Quantum Thermal Transport Beyond Second Order with the Reaction Coordinate Mapping
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Quantum Thermal Transport Beyond Second Order with the Reaction
Coordinate Mapping
Nicholas Anto-Sztrikacs,1 Felix Ivander,2 and Dvira Segal2, 1
1) Department of Physics, 60 Saint George St., University of Toronto, Toronto, Ontario,
Canada M5S 1A7
2) Chemical Physics Theory Group, Department of Chemistry and Centre for Quantum Information and Quantum Control,
University of Toronto, 80 Saint George St., Toronto, Ontario, M5S 3H6, Canada
(*Electronic mail: dvira.segal@utoronto.ca)
(Dated: March 17, 2022)
Standard quantum master equation techniques such as the Redfield or Lindblad equations are perturbative to second
order in the microscopic system-reservoir coupling parameter λ . As a result, characteristics of dissipative systems,
which are beyond second order in λ , are not captured by such tools. Moreover, if the leading order in the studied
arXiv:2203.06165v2 [quant-ph] 16 Mar 2022
effect is higher-than-quadratic in λ , a second-order description fundamentally fails even at weak coupling. Here,
using the reaction coordinate (RC) quantum master equation framework, we are able to investigate and classify higher-
than-second order transport mechanisms. This technique, which relies on the redefinition of the system-environment
boundary, allows for the effects of system-bath coupling to be included to high orders. We study steady-state heat
current beyond second-order in two models: The generalized spin-boson model with non-commuting system-bath
operators and a three-level ladder system. In the latter model heat enters in one transition and it is extracted from a
different one. Crucially, we identify two transport pathways: (i) System’s current, where heat conduction is mediated
by transitions in the system, with the heat current scaling as jq ∝ λ 2 to lowest order in λ . (ii) Inter-bath current,
with the thermal baths directly exchanging energy between them, facilitated by the bridging quantum system. To the
lowest order in λ , this current scales as jq ∝ λ 4 . These mechanisms are uncovered and examined using numerical and
analytical tools. We contend that the RC mapping brings already at the level of the mapped Hamiltonian much insights
on transport characteristics.
I. INTRODUCTION the system-reservoir coupling beyond second order are re-
quired. We stress that this aspect is distinct from the com-
Dissipative quantum impurity systems1 constitute the core monly addressed issue of inaccurate predictions due to strong
of quantum thermal junctions and machines. Moreover, dissi- system-bath couplings. Instead, here we are interested in sys-
pative systems are central to problems of charge transport at tems in the asymptotically weak system-bath coupling regime,
the nanoscale, chemical reactions in the condensed phases2 , yet where standard QMEs completely fail since the lowest
and light-matter interaction effects in quantum optics3 . More order term in the behavior of the heat current goes beyond
recently, dissipative systems have received substantial atten- second-order in λ .
tion in the field of quantum thermodynamics where one is The reaction coordinate mapping method is well-suited
interested in understanding—and drawing on—quantum me- to describe quantum dissipative systems beyond second
chanical effects for the design of thermal machines4–6 . order10–15 . This method redefines the system-environment
Perturbative quantum master equation (QME) techniques boundary with the identification and extraction of a central
are commonly used for studying quantum dissipative systems (collective) degree of freedom of the environment, termed the
due to their simplicity and potential for analytical insights3,7 . reaction coordinate (RC). The quantum system is then ex-
In particular, the popular Redfield approach relies on the panded to become an “extended system”, which comprises the
Born approximation (as well as other assumptions), where the original, pre-mapped quantum system, the RC, and the inter-
system-environment coupling strength λ is assumed weak. In action of the RC with the original system. With the inclusion
such methods, the dynamics and steady state behavior of the of the RC within the system itself, the microscopic coupling
dissipative system is obtained to second order in the micro- parameter (λ ) of the original system to the surroundings is
scopic system-environment coupling parameter, with the heat now included to high orders within the reaction coordinate
current scaling as jq ∝ λ 2 , see8,9 . framework. As such, the RC mapping should be a viable tool
This outlined perturbative approach is valid and accurate in to explore high-order transport phenomena.
the weak coupling regime—as long as the second order ex- The advantage of the exact RC mapping becomes clear
pansion describes the leading-order physics. However, there once adopting a second-order QME technique, such as the
exist systems for which the first non-trivial term in the ex- Born-Markov Redfield (BMR) equation, to simulate the dis-
pansion (say for the heat current) goes beyond second order sipative behavior of the extended system. The resulting
in the system-reservoir coupling parameter. In such cases, a technique, termed the RC-QME method, is nonperturbative
second order QME approach completely fails, and it offers in- in the original coupling parameter λ . Since the method
correct predictions even in the asymptotically weak-coupling enables cheap computations, it has been used in studies
regime where QMEs are typically expected to thrive. To han- involving strongly-coupled system-reservoirs. For exam-
dle such situations, more accurate techniques, which capture ple, it was utilized for studying quantum dynamics and2
steady state behavior of impurity models16–18 , thermal trans- II. CLASSIFICATION OF HEAT CURRENTS
port in nanojunctions19,20 , the operation of quantum thermal
machines21–24 , transport in electronic systems25–27 . Besides Model Hamiltonians for quantum dissipative systems are
capturing system-bath coupling to high orders, the RC-QME typically partitioned into contributions from the system, its
method reports on non-Markovian effects28 . This is because surroundings, and the interaction between them. For a system
correlations building up between the system and reservoirs are coupled to two independent reservoirs, the total Hamiltonian
maintained throughout the evolution of the extended system. may be represented as follows,
The RC method was recently implemented for study-
ing the nonequilibrium spin-boson model (NESB) at strong Ĥ = ĤS + κR ĤI,R + κL ĤI,L + ĤB,R + ĤB,L , (1)
coupling20 , a quintessential model in the quantum thermody- where κR,L are dimensionless parameters used for bookkeep-
namics field. The model includes an impurity two-level sys- ing the coupling between the system and the environments.
