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On the First Law of Thermodynamics in Time-Dependent Open Quantum Systems
Parth Kumar∗and Charles A. Stafford
Department of Physics, University of Arizona, 1118 East Fourth Street, Tucson, Arizona 85721, USA
(Dated: August 16, 2022)
How to rigorously define thermodynamic quantities such as heat, work, and internal energy in
open quantum systems driven far from equilibrium remains a significant open question in quantum
thermodynamics. Heat is a quantity whose fundamental definition applies only to processes in
systems infinitesimally perturbed from equilibrium, and as such, must be accounted for carefully
in strongly-driven systems. In this work, an unambiguous operator for the internal energy of an
interacting time-dependent open quantum system is derived using a key insight from Mesoscopics:
infinitely far from the local driving and coupling of an open quantum system, reservoirs are indeed
only infinitesimally perturbed. Fully general expressions for the heat current and the power delivered
by various agents to the system are derived using the formalism of nonequilibrium Green’s functions,
establishing an experimentally meaningful and quantum mechanically consistent division of the
energy of the system under consideration into Heat flowing out of and Work done on the system.
The spatio-temporal distribution of internal energy in a strongly-driven open quantum system is
also analyzed. This formalism is applied to analyze the thermodynamic performance of a model
quantum machine: a driven two-level quantum system strongly coupled to two metallic reservoirs,
which can operate in several configurations–as a chemical pump/engine or a heat pump/engine.
CONTENTS
I. Introduction 1
II. The Time-Dependent Open Quantum System 2
A. Physical basis of the model Hamiltonian 3
III. Non-Equilibrium Thermodynamic Quantities 3
A. Work done by external forces 3
B. Particle Current 4
C. Heat Currents, Electrochemical Power, and
the Role of Reservoirs 4
D. The First Law of Quantum
Thermodynamics 4
E. Internal Energy Density 5
IV. Evaluating the Time-Dependent Energetics:
NEGF Results 6
V. Proof of Concept: Application to a Driven
Two-Level Quantum Machine 7
A. Machine Setup 7
B. Electrochemical Pump 7
C. Heat Engine 9
D. Spatio-Temporal Distribution of Internal
Energy 10
VI. Conclusions 13
VII. Acknowledgments 14
References 14
A. Derivation of the NEGF formulas for all First
Law quantities 16
∗parthk@arizona.edu
B. Generalization to Time-Dependent reservoirs
and coupling 18
I. INTRODUCTION
Quantum Thermodynamics has seen substantial
progress in recent years in both experimental and the-
oretical directions. These include but are not limited
to quantum thermometry [1–11], quantum and thermal
machines [12–15], quantum stochastic dynamics [16–18]
and quantum information and computation [19]. Despite
these rapid advancements, however, broad consensus on
several basic questions of the field has been elusive. The
reason for this is not entirely mysterious. The field is,
firstly in relative infancy and secondly a melding of two
well-established disciplines of physics. A large subset of
unresolved questions in quantum thermodynamics thus
centers around the need for a careful evaluation of the
foundational concepts and laws that the subject builds
upon.
One of the key open questions has been the possibil-
ity and the formulation of the First Law of Thermody-
namics for an open quantum system (defined as a quan-
tum system statistically and quantum mechanically open
to the environment via some coupling) driven far from
equilibrium. More concretely, we may ask the follow-
ing: how does one establish an experimentally meaning-
ful and quantum mechanically consistent division of the
energy of a far-from-equilibrium quantum system into its
Internal Energy, the Work done by the different agents
acting and boundary conditions imposed on it and the
Heat flowing from it? An even more basic question that
naturally precedes this is whether such a division is even
possible given the fundamental uncertainty constraints
that quantum mechanics places on the measurement of
observables.
arXiv:2208.06544v1 [cond-mat.mes-hall] 13 Aug 2022
2
These questions have been previously addressed in sev-
eral works [20–25], which present highly divergent points
of view on the issue. Much of the controversy revolves
around how to treat the “interface” between the system
and the environment induced by the System-Reservoir
coupling Hamiltonian. An approach based on the ‘Hamil-
tonian of Mean Force’ concept [20] introduces an ambi-
guity in the definition of Internal Energy of the system
and concludes that it might not even be possible to for-
mulate the First Law, even on a quantum-statistical av-
erage level. Another set of analyses [21,22] proposes
dividing up the Coupling energy between the system and
the environment as half of it belonging to the system and
half of it belonging to the environment. Others still, us-
ing the Master equation approach [23] and the so called
Mesoscopic leads approach [24] conclude that the Internal
energy of the nonequilibrium system should be identified
by putting the entire Coupling with the System. A re-
cent study comparing some of these approaches has also
emerged [25].
In this work, we shed light on the problem of for-
mulating the First Law of Thermodynamics for a time-
dependent nonequilibrium quantum system using in-
sights from Mesoscopic physics [26–29]. The mesoscopic
modeling of the Hamiltonian governing the nonequili-
birum dynamics is done such that the model is both
experimentally realistic and computationally tractable
while capturing all the relevant physics of the energy and
particle flows. Specifically, we start by identifying the
conditions under which heat can be unambiguously de-
fined for a strongly-driven system and how one must ac-
count for it by carefully considering what role the Reser-
voirs (which make up the environment) coupled to the
system play. This leads naturally to the identification
that the Internal Energy operator should be the System
Hamiltonian plus the full coupling Hamiltonian.
With the proper thermodynamic identifications made,
we proceed to derive general expressions for all the dif-
ferent possible forms of energy flows out of the system
using the nonequilibrium Green’s functions (NEGF) for-
malism. The NEGF formalism is a powerful framework
for analyzing quantum transport which we utilize to in-
vestigate the thermodynamics of a fairly general struc-
ture, where the system consists of interacting electronic
and phononic degrees of freedom which is strongly cou-
pled to an environment of non-interacting electrons and
phonons. As an application of these formal results, we
simulate a strongly driven quantum machine based on
Rabi oscillations of a double quantum dot system, in-
vestigating in detail its operation as an electrochemical
pump and as a heat engine.
This work is organized as follows: In Section II, we
give a detailed description of the model time-dependent
Hamiltonian that forms the basis of all subsequent deriva-
tions and discussions. Section III identifies all the differ-
ent possible forms of energy flow associated with the sys-
tem, the role of reservoirs, and the consequent form of the
Internal Energy Operator and the First Law. In Section
IV, we derive all the nonequilibrium energy flows in terms
of the System Green’s functions using the NEGF formal-
ism. In Section V, we present results and discussion for
the simulation of the strongly-driven quantum machine
operating as an electrochemical pump and a heat engine,
including an analysis of the spatio-temporal distribution
of the internal energy. Finally, the conclusions of our
work are presented in Section VI.
