# Multi-channel Medium Access Control Protocols for Wireless Networks within
Computing Packages††thanks: This work is supported by the European Commission
under H2020 grant WiPLASH (GA 863337) and HE grant WINC (GA 101042080).

Bernat Ollé1, Pau Talarn1, Albert Cabellos-Aparicio1, Filip Lemic2, Eduard
Alarcón1, and Sergi Abadal1  1NaNoNetworking Center in Catalunya (N3Cat),
Universitat Politècnica de Catalunya, 08034 Barcelona, Spain  2AI-driven
Systems Lab, i2Cat Foundation, 08034 Barcelona, Spain

###### Abstract

Wireless communications at the chip scale emerge as a interesting complement
to traditional wire-based approaches thanks to their low latency, inherent
broadcast nature, and capacity to bypass pin constraints. However, as current
trends push towards massive and bandwidth-hungry processor architectures,
there is a need for wireless chip-scale networks that exploit and share as
many channels as possible. In this context, this work addresses the issue of
channel sharing by exploring the design space of multi-channel Medium Access
Control (MAC) protocols for chip-scale networks. Distinct channel assignment
strategies for both random access and token passing are presented and
evaluated under realistic traffic patterns. It is shown that, even with the
improvements enabled by the multiple channels, both protocols maintain their
intrinsic advantages and disadvantages.

##  I Introduction

Efficient integrated networks at the chip scale within Systems-in-Package
(SiPs) are a prerequisite for high performance in such computing systems.
Currently, most systems incorporate a Network-in-Package (NiP) consisting of a
set of on-chip routers and intra-/inter-chip wired links [1, 2]. However,
recent scaling [3, 4], specialization [5, 6], and disintegration trends [7, 8]
are increasing the pressure placed on the interconnect, to the point that new
communication paradigms may be required [9, 10].

Among the emerging alternatives, wireless chip-scale communications stand as a
promising contender [11, 12, 13, 14]. This communication paradigm relies the
use of modulated electromagnetic waves for data transmission using the chip
package as communications medium (Fig. 1). The resulting wireless in-package
links provide low latency, inherent broadcast capabilities, and global
reconfigurability.

Since the communications medium is shared, wireless in-package communications
require Medium Access Control (MAC) protocols to avoid or manage wasteful
collisions. In this scenario, MAC protocols generally reduce to variants of
multiplexing, random access, or token passing [15, 16, 17, 18, 19, 20, 21].
Even though recent works have demonstrated that computing packages could
support a few frequency [22, 23] and space channels [24, 25], it is still
unclear how MAC protocols can benefit from them. This is because more than a
few channels are needed to implement truly scalable frequency/space
multiplexing techniques [15], and most importantly, because multi-channel
variants of random access and token passing have not been explored yet.

Figure 1: Pictorial view of a wireless chip-to-chip communication link.

This paper aims to bridge this gap by focusing on the study of multi-channel
versions of the two most representative protocol types in chip-scale
scenarios, i.e. random access and token passing. In particular, the main
contributions are as follows. We first describe the different ways we can
extend random access and token passing with a small set of channels in Sec.
II. Then, in Sec. III, we evaluate these protocol variants with traffic models
typically used to mimic multiprocessor workloads [26]. This analysis sheds
light on the impact of channel assignment on the protocol performance, as
summarized in Sec. IV and concluded in Sec. V.

##  II Multi-channel MAC Protocols

In this work, we describe three distinct channel assignment strategies for
random access and token passing. As baselines, we take BRS [18] for random
access and the baseline from [20] for token passing. The strategies presented
here are not provably optimal, but they are simple (as required by the
resource constraints of the chip-scale scenario) and representative of the
potential techniques that can be used.

###  II-A Assignment Methods for BRS

In random access protocols such as BRS [18], nodes contend for channel access
and back off if the channel is busy or there is a collision. Assuming $N$
nodes, we study three ways to reduce the collision probability using $N_{c}$
channels, namely:

AS1: Channels are assigned to nodes individually and randomly. When a node has
a packet to transmit, the node is assigned a random channel. If the channel is
busy or there is a collision, nodes undergo a random back off and also choose
a random channel to use in the next attempt.

AS2: Each channel is assigned to $\tfrac{N}{N_{c}}$ nodes statically following
a uniform distribution, this is, assuming that all nodes have the same load
(see Fig. 2, left). While this is not optimal for spatially unbalanced
traffic, it serves as a baseline.

AS3: Channels are assigned to a variable number of nodes following a
distribution that balances the load in each channel (see Fig. 2, right). To
that end, nodes are ordered based on the expected normalized load and assigned
to each channel in order following a greedy algorithm.

Figure 2: Graphical representations of assignment techniques AS2 (left) and
AS3 (right) for BRS assuming 16 nodes and 4 channels.

###  II-B Assignment Methods for Token Passing

In token passing [27], typically, all $N$ nodes are sorted forming a virtual
ring and the token is passed in order through that ring. In a version with
$N_{c}$ channels, each channel can be a token. The design decisions then lie
on the number of rings and the nodes that form each ring. For instance:

AS1: We assume as many rings as there are channels and map nodes uniformly to
each ring. In other words, we distribute them in rings of $\tfrac{N}{N_{c}}$
nodes, regardless of their expected load.

AS2: We assume a single virtual ring with multiple tokens circulating in it.
In this case, tokens can jump over other tokens: when node $i$ holds a token
for multiple cycles during a transmission, idle tokens that arrive at $i$-1
can jump to $i$+1.

AS3: This strategy is similar to AS1, but nodes are mapped to rings based on
their expected load. This may lead to rings of different sizes, but similar in
the expected overall load.

Figure 3: Graphical representations of the different assignment techniques for
token passing assuming 16 nodes and 4 channels.

##  III Performance Evaluation

The architecture and application parameters are summarized in Table I. We
implement both single-channel baselines and multi-channel versions of BRS and
token passing as finite state machines in a modified version of Multi2sim that
models wireless links and supports collision detection [28]. The protocols are
stressed with synthetic traffic modeling uneven injection distributions
(through the $\sigma$ parameter) and bursty temporal behavior (through the
Hurst exponent $H$) [26]. The default values for the different parameters are
$N=64$ nodes, $N_{c}=4$ channels, $H=0.5$ and $\sigma=1$. Simulations are
cycle-accurate.

