# Ultrafast modulation of vibrational polaritons for controlling the quantum
field statistics at mid-infrared frequencies

Johan F. Triana<EMAIL_ADDRESS>Department of Physics, Universidad de
Santiago de Chile, Av. Ecuador 3493, Estación Central, Santiago, Chile  Felipe
Herrera<EMAIL_ADDRESS>Department of Physics, Universidad de
Santiago de Chile, Av. Ecuador 3493, Estación Central, Santiago, Chile  ANID -
Millennium Science Initiative Program, Millennium Institute for Research in
Optics, Chile

###### Abstract

Controlling the quantum field statistics of confined light is a long-standing
goal in integrated photonics. We show that by coupling molecular vibrations
with a confined mid-infrared cavity vacuum, the photocount and quadrature
field statistics of the cavity field can be reversibly manipulated over sub-
picosecond timescales. The mechanism involves changing the cavity resonance
frequency through a modulation of the dielectric response of the cavity
materials using femtosecond UV pulses. For a single anharmonic molecular
vibration in an infrared cavity under ultrastrong coupling conditions, the
pulsed modulation of the cavity frequency can adiabatically produce mid-
infrared light that is simultaneously sub-Poissonian and quadrature squeezed,
depending on the dipolar behavior of the vibrational mode. For a vibration-
cavity system in strong coupling, non-adiabatic polariton excitations can be
produced after the frequency modulation pulse is over, when the system is
initially prepared in the lower polariton state. We propose design principles
for the generation of mid-infrared quantum light by analyzing the dependence
of the cavity field statistics on the shape of the electric dipole function of
the molecule, the cavity detuning at the modulation peak and the anharmonicity
of the Morse potential. Feasible experimental implementations of the
modulation scheme are suggested. This work paves the way for the development
of molecule-based mid-infrared quantum optical devices at room temperature.

##  I Introduction

Quantum light sources are key to the success of optical quantum technologies
[1]. In the visible spectral range, several mechanisms are available to
produce quantum light, such as single photon emission by atoms in optical
resonators [2, 3], photon blockade in cavity optomechanics [4], or ensemble
strong coupling with atomic emitters [5]. In the near infrared
($0.7-2.5\,\mu{\rm m}$), quantum light generation is a mature technology that
has become the workhorse for applications in quantum communication [6, 7] and
sensing [8, 9]. Quantum-efficient semiconductor [10] and superconductor [11]
photodetectors are broadly available in this frequency range. In contrast,
quantum optical tools in the mid-infrared spectral region ($2.5-25\,\mu{\rm
m}$) are much less developed [12, 13]. Quantum cascade lasers have been used
to produce mid infrared quantum light [14] and early demonstrations of quantum
light detection in this frequency regime are available [15, 16].

In this work, we introduce a new mechanism for the deterministic generation
and dynamical manipulation of quantum light in the mid-infrared frequency
range. The mechanism involves ultrafast modulation of the confined mid
infrared vacuum field under strong and ultrastrong light-matter coupling.
Strong vibration-cavity coupling has been demonstrated with Fabry-Perot
cavities [17, 18, 19, 20, 21, 22, 23, 24, 25], plasmonic resonators [26, 27],
van der Waals resonators [28] and polar dielectric resonators [29].
Ultrastrong coupling [30, 31] has also been demonstrated in the mid-infrared
[32, 33, 34]. The proposed modulation of the light-matter dynamics is based on
the transient modification of the dielectric function of the materials that
confine the infrared vacuum by driving interband transitions with a driving
source. The associated change in boundary conditions for the electromagnetic
field results in a parametric change of the cavity vacuum frequency that is
related with the dynamical Casimir effect [35]. This frequency modulation
mechanism has been demonstrated for mid-infrared resonators using electrical
[36], chemical [37] or electromagnetic [38, 39] driving sources.

We consider a scheme where the mid infrared cavity frequency $\omega_{\rm
c}\approx 1840\,{\rm cm}^{-1}$ ($5.6\,\mu{\rm m}$) is modulated over sub-
picosecond timescales by $200-400\,{\rm cm}^{-1}$ using a single femtosecond
UV pulse that drives a target cavity boundary material to transiently modify
its carrier density. We assume a nanophotonics scenario where the intracavity
medium has a single anharmonic molecular vibration in the ground electronic
state, protected by design of the photonic structure from the UV source. We
show that the photocount and quadrature field statistics of the infrared
cavity field is transiently modified by the frequency modulation pulse and
correlate the predicted changes with the intrinsic properties of the molecular
vibration and the light-matter interaction process.

The article is organized as follows: In Sec. II, we review the theoretical
framework for cavity quantum electrodynamics with anharmonic molecular
vibrations, and the numerically methods used for unitary wavepacket
propagation. We also review the relevant concepts of quantum field statistics.
We study the evolution of the intracavity field statistics for a Gaussian
frequency modulation in vibrational ultrastrong coupling in Sec. III and
discuss strong coupling in Sec. IV. We conclude and discuss possible
experimental realizations in Sec. VI.