tem (spin) interacting with two thermal harmonic-oscillator Without invoking assumptions on the microscopic form of the
environments held at different temperatures, allowing for a model, a formal expression for the average energy (heat, in the
steady state heat current to flow from one reservoir to the present case) current, flowing out of one of the reservoirs may
other. The model has been recently realized experimentally be obtained from Heisenberg’s equation of motion. For exam-
using superconducting quantum circuits demonstrating a ther- ple, the average heat current flowing out of the left reservoir
mal diode effect29 . Furthermore, the NESB model has been is simply (we work in units of h̄ ≡ 1 and kB ≡ 1)
studied extensively using perturbative and numerically-exact
techniques; a partial list includes Refs.20,30–48 . jq ≡ −hĤ˙ B,L i = −ih[Ĥ, ĤB,L ]i, (2)
The spin-boson model has been recently extended to
explore dynamics and transport effects using a general where the dot represents a time derivative. Utilizing the
coupling operator with both diagonal and off-diagonal equation of motion for ĤI,L , and the fact that in steady-state
couplings18,49–59 . For heat transport, the coupling operators to hĤ˙ I,L i = ih[Ĥ, ĤI,L ]i = 0, one obtains a formal expression for
the different reservoirs do not commute. As a result, the model the heat current as62
manifests the emergence of two distinct transport regimes56 :
(i) The current flows sequentially from the hot to the cold jq = −iκL h[ĤS , ĤI,L ]i − iκL κR h[ĤI,R , ĤI,L ]i. (3)
reservoir mediated by the excitation of the central spin. (ii) The above expression contains two contributions to the steady
Heat flows directly from the hot to the cold reservoir, ap- state heat current, both stemming from the non-commutativity
parently without exciting the central spin, in a manner rem- of two components of the total Hamiltonian. The first con-
iniscent of the tunneling-superexchange behavior60,61 . This tribution arises due to the non-commutativity of the system
direct bath-to-bath current between reservoirs in the general- Hamiltonian with the interaction Hamiltonian of the left reser-
ized, non-commutative spin-boson model is an example of a voir, and is proportional only to the coupling parameter to the
transport phenomenon that emerges beyond second order in left reservoir, κL . Motivated by this, one defines this contri-
perturbation theory in the system-bath coupling energy. bution as the system current. The second term originates from
In this paper, we use the RC-QME method to probe beyond- the non-commutativity of both interaction Hamiltonians, and
second-order effects in nonequilibrium steady-state. We ana- it depends on both coupling parameters, κL and κR . This term
lyze two impurity models in which the heat current scales as constitutes a form of higher order transport, which is defined
jq ∝ λ 4 even in the asymptotic weak coupling limit, λ → 0, as the inter-bath current38 .
and provide analytical intuition into the unusual transport phe- Using the different dependencies on the coupling parame-
nomena observed. ters in Eq. (3) as a guide, it was pointed out in Ref.62 that
This work contributes the following: (i) Applying the one can distinguish between two transport mechanisms in nu-
RC mapping we immediately bring—at the level of mapped merical simulations of the heat current, by observing how
Hamiltonian and without performing transport calculations— they scale with the coupling parameters. It is worthy of note,
fundamental understanding of different thermal transport however, that standard QME techniques capture the effects of
mechanisms. (ii) We establish the RC-QME method for system-reservoir interactions only to the lowest order in κL,R .
involved nonequilibrium quantum transport applications by As a result, the inter-bath current, which is higher order in
showing that it realizes both system’s and direct-bath currents. the coupling parameters, cannot be captured by second-order
(iii) We bring forward a new, ladder model for heat transport, QME approaches.
with heat inputted from one step (transition) and outputted With this in mind, in order to probe both transport mecha-
through a different transition. nisms, in this work we focus on models where one term in Eq.
The paper is organized as follows. In Section II, we for- (3) dominates over the other. In what follows, we study two
mally discuss two contributions to the heat current, the sys- models of quantum heat transport: (i) the generalized spin-
tem’s current and the inter-bath current. We analyze the gen- boson model and (ii) a three-level ladder model. In both cases,
eralized spin-boson model in Sec. III using the RC mapping we focus on the regime of weak system-bath couplings, when
based on analytical considerations and numerical simulations. only lowest order terms in the expressions Eq. (3) contribute.
In Sec. IV, we study a three-level ladder model for heat trans- The first model, the generalized spin-boson model, shows rich
port, which relies on an inter-bath transport pathway at weak behavior: by tuning the coupling operators, the relative domi-
coupling. We conclude in Section V. nance of the different contributions in Eq. (3) is controlled. In3
the environments to be a structured, Brownian function
4γα ωΩ2α λα2
JSB,α (ω) = . (5)
(ω 2 − Ω2α )2 + (2πγα Ωα ω)2
Here, λα denotes the system-environment coupling strength,
γα is a measure for the width of the spectral density func-
tion, and Ωα is the frequency about which the spectral density
function is peaked.
The microscopic system-bath coupling parameters are fk,α ,
which are the interaction energies between the system and the
kth mode of the α bath. The spectral density function weights
these coupling strengths squared with the density of states. In
the Brownian model Eq. (5), JSB,α (ω) ∝ λα2 , with λα taking
the role of system-bath coupling parameter now that the bath
is assumed continuous and of a certain density of states. In
what follows, we use λα as a measure for the coupling energy.
Figure 1. Schemes of heat transport models investigated in this work: To go beyond a standard second-order QME treatment—
(a) The generalized NESB model, and the (b) ladder model. We mark and uncover higher order transport phenomena—we imple-
with straight arrows heat exchange processes that involve the system, ment the reaction coordinate QME technique. This method
contributing to the system’s current. The dashed-straight arrows il- builds on an exact Hamiltonian mapping where a collec-
lustrate the inter-bath heat current, of direct bath-to-bath energy ex- tive coordinate (reaction coordinate) is extracted from each
change using the system as a bridge. reservoir and included as part of the system’s Hamiltonian.
This mapping thus redefines the system-environment bound-
ary. Performing a reaction coordinate mapping on both baths
the ladder model, which we construct here, the system’s cur- leads to the reaction coordinate generalized spin-boson (SB-
rent does not contribute to heat flow at weak couplings. Both RC) Hamiltonian,
models and the different currents are illustrated in Figure 1.
∆
ĤSB−RC = σ̂z + ΩL â†L âL + ΩR â†R âR + λL σ̂x (â†L + âL )
2
III. THE GENERALIZED SPIN-BOSON MODEL g2k,α †
+ λR σ̂θ (â†R + âR ) + ∑ (âα + âα )2
ω
k,α k,α
A. Hamiltonian and the reaction coordinate mapping
+ ∑(â†α + âα ) ∑ gk,α (b̂†k,α + b̂k,α ) + ∑ ωk,α b̂†k,α b̂k,α .
