# Double-Sided Beamforming in OWC Systems Using Omni-Digital Reconfigurable
Intelligent Surfaces

Alain R. Ndjiongue, Telex M. N. Ngatched, Octavia A. Dobre, and Harald Haas A.
R. Ndjiongue, T. M. N. Ngatched, and O. A. Dobre are with the Faculty of
Engineering and Applied Science, Memorial University, Canada.
Harald Haas is with the LiFi Research and Development Center, Dpt. Electronic
and Electrical Engineering, the University of Strathclyde, Glasgow, U.K.

###### Abstract

In this paper, we introduce a variant of reconfigurable intelligent surfaces
(RISs) called omni-digital-RISs (DRISs), which allow multiple physical
processes, with application to optical wireless communications systems. The
proposed omni-DRIS contains both reflectors and refractive elements, as well
as elements that perform both simultaneously. We describe and explain the
concept of omni-DRIS, suggest and analyze an omni-DRIS coding structure,
discuss metamaterials to be used, and provide a design example. Furthermore,
we demonstrate that the achievable rate of an omni-DRIS system depends on the
number of omni-DRIS elements, bits per phase shift, and the number of unused
elements. In addition, we show that the achievable rate upper bound is related
to the number of omni-DRIS elements, and conclude by discussing future
research directions.

###### Index Terms:

RIS, digital-RIS (DRIS), omni-DRIS, optical wireless communications (OWC),
double-side beam management, simultaneous transmission and reflection, double-
sided beamforming.

##  I Introduction

The advantages of reconfigurable intelligent surfaces (RISs) extend beyond
solving the skip zone problem in wireless communication systems. Additionally,
they enable wireless channels to be reconfigured and are efficient in emerging
beam management. RISs are used in radio-frequency and optical wireless
communication (OWC) systems ([1, 2, 3, 4, 5], and references therein). In both
reflection and refraction, the RISs can perform beam steering, splitting,
scattering, light polarization, and photon absorption [2]. To solve skip zones
in wireless systems, an RIS is placed over the wireless channel and becomes a
part of it. By adjusting its physical properties, the channel can be
reconfigured. As a result, the RIS can resolve dead zones, create
reconfigurable channels, and manage emerging beams [6]. In this position, the
RIS module is mainly used to reflect the incident signals. Therefore, both
incident and reflected signals remain in the same half-space with respect to
the RIS. New development have proven that the RIS module can be mounted inside
the transmitter or receiver [7, 8]. In this configuration, the RIS allows the
signal to pass through it and acts as a refracting system that can steer the
impinging signal and control the emerging beams. Though the emerging signal
lies in a different half-space, the RISs elements deal with a unique physical
process, and are not digitized.

As a result of the more recent development in the RIS technology, digital-RISs
(DRISs), which allow digital signal processing to be applied to the physical
material used in the reconfigurable elements, have been conceived [9]. The
entire system is governed by a control module, which generates the code
sequence for each element. According to the feedback from users, the control
module of the DRIS transfers a specific code to the targeted element. During
the execution of the code, physical properties of the DRIS elements are
reconfigured to fulfill the new wave propagation required by the users.

Simultaneous transmission and reflection (STAR) for 360o coverage by RISs has
also been demonstrated recently [10]. STAR-RISs allow a double physical
processes at the reconfigurable elements. Some of the elements act as
reflectors, while others guide the signal through the structure. The remaining
elements may reflect and refract light simultaneously or remain unused.
Therefore, to connect users located on both sides, the RIS module must be
equipped with both reflectors and elements that can refract signals, as well
as elements that can perform the two processes simultaneously [10].

In this paper, we introduce a RIS that combines the characteristics of STAR-
RISs and DRISs with application to OWC systems. The proposed RIS, referred to
as omni-DRIS can achieve both reflection and refraction simultaneously, while
enabling digital signal processing in its elements. It is thus able to
spatially direct light toward users located on both sides of the module. We
discuss the omni-DRIS, propose and investigate a coding protocol, review
metamaterials that can be used with an emphasis on liquid crystals (LCs), and
propose double-sided beam management for omni-DRIS-assisted OWC systems. To
conclude, we consider a case study of an omni-DRIS-assisted indoor visible
light communication (VLC) design example, and through the achievable rate, we
analyze the system’s performance with regard to the number of active,
inactive, and total elements of the omni-DRIS.

##  II Ordinary RIS vs. Omni-DRIS

When deployed, the RIS has always been mounted on a building wall, pasted to a
flying vehicle, or incorporated into the device’s transmitter or receiver [4,
5, 7]. In such configurations, the RIS is composed of elements that perform
only one physical process at a time. Specifically, they serve as reflectors or
refractive devices. These configurations prevent the user located at the back
of the RIS module or inside the building from being linked to the network at
the same time as the users on the other side of the RIS [10]. Thus, the RIS
creates a blind spot that may be resolved in two ways: (i) by utilizing a
second RIS to cover the back of the first, or (ii) by using an omni-RIS
module, which incorporates elements that can serve as both reflectors and
refractive devices simultaneously [10, 11]. The latter solution has the
additional advantage of being economical. The proposed omni-DRIS is comprised
of elements that can be in one of four possible states: reflection,
refraction, both reflection and refraction, and unused. Reflectors reflect the
light impinging on the omni-DRIS surface towards the user located on the same
side of the incident light. The element has a reflection coefficient close to
unity and a phase shift between zero and pi. On the other hand, an omni-DRIS
element that acts as a refractive device allows the light ray to pass through
it and is characterized by a high transmission coefficient close to one and a
phase shift ranging between pi and two pi. An omni-DRIS element which
simultaneously reflects and refracts the incoming light ray combines the
characteristics of a reflector and refractive device. Note that in this case,
the sum of the refraction and reflection coefficients equals unity. Lastly,
the omni-DRIS element can remain quite (or turned off).

Figure 1: An illustration of the omni-DRIS operation. A number of elements are
reflective while some are refractive, and a set of elements is off and will be
used as soon as needed. Figure 2: Structure of the omni-DRIS codes (top).
Omni-DRIS mapping example: $k$ = 4, 16 omni-RIS element types (bottom).