In all cases, we compare the packet latency (in cycles) and throughput (in
packets/cycle) of the different options. Given the high number of protocol
strategies and traffic types, instead of plotting the classical
latency–throughput curve, we make use of box plots that summarize the latency
and throughput statistics. In our plots, the X axis shows the parameters under
study. The plots have two Y axis: the left axis represents the latency and
corresponds to the box plot values, whereas the right axis represents the
throughput and corresponds to single-value markers of saturation throughput.
Since a single packet takes 4 cycles in a single channel to be transmitted,
the maximum throughput is 0.25 packets/cycle/channel.

TABLE I: Characteristics of simulated protocols and applications. Application | Synthetic traffic, H=0.5–0.85, $\sigma$=0.05–100
---|---
System |  $N$=64–512 cores, one antenna/core, 1-GHz clock
Network | 80-bit packets (preamble: 20 bits), $N_{c}$=1–4 channels
Link | BRS [18], Token passing
Physical | On-Off Keying, 20 Gb/s

###  III-A Number of Channels

Here, we discuss the results shown in Fig. 4 for BRS and token passing and an
increasing number of channels.

Figure 4: Performance of multi-channel BRS (top) and token passing (bottom)
for an increasing number of channels, $C1$ to $C4$, and different assignments.

Latency. In general, it can be observed that BRS is less stable than token in
terms of latency as the range of values is larger, with a higher number of
outlier points. However, BRS has a much better zero-load latency than token
since, in BRS, the protocol allows nodes to start transmitting immediately
when the channel is sensed idle. This fact also can explain why independently
of the parameters evaluated here (assignment, number of channels) the minimum
latency is quite similar. The worst-case latency, however, clearly improves
when having multiple channels, as the high load is distributed over multiple
channels. On the other hand, in token passing, nodes must wait until they
possess the token to start transmitting. For this reason, when the number of
nodes is large, $N=64$ in this case, the system remains idle much longer.

Figure 5: Performance of multi-channel BRS protocol for an increasing number
of nodes, $N$=64–512 (left graph), different spatial concentration levels,
$\sigma$=0.1–100 (center graph), different temporal burstiness levels,
$H$=0.5–0.85 (right graph), and different assignment techniques.

Figure 6: Performance of multi-channel token passing protocol for an
increasing number of nodes, $N$=64–512 (left graph), different spatial
concentration levels, $\sigma$=0.1–100 (center graph), different temporal
burstiness levels, $H$=0.5–0.85 (right graph), and different assignment
techniques.

Throughput. The results for token passing depict a rather stable increase in
saturation throughput as more channels are added, regardless of the assignment
method. This could be due to the use, by default, of non-bursty and non-
hotspot traffic to evaluate scalability. On the other hand, the results for
BRS illustrate a different behavior than in token passing. Firstly, BRS cannot
reach a saturation throughput as high as token passing. The main reasons are
that channel contention and multiple collisions lead to channel waste and,
hence, to a reduced throughput. Furthermore, BRS is more irregular than token
passing in terms of saturation throughput as it depends on the percentage of
collisions at high loads. As a result, the difference between the saturation
throughput achieved for different assignments increases with the number of
channels.

###  III-B Number of Nodes

Next, we comment on the performance of BRS and token passing for an increasing
number of nodes, with $N_{c}=4$. The results are shown in the left charts of
Fig. 5 and Fig. 6.

Latency. BRS has a much lower latency than token passing due to its ability to
transmit when the channel is idle. The span of the latency values differs
across number of nodes and assignments, but in general are restrained to
similar values because in the end, the same aggregated load ends up being
distributed over more nodes. Static assignment of channels (AS2) works worse
than the other alternatives. On the other hand, from the plot of token
passing, it is clear that more nodes lead to much higher latency due to the
increase of the token turnaround time. In fact, the low-load latency is
proportional to the number of nodes in all cases. The span of the latency
values is similar across the different system sizes.

Throughput. In general, saturation throughput is slightly higher for a lower
number of nodes. In our protocols, having more nodes means having a higher
population and, hence, a higher chance of collisions even for the same load
for BRS, and a higher waiting time (or lower probability of having all nodes
backlogged) in token passing. It seems, in any case, that BRS is more
resilient to the change in the number of nodes as the drop is more subtle,
except for AS3, where possibly the load balancing algorithm is not performing
well when such a large number of nodes has to be classified. Finally, all
three assignments have very similar throughput in all cases for token passing,
whereas AS1 (random channel assignment to individual packets) works better in
BRS.

###  III-C Hotspot Traffic

We next discuss the results shown in the middle plots of Fig. 5 and Fig. 6,
which illustrate the impact of uneven spatial injection distribution on
performance. We remind that low/high values of $\sigma$ mean that traffic is
hotspot/evenly distributed [26].

Latency. In BRS, the hotspot behavior of traffic does not seem to have a large
influence on the performance of the different assignment methods. The outlier,
third quartile, and maximum values within the distribution seem to be mildly
impacted by the hotspot nature of traffic. In general, BRS is resilient to
such variations and actually could benefit from having a lower amount of nodes
contending for the available channels. Still, the results show a small
tendency to worse results when traffic is concentrated around a few nodes,
possibly because of the nodes with higher load reaching higher backoff values.
In AS3, this situation is avoided by proactively placing high-load nodes in
different channels. Similarly, in token passing, latency is affected by the
concentration of traffic around a given set of nodes mostly because the
different assignment methods are able to provide tokens quickly to nodes that
need it, even if they are spaced apart within the ring. This is clearly
visible in the extreme case of $\sigma=0.05$. Similarly, outlier values seem
to be larger when traffic is more hotspot. We also observe how AS2 fails to
provide a good performance at low loads, and this behavior is exacerbated for
very hotspot traffic.

Throughput. The throughput of BRS in its different implementations does not
vary significantly with the type of spatial distribution of traffic, except
for AS3, where a higher concentration of traffic around a few nodes seems to
have a positive effect on the throughput. One reason could be that the most
active nodes are distributed over the different channels so that contention is
minimized. That does not happen in other assignment methods. Different
behavior is observed in token passing, where the hotspot behavior of traffic
modifies the throughput of the different assignment methods, with AS3 being
affected a bit less. This is because if the load is concentrated around a
small set of nodes, a large portion of the airtime is wasted while passing the
token among these nodes.

###  III-D Bursty Traffic

Finally, we present the latency and throughput results for an increasingly
bursty traffic. The results are shown in the right plots of Fig. 5 and Fig. 6
for BRS and token passing, respectively. Temporal injection of traffic is
modeled through the Hurst exponent [26], with higher values indicating more
bursty behavior, i.e. longer bursts followed by longer silences.