II. THE TIME-DEPENDENT OPEN QUANTUM
SYSTEM
A fairly general Hamiltonian for a time-dependent
open quantum system can be written as
H(t) = HS(t) + HR+HSR .(1)
The System Hamiltonian HS(t) has explicit time-
dependence due to an external drive and is written as
the sum of electronic, phononic and electron-phonon cou-
pling terms
HS(t) = HS,el(t) + HS,ph (t) + HS,el−ph .(2)
The electronic part of the system Hamiltonian HS,el(t)
includes electron-electron interactions and has explicit
time-dependence from the external electric drive:
HS,el(t) = X
n,m [H(1)
S,el(t)]nm d†
ndm+[H(2)
S,el]nm d†
nd†
mdmdn!,
(3)
where d†
n(dn) creates (removes) an electron in the
nth orbital in the system and obeys fermionic anti-
commutation {d†
n, dm}=δnm. The phononic part
HS,ph(t) is given as the sum of a harmonic term and
an anharmonic term
HS,ph(t) = Hhm
S,ph(t) + Hahm
S,ph ,(4)
where the time-dependent harmonic part Hhm
S,ph(t) ac-
counts for an external mechanical drive:
Hhm
S,ph(t) = X
s,r
[H(1)
S,ph(t)]sr b†
sbr(5)
where b†
r(br) creates (removes) a phonon in the rth
mode in the system and obey the bosonic commutation
[b†
r, bs] = δrs. The anharmonic part Hahm
S,ph is given by
Hahm
S,ph =H(3)
S,ph +H(4)
S,ph +... , (6)
where the cubic term H(3)
S,ph can be written as
H(3)
S,ph =X
n,m,o
[H(3)
S,ph]nmo qnqmqo,(7)
where the generalized coordinate qncan be written in
terms of normal coordinates as qn=Pr(Qreirn +
3
Q†
re−irn) with Qr=1
p[H(1)
S,ph(0)]rr
(b†
r+br). The quartic
term H(4)
S,ph and all higher-order anharmonic terms can
be written analogously. The electron-phonon coupling is
HS,el−ph =X
r,n,m
[HS,el−ph]rnm (b†
r+br)d†
ndm+... (8)
and may include higher-order terms in the lattice-
electron potential expansion.
The environment is modeled as Mperfectly ordered,
semi-infinite Reservoirs of non-interacting electrons and
phonons, each at Temperature Tαand chemical potential
µα, with the Hamiltonian
HR=
M
X
α=1 X
k∈α
kc†
kck+X
q∈α
ωqa†
qaq+1
2!.(9)
where c†
k(ck) creates (removes) an electron in eigenmode
kin the reservoir and obey {c†
k, cl}=δkl, and a†
q(aq)
creates (removes) a phonon in the eigenmode qin the
reservoir and obey [a†
q, ap] = δqp.
The coupling between the System and the Reservoirs
is given by the Coupling Hamiltonian
HSR =HSR,el +HSR,ph ,(10)
where
HSR,el =
M
X
α=1 X
k∈α,n
(Vel
knc†
kdn+h.c.) (11)
and
HSR,ph =
M
X
α=1 X
q∈α,r
(Vph
qr a†
qbr+h.c.) (12)
are the electronic and phononic parts of the coupling,
respectively.
A. Physical basis of the model Hamiltonian
We note some points here about the modeling of the
Hamiltonian just presented that are important from both
physical and computational points of view.
First, the Reservoirs are modeled as non-interacting,
semi-infinite, and perfectly ordered so that they faith-
fully represent the external macroscopic circuit used to
impose any nonequilibrium bias on the system and to
measure the resultant flows of charge and energy. As
counters of charge and energy flowing out the system,
it makes little sense to include two-body interactions in
the Reservoirs, since that would significantly complicate
the task of calculating these quantities, and many-body
correlations are not relevant in such a macroscopic elec-
tric/thermal circuit in any case. The coupling between
the System and the Reservoir HSR is also taken to be
quadratic. Realistically, this is because the electrons in
the metallic Reservoirs, modeled by HR, are well screened
and hence any interaction between the electrons in the
System and the Reservoir can be treated with the method
of image charges [30], the screening charges in the Reser-
voirs treated as slave degrees of freedom, rather than as
independent quantum modes. This is justified because
the screening charges in the reservoirs respond at the
plasma frequency, which is much greater than typical re-
sponse frequencies of a mesosopic system.
Second, the number of Reservoir degrees of freedom
entering HSR is a set of measure zero [31,32] compared to
the full (infinite) Reservoir Hilbert space, with only a few
frontier orbitals taking part in coupling the environment
to the System. This is because tunnel coupling is local
at the Interface between the System and Reservoir and
the screening charges are also localized at the Interface
[33]. The same holds true for the phononic degrees of
freedom, whose elastic coupling is short-ranged [35]. A
model calculation of the spatio-temporal dynamics of the
Interface, illustrating these principles, is presented in Sec.
V D.
III. NON-EQUILIBRIUM THERMODYNAMIC
QUANTITIES
With the open quantum system defined, we now com-
pute the nonequilibrium thermodynamic quantities of the
system.
A. Work done by external forces
Following the so-called Inclusive definition of work,
standard in the Quantum Thermodynamics literature
[36] (in contrast with the Exclusive definition [37]), we
identify the time rate of change of the expectation value
of the total Hamiltonian as the rate of work done by ex-
ternal forces, ˙
Wext(t), which for the form assumed for the
Hamiltonian, we can compute as
d
dthH(t)i ≡ ˙
Wext(t) = h˙
HS(t)i.(13)
The equality can be proved [38] by noting that
hH(t)i=Tr{ρ(t)H(t)}, with ρ(t) denoting the full den-
sity matrix at time tand Tr{} denoting the Trace oper-
ation, and using the chain rule we can write
d
dthH(t)i=Tr{˙ρ(t)H(t) + ρ(t)˙
H(t)},(14)
which, with the equation of motion of the density matrix
i~˙ρ= [H(t), ρ(t)] ,(15)
and cyclicity of the Trace gives the stated result,
˙
Wext(t) = h˙
HS(t)i, since the time-dependence is only
in the System Hamiltonian HS(t).
4
Furthermore, since the time-dependence in HS(t) sits
only in the one-body terms, we get
˙
Wext(t) = X
n,m,s,r
h[˙
H(1)
S,el(t)]nm d†
ndm+ [ ˙
H(1)
S,ph(t)]sr b†
sbri.
(16)
B. Particle Current
The equation of motion for the expectation value of oc-
cupation number operator Nα=Pk∈αc†
kckof reservoir
αis
d
dthNαi=−i
~h[Nα, H(t)]i=−i
~h[Nα, HSR ]i,(17)
which is just the electronic particle current IN
α,el(t) for
reservoir α. Using the identities [A, BC ] = {A, B}C−
B{C, A}= [A, B]C−B[C, A], we obtain the particle
current as
IN
α,el(t) = −i
~X
k∈α,n
[Vel
knhc†
kdni − (Vel
kn)∗hd†
ncki].(18)
The electrical current is obtained by multiplying the
above equation by the electric charge quantum.
C. Heat Currents, Electrochemical Power, and the
Role of Reservoirs
We now compute the heat current by applying the
fundamental thermodynamic identity at constant volume
dE =T dS +µdN to reservoir α
IQ
α(t)≡Tα
d
dtSα=d
dthHR,αi − µα
d
dthNαi.(19)
Carrying out commutator evaluations similar to that for
the particle current, we get the electronic heat current
[39]
IQ
α,el(t) = −i
~X
k∈α,n
(k−µα)[Vel
knhc†
kdni − (Vel
kn)∗hd†
ncki],
(20)
which can be cast in terms of an energy weighted particle
current (for the kth mode) as
IQ
α,el(t) = X
k∈α,n
(k−µα)IN
k(t),(21)
where IN
k(t) = −i
~[Vel
knhc†
kdni − (Vel
kn)∗hd†
ncki]. The
phononic heat current [40] can be analogously evaluated
as
IQ
α,ph(t) = −i
~X
q∈α,r
ωq[Vph
qr ha†
qbri − (Vph
qr )∗hb†
raqi].(22)
Heat is a quantity whose fundamental definition d¯Q=
T dS is only valid for processes involving systems in-
finitesimally perturbed from equilibrium [41]. Further-
more, it represents an irreversible flow of energy, and as
such, must be accounted for carefully in strongly-driven
systems. These requirements are met exactly by the non-
interacting semi-infinite reservoirs to which the nonequi-
librium system is coupled. These properties of the reser-
voir model have been long established in the Mesoscopics
and Quantum Transport literature [42–44], where they
enforce an Ordering of Limits such that one must take the
limit of the Reservoir size going to infinity L→ ∞ before
taking limit of the adiabatic perturbation switch-on time
going to infinity tadiab → ∞. Physically, tadiab is related
to the timescale of setting up and carrying out the exper-
iment. This order of limits ensures that any electrons or
phonons emitted by the System into the Reservoirs can-
not be coherently backscattered into the system (causing
decoherence of any quanta emitted into the Reservoirs
despite the absence of 2-body interactions) and that the
distributions of charge and energy in the Reservoirs can-
not be depleted due to flows mediated by the System.
