#  Success probabilities in time reversal based hybrid quantum state transfer

Kevin Randles<EMAIL_ADDRESS>S. J. van Enk<EMAIL_ADDRESS>Department of Physics and Center for Optical, Molecular and Quantum Science,
University of Oregon, Eugene, OR 97403, USA

###### Abstract

We consider two memory nodes of a quantum network connected by flying qubits.
We are particularly interested in the case where a flying qubit produced by
one node has to be transformed before it can interface efficiently with the
next node. Such transformations can be utilized as a key part of the
distribution of quantum states and hence entanglement between the nodes of a
hybrid quantum network linking together different quantum technologies. We
show how and why the probability of interfacing successfully is determined by
the overlap of the spectral shape of the actual flying qubit and the ideal
shape. This allows us to analytically and numerically analyze how the
probability of success is impacted by realistic errors, and show the utility
of our scheme (in consonance with known error correction methods) in
connecting hybrid nodes of a quantum network. We focus here on a concrete
implementation in which the memory nodes consist of three-level atoms in
cavities and the flying qubits are photons.

## . eneral introduction

The quantum internet [1] is envisioned to allow the implementation of various
quantum communication, computation and measurement (sensing) tasks that
improve upon their classical counterparts. Scaling the underlying networks to
connect a larger numbers of quantum processor systems, often referred to as
nodes in the context of quantum communication, improving these processors, and
controlling their robustness against noise are among important current
practical issues [2, 3]. The fundamental building blocks of these networks are
the nodes themselves and the quantum transmission lines (also called quantum
channels) that serve as the network edges. Such an edge links together two
nodes so that quantum information, say the state of a qubit, can be sent
between them at will. These nodes may range from simple devices operating on
small numbers of qubits to large-scale quantum computers. When envisaging a
large quantum network, or ultimately the quantum internet, many different
implementations of an edge will be needed, e.g., to reliably connect close-by
units comprising a single computation node, different types of nodes, as well
as distant nodes.

Two promising research avenues for the scalability of quantum networks are in
the development of _distributed_ and _hybrid_ quantum communication and
computation technologies. The utility of distributed quantum technologies lies
in the likely scenario that it is easier to connect many high-functioning
modestly sized, often homogeneous, devices (which have been demonstrated),
using quantum effects like entanglement as a resource, than to scale a single
device to have the same net processing power [4, 5, 6, 7, 8, 9, 10]. A
complementary means of scaling quantum networks is by developing a hybrid
quantum architecture, where either [11] different types of (i) qubits or (ii)
nodes are connected so that tasks can be delegated so as to leverage the
strengths of a given type of qubit or node.

Case (i) can be envisaged as a hybrid device that uses different kinds of
qubits for different purposes. For instance, using separate types of qubits
for local logic operations and for interfacing with quantum channels, which
would allow for communication between nodes with minimal disruption to local
computation or storage processes [12, 13, 14, 15, 16, 17]. In case (ii),
heterogeneous nodes (or smaller intranodal units), that are based on different
quantum technologies, are connected to form the elementary unit of a hybrid
quantum network (or node). Such hybridization would be valuable in making more
powerful large-scale nodes, say a node that integrates solid-state computation
units leveraging fast nanosecond gates (at the expense of relatively short
microsecond coherence times for both depolarization and dephasing) [18, 19,
20, 21] with atom-based memory units with long coherence times on the order of
milliseconds (or seconds for ions though they can be significantly longer
[22], typically limited by dephasing times) at the expense of slower
microsecond gate times [23, 24, 25] for hybrid quantum computation [26, 27,
28]. More broadly, realizing such hybrid links would increase the connectivity
of quantum networks. Especially as different technologies (trapped ions,
superconducting circuits, etc.) become better-established as platforms for
qubit implementation, how to interface them is an important question to
address [29, 30, 31, 32, 33, 34, 35, 36, 37, 38].

###  I.1 Background and scope

The implementation of a specific edge depends on the properties of the nodes
it is linking, including their underlying technological implementation,
natural energy scale, physical separation, and intended function (say
communication or computation). In this work we focus on discrete (as opposed
to continuous [39]) variable quantum communication in which intermediate
‘flying qubits’ carry quantum information along an edge realized by a guided
quantum channel. Photons are the quintessential flying qubit, and the one we
consider here (though phonons could be used, e.g., in some optomechanical
systems [40, 41, 42] and other solid-state systems [43]), as they can
efficiently travel between nodes and they have several degrees of freedom
(mode occupation number, polarization, temporal-mode, etc.) that can be used
to encode quantum information.

We focus on a guided channel (e.g., optical fiber [44, 45, 6, 23] or microwave
coaxial cable [46, 47, 48, 49, 50, 51, 52, 53]) linking close-by nodes, say
within a given lab or device, leaving the consideration of nonguided free-
space channels to other work [54, 55, 56, 57, 58]. The main issue with distant
nodes is that the fidelity of states being transferred (and likewise the
degree of entanglement generated) typically decreases exponentially with the
length of the connecting channel due to photon absorption and noise in the
channel (the rate of this decay can be minimized by using telecom light) [59,
60, 61, 62]. In principle, this issue can be solved using a quantum repeater,
which itself needs to be able to distribute entanglement between close-by
‘repeater stations,’ though further details are beyond the scope of this work
[63, 64, 65]. Accordingly, we will not explicitly address the difficulties of
connecting distant nodes. Note that the error correction protocols we mention
are still relevant for distant nodes though they will have more overhead for
larger distances.

In this paper we focus on the theoretical implementation of a deterministic
quantum state transfer (QST) scheme capable of linking hybrid quantum nodes
[in the sense of case (ii) above] via itinerant photons. In particular, we
consider connecting two different types of spatially separated nodes, with
potentially different spectral properties (resonance frequency and decay
width). The state of a qubit prepared at the ‘sender’ node 1 is mapped to the
state of an emitted photon wave packet (facilitated by local controls) that is
sent to a ‘receiver’ node 2 via a guided channel. To optimally be absorbed at
node 2, and hence to map the photonic state to a material qubit state, this
wave packet must be modified, tailoring its time-frequency shape [66]. In our
previous work [67] we showed how this can be accomplished by incorporating a
unitary transformation, $U$, that time reverses (see SM A4 for some
discussion of why we emphasize time reversal), frequency shifts, and stretches
or compresses the intermediate photon wave packet along the quantum channel
[68]. The original version of this scheme (without the unitary) was proposed
in the seminal work of Cirac and co-workers [69]. In the time domain, $U$ is
given by

$U(t,t^{\prime})=\sqrt{\xi}e^{i\omega_{0}(T-t)}\delta(t^{\prime}-\xi(T-t)).$
(1)

With the inclusion of this unitary transformation, we showed how to design
system controls (laser pulses) that wil transfer the state of a qubit at node
1 to one at node 2 even if the nodes have significantly different resonance
frequencies and decay rates (provided we can implement $U$ correctly). This is
especially important in hybrid cases, where without $U$ the likelihood of node
2 absorbing a photon emitted by node 1, even for well-designed controls, is
very small. A schematic representation of how such a pair of linked nodes
might fit into a larger quantum network is given in Fig. 1.

Figure 1:  Potential subgraph representing part of a quantum network (above)
with a zoomed in focus (below) on two nodes (red points) and the quantum
channel (blue edges) linking them, along which our QST scheme is implemented
(see the main text for details). The graph structure is only for illustration
purposes, showing how our scheme can fit into the bigger picture of quantum
networking. The dashed lines indicate potential continuations of the network
to other nodes. The lower image is modified from figures of Refs. 69 and 67.

In this paper we consider and analyze realistic errors that can occur in the
hybrid QST scheme of our previous work [67], which demonstrated how the scheme
works in ideal conditions. We focus on errors in the implementation of the
unitary transformation, as other common errors, such as those due to photon
absorption in the transmission line, incorrect laser controls, and cavity
loss, are well-known and understood [70, 71, 72, 73, 74]. This lets us show
our protocol’s utility in the presence of these realistic errors especially
when it is supplemented by known error correction protocols [73, 74] that can
correct these common errors as well as errors in the unitary transformation
using local quantum computations with auxiliary quantum emitters at each node
(see Sec. III.4). Nonetheless, error mitigation remains a crucial part of
making such error correction protocols viable, specifically to keep the
expected number of repetitions of a primitive transfer operation low and
moreover to lessen additional overhead.

We also show that the probability of successfully transferring the state from
one node to another, $P_{\textrm{success}}$, is determined (rather
intuitively) by the overlap between the actual and ideal single-photon wave
packets to be incident on node 2 denoted by $\Psi$ and $\Phi$, respectively.
Specifically, we find

$\boxed{P_{\textrm{success}}=\left|\innerproduct{\Phi}{\Psi}\right|^{2}=\left|\int_{-\infty}^{\infty}dt~{}\Phi^{*}(t)\Psi(t)\right|^{2}}$
(2)

(see Sec. III.1 for the corresponding derivation in the particular case we
focus on in this work, where each node is comprised of a three-level atom in a
cavity and see SM A for a general argument for this result, which ultimately
amounts to the Born rule). This can be understood by interpreting the
successful completion of the protocol as a photodetection event in which the
receiving node can be thought of as a photodetector, that (in the case of a
“click”) would project onto a particular single-photon wave packet $\Phi$,
which is determined by the parameters of node 2 and the laser driving it. Note
that this is a thought experiment, we do not actually measure the incident
photon in our scheme as we do not want its superposition state to collapse.

###  I.2 Outline

We outline the model for the interaction of the nodes and present our QST
scheme in Sec. II. This include specifying the corresponding Hamiltonian and
equations of motion (EOMs) for the amplitudes of various excitations. In Sec.
III we consider errors in the unitary transformation. In Sec. III.1, we show
how these errors can be described in the context of quantum measurement theory
in terms of positive operator-valued measures (POVMs). Then in Sec. III.2 we
show how the ideal value of the unitary transformation’s timing parameter can
be determined and in Sec. III.3 we perform some further numerical analysis of
the probability of successful state transfer in the presence of unitary
errors. In Sec. III.4, we highlight how known error correction protocols could
be used to reliably perform this state transfer procedure in the presence of
realistic errors. Finally, in Sec. IV we discuss the utility of our scheme and
provide an outlook of other contexts where it could be employed. Additional
analysis and supporting details regarding the scope of our results are
provided in the Supplemental Material (SM) for this manuscript, which is
appended to the end of the main text.

## I. cheme and model

Here we summarize the model used for two linked nodes of a quantum network
interacting via a quantum channel. For brevity, we leave much of the
corresponding derivations and further details to Refs. 75, 76, the textbook
[77], and our previous work [67]. Starting from the Gardiner-Collett model
[75], one can show that in the quantum trajectory formalism [76], the dynamics
(during time intervals when no quantum jump occurs) are determined by an
effective non-Hermitian Hamiltonian of the form

$H_{\textrm{eff}}=H_{1}+H_{2}+H_{\textrm{tl}}.$ (3)

Here $H_{j}$ is the Hamiltonian for node $j=1,2$ and $H_{\textrm{tl}}$
accounts for the nodes’ interaction via the transmission line.

