Ohmic vs Markovian heat bath — two-page

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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).

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