![FIG. 4. Exact results for the Rabi-Hamiltonian in the weakcoupling limit: (a) Inversion σx(t), (b) density σz(t), and (c) exact Kohn-Sham potential aKS(t) in the case of coherent states (in spirit of panel 3 in Fig. 4 in Ref. [55]).](/figures/fig-4-exact-results-for-the-rabi-hamiltonian-in-the-65t0dtbk.webp)
![FIG. 6. The nonrelativistic (NR) limits do not depend on the order of operations. First performing the limit and then quantizing with the equal-time (anti)commutation relations (ETCR) leads to the same (Pauli-Fierz) Hamiltonian as the opposite ordering [see (a)]. Further, first performing the limit in the current and then calculating the equations of motion (EOM) leads to the same result as performing the limit directly on the relativistic EOM [see (b)].](/figures/fig-6-the-nonrelativistic-nr-limits-do-not-depend-on-the-3vnq3xva.webp)
PHYSICAL REVIEW A 90, 012508 (2014)
Quantum-electrodynamical density-functional theory: Bridging quantum optics
and electronic-structure theory
Michael Ruggenthaler,
1,*
Johannes Flick,
2
Camilla Pellegrini,
3
Heiko Appel,
2
Ilya V. Tokatly,
3,4
and Angel Rubio
2,3,†
1
Institut f
¨
ur Theoretische Physik, Universit
¨
at Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria
2
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin-Dahlem, Germany
3
Nano-Bio Spectroscopy Group and ETSF Scientific Development Centre, Departamento de F
´
ısica de Materiales,
Centro de F
´
ısica de Materiales CSIC-UPV/EHU-MPC and DIPC, Universidad del Pa
´
ıs Vasco UPV/EHU, E-20018 San Sebasti
´
an, Spain
4
IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain
(Received 22 April 2014; published 9 July 2014)
In this work, we give a comprehensive derivation of an exact and numerically feasible method to perform
ab initio calculations of quantum particles interacting with a quantized electromagnetic field. We present a
hierarchy of density-functional-type theories that describe the interaction of charged particles with photons and
introduce the appropriate Kohn-Sham schemes. We show how the evolution of a system described by quantum
electrodynamics in Coulomb gauge is uniquely determined by its initial state and two reduced quantities.
These two fundamental observables, the polarization of the Dirac field and the vector potential of the photon
field, can be calculated by solving two coupled, nonlinear evolution equations without the need to explicitly
determine the (numerically infeasible) many-body wave function of the coupled quantum system. To find
reliable approximations to the implicit functionals, we present the appropriate Kohn-Sham construction. In the
nonrelativistic limit, this density-functional-type theory of quantum electrodynamics reduces to the density-
functional reformulation of the Pauli-Fierz Hamiltonian, which is based on the current density of the electrons
and the vector potential of the photon field. By making further approximations, e.g., restricting the allowed modes
of the photon field, we derive f urther density-functional-type theories of coupled matter-photon systems for the
corresponding approximate Hamiltonians. In the limit of only two sites and one mode we deduce the appropriate
effective theory for the two-site Hubbard model coupled to one photonic mode. This model system is used to
illustrate the basic ideas of a density-functional reformulation in great detail and we present the exact Kohn-Sham
potentials for our coupled matter-photon model system.
DOI: 10.1103/PhysRevA.90.012508 PACS number(s): 31.70.Hq, 71.15.−m, 31.15.ee, 42.50.−p
I. INTRODUCTION
The behavior of elementary charged particles, such as elec-
trons and positrons, is governed by quantum electrodynamics
(QED). In this theory, the quantum particles interact via the
exchange of the quanta of light, i.e., the photons [1–3]. Thus,
in principle we have to consider the quantum nature of the
charged particles as well as of the light field. However, in
several important cases we can focus almost exclusively on
either the charged particles or the photons, while employing
crude approximations for the other degrees of freedom.
In condensed matter physics and quantum chemistry, the
quantum nature of light can usually be ignored and the inter-
action between the charged quantum particles is approximated
by the instantaneous Coulomb interaction. However, even
then the resulting quantum mechanical equations (usually the
many-body Schr
¨
odinger equation), where the electromagnetic
fields are treated classically through the solution of the
Maxwell equations, are solvable only for very simple systems.
