# Adaptive, Continuous Entanglement Generation for Quantum Networks
††thanks: A. K. and A. Z. contributed equally to this work. This material is
partly based upon work supported by the U.S. Department of Energy, Office of
Science, National Quantum Information Science Research Centers.

Alexander Kolar, Allen Zang  University of Chicago
<EMAIL_ADDRESS><EMAIL_ADDRESS>Joaquin Chung, Martin Suchara,
Rajkumar Kettimuthu  Argonne National Laboratory
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Quantum networks, which enable the transfer of quantum information across long
distances, promise to provide exciting benefits and new possibilities in many
areas including communication, computation, security, and metrology. These
networks rely on entanglement between qubits at distant nodes to transmit
information; however, creation of these quantum links is not dependent on the
information to be transmitted. Researchers have explored schemes for
continuous generation of entanglement, where network nodes may generate
entanglement links before receiving user requests. In this paper we present an
adaptive scheme that uses information from previous requests to better guide
the choice of randomly generated quantum links before future requests are
received. We analyze parameter spaces where such a scheme may provide benefit
and observe an increase in performance of up to 75% over other continuous
schemes on single-bottleneck and autonomous systems networks. We also test the
scheme for other parameter choices and observe continued benefits of up to
95%. The power of our adaptive scheme on a randomized request queue is
demonstrated on a single-bottleneck topology. We also explore quantum memory
allocation scenarios, where a difference in latency performance implies the
necessity of optimal allocation of resources for quantum networks.

###### Index Terms:

quantum networks, entanglement generation, quantum teleportation, adaptive
protocol

##  I Introduction

Quantum information science has started another revolution of quantum
computation and communication. _Quantum networks_ [1] which connect quantum
devices and interconnect themselves are expected to become a reality, but
their construction is difficult due to fundamental physical differences
compared to a classical network. Direct transmission of qubits, which encode
quantum information, endures inevitable channel losses and environmental
noise. Even worse, the _no-cloning theorem_ forbids creation of identical
copies of an arbitrary unknown quantum state, which means that no
amplification is possible for a quantum signal. Fortunately, _Quantum
teleportation_ based on maximal _entanglement_ between two parties can
transfer an arbitrary quantum state from one party to the other. Entanglement
thus plays a role analogous to links in classical networks. Moreover,
established entanglement links can be extended using _entanglement swapping_
[2], where multiple shorter-distance entanglement links are consumed to
produce a single long-range link. Usage of quantum teleportation for
transmission thus makes entanglement generation between nodes a prerequisite
for a functional quantum network. However, differences exist from classical
links. Entanglement is consumed after each teleportation, which makes it a
one-time-use link. Additionally, _quantum memories_ storing entangled qubits
have finite (and generally short) coherence times which determine the
timescale of entanglement degradation. As a result, entanglement links are
strongly time-sensitive.

Given the complexity of potential realistic situations and the limitations
imposed on quantum networks, including arbitrary entanglement requests and the
probabilistic nature of near-term entanglement generation methods,
establishing entanglement between arbitrary node pairs in a timely manner is
difficult but important. This problem is two-fold: (1) local entanglement
links must be generated between intermediate nodes and (2) an optimal route
between the nodes in the pair must then be found using these links. The
routing problem (2), known as _entanglement routing_ , is an emerging and
active research topic in the quantum network community with various existent
research papers on entanglement routing protocols and algorithms. However, in
this paper we focus on the entanglement generation problem (1). Specifically,
we explore continuous generation, where entanglement links are generated
regardless of requests for entanglement establishment and which may reduce
latencies for serving such requests. We propose a heuristic adaptive scheme
for continuous entanglement generation. For entanglement routing, we adapt the
proposed local best effort routing algorithm from [3].

The structure of this paper is as follows. In Section II, we introduce our
methodology for quantum network simulation, including assumptions, parameters,
and explanation of our simulation procedure. In Section III, we explain the
mechanism of our adaptive entanglement generation protocol and the setup of
our simulated networks. In Section IV, we present our simulation results with
corresponding discussion. In Section V, we conclude our work.

##  II Methodology

###  II-A Network Assumptions

The simulated networks operate as simplified entanglement distribution
networks. To manage entanglement, a node may choose at each simulation time
step to attempt generation of a new entanglement link with an adjacent node or
extend entanglement by attempting entanglement swapping.

We make the following simulation assumptions:

  * •

The adaptive, continuous entanglement generation protocol only has knowledge
of adjacent nodes.

  * •

Each entanglement generation attempt between adjacent nodes (and swapping
between three nodes) occurs within one simulation time step.

  * •

Entanglement generation and swapping have a fixed (and distinct) probability
of success per attempt.

  * •

Each node contains a fixed number of quantum memories.

  * •

Quantum memories have a fixed storage time, after which stored entanglement is
lost and memories are automatically reset. For real quantum memories, the
initialization (preparation) time is finite and technology-dependent. For
example, the preparation time of atomic frequency comb (AFC)-type absorptive
memories is on the order of 100 ms [4], while could be much shorter for
single-qubit memories (e.g. below 100 µs [5, 6]).

  * •

Entanglement fidelity during generation and swapping is ignored. In realistic
scenarios, imperfect Bell states might be generated and entanglement
purification is needed to increase entanglement fidelity. However, we note
that for protocols like Barrett-Kok for entanglement generation between two
ions the entanglement fidelity can be very high for high Purcell factor and
small detuning between the two ions [7, 8].

  * •

Classical communication and computation are considered “free” operations, as
the development of classical networks is much more advanced than quantum
networks. Issues such as classical network congestion could in principle be
ignored if we consider that a classical network topologically identical to the
quantum network is also implemented, forming a “dual network” [9]. Moreover,
since the classical information communicated is always simple, we ignore
processing time. Therefore, the two major quantum operations we consider
(entanglement generation and swapping) share the same scale of operation time
(the sum of on-node operation time and qubit transmission time over optical
fibers). We then take this characteristic time scale as our simulation time
step.

  * •

A centralized scheduler schedules every request and ensures that only one
request is serviced in the network at any given time.

###  II-B Simulation Parameters

Based on our network assumptions, we identify the following network parameters
(with values discussed in III-B): number of nodes $N$, number of memories per
node $m$, memory lifetime $\tau_{m}$, entanglement generation success
probability $p_{e}$, and entanglement swapping success probability $p_{s}$. We
also maintain an adjacency matrix describing the connection of nodes via
optical fibers for transmitting photonic qubits. From this, we derive the
degree of each node $n$, denoted as $d_{n}$.

