# Effect of Emitters on Quantum State Transfer in Coupled Cavity Arrays

Eli Baum  Department of Physics, University of California, Davis, CA 95616,USA
Amelia Broman  Department of Physics and Astronomy, Carleton College, Olin
Hall, 215 Goodsell Circle, Northfield, MN 55057 USA  Trevor Clarke  Department
of Physics, University of California, Davis, CA 95616,USA  Natanael C. Costa
Instituto de Física, Universidade Federal do Rio de Janeiro, Cx.P. 68.528,
21941-972, Rio de Janeiro RJ, Brazil  Jack Mucciaccio  Department of Physics,
Coe College, Cedar Rapids, IA 52402, USA  Alexander Yue  Department of
Physics, University of California, Davis, CA 95616,USA  Yuxi Zhang  Department
of Physics, University of California, Davis, CA 95616,USA  Victoria Norman
Department of Physics, University of California, Davis, CA 95616,USA
Department of Electrical and Computer Engineering, University of California,
Davis, CA 95616,USA  Jesse Patton  Department of Electrical and Computer
Engineering, University of California, Davis, CA 95616,USA  Marina Radulaski
Department of Electrical and Computer Engineering, University of California,
Davis, CA 95616,USA  Richard T. Scalettar  Department of Physics, University
of California, Davis, CA 95616,USA

###### Abstract

Over the last decade, conditions for perfect state transfer in quantum spin
chains have been discovered, and their experimental realizations addressed. In
this paper, we consider an extension of such studies to quantum state transfer
in a coupled cavity array including the effects of atoms in the cavities which
can absorb and emit photons as they propagate down the array. Our model is
equivalent to previously examined spin chains in the one-excitation sector and
in the absence of emitters. We introduce a Monte Carlo approach to the inverse
eigenvalue problem which allows the determination of the inter-cavity and
cavity-emitter couplings resulting in near-perfect quantum state transfer
fidelity, and examine the time dependent polariton wave function through exact
diagonalization of the resulting Tavis-Cummings-Hubbard Hamiltonian. The
effect of inhomogeneous emitter locations is also evaluated.

##  I Introduction

One of the most important questions in quantum information theory is the
faithful and rapid, transmission of a quantum state from one location to
another. Quantum spin chains have proven to be a very useful and powerful
context in which to explore fundamental issues, including the possibility of
perfect transfer, the effect of disorder, and the interplay between high
fidelity and speed of propagation [1, 2, 3, 4, 5, 6, 7]. In the case where a
single excitation is present (one up spin in a background of down spins) the
resulting Hamiltonian is represented by a tridiagonal (‘Jacobi’) matrix. A
general classification of the eigenspectra of such matrices which result in
perfect Quantum State Transfer (QST) has emerged, as has the determination of
the requisite ‘fully engineered’ intersite exchange constants $J_{i}$ [2, 3].
Interestingly, it was also found that nearly perfect QST could be achieved
with more limited and feasible ‘boundary’ engineering in which the $J_{i}$ are
uniform in the interior, and take special values only at the beginning and end
of the chain [8]. Although boundary engineering has the advantage of requiring
less precise, and therefore less experimentally challenging, tuning, good QST
is achieved only in the limit of weak coupling at the ends, and hence is
compromised by long transfer times.

A subsequent focus was on the effect of disorder on QST, since in any physical
realization a certain degree of randomness is inevitable. There are many
eigenvalue distributions which give rise to perfect QST in the ideal limit,
and therefore one line of investigation concerned the types of such engineered
spectra which are most robust to disorder [9]. A key observation was that once
randomness is present, the resulting degradations of state transfer of fully
and boundary engineered chains are roughly similar, so that there is limited
incentive to attempt full engineering as far as fidelity itself is concerned
[10]. (The problem of longer transfer times in boundary engineered chains,
however, remains.)

In this paper, we consider QST within a different physical and geometric
context, namely when the ‘backbone’ chain also possesses branches to localized
qubits, forming a ‘comb-like’ geometry as illustrated in Fig. 1. We are
motivated by the study of the nature and propagation of excitations in a
coupled cavity array (CCA) [11, 12, 13]. A CCA consists of a chain of optical
cavities, which might be empty or may contain one or more atom-like emitters
coupled to the cavity’s electromagnetic field. Photons hop between adjacent
cavities in the CCA due to the overlap of neighboring resonance modes, and
strong interactions between light and matter can be induced. These emitters
form the ‘rungs’ which dress our one-dimensional chain of cavities.

CCAs have become increasingly experimentally viable in recent years [14, 15],
and have been especially intriguing as possible venues for exploring
superfluid to insulator transitions and other many-body phenomena. However, in
order to observe such effects, the CCA must exist in the strong coupling
regime of cavity quantum electrodynamics, where light-matter interactions are
stronger than losses to the environment. Modern integrated optical cavities
achieve this by localizing light on the (sub)wavelength scale. One of the
commonly used optical resonators for these studies is the photonic crystal
cavity, formed by periodic refractive index alteration at the nanoscale.

Figure 1:  Geometry of the one dimensional Jaynes-Cummings-Hubbard Hamiltonian
with $N=4$ cavities each containing $M_{i}=1$ emitter. Photons can ‘hop’
between a set of cavities $i$ and $i+1$ (squares) via the inter-cavity
coupling rates $J_{i}$. Within each cavity $i$ a photon can be absorbed and
excite the emitter (circles) via $g_{i}$. The same process can be reversed,
traveling from emitter to cavity also via $g_{i}$. The numbers indicate our
(arbitrary) convention for labeling the states in the Hilbert space, i.e. the
rows of our matrix in the single excitation sector.

One of the attractive choices for quasi-atoms which might interact with such
solid-state CCAs are color centers formed as lattice defects in semiconductors
[16]. The defect causes electron wavefunctions to localize at that point,
effectively creating an isolated two-level system within a solid-state
material. The most common material substrates for this purpose are silicon
carbide [17, 18, 19, 20, 21, 22] and diamond [23, 24, 25, 26].