##  II Methodology

###  II.1 MLQR model with cavity frequency modulation

We model light-matter interaction of a confined cavity mode of frequency
$\omega_{\rm c}$ with a single anharmonic vibration mode of fundamental
frequency $\omega_{\rm v}$ in the mid-infrared as a Morse oscillator using the
multi-level quantum Rabi (MLQR) model [40, 41], recently developed to
consistently describe the strong and ultrastrong vibration-cavity coupling
regimes in the electric dipole approximation [42]. Taking into account the
possible time-dependence of the cavity frequency due to external UV pulse
driving, the corresponding MLQR Hamiltonian can be written in coordinate space
as (in atomic units)

$\displaystyle\hat{\mathcal{H}}(t)$ $\displaystyle=$
$\displaystyle\hat{H}_{\mathrm{M}}-\frac{1}{2}\frac{\partial^{2}}{\partial\hat{x}^{2}}+\frac{1}{2}\omega_{\mathrm{c}}(t)^{2}\hat{x}^{2}+\sqrt{2\omega_{\mathrm{c}}(t)}\mathcal{E}_{0}(t)\hat{d}(q)\hat{x},$

where the vibrational Hamiltonian $\hat{H}_{M}$ includes the nuclear kinetic
energy $\hat{T}(q)$ and the potential energy $\hat{V}(q)$ along the mass-
weighted normal mode coordinate $q$. The second and third terms describe the
quantized cavity oscillator with quadrature operator $\hat{x}$ and frequency
$\omega_{\mathrm{c}}(t)$. The last term is the electric dipole light-matter
coupling, proportional to the amplitude of the vacuum fluctuations at the
cavity frequency
$\mathcal{E}_{0}(t)=\lambda_{g}\omega_{\mathrm{c}}(t)/d_{10}$.

The light-matter coupling strength is given in terms of the dimensionless
parameter $\lambda_{g}$, which is equivalent to the conventional Rabi coupling
ratio $g/\omega_{\mathrm{c}}$ [30, 31] for $g\equiv\langle
1|\hat{d}(q)|0\rangle\mathcal{E}_{0}$, where $\left|0\right\rangle$ and
$\left|1\right\rangle$ are the ground and first excited vibrational levels,
$\hat{d}(q)$ is the electric dipole function [43] and $d_{10}=\langle
1|\hat{d}(q)|0\rangle$. The onset of the conventional ultrastrong coupling
regime is thus reached when $\lambda_{g}\sim 0.1$ [31]. Strong coupling
implies $\lambda_{g}\gtrsim 0.01$.

Under UV pulse driving, the cavity frequency is taken to vary according to a
Gaussian function as

$\omega_{\mathrm{c}}(t)=\omega_{\mathrm{c}}\left(1+\eta e^{-(t-t_{\rm
d})^{2}/2\tau^{2}}\right),$ (2)

where the undriven cavity frequency being resonant with the fundamental
vibration frequency ($\omega_{\mathrm{c}}(0)=\omega_{\rm v}$). As anticipated
above, we assume the cavity frequency shifts at the UV pulse peak by up to 20%
($\eta=\pm 0.2$), red-detuned (minus sign) or blue-detuned (plus sign)
relative to the undriven resonance frequency. According to Eq.
(LABEL:eq:hamiltonian), the frequency modulation not only changes the
vibration-cavity detuning, but also the light-matter interaction strength. For
the cases that will be discussed below, the peak modulation $|\eta|=0.2$
corresponds to changes in light-matter coupling energy of $12\%$ for
$\lambda_{g}=0.08$ (strong coupling), and $30\%$ for $\lambda_{g}=0.2$
(ultrastrong coupling). Throughout this work, we assume the pulse is centered
at $t_{\rm d}=250.0$ fs and the width parameter is $\tau=62.5$ fs (FWHM =
$147$ fs).

Table 1:  Parameters of Morse potential energy curves [Eq. (3)] and dipole moment functions [Eq. (4)] employed in this work and plotted in Fig. 1. |  $D_{\mathrm{e}}$ [eV] |  $\alpha$ [a.u.] |  $q_{\mathrm{eq}}$ [bohr] |  $\Delta_{21}$ [cm-1]
---|---|---|---|---
$V_{\mathrm{A}}(q)$ | 6.80 | 1.50 | 4.0 | 31.9
$V_{\mathrm{B}}(q)$ | 9.80 | 1.25 | 4.0 | 22.1
|  $d_{0}$ [Debye] |  $q_{0}$ [bohr] |  $q_{1}$ [bohr] |  $\sigma$ bohr
$d_{\mathrm{NP}}(q)$ | 5.08 | 4.0 | 4.0 | 0.6
$d_{\mathrm{PR}}(q)$ | 2.54 | 2.7 | 4.5 | 0.6
Figure 1: Vibrational potentials and dipole functions. (a) Morse potentials
$V_{\mathrm{A}}$ (solid line) and $V_{\mathrm{B}}$ (dashed line) along the
normal mode coordinate $q$. The harmonic oscillator potential $V_{\mathrm{C}}$
(dot-dashed line) with the same $\omega_{\rm v}$ is also shown. In all cases,
the reduced mass is $\mu=8.5$ AMU and the fundamental frequency is
$\omega_{\rm v}\approx 1838$ cm-1. (b) Dipole moment functions $d(q)$ for
polar-right (dashed line) and a non-polar vibrational modes (solid line).

