# Jaynes-Cummings interaction with a traveling light pulse

Victor Rueskov Christiansen<EMAIL_ADDRESS>Mads Middelhede Lund
<EMAIL_ADDRESS>Center for Complex Quantum Systems, Department of Physics and
Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark  Fan
Yang<EMAIL_ADDRESS>Klaus Mølmer<EMAIL_ADDRESS>Center for
Hybrid Quantum Networks, Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, DK-2100 Copenhagen, Denmark

(August 28, 2024)

###### Abstract

The Jaynes-Cummings model provides a simple and accurate description of the
interaction between a two-level quantum emitter and a single mode of quantum
radiation. Due to the multimode continuum of eigenmodes in free space and in
waveguides, the Jaynes-Cummings model should not be expected to properly
describe the interaction between an emitter and a traveling pulse of quantum
radiation. In this article, we review a cascaded quantum system approach that
accurately describes the interaction of a quantum system with an incident
quantum pulse of radiation. This approach leads to different formulations of
the theory, each of a similar structure as the Jaynes-Cummings model but with
important modifications.

##  I Introduction

In their seminal article, _Comparison of Quantum and Semiclassical Theories
with Application to the Beam Maser_ , E. T. Jaynes and F.W. Cummings outlined
the formalism and basic mechanisms of resonant light-matter interactions [1].
In the quantum theory, the field and its quantum states can be expanded on
multi-mode number states, $\phi(n_{1},n_{2},\ldots)$ with $n_{i}$ photons in
the $i^{\textup{th}}$ field mode (using the notation of [1]). Absorption and
emission processes by a single two-level system (TLS) are characterized by
matrix elements of the dipole operator between the emitter’s ground and
excited states and of the electric field operator between field states
differing in photon number in any of the field modes. The quantized electric
field operator can be written as a linear combination of the annihilation and
creation operators $\hat{a}_{i}$, $\hat{a}^{\dagger}_{i}$ for the different
field modes, and hence couples $\phi(n_{1},n_{2},\ldots,n_{i},\ldots)$ to the
states $\phi(n_{1},n_{2},\ldots,n_{i}+1,\ldots)$ and
$\phi(n_{1},n_{2},\ldots,n_{i}-1,\ldots)$ with strengths
$\propto\sqrt{n_{i}+1}$ and $\sqrt{n_{i}}$, respectively.

The Jaynes-Cummings model (JCM) is the restriction of the light-matter
interaction problem to the case of a two-level system coupled to a single mode
of the quantized field. The JCM applies to the case of a TLS residing inside,
or traveling through, an optical cavity that supports a discrete cavity mode
which is nearly resonant with the TLS transition (and it typically assumes
that all other cavity modes are widely detuned and hence not coupled to the
TLS). The JCM has been implemented with atoms and optical cavities [2, 3],
Rydberg atoms and microwave cavities [4, 5, 6], superconducting qubits and
various microwave resonator designs [7, 8], as well as with superconducting
qubits and acoustic resonators (phonons) [9, 10, 11].

Figure 1: (a) A two-level atom passing through a single-mode cavity.(b) A
single-mode pulse traveling over a two-level atom. (c) A light pulse
interacting with a two-level system leads to emission of photons into a
continuum of field modes.

Using Pauli spin operators $\hat{\sigma}^{\pm}$ to represent excitation and
deexcitation of the TLS, the resonant Jaynes-Cummings Hamiltonian reads

$H_{\textup{JCM}}=i\chi(\hat{a}^{\dagger}\hat{\sigma}^{-}-\hat{a}\hat{\sigma}^{+}).$
(1)

We note that $H_{\textup{JCM}}$ acts separately within two-dimensional
subspaces $\\{\ket{n,g},\ket{n-1,e}\\}$ in which the dynamics periodically
exchanges a single quantum between the cavity field mode and the TLS at
frequency $\chi\sqrt{n}$. If different number states are populated, the
quantum is exchanged at different frequencies leading to so-called collapses
and revivals in the oscillating TLS excited state population [12].

If we want to describe the case of an atom flying through a cavity and thus
traversing the standing wave field mode as depicted in Fig. 1(a), the JCM
Hamiltonian Eq. (1) merely has to be equipped with a time-dependent value
$\chi(t)$ reflecting the amplitude of the field mode at the time-dependent
position of the atom.

A natural question arises: Can we employ a similar time-dependent Jaynes-
Cummings Hamiltonian,

$H_{\textup{JC-
pulse}}(t)\stackrel{{\scriptstyle?}}{{=}}i\sqrt{\gamma}u(t)(\hat{a}^{\dagger}\hat{\sigma}^{-}-\hat{a}\hat{\sigma}^{+}),$
(2)

if a stationary TLS is coupled to an incident pulse of radiation with
amplitude $\sqrt{\gamma}u(t)$, cf., Fig. 1(b)? Certainly, photons may be
absorbed and emitted by the TLS like in the case of a cavity, but the
propagation of the wave packet through space explores a continuum of wave
number and frequency components, which are close to resonance and hence also
available for the photons emitted by the TLS. A single photon may thus be
absorbed from the pulse and be subsequently re-emitted into wave packets of
different temporal shapes, as depicted in Fig. 1(c), and, due to the
saturation nonlinearity of the TLS, scattering of a multiphoton input state
may feed many output modes with different number state contents. Due to its
assumption of a strictly single-mode field, the successes of the JCM within
the fields of cavity QED and circuit QED seem not to extend to scattering of
traveling pulses on a TLS, i.e., to the field of waveguide QED. In this
article, we show that despite the apparent differences, we can apply a
description similar to the JCM for a traveling pulse of radiation incident on
a TLS.

The article is organized as follows. In Sec. II we review efforts to describe
quantized light-matter interactions from a quantum field perspective. For
systems sharing only a single excitation, it is possible to solve the
Schrödinger equation exactly, and for multi-photon scattering on linear
optical components, the Heisenberg equations of motion obey linear equations
that can be solved for large numbers of modes. But the nonlinear scattering of
even a few-photon state remains a computationally challenging problem.

In Sec. III, we introduce an effective description, in which the quantum state
occupying a traveling wave packet pulse is released from a cavity mode. We can
treat this artificial cavity mode and the TLS as a cascaded quantum system,
and it obeys a master equation with a Hamiltonian of the same form as Eq. (1),
but with a time-dependent strength that differs significantly from the pulse
shape that we try to model. The field leaking from the cavity and from the TLS
interfere in the forward scattered radiation continuum and all excitation is
gradually lost, but the master equation yields the correct time evolution of
the TLS.

In Sec. IV, we turn to the quantum state of the scattered light. This is made
possible by restricting the analysis of the output field continuum to a single
or a small number of output temporal modes. Our theoretical description at
this stage comprises the input mode, the TLS, and one or several output mode
oscillators, coupled by effective time-dependent JCM Hamiltonians. This
analysis permits evaluation of the final quantum state of any output mode and
hence of protocols for transforming quantum states of light, that are inspired
by ideal single-mode performance in a cavity. It also reveals genuine, and
useful, multimode capabilities, e.g., for the sorting of different number
states into different modes [13] and for subtraction and transferal of a
single photon from an input pulse to an orthogonal pulse mode [14].

In Sec. V, we adopt an interaction picture, where the TLS couples to time-
dependent linear combinations of the input and output modes. In this
interaction picture, we recover the interaction Hamiltonian proposed in Eq.
(2), supplemented by the coupling of the TLS to the other orthogonal mode
combinations, and inclusion in the master equation of their joint loss to the
continuum of modes.

In Sec. VI, we present three examples of how the most characteristic behaviors
of the Jaynes-Cummings model are recovered or modified in the case of the
interaction with a travelling pulse: Rabi oscillations, collapses and revivals
and photon subtraction.

Sec. VII summarizes the main results of the article.

