Completely Positive, Simple, and Possibly Highly Accurate
Approximation of the Redfield Equation
Dragomir Davidović
Georgia Institute of Technology, Atlanta, Georgia, United States
Here we present a Lindblad master equation that approximates the Redfield equation, a well
known master equation derived from first principles, without significantly compromising the range
of applicability of the Redfield equation. Instead of full-scale coarse-graining, this approximation
only truncates terms in the Redfield equation that average out over a time-scale typical of the
quantum system. The first step in this approximation is to properly renormalize the system
Hamiltonian, to symmetrize the gains and losses of the state due to the environmental coupling.
In the second step, we swap out an arithmetic mean of the spectral density with a geometric
one, in these gains and losses, thereby restoring complete positivity. This completely positive
approximation, GAME (geometric-arithmetic master equation), is adaptable between its time-
independent, time-dependent, and Floquet form. In the exactly solvable, three-level, Jaynes-
Cummings model, we find that the error of the approximate state is almost an order of magnitude
lower than that obtained by solving the coarse-grained stochastic master equation. As a test-bed,
we use a ferromagnetic Heisenberg spin-chain with long-range dipole-dipole coupling between up
to 25-spins, and study the differences between various master equations. We find that GAME
has the highest accuracy per computational resource.
1 Introduction
Quantum correlations between particles in quantum systems are counterintuitive, giving us exotic algorithms
that can greatly accelerate some calculations using quantum computing as opposed to conventional comput-
ing. It remains an unsettled issue, however, to describe correlated quantum dynamics in large many-body
quantum systems, with the precision required for quantum computing, if the systems are coupled to an envi-
ronment. In the Born-Markov approximation, the time dependence of the reduced state of the system can be
approximated by a first-order differential equation, (e.g., the master equation), which greatly simplifies this
description. [1, 2] The Born-Markov approximation is justified if the time in which the environment changes
the system state in the interaction picture, τ
r
, is much longer than the time at which the correlations in
the environment decay, τ
c
. [2] We will use the terms bath and environment interchangeably throughout this
paper.
The master equation does not necessarily preserve the positivity of the reduced quantum state of the
system. There is only one form of the master equation that is a completely positive map, the deeply respected
Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. [24] The GKSL equation is a consequence of
the Dirac–von Neumann axioms, while the physical scope of the equation rests in the coarse-graining of the
reduced system dynamics over a timescale much longer than τ
c
. [5]
It has been very difficult to derive the GKSL equation from first principle calculations using the total
Hamiltonian of the system and environment as a starting point. [6] The operator-projection technique has
been the standard way to derive quantum master equations [2, 79] initially pioneered by Redfield. [10, 11]
The Redfield equation, however, suffers from a lack of positivity preservation leading to negative probabilities
of observables [6] which violates the mathematical definition of probability. Negative states also pose problems
in terms of how to quantify the entanglement of the system, which is important for quantum computing.
One measure of entanglement is the negativity of the state’s partial transpose, [12] which raises the question:
Dragomir Davidović: dragomir.davidovic@physics.gatech.edu
Accepted in Quantum 2020-08-20, click title to verify. Published under CC-BY 4.0. 1
arXiv:2003.09063v4 [quant-ph] 17 Sep 2020
Can we trust negativity of the partial transpose as a genuine measure of entanglement if the approximate
state itself is negative?
Although it has this problem of producing negative probabilities, the Redfield equation has major ad-
vantages as well. It is a highly accurate master equation that is simple to apply and not axiomatic. Thus,
the Redfield equation and its modifications have found significant use in quantum chemistry, condensed
matter physics, and quantum optics. [6, 1320] The issue of negative states and their relevance remains de-
bated. [21, 22] The Redfield equation with time dependent coefficients (TDC) has reduced negativity. The
nonpositivity of that equation can even be seen as a benefit of a red-flag for the breakdown of the master
equation concept. [22] This benefit does not extend well to the Redfield equation with asymptotic coefficients
as we will discuss in this paper. The latter is the widely used master equation that we are approximating
into a GKSL form in this paper.