###  II.1 Node implementation

Here we consider nodes that can encode the state of a qubit in an effective
two-level system with a controllable coupling to a well-defined
electromagnetic field mode. Such a node is readily realized, at least
theoretically, by a three-level $\Lambda$-type atom in a high-$Q$ optical
cavity and so we focus on this case (both here and in our previous work [67])
as an exemplar of the physics underlying our scheme though the scheme applies
more generally. These systems along with the rest of our QST scheme are
highlighted in Fig. 1. In this case, at each node the state of a qubit is
encoded in the ground states of the atom, ${|{e}\rangle}$ and ${|{g}\rangle}$,
which are coupled via a Raman transition through an auxiliary atomic excited
state ${|{r}\rangle}$.

That is, the dynamics of the first Raman qubit are controlled using a laser
pulse that drives a transition from ${|{e_{1}}\rangle}$ to
${|{r_{1}}\rangle}$, which is followed by the transition ${|{r_{1}}\rangle}$
to ${|{g_{1}}\rangle}$ and the emission of a photon into cavity 1, and
meanwhile leaves ${|{g_{1}}\rangle}$ undisturbed with no corresponding
emission. Thus, if atom 1 was in the ‘excited’ state ${|{e_{1}}\rangle}$, an
emitted photon would leak out of cavity 1, propagate down the transmission
line along which it would be transformed via $U$, before its incidence on node
2. Then atom 2, which is prepared in ${|{g_{2}}\rangle}$, can simply undergo
the Raman process analogous to that undergone by atom 1 but in the backwards
order, with the goal of inducing absorption of the photon. This is possible as
by implementing $U$ we have effectively equalized the spectral properties of
the two nodes (analogous to impedance matching).

By linearity, this process can thus be used to implement our QST scheme, where
the state of atom 1, $c_{g}{|{g_{1}}\rangle}+c_{e}{|{e_{1}}\rangle}$, is
transferred to the photon, $c_{g}{|{0}\rangle}+c_{e}{|{1}\rangle}$, and then
to atom 2, $c_{g}{|{g_{2}}\rangle}+c_{e}{|{e_{2}}\rangle}$. Here we take the
photonic qubit to be encoded in the occupation number degree of freedom, i.e.,
either the vacuum ${|{0}\rangle}$ or single-photon ${|{1}\rangle}$ state in a
certain mode. Other encodings could potentially be used with appropriate
modifications to our scheme. There are of course tradeoffs between different
encodings, e.g., the polarization encoding $\\{{|{H}\rangle},{|{V}\rangle}\\}$
is more robust in some scenarios [78, 79] yet the occupation number encoding
is necessary for the known error correction protocols we consider (see SM A6
and D for further discussion).

For these nodes, the effective dynamics in the single (or zero) excitation
subspace are ultimately determined by the Hamiltonian ($\hbar=1$) [69, 67]

$H_{j}=iG_{j}(t)\left(a^{\dagger}_{j}{|{g_{j}}\rangle}\\!{\langle{e_{j}}|}-a_{j}{|{e_{j}}\rangle}\\!{\langle{g_{j}}|}\right).$
(4)

The operators $a^{\dagger}_{j}$ and $a_{j}$ are the creation and annihilation
operators for cavity $j$, respectively. Likewise
${|{e_{j}}\rangle}\\!{\langle{g_{j}}|}$ and
${|{g_{j}}\rangle}\\!{\langle{e_{j}}|}$ are the raising and lowering operators
for the effective two-level atom $j$, respectively. Here
$G_{j}(t)=g_{j}\Omega_{j}(t)/2\Delta_{j}$ is the Jaynes–Cummings interaction
strength between the effective two-level atom and cavity at node $j$, where
(as depicted in Fig. 1) $g_{j}$ is the bare atom cavity coupling,
$\Omega_{j}(t)$ is the user-controlled Rabi frequency envelope of the driving
laser, and $\Delta_{j}=\omega_{Lj}-\omega_{rj}$ is the laser detuning. Note
$G_{j}(t)$ is real with the implicit time-dependent laser phase, of the form
$e^{i\phi_{j}(t)}$, factored out. To obtain this form for $H_{j}$ we selected
specific laser frequencies $\omega_{Lj}$ to eliminate shifts to the cavity
energy states and chose the laser phases $\phi_{j}(t)$ (the chirps
specifically) to compensate for the ac Stark shift to ${|{e_{j}}\rangle}$ (see
equation 34 of Ref. 67 and the surrounding discussion for details as well as
SM B1 ). Additionally, we assumed far off-resonant driving lasers (to
suppress spontaneous emission [80]) so that we could adiabatically eliminate
the excited states ${|{r_{j}}\rangle}$, and we went into a rotating frame at
the frequency of the laser driving each system $\omega_{Lj}$ (in which the
states ${|{e}\rangle}$ and ${|{g}\rangle}$ are to be interpreted).

###  II.2 Interaction model

Assuming a vacuum field input, or less strictly that no light is incident on
the nodes in relevant modes, the connection of the nodes is described by

$H_{\textrm{tl}}=-\frac{i}{2}\left(\gamma_{1}a^{\dagger}_{1}a_{1}+\gamma_{2}a^{\dagger}_{2}a_{2}+2\sqrt{\gamma_{1}\gamma_{2}}e^{i\zeta
t}a^{\dagger}_{2}\tilde{a}_{1}\right),$ (5)

which provides an effective description of the transmission line in which the
nodes are effectively directly coupled. This description highlights the role
of the transmission line as an intermediary for the transfer of excitations
from node 1 to node 2 (via the $a^{\dagger}_{2}\tilde{a}_{1}$ term). Here
$\gamma_{j}$ is the decay rate for cavity $j$ into the transmission line. The
relative phase of $\zeta=\omega_{L2}-\omega_{L1}$ in the excitation transfer
term corresponds to oscillations at the mismatch in the frequencies of the
lasers driving the two systems and is due to going into the aforementioned
rotating frames for each node.

Within this model we assume the coupling to be unidirectional with photons
only propagating from system 1 to 2 (there is no $a_{2}a^{\dagger}_{1}$ term
in $H_{\textrm{tl}}$), at least during the QST procedure [81]. Then $U$ does
not affect system 1 dynamics and $a_{1}$ acts in the Heisenberg picture in a
standard way, i.e., on
${|{C}\rangle}_{1}=c_{0}{|{0}\rangle}_{1}+c_{1}{|{1}\rangle}_{1}$ it acts as

${}_{1}{\langle{0}|}a_{1}(t){|{C}\rangle}_{1}=c_{1}(t).$ (6)

Meanwhile, in this effective description, the second system is effectively
directly coupled to the unitarily transformed output of system 1. This is
accounted for by the $a^{\dagger}_{2}\tilde{a}_{1}$ term of Eq. (5), where
$\tilde{a}_{1}$ encodes the effect of $U$, acting as

${}_{1}{\langle{0}|}\tilde{a}_{1}(t){|{C}\rangle}_{1}=\chi(t)c_{1}(f(t))$ (7)

in contrast to Eq. (6), where [82]

$\displaystyle\chi(t)$ $\displaystyle=\begin{cases}0,&t_{i}<t<t_{s}\\\
\sqrt{\xi}e^{i\omega_{0}(T-t)},&t_{s}<t<t_{f}\\\
1,&\textrm{elsewise}\end{cases}$ (8)

and

$f(t)=\begin{cases}\textrm{undefined},&t_{i}<t<t_{s}\\\
\xi(T-t),&t_{s}<t<t_{f}\\\ t,&\textrm{elsewise}\end{cases}.$ (9)

Here $t_{s}$ is the time at which the transformed field starts to be produced
and it is controlled via the relation $T=t_{s}(1+1/\xi)$. These functions are
broken up into intervals as the part of the photon wave packet to be
transformed, taken to be of duration $t_{l}=l/c$, must pass through the
unitary transformation device (in the time interval $t_{i}\equiv
t_{s}-t_{l}<t<t_{s}$), before the transformed wave packet is produced (in the
interval $t_{s}<t<t_{f}\equiv t_{s}+t_{l}/\xi$). Outside of the time interval
where the transformation is happening $\tilde{a}_{1}$ reduces to the standard
$a_{1}$. The form of these functions for $t_{s}<t<t_{f}$ comes directly from
Eq. (1), assuming that the untransformed pulse is blocked at the
transformation device during the production of the transformed field. [This
assumption is not necessary, but it leads to a simpler description of how the
unitary transformations effect can be encoded in the time argument of a
fictitious system $\tilde{1}$ that is, effectively, directly driving system 2
(see Ref. 67 for further discussion). Furthermore, the distinction of whether
we block the original field for $t_{s}<t<t_{f}$ does not matter if the two
Raman processes are driven by lasers with substantially different frequencies
$|\zeta|=|\omega_{L2}-\omega_{L1}|\gg\gamma_{1,2}$ (see Sec. III.2).] The
crucial part is that the transformed part of the field (which should be the
entirety of the single-photon wave packet assuming the control parameters are
suitably picked and implemented) has the time-reversed and stretched argument
$\xi(T-t)$.

###  II.3 Model generality

Importantly, our focus on a particular kind of node does not limit the scope
of our work, which is meant to concern hybrid links, as the ideas behind our
scheme apply to many other analogous controllable systems that could be used
for either (or both) node(s). In fact, the Hamiltonian governing the system
dynamics we analyze [given in Eqs. (3-5)] is quite general for deterministic
QST schemes like ours, so the physics should be identical after an appropriate
mapping of physical parameters. This is provided the unitary transformation
$U$ can be implemented to transduce the intermediate photon between the
emitting and receiving nodes energy and time scales. [Note that $U$ has a
proposed implementation in the optical regime [83]. In other regimes, say for
microwave systems or hybrid cases, such as the coupling of microwave and
optical nodes, some aspects of such a transformation have been considered yet
to our knowledge the entire unitary we consider has not been (see SM A5 for
further discussion).]

In particular, we assume that we utilize nodes for which individual
excitations can controllably and reversibly be transferred from the material
system to a photonic mode (emission) and vice versa (absorption), often at the
single-photon level, say mediated by a cavity or resonator [84, 69, 79]. It
should then be possible to recast the Hamiltonians for the nodes themselves
into the standard Jaynes–Cummings form [as was done for Eq. (4)] through the
appropriate adiabatic elimination of auxiliary states, selection of unitary
transformations to the node Hamiltonian, and tuning of the system and control
parameters. One benefit of such coherent and reversible interactions is that
you can generate different kinds of target states such as tuning the amplitude
and phase of state, and tune the wave packet shape (see SM B for further
discussion). This is to be contrasted against heralded, probabilistic
approaches such as those based on spontaneous emission. The form of the
Hamiltonian describing the coupling of the nodes, Eq. (5), is also common
within the context of input-output theory in which one eliminates the channel
from the description (via its corresponding continuous mode field operators)
to obtain a simpler description where the two nodes are effectively directly
coupled (see SM A2 for further details and examples).