Figure 1 illustrates an omni-DRIS, showing a double-sided beam generation. It
reflects and refracts light simultaneously within the system in which it is
inserted. Elements marked with an ”R” are total reflectors, while those marked
with a ”T” totally refract the incident light. Elements marked with ”OFF” are
not in use. To avoid having an overcrowded figure, this illustration does not
show reflective-refractive elements. For similar reasons, the incident light
beams are not shown. To implement the omni-DRIS, an algorithm must be
developed that controls each element individually. The generated codes are
transferred to the module as sequences of bits. This aspect is discussed in
the next section.

Figure 3: Illustration of light reflection and refraction through the omni-
DRIS in three dimensions.

##  III Coding Omni-DRIS and Emerging Light Construction

Similar to the DRIS proposed in [9], the omni-DRIS is composed of $2^{k}$
elements, where $k$ refers to the number of bits necessary to define the phase
shifts and transition coefficients. Therefore, the omni-DRIS is composed of
$2^{k}$ phase shifts and transition coefficients. To control the omni-DRIS, a
series of bit sequences, having the same length, is mapped to the phase shifts
and transition coefficients of its elements. Each code sequence is composed of
three blocks. The coding structure is depicted at the top of Fig. 2, where the
first block is $a_{1}a_{2}$, while $b_{1}b_{2}b_{3}\dots b_{k}$ and
$c_{1}c_{2}c_{3}\dots c_{k}$ are the second and third blocks, respectively.
The first block, made of two bits, is used to define the state of the omni-
DRIS element. ”01” and ”10” respectively induce reflection and refraction,
”11” is reserved for an element with two-process capability, i.e.,
simultaneous reflection and refraction, while ”00” is reserved for an omni-
DRIS element that should remain off. The second block, made of $k$ bits,
defines the phase profiles, and the third block, also made of $k$ bits, is
reserved for the transition coefficients. As a result, each code is made of
2($k$ \+ 1) bits. For example, an omni-DRIS with 16 active elements has a code
sequence composed of 10 bits. Two bits to select the element’s mode, four bits
to define the phase shifts, and four bits to represent the transition
coefficients.

The bottom part of Fig. 2 illustrates a mapping table for a coding scheme for
an omni-DRIS, which contains 16 types of active elements. The first column
indicates the code number, the second column contains two bits to induce the
physical process, ”01”, ”10”, ”11”, and ”00”. The third column encloses a
four-bit gray coding arrangement for defining the phase profiles and
transition coefficients. It is pertinent to recall that the expression ”phase
shift” is only used if the reflected or refracted light lies over the same
plane as the incident light ray. Due to the fact that the plane of the
emerging light may differ depending on the position of the user, which may be
mobile, the emerging light is not always in the plane of incidence. As a
result, the values in column three represent the phase profiles of the omni-
DRIS elements. The phase profile is the result of combining the two phase
shifts that represent the off-plane orientation of the emitted light. For
instance, if the incident light lies in the $yz$-plane, the phase profile of
the emerging light is given with reference to that plane. The phase profile,
shown at the bottom of Fig. 2, groups the angle between the emerging light and
the $xz$-plane, as well as the angle formed by the projection of the emerging
light onto the $xz$-plane and the $z$-axis.

Figure 3 depicts an illustration of such three-dimensional light orientation.
The figure shows the transition of a light ray through an omni-DRIS element,
highlighting both reflection and refraction for the same incident light ray.
The values of phase profiles, given in Fig. 2, comprise a combination of
$\theta_{r}$ (angle between the emerging light ray and the $xz$-plane) and
$\varphi_{r}$ (angle between the projection of the emerging light ray to the
$xz$-plane and the $z$-axis) for reflection, as well as a combination of
$\theta_{t}$ (angle between the emerging light ray and the $xz$-plane) and
$\varphi_{t}$ (angle between the projection of emerging light ray to the
$xz$-plane and the $z$-axis) for transmission. Accordingly, when $yz$ is the
plane of incidence, the omni-DRIS element directs the light in a specific
direction defined in relation with the $xz$ and $yz$ planes, when the omni-
DRIS is in the $xy$ plane. Likewise, for the same plane of incidence and omni-
DRIS plane as above, the refracted ray direction is given with respect to both
the $xz$ and $yz$ planes. When the impinging light remains in the $yz$-plane,
the emerging light can be described as either reflection or refraction if it
lies over the omni-DRIS’s plane and forms a 90-degree angle with the $z$-axis.
In this case, the phase profile of the omni-DRIS element is the angle formed
by the emerged ray and one of the $x$ or $y$ axes. As previously presented,
the path of light through an omni-DRIS element and the corresponding phase
profile, taken together with the sequences of control codes, demonstrate that
an omni-DRIS steers the light with respect to a specific three-dimensional
constellation. Since the number of phase profiles equals $2^{k}$, the number
of directions is determined by the value $k$ in this constellation. The
combination of the above description and mapping procedure of the bit
sequences to the omni-DRIS elements, along with the concept of spatial three-
dimensional emerging light, leads to two-sided beam management. An example is
discussed in the following section along with a description of reconfigurable
metamaterials for omni-DRIS.

##  IV Omni-DRIS Design for Double-Sided Beam Management

###  IV-A Reconfigurable Metamaterials for Omni-DRIS

Several metamaterials may be used to manufacture omni-DRIS modules for OWC
systems.

Remote control of omni-DRIS-assisted OWC systems is a desirable feature for a
variety of reasons. This control can be achieved through external parameters
such as an electric or magnetic field. Thus, the omni-DRIS reconfigurable
metamaterials should be selected based on their tunability and reaction to
external parameters applied remotely. Therefore, electro-optical metamaterials
are the perfect candidates. These includes plasmonic metamaterials and two-
state materials, to mention only two.

Plasmonic metamaterials can be classified into passive and active
metamaterials. The application discussed in this paper calls for active
plasmonic metamaterials since they provide the required optical properties.
Active metamaterials are mostly exploited in optical, thermo-optical, and
electro-optical systems [12]. These materials have the properties necessary to
be used in information processing, communication, and data storage. Example of
plasmonic materials include glass, prism, and lenses. Metamaterials with
negative and positive indices are needed to respond to emerging light’s three-
dimensional and spherical orientation. Two-state materials may also be used.
An example of two-state metamaterial is the chalcogenide alloy such as
Germanium telluride, which exhibits high resistance in the off-state and low
resistance in the on-state [2]. Phase transitions in this material can be
controlled by external factors such as electrical current or light. These
materials can be used in systems requiring binary metamaterials that operate
in ”on” and off” phases. Due to their absorption capabilities, two-dimensional
nanosheets such as dichalcogenides are also highly used in optical
applications [13]. On the other hand, surface plasmons exhibit positive real
permittivity, which plays an influential role in enhancing the Raman
spectroscopy [14]. This advantage, in addition to its tunability, justifies
its use in the manipulation of light beams.