Our main focus here is to find a simple, completely positive approximation of the Redfield equation
without significantly compromising its accuracy. Note that in the same weak coupling regime where the
Redfield equation is valid, quantum trajectory description is available which guarantees positivity. [23, 24]
There are also quite a few non-perturbative approaches such as the exact quasi adiabatic path integral
method (QUAPI), [2527] the hierarchical equations of motion (HEOM), [2832], the multiconfiguration
time-dependent Hartree (MCTDH) [33, 34], and the multilayer formulation ML-MCTDH. [35, 36] These are
powerful approaches but can be very challenging. Exact methods aiming at improved efficiency include quan-
tum trajectory based hierarchy of stochastic pure states (HOPS) [3739] and tensor network methods. [40, 41]
Other strong coupling approaches involve unitary transformations that result in a non-perturbative bath-
renormalization of the system Hamiltonian, followed by a quantum master equation in the weak coupling
limit after the renormalization. [42, 43]
There have been several attempts to cure the issue of negative states by modifying the Redfield master
equation into a GKSL form. This was done for the first time by Davies, [44] who gave us the rotating-wave
approximation (RWA), also known as the secular approximation. The RWA amounts to coarse-graining of
the reduced state on a timescale comparable to the Heisenberg time. Since this time scales exponentially
with system size, the RWA does not work well for large many-body quantum systems that generally have
exponentially suppressed level spacings with increasing size of the system. More recently, coarse-grained
GKSL master equations have been derived from first principles [4547] capable of capturing correlation effects
that the RWA cannot. Another notable completely positive approximation is the partial-RWA (PRWA). [48,
49] An alternative approach to restoring complete positivity includes the dynamically coarse-grained (DCG)
master equation. [50, 51] Phenomenological models have also worked well to briskly set up a completely
positive master equation. [5255]
The approximate, completely positive GAME works in two steps. First, it identifies an implicit renormal-
ization of the Hamiltonian (e.g., the Lamb-shift) in the Redfield equation. After separating the Lamb-shift
by adding it to the system Hamiltonian, the second step separates the dissipator into two parts: a completely
positive dissipator and the remainder. The latter has the property that it averages-out on a short time-scale
compared to the system relaxation time, and is therefore dropped, while the renormalized Hamiltonian and
the completely positive dissipator remain intact.
Practically, in terms of the spectral density (SD), we swap out an arithmetic mean of the SD at two
system frequencies, with a geometric mean of the SD. The difference between the two means has the sought
after property that it averages out on a typical time scale of the system. We find that the state error added
by the approximation is linear in system-bath coupling and generally comparable to the accuracy of the
Redfield equation (when applicable).
GAME belongs to a relatively new group of approximations that we refer to here as the
SD-approximations.
[55, 56] In this paper we compare many of the aforementioned GKSL-equations with GAME, by applying
them on a spin-1/2 Heisenberg chain, and have not found a single one that outperforms GAME in terms of
the closeness of its solutions to those of the Redfield equation.
The renormalized Hamiltonian plays the key role and is the primary reason that GAME is capable of
accounting for the correlation effects that the bath introduces into the system dynamics. The effects of the
renormalization are studied in detail in the case of the 3-level Jaynes-Cummings model, where the eigenstates
of the renormalized Hamiltonian are identified as the relaxation modes of the system coupled to the heat
bath, those that decay with a single relaxation time.
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By construction, GAME retains the simplicity of the Redfield equation, which is one of its assets. Even
more important is its combination of both simplicity and accuracy.
The paper is organized as follows. In Sec. 2 we introduce the notation relevant for this work. In Sec. 3, 3.1,
and 3.2 we review the Redfield, Coarse-Grained Redfield, and the Davies-Lindblad master equations, respec-
tively, as a prelude to GAME, which is presented in Sec. 4. In Sec. 5 we implement GAME on an exactly
solvable 3-level Jaynes–Cummings model. We follow by the detailed investigation of a 25-body spin-1/2 fer-
romagnetic spin-chain in Sec. 6. We end the analysis by presenting a detailed comparison with other GKSL
equations in Sec. 7, followed by discussion and conclusion.
2 Introducing the problem
Perturbative master equations have been derived many times in the literature. For a tutorial and some recent
examples we refer the reader to Refs. [21, 22, 47, 57].