###  II.2 Dipolar Morse oscillator

To understand the influence of the spectral anharmonicity and the dipolar
structure of molecular vibrations on the unitary evolution of the frequency-
modulated cavity field, we combine three different vibrational potentials
$V(q)$ and two dipole moment functions $d(q)$ to search for conditions that
are most effective at producing dramatic changes in the quantum field
statistics of the infrared field.

For anharmonic vibrations we use the Morse potential [44]

$V(q)=D_{\mathrm{e}}(1-e^{-\alpha(q-q_{\mathrm{eq}})})^{2}-D_{\mathrm{e}},$
(3)

where $D_{\mathrm{e}}$ is the classical dissociation energy, $q_{\mathrm{eq}}$
the equilibrium mode coordinate, and $\alpha$ the anharmonicity parameter. For
$\alpha\ll 1$, the Morse potential can be accurately truncated at second order
in $q$, giving the harmonic oscillator. In Table 1, we list the parameters for
the two Morse potentials shown in Fig. 1a. To ensure a fair comparison between
anharmonic and harmonic potentials, the parameters in Table 1 give the same
vibrational frequency $\omega_{\rm v}=1838.26$ cm-1, which is representative
of the vibrational modes used for light-matter coupling in experiments [26,
45, 46]. The harmonic potential $V_{\mathrm{C}}$ in Fig. 1 also has the same
$\omega_{\rm v}$.

The electric dipole function along the vibrational coordinate is modeled as
[40, 41]

$\hat{d}(q)=d_{0}(q-q_{0})e^{-(q-q_{1})^{2}/2\sigma^{2}},$ (4)

which accurately captures the dipole nature of diatomic and polyatomic
molecules, depending on the choice of parameters $d_{0}$, $q_{0}$, $q_{1}$ and
$\sigma$ [46]. The set of dipole parameters used in this work are given in
Table 1. These are chosen to describe the behavior of non-polar vibrations
($d_{\rm NP}$), which due to nuclear symmetry have a vanishing dipole moment
at equilibrium ($q_{\mathrm{eq}}$), but acquire a finite dipole moment away
from the equilibrium geometry. Examples of this behavior include the
degenerate CO stretching mode in iron pentacarbonyl [32]. The other dipole
behavior of interest corresponds to polar-right molecules [41], which have a
finite dipole moment at equilibrium that further increases as the mode
distance $q$ increases. Examples of this behavior include lithium fluoride
[47]. In Fig. 1b we show the dipole functions that correspond to the
parameters in Table 1. Below we study combinations of these potentials and
dipole functions to compute the unitary light-matter interaction dynamics of
frequency-driven cavity fields.

###  II.3 MCTDH unitary polariton propagation

For accurately computing the quantum field statistics of cavity photons
subject to ultrafast frequency modulation, we solve the time-dependent
Schrödinger equation in coordinate space with a Hamiltonian given by Eq.
(LABEL:eq:hamiltonian) using the multi-configurational time-dependent Hartree
(MCTDH) method [48, 49], explicitly developed for the accurate and scalable
treatment of strongly coupled anharmonic oscillators, as they commonly occur
in photochemistry [50]. The MCTDH method has been successfully used to study
strong and ultrastrong light-matter coupling of realistic molecules with
quantized optical [51, 47] and infrared cavity fields [41, 52].

In MCTDH, the system wave function $\Psi(t)$ is written in continuous
coordinate space by the ansatz

$\Psi(q,x,t)=\sum_{j_{q}=1}^{n_{q}}\sum_{j_{x}=1}^{n_{x}}A_{j_{q}j_{x}}(t)\phi_{j_{q}}(q,t)\phi_{j_{x}}(x,t),$
(5)

which is an expansion in products of the molecular basis functions
$\phi_{j_{q}}(q,t)$ and cavity mode functions $\phi_{j_{x}}(x,t)$. To obtain
the results below, we set $n_{q}=n_{x}=12$ and represent the vibrational
motion in an uniform grid with $N_{q}=721$ grid points on the interval
$2.5<q<20.5$ Bohr. The cavity mode is described by a harmonic oscillator basis
in a uniform grid with $N_{x}=361$ grid points along the dimensionless
interval $-90<x<90$, which is enough to capture the Fock states that
participate in the dynamics.

The wavefunction ansatz in Eq. (5) is inserted in the Schrödinger equation to
obtain a set of coupled nonlinear equations for the amplitudes
$A_{j_{q}j_{x}}(t)$ and one-dimensional functions $\phi_{j_{q}}(q,t)$ and
$\phi_{j_{x}}(x,t)$, which are solved numerically using Runge-Kutta
propagators. Eigenstates of the coupled system Hamiltonian can also be
obtained in the MCTDH method via propagation in imaginary time [53], which is
used to initialize the coordinate-space wavefunction $\Psi(q,x,t)$ into a
Hilbert space vibrational polariton eigenstate [54, 50, 41].