An immediate question is whether perfect QST is still possible in these more
complex ‘two component’ systems, and, if so, what are the associated cavity-
cavity and cavity-emitter couplings. A second question pertains to the effect
of a fundamentally different type of ‘geometric’ disorder which arises from
inhomogeneity in the emitter numbers and locations, rather than the previously
explored situations where randomness is introduced via bond-dependent
couplings in a fixed and regular geometry. A cavity which is absent an emitter
corresponds, for example, to a missing ‘tooth’ at that location of the comb.
We will describe the consequences of such disorder on QST.

One interesting aspect of such cavity-emitter arrays is as a novel realization
of ‘boundary engineering’. If atoms are placed only in the initial and final
positions of $N-2$ cavities, the geometry is identical as that of an $N$
cavity chain. The emitter-cavity couplings then play the role of the bond
strengths $J_{1}$ and $J_{N-1}$ of a spin chain.

A final avenue of investigation described here concerns the case of multiple
excitations in cavity-emitter systems. In the absence of emitters, the
Hamiltonian is quadratic and describes a set of independent bosonic particles
(photons). As a consequence, perfect QST in the single excitation sector
guarantees the same occurs for multiple excitations. When emitters are
present, the Hamiltonian remains quadratic in the photon and emitter
operators. However the mixed nature of the commutation relations/allowed
‘occupations’ makes the multi-excitation sector fundamentally different from
single excitations. We will describe the prospects for achieving high
fidelities in this situation.

Our paper is organized as follows. In Section II we review the Jaynes-
Cummings-Hubbard Hamiltonian (JCHH) and its matrix representation in the
single excitation sector. We also briefly describe the exact diagonalization
method used in the time evolution of states and the Monte Carlo approach used
to solve the inverse eigenvalue problem. Section III presents evidence for the
possibility of perfect QST in cavity-emitter systems. These results provide
‘full engineering’ solutions to perfect QST in the JCHH, generalizing known
spin chain results. Having established perfect QST in this more complex
setting, we next consider, in Sec. IV, the effects of disorder. Section V and
Sec. VI discuss how cavity emitter systems can provide a novel realization of
boundary engineering, and the nature of QST when multiple excitations are
present, respectively. A brief overview of experimental parameters in CCA in
silicon carbide with color centers serving as emitters is contained in Sec.
VII. Finally our results are summarized in Sec. VIII. Several details are
discussed in the Appendix.

##  II Model and Methodology

The cavity-emitter arrays we will study are described by the Jaynes-Cummings-
Hubbard Hamiltonian,

$\displaystyle{\cal H}$
$\displaystyle=\sum\limits_{i=1}^{N}\Omega_{i}^{\phantom{\dagger}}a_{i}^{\dagger}a_{i}^{\phantom{\dagger}}+\sum\limits_{i=1}^{N-1}J_{i}^{\phantom{\dagger}}\big{(}\,a_{i+1}^{\dagger}a_{i}^{\phantom{\dagger}}+a_{i}^{\dagger}a_{i+1}^{\phantom{\dagger}}\,\big{)}$
(1)
$\displaystyle+\sum\limits_{i=1}^{N}\,\sum\limits_{j=1}^{M_{i}}\omega_{ij}^{\phantom{\dagger}}\sigma_{ij}^{+}\sigma_{ij}^{-}+g_{ij}\big{(}\,a_{i}^{\dagger}\sigma_{ij}^{-}+\sigma_{ij}^{+}a_{i}^{\phantom{\dagger}}\,\big{)}$

Here $N$ is the number of cavities and $\\{M_{i}\\}$ are the numbers of
emitters in cavity $i$.
$a_{i}^{\dagger}\big{(}a_{i}^{\phantom{\dagger}}\big{)}$ are photon creation
(annihilation) operators in cavity $i$, and
$\sigma_{ij}^{+}\big{(}\sigma_{ij}^{-}\big{)}$ are excitation (de-excitation)
operators for emitter $j$ in cavity $i$. The model is parameterized by cavity
energies $\Omega_{i}$, photon hopping rates $J_{i}$, emitter energy levels
$\omega_{ij}$ and photon-emitter coupling rates $g_{ij}$. We focus on the case
when there is at most one emitter per cavity, $M_{i}=0,1$, and hence will
simplify the notation to $g_{i}$ and $\omega_{i}$, dropping the $j$ subscript
which distinguishes different emitters in the same cavity. In cases when the
number of emitters varies, we will refer to the sparse JCHH.

Real cavities and emitters have finite linewidth, representing the possibility
of loss. High quality (small linewidth) cavities and emitters are increasingly
available [16]. Hence these effects are ignored in the present work.

A basis for the Hilbert space in the single excitation sector and in the
absence of emitters is the collection of states $|\,0\,0\,0\,\cdots
0\,1_{i}\,0\cdots 0\,\rangle$ with a single phonon in cavity $i$. The
Hamiltonian is represented by the tridiagonal (‘Jacobi’) matrix,

${{\cal H}=\begin{pmatrix}\Omega_{1}&-J_{1}&0&\ldots&0\\\
-J_{1}&\Omega_{2}&-J_{2}&\ldots&0\\\ 0&-J_{2}&\Omega_{3}&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&-J_{N-1}\\\
0&0&0&-J_{N-1}&\Omega_{N}\,\,.\end{pmatrix}}$ (2)

We compute the time evolution from an initial state $|\,\Psi(t=0)\,\rangle$ by
diagonalizing ${\cal H}=S\,DS^{\dagger}$, exponentiating ${\cal H}$ to obtain
$U=e^{-i{\cal H}t}=S\,e^{-iDt}S^{\dagger}$, thereby finding

$\ket{\psi(t)}=e^{-i{\cal H}t}\ket{\psi(0)}$ (3)

where we take $\hbar=1$. We begin our system with
$\ket{\psi(0)}=\ket{1,0,0,\ldots,0}$ corresponding to a single photon
contained entirely in cavity $i=1$ at time $t=0$ and let the system evolve in
time. We are interested in a final state $\ket{\psi_{\rm
f}}=\ket{0,0,0,\ldots,1}$ with the photon in cavity $i=N$.