The time derivative of the expectation value of the
Reservoir Hamiltonian has a simple interpretation: it
is the total energy current flowing into the Reservoirs
PαIE
α(t) i.e.
d
dthHR(t)i=X
α
IE
α(t).(23)
At infinity, this in turn is just the electrical (or electro-
chemical) power delivered to the reservoir and the heat
current flowing into it
X
α
IE
α(t) = X
α
µαIN
α,el(t) + X
α
IQ
α(t).(24)
where IQ
α(t) = IQ
α,el(t) + IQ
α,ph(t).
D. The First Law of Quantum Thermodynamics
From the discussion in III A, specifically Eq. (13), it
follows that
d
dthHS(t) + HSR +HRi=˙
Wext(t).(25)
Furthermore, with the identification of III C, specifically
Eq. (24), we get
d
dthHS(t)+HSR i=˙
Wext(t)+(X
α
−µαIN
α(t))+(X
α
−IQ
α(t)) .
(26)
It is then clear that the internal energy operator for
the “System” must be recognized as sum of the System
Hamiltonian and full Coupling Hamiltonian. Defining
˙
Welec(t) := Pα−µαIN
α,el(t) as the electrochemical power
delivered to the system and ˙
Q(t) := Pα−IQ
α(t) as the
5
HR
HS(t)HSR
“System” ≡Usys(t) = HS(t) + HS R
FIG. 1: Schematic representation of the energetic partitioning of a time-dependent open quantum system consistent
with First Law of Quantum Thermodynamics [Eq. (27)]. This partitioning requires that the Internal Energy
operator be identified as the sum of the System Hamiltonian HS(t) and the full coupling Hamiltonian HSR
describing the Interface; the spatial extent of this operator is denoted by the blue dashed box in the figure.
heat current flowing into the system allows us to write
the First Law of Quantum Thermodynamics as
d
dthUsys (t)i=˙
Wext(t) + ˙
Welec(t) + ˙
Q(t),(27)
where we have the Internal Energy operator Usys(t) for
the open quantum system
Usys(t) = HS(t) + HS R .(28)
This analysis lays to rest any ambiguity about the inter-
nal energy operator and gives a quantum mechanically
consistent and experimentally meaningful division of the
energetics of a fairly general open quantum system. Fig.
1gives a schematic for this energy partitioning. We also
note that this identification of the First Law and Internal
Energy operator holds true in the even more general case
where both the coupling and reservoirs become explicitly
time-dependent, as discussed in Appendix B. The defi-
nition (28) agrees with that proposed by Strasberg and
Winter [23] and Lacerda et al. [24], but disagrees with
those put forward in Refs. [21,22].
It should be noted, however, that as opposed to Inter-
nal Energy, Work being a path (and not a state) function
cannot, in general, be a Quantum Operator [45,46]. Fur-
thermore, Heat (again a path function) also cannot have
a quantum operator associated with it since it is a statis-
tical quantity (d¯Q=T dS) defined unambiguously only
under the special conditions discussed in III C.
E. Internal Energy Density
To further study the spatio-temporal distribution of
the internal energy—for the special case where 2-body
interactions are absent from the system—we make use of
a general result for the spatial density of any one-body
observable [47,48].
The expectation value of the Internal energy operator
Usys(t) identified in Eq. (28) is
hUsys(t)i=Tr{ρ(t)Usy s(t)},(29)
where ρ(t) is the density matrix. If the position repre-
sentation of the corresponding Hilbert space operator is
hx|usys(t)|yithen the Fock space operator is
Usys(t) = Zdx Zdy ˆ
ψ†(x)hx|usys(t)|yiˆ
ψ(y),(30)
where ˆ
ψ†(x) and ˆ
ψ(y) are Fock space creation and an-
nihilation operators in position representation, and the
spin index has been suppressed for simplicity.
We define the Fock space internal energy density op-
erator as
%usys (x, t)= 1
2Zdy ˆ
ψ†(x)hx|usys(t)|yiˆ
ψ(y)+
1
2Zdy ˆ
ψ†(y)hy|usys(t)|xiˆ
ψ(x),(31)
which evidently satisfies the condition
Usys(t) = Zdx%Usy s (x, t).(32)
The internal energy density is
ρUsys (x, t) = Tr{ρ(t)%Usys (x, t)},(33)
and from Eq. (28) it follows directly that
ρUsys (x, t) = ρHS(x, t) + ρHSR (x, t),(34)
where ρHS(x, t) and ρHSR (x, t) are defined analogously
to ρUsys (x, t), using the one-body Hilbert-space operators
corresponding to HS(t) and HSR , respectively.
6
IV. EVALUATING THE TIME-DEPENDENT
ENERGETICS: NEGF RESULTS
We employ the Nonequilibrum Green’s function
(NEGF) formalism [26,27,49–51] to evaluate all the
terms appearing in the First Law (27), each term on the
LHS of which we have evaluated in terms of the expec-
tation values in Eqs. (16), (18), (21) and (22). Here
we present only the final results of our analysis and
defer the relevant details of the derivations and meth-
ods of evaluation to Appendix A. We also note that
the NEGF formalism has a well-established connection
with the S-Matrix formalism [52]—another technique of-
ten employed in quantum transport and thermodynam-
ics. A key advantage of NEGF is its simplicity and
power in computing one-body (or even few-body) ob-
servables. This is in contrast to the Master equation ap-
proach [26,28,53], which works in the full many-particle
Hilbert space .
The electronic particle current, given by the Jauho-
Wingreen-Meir formula [54,55], is
IN
α,el(t)= 2
~Z∞
−∞
dE
2πZt
−∞
dt1ImTr{e−iE(t1−t)Γel
α(E)
[G<,el(t, t1) + fα(E)GR,el (t, t1)]},
(35)
where Γel
α(E), G<,el(t, t0) and GR,el (t, t0) are understood
to be matrices. The system electronic Lesser and Re-
tarded Green’s functions are defined as (see Appendix A
for the method of their evaluation)
G<,el
nm (t, t1) = ihd†
m(t1)dn(t)i(36)
and
GR,el
nm (t, t1) = −iθ(t−t1)h{dn(t), d†
m(t1)}i ,(37)
respectively, where h i denotes the quantum and statisti-
cal average, and
fα(k) = 1
eβα(k−µ)+ 1 (38)
is the Fermi-Dirac distribution function of reservoir α
with βα= (kBTα)−1, where kBis the Boltzmann
Constant. Following [56], we identify the electronic
tunneling-width matrix Γel
α(E) as
[Γel
α(E)]nm =X
k∈α
2πδ(E−k)Vel
n(E)(Vel
m)∗(E),(39)
with Vel
n(E) = Vel
kn when E=k. The electronic heat
current formula (20) becomes (see Appendix Afor a
derivation)
IQ
α,el(t)= 2
~Z∞
−∞
dE
2πZt
−∞
dt1(E−µα)ImTr{e−iE(t1−t)
Γel
α(E)[G<,el(t, t1) + fα(E)GR,el (t, t1)]},
(40)
which generalizes the Bergfield-Stafford formula [39] to
the case of a time-dependent system. The electronic
(heat and particle) currents flowing into reservoir αcan
be cast in a compact form
I(ν)
α,el(t)= 2
~Z∞
−∞
dE
2π(E−µα)νZt
−∞
dt1ImTr{e−iE(t1−t)
Γel
α(E)[G<,el(t, t1) + fα(E)GR,el (t, t1)]},
(41)
where (ν= 0) gives the electronic particle current and
(ν= 1) gives the electronic heat current.