Figure 2:  Ground state in ultrastrong coupling under frequency modulation.
(a) Evolution of the Mandel $Q$ parameter for a polar-right vibration with a
blue modulation of the cavity frequency ($\eta=+0.2$, solid line), a polar-
right vibration in red detuning ($\eta=-0.2$, dot-dashed line) and a non-polar
vibration in blue detuning ($\eta=+0.2$, dashed line). The shaded region
indicates sub-Poissonian photocount statistics. (b) Evolution of the photon
number variance $\langle\Delta\hat{n}^{2}\rangle$ for the three cases in panel
a. The curve for non-polar vibrations is rescaled by 100. (c) Evolution of the
quadrature variance in the phase space direction $\theta=0$. (d) Evolution of
the quadrature variance at $\theta=\pi/2$. In panels c and d, the shaded
region indicates squeezing relative to the standard vacuum. The light-matter
coupling parameter is $\lambda_{g}=0.2$. Throughout this work, the frequency
modulation pulse is centered at $250$ fs and is 147 fs wide (FWHM).

###  II.4 Cavity Field Statistics

The study of intensity and field quadrature correlations is used in quantum
optics to distinguish classical and non-classical light sources [55]. The
normalized intensity correlation at zero delay $g^{(2)}(0)$ is commonly used
to identify light sources in which photons are bunched or anti-bunched [56].
This photocounting behavior is quantified by the Mandel $Q$ parameter of the
intracavity field, given by [57]

$Q=\frac{\left\langle\Delta\hat{n}^{2}\right\rangle-\left\langle\hat{n}\right\rangle}{\left\langle\hat{n}\right\rangle},$
(6)

where $\langle\hat{n}\rangle\equiv\langle\hat{a}^{\dagger}\hat{a}\rangle$ is
the expectation value of photon number operator and
$\langle\Delta\hat{n}^{2}\rangle$ is the photon number variance. For a
classical laser source that has a Poissonian photocount distribution, the
Mandel parameter is $Q=0$ [56]. For sub-Poissonian light sources we have
$-1<Q<0$ (Fock state, squeezed vacuum) and for super-Poissonian sources we
have $Q>0$ (e.g., thermal light, entangled photon pairs).

Phase space squeezing is an alternative metric for characterizing quantum
light sources. This is done by comparing the variances of the electric field
quadratures $\hat{x}=(\hat{a}+\hat{a}^{\dagger})$ and
$\hat{y}=-i(\hat{a}-\hat{a}^{\dagger})$ relative to the vacuum Fock state
[56]. For vacuum, the normalized quadrature variances are
$\langle\Delta\hat{x}^{2}\rangle=\langle\Delta\hat{y}^{2}\rangle=1$.
Quadrature squeezing implies that $\langle\Delta\hat{x}^{2}\rangle<1$ or
$\langle\Delta\hat{y}^{2}\rangle<1$, i.e., one of the quadratures components
has less quantum noise than the standard vacuum. In terms of the generalized
phase space quadrature
$\hat{X}_{\theta}=\cos\theta\,\hat{x}+\sin{\theta}\,\hat{y}$, the squeezing
factor (in dB) is defined as $\zeta_{\theta}=-10\times{\log}\langle\Delta
X_{\theta}^{2}\rangle$ [58], where $\theta$ is a phase space angle that can be
experimentally tuned using homodyne techniques. We only consider cases with
$\theta=0$ and $\theta=\pi/2$ in this article.

Figure 3:  Ground state photocount distribution under frequency modulation.
(a) Photon number distribution $P(n)$ before modulation (black line) and the
modulation peak (blue line) for a polar-right molecule subject to blue
Gaussian frequency modulation ($\eta=+0.2$); (b) $P(n)$ for polar-right
molecules with red modulation ($\eta=-0.2$); (c) $P(n)$ for non-polar
molecules with blue modulation ($\eta=+0.2$). In all panels, the photon
distribution of a coherent state $|\alpha\rangle$ with the same mean photon
number as the modulated state at peak detuning is also shown (orange line).
The legends indicate the mean photon number and number variance before
modulation (black font) and at the modulation peak (blue font). The light-
matter coupling parameter is $\lambda_{g}=0.2$.

##  III Cavity Frequency Modulation in Ultrastrong Coupling

In this Section we study the evolution of the Mandel $Q$ parameter, the cavity
number variance $\langle \Delta\hat{n}^{2}\rangle$, and the generalized
quadrature variance $\langle\Delta X_{\theta}^{2}\rangle$ of the intracavity
field $\hat{a}$, for a coupled vibration-cavity system with a photon frequency
modulated according to Eq. (2). The ultrastrong coupling regime is assumed
($\lambda_{g}>0.1$) for a system initialized in the ground state
$\left|\Psi_{\rm G}\right\rangle$ or the lower polariton state
$\left|\Psi_{\rm LP}\right\rangle$ prior to the modulation pulse. We compare
the unitary cavity field evolutions obtained for different types of dipole
functions and different signs of the peak modulation parameter $\eta$. In Sec.
IV, a similar analysis is carried out for a strong coupling scenario
($\lambda_{g}\ll 0.1$).