####  II-B1 Parameter Space

Here, we qualitatively analyze which parameter regimes our adaptive protocol
might be helpful for. First, we consider a simple protocol – which is also the
starting phase of our adaptive protocol as described in Section III-A. At each
time step, a node $n$ chooses a neighbor node $k$ at random with uniform
probability to attempt entanglement generation. We then have the expected
number of links formed between neighboring nodes $n$ and $k$ in one time step

$P(L)=p_{e}\left(\frac{1}{d_{n}}+\frac{1}{d_{k}}\right)=p_{e}\frac{d_{n}+d_{k}}{d_{n}d_{k}}$
(1)

We also compute the expected number of links $E(L)$ that node $n$ creates with
all nodes in its neighbor set $K$. This is equivalent to the probability of
any neighbor in $K$ randomly selecting $n$ and succeeding in generating
entanglement, or that node $n$ randomly selects a node in $K$ and succeeds in
generating entanglement. For our simple model, the probability of selecting a
node in $K$ is one; we may then write the expected number of links as

$E(L)=p_{e}+\sum_{k}\frac{p_{e}}{d_{k}}=p_{e}\left(1+\sum_{k}\frac{1}{d_{k}}\right)$
(2)

This number induces a time scale $1/E(L)$, after which we expect one memory
will be occupied. Finally, we consider the expected number of links generated
with a specific neighboring node $k\in K$ after some number of time steps
$\Delta T$. If we assume that the lifetime $\tau_{m}$ is longer than the time
interval $\Delta T$, then the number of links shared with node $k$ is bounded
by the number of memories $m$ under the assumption that one quantum memory can
only be entangled with one other memory (according to monogamy of
entanglement[10]). Combining all results, we have that

$E(L_{k})=p_{e}\frac{d_{n}+d_{k}}{d_{n}d_{k}}\,\text{min}\left(\Delta
T,\frac{m}{E(L)}\right)$ (3)

For parameter regimes where $E(L_{k})$ is small it is more likely that certain
needed entanglement links may not have been generated by a protocol with
uniform probabilities before a request is scheduled. Therefore we expect our
adaptive protocol to be useful in scenarios with small $E(L_{k})$, i.e. with

  * •

Low entanglement generation probability $p_{e}$,

  * •

Low number of memories $m$ with respect to node degree,

  * •

High degree (connectivity) of nodes $d_{n}$, or

  * •

Low time interval between requests $\Delta T$.

In this paper, we focus on the first two items, which correspond to realistic
expectations for near-term quantum networks. Considering the high cost of
building and maintaining quantum memories, the number of memories per node
will not be high in the near future. We also study small-scale networks
(discussed in III-B) with modest average node degrees.

###  II-C Simulation Procedure

We use a traffic matrix $T$ to represent expected average network usage. From
this matrix, we generate a queue of $r$ requests $R(a,b)$ for an entanglement
link between two arbitrary network nodes $a$ and $b$. Each element of the
queue is chosen randomly with a distribution determined by the traffic in the
traffic matrix, such that heavily trafficked connections (with a larger
$T_{ab}\in[0,1]$) have a higher number of corresponding requests in the queue
on average.

At the beginning of each simulation trial, time is allocated for initial
entanglement generation. For our simulations, this period is equal to the
interval between successive requests with a uniform request interval. Then the
first request from the queue is passed to the network. After a request is
completed (an entanglement link has been established between the two nodes),
we wait until the scheduled time of the next request. This period allows
adaptive generation protocol instances to generate new links with an updated
internal state. If a request has not been completed after the scheduled time
for the next request, we start to serve the next request immediately after the
current is completed, leaving no wait time. We record the latency of serving
each request (as the number of time steps between submission of the request
and its completion).

At the scheduled time of a request, a path will be decided based on local
best-effort routing [3], where nodes make path decisions based on existing
entanglement links and network connectivity. To complete the request,
additional entanglement links may be needed. Thus, after a route has been
selected, nodes along the path will prioritize the creation of on-demand
entanglement links to service this request over continuous random generation.
This may overwrite existing links.

Once the request queue has been serviced, we store the service latencies of
each request $\ell(n)$ as a function of $n$ (the index of the request in the
queue). We then reset the simulation by clearing the state of network nodes
and setting the simulation time to 0. We repeat this trial $N_{T}$ times and
average each $\ell(n)$ into the average latency of requests $\ell_{avg}(n)$.
We also record an upper bound to the latency $\ell_{max}(n)$ as the ninety-
fifth percentile of service latencies over all trials. Plots are made with a
moving window average with window size 3.

For further details of our implementation, the source code is available in our
GitHub repository [11] for this project.

###  II-D Request Scenarios

We consider multiple scenarios with increasing complexity:

  1. 1.

Scheduling a queue of equivalent requests (always between one pair of nodes)
with a constant time interval between requests.

  2. 2.

Scheduling a queue of random requests (from the traffic matrix) with a
constant time interval between requests.

  3. 3.

Varying the time between requests.

The first scenario is the simplest and provides basic insight on the behavior
of latencies. The second takes interaction between different requests into
consideration. Different requests might interfere with each other with regards
to the adaptive protocol, and traffic load differences might further affect
the performance of the adaptive protocol. The third scenario is the most
realistic, and it requires the consideration of such effects as resource
reservation and deadlock. In this paper, we focus on the first two scenarios
and leave the third as future work.

##  III Simulation Setup

###  III-A Adaptive Generation Protocol

The adaptive generation protocol will continuously attempt to generate
entanglement with adjacent nodes. It will randomly choose which adjacent node
to attempt to generate entanglement with. Initially, the distribution of nodes
will be uniform, such that a protocol instance on node $n$ may select each
neighboring node $i$ with probability $p^{(i)}_{0}=1/d_{n}$ probability of
being selected (where $d_{n}$ is the degree of node $n$).

Figure 1: Neighbors of a central node $0$ with entanglement links (dotted red)
and part of a found route (solid green). In this scenario, $S=\\{(0,1)\\}$ and
$V=\\{(0,5)\\}$.

As a request is being serviced, the adaptive protocol will track the requested
links and their overlap with the previously generated links; that is, it will
track

  * •

The set of links $S$ which were pre-generated with direct neighbors and
consumed in servicing the requests, and

  * •

The set of links $V$ which were not pre-generated and had to be created on-
demand to service the request.

The relation between these sets is shown in Figure 1. Once the request has
been completed, the adaptive protocol will

  * •

Maintain the probability of selecting nodes corresponding to links in $S$,

  * •

Increase the probability of selecting nodes corresponding to links in $V$,
proportional to 1 minus the sum of probabilities for $S$ and $V$, and

  * •

Decrease the probability of selecting nodes corresponding to unused links;
that is, links that are not in $S$ or $V$.

Formally, this may be written as

$\displaystyle p_{t+1}^{(i)}=\begin{cases}p_{t}^{(i)}&\text{if }i\in S\\\
p_{t}^{(i)}+\frac{\alpha}{|V|}\left(1-\sum_{j\in S\cup
V}p_{t}^{(j)}\right)&\text{if }i\in V\\\ \frac{1}{d_{n}-|S\cup
V|}\left(1-\sum_{j\in S\cup V}p_{t+1}^{(j)}\right)&\text{otherwise
}\end{cases}$ (4)

where we ensure that the sum of probabilities $\sum p^{(i)}$ equals 1 for all
rounds. This protocol description introduces an additional simulation
parameter $\alpha$, describing how quickly we change probabilities. This
parameter has bounds $0\leq\alpha\leq 1$ to keep probability distribution
normalized. These probabilities will then be used until another request passes
through the node.

