# Graphene phase modulators operating in the transparency regime

H. F. Y. Watson  A. Ruocco  M. Tiberi  J. E. Muench  O. Balci  S. M. Shinde
S. Mignuzzi  Cambridge Graphene Centre, University of Cambridge, 9 JJ Thomson
Avenue, CB3 0FA, Cambridge, UK  M. Pantouvaki  D. Van Thourhout  IMEC,
Kapeldreef 75, B-3001 Leuven, Belgium  R. Sordan  Politecnico di Milano, Polo
di Como, Via Anzani 42, 22100 Como, Italy  A. Tomadin  Dipartimento di Fisica,
Università di Pisa, Largo Bruno Pontecorvo 3, 56127 Pisa, Italy  M. Romagnoli
Consorzio Nazionale per le Telecomunicazioni, 56124 Pisa, Italy  A. C. Ferrari
Cambridge Graphene Centre, University of Cambridge, 9 JJ Thomson Avenue, CB3
0FA, Cambridge, UK

###### Abstract

Next-generation data networks need to support Tb/s rates. In-phase and
quadrature (IQ) modulation combine phase and intensity information to increase
the density of encoded data, reduce overall power consumption by minimising
the number of channels, and increase noise tolerance. To reduce errors when
decoding the received signal, intersymbol interference must be minimised. This
is achieved with pure phase modulation, where the phase of the optical signal
is controlled without changing its intensity. Phase modulators are
characterised by the voltage required to achieve a $\pi$ phase shift Vπ, the
device length L, and their product VπL. To reduce power consumption, IQ
modulators are needed with$<$1V drive voltages and compact (sub-cm)
dimensions, which translate in VπL$<$1Vcm. Si and LiNbO3 (LN) IQ modulators do
not currently meet these requirements, because VπL$>$1Vcm. Here, we report a
double single-layer graphene (SLG) Mach-Zehnder modulator (MZM) with pure
phase modulation in the transparent regime, where optical losses are minimised
and remain constant with increasing voltage. Our device has
$V_{\pi}L\sim$0.3Vcm, matching state-of-the-art SLG-based MZMs and plasmonic
LN MZMs, but with pure phase modulation and low insertion loss ($\sim$5dB),
essential for IQ modulation. Our $V_{\pi}L$ is$\sim$5 times lower than the
lowest thin-film LN MZMs, and$\sim$3 times lower than the lowest Si MZMs. This
enables devices with complementary metal-oxide semiconductor compatible VπL
($<$1Vcm) and smaller footprint than LN or Si MZMs, improving circuit density
and reducing power consumption by one order of magnitude.

The global internet traffic was expected to triple between 2019-2024 with the
advent of 5G and the internet of everything[1]. Lockdowns in response to
COVID-19 shifted the distribution of data traffic across the network[2], with
an additional$\sim$20-200% rise[3], due to remote working[3, 4], and increased
use of home entertainment services[5]. This vast amount of data relies on a
backbone of high-density data network infrastructures, with 2018 standards of
400Gb/s[6] to be extended$>$1Tb/s by 2025[7]. To go$>$1Tb/s it is preferable
to increase data rates in a single channel[8, 9], rather than the number of
channels. By minimising the number of channels, the power consumption and
system complexity is reduced, because less electrical drivers and active
optical components are needed[8, 9]. The bandwidth (BW) of a single channel
that uses binary modulation formats is limited by that of the electrical
interfaces used to drive the active optical components[8, 9]. These struggle
to exceed 2023 standards[7] because losses increase with increased
frequencies[10]. Consequently, for data rates$>$100Gb/s, binary modulation
formats[11] have been replaced by 4-level pulse-amplitude modulation (PAM)[6].
PAM uses 4 amplitude levels of the transmitted optical signal, to represent 4
symbols that correspond to 2 bits of information[8]. Other multi-level
modulation schemes, such as quadrature amplitude modulation (QAM)[11], encode
information in both phase and amplitude[11]. Transmission systems that use
only amplitude modulation (AM) are known as direct detection systems[11].
Those that use both phase modulation (PM) and AM are known as coherent,
because the phase difference between two or more signals remains constant over
time[11]. Coherent systems have a higher noise tolerance than direct detection
ones, because the signal degradation from fibre dispersion can be compensated
by the received signal phase[11].