Latency. In BRS, it can be seen that the higher the value of $H$, the higher
the latency in average and also the more unpredictable. This is because with
an $H$ of 0.5, the packets are injected following a random Poisson process,
which minimizes the probability of collisions. However, when increasingly
bursty traffic is considered, the probability of packets being injected (and
nodes trying to transmit) in the same exact cycle increases. The effect is
multiplicative with the burstiness, as the effect of cascading collisions
leads to an exponential increase of the backoff time. This affects the system
at all loads. On the other hand, token passing also suffers when bursty
traffic is served, leading to very high latency especially for high values of
$H$. The latency is a bit more stable than in the case of BRS, mainly because
the protocol does not react with exponential backoffs, but rather with linear
token passings to bursts of traffic. Still, the latency is much higher than
that of BRS, discouraging its use for large number of nodes.

Throughput. On one hand, it can be verified that in BRS, the saturation
throughput remains rather constant across all assignments regardless of the
value of the Hurst exponent. A possible reason could stem from the behavior of
the backoff mechanism; bursty traffic leads to a large number of collisions
which increases latency even for low loads, but the protocol may converge to a
large backoff value that can accommodate the load even if it comes in bursts.
In other works, the backoff mechanisms spreads out the bursts of traffic over
time, until all nodes are backlogged. On the other hand, it can be seen that
in the case of token passing, the saturation throughput seems to drop
significantly for higher numbers of $H$, to a point that the achieved
throughput becomes comparable with that of BRS. A potential reason for this
behavior is the lack of an adaptive mechanism to react to bursts; the token
has to still move around the ring even if bursts of traffic lead to the
generation of multiple packets in a given node, leading to gaps where the
wireless channel remains silent. When traffic is less bursty, the probability
of such events is lower.

##  IV Discussion

Figure 7 plots the performance of all the compared protocols and assignments
representing the zero-load latency (X axis) and saturation throughput (Y axis)
of a particular protocol for a given number of channels and assignment method.

In general terms, BRS is preferred over token in terms of zero-load latency
given its ability to transmit immediately when the channels are idle. Hence,
we see most BRS points located in a _low latency region_. Among the assignment
techniques, AS1 achieves similar results than AS3 and would probably be
preferred as it does not require prior knowledge of the load of each node to
assign the channels. On the downside, the throughput is half of that of token
passing, at most.

On the other hand, token passing can reach high throughput levels in the _high
capacity region_ , close to the maximum total bandwidth of the wireless
network. However, while putting more channels reduces the latency
significantly, the best latency in token passing is still several cycles away
from the BRS values. Finally, we observe that it is hard to provide a good
channel assignment overall: AS3 requires prior knowledge on the traffic
distribution, AS1 does not perform well for hotspot traffic and AS2 has high
latency.

Figure 7: Summary of the latency and throughput results over all the
protocols, assignment methods, and traffic conditions. $B$ and $T$ stand for
BRS (random access) and token passing, $C$ and $N$ denote number of channels
and nodes, whereas $S$ and $H$ represent the different spatial and temporal
injection distributions. For instance, the $B$_$C$ symbols represent the
latency-throughput of all the assignment methods for BRS for different number
of channels. Two desirable design spaces and a Pareto frontier are also given.

##  V Conclusions

This paper has explored several techniques to extend random access and token
passing MAC protocols to multiple channels for wireless chip-scale networks.
In general, more channels alleviate the problems of both types of protocols,
increasing the throughput of random access and cutting down the latency of
token passing to a few tens of cycles. Additionally, random access is more
resilient to hotspot and bursty traffic and more scalable to massive chip-
scale networks. However, the higher throughput achievable with token renders
the decision of the protocol (and assignment) to choose extremely challenging.
Hence, we see a trend similar to that of single-channel protocols: it would be
desirable to develop a multi-channel protocol that is able to seamlessly
obtain the best of both paradigms. This will be explored in future work.