We define the fidelity ${\cal F}$ to be ${\cal F}={\rm max}_{t}\,f(t)$ where
$f(t)\equiv|\matrixelement{\psi_{\rm f}}{e^{-i{\cal H}t}}{\psi(0)}|^{2}$ is
the probability the excitation, beginning in cavity $i=1$, evolves to be in
cavity $i=N$, at time $t$. The arrival time for perfect QST is known in
certain cases, however, more generally, e.g. in the presence of disorder, a
complication is the necessity to search for the time at which $f(t)$ is
maximal.

It is intuitive that solutions to the time evolution equation should usually
spread in time so that the location of the quantum particle becomes less well
known. Indeed, this is also a simple consequence of the uncertainty principle:
a lack of precise knowledge of the momentum implies that the wave packet can
move with different possible speeds and hence as time passes the distribution
of possible locations is increasingly broad. For these reasons it might appear
remarkable that there are solutions of the Schrodinger equation on a lattice
which can begin at a unique location and arrive later at a different unique
location.

Despite this argument, it has been shown [2] that, for a CCA with no emitters
operating in the single excitation sector, there are a variety of
prescriptions for $J_{i}$ which yield perfect QST at a known time. For a
system of $N$ cavities and $N-1$ couplings, one of the simplest arrangements
is:

$J_{i}=\sqrt{i\cdot(N-i)}\,J_{0}$ (4)

The insight here is that the hoppings $J_{i}$ match the Clebsch-Gordon
coefficients for the spin raising operator for spin $N/2-1$. The $N$
associated eigenvalues of the $z$ component of angular momentum are equi-
spaced, allowing for a matching of phases and hence complete re-localization
of the excitation at an appropriate future time. Indeed, with this choice,
perfect QST occurs at $t_{p}=\pi/(2J_{0})$ for any $N$. The surprising feature
that the passage time is independent of chain length $N$ is accounted for by
the fact that the $J_{i}$ increase with $N$. (For example, at the chain
midpoint, $J_{N/2}=\frac{N}{2}$.)

Notice that, although we have labeled the couplings in Fig. 1 completely
generally, the $J_{i}$ of Eq. 4 obey a reflection symmetry about the chain
center. This proves to be a crucial ingredient of perfect QST[27], ensuring
that the ‘return’ transfer from $\ket{\Psi_{\cal B}}$ to $\ket{\Psi_{\cal A}}$
precisely follows the transfer from $\ket{\Psi_{\cal A}}$ to $\ket{\Psi_{\cal
B}}$. We will reproduce these known results in the absence of emitters to
provide a benchmark for our new JCHH results.

The geometry in the presence of emitters is shown by the full structure in
Fig. 1, i.e. including both the cavities, represented by the squares, and the
emitters, by circles. In this situation we will find, unsurprisingly, that the
$J_{i}$ values giving perfect QST are shifted away from those of Eq. 4, which
apply to the cavity-only (spin chain) case. Indeed, the discovery of a
collection of $J_{i},g_{i}$ yielding perfect QST in the presence of emitters
is one of the primary conclusions of this work.

Adding a single emitter to each of the $N$ cavities of our system
($\\{\,M_{i}=1\,\\}$), but remaining in the one excitation sector, the
system’s Hamiltonian doubles in dimension to $2N$. Our convention is that the
first $N$ basis vectors represent photons in cavities $i=1,2,\cdots,N$. We
acquire an additional $N$ basis vectors $i=N+1,N+2,\cdots,2\,N$ for which
there are no photons but instead an emitter is excited. The Hamiltonian matrix
is now, for $N=4$,

${{\cal H}=\begin{pmatrix}\Omega_{1}&-J_{1}&0&0&-g_{1}&0&0&0\\\
-J_{1}&\Omega_{2}&-J_{2}&0&0&-g_{2}&0&0\\\
0&-J_{2}&\Omega_{3}&-J_{3}&0&0&-g_{3}&0\\\
0&0&-J_{3}&\Omega_{4}&0&0&0&-g_{4}\\\ -g_{1}&0&0&0&\omega_{1}&0&0&0\\\
0&-g_{2}&0&0&0&\omega_{2}&0&0\\\ 0&0&-g_{3}&0&0&0&\omega_{3}&0\\\
0&0&0&-g_{4}&0&0&0&\omega_{4}\\\ \end{pmatrix}}$ (5)

This form of ${\cal H}$ has a $2\times 2$ ‘block’ structure reflecting the
presence of two types of ‘sites’ in the lattice.

In the remainder of this paper, we will enforce the reflection symmetry of all
couplings in the JCHH. That is, we will have $J_{i}=J_{N-i}$ and
$g_{i}=g_{N-i}$. In addition, unless otherwise stated, we set the matrix
diagonals to a common value. Since this value corresponds to the arbitrary
choice of a zero of energy, it is set to zero.

While many protocols for the $J_{i}$ yielding perfect QST for the cavity only
(spin chain) geometry are known, the analogous Hamiltonian parameters for
perfect QST in the presence of emitters (the JCHH Hamiltonian) are, to our
knowledge, not yet determined. Here we compute appropriate couplings via a
Monte Carlo procedure. We begin with the assumption that the eigenvalues for a
cavity-only system of length $2N$ which give perfect QST will also give
perfect QST for a JCHH system of $N$ cavities and $N$ emitters. This starting
point is motivated by the insight that the key to perfect QST is in the
(rational fraction) relation between the eigenvalues which allows all
frequencies to be in phase at some future time. We denote these the ‘target’
eigenvalues $\lambda_{n}^{(t)}$and define an action:

${\cal S}=\sum_{n}\big{(}\,\lambda_{n}-\lambda_{n}^{(t)}\,\big{)}^{2}$ (6)