Despite the advantages offered by the above metamaterials, anisotropic
materials may be better suited for omni-DRIS. Unlike isotropic metamaterials,
which exhibit the same properties in all directions, anisotropic metamaterials
exhibit different properties in different directions within the material
structure [15]. The anisotropic behavior is most frequently observed in
crystals of solid or compound materials, but rarely in liquids. Mixtures of
liquids and crystals provide the properties of the liquid, and the resulting
product is anisotropic if the employed crystals are also anisotropic.
Anisotropic materials are characterized by phenomena such as double refraction
or birefringence, a phenomenon that results in the change of light speed along
the axis of the crystal. The anisotropic behavior can be observed in
Selenium’s electrical resistivity, which exhibits high or low values according
to the externally applied field polarity. Liquid crystals (LCs) are
anisotropic materials with a birefringence that can be controlled externally
and remotely.111It should be noted that an LC is a state of matter that has
both liquid and solid crystal properties. Due to its electrically controllable
birefringence, the light’s path through an LC substance may be controlled
smoothly. This control is based on varying the externally applied field, which
changes to a specific value that is allocated to the corresponding element,
leading to a birefringence (refractive index) variation. This procedure
results in the element acquiring a phase profile and transition coefficient,
causing the impinging light to be refracted or reflected in a specific
direction with a specific power.

LC-based metasurfaces possess a unique advantage. The same LC’s substance and
dye can be used in devices doing both reflection and refraction. Specifically,
the bottom glass layer of the LC-based metasurface structure can be replaced
with a silicon to convert it from a refractive device to a reflector.
Furthermore, utilizing transparency-configurable materials such as changing
acrylic metamaterials at the bottom layer of an LC-based metasurface yields a
structure that can simultaneously reflect and refract light. The LCs’
birefringence (refractive index) is sensitive to the variation of the
externally applied electric or magnetic field. As a consequence, it is easy to
control the light’s path through LCs [7, 8]. At a given temperature, varying
the externally applied field induces a retardation angle, which is responsible
for the shift created on the impinging light. The omni-DRIS is structured in
such a way that each element is controlled individually. Therefore, it is
convenient to control each omni-DRIS element using a sequence of bits. In this
manner, the omni-DRIS elements can be identified with a unique code sequence
as explained in Section III. Each code sequence corresponds to a distinct type
of physical process (reflection vs. refraction, or both), a phase profile
(azimuth angle and elevation angle, both from zero to two pi), and a
transition coefficient. The elevation angle in the phase profile varies
between zero and pi for reflection and pi to two pi for refraction, while the
azimuth angle varies from zero to two pi in both reflection and refraction. In
view of this benefit, LCs are adopted in this paper as primary materials for
omni-DRIS. As such, LCs can be used in reflective and refractive elements, and
in elements that simultaneously provide the two processes, as elaborated
subsequently.

Figure 4: An example of scenario where the omni-DRIS can be used. The omni-
DRIS is part of the wall separating the two parts of a room and the
information light source is located in the first part. Four users are located
at A, B, C, and D in Part 1 and four other users are situated at E, F, G, and
H in Part 2. Each omni-DRIS element serves a different user.

As reflective elements: An LC sandwiched between two polarizers is an
effective refractive device. To reflect the incident light, it is necessary to
add an opaque layer after the bottom polarizer [7]. To control the emerged
light and spatially orient it in the desired direction, a discontinuity should
be created on the upper face of the bottom opaque layer. This can be achieved
by using a blazed grating system in three dimensions. Silicon is generally
used as an opaque layer [8]. Thus, smooth control can be realized on both the
phase profile and the reflection coefficient by utilizing both the
birefringence and the grating system. The final material is a pure reflector
and cannot allow the light through the structure.

As transmissive elements: In transmittance, the material is similar to that
used in reflectors, with the exception that the opaque bottom layer is
replaced by a glass. To control the off-axis orientation of the transmitted
light, a discontinuity must be created on either the lower or upper faces of
the glass. Consequently, the emerged light can be easily controlled with the
grating system and by tuning the LC’s birefringence.

As element performing the two processes: An invaluable function of some omni-
DRIS elements is their ability to reflect and refract light at the same time.
These elements combine the features of a reflector and refractive device.
However, it is vital to keep in mind that between reflection and refraction, a
ratio of proportionality should be respected. If, for example, 60% of the
light impinging on the omni-DRIS surface is transmitted, then only 40% of this
light will be reflected. The concern is focused on the bottom layer of the
omni-DRIS element. It should be possible to control the transparency property
of the material that creates the discontinuity in the element’s bottom layer.
Innovative materials, such as smart and switchable glasses, can be used
because they can be electronically adjusted between total transparency and
opacity. The transparency of some materials, such as changing acrylic, can be
controlled by heating. These materials (smart glasses and changing acrylic
materials) display varying levels of transparency and opacity based on the
control parameter selected.

###  IV-B Design Procedure and a Mapping Example

The design involves defining three-dimensional constellations of the omni-DRIS
and determining the corresponding code sequences. Consider an omni-DRIS
composed of $N$ elements that aim to direct an incident light to $N$ points.
This three-dimensional constellation consists of a three-dimensional geometric
figure with $N$ vertices. $N$ is a power-of-two, 2k, as in the modulation
schemes (phase profiles and transition coefficient in the case of omni-DRIS).
This geometric figure is constructed in a three-dimensional space ($x$, $y$,
$z$), where $yz$ is the incident plane and the users are located at its
vertices. $xy$ is the omni-DRIS plane and its elements are anisotropic LCs
made of the same substance and dye. However, the tuning of each element is
based on a unique control code sequence.