## References

  * [1] R. Marculescu, U. Ogras, L.-S. Peh, N. Enright Jerger, and Y. Hoskote, “Outstanding research problems in NoC design: system, microarchitecture, and circuit perspectives,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 28, no. 1, pp. 3–21, 2009.
  * [2] D. Bertozzi, G. Dimitrakopoulos, J. Flich, and S. Sonntag, “The fast evolving landscape of on-chip communication,” Design Automation for Embedded Systems, vol. 19, no. 1, pp. 59–76, 2015.
  * [3] S. Salahuddin, K. Ni, and S. Datta, “The era of hyper-scaling in electronics,” Nature Electronics, vol. 1, no. 8, pp. 442–450, 2018.
  * [4] C. Mayr, S. Hoeppner, and S. Furber, “Spinnaker 2: A 10 million core processor system for brain simulation and machine learning,” arXiv preprint arXiv:1911.02385, 2019.
  * [5] T. Krishna, H. Kwon, A. Parashar, M. Pellauer, and A. Samajdar, “Data orchestration in deep learning accelerators,” Synthesis Lectures on Computer Architecture, vol. 15, no. 3, pp. 1–164, 2020.
  * [6] T. Wang, F. Feng, S. Xiang, Q. Li, and J. Xia, “Application defined on-chip networks for heterogeneous chiplets: An implementation perspective,” in 2022 IEEE International Symposium on High-Performance Computer Architecture (HPCA), pp. 1198–1210, IEEE, 2022.
  * [7] A. Kannan, N. E. Jerger, and G. H. Loh, “Exploiting interposer technologies to disintegrate and reintegrate multicore processors,” Ieee Micro, vol. 36, no. 3, pp. 84–93, 2016.
  * [8] S. Naffziger, N. Beck, T. Burd, K. Lepak, G. H. Loh, M. Subramony, and S. White, “Pioneering chiplet technology and design for the amd epyc™ and ryzen™ processor families: Industrial product,” in 2021 ACM/IEEE 48th Annual International Symposium on Computer Architecture (ISCA), pp. 57–70, IEEE, 2021.
  * [9] A. Karkar, T. Mak, K.-F. Tong, and A. Yakovlev, “A survey of emerging interconnects for on-chip efficient multicast and broadcast in many-cores,” IEEE Circuits and Systems Magazine, vol. 16, no. 1, pp. 58–72, 2016.
  * [10] A. Ganguly, S. Abadal, I. Thakkar, N. E. Jerger, M. Riedel, M. Babaie, R. Balasubramonian, A. Sebastian, S. Pasricha, and B. Taskin, “Interconnects for dna, quantum, in-memory, and optical computing: Insights from a panel discussion,” IEEE micro, vol. 42, no. 3, pp. 40–49, 2022.
  * [11] D. W. Matolak, A. Kodi, S. Kaya, D. Ditomaso, S. Laha, and W. Rayess, “Wireless networks-on-chips: architecture, wireless channel, and devices,” IEEE Wireless Communications, vol. 19, no. 5, pp. 58–65, 2012.
  * [12] S. Laha, S. Kaya, D. W. Matolak, W. Rayess, D. DiTomaso, and A. Kodi, “A new frontier in ultralow power wireless links: Network-on-chip and chip-to-chip interconnects,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 34, no. 2, pp. 186–198, 2014.
  * [13] S. Abadal, R. Guirado, H. Taghvaee, A. Jain, E. P. de Santana, P. H. Bolívar, M. Saeed, R. Negra, Z. Wang, K.-T. Wang, et al., “Graphene-based wireless agile interconnects for massive heterogeneous multi-chip processors,” IEEE Wireless Communications, 2022.
  * [14] R. S. Narde, J. Venkataraman, A. Ganguly, and I. Puchades, “Intra-and inter-chip transmission of millimeter-wave interconnects in noc-based multi-chip systems,” IEEE Access, vol. 7, pp. 112200–112215, 2019.
  * [15] D. DiTomaso, A. Kodi, D. Matolak, S. Kaya, S. Laha, and W. Rayess, “A-WiNoC: Adaptive wireless network-on-chip architecture for chip multiprocessors,” vol. 26, no. 12, pp. 3289–3302, 2015.
  * [16] N. Mansoor and A. Ganguly, “Reconfigurable wireless network-on-chip with a dynamic medium access mechanism,” in Proceedings of the 9th International Symposium on Networks-on-Chip, pp. 1–8, 2015.
  * [17] K. Duraisamy, R. G. Kim, and P. P. Pande, “Enhancing Performance of Wireless NoCs with Distributed MAC Protocols,” in Proceedings of the ISQED ’15, pp. 406–11, 2015.
  * [18] A. Mestres, S. Abadal, J. Torrellas, E. Alarcón, and A. Cabellos-Aparicio, “A MAC protocol for Reliable Broadcast Communications in Wireless Network-on-Chip,” in Proceedings of the NoCArc ’16, pp. 21–26, 2016.
  * [19] S. Jog, Z. Liu, A. Franques, V. Fernando, S. Abadal, J. Torrellas, and H. Hassanieh, “One protocol to rule them all: Wireless network-on-chip using deep reinforcement learning,” in 18th USENIX Symposium on Networked Systems Design and Implementation (NSDI 21), pp. 973–989, 2021.
  * [20] A. Franques, S. Abadal, H. Hassanieh, and J. Torrellas, “Fuzzy-Token: An adaptive mac protocol for wireless-enabled manycores,” in Design, Automation and Test in Europe Conference (DATE), 2021.
  * [21] Y. Ouyang, C. Sun, B. Jia, Q. Wang, and H. Liang, “Architecting a priority-based dynamic media access control mechanism in wireless network-on-chip,” Microelectronics Journal, vol. 116, p. 105218, 2021.
  * [22] W. Rayess, D. W. Matolak, S. Kaya, and A. K. Kodi, “Antennas and Channel Characteristics for Wireless Networks on Chips,” Wireless Personal Communications, vol. 95, no. 4, pp. 5039–5056, 2017.
  * [23] X. Timoneda, A. Cabellos-Aparicio, D. Manessis, E. Alarcón, and S. Abadal, “Channel characterization for chip-scale wireless communications within computing packages,” in 2018 Twelfth IEEE/ACM International Symposium on Networks-on-Chip (NOCS), pp. 1–8, IEEE, 2018.
  * [24] P. Baniya, A. Bisognin, K. L. Melde, and C. Luxey, “Chip-to-Chip Switched Beam 60 GHz Circular Patch Planar Antenna Array and Pattern Considerations,” IEEE Transactions on Antennas and Propagation, vol. 66, no. 4, pp. 1776–1787, 2018.
  * [25] F. Rodríguez-Galán, E. P. de Santana, P. H. Bolívar, S. Abadal, and E. Alarcón, “Towards spatial multiplexing in wireless networks within computing packages,” in Proceedings of the 9th ACM International Conference on Nanoscale Computing and Communication, pp. 1–6, 2022.
  * [26] V. Soteriou, H. Wang, and L. Peh, “A statistical traffic model for on-chip interconnection networks,” in 14th IEEE International Symposium on Modeling, Analysis, and Simulation, pp. 104–116, IEEE, 2006.
  * [27] G. Holland, N. Vaidya, and P. Bahl, “A rate-adaptive mac protocol for multi-hop wireless networks,” in Proceedings of the 7th annual international conference on Mobile computing and networking, pp. 236–251, 2001\.
  * [28] R. Ubal, P. Mistry, D. Schaa, H. Ave, and D. Kaeli, “Multi2Sim: A Simulation Framework for CPU-GPU Computing,” in Proceedings of the PACT’12, 2012\.


# Detecting Gravitational Waves by Twisted Light -
Dipole Interaction of Photons and Gravitational Waves

Longlong Feng1,2<EMAIL_ADDRESS>Qianfan Wu1 1 School of Physics
and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China
2 Purple Mountain Observatory, CAS, No.8 Yuanhua Road, Nanjing 210034, China

###### Abstract

Motivated by the next generation of gravitational wave (GW) detectors, we
study the wave mechanics of a twisted light beam in the GW perturbed
spacetime. We found a new gravitational dipole interaction of photons and
gravitational waves. Physically, this interaction is due to coupling between
the angular momentum of twisted light and the GW polarizations. We demonstrate
that for the higher-order Laguerre-Gauss (LG) modes, this coupling effect
makes photons undergoing dipole transitions between different orbital-angular-
momentum(OAM) eigenstates, and leads to some measurable optical features in
the 2-D intensity pattern. It offers an alternative way to realize precision
measurements of the gravitational waves, and enables us to extract more
information about the physical properties of gravitational waves than the
current interferometry. With a well-designed optical setup, this dipole
interaction is expected to be justified in laboratories.