Here $\lambda_{n}$ are the actual eigenvalues of the matrix ${\cal H}$ of the
JCHH Hamiltonian, Eq. 5, for a given set of $\\{\,J_{i}\,\\}$ and
$\\{\,g_{i}\\}$. We begin with constant $\\{\,J_{i}\,\\}$ and $\\{\,g_{i}\\}$
and propose ‘moves’ which change all the parameters within some ‘step size’.
We accept each move with the ‘heat bath’ probability $e^{-\beta\Delta{\cal
S}}\big{(}1+e^{-\beta\Delta{\cal S}}\big{)}^{-1}$ where $\Delta{\cal S}$ is
the change in the action resulting from the Monte Carlo move. Here $\beta$ is
a parameter 111In statistical mechanics language $\beta=1/T$ is the inverse
temperature, so that $\beta_{\rm initial}=0.1$ corresponds to high temperature
and $\beta_{\rm final}=10^{4}$ corresponds to low temperature. The gradual
increase of $\beta$ (lowering of $T$) allows the Monte Carlo to find the
‘ground state’ ${\cal S}=0$ where the $\\{J_{i}\\}$ and $\\{g_{i}\\}$ give a
Hamiltonian with desired target eigenvalues to high accuracy. which starts at
a small value (e.g. $\beta_{\rm initial}\sim 0.1$) and after $L$ Monte Carlo
sweeps (a typical choice was $L\sim 10^{\,6}$) of all the parameters is
increased by a factor $\alpha$. This process is repeated for $K$ steps until
$\beta_{\rm final}=\alpha^{K}\beta_{\rm initial}$ is large (e.g. $\beta_{\rm
final}=10^{4}$.)

We find that this procedure robustly converges to small values of ${\cal S}$,
corresponding to all the eigenvalues $\lambda_{n}$ of ${\cal H}$ matching
their targets $\lambda_{n}^{(t)}$. For most results presented here we
terminate the Monte Carlo when the eigenvalues match their targets to $0.1\%$,
however, we can continue to run the program with smaller ’step sizes’ until we
reach any desired degree of accuracy. Since the fidelity of the system is
dependant on the eigenvalues, this allows us to reach any desired fidelity. In
this paper, we consider a fidelity of ${\cal F}\gtrsim 0.99$ as an adequate
representation of perfect QST. The time to solution scales with $N^{3}$ owing
to the necessity of repeated diagonalizations of ${\cal H}$ in the computation
of $\Delta{\cal S}$. Since our chain lengths ($N\lesssim 16$) were relatively
small, the Monte Carlo time to solution was quite short. Such calculations can
easily be done in a few minutes to a few hours on a desktop computer,
depending on system size and desired accuracy 222An alternate Monte Carlo
procedure defines an action ${\cal S}$ based on targeting of a desired time
evolution matrix rather than desired eigenvalues. This procedure is useful in
more complicated geometries and will be explored in a subsequent paper..
Larger $N\sim 10^{\,2}$ are similarly quite feasible without resorting to
specialized hardware.

Next we use the Hamiltonian ${\cal H}$ determined by the resulting
$\\{\,J_{i},g_{i}\,\\}$ and find that the time evolution operator $e^{-i{\cal
H}t}$ produces perfect QST for the cavity-emitter geometry. This validates our
assumption that the eigenvalue list is apparently what produces perfect QST,
and the particular tridiagonal structure of the cavity-only (spin chain)
matrix is not essential–it can be generalized to the $2\times 2$ block matrix
structure of Eq. 5 333In future work we will explore yet more general
geometries..

We note that this procedure–the computation of the matrix elements giving a
desired spectrum, or ‘inverse eigenvalue problem’ (IEP)–is of course a well
explored problem in applied mathematics [31]. The IEP is non-trivial only when
the matrix is constrained to have a particular structure. The cavity-only case
is that of a ‘Jacobi matrix’ considered by Hald [32]. Other studied structures
include Toeplitz, Hessenberg, and stochastic matrices [33]. Our work addresses
the IEP for an additional type of matrix structure.

##  III QST in the Uniform JCHH

###  III.1 Background: Limit of No Emitters

inter-cavity |  |
---|---|---
bond $i$ |  $J_{i}\,$ (MC) | $\sqrt{i(N-i)}$
1 | 2.642 | 2.64575
2 | 3.470 | 3.46410
3 | 3.873 | 3.87298
4 | 3.996 | 4.00000
5 | 3.873 | 3.87298
6 | 3.470 | 3.46410
7 | 2.642 | 2.64575
Table 1: Values of the cavity-only (spin chain) couplings determined by the
Monte Carlo for a $N=8$ compared to the known results for perfect QST given by
Eq. 4. Our Monte Carlo enforces the symmetry $J_{i}=J_{N-i}$.

Here we reproduce the known results of Christandl [2] in the absence of
randomness to serve as a point of comparison for our subsequent study of the
JCHH, and to test our Monte Carlo method for the IEP in a situation where a
solution is already established. We therefore consider a cavity-only system
with near neighbor couplings. We confirm rapid and precise convergence to the
known perfect QST values of Eq. 4 from general, random starting configurations
of $\\{J_{i}\\}$. We compare our results in Table 1 to the exact values of Eq.
4 for $N=8$ and target eigenvalues
$\lambda_{n}^{(t)}=\pm\frac{1}{2},\pm\frac{3}{2}$. Figure 2 gives the
resulting time evolution. The heat map of the left-hand panel displays the
probability in each cavity for all times. We supplement this (right-hand
panel) with a fidelity line graph for the first and last cavities, where the
probabilities can be displayed more precisely. The small deviations of $J_{i}$
from the analytic values do not appreciably degrade the fidelity.

Figure 2: We consider an $N=8$ cavity only system with all $\Omega_{i}=0$ and
$J_{i}$ couplings determined by Monte Carlo in 1, which converges to the
values of the analytical solution. In panel (a) we graph the probability that
the photon is in each cavity. The eight columns on the x axis represent the
eight cavities, and time descends from 0 to 8 along the y axis. For every time
and location, the probability is indicated in the color bar. In panel (b) we
display the probabilities in just the originating and receiving cavities as
functions of time. We observe perfect QST at time $\pi/2$ and with period
$\pi$ for a return to the initial state.

###  III.2 QST in the Presence of Emitters

Next we demonstrate the effectiveness of our Monte Carlo solution of the IEP
for determining cavity-cavity and cavity-emitter coupling leading to perfect
QST in the novel context of the JCHH. Our method works only with systems with
an even number of cavities when there is one emitter in every cavity. The
reason is discussed further in the Appendix. However, with this constraint, we
can successfully determine JCHH parameters giving fidelities ${\cal F}\gtrsim
0.99$ for systems with up to $N\sim 10^{\,2}$ cavities.