###  III-B Network Setup

Unless otherwise stated, the network parameters shown in Table I are used for
each simulation. We test the performance of the adaptive protocol on 2 network
topologies, namely an autonomous systems (AS) network and a single-bottleneck
network. The AS-network is generated randomly using the networkx [12] library
to mimic the design of the classical internet, providing interesting
connectivity and paths.

TABLE I: Parameter List with Default Values. Parameter | Value
---|---
NET_SIZE ($N$) | 10
MEMO_SIZE ($m$) | 5
MEMO_LIFETIME ($\tau_{m}$) | 1000 sim. time steps
ENTANGLEMENT_GEN_PROB ($p_{e}$) | 0.01
ENTANGLEMENT_SWAP_PROB ($p_{s}$) | 1
QUEUE_INT ($\Delta T$) | 500 sim. time steps

We expect for near term implementation of local, metropolitan area quantum
networks, the number of nodes will be a few tens or less, and quantum memories
will remain an expensive resource for networking use so each node should also
contain a few tens or less. Time parameters in the simulation are in units of
simulation time step, whose corresponding real-world value is determined such
that it is of the same order as entanglement generation time
$T_{\mathrm{gen}}$ and swapping time $T_{\mathrm{swap}}$, i.e. we define a
time $T_{0}=O(T_{\mathrm{gen}})\approx O(T_{\mathrm{swap}})$ as time unit (the
approximation is justified by the fact that both operations require
information transmission over quantum and classical channels). The time for
one trial of entanglement generation over a physical link is
$L_{0}/c+T_{\mathrm{init}}$ [13], where $L_{0}$ is the length of the link, $c$
is the speed of light in a certain transmission medium, and
$T_{\mathrm{init}}$ is the time to initialize a quantum memory. For example, a
recently proposed protocol based on rare earth ions [8] involves an
initialization time of $T_{\mathrm{init}}\approx 85$ µs, and given that
$L_{0}\approx 10$ km, we have that $T_{\mathrm{gen}}$ is around a few hundred
microseconds. For swapping, the usual procedure requires a Bell state
measurement on the middle node and communication of the result to neighboring
nodes, followed by local operations based on the measurement outcome.
Therefore, the $T_{\mathrm{swap}}$ is also the sum of communication time and
local operation time, being roughly the same order of $T_{\mathrm{gen}}$. If
we take 1 ms as the time step interval, quantum memory lifetime
$\tau_{m}=1000$ corresponds to a lifetime of 1 s, which can be satisfied by
e.g. trapped ions [14], neutral atoms [15], and rare earth ensembles [16, 17].
The unit swapping success probability refers to deterministic swapping, which
was considered in previous simulation work [3]. Entanglement generation
success probability $p_{e}$ can be estimated as
$(\eta_{c}\eta_{t}\eta_{d})^{2}/2$ [5], where $\eta_{c}$ is coupling
efficiency for light emitted by the quantum memory, $\eta_{d}$ is detector
efficiency, and $\eta_{t}\approx e^{L/L_{\mathrm{att}}}$ is transmission
efficiency, with characteristic attenuation length for optical fibers being
$L_{\mathrm{att}}\approx 22$ km. Considering $L\approx 10$ km and with an
optimistic expectation of high detection and optical coupling efficiencies,
$p_{e}$ on the order of 0.1 might be possible for future implementations.

##  IV Results and Discussion

In this section, we present the results of using our adaptive protocol on a
few example network scenarios.

###  IV-A AS Network Topology

We first test the performance of our adaptive scheme on a simple, 10-node
autonomous systems (AS) network. The topology for this setup is displayed in
Figure 2.

Figure 2: Network topology for a 10-node autonomous systems network.

First, we observe the average latency $\ell_{avg}(n)$ and max latency
$\ell_{max}(n)$ for simulations with adaptive parameters ranging from
$\alpha=0$ to $\alpha=0.2$. These results are shown in Figure 3. We note that
the adaptive protocol is able to provide improved performance for scheduled
requests, compared to the $\alpha=0$ case where there is no adaptation.
Additionally, for higher $\alpha$, the adaptive protocol is able to quickly
converge to better performance; however, this has a limit of about
$\alpha=0.1$ after which convergence does not improve significantly. This is
because our adaptive protocol only updates the probability distributions on
nodes in the found path, and thus nodes not in the path will continue to
generate entanglement with nodes in the path and consume memory resources.

Figure 3: Service latency behavior for an AS-net topology with 10 nodes.

We also compare the $\alpha=0.1$ case to two fixed entanglement schemes
mentioned in [3]; these are

  * •

The uniform scheme, where each node may pick any node from across the network
as an entanglement partner with uniform probability, and

  * •

The power law scheme, where each node may pick any node from across the
network as an entanglement partner with probability that decreases with
distance.

Results for this comparison are shown in Figure 4. We see that the performance
of the adaptive scheme is better than both the power law and uniform schemes,
even before the adaptive scheme has reached a convergence in performance. This
is because the power law and uniform schemes allow choosing partners that are
not direct neighbors for entanglement generation; when the partner is not a
direct neighbor, to restrict entanglement generation attempts within one time
step, we consider an attempt successful only if all needed elementary links
are generated within that time step. The success probability is then
approximated as product of success probabilities for each elementary
entanglement link. It is also for this reason that the power law scheme
provides slightly better performance than the uniform scheme.

Figure 4: Service latency behavior for an AS-net topology with 10 nodes and
various schemes for continuous entanglement generation.

###  IV-B Single-Bottleneck Topology

In section IV-A, we focused on identical requests – not to demonstrate our
protocol’s performance in a “realistic” scenario, but only to verify its
concept. In this section, we consider a more realistic scenario where requests
are randomly generated. Bottleneck structures are ubiquitous in various types
of networks, and we envisage that they might also exist in quantum networks.
Therefore, as a starting point of future exploration, we restrict ourselves to
a simplified 8-node topology with a single bottleneck shown in Figure 5.

Figure 5: Network topology for an 8-node network with a bottleneck.

The traffic matrix was then generated such that nodes will only request
entanglement links with nodes on the other end of the bottleneck. This matrix
is available on our GitHub page [11]. We present results for a specific choice
of parameters: adaptive weight $\alpha=0.05$ and request interval $\Delta
T=200$. Note that our simulation is small scale and it is not hard to simulate
for a different parameter choice.

We consider two scenarios: the case of equal memory allocation to each node,
and unequal memory allocation based on the degree of nodes. We expect
decreased latencies from the unequal allocation, analogous to a classical
network. We justify this by considering that during the random generation
phase, each edge node has only one direct neighbor (a bottleneck node) to
attempt entanglement generation with. Thus, their entanglement attempts will
occupy bottleneck node memories. For equal memory allocation, this will
exhaust the memory resources of the bottleneck nodes, preventing entanglement
links across the bottleneck. With unequal allocation, edge node memories will
be exhausted faster, leaving bottleneck nodes memories to generate
entanglement links across the bottleneck with (which are necessary for any
request). This reduces the probability that a bottleneck entanglement link is
missing when serving requests, thus reducing latency.