Ref. | Material | Type | IL [dB] | ER [dB] | VπL [Vcm] | L [cm] | Modulation speed | VπIL [VdB]
---|---|---|---|---|---|---|---|---
[18] | Si | MZM | 5.4 | 3.6 | 1.4 | 0.2 | 55 GHz | 38
[19] | Si | Ring modulator | 3 | 9.8 | 0.52 | 0.2 | 50 GHz | 8
[20] | Si | Thermo-optic PM | 0.23 | - | 0.027 | 0.006 | 130 kHz | 1
[21] | III-V/Si MOS | MZM depletion mode | - | 11 | 0.24 | 0.05 | 27 GHz | -
[22] | III-V/Si MOS | MZM accumulation mode | 1 | 12 | 0.047 | 0.047 | 100 MHz | 1
[23] | III-V/Si MOS | MZM depletion mode | - | 4.4 | 0.3 | 0.03 | 18 GHz | 70
[24] | Thin-film LN | MZM | 0.5 | 30 | 1.4 | 2 |  $>$ 45 GHz | 0.4
[25] | Thin-film LN | MZM | 7.6 | 20 | 6.7 | 0.5 | 106 GHz | 102
[26] | Thin-film LN/Si | MZM | 2.5 | 40 | 2.2 | 0.3 |  $>$ 70 GHz | 18.5
[27] | Thin-film LN/Si | MZM | 15 | 19 | 0.8 | 0.3 |  $>$ 40 GHz | 19.5
[28] | Thin-film LN | Plasmonic MZM | 19.5 | 2.5 | 0.23 | 0.0015 |  $>$ 10 GHz | 2,990
[29] | DSLG | EAM | 20 | 3 | - | 0.01 | 29 GHz | -
[30] | DSLG | Ring modulator | - | 15 | - | 0.003 | 30 GHz | -
[31] | DSLG (flakes) | EAM | 4 | 5 | - | 0.006 | 39 GHz | -
[32] | SLG/Si | MZM | 10 | 35 | 0.28 | 0.03 | 5 GHz | 62
This work | DSLG | MZM | 5.6 | 25 | 0.3 | 0.0075 | 24 GHz | 3
Table 1: Modulators based on Si, III-V (InGaAsP), LN and graphene for IQ
modulators design.

Information is transmitted by electro-optic (EO) modulators that convert an
electrical signal into an optical one[12]. This can be encoded into the
intensity of the transmitted signal, known as AM, or electro-absorption
modulation[12], and into the phase, known as PM or electro-refractive
modulation[12]. In-phase and quadrature (IQ) modulators are interferometric
devices that use pure PM, with no change of amplitude, to generate the
different QAM symbols[11]. No direct AM is required to generate QAM symbols,
because the interferometer converts a phase difference into a change in
amplitude[11]. To reduce intersymbol interference, therefore errors at the
receiver[11], the symbol noise should be minimised, and symbols should be
evenly spaced in the in-plane and quadrature axes[11]. Thus, any unwanted AM
will increase symbol noise, and any non-linear PM will result in irregular
symbol spacing[11].

An important parameter for comparing phase modulators is the product of the
voltage required to achieve a $\pi$ phase shift, $V_{\pi}$, and the device
length, $L$[13]. The additional optical loss resulting from inserting the
device in the transmission line is the insertion loss IL=$\alpha L$, where
$\alpha$ is the absorption coefficient per unit length[14]. In order to reduce
overall power consumption, we need to minimise $V_{\pi}L$ and IL[14], because
a lower $V_{\pi}$L reduces the device area and capacitance, hence reducing the
dynamic energy consumption E=CV2=CV${}_{pp}^{2}$/4[15], i.e. the energy
charged and discharged in a capacitor by an AC voltage with peak-to-peak
voltage Vpp. IL contributes to optical power loss and signal degradation. The
PM figure of merit (FOM${}_{\text{PM}}$) is defined as the product of
$V_{\pi}$ and IL (FOM${}_{\text{PM}}=V_{\pi}\alpha L$)[16], whereby better
phase modulators have a smaller FOM${}_{\text{PM}}$. The modulator BW is
critical for Tb/s data transmission, in order to maximise the data rates that
a single channel can support[14], which is $T=BWlog_{2}(1+\frac{S}{N})$[17],
where BW is in Hz and the signal to noise ratio, S/N, is the ratio of the
received power to the noise power. E.g., a data rate of 100Gb/s in a single
lane with S/N$\geq$20, which is the goal set by the 2023 Ethernet Alliance
roadmap[7], requires BW$\geq$23GHz.