###### pacs:

04.80.Nn, 04.30.Nk, 03.50.De, 42.25.−p

The direct detection of gravitational waves by the advanced LIGO
interferometer LIGO  and the advanced Virgo interferometer Virgo  marks the
beginning of the era of gravitational wave astronomy. These discoveries
stimulate the great efforts devoted to develop more advanced optical detection
technology. With upgrading to next generation of interferometer, it is
expected to move to design sensitivity and observe ever-increasing numbers of
GW sources. The next generation of GW laser-interferometric GW detector will
be upgraded to have an unprecedented sensitivity by minimizing various
technical noises. One limiting noise arises from mirror thermal noise,
producing by Brownian motion of particles in coatings and substrates. Besides
the physically cooling of mirror, an alternative solution is to change the
mode shape of the laser beam inside the interferometer, e.g. resonating the
higher-order Laguerre Gauss (LG${}_{p}^{l}$) modes in the detector arm
cavities to smooth out the thermal noise fluctuations over a bigger portion of
the mirror surfacechelkowski . A well-developed technology has made it
possible to produce higher-order LG-modes with high power output and high mode
purity as required in the GW detectionLGgen . Therefore, it would be crucial
to understand the wave mechanics of the higher-order LG modes interacting with
the passing gravitational wave.

For light propagation in a gravitational field, the curved space background
can be transformed to a linear optical medium, whose optical properties
characterized by the effective dielectric tensor are fully specified by the
spacetime geometriestamm . A typical example is the deflection of light in
gravitational fields, this phenomenon can be illustrated by the correspondence
between the refractive index and the scalar gravitational potential. Again, a
rotating gravitational field exhibits the optical activity attributed to the
vortical dragging vector. Basically, it can be also equivalently described by
the coupling between the macroscopic rotation ${\bf\Omega}$ and the internal
spin ${\bf s}$ of photonmash ; fll . This coupling has a typical form of
$\sim\gamma_{s}{\bf\Omega}\cdot{\bf s}$($\gamma_{s}$ is introduced to quantify
the coupling strength), which may give rise to an extra phase shift for
circular polarized lightcohen ; fll , and the helicity flip for spin-1/2
particlesfll1991 . Moreover, we noted that there exists another classical
phase shift referred as Sagnac factorsagnac , currently explained by the
dragging effect of frame in general relativitylandau , can be understood
alternatively by the coupling between the rotation and the orbital angular
momentum (OAM) of photons, $\sim\gamma_{L}{\bf\Omega}\cdot{\bf L}$,
$\gamma_{L}$ is the coupling strength.

On the other hand, the study of rotational feature in light phenomena has a
long history, even tracking back to nineteen century. In 1898, Sadovsky
sadowsky  gave the first prediction that the light with circular and elliptic
polarization can exert a rotatory action upon material objects in context of
classical Maxwell theory, and the repeated discovery had been made by Poynting
based on analysis of the energy flow of polarized light. Later on, a
circulatory flow of light energy has been realized in the focused flied
structure. Actually, this vortical properties of light is associated with the
OAM of photons. The blooming of researches on the OAM of light has been until
1992, when the angular momentum of a paraxial beam with the Laguerre-Gaussian
modes was explicitly worked out by Allen and co-workersallen . Particularly,
the twisted light propagating in an inhomogeneous optical medium has receive
much attentions, leading to some interesting predictions including optical
Magnus effect Magnus  or optical spin Hall effectHall  etc.

This paper does not focus on the technical aspects of higher-order LG-beams in
optical cavity, but on how the high-mode twisted light propagating in the
ripples of spacetime disturbed by the gravitational waves. To formulate a
possible coupling effect between angular momentum of photon and the GW‘s
polarization, we will take a Schrödinger’s view of the Maxwell theory in
curved space, and explore how this coupling effect manifests its significance
in future GW experiments.

Adopting the Hanni’s 3+1 decompositionhanni , and introducing a complex
wavefunction defined by $|{\bf\Psi}\rangle={\bf D}+i{\bf B}$, the Maxwell
equation in curved space $\\{g_{\mu\nu}\\}$ takes the Schrödinger-like formfll
,

$i\frac{\partial}{\partial
t}|{\bf\Psi}\rangle=\nabla\times\Bigl{(}|{\bf\Psi}\rangle/\sqrt{g^{00}}+i{\bf
g}\times|{\bf\Psi}\rangle\Bigr{)}$ (1)

together with the transverse condition $\nabla\cdot|{\bf\Psi}\rangle=0$. In
Eq.(1), ${\bf g}$ is the 3d dragging vector $g_{i}=g_{0i},g^{i}=g^{i0}/g^{00}$
and the vectorial operations are performed in the 3d curvilinear space
$\gamma_{ij}=-g_{ij},\gamma^{ij}=-g^{ij}+g^{0i}g^{0j}/g^{00}$, and the
electromagnetic vectors are identified by
$D^{i}=F^{0i}/\sqrt{g^{00}},B^{i}=\epsilon^{ijk}F_{jk}/2\sqrt{\gamma}$
,$\gamma$ is the determinant of $\gamma_{ij}$. it is easy to verify that, in a
free space, the Hamiltonian in Eq.(1) reduces to $H={\bf k}\cdot{\bf s}$,
where ${\bf k}=-i\nabla$ is the momentum operator, ${\bf s}$ is the spin-1
operator given by the adjoint representation of group SO(3), i.e.,
$\\{s^{i}\\}^{jk}=-i\epsilon^{ijk}$. Obviously, for a plane wave, it has two
helicity states, the positive helicity with positive energy $\omega=|{\bf
k}|$, and the negative one with $\omega=-|{\bf k}|$, corresponding the right
and left polarization states of photon. The zero helicity state is eliminated
by the transverse condition ${\bf k}\cdot|{\bf\Psi}\rangle=0$. The opposite
helicity states are mutually complex conjugates.

