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Journal of Chemical Theory and Computation

J Chem Theory Comput. 2017 April 11; 13(4): 1616–1625.

Published online 2017 March 9. doi: 10.1021/acs.jctc.6b01126

PMCID: PMC5390309

Johannes Flick,^{}*^{†} Heiko Appel,^{}*^{†} Michael Ruggenthaler,^{}*^{†} and Angel Rubio^{}*^{†}^{‡}

Received 2016 November 19

In this work, we illustrate the recently introduced concept of the cavity Born–Oppenheimer approximation [Flick et al. PNAS 2017, 10.1073/pnas.1615509114] for correlated electron–nuclear-photon problems in detail. We demonstrate how an expansion in terms of conditional electronic and photon-nuclear wave functions accurately describes eigenstates of strongly correlated light-matter systems. For a GaAs quantum ring model in resonance with a photon mode we highlight how the ground-state electronic potential-energy surface changes the usual harmonic potential of the free photon mode to a dressed mode with a double-well structure. This change is accompanied by a splitting of the electronic ground-state density. For a model where the photon mode is in resonance with a vibrational transition, we observe in the excited-state electronic potential-energy surface a splitting from a single minimum to a double minimum. Furthermore, for a time-dependent setup, we show how the dynamics in correlated light-matter systems can be understood in terms of population transfer between potential energy surfaces. This work at the interface of quantum chemistry and quantum optics paves the way for the full ab initio description of matter-photon systems.

Recent experimental progress
has made it possible to study light-matter
interactions in the regime of strong and ultrastrong light-matter
coupling. Experiments from exciton-polariton condensates,^{2,3} near-field spectroscopy,^{4,5} plasmon-mediated single-molecule
strong coupling,^{6} superconducting qubit
circuits,^{7} quantum information,^{8} direct measurements of vacuum fluctuations,^{9} and chemistry in optical cavities^{10−12} open now the path to shape the emerging correlated light-matter
interactions with the goal toward a new control of material properties.
In this new field that has been driven in particular by experiment,
traditional theoretical methods from either quantum chemistry or quantum
optics can lose their applicability. On the one hand, traditional
quantum chemistry concepts such as the Born–Oppenheimer (BO)
approximation^{13,14} or electronic structure methods
such as Hartree–Fock theory,^{15} coupled-cluster
theory,^{16} or density-functional theory
(DFT)^{17} have been originally designed to
treat approximately correlated electron–nuclear problems but
are not capable to correctly account for the quantum nature of light.
On the other hand, concepts from quantum optics typically describe
the quantum nature of the light field in great detail but fail in
describing more complex dynamics of matter due to the often employed
simplification to a few levels.^{18,19} To fill this gap, in
this work, we generalize a well-established concept from quantum chemistry,
namely the Born–Oppenheimer approximation, to the realm of
correlated light-matter interactions for systems in optical high-Q
cavities.

First theoretical studies in similar direction, e.g.
the modification
of the molecular structure under strong light-matter coupling,^{20} the nonadiabatic dynamics of molecules in optical
cavities,^{21,22} or the cavity-controlled chemistry,^{23} have already been conducted.

Since the complexity of an exact
ab initio description of such
correlated many-body systems that contain electronic, nuclear, and
photonic (Fermionic and bosonic) degrees of freedom scales exponentially
with system size, approximate descriptions have to be employed for
any realistic system. Recently, the concept of DFT has been generalized
to electron-photon problems and was termed quantum-electrodynamical
density-functional theory.^{24−27} This theory maps the complicated many-body problem
into a set of nonlinear equations for the electronic and photonic
degrees of the densities/currents that facilitates the treatment of
such complex systems, similarly as standard DFT has done over the
years to deal with correlated electronic systems. Still for this theory
to be applicable, accurate functionals for combined light-matter systems
have to be developed to calculate approximate effective potentials
and observables. In this work, we use an alternative approach, the
cavity Born–Oppenheimer (CBO)^{1} approximation,
that allows to construct approximate wave functions to the exact eigenstates
for such problems. The cavity Born–Oppenheimer approximation
has recently been introduced in ref (1), and in this paper we derive the theory in a
complete manner and give explicit examples to highlight its applicability
for general electron–nuclear-photon systems. This work is structured
into three sections: (i) First, the theoretical framework is introduced
where we demonstrate how the concept of the Born–Oppenheimer
approximation can be generalized to matter-photon coupled systems.
(ii) We apply this theoretical framework to study a prototypical electron-photon
system, where the photon couples resonantly to an electronic transition.
(iii) The last section is devoted to a model system of an electron,
a nuclei, and photons, where a photon mode couples to a vibrational
excitation.

In what follows and without loss of generality, we
describe the electron–nuclear-photon problem in Coulomb gauge,
dipole approximation, and the Power-Zienau-Woolley frame.^{28,29} Our system of interest contains *n*_{e} electrons, *n*_{n} nuclei, and *n*_{p} quantized
photon modes, e.g. the matter is located in an optical high-Q cavity.
Strong light-matter coupling is obtained, once the light-matter coupling
is stronger than the dissipation of the system due to e.g. cavity
losses. For simplicity, we neglect dissipative channels in the following.
[Since in this work dissipation is neglected, we find modifications
of the eigenstates of the matter-photon system with respect to the
bare matter eigenstates for all nonvanishing matter-photon coupling
strengths.] The original derivation of the Born–Oppenheimer
approximation is outlined e.g. in ref (14) for the specific case of electrons and ions,
and here we extend it to the photon case. In general, the correlated
electron–nuclear-photon Hamiltonian^{1,18,25,30,31} can be written as follows [Throughout this work,
we assume SI units, unless stated otherwise.]