##  II Quantum light-matter interaction

A natural starting point to describe the quantum interaction between a
propagating pulse and a TLS, but _not_ the one adopted in this work, is by
introduction of a general representation of the quantum state of the radiation
and its accompanying Schrödinger equation. In a one-dimensional (1D)
waveguide, the continuous field operators $\hat{a}(t)$ evaluated at a fixed
location satisfy the commutation relation
$[\hat{a}(t),\hat{a}^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$, and a pure
state of the field can be expanded in the form

$|\psi\rangle=\sum_{n}\int dt_{1}\cdots
dt_{n}\psi_{n}(t_{1},\cdots,t_{n})\hat{a}^{\dagger}(t_{1})\cdots\hat{a}^{\dagger}(t_{n})|0\rangle,$
(3)

where $\psi_{n}(t_{1},\cdots,t_{n})$ denotes the $n$-photon wave function.
While the scattering of one- and two-photon states on a single emitter can be
solved by standard scattering theory [15, 16, 17, 18], the complex (many-body)
character of the quantum state of the field makes the interaction of just a
few more photons with a two-level scatterer prohibitively complicated to
solve. Despite that several general formalisms have been established for an
arbitrary number of photons [19, 20, 21, 22, 23, 24], their numerical
implementation is restricted to the few-photon regime due to the complexity of
the general problem, as well illustrated by the numerical procedures presented
in [23].

Even in the case where the initial state of the incident field occupies only a
single temporal mode $u(t)$ with creation operator $\hat{a}^{\dagger}_{u}=\int
dtu(t)\hat{a}^{\dagger}(t)$, the exchange of quanta of excitation with the TLS
may readily lead to population of correlated state of the general form Eq.
(3). Using input-output theory, it has been shown, however, that for a single
mode input field the dynamics of the TLS, can be obtained from a hierarchy of
coupled master equations [25]. This theory does not provide the quantum state
of the output field, but it makes it possible to obtain mean values and
correlation functions and to simulate the outcome of continuous measurements
on the output field [26, 27, 28].

All the methods applicable to interactions with pulses seem to be very
different from the solution of the Jaynes-Cummings model, but as we show in
Sec. III, we can cast the problem in such a way from the beginning that we can
employ the cascaded-system-approach [29, 30, 31], and describe the evolution
of the TLS by a JCM-like model where the output field is treated as loss.

In the single-mode JCM, we obtain both the state of the TLS and the field. If
we should be able to obtain the quantum state of the emitted field in the form
of Eq. (3), we may consider a simpler characterization of the scattered field,
i.e., the quantum state of photons in a single temporal mode $v(t)$, described
by the annihilation operator $\hat{a}_{v}=\int dtv^{*}(t)\hat{a}(t)$. The
density matrix for such a single-mode mixed state can be obtained by tracing
out the modes orthogonal to $\hat{a}_{v}$,
$\hat{\rho}_{v}=\mathrm{Tr}_{v_{\perp}}[|\psi\rangle\langle\psi|]$, and is
given by the formal expression,

$\hat{\rho}_{v}=\sum_{m=0}^{\infty}\frac{1}{m!}\int
dt_{1}dt_{2}\cdots{dt_{m}}\langle
0|\hat{b}(t_{m})\cdots\hat{b}(t_{2})\hat{b}(t_{1})|\psi\rangle\langle\psi|\hat{b}^{\dagger}(t_{1})\hat{b}^{\dagger}(t_{2})\cdots\hat{b}^{\dagger}(t_{m})|0\rangle,$
(4)

where $\hat{b}(t)=\hat{a}(t)-v(t)\hat{a}_{v}$. The reduced quantum state
$\hat{\rho}_{v}$ is the relevant quantity of interest for many quantum optical
applications, and it permits, for example, direct experimental distillation
via the quantum pulse gate [32, 33]. In Sec. IV of this article, we show that
the reduced density matrix $\hat{\rho}_{v}$ can be obtained directly and
easily from an extended version of the JCM and cascaded master equation, i.e.,
without ever computing the full output state $|\psi\rangle$.

Figure 2: Illustration of the modeling of a traveling pulse as the emission
from a single mode leaking cavity.  Figure 3: (a) Illustration of our
effective model for describing the interaction between a two-level system and
an input pulse, see Eq. (8). (b) Illustration of our model permitting analysis
of the output state in any specified temporal mode, see Eq. (13). (c) When the
output temporal mode (cf. panel (b)) is chosen identical to the input pulse
mode, we can recover a particularly efficient description in the interaction
picture, see Eq. (24). The bent and curvy arrows represent the Jaynes-
Cummings-like interactions given by Hamiltonians
$\hat{H}_{\textup{JCM-I(II)(III)}}$ and the interfering emission into the
waveguide, given by the Lindblad damping operators
$\hat{L}_{\textup{JCM-I(II)(III)}}$. The combination of the Hermitian
interaction and the damping terms cause excitation to only propagate from left
to right in panels (a) and (b).

##  III Cascaded system description of a quantum pulse incident on a two-level
system

We consider a pulse of radiation with a spatial amplitude $u(x,t)$ incident on
the TLS. The pulse is assumed to move at a constant speed so it is
equivalently specified by its spatial amplitude at a fixed early moment in
time and by the time-dependent amplitude $u(t)$ at a fixed position in space.

In Refs. [34, 35, 36], we argued that the interaction with the pulse is
equivalent to the one obtained by positioning the TLS at a distance downstream
from a cavity that emits its quantum state into a pulse of the shape $u(t)$,
as depicted in Fig. 2. For now, we assume unidirectional propagation of the
field, and the constant propagation delay due to their separation $\Delta x$
can be incorporated by a formal change of time argument as if the state
assigned to the cavity mode at time $t$ in reality applies at the earlier time
$t-\Delta x/c$. The resulting equation treats the TLS as if it is located in
the immediate vicinity outside the cavity, as shown in Fig. 3(a).

To leak a pulse with temporal shape $u(t)$, the cavity in Fig. 2 must have the
coherent out-coupling amplitude

$\displaystyle
g_{u}(t)=\frac{u^{*}(t)}{\sqrt{1-\int_{0}^{t}dt^{\prime}\,|u(t^{\prime})|^{2}}}.$
(5)

According to the theory of cascaded quantum systems [37, 30], this amplitude
governs the Jaynes-Cummings-like interaction Hamiltonian between the cavity
emitter mode and the TLS,

$H_{\textup{JCM-I}}(t)=\frac{i}{2}(\sqrt{\gamma}g_{u}(t)\hat{a}^{\dagger}\hat{\sigma}^{-}-\sqrt{\gamma}g^{*}_{u}(t)\hat{a}\hat{\sigma}^{+}),$
(6)

while the ultimate leakage of all excitation from the two systems into the
waveguide is described by master equation damping terms.

Downstream from the TLS, the radiation field is composed of the vacuum input
to the waveguide, $\hat{b}_{\textup{in,vac}}(t)$, the field amplitude,
$g_{u}(t)\hat{a}$, leaked by the cavity and the amplitude,
$\sqrt{\gamma}\hat{\sigma}^{-}$, leaked by the TLS. These terms interfere, and
downstream detection of a photon in the waveguide is accompanied by the action
of the collective annihilation operator,

$\displaystyle\hat{b}_{\textup{out}}(t)=\hat{b}_{\textup{in,vac}}(t)+g_{u}^{*}(t)\hat{a}_{u}+\sqrt{\gamma}\hat{\sigma}^{-},$
(7)

on the joint state of the input vacuum modes, the cavity mode, and the TLS.

The operation of $\hat{b}_{\textup{in,vac}}(t)$ on vacuum is trivial, and the
radiation loss represented by $\hat{b}_{\textup{out}}(t)$ corresponds to the
action of the jump operator
$\hat{L}_{\textup{JCM-I}}=g_{u}^{*}(t)\hat{a}_{u}+\sqrt{\gamma}\hat{\sigma}^{-}$
in the Lindblad master equation for the joint TLS and single cavity mode
density matrix,

$\displaystyle\frac{d\rho}{dt}=-i[H_{\textup{JCM-I}},\rho]+D[\hat{L}_{\textup{JCM-I}}]\rho,$
(8)

where
$D[\hat{L}]\rho\equiv-\frac{1}{2}(\hat{L}^{\dagger}\hat{L}\rho+\rho\hat{L}^{\dagger}\hat{L})+\hat{L}\rho\hat{L}^{\dagger}$.