###  II.4 System evolution

Starting from Eq. (3), the zero-excitation ground state evolves trivially
${|{gg}\rangle}{|{00}\rangle}\rightarrow{|{gg}\rangle}{|{00}\rangle}$ as

$i\frac{d}{dt}{|{gg}\rangle}{|{00}\rangle}=H_{\textrm{eff}}{|{gg}\rangle}{|{00}\rangle}=0$
(10)

in the interaction picture. Meanwhile the dynamics of a state in the single-
excitation subspace

$\displaystyle{|{\psi(t)}\rangle}=\;$
$\displaystyle\alpha_{1}(t){|{eg}\rangle}{|{00}\rangle}+\alpha_{2}(t){|{ge}\rangle}{|{00}\rangle}$
$\displaystyle+\beta_{1}(t){|{gg}\rangle}{|{10}\rangle}+\beta_{2}(t){|{gg}\rangle}{|{01}\rangle}$
(11)

are encoded by $\alpha_{j}$ and $\beta_{j}$, which are the state amplitudes
for an excitation being in atom $j$ (${|{e_{j}}\rangle}$) and cavity $j$
(${|{1_{j}}\rangle}$), respectively. The corresponding amplitude EOMs are

$\displaystyle\dot{\alpha}_{1}$ $\displaystyle=-G_{1}\beta_{1},$ (12a)
$\displaystyle\dot{\beta}_{1}$
$\displaystyle=G_{1}\alpha_{1}-\frac{\gamma_{1}}{2}\beta_{1},$ (12b)
$\displaystyle\dot{\alpha}_{2}$ $\displaystyle=-G_{2}\beta_{2},$ (12c)
$\displaystyle\dot{\beta}_{2}$
$\displaystyle=G_{2}\alpha_{2}-\frac{\gamma_{2}}{2}\beta_{2}-\sqrt{\gamma_{2}}e^{i\zeta
t}\Psi(t),$ (12d)

where

$\displaystyle\Psi(t)$
$\displaystyle:=\sqrt{\gamma_{1}}\chi(t)\beta_{1}(f(t))$
$\displaystyle\approx\sqrt{\gamma_{1}\xi}e^{i\omega_{0}(T-t)}\beta_{1}(\xi(T-t)),$
(13)

is the transformed wave packet [85] emitted by system 1 that is driving system
2. The approximation in Eq. (II.4) is exact during the transformed field
production $t_{s}<t<t_{f}$ and hence holds in the large $l$ limit. Note that
by tuning the laser frequencies and phases to obtain the simple
Jaynes–Cummings type node Hamiltonians of Eq. (4) we are fixing $\alpha_{1}$
and $\beta_{1}$ to have the same constant phase, so without loss of generality
we take them both to be real. This simplified case, where we tune the control
parameters so that the system 1 amplitudes are real, is of course not general,
though it entirely suffices for our QST scheme and makes our analysis easier.
It is worth mentioning that one can controllably modulate the phase of the
emitted photon wave packet, $\Psi(t)=\sqrt{\gamma_{1}}\beta_{1}(t)$, in time
by adjusting the parameters of the driving laser, namely the phase (though
this can exacerbate non-Markovian effects, which need more careful treatment
[86]), and then $\alpha_{1}$ also acquires a time-dependent phase (see SM B1
for more details).

As we have alluded to, we want to highlight the role of unitary transformation
in our QST scheme. Accordingly, we will make several simplifying assumptions
that allow us to effectively ignore other well-studied kinds of errors. These
assumptions include that: 1) the transmission line is lossless, 2) the
cavities are lossless (this includes loss due to photons exiting the cavities
through the wrong spatial modes, i.e., not the transmission line, as well as
absorption loss at the cavity mirrors), 3) the transmission line is
dispersionless, 4) unit efficiencies of emission and absorption into the
desired cavity mode are achievable, and 5) the laser pulses (controls) are
implemented correctly. Assumptions 1) and 2) are not realistic, though they
can easily be treated post hoc by multiplying our probability of success by
the respective probabilities of a photon propagating fully through the channel
and of the cavity photon exiting node 1 (emission) into the transmission line
as well as the time-reverse process of the photon entering and remaining in
node 2 (absorption) [87]. Dispersion in the channel leads to a change in the
shape and phase of the photon wave packet $\Psi(t)$, its impact can thus be
accounted for by Eq. (2). If the dispersion is known and systematic (say by
characterizing the fiber) it can be (at least partially) compensated for by
adjusting $U$. For instance, the stretching parameter $\xi$ can be used to
mitigate wave packet broadening issues due to dispersion (see SM A4 ).

We distinguish assumption 4) from 2) as even if we assume 2), the maximum
efficiency of being able to emit light into (or absorb light out of) a
specific cavity mode is limited by atomic structure, such as additional
excited states, as well as parameters of the driving laser and cavity relative
to each other. Refs. 88 and 89 show this, that the optimal emission and
absorption efficiencies are bounded above, in contexts similar to ours in
which single photon interacts with $\Lambda$-type atom(s) in a cavity. We do
not expect such a bound to significantly limit us here as we assume that the
decay rate of the optical coherences for the ground-excited state transitions
has been suppressed by using an off-resonant Raman process. However,
significant coupling to other excited states could change this, and indeed
understanding the impact of the multilevel structure of realistic atoms in
cavity quantum electrodynamics (QED) is an active area of research, wherein
optical pumping offers a potential means of controlling decay channels to
other states thereby potentially still achieving controlled interactions of
single-photons with material qubits [89, 90, 91]. This entails that, within
our model, certain wave packets can be produced with unit efficiency, though
not all can (see SM B for details). Assumption 5) is more modest, yet of
course the strength and phase of the drives (laser pulses in our case) used to
control the systems can only themselves be controlled up to some precision set
by their implementation. A full treatment of corrections to assumptions 3)-5)
would require additional model details about the nodes being used and the
drives used to control them and so is not considered here. We will mention the
impacts of these assumptions when applicable.

## II. nitary errors

Here we consider errors in the unitary parameter values of $\omega_{0}$,
$\xi$, and $T$. We focus on cases where the transformation duration, $t_{l}$,
is long enough to transform essentially all of the pulse. That is, we do not
analyze errors due to the unitary not being implemented for long enough, which
(at least in non-linear optical setups) could be due to not using a long
enough medium for the transformation device. In this section, we neglect any
other potential errors in the unitary transformation. Errors such as photon
absorption or dispersion can be corrected for [73, 74], though the
corresponding protocols do not correct for errors due to the undesired
production of a photon. Hence we assume there is no other mechanism for atom 2
to absorb an excitation, which is consistent with the assumption of a vacuum
field input to system 1 needed to obtain the effective Hamiltonian of Eq. (3).
In particular, we assume that there are no thermal excitations, which is only
valid for systems at low temperatures relative to their operation frequency
such that the average thermal occupation number $\mkern
1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern
1.5mu_{\textrm{th}}=1/(e^{\hbar\omega/k_{B}T}-1)$ is nearly zero (it is
desirable to have it be far less than 1) [92]. This is typically a very good
approximation for optical systems though systems that operate at lower
frequencies need to be cooled. For instance, at room temperature $T=293$ K
optical light with wavelength $\lambda\approx 700$ nm will have $\mkern
1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern 1.5mu_{\textrm{th}}\approx
3\cdot 10^{-31}$, whereas microwave light with $\lambda\approx 20$ mm will
have $\mkern 1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern
1.5mu_{\textrm{th}}\approx 400$. Hence adequate cooling is crucial in reducing
thermal noise and loss experienced by guided microwave channels such as
cryogenic microwave links [93, 53].

The ideal values of the frequency and stretching parameters are
$\omega_{0}=\omega_{0i}\equiv\zeta=\omega_{L2}-\omega_{L1}$, which compensates
for the difference in frequencies of the two systems, and
$\xi=\xi_{i}\equiv\gamma_{2}/\gamma_{1}$, which stretches the wave packet to
match the receiving time scale of cavity 2 [67]. Unlike for $\omega_{0}$ and
$\xi$, there is not a similarly ‘nice’ expression for the ideal value of the
timing parameter $T=t_{s}(1+1/\xi)$, which controls the starting time for the
transformation $t_{s}$. However, the goal is simple: for a given $l$ one need
simply select a $t_{s}$ (and hence $T$) such that the largest contributions to
$\beta_{1}(t)$ are transformed. We can be more precise in finding the optimal
value of $T$, though we defer this to Sec. III.2, where the analysis will be
made easier using the machinery we will develop in the following section. For
now we simply note that such an optimal $T$, which we will call $T^{*}$, must
exist.

As we are focusing on errors in the unitary parameters in this section, we
assume that $G_{1}$ is implemented correctly such that $\alpha_{1}$,
$\beta_{1}$ are as desired (see SM B for a discussion of how an appropriate
$\alpha_{1}$ or $\beta_{1}$ can be used to determine the corresponding
$G_{1}$) and that [94]

$G_{2}(t)=\xi_{i}G_{1}(\xi_{i}(T_{i}-t))$ (14)

with $T_{i}=T|_{\xi=\xi_{i}}$ [see assumption 4) above]. Note that with this
choice, the laser phase $\phi_{2}(t)$ for system 2 is also time reversed and
scaled relative to that for system 1, assuming they are implemented in the
prescribed way. Hence the second laser pulse is not affected by an error in
the unitaries’ values of $\xi$ and $T$. (In fact $T_{i}$ need not take on its
optimal value $T_{i}^{*}$ as long as it is consistent in $U$ and $G_{2}$ and
$l$ is long enough for the entire wave packet to be transformed.) One can show
that the above choices for the unitary parameters and $G_{2}$’s relation to
$G_{1}$ are an optimum, in that given solutions $\alpha_{1}$ and $\beta_{1}$,
there are corresponding solutions for system 2

$\alpha_{2}^{i}(t)=e^{i\omega_{0i}T_{i}}\alpha_{1}(\xi_{i}(T_{i}-t))$ (15)

and

$\beta_{2}^{i}(t)=-e^{i\omega_{0i}T_{i}}\beta_{1}(\xi_{i}(T_{i}-t))$ (16)

(assuming $l$ is long enough for the entire wave packet to be transformed),
which act in a time-reversed manner relative to their system 1 counterparts.
Hence, with the above choices and assumptions, if atom 1 loses an excitation,
$\alpha_{1}$ goes from one to zero, then atom 2 will absorb it, as
$|\alpha_{2}^{i}|$ goes from zero to one.