Here we consider a quantum system coupled to a heat-bath, represented by the total Hamiltonian
H
tot
= H
0
1
B
+ 1
S
H
B
+ A B. (1)
H
0
and A are the respective Hamiltonian and arbitrary operators performing in the Hilbert space of the
quantum system, H
B
and B are the corresponding operators for the heat bath, and A B is the interaction
Hamiltonian between the system and the heat bath. All the operators are assumed to be Hermitian. For
notational clarity we only consider one coupling term in the interaction Hamiltonian. The results can be
straightforwardly extended to a sum of interaction terms assuming uncorrelated heat baths.
In the Hilbert space of the quantum system, we use the eigenbasis of H
0
, H
0
|ni = E
n
|ni, where |ni
are the eigenvectors and E
n
are the eigenenergies. Then, in the interaction picture A(t) = e
iH
0
t
Ae
iH
0
t
is
decomposed as
A(t) =
X
n,m
A
nm
e
mn
t
|nihm|, (2)
where A
nm
= hn|A|mi, ω
mn
= E
m
E
n
, t is time, and we use ~ = 1. Operator A(t) can be represented in
the matrix form as
A(t) = A Ω(t), (3)
where is the Hadamard product (the element-wise product) and [Ω(t)]
nm
= e
mn
t
.
The bath correlation function is defined as C(t) = T r[ρ
B
B(t)B(0)], where B(t) = e
iH
B
t
Be
iH
B
t
and
ρ
B
is the reduced density matrix of the heat bath. We work under the premise that the excitations in the
heat-bath decay on a time-scale much smaller than the time scale of the system state in the interaction
picture. In that case, the change in the reduced density matrix of the bath due to the coupling to the system
can be neglected, in the leading order of the approximation, and the density matrix of the environment can
be approximated by the thermal state where C(t) = C
?
(t). [2] In our notation we reserve the symbols ρ
and %, for the density matrix in the Schrödinger and interaction picture, respectively.
3 Redfield equation
The equation is a widely used fundamental master equation describing fluctuations and dissipation in quantal
systems. [9] It is derived from first principles, by applying the Born-Markov approximation and projection
operator technique. We begin here from Eq. 16 from a recent re-derivation, [47]
dt
= i[H
0
, ρ] AA
f
ρ ρA
f
A + A
f
ρA + AρA
f
, (4)
where
A
f
=
Z
0
C(τ)A(τ) A Γ
?
(5)
is termed the "filtered" operator, and adopt the notation from that reference throughout this paper. The
star symbol indicates complex-conjugate. The matrix elements of Γ can be expressed as
Γ
nm
=
Z
0
C(τ)e
nm
τ
=
1
2
γ
nm
+ iS
nm
. (6)
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C(t) γ(ω) S(ω)
1
2π
R
−∞
e
it
γ(Ω)d
R
−∞
e
t
C(t)dt
1
2π
P
R
−∞
γ(Ω)
ω
d
gω
2
c
(1+
c
t)
2
2πgωe
ω
ω
c
Θ(
ω
ω
c
) gω
c
h
1
ω
ω
c
e
ω
ω
c
Ei(
ω
ω
c
)
i
gω
2
c
[ln |ω
c
t| + γ
+i
π
2
sgn(t)] + O[t
2
ln(t)]
2πgω
ω
2
c
ω
2
c
+ω
2
Θ(
ω
ω
c
) gω
c
h
π/2(ω/ω
c
) ln(|ω|
c
)
1+(ω
c
)
2
i
6gω
2
c
(1+
c
t)
4
2πg
ω
3
ω
2
c
e
ω
ω
c
Θ(
ω
ω
c
)
gω
c
h
2 +
ω
ω
c
+ (
ω
ω
c
)
2
(
ω
ω
c
)
3
e
ω
ω
c
Ei(
ω
ω
c
)
i
.
Table 1: Integral transforms between bath correlation function [C(t)], and spectral functions γ(ω) (spectral density) and
S(ω) (principal density) at zero temperature. P indicates the principal value of the integral. The second and third row
display these functions in case of an Ohmic bath with exponential and Drude-Lorentz cutoff at frequency ω
c
. Here, Θ(x)
is the Heaviside step function, Ei(x) is the exponential integral
R
x
−∞
e
t
t
dt, g is the dimensionless system-bath coupling
constant, and γ is the Euler–Mascheroni constant. The forth row exhibits the super-Ohmic spectral density.