Without loss of generality, suppose an incident light propagating in z-axis,
and the wavefunction $|{\bf\Psi}\rangle$ is decomposed into the transverse
component $|{\bf\Psi}\rangle_{\perp}=\Psi_{x}{\bf e}_{x}+\Psi_{y}{\bf e}_{y}$
with respective to the z-component $|{\bf\Psi}\rangle_{z}$. we introduce a
‘helicity basis’ $\\{{\bf e}_{\pm},{\bf e}_{z}\\}$, satisfying $({\bf
e}_{z}\cdot{\bf s}){\bf e}_{\pm}=\pm{\bf e}_{\pm}$, which is related to the
‘Cartesian frame’ explicitly by ${\bf e}_{\pm}=({\bf e}_{x}\pm i{\bf
e}_{y})/\sqrt{2}$. In the ‘helicity basis’ (denoted by superscript c), we
define the transverse component $|{\bf\Psi}\rangle_{\perp}^{h}={\Psi}_{+}{\bf
e}_{+}+\Psi_{-}{\bf e}_{-}$, the differential operator is thus
$\nabla_{\perp}=\nabla_{+}{\bf e}_{+}+\nabla_{-}{\bf e}_{-}$ with
${\bf\nabla}_{\pm}=({\bf\nabla}_{x}\mp i{\bf\nabla}_{y})/\sqrt{2}$,
accordingly. By these conventions, it is easy to write down the Maxwell
equation in the following matric form

$i\frac{\partial}{\partial t}{\bf\Psi}^{h}={\bf\nabla}\times({{\bf
U}{\bf\Psi}^{h}})$ (2)

where ${\bf U}$ is the $3\times 3$ dielectric tensor specified by the space
metric of the GW, under the weak field approximation, ${\bf g}=0$, and
$g_{ij}=\eta_{ij}+h_{ij}$,

${\bf U}=\left(\begin{array}[]{cc}{\bf I}+{\bf Q}&{\bf q}\\\ \\\ {\bf
q}^{\dagger}&1+q_{0}\\\ \end{array}\right)$ (3)

here ${\bf I}$ is the $2\times 2$ unit matrix, ${\bf
q}=\\{Q_{+1},Q_{-1}\\}^{T}$ is the dipole momentum with the components $Q_{\pm
1}=\frac{1}{\sqrt{2}}(h_{13}\mp ih_{23})$, $q_{0}=h_{33}$, and ${\mathbf{Q}}$
is the $2\times 2$ matrix with the matrix elements, the monopoles
$Q_{11}=Q_{22}=\frac{1}{2}(h_{11}+h_{22})$, and the quadrupoles
$Q_{12}=\frac{1}{2}(h_{11}-h_{22})-ih_{12}$, $Q_{21}=Q_{12}^{*}$.

For simplicity, we consider the positive helicity state only, in this case, we
take the wavefunction in the form of
$|{\bf\Psi}\rangle_{\perp}^{h}={\Psi_{0}^{+}}e^{-i\omega t+ikz}{\bf e}_{+}$,
accordingly, the z-component can be derived from the transverse condition,
$\Psi_{z}=\frac{i}{k}\nabla_{\perp}\cdot|{\bf\Psi}\rangle_{\perp}$ where the
paraxial approximation was used. Since GW’s wavelength is always assumed to be
much larger than light beam, the geometric-optics approximation is valid, and
the leading terms at the lowest order gives the eigenstate equation, from
which we have $\omega=(1+Q_{0})k$ and an extra phase shift along the optical
path $\phi=k\int Q_{0}dz$. Keeping the first order terms yields

$\displaystyle i\frac{\partial\Psi_{0}^{+}}{\partial z}=$
$\displaystyle-\displaystyle{\frac{1}{k^{\prime}}}\nabla_{+}\nabla_{-}\Psi_{0}^{+}-({\bf
q}\cdot\frac{1}{i}{\bf\nabla}_{\perp})\Psi_{0}^{+}$ (4)

where $k^{\prime}=k(1+Q_{0})/(1+q_{0})$ resulting from the gravitational
frequency shift. In addition, the GW’s wavelength is also assumed to be much
larger than the optical length $L$, thus the spatial derivative of the GW
fields $\\{h_{ij}\\}$ can been neglected. If we keep the zeroth order only,
Eq.(4) reduces to the familiar paraxial equation in the flat space which has a
set of solutions of the Laguerre-Gaussian modes in the cylindrical coordinates

$LG_{n}^{l}=\frac{a_{n}^{l}}{w(z)}\Bigl{(}\frac{\sqrt{2}\rho}{w(z)}\Bigr{)}^{|l|}L_{n}^{|l|}\Bigl{(}\frac{2\rho^{2}}{w(z)^{2}}\Bigr{)}e^{-\rho^{2}/w(z)^{2}}e^{ik\rho^{2}/2R}e^{il\phi}e^{-i\varphi(z)}$
(5)

where $n$ and $l$ are the radial and the azimuthal indices, the order of the
model is given by $N=2n+|l|$, the constant
$a_{n}^{l}=(2n!/\pi(n+|l|)!)^{1/2}$, $w(z)$ is the width of mode, $R$ the
radius of the wavefront curvature, and the Gouy phase factor
$\varphi(z)=(N+1)\tan^{-1}(z/z_{R})$, $z_{R}=\frac{1}{2}kw_{0}^{2}$, here
$w_{0}=w(z=0)$ is the beam waist.

The second term in Eq.(4) is the perturbation due to the passing GW, which
appears as a dipole interaction,

$H_{I}=-{\bf q}\cdot{\bf k}_{\perp}$ (6)

where ${\bf k}_{\perp}=-i\nabla_{\perp}$ is the transverse momentum operator,
and the dipole momentum ${\bf q}$ is given by

${\bf q}=h^{+}[{\bf e}_{k}-({\bf e}_{g}\cdot{\bf e}_{k}){\bf
e}_{g}]+h^{\times}({\bf e}_{g}\times{\bf e}_{k})$ (7)

where ${\bf e}_{g}$ and ${\bf e}_{k}$ denote for the unit vectors in the
propagation directions of the gravitational wave and light beam respectively.
Following Eq.(7), ${\bf q}\cdot{\bf e}_{g}=0$, which implies the dipole
momentum ${\bf q}$ being in the polarization plane of the GW. if the incident
GW is propagating parallelly with the light beam, the dipole is identically
zero, ${\bf q}=0$. This dipole interaction is physically due to the coupling
of GW polarization and the transverse momentum of photons. The coupling term
actually consists of the two contributions, one is $\propto h^{+}({\bf
e}_{g}\cdot{\bf k}_{\perp})$, another is $\propto h^{\times}[{\bf
e}_{g}\cdot({\bf e}_{k}\times{\bf k}_{\perp})]$. Formally, the former is from
the GW polarization coupled with the OAM density flow of photons, and the
latter from the spin density flow.