###  III.1 Unitary parameter errors

If an excitation is sent from atom 1, we want atom 2 to absorb it and hence to
achieve $|\alpha_{2}(t\rightarrow\infty)|=1$. Thus, we analyze here the
structure of $\alpha_{2}$’s EOM, which, after eliminating $\beta_{2}$ in the
coupled Eqs. (12c–d), can be found to be

$\displaystyle\ddot{\alpha}_{2}$
$\displaystyle=\left(\frac{\dot{G}_{2}}{G_{2}}-\frac{\gamma_{2}}{2}\right)\dot{\alpha}_{2}-G_{2}^{2}\alpha_{2}+\sqrt{\gamma_{2}}e^{i\zeta
t}G_{2}\Psi(t).$ (17)

Rearranging the terms, it follows that

$L(t)\alpha_{2}(t)=\Psi(t)$ (18)

with

$L(t):=\frac{e^{-i\zeta
t}}{\sqrt{\gamma_{2}}G_{2}}\left[\frac{d^{2}}{dt^{2}}-\left(\frac{\dot{G}_{2}}{G_{2}}-\frac{\gamma_{2}}{2}\right)\frac{d}{dt}+G_{2}^{2}\right]$
(19)

a linear operator (we assume it is invertible) and so it has some Green’s
function $\Gamma^{*}(t,t^{\prime})$ [95] such that

$\alpha_{2}(t)=\int_{-\infty}^{\infty}dt^{\prime}~{}\Gamma^{*}(t,t^{\prime})\Psi(t^{\prime}).$
(20)

Here we treat $\Psi(t)$ as some generic (possibly sub-normalized) wave packet
of arbitrary shape. This is justified mathematically as, in terms of the
differential equation Eq. (18), $\Psi(t)$ is just some non-homogeneous source
term, and the Green’s function solution is indifferent to the origin of
$\Psi$. Note we cannot design the first laser pulse $G_{1}$ (even if it is
supplemented by our unitary) to produce arbitrary wave packets from system 1
(see SM B4 for further discussion).

We will now use this Green’s function to derive an expression for the limiting
value of $\alpha_{2}$ at some end time $t_{e}$ by which the amplitudes have
reached steady values (in practice $t_{e}$ can be taken to be $+\infty$
mathematically). Physically, we know that $|\alpha_{2}|\leq 1$, so with the
shorthand $\Gamma(t=t_{e},t^{\prime})=\Gamma_{e}(t^{\prime})$, we have

$1\geq|\alpha_{2}(t_{e})|^{2}=\left|\int_{-\infty}^{\infty}dt^{\prime}~{}\Gamma^{*}_{e}(t^{\prime})\Psi(t^{\prime})\right|^{2}=\left|\innerproduct{\Gamma_{e}}{\Psi}\right|^{2},$
(21)

where we are assuming $\Gamma_{e}$ is well-defined and unique (in practice,
this is typically the case with appropriate boundary conditions, and our
numerics in Sec. III.3 substantiate the validity of the solution we find), yet
we don’t yet know its norm. [We focus on $\Gamma_{e}(t^{\prime})$ because our
primary goal is that the excitation is ultimately transferred (as part of the
QST scheme), not to know the exact dynamics of the excitations. Accordingly,
we do not attempt to compute $\Gamma(t,t^{\prime})$ for all times $t$, though
we do note that it must be causal so $\Gamma(t,t^{\prime}>t)=0$.] This is true
for arbitrary $\Psi$, which will be normalized (in the single-excitation
subspace) unless there are losses, say due to photon absorption. Thus, we can
select

${|{\Psi}\rangle}=\frac{{|{\Gamma_{e}}\rangle}}{\sqrt{\innerproduct{\Gamma_{e}}{\Gamma_{e}}}}$
(22)

(up to a phase) to maximize
$|\alpha_{2}(t_{e})|^{2}=\left|\innerproduct{\Gamma_{e}}{\Psi}\right|^{2}$ via
the Cauchy–Schwarz inequality. With this selection, by Eq. (21) we have

$1\geq|\alpha_{2}(t_{e})|^{2}=\left|\sqrt{\innerproduct{\Gamma_{e}}{\Gamma_{e}}}\right|^{2}=\innerproduct{\Gamma_{e}}{\Gamma_{e}}$
(23)

so ${|{\Gamma_{e}}\rangle}$ is either a normalized or subnormalized quantum
state.

We know that with ideal parameters $\omega_{0}=\omega_{0i}$ and $\xi=\xi_{i}$
(again, for sufficiently long $l$ with appropriate timing $T=T_{i}$) our
unitary will produce the ideal wave packet

$\displaystyle\Phi(t)$
$\displaystyle=\Psi(t)|_{\omega_{0}=\omega_{0i},\xi=\xi_{i},T=T_{i},l\rightarrow\infty}$
$\displaystyle=\sqrt{\gamma_{2}}e^{i\omega_{0i}(T_{i}-t)}\beta_{1}(\xi_{i}(T_{i}-t)),$
(24)

which evidently is the time reversed, stretched, and frequency shifted
counterpart of the wave packet to be emitted by system 1. Crucially, note that
for properly designed $\alpha_{1}$ that start at 1 at some early preparation
time $t_{p}$, from Eq. (15) we have
$|\alpha_{2}^{i}(t=t_{e})|=|e^{i\omega_{0i}T_{i}}|=1$ for $t_{e}\geq
T_{i}-t_{p}/\xi_{i}$ and hence the corresponding normalized $\Phi(t)$,
$\innerproduct{\Phi}{\Phi}=1$, would lead to perfect absorption at system 2
[96]. Thus, again using Cauchy–Schwarz, we have that

$\displaystyle 1$
$\displaystyle=|\alpha_{2}^{i}(t_{e})|^{2}=\left|\innerproduct{\Gamma_{e}}{\Phi}\right|^{2}$
$\displaystyle\leq\innerproduct{\Gamma_{e}}{\Gamma_{e}}\innerproduct{\Phi}{\Phi}=\innerproduct{\Gamma_{e}}{\Gamma_{e}}\leq
1,$ (25)

where we used Eq. (23) in the last step. It clearly follows that

$\innerproduct{\Gamma_{e}}{\Gamma_{e}}\equiv 1$ (26)

and hence ${|{\Gamma_{e}}\rangle}$ corresponds to a normalized state vector,
and moreover

${|{\Gamma_{e}}\rangle}={|{\Phi}\rangle}\Longleftrightarrow\Gamma_{e}(t)=\Phi(t)$
(27)

(again, up to a phase) as the states must be linearly dependent to saturate
Cauchy–Schwarz.

It thus follows that the probability of success is

$\displaystyle P_{\textrm{success}}$
$\displaystyle\equiv|\alpha_{2}(t_{e})|^{2}=\left|\innerproduct{\Phi}{\Psi}\right|^{2}$
$\displaystyle=\left|\int_{-\infty}^{\infty}dt^{\prime}~{}\Phi^{*}(t^{\prime})\Psi(t^{\prime})\right|^{2}$
(28)

as claimed in Eq. (2). That is, the probability of success is given by the
squared overlap of the incident photon wave packet $\Psi(t)$ with the ideal
wave packet $\Phi(t)$. Thus, if no errors occur, the transformed wave packet
$\Psi(t)$ will be equal to the normalized $\Phi(t)$ and hence
$P_{\textrm{success}}=1$. It follows that system 2 can be thought of as a
photodetector that in the case of a click would project on the ideal state via
the POVM element

$\hat{\Pi}_{\textrm{success}}=\outerproduct{\Phi}{\Phi}.$ (29)

Note we would only register such a click (or not) if we appended an atomic
measurement of the receiving qubit state. Then, we see that for the pure input
state $\rho_{\textrm{in}}=\outerproduct{\Psi}{\Psi}$,

$P_{\textrm{success}}=\Tr(\rho_{\textrm{in}}\hat{\Pi}_{\textrm{success}})=\left|\innerproduct{\Phi}{\Psi}\right|^{2},$
(30)

which matches Eq. (III.1). Note that this does not assume that $\Psi$ is
normalized, it could be sub-normalized due to an error at system 1, an error
in the unitary, or loss during the transmission.

More generally, such a POVM would be a weighted sum of projectors (mixed) but
here it is a lone projector (pure) as our process is reversible. For instance,
if the system parameters, such as $g_{j}$, varied due to fluctuations in the
position of the atoms or cavities, then averaging over these fluctuations
would give a mixed POVM [97]. Another relevant situation would be fluctuations
in the unitary transformation parameters ($\omega_{0}$, $\xi$, and $T$), which
would result in the input state $\rho_{\textrm{in}}$ of Eq. (30) being an
incoherent mixture of states $\Psi$ with different parameters according to
some underlying classical probability distribution. Specifically, if there is
classical uncertainty or variation in the unitary transformation parameters,
e.g., variation in $\omega_{0}$ due to the finite linewidth of a control
laser, then the actual probability of success would be
$P_{\textrm{success}}(\omega_{0},\xi,T)$ averaged over the corresponding
classical probability distribution. Note that the ideal wave packet should
remain the same as it is purely determined by the parameters of node 2 and the
laser driving it, $G_{2}$.

Note that $P_{\textrm{success}}$ is actually just the probability that an
excitation from atom 1 is transferred, via the intermediate photonic degree of
freedom, to atom 2. This assumes that no other excitations that can excite
atom 2 are produced during the transmission. We additionally assume that the
phase of the initial state of atom 1 is correctly transferred to atom 2. This
can be accomplished using stable local oscillators as frequency references at
each system, which we assume are used so that we need not worry about such
phase errors [98, 67]. Even if such phase errors do occur, error correction
protocols could be used to eliminate them (see Sec. III.4). We will proceed
under these assumptions so that $P_{\textrm{success}}$, as defined in Eq.
(III.1), is a good measure of the success of the entire QST scheme. Note that
what ‘success’ ultimately means is up to the particular scheme. For instance,
the quantum state fidelity $\mathcal{F}$ may be a more apt measure of
successful QST. Here $\mathcal{F}$ is given by the magnitude squared of the
overlap of the initial (qubit) state of atom 1 with the final state of atom 2.
Notably in the absence of phase errors $\mathcal{F}(|c_{e}|)$ is bounded below
by $P_{\textrm{success}}=|\alpha_{2}(t_{e})|^{2}$ (see SM A3 for details).

###  III.2 Optimal timing

Now that we have shown this new perspective, where system 2 can be thought of
as a single-photon detector, we will derive an expression that the optimal
timing parameter $T^{*}$ must satisfy. (Note this is new within the context of
this problem; that POVMs play a key role in describing photodetection [99,
100, 101, 102, 103, 104] and measurements more generally [105, 106, 107] is
well known.) We compute $T^{*}$ assuming that the transformation is
implemented correctly in all ways except that it has a limited duration
$t_{l}$. Then the probability of success based on the Green’s function
argument above is

$P_{\textrm{success}}(t_{s})=\left|\innerproduct{\Phi}{\Phi_{l}}\right|^{2},$
(31)

where $\Phi_{l}(t)=\Psi(t)|_{\omega_{0}=\omega_{0i},\xi=\xi_{i},T=T_{i}}$ is
the actual transformed wave packet for some finite $l$, assuming the ideal
frequency and stretching parameters as well as consistent timing;
$\Phi_{l}(t)$ is equal to $\Phi(t)$ for $t_{s}<t<t_{f}$. Maximizing
$P_{\textrm{success}}(t_{s})$ can thus be accomplished by solving for the
$t_{s}$ (and hence $T_{i}$) such that

$\frac{d}{dt_{s}}P_{\textrm{success}}(t_{s})=0$ (32)

(and verifying that the optimum is a maximum).