Eq. (8) resembles a damped version of the Jaynes-Cummings model, but our
system differs from the conventional Jaynes-Cummings dynamics in two important
ways: (i) The coupling term has a very different time dependence than
suggested in Eq. (2), and (ii) after the passage of the pulse, both the
(upstream) oscillator mode and the TLS have lost all excitations into
downstream propagation of radiation away from the cavity and TLS.

As a further interesting deviation from the usual JCM dynamics, we note that
terms in the master equation coming from the unitary commutator part and from
the Lindblad anticommutator terms
$\frac{1}{2}(\hat{L}^{\dagger}\hat{L}\rho+\rho\hat{L}^{\dagger}\hat{L})$
conspire so that only $\hat{a}\hat{\sigma}^{+}$
$(\hat{a}^{\dagger}\hat{\sigma}^{-})$ acts on $\rho$ from the left (right).
This implies the cascaded nature of the problem: no excitation propagates
upstream from the TLS to the oscillator mode.

##  IV The quantum state of the scattered light

Since the downstream light field is treated as loss, our master equation does
not reveal its final quantum state in the same way as propagation by the
Hamiltonian Eq. (1) explicitly provides the state of a cavity field mode
during and after the Jaynes-Cummings interaction with a TLS.

The downstream radiation may be multimode and hence it may be prohibitively
complicated to obtain its exact quantum state in the form of Eq. (3). There
is, however, no obstacle to the calculation of the quantum state contents of
any specified single wavepacket mode $v(t)$ in the output field. To that end,
we can apply a similar cavity trick as above and assume a downstream cavity
with a coupling amplitude $g_{v}(t)$ to the waveguided field, that would fully
capture the field populating the wavepacket mode $v(t)$, i.e., the reverse of
the process that we employed for emission of $u(t)$. Such a cavity should
couple to the waveguide by the coupling strength

$\displaystyle
g_{v}(t)=-\frac{v^{*}(t)}{\sqrt{\int_{0}^{t}dt^{\prime}\,|v(t^{\prime})|^{2}}}.$
(9)

The cascaded system of the emitter cavity, the TLS, and the final pick-up
cavity is depicted in Fig. 3(b). Its unidirectional dynamics is described by a
Hamiltonian that exchanges quanta between all three components with their
respective coupling strengths to the waveguide,

$\displaystyle\begin{split}\hat{H}_{\textup{JCM-
II}}(t)=\frac{i}{2}&\left(\sqrt{\gamma}g_{u}(t)\hat{a}_{u}^{\dagger}\hat{\sigma}^{-}+\sqrt{\gamma}g_{v}^{*}(t)\hat{\sigma}^{+}\hat{a}_{v}\right.\\\
&\left.+g_{u}(t)g_{v}^{*}(t)\hat{a}_{u}^{\dagger}\hat{a}_{v}-\mathrm{H.c.}\right).\end{split}$
(10)

The field components that are not retrieved by the pick-up cavity, are
governed by the annihilation operator

$\displaystyle\hat{b}_{\textup{out}}(t)=\hat{b}_{\textup{in,vac}}(t)+g_{u}^{*}(t)\hat{a}_{u}+\sqrt{\gamma}\hat{\sigma}^{-}+g_{v}^{*}(t)\hat{a}_{v}.$
(11)

This residual loss of radiation is represented by the Lindblad damping term,

$\displaystyle\hat{L}_{\textup{JCM-
II}}(t)=\sqrt{\gamma}\hat{\sigma}^{-}+g_{u}^{*}(t)\hat{a}_{u}+g_{v}^{*}(t)\hat{a}_{v},$
(12)

in the master equation for the joint state of our quantum system which now
consists of the TLS and two cavity modes representing the input pulse and a
selected output pulse,

$\displaystyle\frac{d\rho}{dt}=-i[\hat{H}_{\textup{JCM-
II}}(t),\rho]+D[\hat{L}_{\textup{JCM-II}}(t)]\rho.$ (13)

The scattering of a quantum pulse is calculated by assuming any initial
quantum state of the input cavity-mode $(u)$, and the ground and vacuum state
of the TLS and pick-up cavity mode. After solving the master equation (11),
the TLS and input cavity are in the ground state and the reduced density
matrix of the pick-up cavity mode represents the quantum state of the outgoing
wave packet $v(t)$, as in Eq. (4).

##  V Interaction picture with respect to the free propagation of a single
mode pulse

We have established a Hamiltonian and a master equation that describe the
interaction of an incoming pulse of quantum radiation with a quantum system,
as well as the state of the field retrieved in any single mode of the output
field Eq. (10). This description deals with two field modes rather than the
single mode in the Jaynes-Cummings model. In this section, we will show that
it is possible to recover a description where the hypothesized Jaynes-Cummings
Hamiltonian Eq. (2) is the dominant interaction term, but where extra
components take care of the multimode nature of the problem.

To achieve this simple description, we perform a transformation to the
interaction picture defined by the cavity-cavity coupling part of the
Hamiltonian

$\hat{H}_{0}(t)=\frac{i}{2}[g_{u}(t)g_{v}^{\ast}(t)\hat{a}_{u}^{\dagger}\hat{a}_{v}-g_{v}(t)g_{u}^{\ast}(t)\hat{a}_{v}^{\dagger}\hat{a}_{u}].$
(14)

At this point, we assume that the input and output virtual cavities address
the same temporal shape $u(t)=v(t)$. The general case is described in [36].
The equations of motion for the operators in the interaction picture are

$\displaystyle\dot{\hat{a}}_{u,\textup{IP}}(t)$
$\displaystyle=\frac{1}{2}g_{u}(t)g_{v}^{*}(t)\hat{a}_{v,\textup{IP}}(t)$ (15)
$\displaystyle\dot{\hat{a}}_{v,\textup{IP}}(t)$
$\displaystyle=-\frac{1}{2}g_{u}^{*}(t)g_{v}(t)\hat{a}_{u,\textup{IP}}(t),$
(16)

where IP denotes that the operator is described in the interaction picture.
The solution to this coupled set of equations is

$\displaystyle\hat{a}_{u,\textup{IP}}(t)$
$\displaystyle=\cos{\theta(t)}\hat{a}_{u}(0)-\sin{\theta(t)}\hat{a}_{v}(0)$
(17) $\displaystyle\hat{a}_{v,\textup{IP}}(t)$
$\displaystyle=\cos{\theta(t)}\hat{a}_{v}(0)+\sin{\theta(t)}\hat{a}_{u}(0),$
(18)

where $\theta(t)$ is determined by the condition

$\displaystyle\frac{d}{dt}\theta(t)=-\frac{1}{2}g_{u}(t)g_{v}^{*}(t).$ (19)

Our assumption that $v(t)=u(t)$ guarantees that $\theta(t)$ is real, and we
can show that

$\displaystyle\sin^{2}{\theta(t)}=\int_{0}^{t}dt^{\prime}|u(t^{\prime})|^{2},$
(20)

by differentiating both sides of the equation with respect to time. This
yields

$2\sin{\theta(t)}\cos{\theta(t)}\frac{d\theta(t)}{dt}=|u(t)|^{2},$ (21)

and we note that due to (20), the factors $\cos\theta(t)$ and $\sin\theta(t)$
are nothing but the denominators in Eqs. (5) and (9).