####  IV-B1 Equal Memory Number per Node

We first consider the scenario where every node has the same number of quantum
memories. Based on the expectation that near-term networks will have a small
number of quantum memories per node, we arbitrarily choose two different
quantities, 5 and 30 memories, to explore the effect of resource scarcity on
the protocol performance, and we also consider different quantum memory
lifetimes and entanglement generation probabilities. We compare our adaptive
scheme to a non-adaptive scheme (i.e. with $\alpha=0$), with average latencies
shown in Figure 6.

Figure 6: Service latency behavior for a bottleneck topology with various
parameter choices and continuous entanglement generation schemes.

First, we note that the adaptive protocol gives lower latencies for later
requests (even with randomized requests). For different sets of parameters,
more quantum memories per node lead to lower average latencies. Additionally,
we observe that by increasing quantum memory lifetime $\tau_{m}$ and
entanglement generation probability $p_{e}$ the average latencies decrease.
These results are in accordance with the intuition that more resources of
higher quality will lead to better performance.

####  IV-B2 Quantum Memory Allocation

In the network topology from Figure 5, the physical neighbors and traffic are
not uniform. Therefore, it is natural to conjecture that quantum memory
allocation on nodes might also influence network behavior. We explore the
scenario where edge nodes (nodes 0-2, 5-7 from Figure 5) are assigned 5
memories and bottleneck nodes (nodes 3 and 4) are assigned 30. In Figure 7, we
compare this new allocation to the case where every node is assigned 30
memories.

Figure 7: Latency behavior for a bottleneck topology with unequal memory count
and continuous schemes.

We observe significant improvement with unequal quantum memory allocation over
equal allocation of 30 memories for $(\tau_{m}=1000,p_{e}=0.01)$ and
$(\tau_{m}=1000,p_{e}=0.1)$. We note that unequal allocation leads to lower
latencies for both adaptive and non-adaptive protocols (even approaching zero
for $(\tau_{m}=1000,p_{e}=0.1)$). We additionally observe that the adaptive
protocol still provides a decrease in latency over a non-adaptive protocol for
unequal memory allocation (except for the $(\tau_{m}=1000,p_{e}=0.1)$ case,
where latencies are already zero for both schemes).

Finally, we analyze why zero latency for $(\tau_{m}=1000,p_{e}=0.1)$ is
possible in our simulation. Given $p_{e}=0.1$, on average a single
entanglement link can be generated after 10 attempts. With limited memory
$m=5$ for edge nodes, the expected number of entanglement links $E(L)$ is thus
capped at $5$ for $\Delta T>50$ (where a queueing interval $\Delta T=200$ was
used for the simulation). Under the most extreme scenario where both
bottleneck nodes only attempt to generate entanglement with each other, within
50 time steps on average 10 memories will be occupied by entanglement links
over the bottleneck. This leaves enough excess memories for entanglement with
edge nodes. Thus, within $\Delta T=200$ all elementary entanglement links will
be generated for this specific topology and memory allocation. Moreover, with
$\tau_{m}=1000$, pre-generated entanglement links have a long lifetime and may
be used to service future requests. As a result, required entanglement links
for any request may be generated before the request is scheduled, leading to a
latency of zero. On the contrary, if we consider a smaller quantum memory
lifetime (e.g. in our simulation $\tau_{m}=100<\Delta T=200$), the pre-
generated entanglement links are prone to expiration before the submission of
later requests. This is true regardless of memory exhaustion. Therefore, there
will likely exist links which must be generated on demand. This explains the
performance similarity between the two allocation schemes for
$(\tau_{m}=100,p_{e}=0.01)$.

These results confirm our expectation that simply adding more resources will
not always lead to improved performance for quantum networks, as is the case
with classical networks.

##  V Conclusion

We proposed an adaptive protocol that uses pre-generated entanglement links
between neighboring nodes to reduce service latencies for entanglement
requests between arbitrary nodes in a quantum network. We also provided a
simple but effective framework for qualitative analysis of request serving
behavior. We then built a time-step simulator for generation of entanglement
links and request servicing on a quantum network. Our simulation results show
the effectiveness of our adaptive protocol for homogeneous request queues and
certain network parameters, as compared to continuous entanglement generation
schemes relying on static distributions. We also studied latency reduction for
randomized requests on a single-bottleneck topology. Moreover, the difference
in protocol performance for different methods of assigning quantum memories to
nodes confirms that sophisticated resource allocation is required for
optimizing quantum network performance. We leave further study of the network
parameter space, more detailed network simulation, and inclusion of quantum
effects within our protocol for future work.

## Acknowledgment

We thank Manish Kumar Singh for useful insights to improve our simulation. A.
K. and A. Z. thank Ben Y. Zhao, from whose course this project originated.

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supplement.pdf

# Floquet driven frictional effects

Vahid Mosallanejad<EMAIL_ADDRESS>Department of Chemistry, School of
Science, Westlake University, Hangzhou, Zhejiang 310024, China  Institute of
Natural Sciences, Westlake Institute for Advanced Study, Hangzhou, Zhejiang
310024, China  Jingqi Chen  Department of Chemistry, School of Science,
Westlake University, Hangzhou, Zhejiang 310024, China  Institute of Natural
Sciences, Westlake Institute for Advanced Study, Hangzhou, Zhejiang 310024,
China  Wenjie Dou<EMAIL_ADDRESS>Department of Chemistry, School
of Science, Westlake University, Hangzhou, Zhejiang 310024, China  Department
of Physics, School of Science, Westlake University, Hangzhou, Zhejiang 310024,
China  Institute of Natural Sciences, Westlake Institute for Advanced Study,
Hangzhou, Zhejiang 310024, China

###### Abstract

When the coupled electron-nuclear dynamics are subjected to strong Floquet
driving, there is a strong breakdown of the Born-Oppenheimer approximation. In
this article, we derive a Fokker-Planck equation to describe non-adiabatic
molecular dynamics with electronic friction for Floquet driven systems. We
first provide a new derivation of the Floquet quantum-classical Liouville
equation (QCLE) for driven electron-nuclear dynamics. We then transform the
Floquet QCLE into a Fokker-Planck equation with explicit forms of frictional
force and random force. We recast the electronic friction in terms of Floquet
Green’s functions such that we can evaluate the electronic friction
explicitly. We show that the Floquet electronic friction tensor exhibits
antisymmetric terms even at equilibrium for real-valued Hamiltonian,
suggesting that there is a Lorentz-like force in Floquet driven non-Born
Oppenheimer dynamics even without any spin-orbit couplings.