In this work we also use functions Γ(x) =
1
2
γ(x) + iS(x), so that Γ
nm
= Γ(ω
nm
), γ
nm
= γ(ω
nm
), and
S
nm
= S(ω
nm
). Here γ(x) is the spectral density (SD), equal to the Fourier transform of the bath correlation
function, and S(x) is the principal density (PD), equal to the imaginary part of the half-range Fourier
transform of the bath correlation function. The symbols Γ, γ, and S mean matrices with respective matrix
elements Γ
nm
, γ
nm
and S
nm
. Table 1 summarizes key formulae for the bath correlation functions and its
integral transforms, and three examples of heat baths at zero temperature, that we apply in this paper.
Eq. 4 can be recast in a different form,
dt
= i[H
0
, ρ]
1
2
[AA
f
A
f
A, ρ]
1
2
{AA
f
+ A
f
A, ρ}+ A
f
ρA + AρA
f
. (7)
With the renormalization of the system Hamiltonian,
H = H
0
i
2
(AA
f
A
f
A) H
0
+ H
L
, (8)
the Redfield equation becomes
dt
+ i[H, ρ] =
1
2
{AA
f
+ A
f
A, ρ}+ AρA
f
+ A
f
ρA, (9)
or in terms of matrix elements,
nm
dt
+ i[H, ρ]
nm
=
X
ij
A
?
in
ρ
ij
A
jm
γ
in
+ γ
jm
2
i(S
jm
S
in
)
1
2
X
i,j
ρ
ni
A
?
mj
A
ij
γ
ij
+ γ
mj
2
i(S
ij
S
mj
)
1
2
X
i,j
A
?
ji
A
ni
ρ
jm
γ
ni
+ γ
ji
2
i(S
ni
S
ji
)
. (10)
The matrix elements of the renormalized Hamiltonian 8 are
H
nm
= E
n
δ
nm
+
X
i
H(ω
ni
, ω
mi
)A
ni
A
im
, (11)
where we introduce the kernel for the unitary component of the reduced system dynamics (from now on the
unitary kernel):
H(ω, ω
0
) =
1
2
S(ω) + S(ω
0
) + i
γ(ω) γ(ω
0
)
2
. (12)
The role of the renormalization is to rewrite the loss terms in the Redfield equation so that all three
summands in Eq. 10 involve this one and only function of system frequencies,
G(ω, ω
0
) =
1
2
γ(ω) + γ(ω
0
)
i
S(ω
0
) S(ω)
, (13)
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that we refer to here as the dissipative kernel. That is to say, we reformulate the Redfield equation as
nm
dt
+ i[H, ρ]
nm
=
X
ij
A
?
in
ρ
ij
A
jm
G
in,jm
1
2
X
i,j
ρ
ni
A
ij
A
?
mj
G
mj,ij
1
2
X
i,j
A
ni
A
?
ji
ρ
jm
G
ji,ni
, (14)
with one simple kernel
G
in,jm
= G(ω
in
, ω
jm
) = Γ
in
+ Γ
?
jm
. (15)
The importance of this form of the Redfield equation is that we can approximate the kernel only. The
approximation will then carry over to the state gains and losses unbiasedly, maintaining the balance between
them. Furthermore, if the kernel in Eq. 15 were positive-semidefinite, it would lead to a GKSL-equation.
See for example Ref. [50]. So restoring positivity of the Redfield equation now amounts to approximating its
kernel with a positive-semidefinite one.
As written above, the Redfield equation has no dependence on the initial time and the history of the state
is fully encoded in the present state. In the derivation of the Redfield equation, before the last approximation
is applied, there is a master equation with time-dependent coefficients (TDCs), that depend on the choice of
the initial time. The filtered operator in that case has time dependence
A
f
(t) =
Z
t
0
C(τ)A(τ) A Γ
?
(t), (16)
which leads to the time-dependent SDs and PDs vis-à-vis
Γ
t
(ω
nm
) =
1
2
γ
t
(ω
nm
) + iS
t
(ω
nm
), (17)
while the master equation is
dt
= i[H
0
, ρ] AA
f
(t)ρ ρA
f
(t)A + A
f
(t)ρA + AρA
f
(t). (18)
This equation depends on the initial time, but, as t becomes much longer than τ
c
, the coefficients Γ
nm
(t)
approach the asymptotic value, and the equation no longer depends on the history of the system. [2] We also
call this the
R
t
R
approximation.