It is evident from the definition of $\theta(t)$ that $\sin^{2}\theta(t=0)=0$
and $\sin^{2}\theta(t\rightarrow\infty)=1$, such that during the interaction
with the pulse, the cavity operators are completely exchanged. In the
interaction picture the $u$-cavity operator “follows the pulse”, and in the
case of no intermediate emitter, it represents the constant content of the
pulse at all times. When the scatterer is present, the transformed Hamiltonian
in the interaction picture becomes

$\displaystyle\begin{split}&\hat{H}_{\textup{JCM-
III}}(t)=i\sqrt{\gamma}u(t)(\hat{a}_{u,\textup{IP}}(t)^{\dagger}\hat{\sigma}^{-}-\hat{\sigma}^{+}\hat{a}_{u,\textup{IP}}(t))\\\
&+\frac{i}{2}\sqrt{\gamma}u(t)(\cot\theta-\tan\theta)(\hat{a}_{v,\textup{IP}}(t)^{\dagger}\hat{\sigma}^{-}-\hat{\sigma}^{+}\hat{a}_{v,\textup{IP}}(t)),\end{split}$
(22)

which contains in the first line precisely the naive guess, hypothesized in
Eq. (2) in the Introduction. Our analysis, however, also yields the
interaction with a second mode, namely the initially empty downstream cavity
mode, which gets transiently occupied and hence forms a non-Markovian element
in the TLS dynamics in the interaction picture, see Fig. 3(c). The pulse mode
suffers no direct loss, while excitation transferred to the TLS and
subsequently to the second mode leaks as revealed by evaluating the Lindblad
operator in the interaction picture

$\displaystyle\hat{L}_{\textup{JCM-
III}}(t)=\sqrt{\gamma}\hat{\sigma}^{-}-(\tan\theta+\cot\theta)u(t)\hat{a}_{v,\textup{IP}}(t).$
(23)

To summarize, we have found that the interaction between a traveling pulse and
a quantum emitter has a Jaynes-Cummings-like component, which is supplemented
with loss and coupling to a secondary, lossy mode in the interaction picture
master equation,

$\displaystyle\frac{d\rho}{dt}=-i[\hat{H}_{\textup{JCM-
III}}(t),\rho]+D[\hat{L}_{\textup{JCM-III}}(t)]\rho.$ (24)

Figure 4:  Rabi oscillations of a TLS interacting with an $|n=20\rangle$ Fock
state. Panel (a) shows the TLS excitation dynamics (lower curves) and the mean
number of oscillator quanta (upper curves) for the unitary JCM model with
Gaussian time-dependent coupling $\chi(t)=\sqrt{\gamma}u(t)$, with
$\tau=\gamma^{-1}$, and for the model with a TLS with decay rate $\gamma$.
Panel (b) shows the results of Eqs. (8) and (13) for a travelling pulse with
the same Gaussian form of $u(t)$ as in panel (a). The TLS excitation dynamics
is similar but differ from that in panel (a), while the excitation leaks from
the upstream $u$-cavity, and is partly recovered in the downstream $v$-cavity.
Note the left y-axis pertains to the TLS and the right y-axis pertains to the
cavity excitation dynamics. Panel (c) shows the same results but for the
interaction picture modes applied in Eq. (24). The $u_{\textup{IP}}$-mode
represents the traveling pulse and its excitation dynamics is similar to the
one of the oscillator mode in the damped JCM, shown in panel (a). The
ancillary $v_{\textup{IP}}$-mode is weakly excited and causes the small
quantitative discrepancies with the results in panel (a).

##  VI Results

In the previous sections, we have reviewed an efficient description of the
interaction between a TLS and a traveling pulse mode of quantum radiation. We
have shown that despite the multimode character of propagating fields, it is
possible to recast the problem with a Jaynes-Cummings-like interaction
Hamiltonian and Lindblad loss terms acting on only the TLS and one or two
oscillator modes.

Our analysis has identified three different master equations (8), (13) and
(24). In this section, we solve these equations, and we show how the
interaction between a TLS and a traveling pulse has similarities with and
deviates from the time-dependent interaction with a resonant cavity mode, with
and without a dissipating TLS.

Numerical results in this section are obtained by use of the `QuTiP` toolbox
[38, 39].

###  VI.1 Rabi oscillations with a number state pulse

In the master equation Eq. (8), the coupling strength $g_{u}(t)$ differs from
the shape $u(t)$ of the pulse. Note, however, that $H_{\textup{JCM-I}}$ acts
on a quantum state with progressively fewer photons, governed by the same
factor as the denominator in Eq. (5), and hence its strength follows the value
of $u(t)$ in an average sense.

Figure 4(a) shows the photon number and TLS excitation probability according
to the unitary JCM dynamics of Eq. (1) with a time-dependent coupling,
$\chi(t)=\sqrt{\gamma}u(t)$, with a Gaussian temporal width
$\tau=\gamma^{-1}$, for the cavity mode initially in an $|n=20\rangle$ Fock
state. The figure also shows the solution of the master equation incorporating
decay from the excited state of the TLS with rate $\gamma$, where we note a
loss of a little more than $2$ photons.

Figure 4(b) shows the results for an incident Gaussian wave packet, $u(t)$,
with an $|n=20\rangle$ Fock state of radiation scattering on the TLS, as in
Eq. (8). The TLS excited state population yields similar damped Rabi
oscillations as in panel (a), accompanying the rapid release of the field from
the $u$-cavity. Fig. 4(b) also shows the population emerging in the outgoing
wavepacket $v(t)$ [chosen here on the same form as $u(t)$] after the
interaction with the TLS, as in Eq. (13). We note a loss of a little less than
$2$ quanta of radiation to other modes.

Figure 4(c) shows the same TLS excitation dynamics but calculated according to
Eq. (24). The interaction picture annihilation operator
$\hat{a}_{u,\mathrm{IP}}(t)$ represents the $u(t)$ wave packet mode both
before and after the interaction with the TLS, and its population dynamics
(upper dot dashed curve) resembles the conventional JCM dynamics in presence
of TLS decay, shown in panel (a). The interaction picture annihilation
operator $\hat{a}_{v,\mathrm{IP}}(t)$ represents an orthogonal mode that is
weakly excited and thus modifies the TLS dynamics during the interaction.

Although, we obtain results showing similar features with the single-mode
time-dependent Jaynes-Cummings model with damping and with our modified JCM-
like master equations, there are quantitative discrepancies. These are due to
the multimode character of the non-linear scattering of traveling pulse, which
is only taken fully into account in the extended versions of the JCM model
presented here.

###  VI.2 Number state vs. coherent state dynamics. Collapses and revivals.

The cascaded nature of the master equation, Eqs. (8) and (13), ensures that
the combination of the commutator and the Lindblad anti-commutator terms are
responsible for excitations propagating only downstream (as only annihilation
and no creation operators for the upstream cavity act on $\rho$ from the left,
and vice versa from the right). This has another striking consequence: if the
cavity field mode is initially in a coherent state with amplitude
$\alpha_{0}$, it remains a (damped) coherent state $|\alpha(t)\rangle$ for all
later times. The TLS thus evolves as if a classical pulse drives it with a
time-dependent field amplitude $\alpha(t)g_{u}^{*}(t)=\alpha_{0}u(t)$ [26,
35]. In addition, the TLS decays to the waveguide with the rate $\gamma$, and
the system may thus show damped Rabi oscillations but no trace of the collapse
and revival phenomenon, which is a crucial feature of the interaction with a
coherent state in a cavity mode.

Conversely, for an initial number state, which would exhibit perfect Rabi
oscillations in the JCM, our Eq. (8) evolves the system through a gradual loss
of the initial excitation. It hence explores a distribution of excitation
subspaces with a corresponding distribution of Rabi oscillation frequencies
$g\sqrt{n}$. This could potentially lead to prediction and observation of
collapse and revival occurring in the interaction with a traveling pulse in a
number state, which would be the complete reverse of roles of coherent states
and number states in the JCM.

The gradual loss of excitation and broadening of the number state distribution
in the $u$-mode, does not, however, occur to the same extent, when we solve
the dynamics in the interaction picture, cf., the upper curve in Fig. 4(c).
Since our Eqs. (8) and (24) yield identical TLS dynamics, despite the
structure of Eq. (8), we should hence not expect to observe the Rabi
oscillation collapses and revivals, while we should expect to observe damped
oscillations due to the decay of the TLS.

###  VI.3 Further dissipation, non-chiral coupling

In the previous paragraphs, we have mentioned the possibility of incorporating
decay and dissipation in the JCM through damping terms in a master equation.
Similarly, it is straightforward to supplement our master equations with
further damping terms of the same form $D[\hat{L}_{i}]\rho$ as the Lindblad
terms already present in Eqs. (8), (13) and (24).