Table 1 shows silicon photonics (SiP)[18, 19, 20], III-V (InGaAsP)[21, 22,
23], LiNbO3 (LN)[24, 25, 26, 27, 28] and graphene-based Electro-absorption
modulators (EAMs)[29, 30, 31] and PM[32]. SiP offers a cost-effective solution
for integrating electronic and photonic components in the same circuit by
using existing complementary metal oxide semiconductor (CMOS) technology[33].
Pure PM is difficult to achieve with Si modulators based on the plasma
dispersion effect[34, 35, 36, 18, 37, 19] because, due to the Kramers-Kronig
relations[38], any change in carrier concentration results in changes in both
absorption and phase. Even if pure phase modulators in Si were to be achieved,
these devices would rely on doped Si waveguides (WGs), requiring an increased
optical power to overcome the additional optical losses introduced by
dopants[38], when compared to undoped Si WGs. Other modulation mechanisms in
Si can be used, such as the thermo-optic effect[39, 20], changing the Si
optical properties via electrically induced temperature changes. The thermal
time constant of Si is$\sim 1$ms[40] at RT, limiting operating speeds to the
kHz range[20]. Non-linear effects, such as the Kerr effect[12], produce a
change in the refractive index proportional to the product of the nonlinear
refractive index and the intensity of the propagating light[12]. However, at
telecom wavelengths (1.3, 1.5$\mu$m) this is$\sim$3 orders of magnitude weaker
than the plasma dispersion effect[38]. Thus, new materials with higher
nonlinear refractive index are needed.

Hybrid approaches that incorporate III-V compounds[21, 41, 22, 23] with doped
Si WGs reduce $V_{\pi}L$ by utilising other effects, such as band-filling[42],
which results in reduced absorption due to occupied energy states[43].
III-V/Si metal-oxide-semiconductor (MOS) Mach-Zehnder modulators (MZM)
operating in accumulation mode[44], which rely on the change in accumulated
charge carriers within the MOS capacitor by applying a gate voltage, have the
lowest $V_{\pi}L\sim$0.047 V$\cdot$cm to date[22], with IL$\sim$1dB[22], but
are BW limited to$\sim$100MHz[22]. III-V/Si MOS MZMs struggle to maintain
$V_{\pi}L=0.047$Vcm with a higher BW, because of the high
($\sim$3$k\Omega\mu$m[22]) contact resistance[22] to the Si electrode in the
MOS configuration, with more moderate values of
$V_{\pi}L$=0.24-0.3V$\cdot$cm[21, 23] for III-V MZMs operating in depletion
mode with BW up to$\sim$27GHz[21]. III-V based MZMs offer a lower $V_{\pi}L$
compared to Si MZMs, but at the cost of more complex fabrication, with
expensive III-V processing[45]. Cost-effectiveness is determined by the cost
per unit Watt used to manufacture III-V devices, which is $\$$40/W to
$\$$100/W at 2018 prices[46, 47]. This is at least two orders of magnitude
higher than Si manufacturing[46, 47].

Integrating LiNbO3 (LN) on undoped Si WGs enables pure PM, exploiting the
Pockel’s effect[11], producing a change in refractive index proportional to
the electric field. Modulators based on sub-$\mu$m thin-film LN[24, 25, 48,
26, 27] have IL$<$1dB[24] and BW$>$100GHz[24, 25]. Thin-film LN MZMs were
reported with $V_{\pi}L\sim$1.4V$\cdot$cm[24], a factor of 2 larger than
state-of-the-art Si plasma-dispersion MZMs[49]. However, this $V_{\pi}L$ means
that cm long devices are needed to reduce $V_{\pi}$ to CMOS compatible
levels$<$1V[50]. Thin-film LN MZMs with $V_{\pi}L\sim$0.8Vcm[27] were
demonstrated in the visible range, but with IL$\sim$15dB[27]. Plasmonic LN
modulators show $V_{\pi}L\sim$0.23Vcm[28], but with IL$\sim$19.5dB[28].
Modulators with lower $V_{\pi}L$ and IL are essential to increase the density
of SiP integrated circuits, thus reducing power consumption by minimising
electrical interconnects. The interconnect losses are frequency ($f$)
dependent ($\propto\sqrt{f}$[51]) due to increased resistance caused by the
skin effect[51], where more of the current flows at the surface as $f$
increases[52]. Therefore, for electrical interfaces driving Tb/s data rates,
the power consumption of interconnects becomes the limiting factor[51, 10,
53].