In the illustration provided in Fig. 4, $k$ = 3, $N$ = 8, the users are placed
in a way to obtain a tetrahedron or cuboid, in a room divided in two parts.
The omni-DRIS is part of the separating wall, and the light source carrying
the message is located in the first part. The omni-DRIS contains 16 elements,
with eight unused elements (no bit sequences are transferred to them and no
electric field is applied). The remaining eight elements are organized such
that four at the four corners of the arrangement are used as reflectors, while
the four located at the center are utilized to refract the incident light. The
reflective elements are LC on silicons and the upper face of each element is a
grating system with a specific discontinuity. On the other hand, the elements
exploited to target the users at the back of the omni-DRIS have a transparent
glass at their bottom layer with a discontinued grating system. The same LC
and dye are used in all elements; however, each of them is controlled
individually. The tuning is coordinated locally at the omni-DRIS based on the
different code sequences transferred to it. With reference to the order
provided at the bottom of Fig. 2 and the element arrangement in Fig. 4, codes
no. 1, 4, 13, and 16 are transferred to the outer elements, while codes no. 6,
7, 10, and 11 are transferred to the inner elements. These codes are generated
according to the protocol proposed at the top of Fig. 2. Thus different codes
are transferred to the eight active omni-DRIS elements. These codes are
typically designed based on the feedback from the corresponding users. Their
actions on the elements are to adjust the externally applied electric field.
This creates specific values of the element’s temperature and refractive
index, leading to an azimuth deviation and elevation in the emerging light
orientation. The obtained temperature and refractive index yield a phase
profile containing the corresponding elevation and azimuth angles. In this
practical example, codes 0100001000, 0100101000, 0110101000, and 0110001000
respectively yield (31.22, -27.39), (-31.22, -27.39), (31.22, 27.39), and
(-31.22, 27.39) for reflectors, while codes 1001111000, 1001011000,
1011011000, and 1011111000 yield (36.47, -30.33), (36.47, 30.33), (-36.47,
-30.33), and (-36.47, 30.33), respectively, for the refractive
elements.222Note that in the pair ($\theta$, $\varphi$), $\theta$ and
$\varphi$ are the elevation and azimuth angles, respectively. The dimensions
of the beam formed by these omni-DRIS elements depends on the reflective
surface and the discontinuity orientation of the elements. Thus, the
narrowness of the emerged beam relates to the physical structure of the
elements, most precisely the reflecting area of the element. Described above
is an example of double-sided beam management in indoor VLC systems. In this
OWC beamforming, the motion of the user is followed by a change in the omni-
DRIS code, which results in an adjustment of the azimuth and elevation angles
to track the user. In cases where the user is out of reach, beam switching
prevents the user from losing connection.

It is useful to establish threshold ratios between the number of omni-DRIS
elements, the elements that are used, and those that are not used. These
ratios impact the achievable rate of the system as shown in Fig. 5 in terms of
the number of bits per phase shift (transition coefficient). We demonstrate in
this figure that the achievable rate is inversely proportional to the number
of elements used, $2^{k}$, compared with a fixed total number of omni-DRIS
elements. Note that the behavior observed in Fig. 5 relates to the number of
users. Additionally, Fig. 5 implies that as the number of bits per phase shift
(transition coefficient) increases, the system’s achievable rate increases,
which may suggest that the system will be able to perform at a higher rate for
an infinite amount of omni-DRIS elements. However, this is not the case. It is
pertinent to note that there is an upper limit, i.e., there is a number of
omni-DRIS elements which results in the highest achievable rate. Over this
threshold, the achievable rate is reduced as the number of bits per phase
shift increases. This is illustrated in Fig. 6, where the achievable rate is
given in terms of the number of omni-DRIS elements. The figure shows that
there is a value for the number of omni-DRIS elements where the system has the
maximum achievable rate.

Figure 5: Achievable rate of an omni-DRIS assisted indoor VLC system in terms
of bits per phase shift for a variety of noise levels. Figure 6: Achievable
rate of an omni-DRIS assisted indoor VLC system in terms of the number of
omni-DRIS elements for a variety of noise levels.

##  V Future Research Directions and Conclusion

Geometrical analysis of omni-DRIS systems and double-sided beamforming: The
development of an omni-DRIS modules for OWC systems may have a variety of
applications as underlined in [10] for the STAR-RIS. However, this idea is
still in its infancy and therefore requires extensive and intensive
investigation. It is important to identify these applications and analyze
their respective channels and performances. Double-sided beamforming in omni-
DRIS-assisted OWC systems, including the indoor VLC and free-space optical
communication environments should be studied.

Omni-DRIS-assisted integrated sensing and communication (ISAC): The
integration of sensing and communication technologies to utilize wireless
resources efficiently and potentially avail of their mutual benefits is one of
the major features of 5G-advanced and 6G mobile networks. The proposed omni-
DRIS can find a wide range of applications in ISAC, where the sensing signal
can be used to design the control sequence of the different elements.

Omni-DRIS-assisted OWC system models and channel estimation: Modeling an OWC
channel assisted by an LC-based omni-DRIS has many challenges. First, when the
omni-DRIS is exploited as a reflective device, the communication signal
transits through the LC substance with a double run. In addition, the double
refraction at the surface of the omni-DRIS will have an impact on the
transmitted message. Second, the transmissive omni-DRIS element oversees the
light path in a single run and a double phase shift at its input (top) and
output (bottom), leading to the transmitted signal being affected. Third, both
influences related to reflection and refraction will simultaneously happen to
the impinging signal while facing a transmissive-reflective omni-DRIS element.

Omni-DRIS-assisted OWC systems implementations: Practical deployment of an
omni-DRIS-assisted OWC system requires an efficient LC materials selection. An
analysis of the transition time to meet the 5G and beyond transmission speed
requirement is needed. Apart from these, the spherical orientation of the
emerging light in these applications requires specific materials to meet the
omni-orientation of the light beams on both side of the module. Transferring
the code sequences from the control module to the omni-DRIS elements have not
yet been studied. The algorithm that will perform the omni-DRIS element’s
tuning and reconfiguration is also to be designed.

In this paper, we investigated the use of omni-DRIS in OWC systems. Its
concept was described and explained. A number of applications of this type of
DRIS were also highlighted. Additionally, we examined the omni-DRIS coding and
outlined a code structure. We also explored the omni-DRIS design by
emphasizing the materials that could be utilized as reflectors, to transmit a
light, and both phenomena simultaneously. A proposal was made to use an LC-
based metasurface to realize the three types of materials by altering its
bottom layer, and a design example was investigated.