In the ”helicity basis”, the dipole interaction can be alternatively written
by

$H_{I}=i\bigl{[}Q_{-1}\nabla_{+}+Q_{+1}\nabla_{-}\bigr{]}$ (8)

where $Q_{\pm 1}=\pm ihe^{\pm i\beta}({\bf e}_{g}\cdot{\bf e}_{\mp})$, $h$ and
$\beta$ are given by

$h=\sqrt{h_{+}^{2}\cos^{2}\theta_{g}+h_{\times}^{2}}\quad\tan\beta=\frac{h_{+}}{h_{\times}}\cos\theta_{g}$
(9)

The complex differential operators $\nabla_{\pm}$ in the cylindrical
coordinates become $\nabla_{\pm}=\frac{1}{\sqrt{2}}e^{\mp
i\phi}(\partial_{\rho}\pm\frac{1}{\rho}L_{z})$ where $L_{z}=-i\partial_{\phi}$
is the orbital angular momentum operator in z-direction. Obviously,
$\nabla_{\pm}$ play a role similar to the creation and destruction operators.
While operating on the LG modes. $\nabla_{\pm}$ will either lowers or raises
the orbital angular momentum by one unit. It is not difficult to work out the
ladder relation,

$\displaystyle\nabla_{\pm}|n,\pm
l\rangle_{LG}=k_{w}\bigl{[}\sqrt{n+l}|n,\pm(l-1)\rangle_{LG}$
$\displaystyle\quad+\sqrt{n+1}|n+1,\pm(l-1)\rangle_{LG}\bigr{]}$ (10)
$\displaystyle\nabla_{\mp}|n,\pm
l\rangle_{LG}=-k_{w}\bigl{[}\sqrt{n+l+1}|n,\pm(l+1)\rangle_{LG}$
$\displaystyle\quad+\sqrt{n}|n-1,\pm(l+1)\rangle_{LG}\bigr{]}$ (11)

where we define the wavenumber $k_{w}=1/w_{0}$. Combining Eqs.(8) (Detecting
Gravitational Waves by Twisted Light -  Dipole Interaction of Photons and
Gravitational Waves) and (Detecting Gravitational Waves by Twisted Light -
Dipole Interaction of Photons and Gravitational Waves) states that, for a
given LG mode $|n,l\rangle_{LG}$, the dipole interaction will make the initial
mode converting into two modes with the OAM number of $l-1$ and $l+1$, the
difference between them is 2. It can be easily understood by the polarization
states of gravitational wave having helicity $h=\pm 2$. Because $Q_{+1}$ and
$Q_{-1}$ are conjugates of each other, the total orbit angular momentum is
conserved. It’s noted that the system modeled by Eq.(6) is an analogous to
Jaynes-Cummings model describing the interaction of a two-level atom with a
single quantized mode of the radiation field.

Since the Laguerre-Gaussian modes form a complete and orthonormal set with
respect to the mode indices $n$ and $l$ in the polar plane $\\{\rho,\phi\\}$,
we can make a decomposition such that

$\Psi_{0+}=\sum_{m,k}\xi_{m.k}(z)|m,k\rangle_{LG}$ (12)

Inserting this expansion Eq.(12) into Eq.(4), we have

$\frac{d\xi_{n,l}(z)}{dz}=\sum_{m,k}\langle
n,l|Q_{-1}\nabla_{+}+Q_{+1}\nabla_{-}|m,k\rangle$ (13)

$\xi_{n,l}(z)$ can be obtaind by direct integration over the summation in
above equation.

Figure 1:  The transverse self-interference intensity pattern for an incident
Gaussian beam LG${}_{0}^{0}$ (upper left) and the dipole structure with the
rotation angles $\alpha=0^{\circ}$, (upper right) $\alpha=45^{\circ}$ (lower
left) and $\alpha=90^{\circ}$ (lower right), the color bar is labeled by the
relative intensity.

In the following, we will present two typical ideal experiments to demonstrate
the optical features induced by the dipole interaction. As a simplest case,
let the incident beam is a Gaussian beam or $LG_{0}^{0}$ mode with zero OAM.
Under the GW’s perturbation, the dipole transition generates two splitting
modes with the opposite unit OAMs, i.e., $l=\pm 1$,

$|\hbox{out}\rangle=|\hbox{in}\rangle-
Q_{1}k_{w}L\bigl{[}e^{i\alpha}|0,+1\rangle_{LG}+e^{-i\alpha}|0,-1\rangle_{LG}\bigr{]}$
(14)

where we have taken $Q_{\pm 1}=Q_{1}e^{\pm i\alpha}$. The overall amplitude of
$|0,\pm 1\rangle_{LG}$ is proportion to $Q_{1}=\sqrt{h_{13}^{2}+h_{23}^{2}}$.
For the GW propagating in the $\\{\theta_{g},\phi_{g}\\}$ direction,
$Q_{1}=\sin\theta_{g}h(\theta_{g})$, and the rotation angle $\alpha$,
depending on the incident direction of the GW, is related to $\beta$ by

$\alpha+\phi_{g}=\frac{\pi}{2}+\beta$ (15)

Eq.(14) indicates that, for the LG modes with the opposite OAM, the dipole
interaction lead to extra opposite rotations by the angle $\alpha$. This
rotation should be visualized in the intensity pattern of light beam. In the
first order of the GW strain $O(h)$, the readout intensity is

$I\approx
I_{0}\Big{[}1-4\sqrt{2}hk_{w}L\frac{\rho}{w(L)}\sin\theta_{g}\cos\varphi\cos(\phi+\alpha)\Bigr{]}$
(16)

where the strain $h$ is given by Eq.(9), and $\varphi=\varphi(L)$ is the Gouy
phase. Fig.(1) displays clearly the excess dipole components pointing to the
direction $\alpha$ in the transverse intensity frame. In addition, it should
be emphasized here that there exists an addition rotation freedom from the GW
polarization around ${\bf e}_{g}$, with respect to which the polarizations
$\\{h_{+},h_{\times}\\}$ are defined. Performing an rotation in the transverse
place ${\bf e}_{\perp}^{\prime}=R(\psi){\bf e}_{\perp}$, (R is a rotation
transformation with an angle $\psi$), we have
$\\{h^{\prime}_{+},h^{\prime}_{\times}\\}^{T}=R(2\psi)\\{h_{+},h_{\times}\\}^{T}$.

Figure 2:  The normalized transverse intensity pattern for an incident HG10
mode (upper left), and output excess mixed modes in different angles of
$\alpha=0^{\circ}$, (upper right) $\alpha=45^{\circ}$ (lower left) and
$\alpha=90^{\circ}$ (lower right).