###  III.1 Initial Ground State

In Figure 2 we show the evolution of the Mandel $Q$ parameter, number
variance, and quadrature variances at $\theta=0$ and $\pi/2$, for a coupled
vibration-cavity system that is initially in the ground state $\left|\Psi_{\rm
G}\right\rangle$ before modulation. The Rabi coupling parameter is
$\lambda_{g}=0.2$ and the cavity is initially on resonance with the
fundamental vibration frequency $\omega_{\rm v}$. Three cases are highlighted,
differing in the type dipole function $d(q)$ and peak frequency shift $\eta$:
polar-right vibration with blue cavity detuning ($\eta>0$); polar-right
vibration with red cavity detuning ($\eta<0$), and non-polar vibration with
blue cavity detuning ($\eta>0$). We use the Morse potential $V_{A}$ in Table 1
in all cases.

Figure 2 shows that without frequency modulation, the ground state for polar-
right vibrations is slightly super-Poissonian ($Q\approx 0.1$), the number
variance is relatively large ($\approx 4.5$ photons), there is no squeezing in
the $x$-direction, and the $y$-quadrature is only weakly squeezed
($\zeta_{\pi/2}=0.13$ dB) for the chosen $\lambda_{g}$. For non-polar
vibrations, the ground state behaves qualitatively similar to the polar-right
case, but the number variance is two orders of magnitude smaller. Significant
differences between polar and non-polar species become more evident under
frequency modulation.

For polar-right molecules we find an asymmetry in the evolution of the cavity
field observables depending on the sign of the modulation amplitude $\eta$.
For blue frequency modulations ($\eta>0$), the ground state becomes
transiently sub-Poissonian at the modulation peak (see Fig. 2a), largely
driven by a significant decrease of the number variance (see Fig. 2b). The
$x$-quadrature also becomes squeezed by $0.41$ dB at the modulation peak (see
Fig. 2c), accompanied by the increase in the $y$-quadrature variance (see Fig.
2d). In contrast, for red modulations ($\eta<0$) the same polar-right
vibrational mode generates a cavity field with super-Poissonian statistics
($Q\approx 0.5$) at the modulation peak, associated with a significant
increase in the number variance (see Fig. 2b). The $x$-quadrature variance
grows relative to the modulation-free value, and the conjugate $y$-quadrature
becomes squeezed $\zeta_{\pi/2}=1.2$ dB at the modulation peak.

In order to understand this behavior, in Fig. 3 we show the corresponding
photocount distributions $P(n)$ before and during the cavity frequency
modulation. In Fig. 3a, we show the decrease of the average photon number and
variance predicted for polar-right molecules at the peak of a blue modulation,
relative to the initial coupled ground state. This decrease is responsible for
the change in the $Q$ parameter around the Poissonian $Q=1$ limit in Fig. 2a.
For comparison, we also show the exact Poissonian distribution for a coherent
state with the same average photon number as the system at the modulation
peak. For blue modulations the overlap of the driven system with the coherent
state distribution can be significant, but for red modulations the deviations
from a coherent state are large at low $n$ (see Fig. 3b).

The $P(n)$ distributions for non-polar vibrations with and without frequency
modulation are shown in Fig. 3c. Without modulation, the photon number and
number variance of the coupled ground state are orders of magnitude smaller
than the case of polar vibrations. This is consistent with previous works [40,
41]. Despite the small absolute numbers, Fig. 3c shows that the photocount
distribution broadens at the peak of a blue modulation shown, which explains
the increase of the number variance in Fig. 2b.

###  III.2 Initial Lower Polariton

We now carry out the analysis from the previous section on a coupled system
that is initialized in the lower polariton state $\left|\Psi_{\rm
LP}\right\rangle$ at ultrastrong coupling ($\lambda_{g}=0.2$). Excited
polariton population is commonly produced in non-linear cavity infrared
spectroscopy experiments [46]. We again study polar-right and non-polar
vibrations under blue frequency modulation, but now also analyze the dynamics
of non-polar molecules under red modulation $(\eta=-0.2)$.

Figure 4:  Lower polariton in ultrastrong coupling under frequency modulation.
(a) Evolution of the Mandel $Q$ parameter for a polar-right vibration with a
blue modulation of the cavity frequency ($\eta=+0.2$, solid line), a non-polar
vibration with red modulation ($\eta=-0.2$, dot-dashed line) and a non-polar
vibration with blue modulation ($\eta=+0.2$, dashed line). The shaded region
indicates sub-Poissonian photocount statistics. (b) Evolution of the photon
number variance $\langle\Delta\hat{n}^{2}\rangle$ for the three cases in panel
a. The curve for non-polar vibrations is rescaled by 10. (c) Evolution of the
quadrature variance in the phase space direction $\theta=0$. (d) Evolution of
the quadrature variance at $\theta=\pi/2$. In panels c and d, the shaded
region indicates squeezing relative to the standard vacuum. The light-matter
coupling parameter is $\lambda_{g}=0.2$.

Figure 4a shows that lower polariton state for $\lambda_{g}=0.2$ is super-
Poissonian ($Q\approx 0.85$) for polar-right molecules and sub-Poissonian for
non-polar vibrations ($Q\approx-0.45$). The number variance is large for
polar-right vibrations ($\approx 8.5$ photons), but is an order of magnitude
smaller for non-polar species (see Fig. 4b). Moreover, there is no squeezing
in the $x$\- or $y$\- quadrature in either case. Under pulsed frequency
modulation, the cavity observables vary significantly for the molecular
species considered, although the sign of the Mandel $Q$ parameter does not
change for the pulse parameters used. In contrast with the ground state
analysis in Sec. III.1, frequency modulating the lower polariton does not
generate squeezing at the modulation peak (see Figs. 4c and 4d). The
corresponding photocount distributions with and without modulation are shown
in Fig. 5. The deviation of the number distribution from a classical coherent
state is more notorious for both polar and non-polar species. For non-polar
vibrations, Figs. 5b and 5c show that the frequency modulation mainly changes
the contribution of low-$n$ Fock states.