The phononic heat current is obtained similarly in
terms of the system Phononic Green’s functions as
IQ
α,ph(t)= 2
~Z∞
−∞
dE
2πEZt
−∞
dt1ImTr{e−iE(t1−t)
Γph
α(E)[G<,ph(t, t1) + fP lanck
α(E)GR,ph(t, t1)]},
(42)
where we have the system phononic Green’s functions
G<,ph
rs (t, t1) = ihb†
s(t1)br(t)i(43)
and
GR,ph
rs (t, t1) = −iθ(t−t1)h[br(t), b†
s(t1)]i,(44)
and the phononic tunneling-width matrix
[Γph
α(E)]rs =X
q∈α
2πδ(ω−ωq)Vph
r(E)(Vph
s)∗(E),(45)
and
fP lanck
α(q) = 1
eβα(q)−1(46)
is the Planck distribution function of reservoir α.
Finally, the work done by external forces can also be
written in terms of the system Green’s functions as
˙
Wext =−iTr{˙
H(1)
S,el(t)G<,el (t, t) + ˙
H(1)
S,ph(t)G<,ph (t, t)}.
(47)
The expectation value of the internal energy operator
defined in Eq. (28) can also be evaluated in terms of the
system Green’s functions. The first term is
hHS(t)i=−iTr{H(1)
S,el(t)G<,el (t, t) + H(1)
S,ph(t)G<,ph (t, t)
+H(2)
S,elG(2),el (t, t) + H(3)
S,phG(3),ph (t, t)
+HS,el−phG(2),el−ph (t, t)},
(48)
where the 2-body and higher electronic and phononic
Green’s functions are defined as G(2),el
nnmm(t, t) =
ihd†
n(t)d†
m(t)dm(t)dn(t)i,G(2),el−ph
rnm (t, t) = ih[b†
r(t) +
br(t)]d†
n(t)dm(t)i, and G(3),ph
nmo (t, t) = ihqn(t)qm(t)qo(t)i,
respectively (higher order phononic Green’s functions can
7
be constructed analogously). The second term in the in-
ternal energy may be evaluated as (see Appendix Afor
details)
hHSR (t)i=2
~Z∞
−∞
dE
2πZt
−∞
dt1ReTr{e−iE(t1−t)
(Γel
α(E)[G<,el(t, t1) + fα(E)GR,el (t, t1)]+
Γph
α(E)[G<,ph(t, t1) + fP lanck
α(E)GR,ph(t, t1)])}.
(49)
With this, all the terms appearing in the First Law
[Eq. (27)] have now been evaluated in terms of system
Green’s functions, and we identify IN
α,el(t) = I(0)
α,el(t) and
IQ
α(t) = I(1)
α,el(t) + IQ
α,ph(t), so that ˙
Welec(t) and ˙
Q(t) are
defined exactly as in Sec. III D.
Finally, both terms on the RHS of Eq. (34) for the
internal energy density can also be evaluated in terms of
the Green’s functions in a similar manner, giving
ρHS(x, t) = Im[hx|h(1)
S,el(t)G<,el (t, t)+h(1)
S,ph(t)G<,ph (t, t)|xi],
(50)
and
ρHSR (x, t) = Im[hx|hSR,el G<,el
tun (t, t)+hSR,ph G<,ph
tun (t, t)|xi],
(51)
where h(1)
S,el/ph are Hilbert-space operators corresponding
to the one-body part of the Fock-space system Hamil-
toninans defined in Eqs. (3) and (5) and hSR,el/ph are
Hilbert-space operators corresponding to the Fock-space
coupling Hamiltoninans defined in Eqs. (11) and (12),
while G<,el
tun,kn(t, t) = ihd†
n(t)ck(t)iand G<,ph
tun,qr (t, t) =
ihb†
r(t)aq(t)iare the tunneling Green’s functions for the
electrons and phonons respectively.
V. PROOF OF CONCEPT: APPLICATION TO
A DRIVEN TWO-LEVEL QUANTUM MACHINE
We have applied our formal results to perform a com-
plete thermodynamic analysis of a strongly driven two-
level quantum system coupled to two metallic reservoirs.
We first describe the basic setup of the quantum ma-
chine, followed by a detailed analysis of its operation in
two different configurations. This simulation underscores
the utility of our formal results derived in the previous
sections, as tools to investigate the real-time dynamics of
quantum systems driven far from equilibrium.
A. Machine Setup
The quantum machine consists of two (spatially sep-
arated) quantum dots, each with a single active level,
with energies E1and E2, coupled to each other via
inter-dot coupling w. The system is opened to the en-
vironment by coupling each dot to a perfectly ordered,
semi-infinite Fermionic reservoir via energy-independent
tunneling-width matrix elements Γ1= Γ2= Γ. Both
reservoirs are maintained in internal equilibrium charac-
terized by fixed Chemical Potentials and Temperatures
µ1, T1and µ2, T2(see Ref. [57] for an analysis of a similar
quantum machine). The time-dependent drive is applied
only to one of the dots (the dot with energy E1for the
results presented) and is set up as a rectangular pulse of
strength δwhich is active for a duration τ. The pulse
strength δis set equal to the difference between the dot
energies ∆E=E2−E1and brings the two levels into
resonance, allowing the electron density to strongly Rabi
oscillate between them. The pulse duration is tuned as
aπ-pulse by setting τ=π
2w. This maximizes the prob-
ability of transferring the electron density from one dot
to the other at the end of the pulse. We also order the
parameters such that ∆EwΓ to ensure that the
levels remain highly localized in the absence of a drive.
The energy parameters of the system can be tuned rel-
ative to each other such that its Rabi Oscillations can
be leveraged to operate it as an Electrochemical Pump,
an Electrochemical Engine, a Heat Engine, and a Heat
Pump. Here we present a detailed analysis for the Elec-
trochemical Pump and Heat Engine configurations.
B. Electrochemical Pump
For the setup described, if the electrochemical poten-
tials of the reservoirs are biased such that µ1> E1and
µ2< E2, electrons can be pumped uphill from the left
to the right reservoir. A schematic of the chemical pump
configuration is shown in Fig. 2. In this configuration,
the Temperatures T1, T2of the reservoirs are set to very
low values to suppress any thermal excitations from con-
tributing to the operation. The thermodynamic cycle for
the pump involves the electron tunneling from the left
reservoir to dot 1 followed by the pulse raising its energy
to bring it into resonance with dot 2. The π-pulse ensures
a maximum probability of dot 2 capturing the electron at
the end of the pulse which is followed by the electron tun-
neling out into the right reservoir. The machine relaxes
back to its initial state at late times, at which point we
can calculate the efficiency of the electrochemical pump
as
ηEP =|Welec (∞)|
|Wext(∞)|.(52)
The full Hamiltonian for this configuration is given by
Eq. (1), where the System Hamiltonian is given specifi-
cally by
HS(t) = E1(t)d†
1d1+E2d†
2d2+w(d†
1d2+d†
2d1),(53)
with the Reservoir Hamiltonian given by
HR=X
i=1,2X
k∈i
kc†
kck,(54)
8
Pulse Width = τ
Pulse Height = δ
E1
E2
w
µ1
µ2
Γ Γ
FIG. 2: Schematic representation of an Electrochemical Pump based on Rabi Oscillations between states of a double
quantum dot. The left and right dots in the system have on-site energies E1and E2, respectively, and are coupled
by a constant hopping matrix element w. The left dot is driven by a rectangular pulse of width τand height δ.