This lies ultimately in our incapability of handling the huge
number of degrees of freedom of many-particle systems and
consequently in our inability to determine the many-body
states. This so-called many-body problem spawned a lot of
interest into the question as to whether one can devise a
closed set of equations for reduced quantities which do not
*
michael.ruggenthaler@uibk.ac.at
†
angel.rubio@ehu.es
involve the explicit solution of the full quantum mechanical
equations and in which the many-body correlations can be
approximated efficiently. Pursuits in this direction have led to
various approaches such as, among others, many-body Green’s
function theories [4,5], (reduced) density-matrix theories [6],
and density-functional theories [7–10]. These approaches
differ in the complexity of the reduced quantity, which is
used to calculate the various observables of interest. Especially
density-functional theories, which are based on the simplest of
those (functional) variables, the one-particle density (current),
have proven to be exceptionally successful [11]. Their success
can be attributed to the unprecedented balance between
accuracy and numerical feasibility [12], which allows us at
present to treat several thousands of atoms [13]. Although the
different flavors of density-functional theories cover most of
the traditional problems of physics and chemistry (including
approaches that combine classical Maxwell dynamics with the
quantum particles [14–18]), by construction these theories can
not treat problems involving the quantum nature of light.
In quantum optics, on the other hand, the focus is on the
photons, while usually simple approximations for the charged
particles are employed, e.g., a few-level approximation.
However, even in this situation the solution of the resulting
equations [19,20] is only possible in simple cases (again
due to the large number of degrees of freedom) and usually
simplified model Hamiltonians, e.g., the Dicke model realized
in a cavity [21–23], are employed to describe these physical
situations. Already the validity of these effective Hamiltonians
and their properties can be a matter of debate [24–26] and
1050-2947/2014/90(1)/012508(26) 012508-1 ©2014 American Physical Society
MICHAEL RUGGENTHALER et al. PHYSICAL REVIEW A 90, 012508 (2014)
often further simplifications are adopted such as the Jaynes-
Cummings model in the rotating-wave approximation. The
rapid progress in quantum-optical experiments, on the other
hand, especially in the field of cavity QED [27–30] and circuit
QED [31,32], allows us to study and control multiparticle
systems ultrastrongly coupled to photons [33–36], where
such a simple approximative treatment is no longer valid
[37]. This new regime of light-matter interaction is widely
unexplored for, e.g., molecular physics and material sciences
[38]. Possibilities such as altering and strongly influencing the
chemical reactions of a molecule in the presence of a cavity
mode or setting the matter into new nonequilibrium states
with novel properties, e.g., light-induced superconductivity
[39], arise. Specifically in such situations, an oversimplified
treatment of the charged particles may no longer be allowed
and an approach that considers both, the quantum nature of
the light field as well as of t he charged particles, is needed.
In this work, we give a comprehensive derivation of an
exact and numerically feasible method that generalizes ideas
of time-dependent density-functional theory (TDDFT). This
method bridges the gap between the above two extreme cases
and provides a scheme to perform ab initio calculations of
quantum particles coupled to photons. The electron-photon
generalization of TDDFT in describing nonrelativistic many-
electron systems coupled to photon modes of mesoscopic
cavities was introduced in Ref. [40]. Here, we provide a
general framework describing fully coupled electron-photon
systems in most possible regimes and systems ranging from
effective model Hamiltonians to strongly relativistic cases,
which has been introduced in Refs. [41,42]. For clarity,
we divide the following presentation in two parts: We first
demonstrate the basic ideas in a simple model system and
then show how these concepts can be used in the case of
general coupled matter-photon problems. A summary of all
findings of this work for the time-dependent density-functional
description of QED at different levels of approximations,
namely, the basic variables, initial conditions, and funda-
mental Kohn-Sham multicomponent equations is given in
Appendix F.
We start considering a simple model system for charged
matter coupled to photons: the two-site Hubbard model
interacting with one photonic mode. By employing density-
functional ideas, we show how one can solve this quantum-
mechanical problem without the need to explicitly calculate
the complex many-body wave function. Instead, we derive
equations of motion for a pair of reduced quantities from which
all physical observables can by determined. We demonstrate
that these equations have unique solutions and can be used to
calculate the basic reduced quantities (here the basic pair of
reduced quantities is t he charge density of the particle and the
potential induced by the photons) of the coupled problem.