Note that

$\displaystyle\innerproduct{\Phi}{\Phi_{l}}=\sqrt{\gamma_{1}\gamma_{2}}\int_{-\infty}^{\infty}dt~{}\Big{[}$
$\displaystyle e^{-i\zeta(T_{i}-t)}\beta_{1}(\xi_{i}(T_{i}-t))$
$\displaystyle\times\chi(t)\beta_{1}(f(t))\Big{]},$ (33)

where $\chi(t)$ and $f(t)$ characterize the different stages of the
transformation. The parameters $T_{i}$, $t_{i}$, and $t_{f}$ (which are
related to the different stages) all implicitly depend on $t_{s}$ and so Eq.
(32) should be solved numerically in general. However, in typical cases the
contribution to Eq. (III.2) due to the untransformed portion of the wave
packet (i.e., outside the interval $t_{s}<t<t_{f}$) will be negligible. This
is due to some combination of $l$ being long enough for the $\beta_{1}$
product in the integrand to be small over the relevant domain and the term
being far off-resonant, with $|\omega_{0i}|\gg\gamma_{1,2}$, such that the
phase rapidly oscillates and the integrand averages to zero. In such a case,
effectively none of the untransformed wave packet emitted from system 1 will
induce a transition at system 2 and so we have

$\displaystyle\innerproduct{\Phi}{\Phi_{l}}$
$\displaystyle\approx\int_{t_{s}}^{t_{f}}dt~{}|\Phi(t)|^{2}$
$\displaystyle=\gamma_{2}\int_{t_{s}}^{t_{f}}dt~{}\beta_{1}^{2}(\xi_{i}(T_{i}-t))$
$\displaystyle=\gamma_{1}\int_{t_{i}}^{t_{s}}dt~{}\beta_{1}^{2}(t)$ (34)

(the approximation gets better for large $|\omega_{0i}|$ and/or $l$). As this
inner product is real, we can maximize $P_{\textrm{success}}(t_{s})$ by
finding the $t_{s}$ such that

$0=\frac{d}{dt_{s}}\int_{t_{s}-t_{l}}^{t_{s}}dt~{}\beta_{1}^{2}(t)=\beta_{1}^{2}(t_{s})-\beta_{1}^{2}(t_{s}-t_{l}).$
(35)

This is a much simpler condition than in the general case and it can easily be
solved for numerically once $G_{1}$ is specified and hence $\beta_{1}$ is
determined. The corresponding solution is the ideal value of $t_{s}$, which we
will denote by $t_{s}^{*}\equiv T_{i}^{*}/(1+1/\xi_{i})$.

###  III.3 Numerical results

To makes plots illustrating errors in the different unitary parameters, we
must specify the first laser pulse $G_{1}$. A natural case to consider is the
$G_{1}$ such that the amplitude for atom 1, $\alpha_{1}$, logistically
decreases from 1 to 0

$\alpha_{1}(t)=\frac{1+\tanh(-kt)}{2}.$ (36)

Note, for any monotonically decreasing $\alpha_{1}$, one can compute the
corresponding laser pulse $G_{1}$ that would generate it [see Eq. (B7) of SM
B2 ]. Then the amplitude for cavity 1 can be determined as
$\beta_{1}=-\dot{\alpha}_{1}/G_{1}$, which in turn gives the exact form of the
ideal emitted photon wave packet $\Phi$ of Eq. (III.1). (See SM B2 for
$\beta_{1}$ in this logistic $\alpha_{1}$ case.) We can thus compute how
errors in the unitary transformation parameters, i.e., incorrect values of
$\omega_{0}$, $\xi$, and $T$, degrade the transfer resulting in a decreased
$P_{\textrm{success}}$. The impact of such errors on $P_{\textrm{success}}$
can be seen graphically as in Fig. 2, where we compare a wave packet $\Psi$
due to a unitary transformation with stretching and timing errors to the
corresponding ideal wave packet $\Phi$.

Figure 2:  (a) Plot of the ideal wave packet modulus $|\Phi|$ for $k=2$, with
the unitary parameters taking on their ideal values
$\omega_{0}=\omega_{0i}=50,$ $\xi=\xi_{i}=1/2$, $T=T_{i}^{*}=19.7$ for
transformation duration $t_{l}=10$, and a non-ideal wave packet modulus
$|\Psi|$ still with $\omega_{0}=\omega_{0i}$, but with stretching and timing
errors $\xi=0.75$ and $T=17$. (All quantities are in units where
$\gamma_{2}=1$.) (b) Plot of the integrand in Eq. (III.1) for computing
$P_{\textrm{success}}$ in the ideal case (solid line) and the real and
imaginary parts in the non-ideal case (dashed and dotted lines, respectively).
Here, for visualization purposes, we decompose Eq. (III.1) as
$P_{\textrm{success}}=\left[\int
dt\real\left(\Phi^{*}\Psi\right)\right]^{2}+\left[\int
dt\imaginary\left(\Phi^{*}\Psi\right)\right]^{2}=0.186$, whereas
$P_{\textrm{success}}=\int dt|\Phi|^{2}=1$ for the ideal wave packet (which
assumes the entirety of the wave packet emitted from node 1 is transformed).

Furthermore, we can compute how $P_{\textrm{success}}$ varies as a function of
the amount of error in the various transformation parameters. For instance, we
illustrate the effect of errors in the unitary parameters $\omega_{0}$, $T$,
and $\xi$ by plotting $P_{\textrm{success}}$ versus one of these parameters
assuming the other parameters are ideal in Figs. 3, 4, and 5, respectively. We
work with a set of shifted ‘error variables’ that are centered at zero:
$\Delta\omega_{0}\equiv\omega_{0}-\omega_{0i}$,
$\Delta\ell_{\xi}\equiv\log_{2}{\xi}-\log_{2}{\xi_{i}}$, and $\Delta T\equiv
T-T_{i}^{*}$. We consider the logarithm of $\xi$ in most of our plots as its
ideal value is a ratio of two decay widths, $\xi_{i}=\gamma_{2}/\gamma_{1}$,
so errors in $\xi$ should scale multiplicatively, e.g., doubling $\xi$ (with
respect to its ideal value) should be (about) as bad as halving it. In each of
these plots (Figs. 3-5), the solid black lines are a cubic interpolation
between 201 points with the independent variable’s values distributed evenly
over the intervals shown. The corresponding value of $P_{\textrm{success}}$ is
given by the overlap calculation $|\innerproduct{\Phi}{\Psi}|^{2}$. The
overlaid colored points are calculated by numerically solving the coupled
ordinary differential equations (ODEs) of Eqs. (12a-d) for various values of
the independent variable (with larger separations because the relevant
numerics are more computationally expensive).

For concreteness we focus on the specific case of a logistic $\alpha_{1}$ with
$k=2$ and the physical parameters $\omega_{0i}=50$ and $\xi_{i}=1/2$.
Additionally, we assume a long transformation length $l=10$, for which
$T^{*}=19.7$, such that in the absence of errors in the unitary, effectively
all of the wave packet would be transformed with
$P_{\textrm{success}}=0.999995\approx 1$. Here all quantities are given in
natural units in which $\gamma_{2}=c=1$. Importantly, these underlying
parameter values are not crucial as, at least for large $l$ as we have here,
only the error variables (made dimensionless with appropriate factors of
$\gamma_{2}$) and the wave packet shape (as effectively controlled by
$\alpha_{1}$) matter (see SM C ). Hence the results gleaned from this
specific case apply more generally. The largest discrepancy in calculating
$P_{\textrm{success}}$ between the original coupled ODEs solution method and
the POVM wave packet overlap method is $1.1\cdot 10^{-6}$, which we assume to
be numerical error (in particular, we see that this specific error value goes
down if we increase our error tolerance when solving the ODEs). Hence this
numerical comparison serves as a strong indicator of the validity of the POVM
based results we found.

Figure 3:  Plot of $P_{\textrm{success}}$ as a function of the detuning
$\Delta\omega_{0}$ assuming the other unitary parameters are ideal. We take
the underlying parameter values to be the same as in Fig. 2 (that is, $k=2$,
$\omega_{0i}=50$, $\xi_{i}=1/2$, and $t_{l}=10$ in units where
$\gamma_{2}=1$). The probability of success quickly falls off away from
resonance, $\Delta\omega_{0}=0$. The solid line is from a state overlap
calculation using Eq. (III.1) and the overlaid points are from directly
numerically solving coupled EOMs for the state amplitudes (see main text).
Figure 4:  Plot of $P_{\textrm{success}}$ versus $\Delta T$ assuming the other
unitary parameters are ideal (with the same underlying physical parameters as
the previous figures). The solid line is from an overlap calculation and the
overlaid points are from solving coupled amplitude EOMs (see main text).
Figure 5: Plots of $P_{\textrm{success}}$ versus $\xi$ on (a) a linear scale
and (b) a logarithmic scale with $\Delta\ell_{\xi}=\log_{2}{\xi/\xi_{i}}$,
assuming the other unitary parameters are ideal (with the same underlying
physical parameters as the previous figures). The solid line is from an
overlap calculation and the overlaid points are from solving coupled amplitude
EOMs (see main text).

The probability of success quickly falls off away from its peak value near
unity as $\omega_{0}$, $T$, and $\xi$ are shifted from their ideal values. The
narrow peak around $\omega_{0}=\omega_{0i}$ in Fig. 3, which has a full width
at half maximum (FWHM) of $1.4\gamma_{2}$, illustrates the importance of
frequency conversion. We note that the shape and width of the
$P_{\textrm{success}}$ versus $\omega_{0}$ curve do not change appreciably as
the central frequency $|\omega_{0i}|$ is further increased. This is because,
as alluded to above, only the detuning of the transformed wave packet from
resonance with system 2 relative to $\gamma_{2}$,
$|\Delta\omega_{0}|/\gamma_{2}$, significantly matters during the
transformation. Moreover, for large $|\omega_{0i}|$ the untransformed portions
of the wave packet will not induce an excitation in atom 2. Thus, even when
$|\omega_{0i}|$ is very large, e.g., $|\omega_{0i}|/\gamma_{2}\sim 10^{6}$ is
typical, the FWHM will be stable. Here it remains at $1.4\gamma_{2}$ (for this
logistic $\alpha_{1}$ with $k=2\gamma_{2}=\gamma_{1}$), and hence it is
critical to control against frequency errors. Note that wave packets with
narrower temporal shapes are less susceptible to such frequency errors, though
this will be limited by emitting node parameters (see SM B3 ).