Perfect QST for a system of eight cavities with emitters in every cavity is
shown in Fig. 3. Our labeling convention is such that we index of states with
a photon in one of the $N=8$ optical cavities as 1-8, and states with the
corresponding emitter in an excited level as 9-16. As with Fig. 2, the left
panel is the heat map of the probability in all ‘sites’ (cavities and
emitters), whereas the right panel focuses on the originating and receiving
cavities only. We see that perfect QST is obtained in this ‘8+8’ JCHH system.
However, the time evolution is considerably more complex than for the cavity-
only (spin) system of Fig. 2. The transfer time remains $\pi/2$, but the peaks
now form envelopes containing an additional higher frequency structure. This
results from a rapid transfer of probability between each cavity and its
associated emitter which occurs as the overall probability moves, with a
longer time scale, down the cavity backbone.

Table 2 gives the values of the JCHH Hamiltonian parameters determined by our
Monte Carlo and yielding the time evolution of Fig. 3. Values for $J_{i}$ and
$g_{i}$ for several other $N$ are given in the Appendix, as is a discussion of
an empirical formula which gives a reasonable fit to the data.

inter-cavity |
---|---
bond $i$ | $J_{i}\,$
1 | 4.521
2 | 6.158
3 | 7.232
4 | 7.979
5 | 7.232
6 | 6.158
7 | 4.521

cavity-emitter |
---|---
bond $i$ | $g_{i}\,$
1 | 9.558
2 | 7.825
3 | 5.872
4 | 3.234
5 | 3.234
6 | 5.872
7 | 7.825
8 | 9.558

Table 2: Values of the JCHH couplings determined by the Monte Carlo for an
$N=8$ cavity array with an emitter in each cavity. Note that bond $J_{i}$
connects cavities $i$ and $i+1$ whereas bond $g_{i}$ connects cavity $i$ with
its associated emitter. See Fig. 1.  Figure 3: We consider a JCHH system of
eight cavities and eight emitters with all $\Omega=\omega=0$ and couplings
according to 2. Target eigenvalues were chosen to be those giving perfect QST
for a $N=16$ cavity-only chain, i.e. using Eq. 4 with $N=16$. In panel (a) we
graph the probability that the photon is in each cavity and in each emitter
for multiple times. The sixteen columns on the x axis represent the eight
cavities and eight emitters, and time descends from 0 to 8 along the y axis.
For every time and location, the probability is indicated in the color bar. In
panel (b) we display the probability in starting and receiving cavities as a
function of time. We observe perfect QST at time $\pi/2$ and with period
$\pi$.

##  IV Effect of Emitters on Perfect cavity-only QST

In the preceding section we demonstrated that perfect QST is possible for
systems with uniform arrangements of emitters, precisely one per cavity. We
now consider a distinct issue, namely what effect a single ‘impurity’ emitter
would have on the perfect QST which would occur in a cavity-only system. This
explores a different type of ‘disorder’ from that considered previously, and
is experimentally relevant, since in cavity-emitter systems fluctuations in
the numbers of emitters in each cavity are to be expected.

###  IV.1 Background: Limit of No Emitters

Again, we begin by establishing context for our new results on the effect of
disorder in the JCHH by re-examining the cavity-only system previously
considered in [9, 10] . We set $J_{0}=1$ as our scale of energy (time-1) and
add an ‘absolute’ random noise of scale $\Delta J=0.5$ to each of the
engineered $J_{i}$ 444Situations where the randomness is ‘relative’, i.e.
scaled to the $J_{i}$ on each site, have also been studied [10]. We observe in
Fig. 4 that, while we still see the oscillations present in the perfect
system, the added noise significantly degrades QST.

Figure 4: State transfer in a system of eight cavities with couplings
according to Equation 4. A random disorder between $-\Delta J$ and $+\Delta J$
is added to each $J_{i}$. (a): Probability heat map for a single realization
with $\Delta J=0.5$. (b): Associated fidelity line graph. We observe that
perfect QST does not occur, and the fidelity in the last cavity decreases
after each “pass.” (c): we graph the average fidelity for an $N=8$ cavity
perfect QST system with varying levels of disorder $\Delta J$. The fidelity is
measured at the expected transfer time $\frac{\pi}{2}$, and the average is
taken over $10^{\,4}$ disorder realizations.

By calculating the fidelity at $t=\frac{\pi}{2}$ for many values of $\Delta J$
and taking the average fidelity over $10^{\,4}$ realizations of randomized
disorder, we can determine the effect $\Delta J$ has on the fidelity. To
emphasize the distinction from the fidelity for the clean system or for a
single realization, we denote this average as ${\cal P}_{f}$. We obtain ${\cal
P}_{f}$ for the first, second and third passes, where the $n$th pass is the
fidelity taken at $t_{n}=\frac{\pi}{2}+(n-1)\cdot\pi$. The results are
displayed in the bottom right panel of Fig. 4. The fidelity at the first pass
decreases as the disorder increases, and in each successive pass the fidelity
decreases more steeply. The second and third passes undergo a small rise after
their initial declines, but this quickly flattens out. This non-monotonicity
with $\Delta J$ is associated with the way in which the data are extracted: we
measure $f(t)$ for each realization at the fixed clean system transfer time
$t_{n}$. However $\Delta J$ not only disrupts the phase matching of the
engineered $J_{i}$, it also alters the speed of propagation. An alternate (and
computationally time-consuming) protocol would be to search over time for the
optimal fidelity for each $\Delta J$ and for each realization.

We can also quantify the effects of random $\Omega_{i}$ by adding noise so
that the cavity energy levels are uniformly distributed on
$\big{(}-\frac{\Delta\Omega}{2},+\frac{\Delta\Omega}{2}\big{)}$. Such
randomness can arise from variations in the size and shape of the cavities.
Our observations (Fig. 5) are similar to our discussion of hopping disorder:
We see oscillations with peaks that successively decline.

Figure 5:  Analog of Fig. 4 except for disordered cavity energies rather than
inter-cavity hopping. panels (a) and (b) give results for a single realization
with $\Delta\Omega=1$, while panel (c) shows averages over many realizations
for different $\Delta\Omega$. The fidelity loss appears to be roughly linear
in the pass number for small $\Delta\Omega$; that is, the deviation in the
maxima in the fidelity grow roughly linearly with $n$.