Therefore, we here reformulate the coupled matter-photon
problem in terms of an effective theory, that we call in
the following a model of quantum electrodynamical density-
functional theory (QEDFT). Since an explicit calculation of the
coupled wave function is not needed, this approach allows us to
determine properties of the matter-photon system in a numeri-
cally feasible way. We introduce a new Kohn-Sham scheme to
approximate the unknown functionals in the basic equations
of motion and present results for a simple approximation. We
compare these results to the exact Kohn-Sham functionals and
identify shortcomings and indicate improvements.
Based on the ideas developed in the first part of this work,
we repeat the steps illustrated in our example but now we
construct a density-functional reformulation f or the full theory
of QED [41,42]. We show that a straightforward approach
based on the current and the potential leads to problems and
that a consistent density-functional reformulation of QED has
to be based on the polarization and the potential which is gen-
erated by the photons. This approach to the fully coupled QED
problem we denote as relativistic QEDFT, and we present the
corresponding Kohn-Sham construction and give the simplest
approximation to the unknown functionals. In the following,
we then demonstrate how relativistic QEDFT reduces in the
nonrelativistic limit to its nonrelativistic version of the corre-
sponding nonrelativistic Hamiltonian. By employing further
approximations on the matter system or on the photon field, a
family of different approximate QEDFTs is introduced, which
are consistent with their respective approximate Hamiltonians.
At this level, we recover the theory of Ref. [40]. In lowest
order, we rederive the model QEDFT of the first part of this
work. Therefore, we demonstrate how all different flavors of
QEDFT are just approximations to relativistic QEDFT in the
same manner as different physical Hamiltonians are merely
approximations to the QED Hamiltonian. Furthermore, by
ignoring all photonic degrees of freedom, we find the standard
formulations of TDDFT which are extensively used in the
electronic-structure community [9,10].
Outline. In Sec. II, we investigate the QEDFT reformulation
of a simple model of one particle coupled to one mode in great
detail. The developed ideas are then employed in Sec. III to
derive a QEDFT reformulation of QED. In Sec. IV,weshow
how all different QEDFT reformulations are approximations
to relativistic QEDFT. We conclude and give an outlook in
Sec. V.
II. MODEL OF QEDFT
In this section, we introduce the basic formulation and un-
derlying ideas of QEDFT. By employing a model Hamiltonian,
we can almost exclusively focus on the density-functional
ideas that allow a reformulation of the wave-function problem
in terms of simple effective quantities. We first identify the
pair of external and internal variables and then show that both
are connected via a bijective mapping. As a consequence, all
expectation values become functionals of the initial state and
the internal pair. This allows for a reformulation of the problem
in terms of two coupled equations for the internal pair. Then,
we introduce the Kohn-Sham construction as a way to find
approximations to the unknown functionals, and show first
numerical results.
To describe the dynamics of particles coupled to photons,
we solve an evolution equation of the form
ic∂
0
|(t)=
ˆ
H (t)|(t) (1)
for a given initial state |
0
. Here, ∂
0
= ∂/∂x
0
with x
0
= ct
and the standard relativistic (covariant) notation x ≡ (ct,r)
(see also Appendix A for notational conventions). The
012508-2
QUANTUM-ELECTRODYNAMICAL DENSITY-FUNCTIONAL . . . PHYSICAL REVIEW A 90, 012508 (2014)
corresponding Hermitian Hamiltonian has the general form
ˆ
H (t) =
ˆ
H
M
+
ˆ
H
EM
+
1
c
d
3
r
ˆ
J
μ
(x)
ˆ
A
μ
(x)
+
1
c
d
3
r
ˆ
J
μ
(x)a
μ
ext
(x) +
ˆ
A
μ
(x)j
μ
ext
(x)
, (2)
where the dependence of the total Hamiltonian on t indicates
an explicit time dependence. Here, the (time-independent)
Hamiltonian
ˆ
H
M
describes the kinetic energy of the particles,
i.e., how they would evolve without any perturbation, and
ˆ
H
EM
is the energy of the photon field. The third term describes
the coupling between the (charged) particles and the photons
by the charge current
ˆ
J
μ
and the Maxwell-field operators
ˆ
A
μ
(where the Einstein sum convention with the Minkowski
metric g
μν
≡ (1, −1, −1, −1) is implied and Greek letters
refer to four vectors, e.g., μ ∈{0,1,2,3}, while Roman
letters are restricted to spatial vectors only, e.g., k ∈{1,2,3}).