Each dot is coupled with the same Tunneling width matrix element Γ to a Reservoir. The Reservoirs are modeled as
non-interacting, Fermionic, semi-infinite reservoirs at Electrochemical Potentials µ1and µ2for the left and right
reservoirs, respectively, with T2=T1. The cycle of operation of the electrochemical pump is denoted by the dashed
purple arrows.
and the coupling is such that the ith dot is only coupled
to the ith reservoir
HSR =X
k∈1
(Vk1c†
kd1+h.c.) + X
k∈2
(Vk2c†
kd2+h.c.).(55)
Finally, the time dependence of the system is encoded in
the left dot as
E1(t) =
E1t < 0,
E1+δ0< t < τ,
E1t > τ.
(56)
Fig. 3shows plots of all the thermodynamic quanti-
ties entering the First Law as functions of time. We
note that we have plotted the integrated thermodynamic
quantities here instead of the time-derivatives that ap-
pear in our First Law equation [Eq. (27)], so that we have
the total external work done on the system Wext(t) =
−iRt
−∞ dtTr{˙
H(1)
S(t)G<(t, t)}, the total electrical work
done on the system Welec(t) = Rt
−∞ dt[Pα−µαI(0)
α(t)],
and the total heat dissipated into the reservoirs −Q(t) =
Rt
−∞ dt(PαI(1)
α(t)). All the energies reported for this
setup are in units of the on-site energy of the right dot
E2and time is in units of E−1
2.
The total external work done by the drive Wext(t),
(Fig. 3a) follows the expected instantaneous rise of the
pulse at t= 0 to a value of 1.87, indicating that work is
done on the system by the drive. Wext(t) remains at this
constant value for the duration of the pulse τ, followed
by a slight instantaneous decrease at t=τ. This can be
explained by noting that there is a small but finite prob-
ability for the electron being on dot 1 at the end of the
π-pulse. This results in work being done by the system
on the drive as the energy of the first dot is lowered back
to its initial value, appearing as a negative contribution.
The external work done remains constant at a value of
1.78 thereafter since the drive is inactive for t>τ.
As the pulse starts, the electrical work Welec(t), (Fig.
3b) is done on the system as the particle current flows
mostly into the left reservoir initially. At a certain time
t > τ the electrochemical work becomes negative indi-
cating that chemical work is now done by the system as
the particle current starts to flow mostly into the right
reservoir. It increases in magnitude for some time and
eventually attains a constant value of −0.65 at late times
as the particle currents flowing into the reservoirs relax to
0. For the complete cycle, electrical work is thus done by
the system in transferring an electron from the reservoir
with the lower chemical potential µ1to the one with the
higher chemical potential µ2. With the late time values
of Welec(t) and Wext (t), we can see that the efficiency of
the electrochemical pump ηEP is about 36.5%.
The total heat dissipated into the reservoirs −Q(t)
(Fig. 3c) always remains positive implying that heat is
only ever flowing into the reservoirs in the electrochemi-
cal pump configuration for the parameters chosen. This
can be explained by noting that the (steady-state) heat
flowing into the ith reservoir goes as (Ei−µi)vifi(E),
where viis the velocity of the electron [58]. The levels
in the numerics are biased such that (Ei−µi)vialways
remains positive for both reservoirs, no matter which di-
rection the electron is moving. The pronounced knee at
t=τis due to the opening up of another transport chan-
nel at the first dot in the form of a hole channel once the
electron has been captured by the second dot at the end
of the pulse and the left dot is in an empty state. The
total heat dissipated asymptotes to a constant value of
9
0 20 40 60 80 100 120
0.0
0.5
1.0
1.5
Time [E2-1]
Wext [E2]
(a) External Work
0 20 40 60 80 100 120
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
Time [E2-1]
Welec [E2]
(b) Electrochemical Work
0 20 40 60 80 100 120
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Time [E2-1]
-Q[E2]
(c) Total Heat Dissipated
Usys(t) [E2]
ΔUsys(t) [E2]
0 20 40 60 80 100 120
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Time [E2-1]
(d) Internal Energy (and consistency check)
FIG. 3: The time-dependent energetics of the Electrochemical Pump depicted schematically in Fig. 2. (a) Total
work done by external drive versus time, (b) Total electrochemical work done versus time, (c) Total Heat dissipated
into reservoirs versus time, (d) Internal Energy Usys (in orange) and ∆Usys from integration of the RHS of the First
Law equation (27) (in blue) versus time. For the results shown we have used ~= 1 and we set the on-site energy on
the left and right dots {E1, E2}={−1,1}so that the difference in the on-site energies of the two dots is
∆E=E2−E1= 2 in units of E2. The dot hybridization is set to w= ∆E/5 and the coupling to the reservoirs is
taken as Γ = w/5. The chemical potential and temperatures of the reservoirs are set −µ1=µ2= 0.5 in units of E2
and kBT1=kBT2= 0.016 in units of E2, respectively. The pulse amplitude is set equal to the difference in the
on-site energies δ= ∆Eand the pulse duration is set τ=π/2w(π-pulse).
1.17 at late times.
We finally plot the internal energy Usys(t) (Fig. 3d)
and note that it matches up well with the plot of the sum
of all the terms appearing on the right hand side of the
First Law i.e., ∆Usys(t) = Wext (t)+Welec(t)+Q(t). The
offset is expected since we have plotted Usys(t) directly
from the Eqs. (48) and (49), which are expressions for
energy, whereas the plot of ∆Usys (t) was generated by
integration of the current formulas where the constant of
integration was set to 0.
C. Heat Engine
For the heat engine configuration, the temperatures are
raised to significant unequal values (T2> T1) comparable
to that of the system energy levels, and the chemical
potentials of both the reservoirs are set to zero, µ1=
µ2= 0. Furthermore, in addition to the left dot energy,
the inter-dot coupling is now a function of time w→w(t).
Specifically, it is activated to a constant value wonly
during the pulse and is zero before and after it. This is
done to suppress heat flow into the reservoirs before and
after the pulse thus increasing the efficiency of the heat
engine
ηHE =|Wext (∞)|
|Q2(∞)|,(57)
where Q2(∞) is the heat extracted from the right reser-
voir at late times.
Again, the full Hamiltonian for this configuration is
given by Eq. (1) and the System Hamiltonian is given
specifically by
HS(t) = E1(t)d†
1d1+E2d†
2d2+w(t)(d†
1d2+d†
2d1),(58)
10
Pulse Width = τ
Pulse Height = δ
E1
E2
w(t)
T1
T2
Γ Γ
FIG. 4: Schematic representation of a Heat Engine based on Rabi Oscillations between states of a double quantum
dot. The left and right dots in the system have on-site energies E1 and E2, respectively, and are dynamically
coupled by a time-dependent hopping term w(t). The left dot is driven by a rectangular pulse of width τand
strength δ. Each dot is coupled with the same Tunneling width matrix element Γ to a Reservoir. The Reservoirs are
modeled as non-interacting, Fermionic, semi-infinite systems at Temperatures T1and T2for the left and right
reservoirs, respectively, with T2> T1. The cycle of operation of the heat engine is denoted by the dashed purple
arrows.
with the Reservoir and Coupling Hamiltonian exactly the
same as in the electrochemical pump case given by Eqs.
(54) and (55), respectively.
As before, the rectangular pulse acts only on the left
dot with E1(t) given by Eq. (56), and the time-dependent
inter-dot coupling is
w(t) =
0t < 0,
w0< t < τ,
0t > τ.
(59)
In the heat engine configuration, the electron transport
path is essentially reversed from the pump configuration.
The electron tunnels in from the right reservoir on to the
right dot, Rabi Oscillates onto the left dot, is lowered
to energy E1at the end of the pulse and tunnels out
into the left reservoir, accomplishing engine operation.
A schematic for the engine is given in Fig. 4.
The time-dependent energetics shown in Fig. 5, where
all the energies are reported in units of the on-site energy
of the left dot E1and time is in units of E−1
1, demonstrate
that external work Wext(t) (Fig. 5a) is now done by the
engine on the drive as the left dot is lowered at the end
of the pulse. For the parameters chosen Wext(t) at late
times attains a constant value of −0.18. This work is
done by extracting heat Q2(t) (Fig. 5b) from the hot
reservoir at T2, and eventually attains a constant value of
−0.60 at late times after the pulse and inter-dot coupling
are inactivated.