α k k,α
Our first model is the generalized spin-boson model as in-
troduced in Ref.56 . The model consists of a single spin inter- (6)
acting with two bosonic environments maintained at different In this expression, â†α
(âα ) and b̂†k,α
(b̂k,α ) represent the cre-
temperatures. The nontrivial aspect of the generalized model, ation (annihilation) operators of the reaction coordinates and
compared to the “standard" one, is that system’s operators that the residual baths, with frequencies Ωα and ωk,α , respectively.
are coupled to the baths do not commute with each other; the The reaction coordinates couple to the system via coupling
left (hot) bath is coupled to the system via σ̂x while the right parameter λα . In turn, the residual baths couple to the reac-
(cold) reservoir is coupled with σ̂θ = σ̂z cos(θ ) + σ̂x sin(θ ) tion coordinates via a coupling gk,α , which is captured via a
with 0 ≤ θ ≤ π/2 denoting the non-commutativity parameter. spectral density function JSB−RC,α (ω) = ∑k g2k,α δ (ω − ωk,α ).
This model is expressed as It can be shown that if the spectral density function of the
∆ original, pre-mapped reservoir is of Brownian form, Eq. (5),
ĤSB = σ̂z + σ̂x ∑ fk,L (ĉ†k,L + ĉk,L ) then the spectral density function of the residual bath becomes
2 k Ohmic with an infinite high frequency cutoff after the reaction
+ σ̂θ ∑ fk,R (ĉ†k,R + ĉk,R ) + ∑ νk,α ĉ†k,α ĉk,α . (4) coordinate mapping16,20 ,
k k,α∈{R,L}
JSB−RC,α (ω) = γα ωe−|ω|/Λα . (7)
The system includes a spin with energy splitting ∆; σ̂x,y,z In this representation, γα is a dimensionless coupling constant
are the Pauli matrices. ĉ†k,α (ĉk,α ) are the creation (annihi- between the αth reaction coordinate and its residual bath, and
lation) operators of the two bosonic environments enumerated Λα is the cutoff frequency of the αth bath. The first two lines
by α = L, R. fk,α denotes the coupling energies between the of the SB-RC Hamiltonian, Eq. (6), represent the extended
system and the kth mode of the α environment of frequency system, comprising of the spin, the two RC harmonic oscilla-
νk,α . The coupling of the spin to the baths is captured by the tors, and their interactions. Note the quadratic term in the RC,
spectral density function JSB,α (ω) = ∑k fk,α2 δ (ω − ν
k,α ). The which emerges after the mapping. The last line contains the
setup is depicted in Figure 1 (a). For reasons that will soon residual baths and the interaction between the reaction coor-
become evident, we choose the spectral density function of dinates and the residual baths.4
B. Transport mechanisms based on the RC mapping tween the system’s and the inter-bath currents, see Eq. (3), as
each term distinctly control the heat current.
Equation (3) suggests two distinct contributions to the heat For simplicity, the energy spectrum displayed in Figure 2
transfer, which may be formally understood as arising from was calculated after truncating the harmonic manifold of the
the non-commutativity of components of the total Hamilto- reaction coordinates, including only M = 2 levels. However,
nian. However, this formally-exact expression does not pro- the discussion holds analogously for larger M values. The
vide fundamental understanding of the mechanisms behind symmetry presented in the spectrum arises due to the values
different transport pathways. The standard way to approach of the RC frequency Ωα employed, being equal for both reser-
this problem and gain deeper understanding is to preform nu- voirs. Again, however, this is not an essential requirement,
merical simulations of the heat current. This task is typically and asymmetric choices could be adopted as well.
nontrivial: It requires solving quantum equations of motion The key observation from Fig. 2 is that there are three clus-
in steady state, and calculating expectation values of different ters of states, separated by energies of order Ωα (which is the
observables. Nonetheless, the advantage of the RC mapping is frequency of the RCs). The lowest two levels correspond ap-
that already at the level of the mapped Hamiltonian, Eq. (6), proximately to RCs occupying their ground states while the
and without performing transport calculations, we can gain two level system (spin) is either in its up or down state, | ↓ 00i
substantial insights onto transport behavior in the model. and | ↑ 00i. From the other end, the highest two levels roughly
In what follows, we examine contributions to the heat cur- correspond to the spin occupying its up or down states while
rent, both the system’s current and the inter-bath currents, by the RCs now both occupy their first excited states, | ↓ 11i
studying the SB-RC Hamiltonian, Eq. (6). First, we use nu- and | ↑ 11i. The intermediate four levels approximately cor-
merical tools and study the eigenspectrum and the coupling respond to having one RC in its first excited state, and one
pattern of the SB-RC Hamiltonian. We complement this nu- remaining in its ground state, while again the spin can be up
merical investigation with an analytical analysis: We apply or down for both situations, e.g., a state of the form | ↓ 10i.
the small-polaron transformation onto the SB-RC Hamilto-
nian, and interpret the contribution of different terms to the 35
heat current.
| 11
30 | 11
1. Numerical diagonalization 25
| 01
| 01
We diagonalize the extended system [first five terms in Eq. 20 | 10
| 10
(6)] and denote the eigenenergies by En , ĤSD = ∑n En |nihn|
with n the eigenvalue index. We further write down the 15
system-bath coupling (the first term in the last line of Eq. (6))
| 00
in this basis as 10
" #" # | 00
D D † 5
ĤI,α = ∑ (Ŝα )m,n |mihn| ∑ gk,α b̂k,α + b̂k,α . (8)
m,n k
0 2 4 6 8
The coefficients (ŜαD )m,n are dictated by the original form of
Figure 2. Eigenspectrum of the extended system Hamiltonian Eq. (6)
couplings and the diagonalization of (âα + â†α ).
as a function of eigenstate index n for two non-commutativity angles:
Figure 2 displays the eigenspectrum En as a function of the (i) The σx −σx model with θ = π/2. (ii) The σx −σz model with θ =
index n for two angles: (i) The choice θ = π/2 corresponds to 0. Parameters are set to ∆ = 0.1, λL = λR = λ = 0.01, ΩL = ΩR = 10.
the standard spin-boson model, with both baths coupled to the Since λ /∆ 1, we can interpret the eigenstates using the local-site
system via σx . This is the σx − σx model. (ii) With the angle basis representation, where levels are labeled according to the state
θ = 0, the left bath is coupled via σx to the system, while the of the spin and the occupation number of the reaction coordinates,
right bath couples with a so-called diagonal form, to σz . This |s, nR , nL i.
case is labeled as the σx − σz model.