We now turn to analyzing the effects of adding atom-like emitters to our
cavity-only system. We will first consider the effect of adding a single
emitter to a cavity-only geometry with $J_{i}$ engineered to give perfect QST.
We will next consider cases with many periodically placed, but non-uniform,
emitters (random $g_{i}$ and $\omega_{i}$). The subsections below analyze
these two situations.

###  IV.2 Loss of Fidelity due to Emitters

Figure 6 shows the first geometry we consider: a single emitter is added to a
chain of $N$ cavities with couplings $J_{i}$. The position of the emitter is
variable. The left panels of Fig. 7 describe the effects of such an impurity
emitter on a cavity system with $J_{i}$ engineered to perfect QST. Results for
different emitter cavity coupling $g$ and emitter placement are shown. An
emitter at the edge of the chain (i.e. close to either the origin cavity or
the destination cavity) causes the most rapid fidelity loss. It is interesting
that the disruption of QST is less severe as the chain length increases
(bottom left compared to top left). As with the independence of passage time
on $N$, it is possible this greater robustness of perfect QST with $N$ is
associated with the increasing values of $J_{i}$.

The right panels of Figure 7 consider another type of emitter disruption,
namely a situation where an emitter is present in each cavity (all with the
same $g_{i}=g$). The fidelity falls more rapidly with $g$ than for a single
emitter (left panels), but there are periodic fidelity ‘revivals’ which are
associated with the more regular geometric structure of uniform emitter
placement.

Figure 6:  Geometry of the one dimensional (sparse) Jaynes-Cummings-Hubbard
Hamiltonian with a single emitter. We will focus on the emitter’s effect on
the fidelity for values of $J_{i}$ which give perfect QST for a cavity-only
system.  Figure 7: Effect of the addition of a single emitter as a
perturbation to cavity-only perfect QST. (a): Fidelity $f(t=\pi/2)$ as a
function of the coupling $g$ of the (single) ‘impurity’ emitter to its cavity.
Curves for three emitter placements, cavities 1,3,5 are shown. (Reflection
symmetry implies the effect of an emitter in cavity $N-i$ is identical to that
of an emitter in cavity $i$.) The number of cavities $N=9$. (b): Same as (a)
except for $N=17$. (c): Fidelity $f(t=\pi/2)$ as a function the coupling $g$
of a collection of emitters, one in each cavity, as a perturbation to cavity-
only perfect QST. The number of cavities $N=9$. (d): Same as (c) except for
$N=17$.

Finally, we examine disorder which has a similar form to randomness in $J_{i}$
considered in earlier spin-chain studies [9]. Specifically, we consider a
situation of $N$ cavities, each with an emitter, but allow both the
intercavity hoppings to be random on $(J_{i}-\frac{\Delta
J}{2},J_{i}+\frac{\Delta J}{2})$, and the emitter-cavity couplings to be
random on $(g_{i}-\frac{\Delta g}{2},g_{i}+\frac{\Delta g}{2})$, with $J_{i}$
and $g_{i}$ according to Table 2. The heat map of Fig. 8 gives the
realization-averaged fidelity $P_{f}(\Delta g,\Delta J)$. The deterioration of
perfect QST is more rapid here than in Fig. 7 because we not only have
additional transfer paths provided by the emitters, but also these paths
themselves have randomized hopping.

Figure 8:  Fidelity $f(t=\pi/2)$ in the JCHH system of eight cavities and
eight emitters as a function of varied cavity and emitter coupling disorder.
The initial couplings are determined by the Monte Carlo procedure according to
2. For each realization, every coupling $J_{i}$ is randomly disordered between
$J_{i}\pm\frac{\Delta J}{2}$. Similarly, each $g_{i}$ coupling is randomly
disordered between $g_{i}\pm\frac{\Delta g}{2}$. Each data point is taken as
the average fidelity over 200 realizations.

##  V The JCHH as a Realization of Boundary Engineering

This short section mainly makes an observation about an intriguing connection
between ‘boundary engineering’ commonly discussed in spin chains [8] and
cavity emitter systems. Topologically, and in the single excitation sector, a
single emitter in an end cavity behaves identically to an additional cavity
with $g$ playing the role of $J$. Thus there is a precise equivalence between
the Hamitonian matrix and hence QST of systems with $N-2$ cavities and two
‘end’ emitters and ones with $N$ cavities and no emitters.

This mapping is especially interesting in that the known prescription for good
QST when the $J_{i}$ are uniform except at the end requires $J_{1}$ and
$J_{N}$ to be much less than the other, uniform $J_{i}$ in the chain interior.
Such a situation arises very naturally in cavity-emitter systems. Hence this
might be a promising alternate way to construct boundary engineered systems.

Figure 9:  Geometry of the one dimensional Tavis-Cummings-Hubbard Hamiltonian
with emitters only in the first and last cavities. The arrows indicate the
geometric equivalence of this structure to a ‘boundary engineered’ spin chain
for which only the first and last exchange constants $J_{1}=J_{7}=g$ are
different from the bulk value $J$.

##  VI Multiple Excitations

Coupled cavity arrays differ from spin chains if more than one photon is
present in the array, since two photons can occupy the same cavity, whereas an
emitter can only be excited a single time. A final avenue of investigation
described here concerns the case of multiple excitations in cavity-emitter
systems. In the absence of emitters, perfect QST in the single excitation
sector automatically implies perfect QST for multiple excitations: the photons
are non-interacting particles. When emitters are present, this theorem no
longer holds: emitters can only be excited once and hence the two excitation
sector differs in a fundamental way from single excitation sector. Another way
to phrase the non-triviality of multiple excitations is to note that even
though the Hamiltonian is quadratic in the creation and destruction operators,
usually a hallmark of the absence of interactions, the mixed nature of the
allowed occupations introduces an effective ‘many-body’ correlation between
excitations, in the sense that the eigen-energies of the two particle system
are not sums of the single particle eigen-energies, as they would be if the
character of the operators were purely bosonic or purely fermionic. Zhu etal
have considered the contact interaction induced by the non-linearity of the
JCHH in the context of the two-polariton scattering problem[35].

