Ohmic vs Markovian heat bath — two-page-tutorial Lajos Diósi Wigner Research Center for Physics, H-1525 Budapest 114, POB 49, Hungary (Dated: June 29, 2012) In open quantum system theory the Ohmic heat bath and the Markovian heat bath are two different but closely related special cases. We discuss them on a common bases. I. HEAT BATH: COORDINATE COUPLING System-Bath total Hamiltonian: ĤS + ĤB + ĤI . p̂2 + V (q̂), 2M X X p̂2 1 α 2 2 ĤB = + mα ωα x̂α = ~ωα b̂†α b̂α , 2mα 2 X X ĤI = −q̂ cα x̂α = −q̂ gα (b̂α + b̂†α ) = −q̂ X̂, P P where X̂ = cα x̂α = gα (b̂α + b̂†α ) is called the B-field. The two p conventions of coupling constants are related by cα = 2mα ωα /~ gα . Theorem: If at t = 0 the initial states of S and B are uncorrelated and B is in thermal equilibrium (at a certain inverse temperature β = 1/kB T ) then the reduced dynamics of S for t > 0 is completely determined by ĤS and the equilibrium correlation ĤS = CXX (t − u) = hX̂t X̂u iβ where X̂t is the B-field in interaction picture. This correlation is uniquely determined by the effective spectral density πX 2 gα δ(ω − ωα ) J(ω) = ~ which encodes the coupling constants P as well. [With the spectral density itself, n(ω) = π~ δ(ω − ωα ), the effective spectral density takes the form J(ω) = ((g(ω))2 n(ω) where g(ω) is the frequency-smoothened form of gα .] We can express CXX (t) via J(ω): Z ~ ∞ ~βω CXX (t) = J(ω) coth( ) cos(ωt) − i sin(ωt) dω. π 0 2 The imaginary part is purely dynamical, independent of T. To describe the reduced dynamics of S, either the general (non-Markovian) master equation for the reduced density matrix ρ̂ or the Heisenberg equation of q̂ can be used. With the second option, the following nonMarkovian quantum Langevin equation can be derived (‘Lamb-shift’ in ĤS and the ‘initial slip’ are ignored): Z t ¨ = −V 0 (q̂) − M ˙ 0 )dt0 + X̂t M q̂(t) γ(t − t0 )q̂(t 0 where the damping term is determined by the memory kernel γ(t − t0 ) which is independent of ~ and of T : Z 2 ∞ J(ω) M γ(t) = cos(ωt)dω. π 0 ω II. OHMIC DAMPING The Ohmic model applies when damping force is proportional to the instant velocity. Ohm’s Law in electricity results from such microscopic damping force on electrons moving in a potential. If we are interested in such memory-less damping, we must assume the Ohmic effective spectral density J(ω) = ηω (with high-frequency cutoff ωc ) when the memory disappears from the damping kernel: M γ(t) = 2ηδ(t). The quantum Langevin equation of motion becomes M q̂¨ = −V 0 (q̂) − η q̂˙ + X̂t , η is the damping (friction) constant. The fluctuation force X̂t is a colored quantum noise of correlation CXX (t − t0 ) hence the corresponding reduced dynamics remains non-Markovian! However, at higher T the real part of the Ohmic correlation dominates, the imaginary part can be ignored. We can replace the operator force X̂t by the classical colored noise force Xt : M q̂¨ = −V 0 (q̂) − η q̂˙ + Xt . In the high-T limit β → 0, the correlation tends to be time-local: βCXX (t) → 2ηδ(t). Thus the random force Xt becomes a classical white-noise: hXt Xu istoch = 2ηkB T δ(t − u). Now, replacing q̂ by q would yield the classical Langevin equation, its solution q(t) at V = 0 would be the Ornstein-Uhlenbeck stochastic process which is nonMarkovian itself. Fortunately, the pair of phase space coordinates satisfy Markovian equations (let’s go back to the quantum case): q̂˙ = p̂/M p̂˙ = −V 0 (q̂) − η p̂/M + X. Hence the Ohmic (or high-T ) dynamics is often called Markovian. The classical Langevin equations do not preserve the canonical commutation relations between q and p, yet nobody cares because this follows duely from the irreversible modification of the canonical dynamics. In the quantum case, however, the issue [q̂, p̂] 6= i~ is a fatal error, the above quantum Langevin equation with the classical white-noise Xt can be totally incorrect e.g. for certain minimum uncertainty wave packets. 