The coupling parameters to the reaction coordinates, λα , The energy spectrum displays almost identical characteris-
are kept small relative to the internal energy scale ∆ to main- tics for the σx − σx and the σx − σz model. Their crucial dis-
tain an approximate local-basis representation, where states tinction however shows up in the system-bath coupling pattern
are labeled according to their spin state, s, and the reaction at the two ends. Figure 3 depicts the matrix elements (ŜαD )m,n
coordinate excitation numbers, nR , and nL , combined into a of the system’s operators that are coupled to the baths, see
single ket vector as |s, nR , nL i. Working in a small λα regime Eq. (8). Here, the D superscript identifies that |ni and |mi are
explains the significant overlap of eigenvalues of the σx − σx eigenvectors of the extended system. We display the elements
and σx − σz models, as the interaction λα is too weak to dis- coupled to the left (hot) and right (cold) baths and consider the
cern between the two models. We emphasize that maintaining angles θ = π/2 (top) and θ = 0 (bottom). We represent the
the coupling parameters λα small allows us to distinguish be- matrix elements on a logarithmic scale, where larger values,5
1
* *
2
* *
3 -5 -5
4
5 -10 -10
6
7 -15 -15
8
1 x x
2 o o
3 x -5 -5
4 x
5 o -10
-10
6 o
7
-15
8 -15
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
Figure 3. Heat map displaying the logarithm of the absolute value of the extended system’s coupling operator log10 | hn| ŜL/R D |mi |, see Eq. (8),
with light color identifying large couplings. Red (blue) panels show the coupling elements to the hot (cold) bath. The eigenstates enumerated
by n = 1, 2, ...8, are approximately local (spin and RCs) since λ is small; the local basis closely coincides with the energy basis, as written
in Figure 2. We mark by symbols (*,x,o) dominant processes. (a1)-(a2) σx − σx model. Here, heat transfer involves the excitation and de
excitation of the system’s spin, with the dominant process (*) |↓, 0, 0i ↔ |↑, 0, 0i. (b1)-(b2) σx − σz model. Here, the RCs can exchange energy
using the spin as a bridge: The dominant transitions are |↓, 0, 0i ↔ |↓, 0, 1i and |↓, 0, 0i ↔ |↓, 1, 0i (marked by x in (b1) and (b2), respectively)
and similar transitions involving the spin up state |↑, 0, 0i ↔ |↑, 0, 1i and |↑, 0, 0i ↔ |↑, 1, 0i (marked by o in (b1) and (b2), respectively).
Parameters are identical to those used in Figure 2.
and hence, more likely transitions are given by a lighter color. first term in Eq. (3), which is the system’s current, vanishes,
The symbols shown on the grid elements (*,x,o) mark transi- and only inter-bath current terms contribute towards the heat
tions that contribute most significantly to the heat current. current. This is demonstrated in Figure 3 (b1). For example
Figure 3 (a1) and (a2) correspond to the σx − σx model the (x) pathway corresponds to the left (hot) bath allowing
(θ = π/2). These two panels show identical results given the transitions between the approximate states |↓ 00i ↔ |↓ 01i,
symmetry in the coupling scheme. Since the coupling opera- while the right bath drives the transition |↓ 00i ↔ |↓ 10i. As a
tors (σx ) commute, the second term in Eq. (3) is absent. As- whole, the two baths exchange energy through the RCs, rather
suming that temperatures are lower than the baths character- than the system. Similarly, (o) signifies transitions between
istic frequencies, Tα < Ωα (though the temperature could be |↑ 00i ↔ |↑ 01i and |↑ 00i ↔ |↑ 10i, again an inter-bath cur-
comparable to ∆), the dominant transitions are those with the rent.
RCs occupying their ground states and the baths creating exci- Concerning the σx − σz model, our second observation is
tations between the up and down spin states, |↓ 00i ↔ |↑ 00i. that the second term in Eq. (3), the inter-bath current, de-
In Figure 3, we identify these transitions with asterisks (*). scribes coupled transitions of the RCs. It is therefore a higher
We now make our first observation: To low order in λ , the order transport pathway corresponding to transitions in the
system’s heat current, the first term in Eq. (3), involves the baths; after all, the reaction coordinates are reservoirs’ degrees
baths exchanging energy with the spin, or more generally, the of freedom.
central system. As a result, in heat current simulations at weak To summarize our analysis thus far, we diagonalized nu-
coupling we expect the system’s current to be accurately pre- merically the extended Hamiltonian after the RC mapping.
dicted by low-order QME techniques. Given the connectivity nature of the model, even without per-
Figure 3 (b1) and (b2) picture the coupling pattern for the forming transport calculations we are able to identify transport
σx − σz model (θ = 0) with σ̂z coupled to the right bath. The mechanisms and describe their characteristics: The system’s
spatial symmetry in the spin-boson model is now broken. The current involves spin transitions, and it will thus strongly de-6
pend on the spin splitting, ∆. In contrast, in the σx − σz model an electron-phonon cloud (or a dressed electron, a quasipar-
the spin plays an intermediating role, thus (i) we expect the ticle). In the new basis, perturbative master equations in the
current to be largely independent of ∆, but (ii) strongly depen- tunneling splitting can be employed while still preserving as-
dent on the RC frequency Ω, or in other words, the spectral pects of strong system-reservoir couplings. Here, however, we
properties of the bath. employ the polaron transformation in a different context: It is
used to recast interactions in the extended system, between
the spin and the reaction coordinates. We emphasize that we
2. Analysis in the polaron frame are not computing here the heat current in the polaron frame.
Notably, only by inspecting the polaron-transformed Hamil-
Complementing the numerical study, we include here a tonian we gain insight into the two transport contributions.
short analysis that exposes the different transport contribu- We apply a small-polaron transformation to the left reser-
tions (system’s current and inter-bath currents). This is voir in Eq. (6) H̃ˆ SB−RC = ÛP ĤSB−RCÛP† with the unitary
achieved by performing an additional transformation on the transformation63
SB-RC Hamiltonian Eq. (6).