1

consisting of the electronic Hamiltonian *Ĥ*_{e} with *n*_{e} electrons of mass *m*_{e}

2

the nuclear Hamiltonian *Ĥ*_{n} with *n*_{n} nuclei
each with possibly different individual masses *m*_{i} and charges *Z*_{i}

3

where _{n} and *Ŵ*_{n} are the nuclear kinetic
energy and nuclear interaction, respectively.
The electron–nuclear interaction Hamiltonian *Ĥ*_{en} is given by

4

and the cavity photon Hamiltonian *Ĥ*_{p} with *n*_{p} quantized photon modes of frequency
ω_{α} takes the form

5

The displacement field operators consist of the usual photon creation and
annihilation operators and [_{α}, _{α′} ] = *i*δ_{α,α′}. Furthermore,
the _{α} are directly proportional
to the electric displacement field operator of the α-th photon
mode^{30,31} at the charge-center of the system by the
connection _{α} = ϵ_{0}ω_{α}**λ**_{α}_{α} and the _{α} are proportional to the magnetic field. In eq 5, the sum runs from 1 to
2*n*_{p}, to correctly account
for the two possible polarization directions of the electromagnetic
field. The last three terms in eq 1 describe the light-matter interaction Hamiltonian.
The first term is the explicit electron-photon interaction in the
dipole approximation

6

with the total electronic dipole moment **X**_{e} = −∑_{i=1}^{ne}*e***r**_{i} and the matter-photon coupling
strength **λ**_{α}.^{25,31} The second term gives the explicit nuclear-photon interaction, again
in the dipole approximation

7

with the total nuclear dipole moment **X**_{n} = ∑_{i=1}^{nn}*Z*_{i}*e***R**_{i}, and the last term describes
the quadratic dipole-self-interaction
term

8

where **X** now describes
the total
dipole moment of the system, i.e. **X** = **X**_{e} + **X**_{n}. We then introduce the following abbreviations

Under this change of notation, we can rewrite eq 1 in the following form

9

In general, we are interested
in calculating eigenstates Ψ_{i}(**r**, **R**, *q*) and eigenvalues *E*_{i} of the particular problem.
These states then give us access to any observable of interest. To
calculate these quantities, we have to solve the full Schrödinger
equation of the correlated electron–nuclear-photon problem
that is given by

10

where the Hamiltonian *Ĥ* is
given by eq 1. Obtaining
general solutions to the Schrödinger
equation of eq 10 is
an ungrateful task. [We note that in free space eq 10 has no square-integrable eigenstates in
the charge neutral case due to its translational invariance. Hence
one either has to go into a comoving frame, e.g., a center-of-mass
frame, and consider the corresponding reduced Hamiltonian, or one
has to use a confining potential to localize the molecule.] In practice,
the Schrödinger equation is barely solved exactly but only
approximately. One of such approximate methods is the cavity Born–Oppenheimer
approximation^{1} that is capable to partially
decouple the electronic degrees of freedom from the nuclear and photonic
degrees of freedom. In electron–nuclear problems, such an adiabatic
decoupling procedure is commonly assumed^{14} and well justified for low lying states, e.g. the ground state.
However, severe limitations are known that require going beyond the
adiabatic treatment by including nonadiabatic electron–nuclear
terms, e.g. at conical intersections.^{32}