The couplings $\gamma_{j}$ tend to be on the order of kHz-MHz, and are
typically smaller in the microwave regime as compared to the optical. Hence,
typical values of $\gamma_{j}$ can vary by 2 to 3 orders of magnitude between
different systems and so $\xi_{i}=\gamma_{2}/\gamma_{1}\sim 100-1000$ is
reasonable for hybrid interconnects (assuming $\gamma_{2}>\gamma_{1}$,
otherwise $\xi_{i}$ would be the reciprocal of this). Experiments across
different platforms have demonstrated the ability to shape photon wave
packets, which includes stretching and compression by these orders of
magnitude [108, 109, 110, 89]. Note that the results for different $\xi_{i}$
can be mapped between one another because, as mentioned above,
$P_{\textrm{success}}$ is predominantly determined by the error variables,
$\Delta\ell_{\xi}$ in this case. Importantly, the narrow width of
$P_{\textrm{success}}$ as a function of $\Delta\omega_{0}/\gamma_{2}$ does not
doom us as kHz-MHz level precision in frequencies (necessary to obtain
$|\Delta\omega_{0}|\lesssim\gamma_{2}$) is possible using tunable, narrow-
linewidth lasers [111, 112, 113, 114], this includes mid-infrared and
terahertz laser sources based on difference frequency generation [115, 116].

We can quantify how much two of the dimensionless error variables
$\\{\Delta\omega_{0}/\gamma_{2},\Delta\ell_{\xi},\gamma_{2}\Delta T\\}$ depend
on one another by computing their “index of separability” $\mathcal{S}$ [117].
For an $m\times n$ matrix $A$ we define $\mathcal{S}$ in terms of its singular
values $\\{\sigma_{i}(A)\\}$ with maximum $\sigma_{\max}(A)$ as

$\mathcal{S}(A):=\frac{\sigma_{\max}^{2}(A)}{\sum_{i}\sigma_{i}^{2}(A)},$ (37)

which is the square of the ratio of the induced 2-norm and Frobenius norm of
$A$. This index of separability is bounded as
$0<1/\min\\{m,n\\}\leq\mathcal{S}(A)\leq 1$, where the maximum
$\mathcal{S}(A)=1$ entails that $A$ is separable, i.e., can be written as an
outer product of two vectors, and the minimum $\mathcal{S}(A)=1/\min\\{m,n\\}$
(which is nearly zero for large matrices) entails $A$ is full rank with all
equal singular values. However, practically the smallest observed values will
be much larger than this ($\sim{0.75}$ here), which comes from comparing to
random matrices (see SM C ).

For instance, with the same underlying physical parameters as in Figs. 2–5, we
take the joint probability distribution for $\gamma_{2}\Delta T$ and
$\Delta\ell_{\xi}$ assuming $\omega_{0}$ takes on its ideal value,
$P_{\textrm{success}}(\Delta T,\Delta\ell_{\xi})|_{\omega_{0}=\omega_{0i}}$,
compute it on a grid of equally spaced values for $\Delta T$ and
$\Delta\ell_{\xi}$ yielding a matrix, and then compute its index of
separability to be

$S_{T,\xi}\equiv\mathcal{S}[P_{\textrm{success}}(\Delta
T,\Delta\ell_{\xi})|_{\omega_{0}=\omega_{0i}}]=0.87.$ (38)

Similarly, for the other two variable pairs (keeping the third variable at its
ideal value) we find

$S_{\omega_{0},\xi}\equiv\mathcal{S}[P_{\textrm{success}}(\Delta\omega_{0},\Delta\ell_{\xi})|_{T=T^{*}}]=0.87$
(39)

and

$S_{\omega_{0},T}\equiv\mathcal{S}[P_{\textrm{success}}(\Delta\omega_{0},\Delta
T)|_{\xi=\xi_{i}}]=0.998.$ (40)

[Each of these $\mathcal{S}$ values is computed on a 121 by 121 grid (matrix)
with the same spacings as described in the Fig. 6 caption over a rectangular
region twice as large in each direction as those depicted. These reported
values ultimately serve as upper bounds for the separability index for large
grids. Corresponding lower bounds can be calculated using the zero-mean
counterparts of the $P_{\textrm{success}}$ matrices used here, yielding
respective $\mathcal{S}$ values of $0.80$, $0.80$, and $0.97$ compared to Eqs.
(38)–(40). (See SM C  for additional method details.)]

Hence $\omega_{0}$ and $T$ errors are largely independent of one another (as
the corresponding probability distribution is almost separable,
$\mathcal{S}\approx 1$), whereas errors in $\omega_{0}$ and $T$ are distinctly
dependent on what $\xi$ error occurs. The corresponding joint probability
distributions are given in Fig. 6 and can be used to get visual intuition for
the index of separability. For instance, Fig. 6 (a) illustrates that errors in
$T$ and $\xi$ are dependent on each other, which can intuitively be explained
as when the wave packet timing $T$ is off, there will be little overlap, yet
there will be more overlap if you elongate the wave packet in the time domain
by choosing $\xi<\xi_{i}$ ($\Delta\ell_{\xi}<0$). This analysis accounts for
dependencies of the errors on one another that are intrinsic to our model. We
leave considerations of other error dependencies that may be due to a
particular implementation of the transformation $U$ to other work.

Figure 6:  Density plot of $P_{\textrm{success}}$ as a function the error
variable pairs for the transformation parameters (a) $T$ and $\xi$, (b)
$\omega_{0}$ and $\xi$, and (c) $\omega_{0}$ and $T$ assuming the other
unitary parameter is ideal (with the same underlying physical parameters as
the previous figures). Each plot is computed as the cubic interpolation of a
61 by 61 grid of points that are evenly spaced over the region shown and is
colored according to the legend.

###  III.4 Error correction

We have highlighted many things can go wrong in the implementation of the
unitary itself, which is in addition to standard errors due to incorrect
driving laser pulses, incorrect laser frequencies, fluctuating system
parameters, as well as photon absorption or other losses in the transmission
line, cavities, or transformation device. With all of these possibilities, the
presence of errors on a single attempt of a QST scheme like ours is
inescapable, though crucially we can correct for these errors. Accordingly, we
will now discuss the applicability of a few error correction protocols to our
scheme as well as their limitations.

ECZ protocols. By using the occupation number encoding our scheme could
naturally be improved by utilizing one of two relatively low overhead error
correction protocols [73, 74] that capitalize on the feasibility of
suppressing photon production errors ${|{0}\rangle}\rightarrow{|{1}\rangle}$.
We refer to these as the ‘ECZ protocols’ after their authors’ last names.
These protocols can correct for photon loss and phase errors, which are
recurrent in standard transmission lines, e.g., optical fibers and waveguides.
They do so using auxiliary quantum emitters, again envisaged as atoms in a
cavity, that can encode the state of a qubit at each node and serve as a form
of redundancy. They cannot correct for errors caused by the creation of
photons in the relevant mode, which is often reasonable as the production of
photons can be suppressed in a well-isolated setup operating at an appropriate
temperature (such that there is low thermal occupation, $\mkern
1.5mu\overline{\mkern-1.5mun\mkern-1.5mu}\mkern 1.5mu_{\textrm{th}}\ll 1$,
which as previously discussed is more of a problem in microwave-based
systems).

Each ECZ protocol is based on repeating a certain primitive transfer
operation, which we will refer to as a trial, that includes QST attempts,
local single-qubit and two-qubit entanglement gate operations, and
measurements of auxiliary qubits or states. These local gate and measurement
operations are assumed to be implemented perfectly, though in practice these
operations will also be error prone and hence limit how close to unity the
ultimate transfer fidelity can be. In the earlier protocol [73], measurements
are performed on each trial, a successful outcome of which validates the state
transfer. In the later protocol [74], a given trial serves as a ‘purification’
step to be iterated so that a target state is approached exponentially, at a
rate that gets faster for smaller net errors in the protocol, and hence can be
reached up to some error threshold in fidelity. (See SM D  for further
discussion of these protocols.)

Both ECZ protocols consider an effective channel in which on a single
transmission attempt (which forms an integral part of both protocols) the
initial state will evolve for large times (so that no excitations remain in
the cavities) according to a map of the form [118]

$\begin{matrix}{|{g_{1}}\rangle}{|{g_{2}}\rangle}\\\
{|{e_{1}}\rangle}{|{g_{2}}\rangle}\end{matrix}\longrightarrow\begin{matrix}\alpha{|{g_{1}}\rangle}{|{g_{2}}\rangle}\\\
\beta{|{g_{1}}\rangle}{|{e_{2}}\rangle}+\Upsilon_{1}{|{g_{1}}\rangle}{|{g_{2}}\rangle}+\Upsilon_{2}{|{e_{1}}\rangle}{|{g_{2}}\rangle}\end{matrix},$
(41)

where $\alpha,\beta$, and $\Upsilon_{1,2}$ are constants. These are the form
of the long time limit of the solutions to EOMs analogous to Eq. (12) with
$|\beta|=|\alpha_{2}(t_{e})|$ and $|\Upsilon_{2}|=|\alpha_{1}(t_{e})|$, but
with additional possible errors included. Importantly for our scheme, an error
in the unitary transformation falls under this class of channel as it will
contribute to photon loss. Chiefly, an incorrectly shaped incident wave packet
will not be absorbed by node 2 and hence will be directed out a different
spatial mode [81]. This is in addition to loss due to absorption at the
transformation device. Such errors can be accounted for via the amplitudes
$\beta$ and $\Upsilon_{1}$. Thus, our scheme can naturally be incorporated as
an extended version of the transmission steps in the ECZ protocols, which can
aptly be utilized in our context for reliable QST between hybrid nodes.

We acknowledge that such a protocol would slow down the rate of entanglement
generation due to the time it takes to repeat the primitive transfer scheme as
well as to perform the requisite local computations. Importantly, however, if
we can achieve reasonably small error probabilities in the necessary
operations the slowdown in rate can be manageable. We will now illustrate this
by considering the expected number of repetitions $E[n]$ of the earlier ECZ
protocol [73] in the case where $|\alpha|=1$, as we have previously assumed in
our scheme, yet its phase could be nontrivial due to a relative phase error.
Consider an error on a given transmission of
$|\Upsilon_{1}|^{2}+|\Upsilon_{2}|^{2}=\varepsilon$ such that
$|\beta|^{2}=1-\varepsilon$ by normalization. In this model the worst case,
that with the largest $E[n]$, is when we are dominated by $\Upsilon_{1}$
errors (i.e., $|\Upsilon_{1}|^{2}=\varepsilon$ and $\Upsilon_{2}=0$). Note in
fact we expect to be in this regime with photon loss and unitary
transformation errors being dominant. In this worst case we find

$E[n]=\frac{4}{(1-\varepsilon)(2-\varepsilon)^{2}},$ (42)

which starts at 1 for $\varepsilon=0$, corresponding to only needing a single
trial without errors, and monotonically increases as a function of
$\varepsilon\in[0,1]$, tending towards $+\infty$ as $\varepsilon\rightarrow 1$
as the state transfer entirely fails for $\beta=0$ (see SM D for the general
derivation and further details). For a modest but still significant error of
$\varepsilon=0.1$ we find $E[n]=1.23$. Even for a more serious
$\varepsilon=0.5$ we find a relatively modest $E[n]=3.56$. Thus for reasonably
small error probabilities few trials (protocol repetitions) will be needed.