Graphene is ideal for opto-electronics[54, 55, 56, 57] due to its high carrier
mobility ($\mu>$50,000cm2/Vs at room temperature, $RT$[58, 59]), electrically
tunable optical conductivity[60, 61], and wavelength independent absorption in
the visible (500nm) to mid-infrared (10$\mu$m)[62, 61]. The gapless band
structure with massless Dirac Fermions in single-layer graphene (SLG) enables
the optical conductivity to be electrostatically controlled[60, 61], and
absorption to be suppressed[63]. Double SLG (DSLG) phase modulators can reach
a theoretical $V_{\pi}L\sim$0.1V$\cdot$cm[64], which enables mm long devices
with driving voltages$<$1V. When absorption is suppressed, the optical loss
can be reduced by orders of magnitude from$>$1000 dB/cm[64] to$<$10dB/cm[64].
The combination of mm lengths and$<$10dB/cm optical losses, leads to IL$<$1dB,
therefore minimising power consumption. SLG can be produced at wafer scale[65,
56, 66]. Chemical vapour deposition (CVD) can be used to grow polycrystalline
films up to 30”[67] or single crystals[68]. The latter allows one to fabricate
devices at predetermined locations[29, 69]. SLG films can be integrated in the
CMOS back-end-of-line for wafer scale processing after fabrication of the
integrated circuits[70]. This can reduce cost and complexity of fabrication,
by removing the need for doped Si WGs in DSLG designs[71, 72, 30, 73, 29].
EAMs[74, 71, 73, 29, 31] and electro-refractive modulators[72, 30, 75, 32]
(ERMs) based on one or more SLG have been reported, with
$V_{\pi}L\sim$0.28$V\cdot$cm[32] and data transmission rates$\sim$50Gb/s[29].
However, pure PM has not been reported yet, to the best of our knowledge.

The SLG conductivity $\sigma(\omega)$, derived from the Kubo formula[76], is a
function of the angular frequency of light ($\omega$), SLG transport
relaxation time ($\tau$), SLG Fermi level ($E_{\text{F}}$), and temperature
$T$[77, 78, 79]:

$\begin{split}\sigma(\omega)=\frac{\sigma_{0}}{2}\left[\text{tanh}\left(\frac{\hbar\omega+2E_{\text{F}}}{4k_{B}T}\right)+\text{tanh}\left(\frac{\hbar\omega-2E_{\text{F}}}{4k_{B}T}\right)\right]\\\
-i\frac{\sigma_{0}}{2\pi}\text{log}\left[\frac{(\hbar\omega+2E_{\text{F}})^{2}}{(\hbar\omega-2E_{\text{F}})^{2}+(2k_{B}T)^{2}}\right]\\\
+i\frac{4\sigma_{0}}{\pi}\frac{E_{\text{F}}}{\hbar\omega+i\hbar/\tau}\end{split}$
(1)

where $\sigma_{0}=e^{2}/4\hbar$ is the $f$-independent, or universal
conductivity of SLG[80, 62], $\hbar$ is the reduced Planck’s constant, and
$k_{B}$ is the Boltzmann constant. The first two terms represent interband
transitions[81, 82]. The third represents intraband transitions[81, 82], and
it is a function of $\sigma_{0}$, $E_{\text{F}}$, $\omega$ and $\tau$. The
intraband contribution to $\sigma(\omega)$ can be simplified to express the DC
conductivity of SLG ($\sigma_{\text{d.c.}}$) when $\omega\to$0[57]. $\tau$ can
then be related to $\mu$ by using $\sigma_{\text{d.c.}}=ne\mu$[83], where $n$
is the carrier concentration given by $E_{\text{F}}=\hbar
v_{F}\sqrt{n\pi}$[84, 77, 78]. We thus arrive at $\mu\sim e\tau
v_{F}^{2}/E_{\text{F}}$[57] for $E_{\text{F}}\gg k_{B}T$, where $v_{F}\simeq
9.5\times 10^{5}\text{ ms}^{-1}$ is the Fermi velocity[81, 77, 78]. Eq.1
implies that $\sigma(\omega)$ of each SLG depends on $E_{\text{F}}$, and the
energy of the incident light ($E_{\text{in}}=hc/\lambda$). Absorption in
undoped SLG is dominated by interband transitions and is suppressed when
$2E_{\text{F}}>hc/\lambda$, due to Pauli blocking[63]. For
$\lambda$=1.55$\mu$m, or $E_{\text{in}}$=0.8eV, Pauli blocking occurs for
$E_{\text{F}}>$0.4eV.

For Pauli blocking, SLG enters the transparency regime, whereby interband
transitions are suppressed and only intraband transitions occur[63]. Intraband
transitions dominate for low energy photons ($\omega<$2000cm-1[61]) and for
$2E_{\text{F}}>hc/\lambda$. Intraband transitions are dependent on $\tau$
because they depend on scattering centres (e.g. defects) for conservation of
momentum[85]. Therefore, absorption by intraband transitions increases for
shorter $\tau$, which is related to mobility $\mu=\frac{e\tau
v_{F}^{2}}{E_{F}}$[57].

Operating beyond Pauli blocking is essential for pure PM, because in this
regime SLG absorption is minimised and constant with respect to gate voltage,
thus reducing the overall IL. A DSLG modulator can work as EAM or ERM
depending on bias[86]. For EAMs, the onset of Pauli blocking results in the
largest change in absorption[74], hence the bias should be set at the onset of
Pauli blocking. For ERMs, the bias is set beyond the Pauli blocking condition,
where the change in refractive index is quasi-linear[75] and absorption is
minimised[64].