By design, no net electrochemical is work done (Fig.
5c) since we had set the chemical potentials of both the
reservoirs to 0 in this configuration. Finally, the left and
right hand sides of the First Law are again in good agree-
ment (Fig. 5d), up to a constant of integration offset. For
the parameters chosen, the heat engine operates at an
efficiency ηHE of about 46.5% which is close to 53% of
Carnot efficiency.
D. Spatio-Temporal Distribution of Internal
Energy
To investigate the spatio-temporal distribution of en-
ergy in the driven quantum machine, we utilize the the-
ory of internal energy density developed in Sec. III E. We
work with essentially the same setup as in the previous
two subsections, but now with a specific model for the
reservoirs: 1D tight-binding chains. The System Hamil-
tonian HS(t) is chosen the same as in the electrochemical
pump case [Eq. (53)]. The Reservoir Hamiltonian is
HR=X
α=L,R "∞
X
j=1
0c†
jα cjα +
∞
X
j=1
t0(c†
jα cj+1α+h.c.)#,
(60)
where 0is the on-site energy at any given site and t0is
the hopping integral between nearest-neighbor sites on a
given chain. The Coupling Hamiltonian is
HSR =t"(d†
1c1L+h.c.)+(d†
2c1R+h.c.)#,(61)
where tis the coupling between the first site in the
left reservoir 1Land the left dot, and that between the
first site in the right reservoir 1Rand the right dot. A
schematic of this setup is given in Fig. 6(Right Panel).
We study the internal energy density ρUsys of the sys-
tem, given by Eq. (34), as a function of time. Due to
11
0 20 40 60 80 100 120
-0.2
-0.1
0.0
0.1
0.2
Time [E1-1]
Wext [E1]
(a) External Work
0 20 40 60 80 100 120
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time [E1-1]
Q2[E1]
(b) Heat extracted from Right Reservoir
0 20 40 60 80 100 120
-1.0
-0.5
0.0
0.5
1.0
Time [E1-1]
Welec [E1]
(c) Electrochemical Work
Usys(t) [E1]
ΔUsys(t) [E1]
0 20 40 60 80 100 120
0.0
0.5
1.0
Time [E1-1]
(d) Internal Energy (and consistency check)
FIG. 5: The Time-dependent energetics of the Heat Engine depicted schematically in Fig. 4. (a) Total work done by
external drive versus time, (b) Heat extracted from the right (hot) Reservoir, (c) Total electrochemical work done
versus time, (d) Internal Energy (in orange) and RHS of the First Law equation (27) (in blue) versus time. For the
results shown, we have used ~= 1 and we set the on-site energy on the left and right dots to {E1, E2}={1,3}so
that the difference in the on-site energies of the two dots is ∆E=E2−E1= 2 in units of E1. The dot hybridization
(when non-zero during the pulse) is set w= ∆E/5 and the coupling to the reservoirs is taken as Γ = w/5. The
chemical potentials and temperatures of the reservoirs are set to µ1=µ2= 0 and {kBT1, kBT2}={0.5,3.6}in units
of E1, respectively. The pulse amplitude is set equal to the difference in the on-site energies δ= ∆Eand the pulse
duration is set τ=π/2w(π-pulse).
the short-range coupling between the system and reser-
voirs, ρUsys extends spatially only to the first site on the
left reservoir chain (1L), the left dot (D1), the right dot
(D2), and the first site on the right reservoir chain (1R).
All other sites have zero contribution from the internal
energy operator for this model. The tunneling-width ma-
trices for the 1D tight-binding chains coupled to the dou-
ble quantum dot system are [59,60]
[Γα(E)]ij =
2t2
t01−0−E
2t021
2δiαδj α,|E−0|<2t0,
0,otherwise.
(62)
The numerical results for the internal energy density
are shown in the left panel of Fig. 6for the system in
the electrochemical pump configuration (Sec. V B) [61].
The internal energy on the left dot ρUsys (D1, t) (Fig. 6b)
rises instantaneously to its maximum positive value at
the start of the pulse as the external drive does work on
the dot. As the electron Rabi oscillates onto the right
dot, the energy on the left dot begins to decrease and at
the end of the pulse becomes zero and then asymptotes
to its intial value as the initial conditions of the system
are established again at late times. The Rabi oscillations
of the system become more evident for the case of a 3π-
pulse, as seen in Fig. 7, where we have again plotted the
internal energy density ρUsys(x, t) [Eq. (34)] for both the
dots and first sites of both reservoirs.
The energy at the first site in the left reservoir
ρUsys (1L, t) (Fig. 6a) rises and falls in response to changes
in the energy on Dot 1. This “in-phase with D1” behav-
ior can be understood by noting first that ρHS(1L, t)=0
so that ρUsys (1L, t) = ρHSR (1L, t). HSR models the cova-
lent bond between the dot and the first site on the chain
which is equally shared between the two. Negative (posi-
12
(a)ρU(1L,t) [E2]
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
(b)ρU(D1,t) [E2]
-1.0
-0.5
0.0
0.5
(c)ρU(D2,t) [E2]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(d)ρU(1R,t) [E2]
0 20 40 60 80 100 120
0.000
0.005
0.010
0.015
0.020
0.025
Time [E2-1]
2L
∞
1L D1D2 1R2R
∞
1
FIG. 6: Left Panel: Plots of Internal Energy as a function of time for the same double quantum dot system as in
Sec. V B, now coupled specifically to semi-infinite tight-binding chains. The left and right dots are labeled as D1 and
D2, respectively, and the chain sites on the left and right reservoirs are labelled 1L, 2L,... and 1R, 2R,...,
respectively. Only the site 1Lis directly coupled to D1 and only the site 1Ris directly coupled to D2. The Internal
Energy is denoted by ρU(1L, t) on the first site on the left chain, by ρU(D1, t) on the left dot, by ρU(D2, t) on the
right dot, and by ρU(1R, t) on the first site on the right chain. Note that the energy scale for ρU(1L, t) and
ρU(1R, t) is different from that for ρU(D1, t) and ρU(D2, t) in the figure. Right Panel: Schematic representation of
the system. The solid red box denotes the spatial extent of the degrees of freedom of the System Hamiltonian HS(t)
and the dashed blue box denotes the spatial extent of the Internal Energy operator Usys(t) = HS(t) + HS R. The
energies are in units of the on-site energy of the right dot E2and time is in units of E−1
2.
tive) bond energies correspond to bonding (anti-bonding)
character of the Dot-Reservoir interface. The bond en-
ergy tends to zero at long times as the system relaxes to
its initial state wherein the occupancy of Dot 1 is nearly
unity so that the hybridization with the left Reservoir
tends to zero. The oscillatory behavior during the relax-
ation is a signature of the Rabi oscillations between the
two dots D1 and D2, which are enhanced in frequency
(but suppressed in amplitude) when the pulse is inacti-
vated.
The energy on the right dot ρUsys (D2, t) (Fig. 6c) rises
as the electron Rabi oscillates onto it from the left dot
13
(a)ρU(1L,t) [E2]
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
(b)ρU(D1,t) [E2]
-1.0
-0.5
0.0
0.5
(c)ρU(D2,t) [E2]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(d)ρU(1R,t) [E2]
0 20 40 60 80 100 120
0.000
0.005
0.010
0.015
0.020
Time [E2-1]
FIG. 7: The internal energy density ρU(x, t) plotted for the system depicted in the Right Panel of Fig. 6, but with
pulse duration set as a 3π-pulse instead of a π-pulse. The 1.5 periods of Rabi oscillations during the pulse (which is
now active up to a time of 11.78 in units of E−1
2) are quite evident in this case. Note that the energy scale for
ρU(1L, t) and ρU(1R, t) is different from that for ρU(D1, t) and ρU(D2, t) in the figure.
and begins to decrease as the electron tunnels out onto
the first site in the right reservoir, eventually relaxing
back to its zero initial value. The energy on the first
site in the right reservoir ρUsys(1R, t) (Fig. 6d) can be
understood similarly to that on site 1Lwithin the bond-
ing picture discussed in the previous paragraph. As the
electron tunnels onto 1Rfrom the right dot the internal
energy there increases followed by an asymptotic decrease
to its initial value again almost mimicking the temporal
behavior at D2.