Introduction.– The molecular dynamics near metallic surfaces can be non-
adiabatic in nature and hence Born-Oppenheimer (BO) approximation is not
necessary correct Rittmeyer  _et al._ (2018); Galperin and Nitzan (2015);
Juaristi  _et al._ (2008). The electronic friction approach is considered as
the first order correction to the BO approximation Dou and Subotnik (2018a),
which can be understood as quantum mechanical damping force of a manifold of
fast relaxing electronic on classical nuclear motion. Electronic friction
approaches were successful in explaining many experimental results such as
molecular beam experiments Spiering and Meyer (2018); Lü  _et al._ (2019);
Hertl  _et al._ (2021), electrochemistry Lam  _et al._ (2019), charge$/$spin
transport phenomenaDou  _et al._ (2018); Bajpai and Nikolić (2020).
Quantitatively, electronic friction is a tensor which appears on the
generalized Langevin equation Askerka  _et al._ (2016). One of the first
notable quantum mechanical derivations of the electronic friction tensor is
given by Head-Gordon and Tully Head-Gordon and Tully (1995). Later, more
rigorous expressions are derived from Keldysh Green’s function Bode  _et al._
(2012); Kershaw and Kosov (2020), path integral Lü  _et al._ (2012); Chen  _et
al._ (2019), quantum classical Liouville equation (QCLE) Dou and Subotnik
(2016), exact factorization Martinazzo and Burghardt (2022). It has been shown
that there is only one universal electronic friction tensor in the Markovian
limit Dou and Subotnik (2017, 2018b). Furthermore, study shows that the
friction tensor can exhibit antisymmetric terms even at equilibrium when spin-
orbit couplings are involved. Teh  _et al._ (2021)

Now, there are increasing interests in understanding of the dynamics of
molecular systems with strong light-matter interactions, which is helpful for
interpreting photochemistry and spectroscopy Schirò  _et al._ (2021); Lang
_et al._ (2015). In particular, people are interested in how to use
light$/$photon to manipulate chemical reactions where the dynamical interplay
between light and electronic non-adiabatic transitions plays a significant
role. At the same time, active research is currently ongoing to understand the
response of quantum systems to a periodic driving force, or so called “Floquet
driven” systems Sambe (1973); Kohler  _et al._ (2005); Novičenko  _et al._
(2017). Floquet theorem provides a powerful method for the analysis of quantum
systems subjected to periodic external drivings. Effects, such as phase
transitions and pump-probe photoemission can be explained by applying Floquet
theorem in solving quantum mechanical problems Yang  _et al._ (2019); Dehghani
_et al._ (2014); Sentef  _et al._ (2015). The coupled electron-nuclear
dynamics with strong light-matter interactions can be described by the Floquet
quantum classical Liouville equation (QCLE) successfully. Kapral (2006); Kelly
and Kapral (2010); Kim  _et al._ (2008). In this article, we offer a new
derivation for the Floquet QCLE starting from Floquet Liouville equation.
Moreover, we map the Floquet QCLE into a Langevin equation with all non-
adiabatic correction being incorporated into frictional effects. Furthermore,
we demonstrate that the Floquet electronic friction tensor exhibits
antisymmetric terms even at equilibrium for real-valued Hamiltonian.

Liouville-von Neumann equation in the Floquet representation.– For the coupled
electron-nuclear motion, we consider a general Hamiltonian $\hat{H}$ that can
be divided into the electronic Hamiltonian $\hat{H}_{e}$ and the nuclear
kinetic energy:

$\hat{H}=\hat{H}_{e}({\mathbb{R}},t)+\sum_{\alpha}\frac{\hat{P}_{\alpha}^{2}}{2M_{\alpha}}$
(1)

Here ${\mathbf{R}}=\\{{R}_{\alpha}\\}$ and
$\hat{\mathbf{P}}=\\{\hat{P}_{\alpha}\\}$ are position and momentum operators
for the nuclei respectively. We use $\alpha$ to denote nuclear degrees of
freedom. Note that, the electronic Hamiltonian $\hat{H}_{e}({\mathbb{R}},t)$
is considered to be an explicit function of ${\mathbf{R}}$ and time $t$.
Below, we will consider the case that the system is subjected to periodic
driving, such that
$\hat{H}_{e}({\mathbb{R}},t+T)=\hat{H}_{e}({\mathbb{R}},t)$. $T$ is the period
of the driving frequency.

The equation of motion for the density operator follows Liouville-von Neumann
(LvN): $\frac{d\hat{\rho}(t)}{dt}=-\frac{i}{\hbar}[\hat{H}(t),\hat{\rho}(t)]$.
For the periodic driving system, we can derive a Floquet Liouville-von Neumann
(LvN) equation describe the time evolution of the density operator in Floquet
representation. To do so, two transformations are needed to derive Floquet
representation of LvN: (I) Transformation of LvN into the Fourier
representation and (II) transformation from the Fourier representation to the
Floquet representation. The details of part (I) is given in the Supplementary
Information S1. Fourier representation of the LvN equation reads

$\displaystyle\begin{aligned}
&\frac{d\hat{\rho}^{f}(t)}{dt}=-\frac{i}{\hbar}[\hat{H}^{f}(t),\hat{\rho}^{f}(t)],\end{aligned}$
(2)

Here, the Fourier representations of Hamiltonian and density operators
[$\hat{H}^{f}(t)$ and $\hat{\rho}^{f}(t)$] are given by

$\displaystyle\hat{H}^{f}(t)=\sum_{n}\hat{H}^{(n)}\hat{L}_{n}e^{in\omega
t},\hat{\rho}^{f}(t)=\sum_{n}\hat{\rho}^{(n)}(t)\hat{L}_{n}e^{in\omega
t}.~{}~{}$ (3)

The operator $\hat{L}_{n}$ denotes the $n$th ladder operator in Fourier space
(see SI for detailed definition). The Fourier expansion coefficients of the
Hamiltonian is given by $\hat{H}^{(n)}=1/T\int_{0}^{T}H(t)e^{-in\omega t}dt$.
Indeed, Eqs. (3) are Fourier expansions modified by adding the ladder operator
$\hat{L}_{n}$. We stress that, the ladder operator turns the vector-like
Fourier expansion into a matrix-like representation. We then transform the
density operator from the Fourier representation to the Floquet representation
as

$\displaystyle\hat{\rho}_{F}(t)=e^{-i\hat{N}\omega
t}\hat{\rho}^{f}(t)e^{i\hat{N}\omega
t}=\sum_{n}\hat{\rho}^{(n)}(t)\hat{L}_{n},$ (4)

where $\hat{N}$ is the number operator in Floquet representation (see SI for
detailed definition). Employing such a definition, the equation of motion for
$\hat{\rho}_{F}(t)$ now reads

$\displaystyle\frac{d}{dt}\hat{\rho}_{F}(t)=-\frac{i}{\hbar}\left[\hat{H}_{F},\hat{\rho}_{F}(t)\right],$
(5)

where we have defined the following Floquet representation for the Hamiltonian
as

$\displaystyle\begin{aligned}
\hat{H}_{F}&=\sum_{n}\hat{H}^{(n)}\hat{L}_{n}+\hat{N}\hbar\omega.\end{aligned}$
(6)