Figure 10 makes this observation more precise. The left panels are for a
cavity only system with a single excitation at top, and two excitations at
bottom. The same $\\{\,J_{i}\,\\}$ are used in the two cases. Perfect QST is
preserved for multiple excitations[36]. The only difference is that the
arrival time is more narrowly defined for two excitations.

On the other hand, in the two right panels, which are for a cavity-emitter
system, perfect QST occurs in the case of a single excitation but is destroyed
in the case of two excitations. As with the cavity-only geometry our procedure
is to find the $\\{\,J_{i},g_{i}\,\\}$ which work for a single excitation (by
targeting eigenvalues for a $2\,N$ cavity-only system as discussed earlier)
and then simulate what happens for two excitations. We conclude that the
‘effective interaction’ induced by the mixed commutation rules introduces
inter-particle scattering during the propagation.

A possible way to recover perfect QST for multiple excitations in the cavity-
emitter case would be to use a different set of couplings for two excitations
than for one. However, finding such a set is not straightforward. For single
excitation systems, $N$ devices (cavities and emitters) will always have $N$
basis states, thus, given any given configuration of cavities and emitters,
there exists a cavity only system with the same number of basis states. This
means you can always tune these systems as you can create perfect QST systems
with the same number of eigenvalues. This ceases to be the case for more than
one excitation.

Figure 10: In the left column, the fidelity of a cavity-only system with a
single excitation (top) and two excitations (bottom). The hoppings
$\\{\,J_{i}\,\\}$ that give perfect QST for $N_{\rm exc}=1$ also do so for
$N_{\rm exc}=2$, reflecting the non-interacting nature of the system. In the
right column, similar data for the cavity-emitter system as shown in Fig. 1,
with one excitation (top) and multiple excitations (bottom). For the cavity-
emitter system, the JCHH parmeters $\\{\,J_{i},g_{i}\,\\}$ (determined by
Monte Carlo) that give perfect QST for $N_{\rm exc}=1$ fail to give good
fidelity for $N_{\rm exc}=2$. This is a consequence of the mixed
bosonic/fermionic character of the JCHH operators which comes into play when
$N_{\rm exc}>1$.

##  VII Experimental Parameters

We discuss here the typical range of values for the parameters in the JCHH
which would arise in one of its potential realizations.

The proposed coupled cavity arrays with quantum emitters are well suited for
implementations in color center platforms, such as silicon carbide and
diamond. Color centers are quasi-atoms formed within the lattice defects of a
semiconductor emitting at visible and near infra-red frequencies, 200 THz
$<\omega/2\pi<$500 THz [37]. Recently, significant progress has been made in
fabrication of optical cavities in these materials ($\Omega\approx\omega$) and
the engineering of light and matter interaction with rates of $g/2\pi\sim$ 5
GHz [38]. This level of interaction, several orders of magnitude higher than
achievable in atomic cavity QED systems, is a consequence of the large dipole
momentum of color centers and the small mode volume of the cavities. It is
worth noting that the optimal positioning of the color center, resulting in
maximal $g$ value, is at the maximum of the electromagnetic field of the
optical mode. An ensemble integrated into the cavity is likely to have a
variation in individual emitter-cavity coupling rates. Scaling these systems
into an array, photonic designs of coupled cavities have been proposed for a
range of hopping rates 1 GHz $<J/2\pi<$ 200 GHz [22]. Variation of
nanofabrication conditions across the sample may cause a variation in resonant
frequencies of each cavity, however, methods such as photo-oxidation [39]. can
be used to shift resonances and synchronize the system. Finally, intrinsic as
well as fabrication-induced strain in the sample causes spectral disorder
among color centers. This inhomogeneity has typically been in the $\sim$10 GHz
range for a variety of emitters in silicon carbide and diamond [40, 21].

A link between fluctuations in $g_{i}$ and in emitter locations is that in a
cavity with $M_{i}$ emitters there is a renormalization of the emitter-cavity
coupling $g\rightarrow g\sqrt{M}$, or more specifically
$\sqrt{\sum_{j=1}^{M}g_{j}^{2}}$) to form a polariton state. Thus fluctuations
in $\\{\,M_{i}\,\\}$ serve as an additional source of randomness in $g_{i}$.

##  VIII Conclusions

Over the past two decades, the experimental realization of individual optical
cavities, and their assembly into a CCA [41], has allowed for the study of a
wealth of quantum many-body phenomena, including the simulation of strong
correlation phenomena encountered in condensed matter physics [11, 42]. As
with their ultracold atom, optical lattice counterparts [43, 44], cavity QED
systems permit the manipulation of individual system components. This level of
experimental control makes them attractive candidates for performing
simulations of superfluid to Mott insulating behavior, Anderson localization,
etc. When emitters are also present, new effects occur, including the
emergence of polaritons, or quasiparticles consisting of a superposition of
photonic and atomic excitations [45, 11, 46, 47]. The study of polaritons
allows new strongly correlated regimes of light-matter interaction to be
probed. Our study of quantum state transfer in such systems is complementary
to those endeavors.

There are interesting analogies between the geometry considered here and that
of the one dimensional Kondo or Periodic Anderson Hamiltonians. In those
canons of condensed matter physics, electron hopping occurs between sites of a
‘conduction band’ (hence the analog of cavities here) while there are also
‘localized electrons’ which hybridize with their conduction electron partners
but not each other (the analogs of emitters). The single particle physics of
the periodic Anderson Hamiltonian is well understood: a hybridization gap
opens where the flat impurity band crosses the conduction band. Our work
directly connects to the QST problem in a one-dimensional, non-interacting,
periodic Anderson Hamiltonian. It would be interesting to contrast the role of
the ‘induced correlations’ in our cavity-emitter system which arise from mixed
photon and emitter statistics, with the correlations arising from electron-
electron interactions in the periodic Anderson Hamiltonian (which has only
fermionic particles).

Acknowledgements J.M. and A.B. were supported by the Research Experience for
Undergraduates program (NSF grant PHY-1852581). T.C. is a McNair and MURPPS
scholar at the University of California, Davis. R.T.S. was supported by the
grant DE‐SC0014671 funded by the U.S. Department of Energy, Office of Science.
M.R. was supported by the National Science Foundation CAREER award 2047564.