Alternate protocols and their scope. Note that there are many alternate error
correction protocols that can be used for deterministic quantum communication
via photons. Such protocols often use multiple photons as a type of redundancy
(the consideration of heralded cases is beyond the scope of this work). Such
protocols can be realized through mode occupation number encodings using
larger photon number, e.g., using so-called binomial [119] or cat [52] codes,
or using photonic graph states (such as cluster states) generated from a
single emitter [120, 121, 122]. Each of these alternate protocols comes with
their own technical challenges, especially in hybrid interfacing contexts like
ours where we would need to transform all the itinerant photons in a logical
state preserving way. Moreover, both the sender and receiver would need to be
able to interface with these multiphoton logical states. For instance, such
binomial codes are realizable with superconducting circuit technology [47] but
are more challenging for other implementations with less control of the
requisite multiphoton states. Similarly, only a few emitters, including
quantum dots and atoms in cavities, have demonstrated viability in the
generation of such photonic graph states. Accordingly, one advantage of the
ECZ protocols, especially for hybrid interconnects, is their relative
simplicity as the generation of single photon states is a standard, feasible
task for a wide variety of node implementations.

To make such protocols (including those of ECZ) applicable in the NISQ (Noisy
Intermediate-Scale Quantum [123]) era with error prone operations on small
numbers of qubits one has to compromise between the potential noise reduction
of a given error correction protocol versus the realistic ‘cost’ of its
implementation in a given deterministic QST proposal or experiment. This cost
includes additional resources such as local operations and controls as well as
auxiliary emitters that can simultaneously be coupled at each node, which all
come with their own potential errors as well as slower rates. For instance,
the earlier ECZ protocol [73] assumes that the errors in the unitary
parameters, laser pulses, and system parameters [specifically $\alpha$ and
$\beta$ in Eq. (41)] do not vary over subsequent transmissions in a given
trial. If this systematic approximation breaks down, then the later ECZ
protocol [74] should potentially be used as it can iteratively purify more
general random errors in the channel even with potentially non-Markovian
decoherence. However, this more comprehensive nature of the later protocol is
at the expense of additional overhead, in the form of additional local
operations, especially for large errors, so a compromise must be made.

## V. iscussion

In this paper we have demonstrated theoretically how unitary transformations
to the temporal-spectral mode of a photon serving as a flying qubit between
hybrid quantum nodes can be used to drastically improve the probability of
successfully transferring quantum information between the nodes. We showed
that the probability of transferring an excitation from one node to another,
which is a good measure of successful quantum state transfer, is given by the
modulus squared of the overlap of the spectral shapes of the actual and ideal
photon wave packets. Doing so makes the role of the unitary transformation
apparent: it should transform the emitted photon to one with this ideal
spectral shape (which can be calculated). Importantly, our scheme applies
quite generally to any nodes in which controlled quantum light-matter
interaction can be realized to reversibly transfer the state of a material
qubit to a photonic degree of freedom and back, not just for the three-level
atom in a cavity type nodes we focus on.

We analyzed the impact of errors in the implementation of the unitary
transformation. This includes showing how the success of the protocol depends
on the deviations from the ideal parameters as well as quantifying how much
these parameter errors depend on one another. This analysis can be used to
determine what kinds of errors dominate in a given physical setup, say to form
an error budget. Furthermore, we discussed how our scheme can naturally be
incorporated with and aid known error correction protocols to significantly
suppress or potentially eliminate unavoidable errors in deterministic quantum
state transfer, even between hybrid systems. Such an error-corrected
adaptation of our scheme could be used to distribute entanglement in a quantum
network or for distributed quantum computing. As a next step, our methods
could be applied to other material systems and the unitary transformation’s
physical implementation could be further considered, especially for cases
outside of the optical regime.

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Supplemental Material for “Success probabilities in time reversal based hybrid
quantum state transfer”

### Contents

A. Scope of results . Scope of results
1\. Generality of $P_{\textrm{success}}$ 1\. Generality of
$\boldsymbol{P_{\textrm{success}}}$
2\. Generality of model Hamiltonian 2\. Generality of model Hamiltonian
3\. Fidelity as a measure of success 3\. Fidelity as a measure of success
4\. Utility of time reversal 4\. Utility of time reversal
5\. Unitary implementations 5\. Unitary implementations
6\. Photonic qubit encodings 6\. Photonic qubit encodings

B. Determining the wave packet . Determining the wave packet
1\. Node Hamiltonians 1\. Node Hamiltonians
2\. Wave packet for a given $\alpha_{1}$ 2\. Wave packet for a given
$\boldsymbol{\alpha_{1}}$
3\. Formal $\beta_{1}$ and frequency errors 3\. Formal
$\boldsymbol{\beta_{1}}$ and frequency errors
4\. Producible wave packets 4\. Producible wave packets

C. Index of Separability . Index of Separability

D. ECZ error correction protocols . ECZ error correction protocols
1\. Scheme summary 1\. Scheme summary
2\. Scope of both schemes 2\. Scope of both schemes

one´

Here we provide additional discussion, calculations, and background to
supplement the main text. We label the equations presented here by prepending
their section number (A–D), thus distinguishing them from equations in the
main text, which we will refer to directly.

## . Scope of results

The atom in cavity type node we focus on in the main text is meant to be an
elucidating example, which should be contrasted against other implementations
of controlled light-matter interaction of individual quanta. As mentioned in
Sec. II.3 , our general theory aims to capture many of such implementations,
which we will further substantiate here. In this section we discuss the
applicability of our work beyond such atom in a cavity type nodes, the impact
of various choices we made in the QST scheme, and relevant alternatives to
said choices. This includes a more general argument for the probability of
successful excitation transfer formula derived in the main text,
$P_{\textrm{success}}$, in Sec. A1 and moreover a discussion of why our model
Hamiltonians themselves are quite general in Sec. A2. In Sec. A3 we discuss
the quantum fidelity of the initial and final qubit states as a different
measure of a successful QST scheme than $P_{\textrm{success}}$. In Sec. A4 we
discuss the impact of including time reversal as part of the unitary
transformation to the itinerant single photons. In Sec. A5 we highlight
relevant implementations of the proposed unitary transformation. In Sec. A6 we
discuss the impact of using a mode occupation number encoding for the photonic
qubits and contrast it to other commonly used encodings, especially
polarization.

### 1\. Generality of $\boldsymbol{P_{\textrm{success}}}$

To illustrate the wide-reaching applicability of our scheme for quantum
networking and communications we will now show how and why Eq. (2 ) of the
main text applies quite generally, under modest assumptions, for nodes
implemented using other systems including those just mentioned. To see this we
interpret the receiving node to be a binary outcome photodetector (the
simplest possibility), which either “clicks” if the incident light is in a
certain small fraction of all possible modes or does not click, corresponding
to a null response where there was either no light or the incident light was
in a mode the detector was not sensitive to. Note that this is just a thought
experiment as in our actual QST scheme it is crucial that we do not actually
perform a measurement (which would cause the state to collapse out of the
desired superposition state). In particular, we do not check the measurement
outcome, which could be done by appending an atomic measurement of the state
of qubit at node 2 (doing so would turn our QST scheme into an actual
measurement).

As is contemporarily standard (see the clarifying note in Sec. III.2 ), such
a quantum measurement can be mathematically represented as a positive
operator-valued measure (POVM), which is a set $\\{\hat{\Pi}_{n}\\}$, where
one can associate a POVM element $\hat{\Pi}_{n}$ (a positive Hermitian
measurement operator) with each of the possible measurement outcomes labeled
by $n$. The fact that upon measurement, one of the possible outcomes must
occur is encapsulated by the completeness relation
$\sum_{n}\hat{\Pi}_{n}=\mathds{1}$, where the sum runs over all $n$ and
$\mathds{1}$ is the identity operator on the Hilbert space associated with the
system being measured. Here the outcomes are “click” or “no click” with
corresponding POVM elements

$\hat{\Pi}_{\textrm{click}}=\sum_{i}w_{i}{|{\phi_{i}}\rangle}\\!{\langle{\phi_{i}}|}$
(A1)

(this form results from diagonalizing the POVM element, which is possible as
it is positive and Hermitian), where the sum runs over an orthonormal set of
modes $\\{{|{\phi_{i}}\rangle}\\}$ with weights $0\leq w_{i}\leq 1$, and
$\hat{\Pi}_{\textrm{no-click}}=\mathds{1}-\hat{\Pi}_{\textrm{click}}$,
respectively. The probability of a click can then be expressed as

$P_{\textrm{click}}=\Tr(\rho_{\textrm{in}}\hat{\Pi}_{\textrm{click}}).$ (A2)

In our case of single-photon detection, time reversibility guarantees that
(under ideal conditions111 These ideal conditions include that, as previously
assumed, no thermal or other unexpected excitations will interact with the
receiving node, that the system parameters are fixed (do not fluctuate), that
there are no dark counts, which would be represented by an additional term in
the click POVM element that projects onto the vacuum, and moreover we do not
consider multiple incident photons in the detection time. These are realistic
assumptions in situations like ours, where the single photon excitation is
mapped onto a material qubit. This is to be contrasted against standard
photodetectors that are destructive and hence irreversible. Accordingly, we do
not consider them in this work. ) our effective photodetector (the receiving
node) will project onto one single-photon wave packet. That is, the POVM
element (and corresponding ideal wave packet) is pure (rank one). This is
simply because the absorption process is reversible, it is time reversed and
complex conjugated relative to the pure (under the aforementioned ideal
conditions) emission process [69, 88]. Thus, here with the deterministic (and
reversible) production of single-photon wave packets, we have the pure POVM
element

$\hat{\Pi}_{\textrm{click}}={|{\Phi}\rangle}\\!{\langle{\Phi}|},$ (A3)

where ${|{\Phi}\rangle}$ is one of the orthornomal basis vectors. Inserting
this into Eq. (A2) for a pure input state
$\rho_{\textrm{in}}=\outerproduct{\Psi}{\Psi}$ we obtain Eq. (2 ). Note that
this click outcome corresponds to the successful transfer of an excitation,
which is why we call it $P_{\textrm{success}}$ in the main text, though
additional considerations (e.g., the use of local oscillators as phase
references) are needed to ensure that the phase is transferred.