Here, we report DSLG-based MZMs on undoped Si WGs operating beyond Pauli
blocking with $V_{\pi}L\sim$0.3V$\cdot$cm and pure PM. These work at 16V
without dielectric breakdown, enabling access to the transparent regime. This
work represents a key step in the development of graphene-based coherent
integrated transmitters for communication systems.

Figure 1: (a) DSLG modulator scheme, with 20nm Al2O3 between bottom and top
SLG. (b) Equivalent circuit, used to calculate $Z_{\text{T}}(\omega)$,
$Z_{\text{C}}(\omega)$ and 3dB cut-off BW $f_{3dB}$. The components contained
within the dashed line contribute to the impedance of the overlapping SLG
section. (c) Simulated $E_{x}$ of fundamental TE mode confined within a
550x220nm2 Si WG at 1.55$\mu m$. Colour scale indicates the $E_{x}$ amplitude,
scale bar 200nm. (d) Simulated change in n${}_{\text{eff}}$ (blue) and optical
loss (orange) of confined mode due to different $V$-$V_{\text{CNP}}$ across
the DSLG capacitor. Simulations are performed at 1.55$\mu$m ($\sim$0.8eV).
SLGs are separated by 20nm Al2O3. The overlapping SLG region is 550nm, the
ungated SLG region is 1$\mu$m ($E_{\text{F}}$=0.2eV). (e) Simulated
$f_{\text{3\text{dB}}}$ as a function of $\mu$ for a 50$\mu m$ DSLG modulator
with ungated sections of each SLG$\sim$1$\mu$m ($E_{\text{F}}\sim$0.23eV),
gated sections of each SLG$\sim$450nm ($E_{\text{F}}\sim$0.4eV).
$R_{C}\sim$1000$\Omega\cdot\mu$m, 20nm Al2O3 with $\epsilon_{\text{r}}$=8,
$C_{\text{eq}}$ calculated with an additional carrier concentration$\sim
10^{10}\text{cm}^{-2}$ from defects and$\sim 10^{11}\text{cm}^{-2}$ from
charged impurities. $\mu$ calculated at 0.4eV. (f) Simulated $f$ response of
DSLG modulator for different $\mu$ for the same modulator specification as
(e).

##  I Results and Discussion

The design of our DSLG phase modulator is in Fig.1a. It consists of two SLG
encapsulated by Al2O3, overlapping in the region above the Si WG. 10nm Al2O3
encapsulates both SLGs to protect them during subsequent processing steps,
minimise contamination, and preserve $\mu$. The bottom encapsulation is used
to maintain symmetry between the two SLGs, so that both are in the same
environment. Each SLG is contacted by a metal placed on either side of the WG.
The two SLG layers form a capacitor (equivalent electrical circuit in
Fig.1b.), where an applied voltage across the contacts creates a perpendicular
electric field which modulates the carrier density, thus $\sigma(\omega)$ of
each SLG. This, in-turn, modulates the complex effective refractive index,
$n_{\text{eff}}$[82, 81], leading to a change in phase and absorption of light
along the propagation direction[12]. We simulate the optical performance of
our DSLG modulators using the Finite-Difference Eigenmode (FDE) solver in
Lumerical[87]. This uses the expansion method to calculate the eigenmodes and
eigenvalues of Maxwell’s equations in the $f$ domain[88]. Each solution, or
mode, has its own electromagnetic field profile and $n_{\text{eff}}$[12]. The
real component is the effective refractive index, $n_{\text{eff}}$, related to
the phase, $\phi$, of the light along $L$ by $\phi=k_{0}n_{\text{eff}}L$[12],
where $k_{0}$ ($2\pi/\lambda_{0}$) is the wavenumber in free-space. The
imaginary component is the extinction coefficient, $\kappa$, related to
$\alpha$, and $\lambda_{0}$ as $\alpha=\frac{4\pi}{\lambda_{0}}{\kappa}$[12].

The simulated light propagation along the WG shows the electric field profile,
Fig.1c, of the propagating mode. The mode interaction with SLG is increased by
maximizing the overlap between SLG and field profile. The metal contacts for
each SLG are placed 1$\mu$m away from the WG edge to avoid optical losses due
to proximity of field profile and contacts. The EO response is simulated by
varying $E_{\text{F}}$ between $0-0.8$eV and extracting the change in
$n_{\text{eff}}$ as a function of $E_{\text{F}}$. We use
$E_{\text{F}}\sim$0.2eV for ungated SLG, to account for the impurity doping of
as prepared SLG[32, 69]. Simulations are performed at 300K for
$\lambda$=1.55$\mu$m with $\tau$=440,22,11fs, corresponding to
$\mu\sim$10,000, 500, 250cm2/Vs at $E_{\text{F}}\sim$0.4eV. We then calculate
the phase shift $\Delta\phi=k_{0}\Delta n_{\text{eff}}L$ and optical losses
$I/I_{0}=e^{-\frac{4\pi}{k}L}$ induced by SLG for a given $L$[12]. We relate
$E_{\text{F}}$ to the applied voltage, $V$, by considering the sum of the
potential applied across the overlapping SLG regions and the surface voltage
due to the accumulated charges at each SLG electrode[50, 64]:

$|V-V_{\text{CNP}}|=\frac{e}{C_{\text{{eq}}}}\frac{1}{\pi}\left(\frac{E_{\text{F}}}{\hbar
v_{F}}\right)^{2}+2\frac{|E_{\text{F}}|}{e}$ (2)

The potential across the overlapping SLG region is related to the total number
of accumulated charges, $n_{\text{tot}}$, and the equivalent capacitance,
$C_{\text{eq}}$ of the overlapping SLG region. $C_{\text{eq}}$ is the series
combination of the quantum capacitance[89], $C_{Q}$, of each SLG and the
capacitance of the parallel-plate geometry, $C_{\text{ox}}$. The equivalent
electrical circuit of the DSLG modulator is in Fig.1b.
$C_{Q}=2e^{2}\sqrt{n_{\text{tot}}}/\hbar v_{F}\sqrt{\pi}$[90, 91] and
$C_{\text{ox}}=\epsilon_{r}\epsilon_{0}/d$, where $d$ is the thickness of the
gate oxide separating the two SLG, $\epsilon_{r}$ is the relative permittivity
of Al2O3, and $\epsilon_{0}$ is the permittivity of free space. The dielectric
constant of Al2O3 is measured with a Woollam Ellipsometer M-2000 as
$\epsilon_{r}\sim 8$. To account for charged impurities at the SLG-Al2O3
interface and the impurities introduced during growth or device fabrication,
we model the total charge density $n_{\text{tot}}$ as the sum of a carrier
concentration from electrostatic doping $n(V_{G})$, where VG is gate voltage,
and an additional concentration from charged impurities $n_{\text{imp}}$[90].
A charged-impurity density$\sim 10^{12}\text{cm}^{-2}$[92], leads to an
increase in carrier concentration of SLG$\sim 10^{11}\text{cm}^{-2}$[90].

Fig.1d plots the simulated EO response at 1550nm in terms of $\Delta
n_{\text{eff}}$ and associated optical losses per $\mu$m with increasing
$V_{G}$ and $E_{\text{F}}$. Optical losses decrease when $E_{\text{F}}>$0.2eV,
corresponding to intraband transitions and the onset of Pauli blocking. For
$E_{\text{F}}>$0.6eV, SLG enters the transparency regime where interband
transitions are blocked such that optical losses are minimised and do not
change as $E_{\text{F}}$ is further increased. $\Delta n_{\text{eff}}$ changes
sign with increasing $V_{G}$, giving a positive or negative $\Delta\phi$ for
the modulated signal. A bias voltage can be applied to the DSLG modulator to
define the operating point on the EO response curve in Fig.1d. The amplitude
of the driving voltage defines the operating range around the operating point.
The ideal working point for pure PM is in the transparent region where $\Delta
n_{\text{eff}}$ changes quasi-linearly, whilst optical losses remain constant.
This also minimises power consumption because optical losses are at their
lowest. Optical losses depend on $\tau$, as plotted in Fig.1d for $\tau=$440,
22, 11fs. Low scattering time $\tau$ is associated with high scattering rate
$\Gamma$ via $\tau=\hbar/\Gamma$[77], leading to increased absorption via
intraband transitions and reduced absorption via interband transitions. As
$E_{\text{F}}$ approaches 0.4eV, optical losses are reduced for a lower $\tau$
because absorption via interband transitions is reduced. However, in
transparency, increased intraband transitions lead to optical losses over 3
times greater for $\tau$=11fs, when compared to 440fs.

Figure 2: (a) DSLG modulator fabrication schematic: 10nm Al2O3 deposition on
Si WG with 20nm Al2O3 between bottom and top SLG. 10nm Al2O3 is used to
encapsulate both SLGs to protect them during subsequent processing steps,
minimise contamination, and preserve $\mu$. The bottom encapsulation is used
to maintain symmetry between the two SLGs, so that both are in the same
environment. (b) Raman spectra at 514nm for the SLG closest to the WG (SLG1)
and that farthest from the WG (SLG2), as-grown on Cu, after transfer, after
device fabrication. The spectra are normalised to I(G), with Cu background PL
removal[95]. (c) Schematic of SLG (grey) on SiO2 (green) top-gated Hall bar
device with Ni/Au (yellow) contacts. d) Measured RS (black) and calculated
$\sigma_{\text{d.c.}}$ ($\sigma_{\text{d.c.}}=1/R_{S}$) (blue). Red dashed
line is the $\sigma_{\text{d.c.}}$ linear fit for$V>$0, showing the transition
from the linear to sub-linear regime for $V>$5V.