In this paper we refer to equations 18 and 4 as the TDC-Redfield equation and the Redfield equation,
respectively. The Redfield equation does not resolve the system dynamics over a time scale of the bath
correlation time. [2]
A time-dependent Redfield equation in form equivalent to 9 has been studied recently in context of
embedding of the quantum system into an expanded system that displays completely positive quantum
dynamics by Breuer. [58] The density matrix of the subsystem, after the embedding, is presented by the
off-diagonal block of the expanded state, and such embedding does not resolve negativity of the reduced
state. The renormalization of the Hamiltonian is mentioned, but not written down explicitly.
The range of applicability of the Redfield equation is given by the condition of the validity of the Born-
Markov approximation. The failure of the Born-Markov approximation can be identified by comparing the
second and fourth order terms in the Dyson expansion of the system state, [59, 60] which leads to the condition
τ
r
τ
c
. The Born-Markov approximation is justified if τ
r
τ
c
.
For any given initial state of the system, the error bound of the Redfield equation can be expressed in
terms of the properties of the heat bath only, [47]
1/2||ρ
RED
(t) ρ
exact
(t)||
1
O
τ
b
τ
sb
e
12t
τ
sb
ln
τ
sb
τ
b
. (19)
Here ρ
exact
(t) is the unapproximated quantum state, τ
b
and τ
sb
are the bath correlation time and the smallest
system relaxation time, respectively, and ||...||
1
is the trace-distance. τ
sb
is roughly τ
b
/g. Ignoring the
logarithmic correction, the error bound is linear in system-bath coupling constant g. Recent simulations on
a two qubit-system find linear scaling of the actual error (as opposed to the error bound) with g. [22]
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In the numerical simulations in this paper, the state relaxation time will be significantly longer than
τ
sb
, as it is in many cases in quantum optics and atomic physics. [61, 62] In that case the error bound 19 is
exponentially large at the system relaxation time. Only if the system relaxation time is comparable to τ
sb
the
bound will be tight enough to present a good error estimate. So here we adopt a system-dependent criterion
to determine the range of applicability of the Redfield master equation. If, for example, we find that the
Redfield states become too negative, we consider it as a sign that we are outside the range of applicability. A
similar point of view has been expressed in Ref. [22]. In addition, if the Redfield equation has an instability,
which it usually does at large system-bath coupling, then it will clearly not be applicable.
3.1 Coarse-grained Redfield Equation
Here we coarse-grain Eq. 14, as follows. First, we transform to the interaction picture [%
nm
= exp(
nm
t)ρ
nm
],
where the Redfield equation reads
d%
nm
dt
= i[e
iH
0
t
H
L
e
iH
0
t
, %]
nm
+
X
ij
A
?
in
%
ij
A
jm
e
i(ω
jm
ω
in
)t
G(ω
in
, ω
jm
)
1
2
X
i,j
h
%
ni
A
ij
A
?
mj
e
im
t
G(ω
mj
, ω
ij
) + A
ni
A
?
ji
%
jm
e
nj
t
G(ω
ji
, ω
ni
)
i
. (20)
Next, we change the time symbol from t to τ, and time average on a rectangular window. That is, we apply
the integral operator to the equation,
ˆ
G =
1
T
0
Z
t+T
0
/2
tT
0
/2
(21)
while keeping % at time t. We will refer to T
0
as the coarse-graining time. This leads to
d%
nm
dt
+ i[
˜
H
L
(t), %]
nm
=
X
ij
A
?
in
%
ij
A
jm
e
i(ω
jm
ω
in
)t
˜
G(ω
in
, ω
jm
)
1
2
X
i,j
h
%
ni
A
ij
A
?
mj
e
im
t
˜
G(ω
mj
, ω
ij
) + A
ni
A
?
ji
%
jm
e
nj
t
˜
G(ω
ji
, ω
ni
)
i
, (22)
where
˜
G(ω, ω
0
) =
1
2
γ(ω) + γ(ω
0
)
+ i
S(ω) S(ω
0
)
sinc
(ω ω
0
)T
0
2
. (23)
Here sinc(x) = [sin(x)]/x, and
˜
H
L
(t) =
ˆ
G e
iH
0
τ
H
L
e
iH
0
τ
. (24)
In the third step, we transform out of the interaction picture, [ρ
nm
= exp(
nm
t)%
nm
], and obtain the
coarse-grained Redfield equation,
nm
dt
+ i[
˜
H, ρ]
nm
=
X
ij
A
?