We can define a probability of success for a premature detection time
$t_{p}\leq t_{d}\leq t_{e}$ as

$P_{\textrm{success}}(t_{d}):=\left|\int_{t_{p}}^{t_{d}}dt~{}\Phi^{*}(t)\Psi(t)\right|^{2},$
(A4)

which is zero at the preparation time $t=t_{p}$ and approaches the probability
of success $P_{\textrm{success}}(t_{e})$ as $t_{d}$ increases towards $t_{e}$.
This can be thought of as the overlap of our state $\Psi$ with the ideal state
up until time $t_{d}$, that is, with the sub-normalized state
$\Phi_{t_{d}}(t^{\prime}):=\Phi(t^{\prime})\Theta(t_{d}-t^{\prime})$ with
$\Theta$ the Heaviside step function. Thus we can consider the corresponding
(unnormalized) POVM element

$\hat{\Pi}_{\textrm{success}}(t_{d})=\outerproduct{\Phi_{t_{d}}}{\Phi_{t_{d}}}.$
(A5)

Equivalently, we can factor out the subnormalization and write the above POVM
element as

$\hat{\Pi}_{\textrm{success}}(t_{d})=W(t_{d})\outerproduct{\tilde{\Phi}_{t_{d}}}{\tilde{\Phi}_{t_{d}}}$
(A6)

with

$W(t):=\int_{t_{p}}^{t}dt^{\prime}~{}|\Phi(t^{\prime})|^{2}=\int_{t_{p}}^{\infty}dt^{\prime}~{}|\Phi(t^{\prime})\Theta(t-t^{\prime})|^{2}=\innerproduct{\Phi_{t}}{\Phi_{t}}$
(A7)

interpreted as an efficiency $0\leq W\leq 1$ and
${|{\tilde{\Phi}_{t}}\rangle}=W^{-1/2}(t){|{\Phi_{t}}\rangle}$ a normalized
single-photon state [131]. Importantly, no matter how we write the success
POVM element describing a click, the effective photodetector is (and in fact
must be222 Just as a POVM element of a general measurement never depends on
the input state on which the measurement is performed. ) independent of the
actual incident wave packet $\Psi$ and only depends on the detection time
$t_{d}$ and on the measurement parameters (those of the receiving system and
the controls used to manipulate it), which determine the ideal wave packet
$\Phi$. However, we cannot specify the form of the ideal wave packet $\Phi(t)$
without further system details, namely, the particular form of the effective
Hamiltonian describing our scheme, which can be obtained by focusing on
specific kinds of nodes. In this work we focus on nodes comprised of a three-
level atom in an optical cavity, though, as we will next discuss, the
corresponding Jaynes–Cummings interaction Hamiltonian is quite general. This
allows us to explicitly determine the ideal wave packet ${|{\Phi}\rangle}$ in
the temporal mode $\Phi(t)$ as in Sec. III.1 .

### 2\. Generality of model Hamiltonian

As we have discussed, the broad scope of Eq. (2 ) is quite intuitive.
Potentially more surprising (at least at first thought) is that our
Hamiltonian itself $H_{\textrm{eff}}$, Eq. (3 ), is quite general. To obtain
$H_{\textrm{eff}}$ one starts by considering two spatially separated systems
(nodes) interacting via an intermediate channel that hosts a quasi-one-
dimensional electromagnetic field [76] for which the only propagating degree
of freedom of light in the channel is the longitudinal mode (the polarization
and two transverse spatial modes are fixed, though this can straightforwardly
be extended). Then, by employing the standard rotating-wave approximation and
a Markov approximation (detailed in [76]) that includes assuming a flat
coupling333 This use of $\kappa$ and $\gamma$ follows the notation of Refs. 76
and 127. When comparing to other references, note that the coupling of node
$j$ to the transmission line, $\gamma_{j}$ here, is often denoted by $\kappa$,
though definitions often differ by a constant factor (typically 2) between
references, and it may be referred to as a cavity decay rate, photon exit
rate, etc.  $\kappa_{j}(\omega)=\sqrt{\gamma_{j}/2\pi}$ within a narrow
bandwidth of positive frequencies, one can eliminate the continuum mode
bosonic bath operators of the channel, as is standard in input-output theory
[76, 127, 132], to obtain the Heisenberg EOMs for system operators. The
corresponding dynamics of the state of both systems in the Schrödinger picture
can then be described by a Lindblad master equation or equivalently in the
quantum trajectory formalism [133, 76, 134] in which a given pure state
trajectory of the effectively directly coupled systems is induced by an
effective Hamiltonian. This Hamiltonian will be of the same three part form as
$H_{\textrm{eff}}$ of Eq. (3 ) (see Appendix I of Ref. 67), where $H_{1,2}$
are the node Hamiltonians and $H_{\textrm{tl}}$ accounts for their interaction
[it takes on the specific form of Eq. (5 ) assuming a unidirectional
coupling [81] and a vacuum field input to system 1; the phase $\zeta$ depends
on what rotating frames we go into for each node].

In our QST scheme and often in quantum networking, it is desirable to use
nodes in which some internal state of a material encoding a qubit can be
reversibly converted into a quantum excitation (often a single photon). Hence,
almost by construction, each node must be describable by a standard
Jaynes–Cummings-like interaction, i.e., via a Hamiltonian of the general form

$H_{\textrm{a}\sigma}=g(t)a^{\dagger}\sigma^{-}+g^{*}(t)a\sigma^{+}$ (A8)

in the interaction picture, under the rotating wave approximation (as was
already assumed to obtain $H_{\textrm{tl}}$ via input-output theory). Here $a$
and $\sigma^{-}$ are annihilation operators ($a^{\dagger}$ and $\sigma^{+}$
are their Hermitian conjugate creation operators) for a photonic and material
excitation (the cavity and atom in our case), respectively, and $g(t)$ is a
time-dependent coupling between them. This Hamiltonian describes a system in
which excitations with similar energies are transferred between light and
matter. When an excitation is removed from the material, a photon is produced,
$a^{\dagger}\sigma^{-}$, (though other excitations of light could potentially
be used, say in a two-photon transition) or vice versa during absorption,
$a\sigma^{+}$. This form assumes we are in the single-excitation regime as we
focus on material dynamics in the two-level subspace encoding the state of the
qubit, $\\{{|{g}\rangle},{|{e}\rangle}\\}$ here. As emphasized by Ref. 1, for
quantum networking applications it is desirable for the light-matter coupling
$g(t)$ to be ‘user controlled’ so that one can tune its amplitude and phase
(say using lasers driving the system) so as to deterministically induce
emission and absorption of photon wave packets of particular shapes. Our
effective node Hamiltonians Eq. (4 ), obtained in the case where each node
is comprised of an atom in cavity, are of this same form with
$\sigma^{-}={|{g}\rangle}\\!{\langle{e}|}$ and $g(t)=iG(t)$, though the phase
can be tuned more generally, see Sec. B1.

The generality of the Jaynes–Cummings (JC) model beyond systems of atoms or
molecules in a cavity (our setting here) was not obvious from its conception
[135]. In fact it took decades for the strong coupling regime, which is
necessary to realize JC physics, to be reached in cavity QED experiments [136,
72]. Since then, additional groundbreaking cavity QED experiments have
demonstrated the ability to engineer highly-controllable interactions of
individual photons with a small number of quantum emitters (single atoms
interacting with single photons in our case) with the corresponding weak
light-matter interaction enhanced by a high-$Q$ cavity. Such advances are
relevant for our scheme, which is a modified and extended version of a seminal
cavity QED based QST protocol [69], and for quantum networking more generally.
See Refs. 79 and 137 for relevant reviews (we point to Refs. 138, 139, 140, 90
for some innovative demonstrations). Meanwhile, other technologies have
emerged that can similarly facilitate such highly-controllable quantum light-
matter interactions to implement fundamental primitives like QST and
entanglement generation (which in some cases are heralded as opposed to
deterministic). These include experimental implementations (and accompanying
theory) using trapped ion qubits [141, 142, 143, 144], collective atomic
excitations [145], various forms of superconducting qubits [146, 46, 48, 47,
51, 52], and electronic spin qubits [nitrogen-vacancy (NV) centers in diamond
are a common realization] [147]. These are not limited to photonic flying
qubits; notably see Ref. 148 which uses an intermediate phonon for hybrid
quantum state transfer between a superconducting microwave qubit and a solid-
state spin qubit. In their main text they do this using a direct coupling,
though in their Supplementary Note 1 they also consider a ‘pitch-and-catch’
protocol with the phonon serving as a flying qubit that could propagate
between distant versions of such nodes connected by a waveguide. These many
implementations make the generality of JC physics manifest. See Ref. 149 for
further exposition of the JC model including its applicability in many
experimental settings, including those just mentioned, as well as its history,
theoretical details, and extensions.

The precise means of obtaining a Hamiltonian of this form depend on the nodes’
physical implementation, though some common steps (as mentioned in the main
text) include adiabatically eliminating other states, selecting an appropriate
rotating frame to go into (i.e., selecting which unitary transformations to
perform on the node Hamiltonians), and tuning the system parameters to
eliminate undesired terms. The physical interpretation of the operators in
$H_{\textrm{a}\sigma}$ and the methods of controlling the coupling $g(t)$
likewise depend on the node. However, as long as such a Hamiltonian is
realizable with a given node implementation (as it often is for deterministic
QST schemes), then the analysis presented here and in Ref. 67 should be
applicable. In particular, the equations we derive, such as the excitation
amplitude EOMs Eq. (12 ) and other subsequently derived expressions (or
similar variations thereof), should apply more generally to these other types
of nodes. Thus, even though we emphasize connecting two nodes that are each
comprised of an atom in a cavity, our results are relevant to hybrid cases
where two different types of nodes are connected.

### 3\. Fidelity as a measure of success

The probability of successfully transmitting an excitation between the systems
$P_{\textrm{success}}=|\alpha_{2}(t_{e})|^{2}$ [given in Eq. (III.1 ) of the
main text] is a useful measure of the success of a state transfer attempt.
This can be contrasted to the quantum state fidelity

$\mathcal{F}=|\innerproduct{\psi_{i}}{\psi_{f}}|^{2}$ (A9)

(some authors refer to the square root of this quantity as ‘the fidelity’),
which accounts for impact of phase errors in addition to excitation loss. Here
the initial state (of atom 1) is
${|{\psi_{i}}\rangle}=c_{g}{|{g}\rangle}+c_{e}{|{e}\rangle}$ (using the same
notation for the coefficients as Sec. II.1  of the main text) and the final
state (of atom 2) is of the form

${|{\psi_{f}}\rangle}=e^{i\theta_{f}}\sqrt{1-|c_{e}|^{2}|\alpha_{2}(t_{e})|^{2}}{|{g}\rangle}+c_{e}|\alpha_{2}(t_{e})|{|{e}\rangle},$
(A10)