We have used the commutation relations between the ladder and number
operators, $[\hat{N},\hat{L}_{n}]=n\hat{L}_{n}$ and $e^{-i\hat{N}\omega
t}\hat{L}_{n}e^{i\hat{N}\omega t}=\hat{L}_{n}e^{-in\omega t}$ to derive the
above equations. We note that the Floquet LvN equation have the same structure
as the traditional LvN. The advantage of the Floquet LvN is to allow us to
program the dynamics using the time independent Hamiltonian. Floquet QCLE.– To
derive the Floquet QCLE, we perform the partial Wigner transformation with
respect to the nuclear degrees of freedom on the Floquet LvN equation, Eq.
(5), as

$\displaystyle\frac{d}{dt}(\hat{\rho}_{F})_{W}(\mathbb{R},\mathbb{P},t)=-\frac{i}{\hbar}\left((\hat{H}_{F}\hat{\rho}_{F})_{W}-(\hat{\rho}_{F}\hat{H}_{F})_{W}\right).~{}$
(7)

We have used subscript $W$ to denote the Wigner transformation. The Wigner
transformation is given by

$\displaystyle\begin{aligned} \hat{O}_{\rm
W}\left(\bm{R},\bm{P},t\right)\equiv\int
d\bm{Y}e^{\frac{-i\bm{R}\cdot\bm{P}}{\hbar}}\langle\bm{R}-\frac{\bm{Y}}{2}|\hat{O}(t)|\bm{R}+\frac{\bm{Y}}{2}\rangle,~{}~{}~{}~{}\end{aligned}$
(8)

where $\hat{O}(t)$ is an arbitrary operator and $|\mathbb{R}\rangle$ is the
real space representation of the nuclear degree of freedom. As a result of
this transformation, $\mathbb{R}$ and $\mathbb{P}$ can be interpreted as
position and momentum variables in the classical limit. Note that, the Wigner-
Moyal operator can be used to express the partial Wigner transform of the
product of operator $\hat{A}$ and $\hat{B}$:

$\displaystyle\begin{aligned}
&(\hat{A}\hat{B})_{W}(\mathbb{R},\mathbb{P})=\hat{A}_{W}(\mathbb{R},\mathbb{P})e^{-i\hbar\overleftrightarrow{\Lambda}/2}\hat{B}_{W}(\mathbb{R},\mathbb{P}),\\\
&\overleftrightarrow{\Lambda}=\sum_{\alpha}\overleftarrow{\frac{\partial}{\partial
P^{\alpha}}}\overrightarrow{\frac{\partial}{\partial
R^{\alpha}}}-\overleftarrow{\frac{\partial}{\partial
R^{\alpha}}}\overrightarrow{\frac{\partial}{\partial
P^{\alpha}}}.\end{aligned}$ (9)

When truncating the Wigner-Moyal operator to the first order in the Tayler
expansion,
$e^{-i\hbar\overleftrightarrow{\Lambda}/2}\approx(1-i\hbar\overleftrightarrow{\Lambda}/2)$,
we arrive at the Floquet QCLE as

$\displaystyle\begin{aligned}
&\frac{d}{dt}\hat{\rho}_{WF}(\mathbb{R},\mathbb{P},t)=-i/\hbar\left[\hat{H}_{WF},\hat{\rho}_{WF}(t)\right]\\\
&-\frac{1}{2}\left(\hat{H}_{WF}\overleftrightarrow{\Lambda}\hat{\rho}_{WF}-\hat{\rho}_{WF}\overleftrightarrow{\Lambda}\hat{H}_{WF}\right),\end{aligned}$
(10)

Here, we have denoted
$(\hat{O}_{F})_{W}(\mathbb{R},\mathbb{P})\equiv\hat{O}_{WF}({\mathbb{R}},{\mathbb{P}})$.
The subscript $\mathop{WF}$ indicates that the Wigner transformation performed
after the Floquet transformation. For the coupled electron-nuclear Hamiltonian
in Eq. (1), we can rewrite the Floquet QCLE as follows

$\displaystyle\begin{aligned} &\frac{\partial}{\partial
t}\hat{\rho}_{WF}(t)=-\hat{\hat{\mathcal{L}}}_{WF}(\hat{\rho}_{WF}(t))-\sum_{\alpha}\frac{P_{\alpha}}{M_{\alpha}}\frac{\partial\hat{\rho}_{WF}(t)}{\partial
R_{\alpha}}\\\
&+\frac{1}{2}\sum_{\alpha}\left\\{\frac{\partial\hat{H}^{e}_{WF}}{\partial
R_{\alpha}},\frac{\partial\hat{\rho}_{WF}(t)}{\partial
P_{\alpha}}\right\\}.\end{aligned}$ (11)

Here $\hat{\hat{\mathcal{L}}}_{WF}(\hat{\rho}_{WF}(t))\equiv
i/\hbar[\hat{H}^{e}_{WF},\hat{\rho}_{WF}(t)]$. $\hat{H}^{e}_{WF}$ is the
Floquet-Wigner transformed electronic Hamiltonian $H_{e}$. We have also
denoted the anti-commutator as
$\\{\hat{A},\hat{B}\\}=\hat{A}\hat{B}+\hat{A}\hat{B}$. This Floquet QCLE is
consistent with the recently published work (see Eq. 14 in Ref. Chen  _et al._
(2020)). Such a Floquet QCLE represents the non-adiabatic dynamics of the
coupled electron-nuclear motion subjected to periodic driving. The Fokker-
Planck equation.– In the limit when the nuclear motion is slow as compared to
electronic motion as well as the driving speed, we can trace out all
electronic degrees of freedom and Floquet levels, such that we are left with
the pure nuclear density. To be more explicit,
$\mathcal{A}(\mathbb{R},\mathbb{P},t)=Tr_{e,F}\hat{\rho}_{WF}$. Here,
$Tr_{e,F}$ denotes trace over both many-body electronic states and Fourier
space. To the first order in the correction to the BO approximation, we arrive
at a Fokker-Planck equation for the pure nuclear density $A$:

$\displaystyle\begin{aligned} &\frac{\partial}{\partial
t}\mathcal{A}=-\sum_{\alpha}\frac{P_{\alpha}}{m_{\alpha}}\frac{\partial\mathcal{A}}{\partial
R_{\alpha}}-\sum_{\alpha}F_{\alpha}\frac{\partial\mathcal{A}}{\partial
P_{\alpha}}+\\\
&\sum_{\alpha\beta}\gamma_{\alpha\beta}\frac{\partial}{\partial
P_{\alpha}}\left(\frac{P_{\beta}}{m_{\beta}}\mathcal{A}\right)+\sum_{\alpha\beta}\bar{D}_{\alpha\beta}^{S}\frac{\partial^{2}\mathcal{A}}{\partial
P_{\alpha}\partial P_{\beta}}.\end{aligned}$ (12)