VI. CONCLUSIONS
In this work, we have clarified a key question in Quan-
tum Thermodynamics: Can the First Law of Thermo-
dynamics be formulated in an unambiguous, experimen-
tally meaningful, and quantum mechanically consistent
fashion for open quantum systems driven far from equi-
librium? We shed light on this question by employing an
insight from Mesoscopics—that far away from the local
driving and coupling of the system, the reservoirs stay
arbitrarily close to equilibrium—allowing us to unam-
biguously identify an internal energy operator and con-
sequently a First Law for such systems. The thermody-
namic partitioning so introduced identifies the internal
14
energy of the system as the sum of the system Hamil-
tonian and the full coupling Hamiltonian describing the
System-Reservoir interface Usys(t) = HS(t) + HSR .
We then derived fully general expressions for all the
terms appearing in the First Law using the formalism
of non-equilibrium Green’s functions (NEGF). In our
analysis, the system-reservoir coupling can be arbitrarily
strong, and our analysis can be readily extended to the
general case where the system, coupling, and reservoir
are all explicitly time-dependent (see Appendix B). Fur-
thermore, our formal results incorporate electronic and
phononic degrees of freedom in both the system and the
reservoir as well as electron-electron, electron-phonon,
and phonon-phonon interactions in the system. In our
analysis, the external electromagnetic field coupling to
the electron-phonon system is treated classically, which
allows for absorption and stimulated emission of energy
quanta, but does not allow spontaneous emission or ra-
diative heat transfer [62].
We also studied the spatio-temporal distribution of the
internal energy of a strongly-driven time-dependent open
quantum system, shedding light on the internal dynamics
of the system as well as on the evolution of the system-
reservoir interface. The interface contribution to the in-
ternal energy of the system was shown to be localized at
the junctions between the system and the reservoirs. For
the case of a metallic reservoir in the nearest-neighbor
tight-binding model, with only nearest-neighbor bonds
to the system, the interface energy was found to be en-
tirely localized on the system-reservoir bonds. The sign
of the interface energy can vary from negative to posi-
tive, corresponding to bonding or antibonding character
of the interface, respectively.
We applied our formal results to a strongly driven
quantum machine utilizing Rabi Oscillations between
states of a double quantum dot coupled to metallic reser-
voirs. We presented a full thermodynamic analysis for
the machine’s operation as both an electrochemical pump
and a heat engine, illustrating the utility of our theo-
retical framework and the power of our computational
approach based on NEGF.
It should be remarked that although the First Law
clearly holds at the level of quantum statistical averages,
provided the energetics are partitioned properly, one can-
not expect it to hold at the level individual quantum
trajectories, since the operators for internal energy, heat
current, and chemical power (among others) do not com-
mute, and hence these are incompatible observables [20].
It is hoped that our very general derivation of the First
Law of Thermodynamics in time-dependent open quan-
tum systems, illustrated with applications of our theory
to specific examples of model quantum thermal machines,
will help to resolve once and for all the controversy over
the validity of First Law of Thermodynamics in open
quantum systems.
VII. ACKNOWLEDGMENTS
It is a pleasure to acknowledge several helpful discus-
sions with Caleb M. Webb, Marco A. Jimenez-Valencia,
and Carter S. Eckel during various stages of this work.
We also thank J. M. Van Ruitenbeek for suggesting the
valuable generalization of incorporating phonons in our
model. This work was partially supported by the U.S.
Department of Energy (DOE), Office of Science under
Award No. DE-SC0006699.
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Appendix A: Derivation of the NEGF formulas for all First Law quantities
In this appendix, we give the details of the derivation of the electronic heat current formula in terms of the System
Green’s functions using the Nonequilibrum Green’s function (NEGF) formalism [26,27,49–51]. (The calculation of
the interface internal energy [Eq. (49)] proceeds in a nearly parallel fashion.)
To derive the heat current formula [Eq. (40)], we begin with the electronic heat current given in terms of expectation
values in Eq. (20). We then define the lesser Tunneling Green’s functions [the equal time version of these was defined
in Sec. IV as G<,el
tun,kn(t, t)]:
G<,el
nk (t, t1) = ihc†
k(t1)dn(t)i,(A1)
and
G<,el
kn (t, t1) = ihd†
n(t1)ck(t)i,(A2)
noting that at equal times G<,el
nk (t, t) = −[G<,el
kn (t, t)]∗. With this, we can write the heat current as
IQ
α,el(t) = −2
~Re"X
k∈α,n
(k−µα)Vel
knG<,el
nk (t, t)#.(A3)
The NEGF prescription tells us that G<,el
nk (t, t0) will be a component of the Contour-Ordered Tunneling Green’s
function
GT
nk(τ , τ0) = −ihT (dn(τ)c†
k(τ0))i,(A4)
where τis the contour time. The equation of motion of the Contour-Ordered Tunneling Green’s function is
−i∂
∂τ 0GT
nk(τ , τ0) = kGT
nk(τ , τ0) + X
m
GT
nm(τ , τ0)(Vel
km)∗,(A5)
where GT
nm(τ , τ0) = −ihT (dn(τ)d†
m(τ0))iis the Contour-Ordered System Green’s function and we get
GT
nk(τ , τ0) = X
mZdτ1GT
nm(τ , τ1)(Vel
km)∗gT
k(τ1, τ 0),(A6)
17
by using the equation of motion of the uncoupled Contour-Ordered reservoir Green’s function gT
k(τ, τ 0) =
−ihck(τ), c†
k(τ0)i. The physical lesser component of the Contour-Ordered Tunneling Green’s function [Eq. (A6)]
can be found using the so-called Langreth rules [26,27], giving
G<,el
nk (t, t0) = X
mZdt1(Vel
km)∗[GR,el
nm (t, t1)g<
k(t1, t0) + G<,el
nm (t, t1)gA
k(t1, t0)] .
(A7)
The lesser and advanced uncoupled Reservoir Green’s functions are given by
g<
k(t, t0) = ihc†
k(t0)ck(t)i=if(0
k) exp(−ik(t−t0)) (A8)
and
gA
k(t, t0) = iθ(t0−t)h{ck(t), c†
k(t0)}i =iθ(t0−t) exp(−ik(t−t0)) ,(A9)
respectively, where
f(0
k) = 1
eβα(0
k−µ)+ 1 (A10)
is the Fermi-Dirac distribution function. Putting this all together, we can write
IQ
α,el(t) = 2
~Im"X
k∈α,n,m
(k−µα)Zt
−∞
dt1{e−ik(t1−t)Vel
kn(Vel
km)∗[G<,el
nm (t, t1) + fα(k)GR,el
nm (t, t1)]}#,
(A11)
which, upon defining the tunneling width-matrix as in Eq. (39), allows us to turn the momentum sum into an energy
integral and can be written, by going over to the matrix notation, as
IQ
α,el(t) = 2
~Z∞
−∞
dE
2πZt
−∞
dt1(E−µα)ImTr{e−iE(t1−t)Γel
α(E)[G<,el(t, t1) + fα(E)GR,el (t, t1)]},(A12)
which is exactly Eq. (40). The phononic heat current Eq. (42) can be obtained in an analogous manner in terms
of phononic Green’s functions G<,ph and GR,ph, starting from Eq. (22). The interfacial contribution to the internal
energy, hHSR i[Eq. (49)], can be derived in a similar fashion.