##  IV Non-Adiabatic Excitations in Vibrational Strong Coupling

In Sec. III we studied the ultrastrong coupling regime of Eq.
(LABEL:eq:hamiltonian) by setting $\lambda_{g}=0.2$. For the Morse potential
$V_{\mathrm{A}}$ (see Table 1), this corresponds to the Rabi coupling strength
$g_{10}=\lambda_{g}\omega_{10}\approx 360$ cm-1 and a Rabi splitting of $973$
cm-1 in linear transmission for polar-right vibrations (greater than
$2g_{10}$). Although such splitting magnitudes are currently beyond
experimental reach with Fabry-Perot resonators [32], expected improvements in
mid-infrared resonator technology may facilitate ultrastrong coupling studies
in the near future [33].

In this Section we explore the cavity field statistics subject to a photon
frequency modulation under strong coupling conditions, which are
experimentally accessible with current resonator technology [19, 20]. For this
we set $\lambda_{g}\sim 10^{-2}$, corresponding to Rabi splittings no greater
than $300$ cm-1. The main qualitative difference expected in strong coupling
is that the polariton spectrum can have energy gaps $\Delta E$ that are
comparable with the bandwidth of modulation pulse $\Delta\omega=2\sqrt{2\ln
2}/\tau$. For the modulation pulse assumed in this work ($\tau=62.5$ fs) the
bandwidth is $\Delta\omega=226.6$ cm-1. Whenever an eigenstate state that has
a nearby energy gap comparable or smaller than $\Delta\omega$ is driven in
frequency, non-adiabatic excitations can be expected to remain in the system
after the pulse is over [59].

Figure 5:  Lower polariton photocount distribution under frequency modulation.
(a) Photon number distribution $P(n)$ before modulation (black line) and the
modulation peak (blue line) for a polar-right molecule subject to blue
Gaussian frequency modulation ($\eta=+0.2$); (b) $P(n)$ for non-polar
molecules with blue modulation ($\eta=+0.2$); (c) $P(n)$ for non-polar
molecules with red modulation ($\eta=-0.2$). In all panels, the photon
distribution of a coherent state $|\alpha\rangle$ with the same mean photon
number as the modulated state at peak detuning is also shown (orange line).
The legends indicate the mean photon number and number variance before
modulation (black font) and at the modulation peak (blue font). The light-
matter coupling parameter is $\lambda_{g}=0.2$.  Figure 6:  Cavity field
statistics in vibrational strong coupling. (a) Evolution of the Mandel $Q$
parameter for a vibration-cavity system initialized in the ground state. (b)
Coupled system initialized in the lower polariton state. In all panels, curves
correspond to a polar-right vibration with blue modulation ($\eta=+0.2$, solid
blue line) and red modulation ($\eta=-0.2$, dot-dashed red line) of the cavity
frequency, and for a non-polar vibration with blue detuning ($\eta=+0.2$,
dashed black line) and red detuning ($\eta=-0.2$, dot-dashed green line)
respect to the undriven cavity frequency. The shaded region indicates sub-
Poissonian statistics. The light-matter coupling parameter is
$\lambda_{g}=0.08$.

We illustrate this effect in Figure 6, where we compare the evolution of the
Mandel $Q$ parameter a vibration-cavity system initialized either in the
ground state (Fig. 6a) or the lower polariton eigenstate (Fig. 6b) before the
modulation pulse. The coupling parameter is $\lambda_{g}=0.08$, which would
give a Rabi splitting of 288 cm-1 in linear transmission. Curves are shown for
polar-right and non-polar vibrations under red and blue modulations ($\eta=\pm
0.2$).

Figure 6a shows that an initial ground state evolves in a qualitatively
similar way to the ultrastrong coupling case in Fig. 2a, with small
differences in the values of $Q$ before and during the modulation pulse. The
driven wavefunction returns adiabatically to initial ground state after the
pulse is over ($t\approx 0.5$ ps), because the energy gap $\Delta E$ to the
first excited (lower) polariton level (1648 cm-1 for polar-right vibrations)
far exceeds the modulation bandwitdh $\Delta\omega$. In other words, no
residual post-pulse excitations remain in the system.

In contrast, Figure 6b shows that although starting from the lower polariton
state can lead to an evolution of the $Q$ parameter with the same trends in
Fig. 4a for ultrastrong coupling, the pulse turn-off dynamics is qualitatively
different. For $\lambda_{g}=0.08$, the energy gap $\Delta E$ between the lower
and upper polariton eigenstates is $307$ cm-1 for polar vibrations and $295$
cm-1 for non-polar molecules, The latter is not much greater than the
modulation bandwidth $\Delta\omega$. We therefore expect the system
wavefunction to exhibit non-adiabatic excitations after the modulation pulse
is over. This manifests in Fig. 6b in an oscillatory behavior of the Mandel
$Q$ parameter as the pulse turns off.