Polaron theory is used to describe electron-phonon cou- λL
σ̂x (â†L −âL )
pling effects by applying a unitary transformation onto the ÛPL = e ΩL . (9)
Hamiltonian. This results in the dressing of electronic de-
grees of freedom by the phonon degrees of freedom, forming This results in
λL †
− ΩL (â†L −âL )
∆ λ
ˆ (âL −âL )
H̃SB−RC = (σ̂z − iσ̂y ) e ΩL + (σ̂ z + iσ̂y ) e L + ΩR â†R âR + ΩL â†L âL
4
† † 1 λL †
(âL −âL ) − ΩL (â†L −âL )
λ
+ λR sin θ (âR + âR )σ̂x + λR cos θ (âR + âR ) (σ̂z − iσ̂y ) e Ω L + (σ̂ z + iσ̂y ) e L
2
2λL 2λL g2k,L g2k,R
− σ̂x ∑ gk,L (b̂†k,L + b̂k,L ) + (â†L + âL − σ̂x )2 ∑ + (â†R + âR )2 ∑
ΩL k ΩL k ωk,L k ωk,R
+ ∑(â†α + âα ) ∑ gk,α (b̂†k,α + b̂k,α ) + ∑ ωk,α b̂†k,α b̂k,α . (10)
α k k,α
The terms in Eq. (10) are organized as follows: The first line end up with (α = L, R and ignoring constant terms),
includes the contribution from the spin and the two bare re-
action coordinates. At low temperature when the RCs are in H̃ˆ SB−RC
σx −σx
=
their ground state, the impact of the polaron dressing is to ∆ λα
(â†α −âα ) λα
(â†α −âα )
2 (σ̂z − iσ̂y ) e∑α Ωα + (σ̂z + iσ̂y ) e− ∑α Ωα
− λ2 20 . 4
renormalize the spin splitting ∆ → ∆e 2Ω
2λα g2k,α
+ΩR â†R âR + ΩL â†L âL + ∑(â†α + âα − σ̂x )2 ∑
α Ωα k ωk,α
The second line includes the interaction between the right 2λα
RC and the spin, part of it is unaltered from the mapping since −∑ σ̂x gk,α (b̂†k,α + b̂k,α )
Ωα ∑ k
[ÛP , σ̂x ] = 0, while the second part arises due to [ÛP , σ̂z ] 6= 0. α
These two terms are central to uncovering transport mecha- + ∑(â†α + âα ) ∑ gk,α (b̂†k,α + b̂k,α ) + ∑ ωk,α b̂†k,α b̂k,α (11)
nisms, and will be discussed below. The third line includes α k k,α
the coupling between the spin and the left residual reservoir, To the lowest order in λ , we get (ignoring constant shifts),
as well as the quadratic term, which represents the reorgani-
zation energy of the reservoirs. The last line describes the ∆ ∆ λα †
H̃ˆ SB−RC
σx −σx
= σ̂z − iσ̂y ∑ (âα − âα ) + ΩR â†R âR + ΩL â†L âL
coupling between the RCs and the residual baths, including 2 2 α Ωα
the bath Hamiltonians themselves. We now separately ana- "
g2k,α
#
† −2λα †
lyze two limiting models, θ = π/2 and θ = 0. + ∑(âα + âα ) σ̂x ∑ ωk,α + ∑ gk,α (b̂k,α + b̂k,α )
α Ωα k k
2λα
−∑ σ̂x gk,α (b̂†k,α + b̂k,α )
(i) σx − σx model. When θ = π/2, the model Eq. (4) re- α Ωα ∑ k
duces to the σx − σx model. In this symmetric case, it is ben-
g2k,α
eficial to perform an additional polaron transformation on the + ∑ ωk,α b̂†k,α b̂k,α + ∑(â†α + âα )2 ∑ . (12)
right side, applying ÛPR , in a complete analogy to Eq. (9). We k,α α k ωk,α7
Assuming for simplicity that temperature is not high enough the extended system and study effects beyond second order in
to excite the RCs, it is clear from the third line that heat trans- λ , yet in a computationally cheap manner.
port takes place through the system’s current: The left hot Employing the RC-QME method follows three steps: (i)
bath excites the the spin with the transition σ̂x , and this sys- We truncate the reaction coordinates and include M levels
tem’s operator is further coupled to the right residual bath. Eq. each. The value of M is chosen large enough to converge nu-
(12) in fact provides the foundation for the effective RC model merical simulations with respect to this parameter. After the
described in Ref.20 . At weak coupling, the σx − σx model thus truncation, the extended system is of an 2 × M 2 dimension.
describes sequential heat transport, from bath to spin, and spin (ii) The truncated SB-RC Hamiltonian is diagonalized numer-
to bath, and it only accounts for system’s current. ically. The operators of the system that are coupled to the
(ii) σx − σz model. We return to Eq. (10), but now use baths are transformed into the diagonal representation. (iii)
θ = 0. After expanding the exponents to lowest order in λ , The Redfield equation is solved in steady state in the energy
the Hamiltonian reduces to eigenbasis of the extended system of the SB-RC Hamiltonian.
The dynamics is assumed Markovian, and the initial state of
σx −σz ∆ iλL ∆ the open system is a product state of the extended system and
H̃ˆ SB−RC = σ̂z − σ̂y (â†L − âL ) + ∑ ωk,α b̂†k,α b̂k,α
2 2ΩL k,α baths, the latter prepared in canonical states at a given tem-
perature. We further ignore the principal values (imaginary
+ΩR â†R âR + ΩL â†L âL + λR σ̂z (â†R + âR ) terms) in the Fourier-transformed bath-bath correlation func-
λR λL † tions, see Appendix A.
−i (â + âR )σ̂y (â†L − âL )
ΩL R Formally, the Redfield equation is given by ρ̇ES (t) =
2λL 2λL g2k,L −i[ĤESD ,ρ ] +
ES ∑α Dα (ρES )2 . The dissipators to the residual
− σ̂x ∑ gk,L (b̂†k,L + b̂k,L ) + (â†L + âL − σ̂x )2 ∑ baths, Dα (ρES ) are additive, given the weak coupling approx-
ΩL k ΩL k ωk,L imation of the RCs to their residual baths. However, the dis-
g2k,R sipators depend in a nonadditive manner on the original cou-
+(â†R + âR )2 ∑ + ∑(â†α + âα ) ∑ gk,α (b̂†k,α + b̂k,α ).(13) pling parameters, λα . We solve the Redfield equation in the
k ωk,R α k energy basis under the constraint of Tr[ρES (t)] = 1 and ob-
tain the steady state density matrix of the extended system,
To the lowest order in the interaction energy, we identify the ss . The steady state heat current at the α contact is given
ρES
term responsible for heat transport as −iλΩRLλL (â†R + âR )σ̂y (â†L −
by jα = Tr Dα (ρES ss )Ĥ D ; the heat current is defined positive
ES
âL ). It captures the main physics: Heat is exchanged in the when flowing from the bath towards the system.
σx − σz model due to the two reaction coordinates becoming
effectively coupled through the spin, an effect reminiscent of
“superexchange" for charge current60,61 . Since the RCs are in D. Simulations
fact part of the heat baths, this term corresponds to the direct,
inter-bath current as described by Eq. (3).