## References

  * [1] G. Alexandropoulos et al., “Reconfigurable intelligent surfaces for rich scattering wireless communications: Recent experiments, challenges, and opportunities,” _IEEE Commun. Mag._ , vol. 59, no. 6, pp. 28–34, Jun. 2021.
  * [2] C. Molero et al., “Metamaterial-based reconfigurable intelligent surface: 3D meta-atoms controlled by graphene structures,” _IEEE Commun. Mag._ , vol. 59, no. 6, pp. 42–48, Jun. 2021.
  * [3] M. Najafi et al., “Intelligent reflecting surfaces for free space optical communication systems,” _IEEE Trans. Commun._ , vol. 69, no. 9, pp. 6134–6151, Sep. 2021.
  * [4] H. Abumarshoud et al., “LiFi through reconfigurable intelligent surfaces: A new frontier for 6G?” _IEEE Veh. Technol. Mag._ , pp. 2–11, Nov. 2021.
  * [5] A. R. Ndjiongue et al., “Toward the use of re-configurable intelligent surfaces in VLC systems: Beam steering,” _IEEE Wireless Commun. Mag._ , vol. 28, no. 3, pp. 156–162, Jun. 2021.
  * [6] C. Pan et al., “Reconfigurable intelligent surfaces for 6G systems: Principles, applications, and research directions,” _IEEE Commun. Mag._ , vol. 59, no. 6, pp. 14–20, Jun. 2021.
  * [7] A. R. Ndjiongue et al., “Re-configurable intelligent surface-based VLC receivers using tunable liquid-crystals: The concept,” _IEEE/OSA J. Lightw. Technol._ , vol. 39, no. 10, pp. 3193–3200, May 2021\.
  * [8] ——, “Design of a power amplifying-RIS for free-space optical communication systems,” _IEEE Wireless Commun. Mag._ , vol. 28, no. 6, pp. 152–159, Dec. 2021.
  * [9] ——, “Digital RIS (DRIS): The future of digital beam management in RIS-assisted OWC systems,” _arXiv_ , Dec. 2021.
  * [10] Y. Liu et al., “STAR: Simultaneous transmission and reflection for 360° coverage by intelligent surfaces,” _IEEE Wireless Commun. Mag._ , vol. 28, no. 6, pp. 102–109, Dec. 2021.
  * [11] S. Zhang et al., “Intelligent omni-surfaces: Ubiquitous wireless transmission by reflective-refractive metasurfaces,” _IEEE Trans. Wireless Commun._ , vol. 21, no. 1, pp. 219–233, Jan. 2022.
  * [12] I. C. Benea-Chelmus et al., “Electro-optic spatial light modulator from an engineered organic layer,” _Nat. Commun._ , vol. 12, no. 1, pp. 1–10, Oct. 2021.
  * [13] T. Mueller et al., “Exciton physics and device application of two-dimensional transition metal dichalcogenide semiconductors,” _Nat. 2D Mater Appl._ , vol. 2, no. 1, pp. 1–12, Sep. 2018.
  * [14] X. Wang et al., “Fundamental understanding and applications of plasmon-enhanced Raman spectroscopy,” _Nat. Rev. Phy._ , vol. 2, no. 5, pp. 253–271, Sep. 2020.
  * [15] M. Bullo et al., “Isotropic and anisotropic electrostatic field computation by means of the cell method,” _IEEE Trans. Magn._ , vol. 40, no. 2, pp. 1013–1016, Mar. 2004.

Alain R. Ndjiongue [S’14, M’18, SM’20] is a senior researcher at Memorial
University of Newfoundland, Canada.
---
Telex. M. N. Ngatched [M’05, SM’17] is an associate professor at Memorial
University, Canada.
---
Octavia A. Dobre [M’05, SM’07, F’20] is a professor and research chair at
Memorial University, Canada.
---
Harald Haas [S’98, AM’00, M’03, SM’16, F’17] holds the Chair of Mobile
Communications at the University of Strathclyde.
---


# Signatures of discretization in quantum black hole spectra

Joshua Foo111Corresponding Author<EMAIL_ADDRESS>Centre for
Quantum Computation & Communication Technology, School of Mathematics &
Physics, The University of Queensland, St. Lucia, Queensland, 4072, Australia
Robert B. Mann<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Perimeter Institute
for Theoretical Physics, Waterloo, Ontario N2L 6B9, Canada  Magdalena Zych
<EMAIL_ADDRESS>Department of Physics, Stockholm University,
AlbaNova University Center, SE-106 91 Stockholm, Sweden  Centre for Engineered
Quantum Systems, School of Mathematics and Physics, The University of
Queensland, St. Lucia, Queensland, 4072, Australia

###### Abstract

The quantum superposition principle states that quantum-mechanical systems
such as atoms can be placed in a superposition of mass-energy eigenstates.
Inspired by this idea and the seminal conjecture of Bekenstein, who proposed
that black holes in quantum gravity must possess a discrete mass
eigenspectrum, here we analyze the effects produced by a black hole in a
superposition of masses. Analogous to using the electromagnetic field to probe
atoms, we consider a quantum scalar field on the spacetime background sourced
by the black hole mass superposition. From the resulting spectra, as measured
by a hypothetical two-level system interacting with the field, we infer
signatures of discretization of the black hole mass in support of Bekenstein’s
conjecture.

Essay written for the Gravity Research Foundation 2023 Awards for Essays on
Gravitation.

Submitted on March 29, 2023

Black holes have long captivated physicists from a diverse array of
backgrounds, from cosmology and astroparticle physics to quantum field theory
and general relativity. Because of the extreme gravitational environments they
generate, they are considered primary candidates for studying regimes in which
quantum-gravitational effects are present Gibbons and Hawking (1993); Wald
(1984); Nicolini (2009). Indeed, the discoveries of Hawking radiation Hawking
(1974) and black hole evaporation Hawking (1975) gave rise to the well-known
information paradox Hawking (1976); Giddings (1995); Mathur (2009); Harvey and
Strominger (1993) and an entire field seeking its resolution, which aptly
illustrates the existing conflicts between quantum theory and general
relativity.

