. Lindblad, Comm. Math. Phys. 48, 119 (1976) 2 V. Gorini, A. Kossakowski, E.C.G. Sudarshan, J. Math. Phys. 17, 821 (1976)

Positivity and Complete positivity probabilistic interpretation of the density matrix COMPLETE POSITIVITY Less intuitive property! Violation of CP incompatible with the assumption of a total closed system (for factorized initial condition) Complete positivity of the dynamical map Λt guarantees that the eigenvalues of any entangled state ρS+Sn of S + Sn remain positive at any time S system Sn n-level No dynamical coupling

Importance of CP A consistent physical description of an open quantum system must be not only positive but also completely positive 1 The Lindblad form of the master equation is the only possible form of first-order linear differential equation for a completely positive semigroup having bounded generator This is valid whenever What about non-Lindblad-type master equations? [ ] CP is not guaranteed and unphysical situations, showing that the model we are using is not appropriate, may show up in the dynamics 1 K. Kraus, States, Effects and Operations, Fundamental Notions of Quantum Theory (Academic, Berlin, 1983).

Non-Markovian master equations Non-Lindblad ME No man’s land Non-Markovian master equations need not be in the Lindblad form, and usually they are not. Why to study them ? [ ] photonic band-gap materials 1 quantum dots atom lasers 2 non-Markovian quantum information processing 3 QUANTUM NANOTECHNOLOGIES 4,5,6,7 4 A. Micheli et al, Single Atom Transistor in a 1D Optical Lattice, Phys. Rev. Lett. 93,140408 (2004) 5 L. Tian et al, Interfacing Quantum-Optical and Solid-State Qubits, Phys. Rev. Lett. 92, 247902 (2004) 6 L. Florescu et al, Theory of a one-atom laser in a photonic band-gap microchip, Phys. Rev. A 69, 013816 (2004) 7 L. Tian and P. Zoller, Coupled ion-nanomechanical systems, Phys. Rev. Lett. 93, 266403 (2004) 1 S. John and T. Quang, Phys. Rev. Lett. 78, 1888 (1997) 2 J.J. Hope et al., Phys. Rev. A 61, 023603 (2000) 3 R. Alicki et al., Phys. Rev A 65, 062101 (2002); R. Alicki et al., ibid. 70, 010501 (2004)

Damped harmonic oscillator or Quantum Brownian Motion in a harmonic potential Paradigmatic model of the theory of open quantum systems Ubiquitous model may be solved exactly Quantum Optics, Quantum Field Theory, Solid State Physics system Htot= Hsys + Hres + Hint Microscopic model environment coupling Spectral density of the reservoir

Two approaches: microscopic and phenomenologic 1 GOAL: To study the dynamics of the system oscillator, in presence of coupling with the reservoir beyond the Markovian approximation Exact Master Equation Time-convolutionless approach Phenomenological master equation containing a memory kernel generalized master equation local in time generalized master equation non-local in time 1 H.-P. Breuer and F. Petruccione, "The theory of open quantum systems", Oxford University Press, Oxford (2002)

Exact Master Equation 1,2 (time convolutionless approach) HU-PAZ-ZHANG MASTER EQUATION SUPEROPERATORS TIME-DEPENDENT COEFFICIENTS Time dependent coefficients have the form of series in the coupling constant α to the second order in the coupling constant we have DIFFUSION TERMS (decoherence) noise kernel DAMPING TERM (dissipation) dissipation kernel 1 J.P. Paz and W.H. Zurek, Environment-induced decoherence and the transition from quantum to classical, Proceedings of the 72nd Les Houches Summer School on Condensed Matter Waves, July-August 1999, quant-ph/0010011. 2 F. Intravaia, S.M., A. Messina, Eur. Phys. J. B 32, 109 (2003).

Secular approximated master equation (and applications) LINDBLAD TYPE NON-LINDBLAD TYPE otherwise Trapped ions F. Intravaia, S. M., J. Piilo and A. Messina, "Quantum theory of heating of a single trapped ion", Phys. Lett. A 308 (2003) 6. S. M., J. Piilo, F. Intravaia, F. Petruccione and A. Messina, “Simulating Quantum Brownian Motion with single trapped ions ”, Phys. Rev. A 69 (2004) 052101. S. M., J. Piilo, F. Intravaia, F. Petruccione and A. Messina, “Lindblad and non-Lindblad type dynamics of a quantum Brownian particle”, Phys. Rev. A. 70 (2004) 032113. S. M. "Revealing virtual processes in the phase space", J.Opt.B: Quantum and Semiclass. Opt. 7 S398–S402 (2005) * Linear amplifier * S.M., J. Piilo, N. Vitanov, and S. Stenholm, “Transient dynamics of quantum linear amplifiers”, Eur. Phys. J. D 36, 329–338 (2005)

Exact solution1 The Quantum Characteristic Function (QCF)2 [NO MARKOVIAN APPROX, NO WEAK COUPLING APPROX, NO RWA APPROX] The Quantum Characteristic Function (QCF)2 1 F. Intravaia, S. M. and A. Messina, Phys. Rev. A 67, 042108 (2003) 2 S.M.Barnett and P.M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, Oxford, 1997)

Example of non-Markovian dynamics The risks of working with non-Lindblad Master Equations

Master equation with memory kernel EXAMPLE: Phenomenological non-Markovian master equation describing an harmonic oscillator interacting with a zero T reservoir Markovian Liouvillian Exponential memory kernel g coupling strength decay constant of system-reservoir correlations Non-Lindblad Master Equation: CP and positivity are not guaranteed!