This is not the place for an exhaustive review of such damping terms, but let
us consider here what may have seemed a severe restriction of our analyses - a
perfect unidirectional interaction between the TLS and the propagating fields.
Most emitters couple to fields propagating in both directions along
waveguides, and if we want to incorporate the possibility of reflection from
the emitter, it is only a matter of adding a corresponding Lindblad term
$D[\hat{L}_{\mathrm{refl}}]\rho$ with
$\hat{L}_{\mathrm{refl}}=\sqrt{\gamma^{\prime}}\hat{\sigma}^{-}$ to our master
equations. The analysis of the TLS dynamics and the analysis of the forward
propagating field proceed by solving the master equations with this extra
term.

Figure 5: Panel (a) presents the solution of Eq. (24) for the interaction of a
TLS with a Gaussian pulse of width $\tau=0.3799\gamma^{-1}$ and two quanta of
excitation. The calculations show the removal of one quantum from the input
mode and the emergence of one quantum in a new, orthogonal temporal mode
(these modes are shown in the inset). The density matrix elements for the
modes (not shown) reveal that the two output modes are, indeed, populating
$|n=1\rangle$ Fock states. Panel (b) shows the fidelity of the subtraction of
a single quantum from the incident pulse as function of the pulse duration,
and for different values of the TLS emission rate in the direction opposite to
the pulse propagation. Perfect subtraction only occurs in the case of perfect
chiral coupling of the TLS to the travelling pulse.

In Fig. 5(a), we show a particular process that occurs in the case of perfect
chiral coupling of a TLS to a two-photon wave packet, namely the very precise
removal of exactly one photon from the incident pulse $u(t)$ (black curve)
[14]. The dynamics is obtained in the interaction picture, using Eq. (24).
Naively, this process is well understood from the JCM Rabi oscillation
dynamics, causing excitation of the TLS with a $\pi$-pulse and subsequent
emission of a single quantum of excitation. However, in our case, the TLS is
constantly coupled to and emitting into the waveguide modes. As shown by the
red curve in Fig. 5(a), the TLS excitation is not complete, but for the right
parameter values, over time it emits one quantum of excitation (blue dotted
curve) into a temporal pulse mode $v_{2}(t)$ which is orthogonal to $u(t)$
(see the modes in the inset). This effective splitting of a number state pulse
into two one-photon states may have useful implications in quantum information
processing with light, but will it only work if we can ensure the
unidirectional coupling of the TLS to the traveling fields? The answer is
given in Fig. 5(b), where we show the fidelity of the photon subtraction
process as a function of the duration of the incident Gaussian 2-photon
pulses. The different curves are obtained for different values of the emission
rate in the direction of reflection from the TLS. Only for the perfect chiral
coupling do we find a pulse duration with perfect subtraction.

##  VII Conclusion

The textbook Jaynes-Cummings model assumes a single quantized field mode and,
hence, it does not describe the interaction between a two-level system and a
traveling pulse of light due to the multi-mode character of the field
continuum available for the scattered photons. We have shown that in a
cascaded open-system approach it is possible to obtain master equations (8),
(13) and (24) of similar form and complexity as the ones describing the
Jaynes-Cummings model. We remind the reader that the cavities depicted in
Figs. 2 and 3 are only introduced to aid the theoretical description of what
are propagating temporal modes in the actual experiment. It is through that
effective description that we recover single- and few-mode variants of the
Jaynes-Cummings interaction. With the assumption of the validity of the Born-
Markov approximation of the emission of the TLS into the waveguide, our master
equations are exact.

By our discussions of the different forms of the master equations and by our
presentation of results for specific processes we have demonstrated the most
important consequences of the modifications to the Jaynes-Cummings model in
the case of propagating light. We are delighted that by so simple
modifications of the theory, the 60-year-old Jaynes-Cummings model may still
be counted on and provide intuition and quantitative predictions for a wide
range of future applications.

##  VIII Acknowledgements

We thank Anton L. Andersen and Danil Kornovan for their helpful discussions.
The authors acknowledge support from the Danish National Research Foundation
through the Center of Excellence for Complex Quantum Systems and the Center of
Excellence for Hybrid Quantum Networks (Grant agreements No. DNRF156 and
DNRF139) and the Carlsberg Foundation through the Semper Ardens project QCooL.

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$L^{\prime\prime}:c_{2}z_{1}-c_{1}z_{2}=0$. By elementary algebra, we can see
that the Euclidean distance $d^{\prime}:=d^{\prime}(y_{1},y_{2})$ (resp. the
Euclidean distance $d^{\prime\prime}:=d^{\prime\prime}(y_{1},y_{2})$) from a
given point $(y_{1},y_{2})\in\mathbb{R}$ to straight line $L^{\prime}$ (resp.
straight line $L^{\prime\prime}$) is given by

$\displaystyle
d^{\prime}=\frac{|c_{1}y_{1}+c_{2}y_{2}|}{\sqrt{c_{1}^{2}+c_{2}^{2}}}\qquad\mbox{and}\qquad
d^{\prime\prime}=\frac{|c_{2}y_{1}-c_{1}y_{2}|}{\sqrt{c_{1}^{2}+c_{2}^{2}}}.$
(37)

Moreover, $L^{\prime}$ and $L^{\prime\prime}$ are perpendicular (by
construction), so the Pythagorean theorem ensures that $d^{\prime
2}+d^{\prime\prime 2}=y_{1}^{2}+y_{2}^{2}$. Accordingly, the larger distance
$\max\\{d^{\prime},\;d^{\prime\prime}\\}$ is lower bounded by

$\displaystyle\max\\{d^{\prime},\;d^{\prime\prime}\\}\geq\sqrt{(d^{\prime
2}+d^{\prime\prime
2})/2}=\sqrt{(y_{1}^{2}+y_{1}^{2})/2}\geq\sqrt{32\ln(n)/n},$ (38)

where the first step uses the AM-GM inequality, and the last step holds since
$(y_{1},y_{2})\notin
R^{\prime}_{y}=[-\sqrt{64\ln(n)/n},\;\sqrt{64\ln(n)/n}]^{2}$ (see Figure 3).

Putting everything together results in

$\displaystyle|G_{c_{1},c_{2}}(y_{1},y_{2})|\leq\eqref{eq:G-bound4}$
$\displaystyle=\frac{1}{2}\cdot\Big{(}e^{-\frac{c_{1}^{2}+c_{2}^{2}}{8}\cdot
d^{\prime 2}}+e^{-\frac{c_{1}^{2}+c_{2}^{2}}{8}\cdot d^{\prime\prime
2}}\Big{)}$ $\displaystyle\leq\frac{1}{2}\cdot\Big{(}e^{-\frac{1}{4}\cdot
d^{\prime 2}}+e^{-\frac{1}{4}\cdot d^{\prime\prime 2}}\Big{)}$
$\displaystyle\leq\frac{1}{2}\cdot\Big{(}e^{-\frac{1}{4}\cdot
0}+e^{-\frac{1}{4}\cdot 32\ln(n)/n}\Big{)}$
$\displaystyle=\frac{1}{2}\cdot\Big{(}1+\frac{1}{\sqrt[n]{n^{8}}}\Big{)}$
$\displaystyle\leq\frac{1}{\sqrt[n]{n^{2}}},$

where the first step applies (37), the second step follows since the
coefficients $c_{1},c_{2}\in C$ are nonzero integers, the third step applies
(38) and the fact that $\min\\{d^{\prime},\;d^{\prime\prime}\\}\geq 0$, the
fourth step is elementary algebra, and the last step holds whenever $n\geq 6$
(see (17)).

Combining all the three cases together gives (32). This completes the proof of
Claim 7. ∎

Below we use Claims 8, 10 and 9 (as auxiliaries) to evaluate (Part 3) in
Corollary 5. This evaluation together with Claims 6 and 7 immediately gives
Lemma 6.