Bekenstein Bekenstein (1973, 2020); Regge and Wheeler (1957) was among the
first to recognise that a complete theory of quantum gravity must account for
the treatment of black holes as quantum objects. One of his key insights was
in recognising that the black hole horizon area, and hence its mass, is an
adiabatic invariant, $I$, with an associated discrete quantization Hod (1998)
of evenly spaced energy levels Bekenstein and Mukhanov (1995). That is,

$\displaystyle I$
$\displaystyle=\int\mathrm{d}A\sim\int\>\mathrm{d}M\>f(d,M)\equiv
n\hslash\omega,$ (1)

where $A$ is the horizon area, $f(d,M)$ is some function of the mass $M$ and
spacetime dimension $d$, $\omega$ is the transition frequency of the black
hole, and $n\in\mathbb{Z}$. The first equality of Eq. (1) is Bekenstein’s
identification of the horizon area as an adiabatic invariant; this gives rise,
via the first law of black hole thermodynamics, to the middle relationship,
while the last equality describes the equally-spaced discretization of the
horizon area and mass.

Following Bekenstein’s analogical treatment of black holes as atomic systems
motivates us to consider them in quantum superpositions of mass-energy
eigenstates. A mass-superposed black hole is an example of a non-classical
spacetime; since the masses individually characterize a unique solution to
Einstein’s field equations in classical general relativity, the resulting
scenario can be described as an entangled state where the masses and the
associated states of spacetime are jointly superposed Foo  _et al._ (2022a,
2023, 2021, b). Understanding the effects that arise in such spacetime
superpositions is an important stepping stone towards developing a complete
description of a quantum spacetime. Formal approaches for quantizing general
relativity date back to the Wheeler-deWitt equation Wheeler (1968); DeWitt
(1967), which governed a wave functional of the metric and offered a
conceptual scheme for dealing with superpositions of spacetime geometries
Rovelli and Vidotto (2015); Rovelli (2008). Building upon this work, canonical
quantization techniques were applied to the metric variables Kuchař (1994);
Thiemann and Kastrup (1993) leading to the development of loop quantization
Ashtekar (1986); Ashtekar  _et al._ (1998), which in turn has yielded
solutions including black holes in a superposition of masses Rovelli (2004);
Kastrup and Thiemann (1994); Campiglia  _et al._ (2007); Gambini  _et al._
(2014); Gambini and Pullin (2013); Demers and Kiefer (1996); Kiefer (2013).

In this essay, we outline an operational approach for studying the
phenomenology of black hole mass superpositions. By operational, we mean that
our approach is grounded in effects that can in-principle be measured by
physical systems such as rods, clocks, and other measurement devices. In
particular, we consider a two-level system that interacts with a scalar
quantum field on a background sourced by a (2+1)-dimensional black hole in a
superposition of masses. We show how the dynamics of this system can probe the
black hole’s “spectrum,” in analogy with spectroscopic experiments that probe
atomic systems via the electromagnetic field. As we will demonstrate, the
spectrum measured by the two-level system elicits signatures of
discretization, in accordance with Bekenstein’s original conjecture for
quantum black holes.

Let us now construct the scenario described above. We focus on the
(2+1)-dimensional Banados-Teitelboim-Zanelli (BTZ) black hole of mass $M$,
whose line element is given by Bañados  _et al._ (1992); Carlip (1995); Carlip
and Carlip (2003)

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-\left(\frac{r^{2}}{l^{2}}-M\right)\mathrm{d}t^{2}+\left(\frac{r^{2}}{l^{2}}-M\right)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\phi^{2},$
(2)

where $-l^{2}=\Lambda$ is the AdS length (corresponding to a negative
cosmological constant $\Lambda$). Let us consider the two-level probe system
situated at a fixed coordinate distance $r=R_{D}$ from the origin and prepared
in its ground state, $|g\rangle$. The scalar field is assumed to be initially
in the AdS vacuum, $|0\rangle$, while the black hole is initially in a
superposition of two different masses $M_{A},M_{B}$. Note that Eq. (2) applies
for any value of the mass $M$, and thus we can use the same coordinates $t$,
$r$, $\phi$ for both masses. Within quantum theory, we can denote the mass
eigenstates of the black hole as $|M_{A}\rangle$, $|M_{B}\rangle$, and we
assume these states are orthogonal (as different black hole masses correspond
to classically distinguishable situations). The state of the entire system at
some initial coordinate time $t=t_{i}$ is thus given by,

$\displaystyle|\psi(t_{i})\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|M_{A}\rangle+|M_{B}\rangle)\otimes|0\rangle\otimes|g\rangle.$
(3)

The interaction between the black hole, quantum field, and the probe can be
described with a Hamiltonian of the form Foo  _et al._ (2022a)

$\displaystyle\hat{H}_{\mathrm{int}}$
$\displaystyle=\hat{H}_{A}\otimes|M_{A}\rangle\langle
M_{A}|+\hat{H}_{B}\otimes|M_{B}\rangle\langle M_{B}|,$ (4)

where $\hat{H}_{D}$ $(D=A,B)$ is a Hamiltonian describing the probe system and
the field in the spacetime generated by the black hole of mass $M_{D}$, in
particular with the line element in Eq. (2) with $M=M_{D}$. Specifically, we
consider a linear coupling between the probe and the field
$\hat{H}_{D}\propto\hat{\phi}(\mathbf{x}_{D})\otimes(|e\rangle\langle
g|e^{i\Omega\tau}+\mathrm{H.c.})$ where $\Omega$ is the internal energy gap of
the two-level probe and $\hat{\phi}(\mathbf{x}_{D})$ is the field operator
pulled back to the worldline $\mathbf{x}_{D}$ in the spacetime generated by
mass $M_{D}$. The Hamiltonian Eq. (4) describes how for each of the black hole
masses, the field and probe interact as in a fixed spacetime characterized by
that mass.