To organize our numerical results and polaron analysis: We use the RC-QME method and simulate the steady state
There are two types of transport pathways. The system’s cur- heat current. Our objective is to exemplify and pinpoint the
rent arises due to internal transitions in the spin. At weak cou- differences between the system and inter-bath transport mech-
pling, this contribution can be captured by standard second- anisms. For simplicity, we consider symmetric RC parame-
order QME approaches. On the other hand, the inter-bath cur- ters: λ = λL = λR , Ω = ΩL = ΩR , and γ = γL = γR .
rent arises due to transitions in the RCs (bath), with inter-bath Figure 4 (a) manifests the scaling properties of the steady
transitions mediated by the spin. This form of current is im- state heat current with the coupling strength λ , in the weak
mediately of higher order in the system-bath coupling strength coupling regime. We use three different angles θ . The solid
as its prefactor is λL λR . To simulate it one needs to use meth- line (θ = π/2) corresponds to the case where the two system
ods capturing effects beyond second order transport character- operators commute (σx −σx model). According to Eq. (3) and
istics. In what follows, we present simulations of the steady the analysis of Sec. III B, in this situation at weak coupling
state heat current using the RC-QME method. we should observe the system’s current, which is mediated
by transitions in the spin, as the sole mechanism of transport.
This current scales as jq ∝ λ 2 .
On the contrary, the solid-dashed line (θ = 0) corresponds
C. RC-Quantum Master Equation Method to the other extreme, where the system Hamiltonian and the
right system operator commute (σx − σz model). As a re-
We outline the principles of the RC-QME method, which sult, the system’s current nullifies and heat transport must go
we employ next to study heat current in the RC-SB Hamilto- beyond second order in the system-bath coupling parameter.
nian, Eq. (6). For more details, we refer readers to Ref.20 . This is shown by the scaling relation jq ∝ λ 4 , which corre-
The key point is that after the reaction coordinate mapping is sponds to the inter-bath current as described in Sec. III B.
performed, one assumes that the coupling between the RCs The dashed line (θ = π/4) pictures an intermediate case,
and the residual baths is weak. This amounts to γα 1; the when both system’s and inter-bath currents contribute at weak
Brownian spectral function is assumed narrow. As a result, coupling. As a result, at the ultra-weak coupling the cur-
one can employ the standard Born-Markov QME method on rent scales as λ 2 (system’s current), but as the coupling is8
10-4
10-6
10-6
10-8 10-8
10-10
10-10
10-12
10-2 10-1 100 10-2 10-1 100
Figure 4. Steady state heat current as a function of (a) coupling strength λ and (b) spin splitting ∆. The currents are plotted for three different
angles, θ = π/2 (solid), θ = π/4 (dashed), and θ = 0 (dashed-dotted). The dark solid lines in panel (a) indicate the scaling behavior of the
system’s current (λ 2 ) and the inter-bath current (λ 4 ). Parameters are Ω = 10, γ = 0.0071/π, Th = 1, Tc = 0.5, Λ = 1000, M = 4 as well as
∆ = 0.1, (a) and λ = 0.1 (b).
10-3 10-3
10-5 10-5
10-7 10-7 10-3
10-9 10-9 -5
10
10-11 10-11 10-1 100
10-1 100 101 10-1 100 101
Figure 5. Steady state heat current as a function of the RC frequency, Ω, plotted with three orientation angles θ = π/2 (solid), θ = π/4
(dashed), and θ = 0 (dashed-dotted). (a) ∆ = 0.1 and λ = 0.1. (b) ∆ = 0 and λ = 0.1. Other parameters are the same as in Figure 4, except
M = 7. Panel (c) shows the convergence of the steady state current with M = 5 (circle), M = 6 (triangle) and M = 7 (square) for the case of
∆ = 0.
enhanced (yet it is still maintained small) there is a smooth ent technique, captures the correct scaling behavior and the
transition, and the inter-bath current takes over showing the continuous transition between them.
scaling jq ∝ λ 4 . These scaling relations, and the turnover be- To strengthen the argument that the system’s current arises
havior for the intermediate angle θ = π/4 were demonstrated from transitions caused in the spin, while the inter-bath cur-
in Ref.56 by using the extended hierarchical equations of mo- rent is a reservoir effect, we show in Figure 4 (b) the steady
tion (HEOM) technique64 (albeit using the super-Ohmic spec- state current as a function of the spin splitting ∆. We find that
tral function for the baths). It is significant to confirm here for θ = π/2 the current diminishes as the spin splitting goes
that the RC-QME method, a rather economical and transpar- to zero, confirming the critical role of transitions in the spin9
system (in sequential transport, which is the dominant sys- the thermal rectification effect, which we do not examine here.
tem’s current at weak coupling, heat is transported in quanta In our work here, in contrast, we focus on physical mecha-
of ∆). On the contrary, when θ = 0 we observe a saturation of nisms underling the two contributions to the current. We aim
the current as the splitting approaches zero. This supports the to establish the RC-QME method, and we show that the RC-
point that in this model heat current is caused by inter-bath ef- mapped Hamiltonian, in particular after the polaron transfor-
fects. Although, we note that the spin still plays a role in this mation over the RCs, offers a transparent starting point for
transport pathway as there is still non-trivial dependence on studying transport mechanisms.
the splitting as ∆ varies. Finally, similarly to Figure 4 (a), for Compared to the extended HEOM, which is numerically-
the intermediate model (θ = π/4) the current smoothly transi- exact, the RC-QME offers cheap computations and a deeper
tions between the two mechanisms, based on which pathway understanding. Compared to the NE-PTRE method, the RC-
provides a larger magnitude in heat current. The ∆ depen- QME offers more generality: It is cumbersome to perform the
dence of the energy current was studied in Ref.56 using the polaron transformation on multiple baths for non-commuting
extended HEOM method. It is substantial to note that the RC- operators, the starting point of the NE-PTRE method. Thus,
QME method can capture the correct behavior. unless the model is symmetric, the NE-PTRE can handle
Next, to further confirm the inter-bath nature of transport in strong coupling at one bath only. In contrast, extracting de-
the σx − σz model, we investigate the dependence of the heat grees of freedom from several thermal baths and adding them
current on ∆ and the RC frequency Ω. Figure 5 shows the to the system can be readily done for general coupling models
steady state heat current as a function Ω for the same three with different θ , and for multiple baths making the RC-QME
angles used in Figure 4. In Figure 5 (a) we study behavior method a useful tool in quantum thermodynamics20,21 .
for ∆ 6= 0, while in Figure 5 (b) we use ∆ = 0. Our main ob-
servations are: (i) The spin-boson model (θ = π/2) supports
current only when ∆ 6= 0, as is expected from system’s current IV. HEAT TRANSPORT IN THE LADDER MODEL
(this current is null in panel (b)). (ii) The current flowing in
the σx − σz model (θ = 0) barely changes between the two A. Model
panels. This is expected as heat flows in this model arises
from transitions of the RCs (reservoirs). When increasing Ω,
the inter-bath current grows, reaches a maximum, and decays Building on our knowledge of different transport mecha-
to zero for large Ω. This is because as Ω grows, the reaction nisms, we now propose a three-level, ladder quantum system.