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##  IX Appendix

i. Constraint on parity of $N$ to solve the IEP

Our Monte Carlo solution to the IEP to determine $\\{\,J_{i}\,\\}$ and
$\\{\,g_{i}\,\\}$ for $N$ cavities each with one emitter worked only for $N$
even. This is because for odd $N$ the parities of the number of cavities, $N$,
and the number of cavities+emitters, $2N$, are different. More precisely, when
$N$ is odd there is a zero eigenvalue in the spectrum of Eq. 4. We cannot
reproduce this zero with our procedure of using the the cavity-only $2N$
spectrum as the target for the $N+N$ cavity-emitter spectrum.

To test this constraint on solvability further, we attempt a Monte Carlo
solution for odd $N$, but removing the emitter in the central cavity, so that
the number of cavities+emitters, $2N-1$ is now also odd. Results are given in
Fig. 11 and demonstrate that (near) perfect QST is recovered.

Figure 11: We consider a sparse JCHH system of nine cavities and eight emitters, in which all cavities except the center cavity contain an emitter. Target eigenvalues were chosen to be those giving perfect QST for a $N=17$ cavity-only chain. In panel (a) we graph the probability that the photon is in each cavity and in each emitter for multiple times. The eighteen columns on the x axis represent the nine cavities and eight emitters, with the center emitter column left as zero. Time descends from 0 to 8 along the y axis. For every time and location, the probability is indicated in the color bar. In panel (b) we display the probability in starting and receiving cavities as a function of time. We observe perfect QST at time $\pi/2$ and with period $\pi$. | $N=N_{e}=12$ | $N=N_{e}=16$
---|---|---
Bond $i$ | $J_{i}$ | $g_{i}$ | $J_{i}$ | $g_{i}$
1 | 5.597 | 14.755 | 6.519 | 19.929
2 | 7.712 | 13.056 | 9.030 | 18.264
3 | 9.255 | 11.278 | 10.876 | 16.511
4 | 10.322 | 9.261 | 12.310 | 14.708
5 | 11.355 | 7.025 | 13.515 | 12.753
6 | 12.007 | 3.987 | 14.461 | 10.581
7 | 11.355 | 3.987 | 15.294 | 8.0311
8 | 10.322 | 7.025 | 16.003 | 4.6159
9 | 9.255 | 9.261 | 15.294 | 4.6159
10 | 7.712 | 11.278 | 14.461 | 8.0311
11 | 5.597 | 13.056 | 13.515 | 10.581
12 |  | 14.755 | 12.310 | 12.753
13 |  |  | 10.876 | 14.708
14 |  |  | 9.0301 | 16.511
15 |  |  | 6.519 | 18.264
16 |  |  |  | 19.929
Table 3:  Intercavity hoppings $J_{i}$ and cavity-emitter couplings $g_{i}$ which give perfect QST for $N=12$ and $N=16$ length cavity arrays with a single emitter in each cavity.  Bond $i$ |  $J_{i}\,$: MC |  $J_{i}$: Eq. 7 |  $g_{i}\,$: MC |  $g_{i}$: Eq. 8
---|---|---|---|---
1 | 4.521 | 4.527 | 9.558 | 9.562
2 | 6.158 | 6.164 | 7.824 | 7.806
3 | 7.232 | 7.246 | 5.872 | 5.826
4 | 7.979 | 8.000 | 3.234 | 3.231
5 | 7.232 | 7.246 | 3.234 | 3.231
6 | 6.158 | 6.164 | 5.872 | 5.826
7 | 4.521 | 4.527 | 7.824 | 7.806
8 |  |  | 9.558 | 9.562
Table 4: Comparison of the Monte Carlo (MC) and empirical cavity-cavity
hoppings. We can see that the empirical formula, Eq. 7, for $J_{i}$ is
extremely accurate to the Monte Carlo derived $J_{i}$, but the formula for
$g_{i}$, Eq. 8, is less accurate. This may be because the actual form for
$g_{i}$ is more complex than our current fitting form.

ii. Additional data for perfect QST in the JCHH

Since a primary result of this paper is the computation of
$\\{\,J_{i},g_{i}\,\\}$ which result in perfect QST for cavity-emitter
systems, described by the JCHH, we provide in Table 3 some additional results
for large size systems, $N=12,16$ to complement the $N=8$ data provided in the
main text.

iii. Functional form for perfect QST JCHH couplings

In the case of the cavity-only (spin chain), precise formulae for the
intercavity-hopping (Heisenberg exchange) constants to achieve perfect QST are
known. The earliest example is that of Christandl and given by Eq. 4. In the
main manuscript we described a Monte Carlo process which works in the more
general cavity-emitter geometry. However, this solution is a ‘black box’ in
the sense that it produces raw numbers which achieve (near) perfect QST
without providing analytic insight or a formula.

We have attempted to fit the raw data produced by the simulation to simple
functional forms. We mimic the spin-chain solution [2] with an ansatz of the
square root of a polynomial function on $N$ and $i$. Indeed, the data
collected allows a good fit to the empirical formulae:

$J_{i}=\frac{\sqrt{i(11N-6i)}}{2}$ (7)
$g_{i}=\frac{\sqrt{(2i-N-1)(14i-27N-7)}}{4}$ (8)

Table 4 compares the Monte Carlo values with these empirical formulae.

iv. Criterion for perfect QST

We note that it is non-trivial to distinguish whether small deviations from
fidelity ${\cal F}\equiv 1$ arise from a fundamental inability to achieve
perfect QST or from small randomness in the Monte Carlo evaluation of the
couplings. We use the term ‘perfect QST’ when our numerics indicate that by
systematically running longer we can achieve arbitrarily close to ${\cal
F}\equiv 1$. In principle an extrapolation of ${\cal F}$ as a function of
simulation time would provide a more rigorous analysis. We do not do this
here, because such an extrapolation is complicated by the necessity to tune
the annealing protocol, i.e. the manner in which $\beta$ is increased, as well
as the choices for the initial $\beta_{i}$ and final $\beta_{f}$. We therefore
elect to use a more loose definition of ‘perfect QST’, ${\cal F}$ very close
to 1 and systematically improvable.