######  Claim 8.

There exists some constant $\epsilon_{2}:=\epsilon_{2}(|C|,a)>0$ such that,
for any $(\theta_{1},\theta_{2})\in[-\pi,\;\pi]^{2}\setminus
R^{\prime\prime}_{\theta}$,

$\displaystyle|G(\theta_{1},\theta_{2}))|\leq|C|^{2}-\frac{1-1/a}{2}\leq|C|^{(2-\epsilon_{2})}.$

$R^{\prime\prime}_{\theta}$$z_{1}$$z_{2}$$0$$-\pi$$\pi$$-\pi$$\pi$ (a) When
$(c_{1},c_{2})=(1,1)$

$R^{\prime\prime}_{\theta}$$z_{1}$$z_{2}$$0$$-\pi$$\pi$$-\pi$$\pi$ (b) When
$(c_{1},c_{2})=(2,1)$

Figure 4: The subregion
$R_{c_{1},c_{2}}:=\\{(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}:\mbox{$(c_{1}\theta_{1}+c_{2}\theta_{2}),(c_{2}\theta_{1}-c_{1}\theta_{2})\in[-\frac{b_{0}}{M},\;\frac{b_{0}}{M}]_{\bigcirc}$}\\}$
is the intersection of the red area (on which
$(c_{1}\theta_{1}+c_{2}\theta_{2})\in[-\frac{b_{0}}{M},\;\frac{b_{0}}{M}]_{\bigcirc}$)
and the blue area (on which
$(c_{2}\theta_{1}-c_{1}\theta_{2})\in[-\frac{b_{0}}{M},\;\frac{b_{0}}{M}]_{\bigcirc}$).
Given any parameters $t_{R},t_{B}\in\mathbb{R}$, by construction a (red)
straight line $c_{1}\theta_{1}+c_{2}\theta_{2}=t_{R}$ and a (blue) straight
line $c_{2}\theta_{1}-c_{1}\theta_{2}=t_{B}$ are vertical. The subregion
$R^{\prime\prime}_{\theta}$ (namely the dashed square) is excluded from
$[-\pi,\pi]^{2}$. Figures 4a and 4b respectively choose $(c_{1},c_{2})=(1,1)$
and $(c_{1},c_{2})=(2,1)$.

###### Proof.

Define the function
$f_{c_{1},c_{2}}(\theta_{1},\theta_{2}):=\frac{1}{2}(f(c_{1}\theta_{1}+c_{2}\theta_{2})+f(c_{2}\theta_{1}-c_{1}\theta_{2}))$
for each pair $(c_{1},c_{2})\in C^{2}$. Because $|f(\theta)|\leq 1$ for any
$\theta\in\mathbb{R}$ (see (15)), we also have
$|f_{c_{1},c_{2}}(\theta_{1},\theta_{2})|\leq 1$ for any
$(\theta_{1},\theta_{2})\in\mathbb{R}^{2}$. Recall that
$|f(\theta)|\leq\frac{1}{a}$ for any $|\theta|\in[\frac{b_{0}}{M},\;\pi]$ (see
(26)) and $\frac{b_{0}}{M}\leq\frac{(\pi/2)\cdot b}{8d\cdot
b}\leq\frac{\pi}{16}$ (see (17) and (18)). Since $f$ is $2\pi$-periodic (see
(15)), only on the $2\pi$-periodic range
$[-\frac{b_{0}}{M},\;\frac{b_{0}}{M}]_{\bigcirc}:=\bigcup_{k\in\mathbb{Z}}[-\frac{b_{0}}{M}+2k\pi,\;\frac{b_{0}}{M}+2k\pi]$
can we possibly have $|f(\theta)|>\frac{1}{a}$. Regarding the function
$f_{c_{1},c_{2}}$, we consider the analogous subregion

$R_{c_{1},c_{2}}:=\Big{\\{}(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}:\mbox{$(c_{1}\theta_{1}+c_{2}\theta_{2}),(c_{2}\theta_{1}-c_{1}\theta_{2})\in[-\frac{b_{0}}{M},\;\frac{b_{0}}{M}]_{\bigcirc}$}\Big{\\}}.$

For any point $(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}\setminus
R_{c_{1},c_{2}}$, we have either
$|f(c_{1}\theta_{1}+c_{2}\theta_{2})|\leq\frac{1}{a}$ or
$|f(c_{2}\theta_{1}-c_{1}\theta_{2})|\leq\frac{1}{a}$ or both, which (together
with the fact that $|f(\theta)|\leq 1$ for any $\theta\in\mathbb{R}$) implies

$\displaystyle|f_{c_{1},c_{2}}(\theta_{1},\theta_{2})|\leq\frac{|f(c_{1}\theta_{1}+c_{2}\theta_{2})|+|f(c_{2}\theta_{1}-c_{1}\theta_{2})|}{2}\leq\frac{1+1/a}{2}.$
(39)

Because $C=C(d)=\\{\pm 1,\pm 2,\dots,\pm d\\}$ for a fixed integer $d>1$, the
considered pair-indices $(c_{1},c_{2})\in C^{2}$ include both
$(c_{1},c_{2})=(1,1)$ and $(c_{1},c_{2})=(2,1)$. As shown in Figure 4a, the
union of the $\frac{b_{0}}{M}$-neighborhoods of four points $(\pm\pi,\pm\pi)$
covers $R_{1,1}$. Also, as shown in Figure 4b, the union of the
$\frac{b_{0}}{M}$-neighborhoods of four points
$(\frac{4\pi}{5},\frac{2\pi}{5})$, $(-\frac{2\pi}{5},\frac{4\pi}{5})$,
$(-\frac{4\pi}{5},-\frac{2\pi}{5})$ and $(\frac{2\pi}{5},-\frac{4\pi}{5})$
covers $R_{2,1}$. We must have $R_{1,1}\cap R_{2,1}=\emptyset$, because the
$\frac{b_{0}}{M}\leq\frac{\pi}{16}$ is small enough.

Given the above arguments, we deduce that for any
$(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}\setminus R^{\prime\prime}_{\theta}$,

$\displaystyle|G(\theta_{1},\theta_{2})|$
$\displaystyle=\Bigg{|}\sum_{(c_{1},c_{2})\in
C^{2}}f(c_{1}\theta_{1}+c_{2}\theta_{2})\Bigg{|}$
$\displaystyle=\Bigg{|}\sum_{(c_{1},c_{2})\in
C^{2}}f_{c_{1},c_{2}}(\theta_{1},\theta_{2})\Bigg{|}$
$\displaystyle\leq\frac{1+1/a}{2}+|C|^{2}-1$
$\displaystyle=|C|^{2}-\frac{1-1/a}{2}.$

Here the second step holds since, regarding the definition of
$f_{c_{1},c_{2}}$, the mapping $(c_{1},c_{2})\mapsto(c_{2},-c_{1})$ is one-to-
one. The third step uses (39), the fact that
$|f_{c_{1},c_{2}}(\theta_{1},\theta_{2})|\leq 1$ for each pair
$(c_{1},c_{2})\in C^{2}$ and any $(\theta_{1},\theta_{2})\in\mathbb{R}^{2}$,
and the fact that $R_{1,1}\cap R_{2,1}=\emptyset$.

Because $a>|C|^{2}/\epsilon>1$ (see (17)), there must exist some constant
$\epsilon_{2}:=\epsilon_{2}(|C|,a)>0$ such that
$|C|^{2}-(1-1/a)/2\leq|C|^{(2-\epsilon_{2})}$. This finishes the proof of
Claim 8. ∎

######  Claim 9.

The subregion $Q^{\prime}\subseteq[-\pi,\pi]^{2}$ defined below has measure at
most $|Q^{\prime}|\leq|C|^{2}\cdot\frac{4\pi b_{0}}{M}$.

$Q^{\prime}:=\Big{\\{}(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}:|G(\theta_{1},\theta_{2})|>\frac{|C|^{2}}{a}\Big{\\}}.$

###### Proof.

To begin with, we introduce an auxiliary set $A_{c_{1},w}$ for each $c_{1}\in
C$ and any offset $w\in\mathbb{R}$.