In this work we decouple the electronic degrees of freedom
from the nuclear and photon degrees of freedom. This allows us, on
the one hand, to simplify the problem much more than if we decoupled
the nuclear from the electronic and photonic degrees of freedom, as
has been done in refs (20 and 33), and the
additional photonic degree of freedom becomes formally equivalent
to a nuclear (phononic) degree of freedom. The latter can be understood
as follows: in eq 5,
the photon degree of freedom is written in terms of a quantum harmonic
oscillator that contains a kinetic energy term _{p} and a potential term *Ŵ*_{p} that are both connected via the virial
theorem.^{34} In this sense, we can regard
the description of the photon modes as formally equivalent to the
description of the nuclei. As a consequence, we can apply traditional
methods to solve the electron–nuclear problem to the generalized
electron–nuclear-photon problem. One of these methods is the
Born–Oppenheimer approach that in the cavity accumulates an
additional photonic degree of freedom, reminiscent of the nuclear
degree of freedom. Thus, the same arguments for the validity of the
usual Born–Oppenheimer approximation that apply in the case
of nuclear motion also apply for any extended system, as they do not
depend on the details of the interactions that produce the potential-energy
surfaces. In practice, the main problem for the standard Born–Oppenheimer
approximation is to solve the resulting electronic equation, while
simple approximations to the nuclear equation, such as harmonic approximations,
are often sufficient. On the other hand, a decoupling of the electronic
degrees of freedom provides most flexibility for the applications
that we consider, e.g. a single electron coupled to one mode. From
a physical perspective, however, this decoupling scheme seems counterintuitive
at a first glance. The usual simplified argument for the decoupling
of the nuclear from the electronic degrees of freedom is that the
nuclei move “slowly” compared to the electrons, i.e.,
the kinetic-energy contribution is negligible, and hence a classical
approximation seems reasonable. In the case of quantized photons,
the term _{p} in eq 5 is related to the square
of the magnetic field operator, thus _{α} is proportional to the magnetic field. Therefore, the
magnetic field can be interpreted as an analogue to the nuclear velocity
in real-space, although the conjugate momentum is defined in the *q*_{α}-space of the harmonic oscillator. The coordinate *q*_{α} describes the displacement of the harmonic
oscillator of the photon mode with specific energy ω_{α}. In this sense, while the usual Born–Oppenheimer approximation
is justified by “slow” nuclei, we can justify the cavity
Born–Oppenheimer approximation, if the magnetic field in the
photon mode is “small”. This is in particular the case
for all eigenstates, due to _{t}*q*_{α} = *p*_{α}. Along these lines, we conclude that the cavity Born–Oppenheimer
approximation is applicable, if *p*_{α} remains small, thus the magnetic field remains “small”.
If this is the case, the time-derivative of *q*_{α} remains “small”, thus the electric displacement
field changes only “slowly” over time, and the electrons
can adapt “quasi-instantaneously” to these “slow”
changes of the electric displacement field. That this approach can
indeed give highly accurate results will be demonstrated in the following.

In this section, we derive the approximate cavity Born–Oppenheimer states to eq 10. This goal is achieved in three successive steps. First, we solve the electronic part of the eq 10, where we consider explicitly all terms containing an explicit electronic contribution. This electronic Schrödinger equation has only a parametric (conditional) dependence on the nuclear and field degrees of freedom, or alternatively nuclear and field coordinates enter the electronic equation as c-numbers. In principle, the electronic Schrödinger equation has to be solved for every possible combined nuclear and photon-field configuration, and the eigenvalues of the electronic Schrödinger equation then enter the nuclear and photon-field Schrödinger equation through the emerging potential-energy surfaces. Having solved both equations, we can then construct the approximate cavity Born–Oppenheimer states in a factorized manner. To obtain the approximate cavity Born–Oppenheimer states, as a first step, we solve the electronic Schrödinger equation

11

for each fixed set of nuclear coordinates **R** and photon displacement coordinates *q*. For each fixed set of (**R**, *q*), the electronic
eigenfunctions of eq 11 {ψ_{j}(**r**, **R**, *q*)} form a complete basis
in the electron many-particle Hilbert space. In the electronic Schrödinger
equation of eq 11, (**R**, *q*) enter the electronic cavity Born–Oppenheimer Hamiltonian
as (classical) parameters, thus the eigenvalues ϵ_{j} also parametrically depend on **R**, *q*. For each
fixed set of (**R**, *q*), we can then expand (also known as the Born-Huang
expansion^{35}) the exact many-body wave function
Ψ_{i}(**r**, **R**, *q*) that is a solution to the full Schrödinger equation
of eq 10 as

12

Here, the
exact wave function is decomposed
into sums of product states consisting of an electronic wave function
ψ_{j}(**r**, **R**, *q*) and a nuclear-photon wave function χ_{ij}(**R**, *q*). The latter is obtained by solving
the following equation

13

where _{n}(**R**) and _{p}(*q*) are given by eqs 3 and 5, respectively.
The eigenvalues *E*_{i} of eq 13 are the exact correlated
eigenvalues of eq 10. The term in the second line of eq 13 describes the nonadiabatic coupling between cavity
Born–Oppenheimer potential energy surfaces (PES). The cavity
Born–Oppenheimer approximation now neglects the offdiagonal
elements in the nonadiabatic coupling terms of eq 13. Then eq 13 can be rewritten in a much simpler form

14

where the newly generalized cavity
PES *V*_{j}(**R**, *q*)
are given
explicitly by

15

The
first two terms are the nuclear and the
photon potentials of eqs 3 and 5, and all anharmonicity in the PES can
be attributed to the electron-photon, electron–nuclear, nuclear–nuclear,
and nuclear-photon interaction contained in eq 1. Furthermore, the eigenvalues *E*_{i} of eq 14 are an approximation to the exact correlated eigenvalues
and provide by the variational principle an upper bound. With this
reformulation, we have the advantage that we can solve the electronic
Schrödinger equation of eq 11 and the nuclear-photon Schrödinger eq 14 separately. The ground-state
Ψ_{0} in the cavity Born–Oppenheimer approximation
then becomes

16

and accordingly for the excited states. In
Born–Oppenheimer calculations for systems that only contain
electrons and nuclei often the harmonic Born–Oppenheimer approximation
is carried out^{14} that can be realized by
expanding *V*_{j}(**R**, *q*) around its minimum value and in this way even simplifies the problem
further. In the harmonic approximation, we have to solve eq 11 not for all possible
values of (**R**, *q*), but only at the minimum of ϵ_{j}(**R**, *q*). However, in this work, we do not
apply the harmonic approximation to correctly demonstrate the full
capacity of the cavity Born–Oppenheimer concept.