The speed of the DSLG phase modulator is defined by the cut-off frequency,
$f_{3\text{dB}}$, at which the power of the modulated signal has decreased by
half (3dB)[14]. The dominant factor that limits $f_{3\text{dB}}$ is the
product of the circuit resistance, $R$, and capacitance, $C$, known as the RC
response[51]. We estimate this by electrical modelling, considering the
different contributions to the total circuit impedance $Z_{\text{T}}(\omega)$,
coming from each contact, $R_{C}$, ungated SLG sections, $R_{\text{ungated}}$,
and gated SLG sections, $R_{\text{gated}}$. The equivalent circuit, Fig.1b,
contains these components in series:

$Z_{\text{T}}(\omega)=2(R_{\text{C}}+R_{\text{{ungated}}})+Z_{\text{C}}(\omega)$
(3)

Where $Z_{\text{C}}(\omega)$ is the impedance of the overlapping SLG regions.
$Z_{\text{C}}(\omega)$ is given by $C_{\text{eq}}$ in series with
$R_{\text{{gated}}}$ for each SLG electrode. The resistance ($R$) of SLG can
be related to the sheet resistance ($R_{S}$) of SLG as $R=LR_{S}/w$ [54] where
$L$ and $w$ the length and width of SLG, respectively. By considering
$\omega\to$0, $R_{S}$ can be related to the electrical conductivity
$\sigma_{\text{d.c.}}$, as $R_{S}=1/\sigma_{\text{d.c.}}$[83]. $R_{S}$ is
calculated for different $E_{\text{F}}$ and $\tau$ from
$\sigma_{\text{d.c.}}=ne\mu$[54]. $R_{S}$ depends on $E_{\text{F}}$, therefore
on the voltage applied across the DSLG modulator. From Ohm’s law and
$Z_{\text{T}}(\omega)$, we calculate the frequency dependent current,
$I(\omega)$, flowing through the circuit at a nominal drive voltage,
$V_{\text{D}}$, $I(\omega)=V_{\text{D}}/Z_{\text{T}}(\omega)$. The voltage
drop across the DSLG modulator, $V_{\text{M}}$, is
$V_{\text{M}}(\omega)=I(\omega)/j\omega C_{\text{eq}}$. When $I(\omega)$
starts to flow and $V_{\text{M}}(\omega)$ goes to 0, the modulator is no
longer operational. Therefore, by substituting $I(\omega)$ into
$V_{\text{M}}(\omega)$, we get:

$\frac{V_{\text{M}}(\omega)}{V_{\text{D}}}=\frac{1}{1+j\omega RC_{\text{eq}}}$
(4)

Samples | SLG1 | Encapsulated SLG1 | SLG2 | Encapsulated SLG2
---|---|---|---|---
Pos(G) (cm-1) | 1592 $\pm$ 3 | 1596 $\pm$ 1 | 1596 $\pm$ 1 | 1585 $\pm$ 5
FWHM(G) (cm-1) | 14 $\pm$ 3 | 12 $\pm$ 2 | 11 $\pm$ 2 | 17 $\pm$ 2
Pos(2D) (cm-1) | 2692 $\pm$ 3 | 2691 $\pm$ 1 | 2694 $\pm$ 2 | 2689 $\pm$ 2
FWHM(2D) (cm-1) | 31 $\pm$ 1 | 30 $\pm$ 1 | 29 $\pm$ 3 | 30 $\pm$ 4
A(2D)/A(G) (cm-1) | 2.2 $\pm$ 0.4 | 5.6 $\pm$ 1.7 | 2.7 $\pm$ 0.1 | 6.3 $\pm$ 0.7
I(2D)/I(G) (cm-1) | 2.7 $\pm$ 0.7 | 2.4 $\pm$ 1 | 1.8 $\pm$ 0.1 | 3.5 $\pm$ 0.5
I(D)/I(G) (cm-1) | 0.07 $\pm$ 0.03 | 0.03 $\pm$ 0.04 | 0.05 $\pm$ 0.04 | 0.13 $\pm$ 0.13
EF (meV) | 190 $\pm$ 80 | 276 $\pm$ 158 | 292 $\pm$ 87 | 180 $\pm$ 130
Doping type | p | p | p | p
n (x 1012) (cm-2) | 2.6 $\pm$ 2.0 | 8.5 $\pm$ 9.5 | 5.8 $\pm$ 3.2 | 3.9 $\pm$ 3.8
Uniaxial strain ($\%$) | -0.20 $\pm$ 0.32 | -0.15 $\pm$ 0.18 | -0.07 $\pm$ 0.17 | 0.08 $\pm$ 0.07
Biaxial strain ($\%$) | -0.08 $\pm$ 0.14 | -0.06 $\pm$ 0.07 | -0.02 $\pm$ 0.06 | 0.003 $\pm$ 0.02
nD (x 1010) (cm-2) | 2.6 $\pm$ 0.4 | 1.6 $\pm$ 0.9 | 2.5 $\pm$ 0.6 | 4.2 $\pm$ 2.7
Table 2: Raman fit parameters and corresponding EF, doping type, $n$, strain,
nD, and error bars.