This term is frequently called the minimal-coupling term and
arises due to the requirement of a gauge-invariant coupling
between the particles and the photon field. The specific form
of the operators
ˆ
J
μ
and
ˆ
A
μ
depends on the details of the
physical situation. Finally, the last term describes how the
particles interact with a (in general time-dependent) classical
external vector potential a
μ
ext
and how the photons couple to a
(in general time-dependent) classical external current j
μ
ext
.
While we usually have no control over how the particles and
photons evolve freely or interact, i.e., the first three terms of the
Hamiltonian (2), we have control over the preparation of the
initial state |
0
and the external fields (a
μ
ext
,j
μ
ext
). Therefore,
all physical wave functions, i.e., found by solving Eq. (1), can
be labeled by their initial state and external pair (a
μ
ext
,j
μ
ext
):
0
,a
μ
ext
,j
μ
ext
; t
.
However, for any but the simplest systems the (numerically
exact) solution of Eq. ( 1) is not feasible. Even if we decouple
the matter part from the photons by employing the Coulomb
approximation (i.e., describing the exchange of photons by the
respective lowest-order propagator), the resulting problem is
far from trivial.
A. Two-level system coupled to one mode
In this section, we introduce a simple model of charged
particles coupled to photons. We discuss the basic concepts
of a density-functional-type reformulation, identify the pair of
conjugate variables, and then deduce the fundamental equa-
tions of motion on which we base our QEDFT reformulation.
In order to demonstrate the basic ideas of a QEDFT,
we employ the simplest yet nontrivial realization of one
charged particle coupled to photons: a two-site Hubbard model
coupled to one photonic mode. The resulting Hamiltonian
(see Appendix E for a detailed derivation) reads as
ˆ
H (t) =
ˆ
H
M
+
ˆ
H
EM
−
λ
c
ˆ
J
ˆ
A −
1
c
[
ˆ
Ja
ext
(t) +
ˆ
Aj
ext
(t)], (3)
where the kinetic energy of the charged particle is given by
ˆ
H
M
=−t
kin
ˆσ
x
,
and the energy of the photon mode reads as
ˆ
H
EM
= ω
ˆ
a
†
ˆ
a.
Here, t
kin
is the hopping parameter between the two sites, ω
is the frequency of the photonic mode, and ( ˆσ
x
, ˆσ
y
, ˆσ
z
)arethe
Pauli matrices that obey the usual fermionic anticommutation
relations. The photon creation and annihilation operators (
ˆ
a
†
and
ˆ
a, respectively) obey the usual bosonic commutation
relations. The current operator
1
is defined by
ˆ
J = eωl ˆσ
z
,
where l is a characteristic length scale of the matter part and λ
is a dimensionless coupling constant.
The operator for the conjugate potential
2
is given by
ˆ
A =
c
2
0
L
3
1/2
(
ˆ
a +
ˆ
a
†
)
√
2ω
,
where L is the l ength of the cubic cavity. Further, the current
operator couples to the external potential a
ext
(t) and the
potential operator to the external current j
ext
(t). These are
the two (classical) external fields that we can use to control the
dynamics.
If we then fix an initial state |
0
and choose an external
pair (a
ext
,j
ext
), we usually want to solve Eq. (1) with the
Hamiltonian given by Eq. (3). The resulting wave function,
given in a site basis |x for the charged particle and a Fock
number-state basis |n for the photons
|([
0
,a
ext
,j
ext
]; t)=
2
x=1
∞
n=0
c
xn
(t)|x⊗|n,
depends on the initial state and the external pair (a
ext
,j
ext
).