The detailed derivation can be found in the SM. Here
$F_{\alpha}=Tr_{e,F}({\partial\hat{H}_{F}^{e}}/{\partial
R_{\alpha}}\hat{\rho}_{ssF})$ is the mean force. $\hat{\rho}_{ssF}$ is the
steady state Floquet electronic density. The Fokker-Planck equation is
equivalent to the Langevin equation

$m_{\alpha}\ddot{R}_{\alpha}=F_{\alpha}-\sum_{\beta}\gamma_{\alpha\beta}\dot{R}_{\beta}+\delta
F_{\alpha}.$ (13)

Here $\delta F_{\alpha}$ is the random force, which satisfies
$\frac{1}{2}\langle\delta F_{\alpha}(0)\delta F_{\beta}(t)+\delta
F_{\beta}(0)\delta F_{\alpha}(t)\rangle=\bar{D}_{\alpha\beta}^{S}\delta(t)$.
$\bar{D}_{\alpha\beta}^{S}$ is the correlation function of the random force.
$\gamma_{\alpha\beta}$ is the friction coefficient

$\displaystyle\gamma_{\alpha\beta}=-\int_{0}^{\infty}dtTr_{e,F}\left(\frac{\partial\hat{H}_{F}^{e}}{\partial
R_{\alpha}}e^{\frac{-i\hat{H}_{F}^{e}t}{\hbar}}\frac{\partial\hat{\rho}_{ssF}}{\partial
R_{\beta}}e^{\frac{i\hat{H}_{F}^{e}t}{\hbar}}\right).~{}~{}$ (14)

Up to now, we have successfully transformed the coupled electron-nuclear
motion subjected to periodic driving into a Langevin equation for the pure
nuclear motion with all electronic motion and Floquet driving being
incorporated into frictional force and random force. We now proceed to
evaluate the frictional force in terms of Green’s function.

Quadratic electronic Hamiltonian.– Our derivation above is general as long as
the nuclear motion is slow as compared to electronic motion and Floquet
driving. We now consider quadratic electronic Hamiltonian,

$\displaystyle\hat{H}_{e}({\mathbb{R}},t)=\sum_{ab}\mathcal{H}_{ab}({\mathbb{R}},t)\hat{b}^{\dagger}_{a}\hat{b}_{b}.$
(15)

One can then transform the Floquet electronic friction into the single
particle representation as

$\displaystyle\begin{aligned}
\gamma_{\alpha\beta}=-\hbar\int_{-\infty}^{\infty}\frac{d\epsilon}{2\pi}~{}Tr_{m,F}\left(\frac{\partial\mathcal{H}_{F}}{\partial
R_{\alpha}}G_{F}^{R}\frac{\partial\sigma_{ssF}}{\partial
R_{\beta}}G_{F}^{A}\right),\end{aligned}$ (16)

where $Tr_{m,F}$ denotes the trace over both single particle electronic DoFs
and Fourier space. We have defined the Floquet Retarded and Advance Green’s
function: $G_{F}^{R/A}=(\epsilon\pm i\eta-\mathcal{H}_{F})^{-1}$,
$\eta\rightarrow 0^{+}$. Here, $\mathcal{H}_{F}$ is the Floquet single
particle electronic Hamiltonian, and ${\sigma}_{{ssF}}$ denotes the Floquet
single-particle density matrix, which is defined as
$[{\sigma}_{ssF}]_{ab}=Tr_{e}(\hat{b}_{b}^{\dagger}\hat{b}_{a}\hat{\rho}_{ssF})$.
The Floquet single-particle density matrix can be further expressed in terms
of Floquet lesser Green’s function, such that the final expression for the
Floquet electronic friction is given by:

$\displaystyle\begin{aligned}
\gamma_{\alpha\beta}=\hbar\int_{-\infty}^{\infty}\frac{d\epsilon}{2\pi}~{}Tr_{m,F}\left(\frac{\partial\mathcal{H}_{F}}{\partial
R_{\alpha}}\frac{\partial
G_{F}^{R}}{\partial\epsilon}\frac{\partial\mathcal{H}_{F}}{\partial
R_{\beta}}G_{F}^{<}\right)+\textrm{h.c}.~{}~{}~{}~{}\end{aligned}$ (17)

See the Supplementary Information for the details of derivation. $G_{F}^{<}$
is the lesser Floquet Green’s function. Note that the Floquet electronic
friction is the same as non-Floquet electronic friction, except Green’s
functions are now the Floquet version of the corresponding Green’s function.
Dot-lead separation.– We now consider a specific model, such that we can
calculate the Floquet electronic friction explicitly. We will demonstrate that
the Floquet driving electronic friction exhibits anti-symmetric terms for real
Hamiltonian even without any current. To be more specific, we consider a
Hamiltonian with dot-lead separation:

$\displaystyle\hat{H}_{e}$ $\displaystyle=$
$\displaystyle\hat{H}_{s}+\hat{H}_{b}+\hat{H}_{v}$ (18)
$\displaystyle\hat{H}_{s}$ $\displaystyle=$
$\displaystyle\sum_{ij}[h^{s}]_{ij}(\mathbb{R},t)\hat{d}_{i}^{\dagger}\hat{d}_{j}+U(\mathbb{R})$
(19) $\displaystyle\hat{H}_{b}$ $\displaystyle=$ $\displaystyle\sum_{\zeta
k}\epsilon_{\zeta k}\hat{c}_{\zeta k}^{\dagger}\hat{c}_{\zeta k}$ (20)
$\displaystyle\hat{H}_{v}$ $\displaystyle=$ $\displaystyle\sum_{\zeta
k,i}V_{\zeta k,i}(\hat{c}_{\zeta
k}^{\dagger}\hat{d}_{i}+\hat{d}^{\dagger}_{i}\hat{c}_{\zeta k})$ (21)

Here, $\hat{H}_{s}$ is the dot Hamiltonian. The bath Hamiltonian consists of
the left and right ($\zeta=L,R$) leads. $\hat{H}_{v}$ describes the system-
bath couplings. $U(\mathbb{R})$ is the potential for the nuclei.