The internal energy density ρUsys(x, t) [Eq. (34)] can also be evaluated in terms of the system Green’s function. Since
ρHS(x, t) is already given in terms of the system Green’s function in Eq. (50), we need to evaluate only ρHSR (x, t) [Eq.
(51)]. Again, by employing the same procedure used for deriving the heat current equation [Eq. (A12)] just discussed,
we get
ρHSR (x, t) = Z∞
−∞
dE
2πZt
−∞
dt1Im{hx|e−iE(t1−t)Γel
α(E)[G<,el(t, t1) + fα(E)GR,el (t, t1)]|xi} .(A13)
Within the framework of NEGF, the Retarded Green’s function GR(t, t0), which describes the dynamics of the
system, can be computed using the Dyson Equation
GR(t, t0) = GR
0(t, t0) + Z∞
−∞
dt1Z∞
−∞
dt2GR
0(t, t1)ΣR(t1, t2)GR(t2, t0),(A14)
where GR
0(t, t0) is the Green’s function for the non-interacting and uncoupled system and the Self-Energy function
which encodes the interactions and tunneling is given by
ΣR(t, t0)=ΣR
int + ΣR
tun(t, t0),(A15)
where ΣR
int is the retarded interaction self-energy of the system (for example, the Coulomb self-energy) and the
retarded tunneling self-energy is given by
ΣR
tun(t, t0) = X
αZ∞
−∞
dE
2πe−iE(t−t0)[Λα(E)−i
2Γα(E)] ,(A16)
18
with Λα(E) = PR∞
−∞
dE0
2π
Γα(E0)
E−E0, where Pdenotes the Principal Value [see, for instance, Eq. (2.26) in [27]]. The
Lesser Green’s function G<(t, t0), which describes the occupation of the system, can be computed using the Keldysh
Equation
G<(t, t0) = Z∞
−∞
dt1Z∞
−∞
dt2GR(t, t1)Σ<(t1, t2)GA(t2, t0),(A17)
where we have omitted the so-called memory term [see, for instance, Eq. (5.11) in [26]] since it is not relevant for the
protocols we consider in this article, and the lesser Self-Energy function is
Σ<(t, t0)=Σ<
int +iX
αZ∞
−∞
dE
2πe−iE(t−t0)fα(E)Γα(E).(A18)
The two-body Green’s function G(2),el(t, t) appearing in Eq. (48) can be evaluated directly using the Bethe-Salpeter
equation [27], or the contribution from electron-electron interactions to Eq. (48) can be evaluated using the Coulomb
self-energy and the one-body Green’s function. The electron-phonon Green’s function G(2),el−ph(t, t) and the higher-
order phonon Green’s functions such as G(3),ph(t, t), also appearing in Eq. (48), can be evaluated using the well-
developed theory of electron-phonon and phonon-phonon interactions [63,64].
Appendix B: Generalization to Time-Dependent reservoirs and coupling
We generalize our results from section IV to the case of a fully time-dependent Hamiltonian H(t) where, in addition
to the System, the Coupling and Reservoirs also acquire explicit time-dependence i.e. HS R →HSR (t) and HR→
HR(t) so that
H(t) = HS(t) + HSR (t) + HR(t).(B1)
For notational simplicity, we will again treat only electronic degrees of freedom here, but our results hold with full
generality for phononic degrees of freedom as well. The time-dependent System Hamiltonian is then given by
HS(t) = X
n,m [H(1)
S(t)]nmd†
ndm+ [H(2)
S]nmd†
nd†
mdmdn!.(B2)
The Reservoirs are modeled with
HR(t) = X
k∈α,α
k(t)c†
kck,(B3)
where, following [54], the time dependence of the Reservoir energy levels is assumed to take the form k(t) = 0
k+∆α(t),
where 0
kis the unperturbed reservoir energy and ∆α(t) is a rigid time-dependent energy shift. The System-Reservoir
Coupling is given by
HSR (t) = X
k∈α,α,n
(Vkn(t)c†
kdn+h.c.).(B4)
The aim again is to compute the particle and heat currents flowing into the reservoirs and see what the First
Law of thermodynamics looks like for this setup. The particle current was worked out as a generalization of the
Meir-Wingreen formula in [54] and is given by
IN
α(t) = 2
~Z∞
−∞
dE
2πZt
−∞
dt1ImTr{e−iE(t1−t)Γα(E;t1, t)[G<(t, t1) + fα(E)GR(t, t1)]},(B5)
where a generalized time-dependent tunneling width matrix Γα(E;t1, t) is identified as
[Γα(E;t1, t)]nm=X
k∈α
2πδ(E−k)Vn(E , t)V∗
m(E, t1)eiRt1
tdt2∆α(t2),
(B6)
19
with Vn(E, t) = Vkn(t) when E=k.
Just as in Sec. III [Eq. (19)], the heat current flowing into reservoir αcan be evaluated by applying the fundamental
thermodynamic identity dE =T dS+µdN to that reservoir. This identity still applies since the external driving voltage
applied to the reservoir ∆α(t) is at microwave frequencies or below, so that the timescale of variation is long compared
to the signal propagation time from the system into the reservoir, for typical experimental setups. The reservoirs can
thus be assumed to be in quasi-equilibrium and the identification of heat as d¯Q=T dS in the driven reservoirs is still
valid:
IQ
α(t)≡Tα
d
dtSα=d
dthHR,α(t)i − µα(t)d
dthNαi,(B7)
where µα(t) = µ0
α+ ∆α(t) (with µ0
αdenoting the chemical potential of the reservoir in steady state). Following the
same steps that lead to Eq.(A3) in Appendix A, we obtain
IQ
α(t) = −2
~Re"X
k∈α,n
(0
k−µ0
α)Vkn(t)G<
nk(t, t)#,(B8)
which upon evaluating G<
nk(t, t0) exactly as in appendix A, gives
IQ
α(t)= 2
~Im"X
k∈α,n,m
(0
k−µ0
α)Zt
−∞
dt1{e−i0
k(t1−t)Vkn(t)V∗
km(t1)eiRt1
tdt2∆α(t2)[G<
nm(t, t1) + fα(E)GR
nm(t, t1)]}#,
(B9)
where we have used the fact that the uncoupled (but driven) reservoir Green’s functions are now given by
g<
k(t, t0) = if(0
k) exp(−iZt
t0
dt1k(t1)) (B10)
and
gA
k(t, t0) = iθ(t0−t) exp(−iZt
t0
dt1k(t1)) .(B11)
The heat current is thus given, upon going over to the matrix notation, by
IQ
α(t) = 2
~Z∞
−∞
dE
2πZt
−∞
dt1(E−µα)ImTr{e−iE(t1−t)Γα(E;t1, t)[G<(t, t1) + fα(E)GR(t, t1)]},(B12)
where we have relabeled µ0
αas µα. Again, the particle and heat currents flowing into reservoir αcan be written
compactly as
I(ν)
α(t) = 2
~Z∞
−∞
dE
2πZt
−∞
dt1(E−µα)νImTr{e−iE(t1−t)Γα(E;t1, t)[G<(t, t1) + fα(E)GR(t, t1)]},(B13)
where ν= 0 gives the particle current and ν= 1 gives the heat current.
The rate of external work done ˙
Wext is now given by
˙
Wext(t) = −i
~Tr[˙
HS(t)G<(t, t) + ˙
HR(t)g<(t, t) + ˙
HSR (t)G<
tun(t, t)] .(B14)
We can then see that the Internal Energy operator is still identified as in Eq. (28) (with HSR replaced with HSR (t))
and the First Law Equation [Eq. (27)] still holds exactly, even when the Reservoirs and Coupling are explicitly
time-dependent.