The second example illustrates an opposite case. Let us consider an incident
beam of the Hermite-Gauss mode $HG_{10}$, which can be written by a
superposition of two LG modes with the opposite OAM, $l=\pm 1$,
$|\hbox{in}\rangle=HG_{10}=\frac{1}{\sqrt{2}}(|0,+1\rangle_{LG}+|0,-1\rangle)_{LG}$
In the gravitational wave, there will appear extra four modes, we have

$\displaystyle|\hbox{out}\rangle=$
$\displaystyle|\hbox{in}\rangle+Q_{1}k_{w}L\cos\alpha\bigl{[}|0,0\rangle_{LG}+|1,0\rangle_{LG}\bigr{]}$
$\displaystyle-\displaystyle{\frac{1}{\sqrt{2}}}Q_{1}k_{w}L\bigl{[}e^{i\alpha}|0,+2\rangle_{LG}+e^{-i\alpha}|0,-2\rangle_{LG}\bigr{]}$

In this numerical experiment, after substracting the incident light intensity,
we focus on the intensity pattern of the dipole induced modes only, which are
thus quadratic in the strain $O(h^{2})$. Clearly, one significant feature is
brighten of the central intensity. For the LG${}_{0}^{\pm 1}$ modes, it
carries $\hbar$ of OAM per photon and has a well-known donut-like intensity
profile since the amplitude goes to zero at the center $\rho=0$. Due to the
dipole interaction, the induced mode involves one by lowering one unit of the
OAM, which inevitably give rise to the vortex-free $|0,0\rangle_{LG}$ mode,
i.e, the Gaussian beam with the bright central spot. Fig.(2) demonstrates the
normalized transverse intensity distribution in this experiment. Though this
second order effect is quite small, it will be hopeful to detect it using the
well-developed OAM sorting technique in futuresort .

We summary the paper and make some concluding remarks as follows. Taking a
Schrödinger view of the Maxwell theory in curved space, we found a new
gravitational dipole interaction between photons and GWs. We demonstrate that
this dipole interaction can produce some striking optical features in the 2-D
intensity pattern, including (1) the induced dipole structure for the Gaussian
beams, (2) a macroscopic rotation of the intensity pattern, depending not only
on the GW incident direction, also on the polarization in the transverse plane
as well as the ratio of cross and plus components of the GW, and (3) the
central intensity brighten adding on the donut-like intensity profile for the
lowest-order LG mode ($l=\pm 1$). Obviously, these features suggest an
alternative way to measure GWs in the 2-D intensity space. Unlike the
classical interference experiments GWdect , the dipole interaction makes the
incident twisted light beam to form a multi-mode intensity profile in the
image plane. In the first order of $O(h)$, the readout signal is weaker than
the current interference experiments by a factor of $\lambda/w_{0}$, where
$\lambda$ is the wavelength of the laser beam, and $w_{0}$ the beam waist.
This ratio may take a value of, e.g., $\sim O(10^{-1})$ or even less. However,
the compensation for this decreasing signal strength can be made by extending
the entire optical-path length of the 2-D coherent imaging. Most importantly,
the benefit is clearly more physical information of the GWs inferred from the
2-D imaging. Finally, it is noted that the newly found dipole interaction,
only existing beyond the plane wave approximation and having been unknown in
the previous studiesGWdect ; harte , must be important for quantifying
precisely the wave behaviors of the higher-order LG modes in optical cavity
for the next generation of GW detectors.

Acknowledgments -  FLL is supported by the Key Program of NFSC through grant
11733010 and 11333008, and the State Key Development Program for Basic
Research of China (2015CB857000)

## References

  * (1) J. Aasi et al. (LIGO Scientific), Class. Quant. Grav. 32, 074001 (2015)
  * (2) F. Acernese et al. (Virgo), Class. Quant. Grav. 32, 024001 (2015)
  * (3) S. Chelkowski, S. Hild & A. Freise, Phys. Rev. D79, 122002 (2009)
  * (4) M. Granata, C. Buy, R. Ward & M. Barsuglia, Phys. Rev. Lett 105, 231102 (2010); L. Carbone, C. Bogan, P. Fulda, A. Freise & B. Willke, Phys. Rev. Lett. 110, 251101(2013)
  * (5) I. Tamm, J. Russ, Phys.-Chem. Soc., Phys. 56, 248 (1924)
  * (6) L. L. Feng, M. Li and R. Ruffini, Nucl. Phys. (Proc. Suppl.) B6, 314 (1989); P. Carini, L.L. Feng, M. Li & R. Ruffini, Phys. Rev. D46, 5407 (1992); L.L. Feng & W.L. Lee, Int. J. Mod. Phys. 10, 961 (2001)
  * (7) B. Mashhoon, Phys. Rev. Lett., 61, 2639 (1988); B. Mashhoon, Phys. Lett. A173, 347 (1993); J.C. Hauck & B. Mashhoon, Annalen der Physik, 515 (Series 8, Vol. 12), Issue 5-275 (2003)
  * (8) J. M. Cohen and D. R. Brill, Nuovo Cimento B56, 209 (1968)
  * (9) L.L.Feng & T. Lu, Class. Quant. Grav. 8, 851 (1991)
  * (10) G. Sagnac, Comput. Rend 157 7-8 (1913)
  * (11) L. Landau & E. Lifshitz, The Classical Theory of Fields, 3rd ed. (Addison-Wesley, New York, 1972)
  * (12) A. Sadowsky, Acta et Commentationes Imp. Universitatis Jurievensis, 7 1 (1899)
  * (13) L. Allen, M.V. Beijersbergen, R.J.C. Spreeuw, & J.P. Woerdman, Phys. Rev. A45, 8185 (1992); J.P. Torres & L. Torner (eds.), Twisted Photons, Wiley-VCH, (2011)
  * (14) A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zeldovich, Phys. Rev. A45, 8204 (1992).
  * (15) A. V. Kavokin, G. Malpuech, & M. M. Glazov, Phys. Rev. Lett. 95, 136601 (2005)
  * (16) R.S. Hanni et al., Phys. Rev. D16, 933 (1977)
  * (17) F.X Wang, W. Chen, Z.Q. Yin, S. Wang, G.C. Guo & Z.F. Han, Phys. Rev. A94, 033847 (2016), and references therein
  * (18) C. Bond, D. Brown, A. Freise & K.A. Strain, Living Rev. Relativ, 19:3 (2016)
  * (19) A. Harte, Class. Quant. Grav., 32, 175017 (2015)