Solution using the QCF What is the QCF? p = p = 0 symmetric ordering p = -1 antinormal ordering p = 1, normal ordering Quantum Characteristic Function Fourier transform p = 1 P function p = -1 Q function p = 0 Wigner function defining properties useful property

Solution using the QCF Example: g /  = 1  =  t defining properties The defining properties are always satisfied No problem with the dynamics

Density matrix solution The quantum characteristic function contain all the information necessary to reconstruct the density matrix, and therefore is an alternative complete description of the state of the system1 we look at the time evolution of the population of the initial state Example: g /  = 1  =  t problem of positivity firstly noted by Barnett&Stenholm2 1 S.M.Barnett and P.M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, Oxford, 1997) 2 S.M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001).

Limits in the QCF description The QCF and the density matrix are not “operatively” equivalent descriptions of the dynamics, in the sense that the QCF may fail in discriminating when non-physical conditions (negativity of the density matrix eigenvalues) show up. The defining properties of the QCF are only necessary conditions defining properties The additional condition to be imposed on the QCF in order to ensure the positivity of the density matrix does not seem to have a simple form S. M., "Limits in the quantum characteristic function description of open quantum systems", Phys. Rev. A 72, 024103 (2005)

Non-Markovian dynamics of a qubit Phenomenological model: Non-Markovian master equation with exponential memory Markovian Liouvillian memory kernel Solution in terms of the Bloch vector Bloch vector

Positivity and Complete Positivity THE BLOCH SPHERE Condition for positivity: The dynamical map  maps a density matrix into another density matrix if and only if the Bloch vector describing the initial state is transformed into a vector contained in the interior of the Bloch sphere, i.e. the Bloch ball. 4 R ≤ 1 A largely unexplored region: The positivity – complete positivity region Are there non-Markovian systems which are positive but not CP? THIS RESULT PROVIDES THE EXPLICIT CONDITIONS OF VALIDITY OF A PARADIGMATIC PHENOMENOLOGICAL MODEL OF THE THEORY OF OPEN QUANTUM SYSTEMS, NAMELY THE SPIN-BOSON MODEL WITH EXPONENTIAL MEMORY. Applying the criterion for CP demonstrated in [1], and using the analytical solutions we have derived for our model, we have proved that, in the case of exponential memory and for the non-Markovian model here considered positivity is a necessary and sufficient condition for complete positivity [2] [1] M.B. Ruskai, S. Szarek, and E. Werner, Lin. Alg. Appl. 347, 159 (2002). [2] S. M., “Complete positivity of the spin-boson model with exponential memory”, submitted for publication

Summary Open quantum systems Markovian master equation Non-Markovian master equation Lindblad form non-Lindblad form Beyond the Lindblad form Two approaches: microscopic and phenomenological Paradigmatic model: QBM or damped harmonic oscillator TCL (microscopic exact approach) Memory kernel Generalized master equation QCF and density matrix solutions Solution based on algebraic method Positivity violation Applications: trapped ions, linear amplifier Limits in the use of the QCF

References Quantum Brownian Motion Trapped ion simulator F. Intravaia, S. Maniscalco, J. Piilo and A. Messina, "Quantum theory of heating of a single trapped ion", Phys. Lett. A 308, 6 (2003). F. Intravaia, S. Maniscalco and A. Messina "Comparison between the rotating wave and Feynman-Vernon system-reservoir couplings in the non-Markovian regime", Eur. Phys. J. B 32, 109 (2003). F. Intravaia, S. Maniscalco and A. Messina, "Density Matrix operatorial solution of the non-Markovian Master Equation for Quantum Brownian Motion", Phys. Rev. A 67, 042108 (2003) . S. Maniscalco, F. Intravaia, J. Piilo and A. Messina, “Misbelief and misunderstandings on the non--Markovian dynamics of a damped harmonic oscillator”, J.Opt.B: Quantum and Semiclass. Opt. 6, S98 (2004) . Trapped ion simulator S. Maniscalco, J. Piilo, F. Intravaia, F. Petruccione and A. Messina, “Simulating Quantum Brownian Motion with single trapped ions ”, Phys. Rev. A 69, 052101 (2004) . Lindblad non-Lindblad border and the existence of a continuous measurement interpretation for non-Markovian stochastic processes S. Maniscalco, J. Piilo, F. Intravaia, F. Petruccione and A. Messina, Lindblad and non-Lindblad type dynamics of a quantum Brownian particle”, Phys. Rev. A. 70, 032113 (2004) . S. Maniscalco "Revealing virtual processes in the phase space", J.Opt.B: Quantum and Semiclass. Opt. 7 S398–S402 (2005). Positivity and complete positivity S. Maniscalco, "Limits in the quantum characteristic function description of open quantum systems", Phys. Rev. A 72, 024103 (2005) S. Maniscalco and F. Petruccione “Non-Markovian dynamics of a qubit”, Phys. Rev. A. 73, 012111 (2006) S. Maniscalco “Complete positivity of the spin-boson model with exponential memory”, submitted for publication Non-Markovian wavefunction method J. Piilo, S. Maniscalco, A. Messina, and F. Petruccione "Scaling of Monte Carlo wavefunction simulations for non-Markovian systems", Phys. Rev. E 71 056701 (2005).