Time-evolution of the system in the interaction picture up to some final time
$t=t_{f}$ takes the form,

$\displaystyle|\psi(t_{f})\rangle$
$\displaystyle=e^{-i\hat{H}_{0,S}t_{f}}\hat{U}(t_{i},t_{f})e^{i\hat{H}_{0,S}t_{i}}|\psi(t_{i})\rangle$
(5)

where $\hat{H}_{0,S}$ is the free Hamiltonian of the system, including the
free evolution of the mass-energy eigenstates of the black hole. Here, the
coordinate time $t$ used to parametrize the evolution of all systems can be
defined as the proper time of a clock at sufficiently large $r$, such that the
fractional difference in proper times between the two spacetimes,
$(M_{B}-M_{A})l^{2}/r$, is negligible. This choice assumes that the state of
the clock is factorizable, i.e. independent of the black hole mass. One could
alternatively choose a clock state that is entangled with the mass; in an
anticipated theory of quantum gravity, one expects both scenarios to be
physically admissible.

Now, recall that we are motivated by the notion of operationalism. In the
present case, we seek to describe effects that are in-principle detectable via
measurements of a probe due to its interaction with a field quantized on a
spacetime in a superposition of masses. In the absence of such an interaction,
one could hypothetically complete a “double-slit experiment” with the black
hole masses, leading to the usual interference fringes between them. However
this would reveal little information about the structure of the non-classical
spacetime sourced by the black hole, and the effect of such an environment on
physical systems residing within. Again drawing an analogy with spectroscopic
experiments used to image atomic wavefunctions, here we propose to “image” the
spacetime produced by the black hole through its coupling with the field and
probe. This is the crux of what makes our approach an operational one.

To this end, let us calculate the transition probability of the two-level
probe from its ground to excited state $P(|g\rangle\to|e\rangle)$. One way to
obtain this is to trace out both the field and black hole degrees of freedom.
If we apply this to the final state in Eq. (5), we obtain

$\displaystyle\mathrm{Tr}_{\phi,M}\Big{[}|\psi(t_{f})\rangle\langle\psi(t_{f})|\Big{]}$
$\displaystyle=\frac{1}{2}(P_{A}+P_{B}),$ (6)

which is a statistical mixture of the transition probabilities $P_{A}$,
$P_{B}$ for a detector situated in the classical spacetime sourced by a black
hole of either mass $M_{A}$ or $M_{B}$. In this case, the quantum features of
the spacetime superposition are washed out.

A more interesting case for us is when the black hole mass is assumed to be
measured in a basis consisting of two orthogonal mass eigenstates, e.g.
$|\pm\rangle=(|M_{A}\rangle\pm|M_{B}\rangle)/\sqrt{2}$, and tracing out the
field. This amounts to completing a Mach-Zehnder-type interferometric protocol
on the black hole masses, with the field and probe interacting with the black
hole in each branch of the interferometer. A measurement of the black hole
mass could be achieved by an interaction of an ancilla with the black hole and
later measuring the ancilla in a superposition basis. More specifically, one
could prepare two qubits entangled in their internal energies, before sending
one into the black hole thereby entangling its mass with the state of other
qubit. Projecting this qubit onto a superposition state will in turn project
the black hole onto a mass superposition state. The result of such a protocol
gives

$\displaystyle\mathrm{Tr}_{\phi}\Big{[}\langle\pm|\psi(t_{f})\rangle\langle\psi(t_{f})|\pm\rangle\Big{]}$
$\displaystyle=\frac{1}{4}(P_{A}+P_{B}\pm 2\cos(\Delta E\Delta t)L_{AB}),$ (7)

where $\Delta t=t_{f}-t_{i}$ is the coordinate time elapsed, $\Delta
E=|\sqrt{M_{B}}-\sqrt{M_{A}}|$ is the energy gap of the black hole
eigenstates, and we refer to $L_{AB}$ as the cross-term, which we discuss in
detail below. We note that the above quantity is a conditional transition
probability of the probe, conditioned on the outcome $\pm$ of the measurement
of the black hole mass in the basis $|\pm\rangle$.

Equation (7) shows that the probe transition probability is now sensitive to
interference between the different spacetimes in superposition, information of
which is contained in the cross-term $L_{AB}$. Notably, this cross-term takes
the form Foo  _et al._ (2022a)

$\displaystyle L_{AB}$
$\displaystyle\propto\sum_{n,m}\mathrm{Re}\int_{0}^{t_{f}/l}\mathrm{d}z\>F\left(z,n\sqrt{M_{A}}-m\sqrt{M_{B}}\right),\>n,m\in\mathbb{Z},$
(8)

where $F(\cdot,\cdot)$ is a complicated function of its arguments. The
dependence of $L_{AB}$ (and hence the transition probability) on
$n\sqrt{M_{A}}-m\sqrt{M_{B}}$ is suggestive of nontrivial behaviour when
$\sqrt{M}_{A}$, $\sqrt{M}_{B}$ are integer-valued. Indeed, this is what we
find. Figure. 1 displays the transition probability as a function of the ratio
$\sqrt{M_{B}/M_{A}}$, where resonances are clearly seen at rational values of
$\sqrt{M_{B}/M_{A}}$; for example $\sqrt{M_{B}/M_{A}}=1/4,1/3$ and so on.

Figure 1: Transition probability of a two-level detector situated outside a
black hole in a superposition of masses. The blue and yellow lines correspond
to measurements of the black hole in the $|+\rangle$ and $|-\rangle$ state
respectively. The transition probability is normalized by $\sigma$, the
timescale of the interaction.

Remarkably, the location of these resonances corresponds with values arising
from Bekenstein’s conjecture that the mass spectrum of a black hole is
discrete. In particular, for the case of the BTZ black hole Kwon and Nam
(2010),

$\displaystyle I=l\sqrt{M}=n\hslash,\>n\in\mathbb{Z},$ (9)

implying, that for any two black hole masses $M_{A}$ and $M_{B}$,
$\sqrt{M_{B}/M_{A}}\in\mathbb{Q}$. Importantly, our construction does not make
any assumptions about the masses $M_{A},M_{B}$, but the form of the resulting
transition probability, Eq. (8), gives rise to a spectrum–as detected by a
two-level system coupled to the field–that encodes a novel signature of mass
discretization. We therefore suggest that the result above constitutes an
independent verification of–indeed an extension to (for superpositions of
masses)–Bekenstein’s conjecture. More significantly, it is also a
demonstration of how operational techniques can be used to reveal genuinely
quantum-gravitational properties of a black hole.