Before we introduce our examples, let us comment on the expectable accuracy of the cavity Born–Oppenheimer states when decoupling electronic from photonic and nuclear degrees of freedom. Our simplified physical arguments for the decoupling scheme so far have been that the nuclei are “slow” and the magnetic-field contribution small, such that we can neglect the corresponding kinetic terms in the equation for the electronic subsystem. However, the decisive quantities that indicate the quality of this approach are the nonadiabatic coupling elements of eq 13 and the distance between the potential-energy surfaces. If these elements are small and the potential-energy surfaces are far apart, we can expect a good quality of the approximate cavity Born–Oppenheimer states. This argument is similar to standard Born–Oppenheimer treatment that loses its validity at crossing of eigenvalues, i.e. conical intersections.

In the following, we now want to illustrate the concept of the cavity Born–Oppenheimer approximation for two specific setups. We numerically analyze first a model system consisting of a single electron coupled resonantly to a photon mode. In this example, the nuclei can be understood as frozen, leading to an external potential acting on the electronic degrees of freedom. This model will allow us to study the decoupling mechanism introduced for the correlated electron-photon interaction in detail. In the second example, we then analyze a model system that contains electron–nuclear-field degrees of freedom. Here, potential-energy surfaces emerge that have nuclear-photon nature.

In this section, we illustrate the
concept of the cavity Born–Oppenheimer
approximation for a simple coupled electron-photon model system. The
system of interest is a model system for a GaAs quantum ring^{36} that is located in an optical cavity and thus
coupled to a single photon mode.^{31} The
model features a single electron confined in two-dimensions in real-space
(**r** = *r*_{x}**e**_{x} + *r*_{y}**e**_{y}) interacting
with the single photon mode with frequency ω_{α} = 1.41 meV and polarization direction **e**_{α}= (1,1). The polarization direction enters via the electron-photon
coupling strength, i.e. **λ**_{α} = λ_{α}**e**_{α} and depends on the
specific experimental setup. The photon mode frequency is chosen to
be in resonance with the first electronic transition. We depict the
model schematically in Figure Figure11 (a). The bare electron ground-state *n*_{λ=0}(**r**) has a ringlike structure shown in Figure Figure11 (b) due to the Mexican-hat-like
external potential that is given by

17

with parameters ω_{0} = 10 meV, *V*_{0} = 200 meV, *d* = 10 nm, and *m*_{0} = 0.067*m*_{e}^{36} and shown
in Figure Figure11 (c). For
the single electron, we employ a two-dimensional grid of *N* = 127 grid points in each direction with *Δx* = 0.7052 nm. In contrast, we include the photons for the exact calculation
in the photon number eigenbasis, where we include up to 41 photons
in the photon mode.

(a) Model for the GaAs quantum ring in an optical cavity.
(b) Bare
ground-state electron density *n*_{λ=0} in the external potential that is shown in (c).

For the cavity Born–Oppenheimer calculations, we calculate
the photons also on an uniform real-space grid (q-representation)
with *N* = 41 with *Δq* = 6.77 fs^{2} and construct the projector
from the uniform real-space grid to the photon number states basis
explicitly. This projector can be calculated by employing the eigenstates
of the quantum harmonic oscillator in real-space. For a more detailed
discussion of the model system, we refer the reader to refs (31 and 36). Since this model can be solved by exact diagonalization in full
Fock space,^{37} all exact results shown in
the following have been calculated employing the full correlated electron-photon
Hamiltonian.^{1,25,30,31} For this model, the potential-energy surfaces
from eq 15 can be calculated
explicitly as

18

In Figure Figure22 (a), we show the PES surfaces *V*_{j}({*q*_{α}})
for the weak-coupling regime of λ_{α} = 0.0034
meV^{1/2}/nm. We find that all PES have a strong harmonic
nature, due to the dominant _{α}^{2} term in eq 18. The eigenvalues ϵ_{j} and the integral in the last line of eq 18 are the corrections
to the harmonic potential. In this case, both are rather small for
all excited-state surfaces in the weak-coupling regime, i.e. for the
ground-state surface adiabatic term in the last line of eq 18 is around 2 orders of magnitude
smaller than ϵ_{0}. In general, a harmonic correction
that can be obtained by calculating the second derivative at the minimum
value will shift the frequency of the photon mode. We define as harmonic
approximation to eq 18

19

where *q*_{j,0} is the minimum value of the *j*-th PES eq 18. In the weak-coupling
regime, we find _{α} ≈ ω_{j,α}. All corrections beyond the second
derivative of these terms are then called the anharmonic corrections.

Born–Oppenheimer
potential energy surfaces *V*_{j} for a correlated electron-photon problem
in (a) weak coupling with λ_{α} = 0.0034 meV^{1/2}/nm and (b) strong coupling λ_{α} =
0.1342 meV^{1/2}/nm.