Thus, by varying over all possible combinations of pairs
(a
ext
,j
ext
), we scan through all physically allowed wave
functions starting from a given initial state. Hence, we
parametrize the relevant, i.e., physical, time-dependent wave
functions by |
0
and (a
ext
,j
ext
). Since the wave functions
have these dependencies, also all derived expressions, e.g., the
expectation values for general operators
ˆ
O
O([
0
,a
ext
,j
ext
],t) =(t)|
ˆ
O|(t),
are determined by the initial state and the external pair
(a
ext
,j
ext
).
The idea of an exact effective theory such as QEDFT is
now that we identify a different set of fundamental variables,
which also allow us to label the physical wave functions (and
their respective observables), and that we have a closed set of
1
To be precise,
ˆ
J is proportional to the dipole-moment operator, i.e.,
it is connected to the zero component
ˆ
J
0
of the general four-current
operator
ˆ
J
μ
. To highlight the analogy in structure to the general case
discussed in the later sections, we give it the units of a current and
denote it by
ˆ
J .
2
To be precise,
ˆ
A is actually proportional to the electric field as
can be seen from the derivations in Appendix E. This is because
in the course of approximations, one employs the length gauge and
thus transforms from the potential to the electric field. However, to
highlight the analogy in structure to the general case discussed in the
later sections, we give it the units of the potential and denote it by
ˆ
A.
012508-3
MICHAEL RUGGENTHALER et al. PHYSICAL REVIEW A 90, 012508 (2014)
equations for these new (functional) variables, which do not
involve the full wave functions explicitly. Such a functional-
variable change is similar to a coordinate transformation,
say from Cartesian coordinates to spherical coordinates. This
can only be done if every point in one coordinate system is
mapped uniquely to a point in the other coordinate system.
For a functional-variable change, we thus need to have a
one-to-one correspondence, i.e., bijective mapping, between
the set of (allowed) pairs (a
ext
,j
ext
) and some other set of
functions (while we keep the initial state fixed). To identify
the simplest new functional variables, one usually employs
arguments based on the Legendre transformation [43]. That is
why these new functional variables are often called conjugate
variables. We will consider this approach in the next sections
where we investigate general QEDFT, and also show how one
can determine the conjugate variables of this model system
from more general formulations of QEDFT. For this simple
model, we simply state that a possible pair of conjugate
variables is (J,A). In the next subsection, we show that this
functional variable-transformation is indeed allowed, i.e.,
|([
0
,J,A]; t).
The main consequence of this result is that from only
knowing these three basic quantities, we can (in principle)
uniquely determine the full wave function. Accordingly, every
expectation value becomes a unique functional of |
0
and
(J,A). Thus, instead of trying to calculate the (numerically
expensive) wave function, it is enough to determine the
internal pair (J,A) for a given initial state. An obvious route to
then also find a closed set of equations for these new variables is
via their respective equations of motion. These equations will
at the same time be used to prove the existence of the above
change of variables, i.e., that the wave function is a unique
functional of the initial state and the internal pair (J,A).
To find appropriate equations, we first apply the Heisenberg
equation of motion once and find
i∂
0
ˆ
J =−i
2t
kin
eωl
c
ˆσ
y
,
i∂
0
ˆ
A =−i
ˆ
E,
where
ˆ
E = i
√
ω
2
0
L
3
(
ˆ
a −
ˆ
a
†
). Yet, these two equations are not
sufficient for our purposes: we need equations that explicitly
connect (a
ext
,j
ext
) and (J,A). Therefore, we have to go to the
second order in time
(
i∂
0
)
2
ˆ
J =
4t
2
kin
2
c
2
ˆ
J − λ
ˆ
n
ˆ
A −
ˆ
na
ext
(t), (4)
(
i∂
0
)
2
ˆ
A = k
2
ˆ
A −
μ
0
c
L
3
(λ
ˆ
J + j
ext
(t)), (5)
where
ˆ
n =
4t
kin
(eωl)
2
2
c
3
ˆσ
x
, (6)
k =
ω
c
, and
0
=
1
μ
0
c
2
. Here, Eq. (4) is the discretized version
of ∂
2
t
n of standard TDDFT [44,45], and Eq. (5)isthe
inhomogeneous Maxwell equation for one-photon mode [40].