in
ρ
ij
A
jm
˜
G(ω
in
, ω
jm
)
1
2
X
i,j
h
ρ
ni
A
ij
A
?
mj
˜
G(ω
mj
, ω
ij
) + A
ni
A
?
ji
ρ
jm
˜
G(ω
ji
, ω
ni
)
i
. (25)
Similarly, the matrix elements of the coarse-grained renormalized Hamiltonian are
˜
H
nm
E
n
δ
nm
+
˜
H
L,nm
= E
n
δ
nm
+
X
i
˜
H(ω
ni
, ω
mi
)A
ni
A
im
, (26)
where
˜
H(ω, ω
0
) = H(ω, ω
0
)sinc
(ω ω
0
)T
0
2
. (27)
Coarse-graining increases the error of the approximate state. Error bound exists for the state error added
by coarse-graining, and increases linearly with the coarsegraining time. [47]
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3.2 Rotating wave and CGSE approximations
The RWA follows from Eqs. 23, 25 and 26 in the limit T
0
1/|ω
nm
|, ω
nm
6= 0. In that limit the sinc-function
becomes the Kronecker-delta thereby truncating the coarse-grained Redfield equation. The truncation causes
cancellations of the principle densities S on the RHS of Eq. 10, leading to the historically important Davies-
Lindblad master equation:
nm
dt
+ i[
˜
˜
H, ρ]
nm
=
X
i,j
δ
ω
in
jm
γ
in
A
?
in
ρ
ij
A
jm
1
2
X
ij
γ
ij
(ρ
ni
A
ij
A
jm
δ
E
i
,E
m
+ A
nj
A
ji
δ
E
n
,E
i
ρ
im
) (28)
where
˜
˜
H is the system Hamiltonian renormalized due to the environmental coupling,
˜
˜
H
nm
= E
n
δ
n,m
+
X
i
˜
˜
H(ω
ni
, ω
mi
)A
ni
A
im
, (29)
which is also known as the Lamb-shift. The unitary kernel in the RWA is
˜
˜
H(ω, ω
0
) =
δ
ω
0
2
S(ω) + S(ω
0
)
. (30)
In the interaction picture, the Davies-Lindblad master equation has time independent coefficients. The
equation and the Lamb-shift simplify further, respectively as
nm
dt
= i
nm
ρ
nm
+
X
i,j
δ
ω
in
jm
γ
in
A
?
in
ρ
ij
A
jm
1
2
ρ
nm
X
i
|A
ni
|
2
γ
ni
+ |A
mi
|
2
γ
mi
, (31)
and
n
= E
n
+
X
i
|A
n,i
|
2
S
n,i
, (32)
if the energy levels of the system are non-degenerate.
RWA is valid when the coarse-graining time is longer than the Heisenberg time τ
h
= 1E, where δE is
the smallest level spacing in the system. In that respect, the system relaxation time also needs to be of that
order or longer, so that the coarse-grained equation can be of value. Therefore, the main disadvantage to
the RWA is in its highly limited range of applicability. Note that at zero temperature, the dissipative part
of the system dynamics under the RWA only involves positive frequencies that correspond to the emission of
quanta from the system into the bath (since the SD at negative frequency is zero). [61]
More recently, a coarse-grained stochastic equation was derived [45] with better error and range of appli-
cability than that of Davies’,
1/2||ρ
cgse
(t) ρ
exact
(t)||
1
O
r
τ
c
τ
sb
, (33)
but not as low as the Redfield error bound given by Eq. 19.