$\displaystyle
A_{c_{1},w}:=\Big{\\{}\theta_{1}\in[-\pi,\pi]:|f(c_{1}\theta_{1}+w)|>\frac{1}{a}\Big{\\}}.$
(40)

Recall (28) that $|A_{c_{1},0}|\leq\frac{2b_{0}}{M}$. Because $f$ is
$2\pi$-periodic, regarding any integer $c_{1}\in C$, the function
$f_{c_{1}}(\theta_{1}):=f(c_{1}\theta_{1})$ is also $2\pi$-periodic. Thus, we
also have

$\displaystyle|A_{c_{1},w}|\leq\frac{2b_{0}}{M}$ (41)

for any $c_{1}\in C$, $w\in\mathbb{R}$. It follows that, for any pair
$(c_{1},c_{2})\in C^{2}$, we have

$\displaystyle\Big{|}\Big{\\{}(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}:|f(c_{1}\theta_{1}+c_{2}\theta_{2})|>\frac{1}{a}\Big{\\}}\Big{|}\leq
2\pi\cdot\frac{2b_{0}}{M}=\frac{4\pi b_{0}}{M}.$ (42)

Recall that $G(\theta_{1},\theta_{2})=\sum_{(c_{1},c_{2})\in
C^{2}}f(c_{1}\theta_{1}+c_{2}\theta_{2})$. Suppose
$|G(\theta_{1},\theta_{2})|>|C|^{2}/a$, then we must have
$|f(c_{1}\theta_{1}+c_{2}\theta_{2})|>1/a$ for at least one pair
$(c_{1},c_{2})\in C^{2}$. Union-bounding the measure (42) over all pairs
$(c_{1},c_{2})\in C^{2}$ concludes the proof of Claim 9. ∎

######  Claim 10.

The subregion $Q^{\prime\prime}\subseteq[-\pi,\pi]^{2}$ defined below has
measure at most $|Q^{\prime\prime}|\leq|C|^{4}\cdot(\frac{2b_{0}}{M})^{2}$.

$Q^{\prime\prime}:=\Big{\\{}(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}:|G(\theta_{1},\theta_{2})|>|C|+\epsilon\Big{\\}}.$

###### Proof.

For this proof, we introduce a convenient decomposition of the function $G$.
Observe that $G(\theta_{1},\theta_{2})=\sum_{c_{2}\in
C}q(\theta_{1},c_{2}\theta_{2})$, where

$q(\theta_{1},w):=\sum_{c_{1}\in C}f(c_{1}\theta_{1}+w).$

This allows us to rewrite $Q^{\prime\prime}$ as

$Q^{\prime\prime}=\Big{\\{}(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}:\Big{|}\sum_{c_{2}\in
C}q(\theta_{1},c_{2}\theta_{2})\Big{|}>|C|+\epsilon\Big{\\}}.$

As $\max_{x}q(x)\leq|C|\cdot\max_{x}f(x)\leq|C|$, $q$ is upper-bounded by
$|C|$. Thus for any point $(\theta_{1},\theta_{2})\in Q^{\prime\prime}$, there
must exist two distinct coefficients $c_{2},c^{\prime}_{2}\in C$ such that
$|q(\theta_{1},c_{2}\theta_{2})|,|q(\theta_{1},c^{\prime}_{2}\theta_{2})|>\frac{\epsilon}{|C|}$.
In other words, the considered set $Q^{\prime\prime}$ is covered by

$\displaystyle Q^{\prime\prime}\subseteq\bigcup_{c_{2}\neq c^{\prime}_{2}\in
C}Q_{c_{2},c^{\prime}_{2}},$ (43)

where

$\displaystyle
Q_{\alpha,\beta}:=\Big{\\{}(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}:|q(\theta_{1},\alpha\theta_{2})|,|q(\theta_{1},\beta\theta_{2})|\geq\frac{|C|}{a}\Big{\\}}.$

This definition of $Q_{\alpha,\beta}$ suffices because $a>|C|^{2}/\epsilon$
(see (17)) and thus $\frac{|C|}{a}<\frac{\epsilon}{|C|}$.

To complete the proof, it suffices to show that

$\displaystyle|Q_{c_{2},c^{\prime}_{2}}|\leq|C|^{2}\cdot\Big{(}\frac{2b_{0}}{M}\Big{)}^{2}$
(44)

for each pair $(c_{2},c^{\prime}_{2})\in C^{2}$ with $c_{2}\neq
c^{\prime}_{2}$. (Suppose so, then the measure $|Q^{\prime\prime}|$ can be
upper-bounded by $|C|^{2}\cdot\max_{c_{2}\neq
c^{\prime}_{2}}|Q_{c_{2},c^{\prime}_{2}}|\leq|C|^{4}\cdot(\frac{2b_{0}}{M})^{2}$
using (43).)

To show (44), fix a pair $(c_{2},c^{\prime}_{2})\in C^{2}$ such that
$c_{2}\neq c^{\prime}_{2}$. We observe that

$\displaystyle
Q_{c_{2},c^{\prime}_{2}}=\Big{\\{}(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}:\theta_{1}\in\big{(}B_{c_{2}\theta_{2}}\cap
B_{c^{\prime}_{2}\theta_{2}}\big{)}\Big{\\}},$

for $B_{w}:=\\{\theta_{1}\in[-\pi,\pi]:|q(\theta_{1},w)|>\frac{|C|}{a}\\}$.
Intuitively, $B_{w}$ captures the values for which $|q(\theta_{1},w)|$ is
large. Because $B_{w}$ is determined by the function
$q(\theta_{1},w)=\sum_{c\in C}f(c_{1}\theta_{1}+w)$, it has a convenient
property: for any $w,z\in\mathbb{R}$, $B_{w+z}$ is just the set $B_{w}$
translated by $z\pmod{2\pi}$. Figure 5 provides a visual aid for the structure
of this set and the remainder of the proof.

It will be convenient to write $|Q_{c_{2},c^{\prime}_{2}}|$ in terms of the
probability that a random point in $[-\pi,\pi]^{2}$ is contained in
$Q_{c_{2},c^{\prime}_{2}}$. We have

$\displaystyle|Q_{c_{2},c^{\prime}_{2}}|$
$\displaystyle=4\pi^{2}\cdot\mathop{{\bf
Pr}\/}_{\bm{\theta}_{1},\bm{\theta_{2}}\sim[-\pi,\pi]}\Big{[}\bm{\theta}_{1}\in\big{(}B_{c_{2}\bm{\theta}_{2}}\cap
B_{c^{\prime}_{2}\bm{\theta}_{2}}\big{)}\Big{]}$
$\displaystyle=4\pi^{2}\cdot\mathop{{\bf
Pr}\/}_{\bm{\theta}_{1},\bm{\theta_{2}}\sim[-\pi,\pi]}\Big{[}\bm{\theta}_{1}\in\big{(}B_{0}\cap
B_{(c^{\prime}_{2}-c_{2})\bm{\theta}_{2}}\big{)}\Big{]}$
$\displaystyle=4\pi^{2}\cdot\mathop{{\bf
Pr}\/}_{\bm{\theta}_{1}\sim[-\pi,\pi]}\Big{[}\bm{\theta}_{1}\in
B_{0}\Big{]}\cdot\mathop{{\bf
Pr}\/}_{\bm{\theta}_{1},\bm{\theta_{2}}\sim[-\pi,\pi]}\Big{[}\bm{\theta}_{1}\in
B_{(c^{\prime}_{2}-c_{2})\bm{\theta}_{2}}~{}\Big{|}~{}\bm{\theta}_{1}\in
B_{0}\Big{]}.$ (45)

Here the second step follows from translating both sets by
$c_{2}\bm{\theta}_{2}\pmod{2\pi}$, which does not change the size of the
intersection. And the last step uses the identity $\operatorname{{\bf
Pr}}[A\cap B]=\operatorname{{\bf Pr}}[A]\cdot\operatorname{{\bf Pr}}[B\mid
A]$.