We quantify the non-adiabatic excitations generated on the initial lower-
polariton wavefunction in Fig. 7 for polar-right vibrations under red
frequency modulation. Analogue results are found for other dipole functions
and values of $\eta$. The degree of adiabaticity of the system evolution is
quantified by the autocorrelation function
$|\langle\Psi(t)|\Psi(0)\rangle|^{2}$. For an adiabatic pulse modulation, the
post-pulse autocorrelation should not differ from unity. Figure 7 shows that
this is indeed the case for a driven ground state (solid line), which as
mentioned above is associated with a large energy gap to the first excited
polariton level ($\Delta E/\Delta\omega\approx 7.3$). In contrast, for a
driven lower polariton state the post-pulse autocorrelation ($t\geq 0.5$) can
be significantly different from unity, depending on the magnitude of the Rabi
splitting. For example, while for $\lambda_{g}=0.08$ the system wavefunction
is left with about 0.1% of non-adiabatic excitations after the pulse is over
(see Fig. 7, dashed line), for $\lambda_{g}=0.05$ the residual population
outside the lower polariton after the pulse is over is already a few percent
(Fig. 7, dot-dashed line), which is feasible to detect in nonlinear infrared
spectroscopy [46]. This non-adiabatic excitations occur due to the small
energy gap between the lower and upper polariton levels, relative to the
modulation bandwidth ($\Delta E/\Delta\omega\approx 0.83$ for polar right
vibrations).

Figure 7: Non-adiabatic dynamics in vibrational strong coupling. State
autocorrelation for a polar-right vibration subject to cavity frequency
modulation for a system initialized in the ground state (solid line), the
lower polariton for $\lambda_{g}=0.08$ (dashed line) and the lower polariton
with $\lambda_{g}=0.05$ (dot-dashed line).

##  V Dependence on the vibrational anharmonicity

Molecular vibrations can in principle have similar electric dipole behavior
$d(q)$ but differ in the spectral anharmonicity of the nuclear potential. Here
we explore the evolution of the Mandel $Q$ parameter in ultrastrong coupling
($\lambda_{g}=0.2$), for the same frequency modulation in the previous
sections. We compare the Morse potentials Morse potentials $V_{\mathrm{A}}$
and $V_{\mathrm{B}}$ in Table 1, which have the same fundamental frequency
$\omega_{\rm v}$ but different anharmonicity parameters. The latter is
quantified by the anharmonic shift $\Delta_{21}\equiv\omega_{\rm v}-\Delta
E_{21}$, where $\Delta E_{21}$ is the energy gap between the $v=1$ and $v=2$
vibrational levels. For $V_{\mathrm{A}}$ we have $\Delta E_{21}=32$ cm-1 and
for $V_{\mathrm{B}}$ we have $\Delta_{21}=22$ cm-1. For comparison we also
consider the purely harmonic case ($\Delta E_{21}=0$). We focus on a system
initialized in the ground state, but similar conclusions are found for excited
initial states.

In Fig. 8a we show the evolution of the $Q$ parameter for polar-right
molecules in ultrastrong coupling. The curves for $V_{\mathrm{A}}$ under blue
modulation (blue solid line) and red modulation (red solid line) of the cavity
frequency simply reproduce the results in Fig. 2a. Reducing the anharmonicity
does not qualitatively change the dynamics of the field statistics, i.e., the
state can still shift from super-Poissonian to sub-Poissonian for blue
modulations ($\eta=0.2$), but the magnitude of the changes are slightly
different. For example, the ground state $Q$ parameter before modulation
follows the same hierarchy as the degree of anharmonicity, i.e.,
$Q_{\mathrm{A}}>Q_{\mathrm{B}}\geq Q_{\rm HO}$, where $Q_{\rm HO}$ is the
harmonic oscillator limit of the ground state at this value of $\lambda_{g}$.
The figure also shows that the relative variations $\Delta Q/Q$ at the
modulation peak relative to the initial state, are fairly insensitive to the
anharmonicity parameter $\Delta_{21}$. Additional calculations carried out
over a broader range values of $\Delta_{21}$ confirm this statement.

Figure 8b finally shows that at least for a frequency-modulated ground state,
no significant dependence with the spectral anharmonicity is expected for the
evolution of the $Q$ parameter for non-polar molecular vibrations. These
results are consistent with previous works [40, 41], which show that light-
matter systems with non-polar molecules are similar to two-level systems and
harmonic oscillators with vanishing permanent dipole moments.

Figure 8: Cavity field statistics and vibrational anharmonicity. (a) Evolution
of the Mandel $Q$ parameter for a vibration-cavity system initialized in the
ground state for a polar-right vibration. (b) Evolution of the $Q$ parameter
for non-polar vibrations. In both panels, curves correspond to a molecular
system with Morse potentials $V_{\mathrm{A}}$ (solid lines) and
$V_{\mathrm{B}}$ (dashed lines), and the harmonic potential $V_{\rm C}$ (dot-
dashed lines). Red lines correspond to frequency modulations with $\eta=-0.2$
and blue lines to $\eta=+0.2$.