B. Foundations of the model QEDFT
In the previous section, we have stated that (J,A) and
(a
ext
,j
ext
) are the possible conjugate pair of the model
Hamiltonian (3). In this section, we want to demonstrate that
indeed this holds true and that we can perform a variable
transformation from the external pair (a
ext
,j
ext
)
3
to the internal
pair (J,A). What we need to show is, that for a fixed initial
state |
0
, the mapping
(a
ext
,j
ext
)
1:1
↔ (J,A)(7)
is bijective, i.e., if (a
ext
,j
ext
) = (
˜
a
ext
,
˜
j
ext
) then necessarily for
the corresponding expectation values (J,A) = (
˜
J,
˜
A). To do
so, we first note that in the above equations of motion every
expectation value is by construction a functional of (a
ext
,j
ext
)
for a fixed initial state
∂
2
0
J ([a
ext
,j
ext
]; t)
=−
4t
2
kin
2
c
2
J ([a
ext
,j
ext
]; t) + λ
ˆ
n
ˆ
A([a
ext
,j
ext
]; t)
+n([a
ext
,j
ext
]; t)a
ext
(t), (8)
∂
2
0
A([a
ext
,j
ext
]; t) =−k
2
A([a
ext
,j
ext
]; t)
+
μ
0
c
L
3
[
λJ ([a
ext
,j
ext
]; t) + j
ext
(t)
]
,
(9)
i.e., they are generated by a time propagation of |
0
with
a given external pair (a
ext
,j
ext
). Suppose now that we fix the
expectation values of the internal variables (J,A), i.e., we
do not regard them as functionals but rather as functional
variables. Then, the above Eqs. (8) and (9) become equations
for the pair ( a
ext
,j
ext
) that produce the given internal pair (J,A)
via propagation of the initial state |
0
, i.e.,
∂
2
0
J (t) =−
4t
2
kin
2
c
2
J (t) + λ
ˆ
n
ˆ
A([a
ext
,j
ext
]; t)
+ n([a
ext
,j
ext
]; t)a
ext
(t), (10)
∂
2
0
A(t) =−k
2
A(t) +
μ
0
c
L
3
[
λJ (t) + j
ext
(t)
]
. (11)
Obviously, these equations can only have a solution, if the
given internal variables are consistent with the initial state,
i.e.,
J
(0)
=
0
|
ˆ
J |
0
,J
(1)
=−
2t
kin
eωl
c
0
|ˆσ
y
|
0
, (12)
A
(0)
=
0
|
ˆ
A|
0
,A
(1)
=−
0
|
ˆ
E|
0
. (13)
3
In the general case, different external pairs can be physically
equivalent, and thus one usually considers equivalence classes of
external pairs. In this model system, however, we have already fixed
these degrees of freedom. First, we have fixed the gauge of a
ext
(a purely time-dependent constant), since a
ext
corresponds to the
potential difference of sites 1 and 2. Second, any freedom with respect
to the external current has been fixed since j
ext
corresponds to the
spatial integral of the current (which makes any divergence zero). A
detailed discussion of these points can be found in Sec. III B.
012508-4
QUANTUM-ELECTRODYNAMICAL DENSITY-FUNCTIONAL . . . PHYSICAL REVIEW A 90, 012508 (2014)
Here, we have used the definition
A
(α)
= ∂
α
0
A(t)
t=0
, (14)
and every internal pair (J,A) that we consider is subject to
these boundary conditions. Thus, the mapping (7) is bijective,
if the corresponding Eqs. (10) and (11), which connect the
internal pair (J,A) with the external pair (a
ext
,j
ext
), allow for
one and only one solution pair.
Let us first note that for a given pair (J,A), Eq. (11) uniquely
determines
4
the external current j
ext
by
j
ext
(t) =
L
3
μ
0
c
∂
2
0
+ k
2
A(t) − λJ (t). (15)
Thus, the original problem reduces to the question as to
whether Eq. (10) determines a
ext
(t) uniquely. The most general
approach to answer this question is via a fixed-point procedure
similar to Ref. [46]. In the case of a discretized Schr
¨
odinger
equation such as Eq. (3), it should also be possible to apply
a rigorous approach based on the well-established theory
of nonlinear ordinary differential equations [45]. However,
for simplicity we follow Ref. [40] and employ the standard
strategy of [47] which restricts the allowed external potentials
a
ext
to being Taylor expandable in time, i.e.,
a
ext
(t) =
∞
α=0
a
(α)
ext
α!