We find the lowest cavity PES that is the ground-state PES
shown
in black, well separated from the first and second excited cavity
PES that are shown in solid red and dotted blue. The first and second
excited cavity PES are close to being degenerate. This 2-fold degeneracy
has its origin in the two-dimensional external potential, similar
to the *s*/*p* degeneracy in the hydrogen
atom. In Figure Figure22 (b),
we show the cavity PES surfaces in the strong-coupling regime with
λ_{α} = 0.134 meV^{1/2}/nm. [In this work,
the weak and strong coupling regimes are defined by the relation of
the Rabi splitting to the photon frequency ω_{α}, thus accordingly to the definitions in the Rabi model, see e.g.
ref (38) and references
therein.] While the second PES shown in blue and the fourth potential
energy surface shown in yellow keep the harmonic shape, in the lowest
cavity PES shown in black and the third cavity PES shown in solid
red, two new minima with a double-well structure appear. [Note that
if we would like to express this electron-dressed photon system in
terms of the original creation and annihilation operators, we will
need new combinations of these operators, i.e., photon-interaction
terms. Physically these interaction terms describe the coupling between
photons mediated via the electron.] The minima of the cavity PES are
strongly shifted away from the equilibrium position at the origin.
This electron-dressed potential for the photon modes induces a new
vacuum state with two maxima. Since the cavity PES is symmetric, the
vacuum state still has a displacement observable of *q*_{α}=0, i.e., we have a stable vacuum
with zero field. However, with respect to the bare vacuum the other
observables, e.g., the vacuum fluctuations, will clearly change. Furthermore,
we find for the harmonic approximation in the ground-state cavity
PES, _{0,α} ≈ 0.8 ω_{α}, hence an effective softening of the photon mode in the ground-state
cavity PES with the strong displacement of *q*_{0,0} = 18.85 fs^{2}. A similar behavior has
been observed before in the context of polaron physics in the Holstein
Hamiltonian.^{39,40} We further analyze this transition
in Figure Figure33. In Figure Figure33 (a), we show how
the ground-state PES depends on the electron-photon coupling strength
λ_{α}. We find that for absent and weak coupling,
the ground-state surface can be well described by a single harmonic
potential that has the minimum at *q*_{α} = 0. If we increase the electron-photon coupling to strong coupling,
we find around λ_{α} = 0.044 meV^{1/2}/nm the splitting of the single-well structure to a double-well structure.
For strong coupling, e.g. λ_{α} = 0.1342 meV^{1/2}/nm this double-well structure becomes strongly pronounced.
In Figure Figure33 (b) and
(c), we plot the corresponding electron density *n*_{λ}(**r**) = ∫*dq*_{α}Ψ_{0,λ}^{*}(**r**,*q*_{α}) ×Ψ_{0,λ}(**r**,*q*_{α}) of the exact correlated ground state Ψ_{0,λ}(**r**,*q*_{α}) for different values of λ. In the weak-coupling regime, shown
in Figure Figure33 (b), we
find that the electron is only slightly distorted in comparison to
the ringlike structure of the bare electron ground state^{31} shown in Figure Figure11 (b). In contrast, in the strong coupling regime, shown
in Figure Figure33 (c), the
electron density becomes spatially separated and localized in direction
of the polarization direction of the quantized photon mode.

Left: (a) Ground-state
cavity PES for different coupling strengths
show an emerging displacement of the photon states. Right: electron
density in (a) the weak coupling regime for λ_{α} = 0.0034 meV^{1/2}/nm and (b) strong coupling for λ **...**

The consequences of the ground-state transition
identified in Figure Figure33 become also apparent
if we study the difference of the correlated and bare electron density.
Let us define the bare electron density. Here, we refer to the electron
density that is the ground-state of the external potential without
coupling to the photon mode or alternatively λ_{α} = 0, thus *n*_{λ=0}(**r**).
This density is shown in Figure Figure11 (b). Then we define *Δn*_{λ}(**r**) = *n*_{λ}(**r**) – *n*_{λ=0}(**r**). In Figure Figure44, we plot *Δn*_{λ}(**r**) as a function of the electron-photon coupling strength λ_{α}. In the weak-coupling limit, shown in Figure Figure44 (a) for λ_{α} = 0.0034 meV^{1/2}/nm, we find that the electron density
is slightly distorted such that in the correlated density more density
is accumulated perpendicular to the polarization direction of the
photon mode compared to the bare electron density. However, once the
strong-coupling regime is approached, we also identify a transition
in *Δn*_{λ}(**r**). In
the strong coupling regime, that is entered in Figure Figure44 (b)-(d), the ground-state electron density
is reoriented until ultimately in Figure Figure44 (e) the electron density is arranged in
direction of the polarization direction of the photon mode, up to
higher strong-coupling regions shown in Figure Figure44 (f).