For such a model, we can calculate Floquet Green’s function exactly. In
particular, the Retarded Green’s function for the system is given by:

$G_{sF}^{R}(\epsilon)=\left(\epsilon-\Sigma^{R}_{F}(\epsilon)-h^{s}_{F}\right)^{-1},$
(22)

$\Sigma^{R}_{F}(\epsilon)=\sum_{\zeta=L,R}\Sigma_{\zeta F}^{R}$ is the total
self energy in Floquet representation. The elements of the self energy is
given by $[\Sigma_{\zeta F}^{R}]_{ij}(\epsilon)=\sum_{k}V_{\zeta
k,i}g_{F,\zeta k}^{R}(\epsilon)V_{\zeta k,j}$, where the $g_{F,\zeta
k}^{R}(\epsilon)=(\epsilon+i0^{+}-\epsilon_{\zeta k}-\hat{N}\hbar\omega)^{-1}$
is the $k$-th element of the Retarded Green’s function of the isolated lead
$\zeta$. $h^{s}_{F}$ is the Floquet representation of the dot energy level.
The lesser Green’s function for the system is then given by

$\displaystyle
G_{sF}^{<}(\epsilon)=G_{sF}^{R}(\epsilon)\Sigma_{F}^{<}(\epsilon)G_{sF}^{A}(\epsilon),$
(23)

Here, $\Sigma_{F}^{<}(\epsilon)=\sum_{\zeta=L,R}\Sigma_{\zeta
F}^{<}(\epsilon)$ is the lesser Green’s function, which can be evaluated as
$[\Sigma_{\zeta F}^{<}]_{ij}(\epsilon)\equiv\sum_{k}V_{{\zeta k},i}g_{F,{\zeta
k}}^{<}(\epsilon)V_{{\zeta k},j}$. Here, $g_{F,{\zeta k}}^{<}(\epsilon)$ is
the lesser green’s function for the $\zeta$ lead. $g_{F,{\zeta
k}}^{<}(\epsilon)=i2\pi
f(\epsilon-\hat{N}\hbar\omega-\mu_{\zeta})\delta(\epsilon-\epsilon_{\zeta
k}-\hat{N}\hbar\omega)$ where $f$ is the Fermi function. In what following, we
will invoke the wide band approximation, such that $[\Sigma_{\zeta
F}^{R}]_{ij}(\epsilon)=-\frac{i}{2}\Gamma_{ij}$, and $[\Sigma_{\zeta
F}^{<}]_{ij}(\epsilon)=i\Gamma_{ij}f(\epsilon-\hat{N}\hbar\omega-\mu_{\zeta})$.
We can then proceed to calculate Floquet electronic friction using these
Green’s functions.

Results and Discussions.– We will now consider a two-level and two nuclear
DoFs model:

$\displaystyle[h^{s}](x,y,t)=\left(\begin{array}[]{cc}x+\Delta&Ay+Bcos(\omega
t)\\\ Ay+Bcos(\omega t)&-x-\Delta\end{array}\right).~{}~{}$ (26)

The nuclear potential $U(\mathbb{R})$ is taken to be harmonic oscillators in
both $x$ and $y$ dimensions. The diagonal terms of Hamiltonian represent two
shifted parabolas in $x$ direction with a driving force of $2\Delta$. The off-
diagonal couplings depend on displacement in $y$ direction as well as external
time-periodic driving $Bcos(\omega t)$ from a monochromatic light source. $B$
represents the strength of the external driving (e.g., the intensity of light)
and $\omega$ is the frequency of the time-periodic driving. Below, we consider
the case where the first level couples to the left lead and the second level
couples to the right lead, and we set $\Gamma_{11}=\Gamma_{22}=\Gamma$. In the
equilibrium case (where $\mu_{L}=\mu_{R}$) and without any driving, the
electronic friction is shown to be symmetric along nuclear DoFs provided the
Hamiltonian is real Teh  _et al._ (2021). In Fig. 1, we plot the friction
tensor as a function of the nuclear coordinates $(x,y)$. In particular, we
define the symmetric and antisymmetric components
[$\gamma^{S}_{xy}=(\gamma_{xy}+\gamma_{yx})/2,\gamma^{A}_{xy}=(\gamma_{xy}-\gamma_{yx})/2$]
of the friction tensor. In the absence of external driving ($B=0$), the
antisymmetric component is indeed vanished (as predicted). The frictions
tensors $\gamma_{xx}$ and $\gamma_{yy}$ consists of two Gaussian curves which
are merged along the orientations of the nuclear coordinate. This results
agree with previous findings for real Hamiltonian without any driving Teh  _et
al._ (2021).

Figure 1:  Floquet friction tensors in absence of external driving $B=0$:
$\gamma_{xx}$ (top left), $\gamma_{xy}^{S}$ (top right), $\gamma_{xy}^{A}$
(bottom left) and $\gamma_{yy}$ (bottom right). Parameters: $\Gamma$=1,
$\mu_{R,L}=0$, $\beta=2$, $A=1$, $\Delta=3$, $\omega=0.5$. We have used $N=5$
Floquet levels to converge the results. Figure 2:  Floquet friction tensors in
presence of external driving: $\gamma_{xx}$ (top left), $\gamma_{xy}^{S}$ (top
right), $\gamma_{xy}^{A}$ (bottom left) and $\gamma_{yy}$ (bottom right).
Parameters: $\Gamma$=1, $\mu_{R,L}=0$, $\beta=2$, $A=1$, $\Delta=3$,
$\omega=0.5$, $B=1$, $N=5$.

We now turn on time-periodic off-diagonal coupling by setting $B=1$. As shown
in Fig. 2, the antisymmetric term $\gamma_{xy}^{A}$ is no longer zero when
Floquet driving is turning on. Moreover, the distributions of $\gamma_{xx}$,
$\gamma_{yy}$, and $\gamma_{xy}^{S}$ in the real space is enlarged as compared
to the non-Floquet case. The magnitude of $\gamma_{xx}$ and $\gamma_{xy}^{S}$
are also increased by almost factor of 2. Finally, in Fig. 3 we plot the
frictional terms for the increased driving frequency ($\omega=1$). In such a
case, the magnitude of the antisymmetric terms (the Lorentz force) is notably
increased, whereas the magnitudes of the other terms do not change
significantly. Interestingly, the shape of $\gamma_{xx}$ is composed of two
large ellipses and two small ones. The central distance between the larger
ellipse and the smaller one in $x$ axis is about $\omega$. This is consistent
with the picture of Floquet replica of the potential surfaces separated by
$\omega$. Finally, note that all friction terms have mirror symmetry around
the avoided crossing point ($x=-\Delta$ and $y=0$) and magnitudes of
$\gamma_{xy}^{A}$ and $\gamma_{xy}^{S}$ are always maximized far from the
avoided crossing.

Figure 3:  Floquet Friction tensors in presence of external driving with a
larger frequency: $\gamma_{xx}$ (top left), $\gamma_{xy}^{S}$ (top right),
$\gamma_{xy}^{A}$ (bottom left) and $\gamma_{yy}$ (bottom right). Parameters:
$\Gamma$=1, $\mu_{R,L}=0$, $\beta=2$, $A=1$, $\Delta=3$, $\omega=1$, $B=1$,
$N=5$.

Conclusion.– We have formulated quantum–classical Liouville equation in
Floquet representation to describe non-adiabatic dynamics with light-matter
interactions. We have further mapped the Floquet QCLE into a Langevin dynamics
where all electronic DoFs and light-matter interactions are incorporated into
a friction tensor. We then recast the friction tensor into the form of Floquet
Green’s functions such that we can evaluate the friction tensor explicitly. We
show that the light-matter interactions can introduce anti-symmetric friction
tensor even at equilibrium without any spin-orbit couplings. Future work must
explore how the Lorentz-like force affects the dynamics in a realistic
situation. We acknowledge the startup funding from Westlake University.

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