(ct)
α
. (16)
From Eq. (10) we can find the Taylor coefficients of J (if they
exist) by
J
(α+2)
=−
4t
2
kin
c
2
2
J
(α)
+λ
ˆ
n
ˆ
A
(α)
+
α
β=0
α
β
n
(α−β)
a
(β)
ext
, (17)
where the terms
ˆ
n
ˆ
A
(α)
and n
(α)
are given by their respec-
tive Heisenberg equations at t = 0 and only contain Taylor
coefficients of a
(β)
ext
for β<α.
Now, assume that we have two different external potentials
a
ext
(t) =
˜
a
ext
(t). This implies, since we assumed Taylor ex-
pandability of a
ext
and
˜
a
ext
, that there is a lowest order α for
which
a
(α)
ext
=
˜
a
(α)
ext
. (18)
For all orders β<α(even though the individual J
(β)
and
˜
J
(β)
might not exist), it necessarily holds that
J
(β+2)
−
˜
J
(β+2)
= 0. (19)
But, for α we accordingly find that
J
(α+2)
−
˜
J
(α+2)
= n
(0)
a
(α)
ext
−
˜
a
(α)
ext
= 0, (20)
provided we choose the initial state such that n
(0)
= 0. Con-
sequently, J (t) =
˜
J (t) infinitesimally later for two different
external potentials a
ext
(t) =
˜
a
ext
(t). Therefore, Eq. (10) allows
4
Note that due to the initial conditions A
(0)
and A
(1)
, one can not add
a nonzero homogeneous solution (∂
2
0
+ k
2
)f (t) = 0totheexternal
current.
only one solution and the mapping (a
ext
,j
ext
) → (A,J )is
bijective.
As a consequence, since every expectation value of the
quantum system becomes a functional of the internal pair
(J,A), in the above Eqs. (10) and (11), we can perform a
change of variables and find
∂
2
0
J (t) =−
4t
2
kin
2
c
2
J (t) + λ
ˆ
n
ˆ
A([J,A]; t ) + n([J,A]; t)a
ext
(t),
(21)
∂
2
0
A(t) =−k
2
A(t) +
μ
0
c
L
3
[
λJ (t) + j
ext
(t)
]
. (22)
These coupled evolution equations have unique solutions
(J,A) for the above initial conditions (12) and (13). Therefore,
we can, instead of solving for the many-body wave function,
solve these nonlinear coupled evolution equations for a given
initial state and external pair (a
ext
,j
ext
), and determine the
current and the potential of the combined matter-photon
system from which all observables could be computed. This
is an exact reformulation of t he model in terms of the current
and the potential of the combined system only.
C. Kohn-Sham approach to the model QEDFT
In the previous section, we have derived a QEDFT refor-
mulation in terms of the current and the potential. While the
equation that determines the potential A is merely the classical
Maxwell equation, and every term is known explicitly, the
equation for the current contains implicit terms. Therefore, to
solve these coupled equations in practice, we need to give
appropriate explicit approximations for t he implicit terms.
Approximations based on (J,A) directly would correspond
to a Thomas-Fermi–type approach to the model. As known
from standard density-functional theory, such approximations
are in general very crude and hard to improve upon. A more
practical scheme i s based on the Kohn-Sham construction,
where an auxiliary quantum system is used to prescribe
explicit approximations. However, t he numerical costs of
a Kohn-Sham approach compared to a Thomas-Fermi–type
approach are increased.
The details of the Kohn-Sham construction depend on the
actual auxiliary quantum system one wants to employ. The
only restriction of the auxiliary system is that one can control
the current and the potential by some external variables. Thus,
one could even add further (unphysical) external fields to
make approximations of the coupled quantum system easier.
However, here we only present the simplest and most natural
Kohn-Sham scheme, which i s to describe the coupled quantum
system by an uncoupled quantum system. To this end, we
assume that we can find a factorized initial state
|
0
=|M
0
⊗|EM
0
that obeys the same initial conditions as the coupled problem
(12) and (13). Especially, if the initial state of the coupled
system is the same as in the uncoupled problem, then this
condition is trivially fulfilled. In a next step, we note that for
the uncoupled system subject to the external pair (a
eff
,j
eff
), the
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