coordinate levels become inaccessible to be thermally pop- Here, at weak coupling, heat current takes place solely due
ulated, resulting in a suppressed heat current. We therefore to inter-bath processes, rather than through the excitation of
find that the role of Ω in the inter-bath transport is analogous the system. The goal of this section is to bring forward this
to the role of ∆ for the system’s current. Namely, once Ω unique model. We show that the current scales as jq ∝ λ 4 at
becomes approximately resonant with the temperature of the weak system-bath coupling λ , and that it can be largely tuned
baths, inter-bath transport is maximized. When Ω is large and by modifying the spectral properties of the bath (frequency
excited states cannot be thermally excited, a rapid suppression Ω). The ladder model is given by the following Hamiltonian
of the current is observed. Figure 5 panel (c) exemplifies the (α = L, R),
convergence of the heat current with respect to the number 2 2
fk,α
of RC levels, M. We notice that once λ /Ω < 1, convergence ĤLD = ∑ εi |ii hi| + ∑ Ŝα2 + ∑ Ŝα fk,α (ĉ†k,α + ĉk,α )
is achieved. Surprisingly, however, even in situations where i=0 k,α νk,α k,α
λ /Ω ≈ 1, convergence with respect to M is still good: The
+∑ νk,α ĉ†k,α ĉk,α . (14)
overall trend of an increase in the magnitude of the current
k,α
with Ω is followed.
Before moving on to the three-level ladder model, we Here, εi is the energy of the ith level of the ladder. For sim-
discuss in more details the relation of our work to Ref.56 . plicity, we set the ground state at ε0 = 0, define ε1 = ∆ and
There, the heat current characteristics in the generalized fix ε2 = 1. fk,α denotes the coupling strength of an oscillator
spin-boson model were studied using the numerically ex- mode in the αth bath with frequency νk,α to the system. The
act extended HEOM technique. Using that method, differ- two baths couples to two distinct transitions via the system op-
ent scaling relations of the heat current were observed, as erators ŜH = |1i h2| + h.c. and ŜC = |0i h1| + h.c.. The setup is
we also show in Figure 4. Furthermore, the nonequilibrium depicted in Figure 1 (b). The system-bath interactions are cap-
polaron-transformed Redfield equation (NE-PTRE)37 was im- tured by a Brownian spectral density functions, as indicated in
plemented in Ref.56 to explain the behavior of the current in Eq. (5).
the σx − σz model at ∆ → 0. It should be stressed that the At weak coupling at the level of the second-order BMR
polaron transformation was performed there on the original, QME, the ladder model cannot conduct heat in steady state.
pre-RC mapped model encompassing all bath modes, Eq. (4). We derive this result in the Appendix. However, in this limit
In contrast, here we perform the polaron transformation af- the ladder model conducts through inter-bath transitions, and
ter the RC mapping, and only on the individual RCs. Ref.56 it therefore scales as jq ∝ λ 4 . To study this effect, We once
further focused on the enhancement that the non-commuting again must implement the RC transformation—in analogy to
system operators provide to the steady state heat flux and to what was done for the generalized spin-boson model. After10
extracting one oscillator from each reservoir we reach the fol-
lowing Hamiltonian, 10-6
2
λα2
ĤLD−RC = ∑ εi |ii hi| + ∑ Ŝα2 + Ωâ†α âα
i=0 α Ωα ∑ α
10-3
g2k,α -8 1
+ ∑ λα Ŝα (â†α + âα ) + ∑ (â†α + âα )2 10
α k,α ωk,α 0.5
+∑ (â†α + âα ) ∑ gk,α (b̂†k,α + b̂k,α ) + ∑ ωk,α b̂†k,α b̂k,α .
0
α k k,α
0 10 20
(15) 10 -10
The coupling between the RCs and the residual baths are given 10-1 100
by Ohmic spectral density functions post mapping, see Eq.
(7). Furthermore, all parameters carry the same meaning as in
the generalized spin-boson Hamiltonian, Eq. (6). In particu- Figure 6. Steady-state heat current in the three-level ladder model as
lar, λ denotes the system-bath coupling strength as it appears a function of coupling parameter λ on (a) a logarithmic scale and (b)
in the spectral function Eq. (5). a linear scale for three different splittings, ∆ = 0.25 (orange), ∆ = 0.5
(black), and ∆ = 0.75 (blue). The dark solid line in panel (a) indicates
the scaling behavior of the inter-bath current. Parameters are ε0 = 0,
ε2 = 1, Ω = 10, γ = 0.0071/π, Th = 1, Tc = 0.5, Λ = 1000, M =
B. Simulations 5. Panel (b) illustrates the suppression of the heat current at strong
coupling.
We simulate the heat current using the Redfield QME, af-
ter a reaction coordinate transformation is applied, Eq. (15).
We take the RC parameters equal in numerical simulations,
λ = λα , and Ω = Ωα . Figure 6 displays the heat current as a
function of the coupling strength, λ , for three different values
of ∆ (the position of the intermediate level). Results are pre- -5 10-3
10 3
sented on a logarithmic scale, exposing the scaling relation in
the weak coupling regime, jq ∝ λ 4 , characteristic to the inter- 2
bath current. The absence of a lower order scaling confirms
that the inter-bath current is the leading mechanism of trans- 1
port at weak coupling. We also find that the value of ∆ has an 0
insignificant impact on the current, supporting the argument 10-10 0 5 10
that heat exchange between baths is the dominant transport
pathway. 10-1 100 101
So far, we focused on the weak coupling limit with λ Ω.
Complementing this, we further demonstrate in Figure 6 (b)
the behavior of he heat current as we push λ into the strong Figure 7. Steady-state heat current in the three-level ladder model as
coupling regime, λ & Ω65 . The heat current rises with λ , a function of the RC frequency Ω for ∆ = 0 (green), ∆ = 0.25 (or-
reaches a maximum, and then decays. This trend of the ladder ange), ∆ = 0.5 (black), and ∆ = 0.75 (blue), showing that the current
model is parallel to the behavior of the spin-boson model20 , does not depend on the internal energetics, ∆. a) The current on loga-
albeit here originating from the inter-bath current. Results rithmic and (b) linear scales. Other parameters are identical to Figure
beyond λ = 10 were not fully converged, and they required 6, with λ = 1 and M = 5.
simulations with higher M values, which posed a computa-
tional challenge. These results are presented to provide the
broad picture of transport in the model. Nevertheless, while 5. Beyond Ω ≈ λ , results are converged whereas for Ω < λ
the magnitude of the current was still changing with M, trends the magnitude of the current was still visibly changing with M
and the peak position stayed intact. on the presented scale.
