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Open quantum systems Marko Ljubotina Mentor: prof. dr. Tomaž Prosen Co-mentor: asist. dr. Ugo Marzolino Ljubljana, 1.3.2016 Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 1 / 23 Overview • Introduction • Mixed states • The density matrix • Closed quantum systems • Von Neumann equation • Open quantum systems • • • • • • Dynamical maps Quantum dynamical semigroups Lindblad equation Jump operators Nonequilibrium steady state Example • Conclusion Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 2 / 23 Introduction • Closed quantum systems are an idealization • Systems interact with the environment • We deal with statistical ensembles • The Markov approximation • Extensive research in the 1970s • Recently regained popularity with the advances in quantum information theory Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 3 / 23 Mixed states • When dealing with statistical quantum mechanics or dissipative dynamics the wave function is not sufficient • A wave function |ψi always represents a pure state of the system • An example of a pure state of a spin 1/2 particle: 1 |ψi = √ (|↑i + |↓i) 2 Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 4 / 23 Example - Polarization of a photon • A single photon can have two helicities: right circular polarized |Ri or left circular polarized |Li (or a superposition of the two) • State of a (vertically polarized) photon exiting a laser 1 |ψi = √ (|Ri + |Li) 2 • A photon from a conventional lightbulb: 50% chance of being in the state |Ri and 50% chance of being in the state |Li • Measuring the probability that the photon is in the state |Ri repeatedly would give us a 50% probability in both cases • We can block the first photon using a polarizer but we can not block the other photon with any polarizer with 100% efficiency • The first is in a superposition of states, while the second can be described by some statistical ensemble, but not with a wave function Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 5 / 23 Mixed states Mixed states arise when • Dealing with systems in thermodynamic or chemical equilibrium • Observing only a part of a larger entangled system • Considering systems with dissipative dynamics Any mixed state can be represented as a pure state of a larger system. • The process is called purification • Second photon - a simultaneous emission of two photons 1 |ψi = √ (|R, Li + |L, Ri) 2 Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 6 / 23 The density matrix • Describes pure and mixed states with the same formalism • The density operator of a pure state: ρ̂ = |ψihψ| • Generalising to mixed states: ρ̂ = X pi |ψi ihψi | i • pi are the probabilites that our system is in the state |ψi i Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 7 / 23 The density matrix - properties • Diagonal elements correspond to probabilites • Offdiagonal elements correspond to correlations • A trace of 1 Tr(ρ) = 1 • Self-adjoint ρ = ρ† • Positive-semidefinite - all it’s eigenvalues are greater or equal to 0 λi ≥ 0 • Expectation value of observables hAi = Tr(ρA) Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 8 / 23 Example - Polarization of a photon Density matrices: • In the first case ρ = |ψihψ| = with |ψi = √1 (|Ri 2 1 1 1 2 1 1 + |Li) • And in the second 1 1 1 1 0 ρ = |RihR| + |LihL| = 2 2 2 0 1 Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 9 / 23 Closed quantum systems • The Schrödinger equation for the dynamics of wave functions ∂ ψ(r, t) = −iHψ(r, t) ∂t • The von Neumann equation for the Hamiltonian dynamics of density matrices ∂ ρ = −i[H, ρ] ∂t • The solution of the von Neumann equation is usually in the form ρ(t0 + t) = e −iHt ρ(t0 )e iHt • Alternatively, we define a superoperator Hρ = [H, ρ] which gives the following solution ρ(t) = e −iHt ρ(0) Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 10 / 23 Closed quantum systems • The Hamiltonian H is self-adjoint → it’s eigenvalues are all real • Hence the eigenvalues of U(t) = e −iHt are of the form λi = e iϕi → ||λi || = 1 • States are preserved with time - the same state will be reached again after a sufficient amount of time has passed Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 11 / 23 Open quantum systems • Account for the interaction with the environment • A different equation is needed to describe these systems A classical example: a metallic bar connected two heat reservoirs at different temperatures. • The steady state is the same regardless of the initial state of the bar • An exchange of energy between our system and the environment Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 12 / 23 Example - spin chain Consider a chain of N spin 1/2 particles interacting with eachother by the Heisenberg interaction H = −J N−1 X ~σi · ~σi+1 i=1 We can couple this to two more spins in states of themal equilibrium ρT = e −βH Tr(e −βH ) using the same interaction Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 13 / 23 Example - spin chain Issues • The state of the spins in thermal equilibirum should not change if they are to represent heat reservoirs at a constant temperature • How does one define a temperature of a single spin Alternative • Creating a state with a spin flow instead • Each spin is in a linear combination of states |↑i and |↓i • A spin sink on the left and a spin source on the right But we still don’t know how to approach this problem Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 14 / 23 Dynamical maps • Open quantum system S • Environment system R • Propagator U = e −iHtot t • We define a dynamical map Λ as ρ → Λρ = TrR (U(ρS ⊗ ρR )U † ) Properties • Trace preserving • Completely positive Λ†n = Λ† ⊗ 1n Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 15 / 23 Quantum dynamical semigroups A family of dynamical maps describing a system forms a semigroup: • Λt Λs = Λt+s - the so-called semigroup condition or Markov property • Tr((Λt ρ)A) is a continuous function of t for any density matrix ρ and observable A • The generator of any quantum dynamical semigroup: ∂ ρ = Lρ ∂t • Λt = e Lt Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 16 / 23 Lindblad equation Using this we derive the Lindblad equation X d 1 † 1 † † ρ = Lρ = −i[H, ρ] + Li ρLi − Li Li ρ − ρLi Li dt 2 2 i • H is the hamiltonian of the open quantum system S • L are the jump operators describing the permitted incoherent changes in the system Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 17 / 23 Jump operators • A spin sink and source + L1 = λ1 σN L2 = λ2 σ1− • A totally asymmetric simple exclusion process − Li = λσi+ σi+1 • Dephasing noise Li = λσiz Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 18 / 23 Nonequilibrium steady state • Describing the complete dynamics of a system can be challenging • Steady states represent the state of the system after a long time LρNESS = 0 • Depending on the system they may either be unique or form a smaller subspace within the phase-space Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 19 / 23 Example - spin chain • N spins interacting through the Heisenberg interaction H = −J N−1 X ~σi · ~σi+1 i=1 • A spin sink and source + L1 = λ1 σN L2 = λ2 σ1− • Expand the operator L over the basis of all 2N × 2N matrices • Solve for the eigensystem of the resulting 22N × 22N matrix and find the eigenstates with λi = 0 Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 20 / 23 Example - spin chain The density matrix corresponding to the unique steady state of the system with N = 4, J = 1 and λ1 = λ2 = 0.5. Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 21 / 23 Example - spin chain Expected values of individual spins Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 22 / 23 Conclusion • Density matrices are needed in statistical quantum mechanics, open quantum systems and other areas • The von Neumann equation was presented • The Lindblad equation was shown and some of the underlying mathematical theory was outlined The field of open quantum systems is particularly important in quantum information theory, quantum optics and condensed matter physics. Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 23 / 23 Thank you! Questions? Marko Ljubotina Open quantum systems Ljubljana, 1.3.2016 23 / 23