Difference of the correlated ground-state
electron density to the
bare electron density (*Δn*_{λ} = *n*_{λ} – *n*_{λ=0}) from the weak- to the strong-coupling limit.
The red color refers to surplus density regions, while the blue **...**

The additional insights from the ground-state transition
can be
obtained by evaluating the exact correlated electron-photon eigenvalues.
In Figure Figure55, we plot
the exact eigenvalues from the weak- to the strong-coupling regime.
The ground-state energies are plotted by the black line and are increasing
for stronger coupling.^{1} For the first excited
state in the case of λ_{α} = 0 coupling, we find
a 3-fold degeneracy that is split once the electron-photon coupling
is introduced. For strong coupling the first-excited state (shown
in blue) and the ground-state become close leading to the splitting
of the electron-density shown in Figure Figure44. [We emphasize that this behavior is similar
to what is in molecular systems known as *static correlation* for e.g. stretched molecules.^{41}] Higher-lying
states show energy crossings that are typical for electron-photon
problems and have been previously observed e.g. in the Rabi model.^{38,42,43} We find not only allowed level
crossings at λ_{α} ≈ 0.031, 0.067, 0.113
meV^{1/2}/nm but also an avoided level crossing at λ_{α} ≈ 0.055 meV^{1/2}/nm between the fifth
and sixth eigenvalue surface. In the Rabi model, level crossings are
used to define transition from the weak, strong, ultrastrong,^{44} and deep-strong coupling regime.^{45} Similarly to the Rabi model,^{42} we find in the strong coupling regime a pairing of states
in terms of the energy. Two states each with different parity become
close to degeneracy. Since in the strong-coupling regime the interaction
terms in the Hamiltonian become dominant and we apply the interaction
in dipole coupling, the eigenstates of the full Hamiltonian become
close to the eigenstates of the dipole operator that are the parity
eigenstates. We can expect a different behavior beyond the dipole
coupling, e.g. if electric quadrupole and magnetic dipole coupling,
or higher multipolar coupling terms are also considered. In contrast
to the Rabi model,^{42} we find an overall
increase of the ground-state energy for increasing coupling strength.
This behavior is due to the inclusion of the quadratic dipole self-interaction
term of eq 8. In Figure Figure55, we indicate by
the dashed line, the ground-state transition discussed before. In
the coupling region indicated by (I), we find a single minimum in
the PES and *Δn* is located perpendicular to
the polarization direction, while in the coupling regime (II), we
find two minima and a double-well structure in the PES and *Δn* are located along the direction of the polarization
of the photon mode. The quality of the cavity Born–Oppenheimer
approximation is shown in Table 1 in terms of overlaps Ψ_{j}|Ψ_{j,CBO}^{2} between approximate and exact states. If the eigenenergies shown
in Figure Figure55 are well
separated as in the strong coupling regime for λ_{α} = 0.1342 meV^{1/2}/nm, then the cavity Born–Oppenheimer
approximation is well justified. For states that are close to degeneracy,
as e.g. the states #2 and #4 in the weak-coupling for λ_{α} = 0.0034 meV^{1/2}/nm, we find a lower quality.
However, this low quality could be improved by symmetry considerations.
Overall, we find a very high and sufficient quality of the approximate
energies and states in comparison to its corresponding exact values.

Exact
eigenvalues of the correlated electron-photon Hamiltonian
as a function of the electron-photon coupling parameter λ_{α}. The dashed line indicates the transition of *Δn*_{λ}(**r**) as discussed
in the main text.

**Exact Correlated
Energies E^{exact} (eV), Cavity BO Energies E_{CBO} (eV), and Overlap between Exact and Cavity
BO States
Depending on the Electron-Photon Coupling Strength λ_{α} Given in meV^{1/2}/nm^{a}**

The remaining
part of this section is concerned with the time-dependent
case. Here, we employ the full correlated electron-photon Hamiltonian
and choose as initial state a factorized initial state that consists
of the bare electronic ground state and a bare photon field in a coherent
state with *â*^{†}*â* = 4 where λ_{α} = 0.0034
meV^{1/2}/nm. This example is also the first time-dependent
example studied in ref (31). To numerically propagate the system, we use a Lanczos scheme and
propagate the initial state in 160000 time steps with *Δt* = 0.146 fs. In Figure Figure66, we briefly analyze this setup by evaluating the dipole moment *$\widehat{x}$* + *ŷ* in Figure Figure66 (a), the purity
γ = Tr(ρ_{ph}^{2}) that contains the reduced photon density
matrix ρ_{ph} and the Mandel *Q* parameter^{46} that is defined
as