To further illustrate that heat flow in the ladder model is due
to the inter-bath mechanism, we simulate the heat current as a
function of the reaction coordinate frequency, Ω in Figure 7. V. SUMMARY
We observe similar trends as found in Figure 5. Namely, an
increase of the current at small Ω followed by a rapid sup- The goal of this work has been threefold:
pression as Ω grows. This suppression is attributed to the (i) To establish the RC mapping and the RC-QME method
small population of excited states of the RC as one increases as an effective tool for studying steady-state quantum trans-
their frequency, thereby reducing the inter-bath current. Test- port characteristics. This study goes beyond earlier bench-
ing convergence of results is performed analogously to Figure marking that focused on the equilibration problem with re-11
spect to a single heat bath7 , as well as previous studies that APPENDIX: ABSENCE OF SYSTEM’S CURRENT IN THE
demonstrated energy renormalization of the system’s current LADDER MODEL UNDER THE BMR QME
as a key signature of strong system-bath couplings20,21 . In
contrast, here we demonstrate that more fundamentally, the In this appendix, we focus on the ladder model at weak cou-
exact RC mapping (as well as the approximate RC-QME im- pling. We show that heat cannot be conducted in the system if
plementation) can realize two distinct transport mechanisms: the study is carried out using the second order Born-Markov
a system’s current, which at weak coupling reduces to sequen- Redfield QME, a method that only captures system’s current.
tial transport, and an inter-bath current, with bath modes di- Indeed, as we show in the main text, the heat current scales as
rectly exchanging energy between them, and the quantum sys- jq ∝ λ 4 at weak coupling, pointing to the inter-bath current as
tem acting as a bridge. the transport mechanism at small λ .
(ii) To realize and tune different heat transport mecha- The ladder model is given by Eq. (14). In the Schrödinger
nisms. Significantly, we gained understanding on transport picture, the BMR QME for the reduced density operator of the
mechanisms and their characteristics without explicitly per- system is given by2 ,
forming transport calculations, only by inspecting the RC-
mapped Hamiltonian, especially after doing an additional po- ρ̇mn (t) = −iωmn ρmn (t)
laron transformation. − ∑ ∑[Rαm j, jk (ωk j )ρkn (t) + Rα∗
nk,k j (ω jk )ρm j (t)
(iii) To bring forward models that exclusively enact a direct- α j,k
bath current at weak coupling. Besides the σx − σz model, − Rαkn,m j (ω jm )ρ jk (t) − Rα∗
jm,nk (ωkn )ρ jk (t)]. (A1)
which was introduced before, we constructed and simulated
the ladder model. We showed that heat current characteristics Here, ωmn = εm − εn are the eigenenergies of the three-level
in the ladder model are of an inter-bath nature at weak cou- system. The elements of the R super-operator are given by
pling with the current scaling quartically with the system-bath half Fourier transform of bath autocorrelation functions,
coupling strength. Z ∞
The main practical message of our work is that the re- Rαmn, jk (ω) = Ŝm,n
α
Ŝαj,k dτeiωτ hB̂α (τ)B̂α i
0
action coordinate mapping technique captures nontrivial as- α
pects of system-bath coupling: Beyond bath-induced lev- = Ŝm,n Ŝαj,k [Γα (ω) + i∆α (ω)]. (A2)
els renormalization20,21 , the method captures inter-bath heat
These functions are evaluated with respect to the thermal state
transfer effects. As for physical results, by inspecting the
of their respective baths. In what follows, we neglect the
RC-mapped Hamiltonian numerically and analytically, and by
imaginary component of the dissipators, ∆α , for simplicity.
performing simulations extracting different scaling relations,
Working in the energy basis of the Hamiltonian Eq. (14), the
we identified two heat transport mechanisms: system’s current
equations of motion for the populations of the reduced density
and inter-bath current. The system’s current is due to the baths
matrix ρii (t) = hi| ρ(t) |ii become
thermally exciting and de-exciting the system, with sequential
transport dominating at weak coupling. The inter-bath current ρ̇00 (t) = − RC01,10 (ω01 ) + R∗C
01,10 (ω01 ) ρ00 (t)
corresponds to direct energy exchange between the modes of
+ RC10,01 (ω10 ) + R∗C
the baths, utilizing the system to build this exchange. These 10,01 (ω10 ) ρ11 (t)
results were demonstrated in both the generalized spin-boson ρ̇11 (t) = − RC10,01 (ω10 ) + R∗C
10,01 (ω10 ) ρ11 (t)
model and the proposed ladder model. The inter-bath cur- ∗H
− RH
rent exists at weak coupling once the system is coupled to the 12,21 (ω12 ) + R12,21 (ω12 ) ρ11 (t)
bath with non-commuting operators, a prevalent scenario in + RC01,10 (ω01 ) + R∗C
01,10 (ω01 ) ρ00 (t)
quantum thermodynamics. For example, in the commonly ex- ∗H
+ RH
plored three-level quantum absorption refrigerator66 system’s 21,12 (ω21 ) + R21,12 (ω21 ) ρ22 (t)
∗H
ρ̇22 (t) = − RH
coupling operators do not commute, giving rise to inter-bath 21,12 (ω21 ) + R21,12 (ω21 ) ρ22 (t)
leakage currents21 . ∗H
+ RH
Combining the reaction coordinate mapping with a polaron 12,21 (ω12 ) + R12,21 (ω12 ) ρ11 (t) (A3)
transformation proved to prepare the Hamiltonian in a highly These equation can be simplified. Each pair of terms com-
interpretive form, exposing transport pathways. Future work bine, and the half Fourier transform becomes a full integral.
will be focused on studies of impurity dynamics at strong cou- We identify those terms as transition rates, providing the equa-
pling using the RC-polaron mapping approach. tions,
ρ̇00 (t)= −kC0→ C
− 1 ρ00 (t) + k1→
− 0 ρ11 (t)
ACKNOWLEDGMENTS ρ̇11 (t)= −(kC1→ H
− 0 + k1→
C
− 2 )ρ11 (t) + k0→
H
− 1 ρ00 (t) + k2→
− 1 ρ22 (t)
ρ̇22 (t)= −k2H→ H
− 1 ρ22 (t) + k1→
− 2 ρ11 (t) (A4)
We acknowledge fruitful discussions with Junjie Liu. DS
acknowledges the NSERC discovery grant and the Canada Due to the natural decoupling between populations and co-
Research Chair Program. The work of FI was partially funded herences in this system, the above equation can be recast
by the Centre for Quantum Information and Quantum Control as a rate equation for the population of each state. In a
at the University of Toronto. compact notation, the kinetic-type equation is of the formYou can also read