Accepted in Quantum 2020-08-20, click title to verify. Published under CC-BY 4.0. 7
4 Geometric Arithmetic Master Equation (GAME)
Since coarse-graining increases the approximate state error, as discussed at the end of Sec. 3.1, here we will
attempt to restore positivity by picking and choosing which part of the Redfield equation to coarse-grain and
by how much. First, we rewrite the Redfield equation (Eq. 10) in terms of the geometric mean SD, instead
of the arithmetic one. For example, we write (γ
in
+ γ
jm
)/2 =
γ
in
γ
jm
+ (
γ
in
γ
jn
)
2
/2. The Redfield
equation 10 becomes
nm
dt
= i[H, ρ]
nm
+
X
i,j
ρ
ij
A
?
in
A
jm
γ
in
γ
jm
1
2
ρ
ni
A
ij
A
?
mj
γ
ij
γ
mj
1
2
A
?
ji
A
ni
ρ
jm
γ
ji
γ
ni
+
X
ij
ρ
ij
A
?
in
A
jm
f (ω
in
, ω
jm
)
1
2
ρ
ni
A
ij
A
?
mj
f(ω
ij
, ω
mj
)
1
2
A
?
ji
A
ni
ρ
jm
f(ω
ji
, ω
ni
)
, (34)
where we introduce the "detuning" function,
f(ω, ω
0
) =
1
2
q
γ(ω)
q
γ(ω
0
)
2
+ i
S(ω) S(ω
0
)
. (35)
Note that f(ω, ω) = 0, which is why we call it the detuning function. We can write ω = ω
a
+ (ω ω
0
)/2 and
ω
0
= ω
a
(ω ω
0
)/2, where ω
a
= (ω + ω
0
)/2 is the center frequency and ω ω
0
is the detuning. For small
detuning, defined as |ω ω
0
| < |ω
a
|, we apply the Tyler expansion and find
f(ω, ω
0
) = i(ω ω
0
)
S(ω
a
)
ω
a
+ O
h
(ω ω
0
)
2
i
. (36)
For Ohmic bath at zero temperature and exponential cutoff, the images in Fig. 1(a) and (d) display
the geometric mean g(ω, ω
0
) =
p
γ(ω)γ(ω
0
) and the detuning function magnitude versus system frequencies,
respectively. The entities f and g are independent of the system operator A. Without any coarse-graining,
f and g are overall comparable in magnitude, which is unfortunate, because, if the detuning function were
negligibly small compared to the geometric mean, we could neglect the second line in Eq. 34, and end up
with a GKSL master equation.
In spite of the significant value of the detuning function, we can make a compromise following the steps
discussed next. We recast the coarse-grained Eq. 25 as
nm
dt
= i[
˜
H, ρ]
nm
+
X
i,j
ρ
ij
A
?
in
A
jm
˜g(ω
in
, ω
jm
)
1
2
ρ
ni
A
ij
A
?
mj
˜g(ω
ij
, ω
mj
)
1
2
A
?
ji
A
ni
ρ
jm
˜g(ω
ji
, ω
ni
)
+
X
ij
ρ
ij
A
?
in
A
jm
˜
f (ω
in
, ω
jm
)
1
2
ρ
ni
A
ij
A
?
mj
˜
f(ω
ij
, ω
mj
)
1
2
A
?
ji
A
ni
ρ
jm
˜
f(ω
ji
, ω
ni
)
, (37)
where ˜g and
˜
f are the coarse-grained geometric mean and detuning function,
˜g(ω, ω
0
) =
q
γ(ω)γ(ω
0
) sinc
(ω ω
0
)T
0
2
, (38)
˜
f(ω, ω
0
) =
n
1
2
q
γ(ω)
q
γ(ω
0
)
2
+ i
S(ω) S(ω
0
)
o
sinc
(ω ω
0
)T
0
2
. (39)
The characteristic detuning frequency of these functions is governed by the sinc function. Irrespective of
the center frequency, if |ω ω
0
| 1/T
0
, then the sinc function will suppress
˜
f in Eq. 39 by prefactor of 1/T
0
.
Lets us next consider the other frequency range, |ω ω
0
| < 1/T
0
. If in this range |ω
a
| 1/T
0
, then the
condition |ω ω
0
| |ω
a
| will hold. Eq. 36 is then applicable and leads to
˜
f(ω, ω
0
) i(ω ω
0
)
×
S(ω
a
)
ω
a
sinc
(ωω
0
)T
0
2
2i
T
0
S(ω
a
)
ω
a
sinc
(ωω
0
)T
0
2
.
(40)
So the corresponding terms in Eq. 37 are also suppressed by the inverse coarse-grain time. Vice-versa,
if |ω
a
| < 1/T
0
, then Eq. 36 may not hold, and the corresponding terms in Eq. 37 are not suppressed.
Accepted in Quantum 2020-08-20, click title to verify. Published under CC-BY 4.0. 8