##  VI Conclusions

Table 2:  Summary of predicted cavity field statistics variations. Values of the Mandel $Q$ parameter and squeezing factor $\zeta_{\theta}$ at $t=0$ and $t=t_{\mathrm{d}}$ for different dipole moment functions $d(q)$, initial states $|\Psi(0)\rangle$ and cavity frequency modulations ($\eta=\pm 0.2$), in the VUSC regime with the light-matter coupling $\lambda_{g}=0.2$. PR and NP correspond to a polar-right and non-polar vibrations, respectively. GS and LP correspond to ground and lower polariton states, and the positive and negative sign of $\eta$ represent blue and red modulation, respectively.  Case |  $d(q)$ | $|\Psi(0)\rangle$ |  $\eta$ | $Q(0)$ | $\zeta_{0}(0)$ | $\zeta_{\pi/2}(0)$ | $Q(t_{\rm d})$ | $\zeta_{0}(t_{\mathrm{d}})$ | $\zeta_{\pi/2}(t_{\mathrm{d}})$
---|---|---|---|---|---|---|---|---|---
i | PR | GS | $+$ | 0.13 | -0.51 | 0.13 | -0.09 | 0.45 | -0.71
ii | PR | GS | $-$ | 0.13 | -0.51 | 0.13 | 0.51 | -1.78 | 1.18
iii | PR | LP | $+$ | 0.88 | -3.06 | -1.01 | 0.31 | -1.44 | -1.44
iv | PR | LP | $-$ | 0.88 | -3.06 | -1.01 | 1.99 | -4.96 | -0.40
v | NP | GS | $+$ | 0.06 | -0.29 | 0.09 | 0.38 | 0.57 | -0.73
vi | NP | GS | $-$ | 0.06 | -0.29 | 0.09 | 0.64 | -1.37 | 1.12
vii | NP | LP | $+$ | -0.46 | -3.70 | -2.40 | -0.23 | -1.79 | -2.15
viii | NP | LP | $-$ | -0.46 | -3.70 | -2.40 | -0.53 | -5.56 | -2.31

We have shown that the quantum statistics of mid-infrared cavity fields can be
dynamically controlled with an ultrafast modulation of the cavity frequency
over sub-picosecond timescales, by changing the reflectivity of the conducting
cavity mirrors using femtosecond UV pulses. For $20\%$ shifts of the cavity
frequency at the modulation peak, we predict the expected changes in the
photocount and quadrature field statistics of the intracavity field by
computing the sub-picosecond unitary dynamics of the underlying cavity-
vibration polariton Hamiltonian using numerically exact techniques (see Sec.
II).

The dynamics of quantum field statistics is described in terms of the shape of
electric dipole function, the anharmonicity of the molecular system, light-
matter coupling strength, and whether the system is initially prepared in the
ground or lower polariton state before the frequency modulation pulse. For a
coupled system in the vibrational ultrastrong coupling (VUSC) regime the
cavity field dynamics is adiabatic, while under vibrational strong coupling
(VSC) non-adiabatic excitations are possible, provided that the energy gap
between the lower and upper polariton levels is comparable with the bandwidth
of the frequency modulation pulse.

The results obtained in this work are summarized in Table 2, where predicted
values of the Mandel $Q$ parameter and squeezing factor $\zeta_{\theta}$
before the modulation ($t=0$) and at the modulation peak ($t=t_{\mathrm{d}}$)
are given for a single Morse potential ($V_{\mathrm{A}}$) in ultrastrong
coupling ($\lambda_{g}=0.2$), for different polarities of the vibrational
mode, different initial states, and different cavity frequency modulations.

As a general rule, we find that the dipolar nature of molecular vibrations is
more important descriptor of the degree of control that can be expected for
the quantum field statistics in mid-infrared cavities than the anharmonicity
of vibrational spectrum. We also find that the photon statistics for polar-
left vibrations [41] follows the same qualitative behavior with smaller values
of $Q$ and $\zeta_{\theta}$ than in the case of polar-right species.
Similarly, if the upper polariton (UP) is considered as the initial state
before modulation, the Mandel parameter and squeezing factor behaves similar
to the LP case, but the dynamics is reversed, i.e., evolution of the photon
statistics under blue modulation for LP corresponds to dynamics with red
modulation for the UP and viceversa.

Cavity designs for implementing the proposed scheme would build on Ref. [60],
where the reflectivity of the conducting cavity mirrors is modulated by
promoting interband transitions with ultrafast UV pulses. This mechanism was
recently used to shift the infrared resonance frequency of a
semiconducting/insulator nanoresonator structure [39]. Similar design
principles could be used with infrared Fabry-Perot cavities with conducting
mirror layers [61], including protecting layers that prevent UV photons from
promoting intracavity molecules to electronic excited states. Given the
feasibility of the proposed scheme, this work paves the way for the
development of integrated molecule-assisted quantum light sources for
applications in quantum metrology and photonic quantum information processing
in the mid-infrared frequency range.

###### Acknowledgements.

We thank B. Simpkins, A. Dunkelberger and J. Owrutsky for technical
discussions. J.F.T. is supported by ANID Postdoctoral Fellowship No. 3200565.
F.H. is supported by ANID – FONDECYT Regular No. 1181743 and ANID – Millennium
Science Initiative Program ICN17_012.

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