2 III. HEAT BATH: GENERAL COUPLING ĤS is arbitrary, ĤB is the same as before, X X ĤI = ŝ† gα b̂α + s gα b̂†α = ŝ† B + h.c. where B̂ is the non-Hermitian bosonic B-field: X B̂ = gα b̂α . E.g.: ŝ = −q̂ − iχp̂ yields ĤI = −q̂ X̂ − χp̂Ŷ where X̂ = B̂ + B̂ † , Ŷ = −i(B̂ − B̂ † ), i.e., the coordinate and the momentum of S couple to the coordinates and momenta of B. [We could have considered complex couplings gα 6= gα∗ but it turns out that the reduced dynamics of S wouldn’t depend on the phases of gα .] The same Theorem holds as before. Starting from uncorrelated S and B, the equilibrium correlations of the B-fields B̂, B̂ † (or X̂, Ŷ ), together with ĤS and β, will fully determine the reduced dynamics of S. All non-vanishing correlations are determined by the effective spectral density and the temperature. R exp(iωt) dω CB † B (t) = hB̂t† B̂iβ = π~ J(ω) exp(~βω)−1 CBB † (t) = hB̂t B̂ † iβ = IV. ~ π R exp(−iωt) J(ω) 1−exp(−~βω) dω That’s the standard Markovian master equation in the Lindblad form. Second, we consider Markovianity at finite T as well. We assume discrete spectrum of ĤS and, for simplicity, we couple a single transition to B: ŝ = |1ih2|, ω2 − ω1 = /~ > 0. We retain the flat Markovian effective spectrum J(ω) = J as before and, as a further approximation, we ignore the frequency dependence of the thermal factors in the relevant vicinity of ω = /~. Then both correlation functions become time-local: CB † B (t) = e−β CBB † (t) = 2~J δ(t). exp(β) − 1 They contribute to the following master equation: ρ̂˙ = − ~i [ĤS , ρ̂] + Γ ŝρ̂ŝ† − 12 {ŝ† ŝ, ρ̂} + + e−β Γ ŝ† ρ̂ŝ − 12 {ŝŝ† , ρ̂} . Γ = ~−1 J/(1 − e−β ) is the decay constant. If more than a single transition is coupled to B, the extension of the model is possible just by adding similar terms to ĤI , yielding similar Lindblad terms in the above Markovian master equation. MARKOVIAN CASE At T > 0 the correlations cannot become time-local in general. If, however, the range of the relevant (coupled) part of the spectrum of ĤS is finite then we can introduce Markovian effective spectral densities. First, we assume zero temperature (β = ∞) where CB † B vanishes while CBB † becomes time-local, CBB † (t) = 2~Jδ(t) provided we extend the spectrum of B for negative frequencies as well and choose flat effective spectral density J(ω) = J. This is correct if the true effective spectral density is unstructured (flat) over the finite range of the relevant frequencies. The chosen abstract B with J(ω) = J may be called Markovian. The reduced dynamics of S becomes Markovian. If this time, instead of the Langevin equation, we use the alternative math to describe the reduced dynamics of S, we can derive the following master equation: i J ρ̂˙ = − [ĤS , ρ̂] − 2ŝρ̂ŝ† − {ŝ† ŝ, ρ̂} . ~ ~ [1] U. Weiss: Quantum Dissipative Systems (World Scientific, Singapore, 1999). [2] C. W. Gardiner and P. Zoller: Quantum Noise (Springer, V. OHMIC VS MARKOVIAN We saw the special case of coordinate coupling ŝ = −q̂ in Ohmic effective spectrum J(ω) = ηω, 0 < ω < ωc which case becomes Markovian asymptotically for ~β → 0, i.e., in the high-T limit. When we retain couplings of both B-coordinates and B-momenta to S, we can achieve quantum Markovianity at any T at a radically different choice J(ω) = J, −∞<ω <∞ called Markovian effective spectrum. The two mathematical models of Markovianity apply in two different physical situations respectively, whose relationship is yet to be clarified. Berlin, 2004).