20

in Figure Figure66 (b) and the photon occupation *â*^{†}*â*
in Figure Figure66 (c). In
the case
of the dipole moment of this example shown in Figure Figure66 (a), we find first regular Rabi oscillations
up to the maximum at *t* = 5 ps and around *t* = 10 ps, and we find the necklike feature^{47} typical for Rabi oscillations. In Figure Figure66 (b), we show the purity γ in dashed
black lines. The purity γ is a measure for the separability
of the many-body wave function into a product of an electronic and
a photon wave function. We find that γ is close to 1 up to *t* = 5 ps, which means that the many-body wave function is
close to a factorizable state. After *t* = 5 ps, γ
deviates strongly from 1, and the system is not factorizable anymore.
This dynamical buildup of correlation has also an effect on the nonclassicality
of the light-field visible in the Mandel *Q*-parameter
shown in Figure Figure66 (b)
in solid black lines. While initially *Q* ≈
0 that indicates the coherent statistics of the photon mode, after *t* = 5 ps also this observable deviates from 0 and nonclassicality
shows up. From Figure Figure66 (c), where we plot the photon number, we see that until *t* = 5 ps a photon is absorbed that is later re-emitted,
and, after *t* = 15 ps, we again observe photon absorption
processes. In the following, we analyze this dynamics of the correlated
electron-photon problem in terms of population in the cavity Born–Oppenheimer
surfaces calculated in Figure Figure22 (a). In Figure Figure77, we show the occupation of the photon number states in the first
cavity PES in (a) and the third cavity PES in (b). The values (P1,P3)
give the population of the first cavity PES and the third cavity PES,
respectively. All other cavity PES have populations which are an order
of magnitude smaller, since P1+P3 is close to 1 for all times. In Figure Figure77 (a), we find that
at the initial time *t* = 0 ps, the first cavity PES
is populated with a photon state, which has a coherent distribution
with *â*_{α}^{†}*â*_{α} = 4, which is in agreement with our initial
condition. During the time propagation, we observe a transfer of population
from the first cavity PES to the third cavity PES. In the first cavity
PES, we see until *t* = 9.3 ps a depletion of population,
while in the third cavity PES (Figure Figure77 (b)), we observe an increase of the population. After
this time, the population is again transferred back from the third
cavity PES to the first cavity PES (Rabi oscillation). However, not
only the amplitude of the population is changing but also the center
of the wave packets. In principle, if the same photon state would
be populated in the two different cavity PES, the system could still
be factorizable. For small times, up to *t* = 5 ps
the center of the wave packet in the first cavity PES remains close
to its initial value. Later it changes to smaller photon numbers,
which indicates photon absorption. We can conclude that the dynamics
of the many-body system is dominated by the population transfer from
the first cavity PES to the third cavity PES and vice versa. While
for this example, a good approximate description may be a two-surface
approximation reminiscent of the Rabi model,^{42} we expect a different behavior for more complex cavity Born–Oppenheimer
surfaces e.g. in many-electron problems, multiphoton modes, or strong-coupling
situations.

The second system that
we analyze is the Shin-Metiu model^{48,49} coupled to
cavity photons. Without coupling to photon modes, this
system exhibits a conical intersection between Born–Oppenheimer
surfaces and has been analyzed heavily in the context of correlated
electron–nuclear dynamics,^{50} exact
forces in nonadiabatic charge transfer,^{51} or nonadiabatic effects in quantum reactive scattering,^{52} to mention a few. In our case, we place the
system, consisting of three nuclei and a single electron into a optical
cavity, where it is coupled to a single mode that is in resonance
with the first vibrational excitation. The outer two nuclei are fixed,
and the free electron and the nuclei are restricted to one-dimension.
The model is schematically depicted in Figure Figure88. The Hamiltonian of such a system is given
by^{48,49}

21

where *Ĥ*_{p}, *Ĥ*_{pe}, *Ĥ*_{pn},
and *Ĥ*_{pen} are given
by eqs 5, 6, 7, and 8, respectively.
The electronic Hamiltonian reads

22

where *V*_{n}(*R*) is the
Coulomb interaction of the free
nuclei with the two fixed nuclei, *r* is the electronic
coordinate, and *R* is the nuclear coordinate. *V*_{e} is the sum of the electron
interaction with the three nuclei, i.e. three terms each of which
is of the following form^{49}

23

where *x* is the electron–nuclear
distance, and erf describes the error-function. We fix the nuclear
mass *M* to the mass of a hydrogen atom, choose *Z* = 1, and set the length *L* = 10 Å.
Furthermore, we use the dipole operators *X*_{e} = −*er* and *X*_{n} = *eR*. In the nuclear
dipole moment operator *X*_{n}, the two outer nuclei cancel each other due to their fixed positions
at ±*L*/2. Further *R*_{c} can be used to tune the energy difference Δ
between the ground-state and the first-excited state potential energy
surface. For the cavity Shin-Metiu model, we represent the electron
on a grid of dimension *N*_{r} = 140 with *Δr* = 0.4233 Å, and the nuclear
coordinate on a grid of dimension *N*_{R} = 280 with *ΔR* = 0.0265 Å, while
the photon wave function is expanded in the photon number eigenbasis,
where the mode can host up to 81 photons in the photon mode. To get
first insights on how the light-matter coupling is capable of changing
the chemical landscape of the system, in Figure Figure99, we calculate the ordinary PES surfaces
of eq 15 for the case
of *q*_{α} = 0. The solid red line shows
the ground-state energy surface, while the blue line shows the excited
state energy surface for *R*_{c} = 1.5 Å with ω_{α} = 72.5
meV and *R*_{c} = 1.75 Å
with ω_{α} = 69.3 meV. In both examples,
the photon frequencies ω_{α} correspond to the
first vibrational transition of the exact bare Hamiltonian. Next,
we tune the matter-photon coupling strength λ_{α} from the weak-coupling regime to the strong-coupling regime. The
corresponding cavity PES are shown in gray in Figure Figure99. The inset in the figures shows the energy
gap Δ depending on the matter-photon coupling strength λ_{α}. In the left figure, we choose the value *R*_{c} = 1.5 Å, and, in the case of λ_{α} = 0, we find well separated cavity Born–Oppenheimer
surfaces. The matter-photon coupling (chosen here from λ_{α} = 0 to λ_{α} = 82.55 eV^{1/2}/nm with a Rabi splitting Ω_{R} =
(*E*_{5}–*E*_{3})/ω_{α} = 43.81%) opens the gap significantly,
as shown in the inset. Additionally, for *R*_{c} = 1.5 Å, we find that the double-well structure
visible in the first-excited state becomes more pronounced for stronger
light-matter coupling. The right figure shows the results for *R*_{c} = 1.75 Å, where in the
field-free case a much narrower gap Δ is found. Introducing
the matter-photon coupling in the system from λ_{α}= 0 to λ_{α}= 84.48 eV^{1/2}/nm with
Ω_{R} = 64.04% also opens the gap significantly,
and we find a similar qualitative behavior as in the previous example
with the notable difference, that we observe in the present example
a similar single-well to double-well transition but now in the first-excited
state. However, since we restricted ourselves to a specific cut in
the full two-dimensional cavity Born–Oppenheimer surface by
choosing *q*_{α} = 0, Figure Figure99 does not show the full picture.
Therefore, in Figure Figure1010, we show the full two-dimensional cavity PES for *R*_{c} = 1.75 Å. In the figure, the *x*-axis shows the nuclear degree of freedom (*R*), while the *y*-axis shows the photonic degree of
freedom *q*_{α}. In the case of λ_{α} = 0, that is the upper panel in the figure, we find
that the photonic degree of freedom introduces harmonicity into the
surface. We also indicate the minima in the surfaces by white crosses.
In agreement with Figure Figure99, we find a double minimum for the ground-state cavity PES
and a single minimum for the excited state cavity PES. In the case
of strong-coupling that is shown in the lower panel of the figure,
we observe new emerging normal modes. These new normal modes are caused
by the entanglement of the matter and photon degrees of freedom and
are manifest in the displacement of the minima out of the equilibrium
positions. In the first-excited state surface in strong coupling,
we also observe a single-well to double-well transition, as observed
in the coupling to the electronic excitation and discussed in the
first part of this work. Here, we find that now two minima appear
in the first-excited state surface. If we adopt an adiabatic picture
we can conclude that now two new reaction pathways are possible from
the first excited state surface to the ground-state surface.