Given any nonzero integer $k$ and any fixed offset $w\in[-\pi,\pi]$, we
observe that

$\displaystyle\mathop{{\bf Pr}\/}_{\bm{\theta}_{2}\sim[-\pi,\pi]}[w\in
B_{k\bm{\theta}_{2}}]=\mathop{{\bf Pr}\/}_{\bm{y}\sim[-\pi,\pi]}[\bm{y}\in
B_{0}]=\frac{|B_{0}|}{2\pi},$ (46)

because sampling $\bm{\theta_{2}}$ uniformly from $[-\pi,\pi]$ distributes
$k\bm{\theta}_{2}\pmod{2\pi}$ uniformly over $[-\pi,\pi]$. (46) holds for any
nonzero integer $k\neq 0$ and any offset $w\in[-\pi,\pi]$, so both
(conditional) probabilities in (45) can be substituted with
$\eqref{eq:discard-sub2-4:6}=|B_{0}|/(2\pi)$.

Recall the definition of the set family $A_{c_{1},w}$ in (40). Using (41) to
union-bound (45) yields (44):

$|Q_{c_{2},c^{\prime}_{2}}|=4\pi^{2}\cdot\Big{(}\frac{|B_{0}|}{2\pi}\Big{)}^{2}\leq\Big{(}\sum_{c\in
C}|A_{c,0}|\Big{)}^{2}\leq|C|^{2}\cdot\Big{(}\frac{2b_{0}}{M}\Big{)}^{2}.\qed$

$B_{0}$$\theta_{2}$$\theta_{1}$$0$$-\pi$$-2\pi$$-3\pi$$\pi$$2\pi$$3\pi$ Figure
5: Visual aid for the proof of Claim 10. For any offset
$\theta_{2}\in[-\pi,\pi]$, the range $B_{(c^{\prime}_{2}-c_{2})\theta_{2}}$ is
the intersection of the line segment $(-\pi,\theta_{2})$–$(\pi,\theta_{2})$
with the gray regions. The red regions indicate
$\\{(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}:\theta_{1}\in B_{0}\cap
B_{(c^{\prime}_{2}-c_{2})\theta_{2}}\\}$.

######  Corollary 5.

$\eqref{eq:ez2-part3}=o(\rho_{n}^{2})$.

###### Proof.

For ease of reference, let us restate the results in Claims 8, 10 and 9:

  * •

Claim 8: $|G(\theta_{1},\theta_{2}))|\leq|C|^{(2-\epsilon_{2})}$ for any point
$(\theta_{1},\theta_{2})\in[-\pi,\;\pi]^{2}\setminus
R^{\prime\prime}_{\theta}$.

  * •

Claim 9: The subregion $Q^{\prime}$ on which
$|G(\theta_{1},\theta_{2})|>|C|^{2}/a$ has measure at most
$|C|^{2}\cdot\frac{4\pi b_{0}}{M}$.

  * •

Claim 10: The subregion $Q^{\prime\prime}$ on which
$|G(\theta_{1},\theta_{2})|>|C|+\epsilon$ has measure at most
$|C|^{4}\cdot(\frac{2b_{0}}{M})^{2}$.

Obviously, the three subregions $(Q^{\prime\prime}\setminus
R^{\prime\prime}_{\theta})$ and $(Q^{\prime}\setminus Q^{\prime\prime})$ and
$[-\pi,\;\pi]^{2}\setminus Q^{\prime}$ together covers the domain of
integration for (Part 3), namely $[-\pi,\;\pi]^{2}\setminus
R^{\prime\prime}_{\theta}$. As a consequence, we have

$\displaystyle\eqref{eq:ez2-part3}$
$\displaystyle\leq\frac{1}{4\pi^{2}}\bigg{(}\iint\limits_{(\theta_{1},\theta_{2})\in[-\pi,\;\pi]^{2}\setminus
Q^{\prime}}+\iint\limits_{(\theta_{1},\theta_{2})\in Q^{\prime}\setminus
Q^{\prime\prime}}+\iint\limits_{(\theta_{1},\theta_{2})\in
Q^{\prime\prime}\setminus
R^{\prime\prime}_{\theta}}\bigg{)}|G(\theta_{1},\theta_{2})|^{n}\cdot\mathrm{d}\theta_{1}\mathrm{d}\theta_{2}$
$\displaystyle\leq\frac{1}{4\pi^{2}}\bigg{(}\Big{(}\frac{|C|^{2}}{a}\Big{)}^{n}\cdot
4\pi^{2}+(|C|+\epsilon)^{n}\cdot|Q^{\prime}|+|C|^{(2-\epsilon_{2})n}\cdot|Q^{\prime\prime}|\bigg{)}$
$\displaystyle\leq\frac{1}{4\pi^{2}}\bigg{(}\Big{(}\frac{|C|^{2}}{a}\Big{)}^{n}\cdot
4\pi^{2}+(|C|+\epsilon)^{n}\cdot|C|^{2}\cdot\frac{4\pi
b_{0}}{M}+|C|^{(2-\epsilon_{2})n}\cdot|C|^{4}\cdot\Big{(}\frac{2b_{0}}{M}\Big{)}^{2}\bigg{)}$
$\displaystyle=O(\epsilon^{n})+O\Big{(}\frac{(|C|+\epsilon)^{n}}{M}\Big{)}+O\Big{(}\frac{|C|^{(2-\epsilon_{2})n}}{M^{2}}\Big{)}.$

Here the second step follows from Claim 8 and the definitions of $Q^{\prime}$
and $Q^{\prime\prime}$. The third step applies Claims 10 and 9. The fourth
step holds since (for the first term) $a>|C|^{2}/\epsilon$ using (17) and (for
the second/third terms) both $|C|$ and $b_{0}$ are constants. Since
$M=O^{*}(|C|^{(1-\epsilon)n})$ for a constant $\epsilon>0$, we know from (14)
that $\rho_{n}^{2}=\Theta(|C|^{2n}/(M^{2}n))=\Omega(|C|^{\epsilon n})$.
Observe that:

  * •

The first term $O(\epsilon^{n})=o_{n}(1)=o(\rho_{n}^{2})$.

  * •

The second term $O((|C|+\epsilon)^{n}\cdot
M^{-1})=O(\rho_{n}^{2})\cdot(|C|+\epsilon)^{n}\cdot\frac{Mn}{|C|^{2n}}$, by
the definition of $\rho_{n}$ (14). Then an upper bound
$O(\rho_{n}^{2})\cdot|C|^{-(\epsilon/4)n}=o(\rho_{n}^{2})$ can be inferred
from the calculation below.

$\displaystyle(|C|+\epsilon)^{n}\cdot\frac{Mn}{|C|^{2n}}$
$\displaystyle=O\Big{(}(|C|+\epsilon)^{n}\cdot\frac{1}{|C|^{(1+\epsilon/2)n}}\Big{)}$
$\displaystyle=O\Big{(}|C|^{(1+\epsilon/4)n}\cdot\frac{1}{|C|^{(1+\epsilon/2)n}}\Big{)}$
$\displaystyle=O(|C|^{-(\epsilon/4)n}).$

Here the first step holds because $M=O^{*}(|C|^{(1-\epsilon)n})$ and then
(given that $\epsilon>0$ is a constant) $Mn=O(|C|^{(1-\epsilon/2)n})$. The
second step holds since
$|C|+\epsilon\leq|C|\cdot(1+\epsilon/4)<|C|^{1+\epsilon/4}$, given that
$|C|\geq 4$ (namely $C=C(d)=\\{\pm 1,\dots,\pm d\\}$ for some integer $d>1$).

  * •

The third term $O(|C|^{(2-\epsilon_{2})n}\cdot M^{-2})=O(\rho_{n}^{2})\cdot
n\cdot|C|^{-\epsilon_{2}n}=o(\rho_{n}^{2})$, given that
$\epsilon_{2}=\epsilon_{2}(|C|,n)>0$ is a constant (see the statement of Claim
8).

Putting everything together completes the proof of Corollary 5. ∎

###### Proof of Lemma 6.

Putting Claims 6, 7 and 5 together, we have

$\displaystyle\operatorname{{\bf
E}}_{\vec{\bm{x}}}[\bm{\mathcal{Z}}^{2}]~{}\leq~{}\eqref{eq:ez2-part1}~{}+~{}\eqref{eq:ez2-part2}~{}+~{}\eqref{eq:ez2-part3}~{}\leq~{}\rho_{n}\cdot(1+o_{n}(1)).$

This finishes the proof of Lemma 6. ∎