The scheme presented here opens a path to understanding the effects of non-
classical spacetimes on important notions such as causal structure or
reference frames in quantum gravity. Concerning the former, our approach
provides techniques for studying quantum communication protocols in such
spacetimes, and the possibility of violating classical constraints on
signalling (for example, considering multiple probes situated outside a mass-
superposed black hole and signalling between them through the field).
Concerning the latter, the mass-quantised black hole considered here naturally
gives rise to two distinct sets of coordinates–correlated as well as
uncorrelated with the mass. An open question remains as to how one
appropriately defines the resulting transformations between them, and how they
act on the systems in question (i.e. the field and the probe system).

Our results also motivate the development of experimental setups that can
simulate the operational effects of spacetime in quantum superposition,
utilizing for example, advances in optomechanical technologies. The line
element of Eq. (2) is obtained by imposing a periodic boundary condition on
AdS spacetime; in this sense, the mass-superposed BTZ black hole is an example
of a quantum superposition of non-trivial topologies. Such topologies have
been realized in superfluid Helium condensates on toroidal cavities. The
fundamental mode providing the analog metric for phonons in the condensate
therein can interact with incident photons, thus suggesting an opportunity to
study new analog quantum-gravitational systems, in which a quantum field
evolves in the presence of quantum-controlled backgrounds Bowen and Milburn
(2015).

## References

  * Gibbons and Hawking (1993) G. W. Gibbons and S. W. Hawking,  _Euclidean quantum gravity_ (World Scientific, 1993).
  * Wald (1984) R. M. Wald, in  _Quantum theory of gravity. Essays in honor of the 60th birthday of Bryce S. Dewitt_ (1984).
  * Nicolini (2009) P. Nicolini, International Journal of Modern Physics A 24, 1229 (2009), https://doi.org/10.1142/S0217751X09043353 .
  * Hawking (1974) S. W. Hawking, Nature 248, 30 (1974).
  * Hawking (1975) S. W. Hawking, Communications in Mathematical Physics 43, 199 (1975).
  * Hawking (1976) S. W. Hawking, Phys. Rev. D 14, 2460 (1976).
  * Giddings (1995) S. B. Giddings, in  _PASCOS / HOPKINS 1995 (Joint Meeting of the International Symposium on Particles, Strings and Cosmology and the 19th Johns Hopkins Workshop on Current Problems in Particle Theory)_ (1995) pp. 415–428, arXiv:hep-th/9508151 .
  * Mathur (2009) S. D. Mathur, Classical and Quantum Gravity 26, 224001 (2009).
  * Harvey and Strominger (1993) J. A. Harvey and A. Strominger, in  _Spring School on String Theory and Quantum Gravity (To be followed by Workshop on String Theory 8-10 Apr)_ (1993) pp. 175–223, arXiv:hep-th/9209055 .
  * Bekenstein (1973) J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
  * Bekenstein (2020) J. D. Bekenstein, in  _JACOB BEKENSTEIN: The Conservative Revolutionary_ (World Scientific, 2020) pp. 331–334.
  * Regge and Wheeler (1957) T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).
  * Hod (1998) S. Hod, Phys. Rev. Lett. 81, 4293 (1998).
  * Bekenstein and Mukhanov (1995) J. D. Bekenstein and V. Mukhanov, Physics Letters B 360, 7 (1995).
  * Foo  _et al._ (2022a) J. Foo, C. S. Arabaci, M. Zych, and R. B. Mann, Phys. Rev. Lett. 129, 181301 (2022a).
  * Foo  _et al._ (2023) J. Foo, C. S. Arabaci, M. Zych, and R. B. Mann, Phys. Rev. D 107, 045014 (2023).
  * Foo  _et al._ (2021) J. Foo, R. B. Mann, and M. Zych, Classical and Quantum Gravity 38, 115010 (2021).
  * Foo  _et al._ (2022b) J. Foo, R. B. Mann, and M. Zych, International Journal of Modern Physics D 31, 2242016 (2022b), https://doi.org/10.1142/S0218271822420160 .
  * Wheeler (1968) J. A. Wheeler (1968).
  * DeWitt (1967) B. S. DeWitt, Phys. Rev. 160, 1113 (1967).
  * Rovelli and Vidotto (2015) C. Rovelli and F. Vidotto,  _Covariant loop quantum gravity: an elementary introduction to quantum gravity and spinfoam theory_ (Cambridge University Press, 2015).
  * Rovelli (2008) C. Rovelli, Living reviews in relativity 11, 1 (2008).
  * Kuchař (1994) K. V. Kuchař, Phys. Rev. D 50, 3961 (1994).
  * Thiemann and Kastrup (1993) T. Thiemann and H. Kastrup, Nuclear Physics B 399, 211 (1993).
  * Ashtekar (1986) A. Ashtekar, Phys. Rev. Lett. 57, 2244 (1986).
  * Ashtekar  _et al._ (1998) A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998).
  * Rovelli (2004) C. Rovelli,  _Quantum gravity_ (Cambridge university press, 2004).
  * Kastrup and Thiemann (1994) H. Kastrup and T. Thiemann, Nuclear Physics B 425, 665 (1994).
  * Campiglia  _et al._ (2007) M. Campiglia, R. Gambini, and J. Pullin, Classical and Quantum Gravity 24, 3649 (2007).
  * Gambini  _et al._ (2014) R. Gambini, J. Olmedo, and J. Pullin, Classical and Quantum Gravity 31, 095009 (2014).
  * Gambini and Pullin (2013) R. Gambini and J. Pullin, Phys. Rev. Lett. 110, 211301 (2013).
  * Demers and Kiefer (1996) J.-G. Demers and C. Kiefer, Physical Review D 53, 7050–7061 (1996).
  * Kiefer (2013) C. Kiefer, ISRN Mathematical Physics 2013, 1–17 (2013).
  * Bañados  _et al._ (1992) M. Bañados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992).
  * Carlip (1995) S. Carlip, Class. Quant. Grav. 12, 2853 (1995), arXiv:gr-qc/9506079 .
  * Carlip and Carlip (2003) S. Carlip and S. J. Carlip,  _Quantum gravity in 2+ 1 dimensions_ , Vol. 50 (Cambridge University Press, 2003).
  * Kwon and Nam (2010) Y. Kwon and S. Nam, Classical and Quantum Gravity 27, 125007 (2010).
  * Bowen and Milburn (2015) W. P. Bowen and G. J. Milburn,  _Quantum optomechanics_ (CRC press, 2015).