Molecule in
an optical cavity. The molecule is modeled by the Shin-Metiu
model^{48,49} that consists of three nuclei and a single
electron. Two of the nuclei are frozen at position *L*/2 and −*L*/2, respectively.

Potential energy surfaces in the cavity Born–Oppenheimer
approximation for the Shin-Metiu model. Increasing matter-photon coupling
strength opens the gap Δ between the ground-state cavity PES
and the first-excited cavity PES. Both plots **...**

Two-dimensional ground-state and first-excited state potential
energy surfaces in the cavity Born–Oppenheimer approximation
for the Shin-Metiu model in the case of λ_{α} =
0 (upper panel) and strong-coupling λ_{α} = 79.20
eV **...**

To conclude, we have seen how the photonic degrees of freedom
alter
considerably chemical properties in a model system containing electronic,
nuclear, and photonic degrees of freedom. We have identified the change
of traditional Born–Oppenheimer surfaces, gap opening, and
transitions from single-well structures to double-well structures
in the first-excited state surface from first principles. The gap
opening can be connected to recent experiments,^{53} where a reduction in chemical activity has been observed
for vibrational strong coupling.

In this paper, we introduced the concept of the
cavity Born–Oppenheimer
approximation for electron–nuclear-photon systems. We used
the cavity Born–Oppenheimer approximation to analyze the ground-state
transition in the system that emerges in the strong-coupling limit.
During this transition the ground-state electron density is split,
and the ground-state cavity PES obtains a double-well structure featuring
finite displacements of the photon coordinate. Furthermore, we illustrated
for a time-dependent situation with a factorizable initial state how
the complex correlated electron-photon dynamics can be interpreted
by an underlying back-and-forth photon population transfer from the
ground-state cavity PES to an excited-state cavity PES. In the last
section, we have demonstrated how this transition can also appear
in case of strong-coupling and vibrational resonance. Here, we find
that the first-excited state surface can obtain a double-well structure
leading to new reaction pathways in an adiabatic picture. In future
studies toward a full ab initio description for cavity light-matter
systems, where solving the electronic Schrödinger equation
of eq 11 by exact diagonalization
is not feasible, the density-functional theory for electron-photon
systems can be used.^{25,26} The discussed methods can be
still improved, e.g. along the lines of a more accurate factorization
method such as the exact factorization^{54−56} known for electron–nuclear
problems, or trajectory based methods^{50,57} can be applied
to simulate such systems dynamically. This work has direct implications
on more complex correlated matter-photon problems that can be approximately
solved employing the cavity Born–Oppenheimer approximation
to better understand complex correlated light-matter coupled systems.

We would like to thank C. Schäfer for a careful reading of the manuscript, and M.R. acknowledges insightful discussions with F.G. Eich. In addition, we thank the referees for contributing to significant improvements in the manuscript.

We acknowledge financial support from the European Research Council (ERC-2015-AdG-694097), Grupos Consolidados (IT578-13), by the European Union’s H2020 program under GA no. 676580 (NOMAD), COST Action MP1306 (EUSpec), and the Austrian Science Fund (FWF P25739-N27).

The authors declare no competing financial interest.

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