The simulated $f$ response of our DSLG modulators is in Fig.1e.
$f_{\text{3\text{dB}}}$ increases with $\mu$ in Fig.1f due to a reduction in
$R_{\text{ungated}}$ and $R_{\text{gated}}$ for SLG with higher $\mu$.
Assuming a constant $\mu$, $f_{\text{3\text{dB}}}$ can be increased by
reducing $C$ and $R$. However, there is a trade-off between minimising ungated
SLG length, to reduce $R$, and minimising the gated SLG length, to reduce $C$.
Even though $R_{\text{ungated}}>R_{\text{gated}}$, $R_{\text{gated}}$ is of
the same order of magnitude as $R_{C}$. $f_{\text{3\text{dB}}}$ can be further
increased by minimising the distance between contacts and WG, to minimise the
impact from ungated regions. There is a trade-off between minimising the
required $V$ to reach Pauli blocking and maximising $f_{\text{3\text{dB}}}$.
To reduce $V$, $C_{\text{ox}}$ should be maximised by using a dielectric with
the highest $\epsilon_{r}$ or reducing $d$. However, to increase
$f_{\text{3\text{dB}}}$, $C_{\text{ox}}$ should be reduced by increasing $d$
and reducing the size of overlapping SLG region. We limit the size of
overlapping SLG to the WG width and use 20nm Al2O3 to maximise
$f_{\text{3\text{dB}}}$ and limit $V<$15V. To operate in the transparency
regime, the dielectric should support the required $V$ to reach
$E_{\text{F}}>$0.4eV without breakdown. Minimising the size of the overlapping
SLG region, we reduce the probability of breakdown due to pinholes in the
dielectric.

The DSLG modulators are then fabricated as for Fig.2a. We use the IMEC
silicon-on-insulator (SOI) platform because of the low (2.3dB per grating)
coupling losses[93]. 10nm Al2O3 is deposited on SOI by atomic layer deposition
(ALD, Cambridge Nanotech Savannah S100 G1) at 120oC. After a 10min purge of N2
for contaminants removal, we apply 238 consecutive cycles of 22ms pulses of
deionized water and 17 ms pulses of trimethylaluminum precursors to reach the
required 10nm thickness, as measured with a Woollam Spectroscopic Ellipsometer
M-2000XI. Continuous SLG is grown on Cu by chemical vapor deposition (CVD).
The Cu foil is first annealed at 1050oC under 90% H2 and 10% Ar at 760torr for
2h and cooled to $RT$. To grow SLG, the annealed Cu foil is heated to 1050oC
with 40sccm H2 at 0.4Torr and annealed for 2h. Growth is initiated by
introducing 5sccm CH4 and the CH4 flow is stopped to terminate growth after
30mins, and SLG/Cu is cooled to $RT$[94]. SLG is then wet-transferred using
polymethyl-methacrylate (PMMA) as a supporting layer and Cu etching in
ammonium persulfate[65].

Figure 3: (a) Optical micrograph of EAM consisting of DSLG modulator on
straight Si WGs. (b) EO response of a 75$\mu$m DSLG EAM showing the different
regimes depending on $E_{\text{F}}$. At 1.55$\mu$m, transmission is lowest in
region I (white), when $E_{\text{F}}<0.4$eV. It increases in region II (blue)
due to the onset of Pauli blocking, when $E_{\text{F}}$ approaches 0.4eV,
before transitioning to the minimum-loss regime in region III (grey)$>$10V,
where $E_{\text{F}}>$0.4eV. The Si WG is 450x220nm2. (c) EO frequency response
for a 50$\mu$m DSLG EAM at different DC biases with a 1V peak-to-peak driving
voltage. $f_{\text{3\text{dB}}}\sim$13GHz for 3V, 24GHz for 5V, 12GHz for 7V,
and 8GHz for 11V.

As-grown and transferred SLG are characterised by Raman spectroscopy with a
Renishaw InVia spectrometer equipped with 50x objective at 514.5nm. 6 spectra
are collected from both as grown SLG on Cu and transferred SLG to estimate
doping and defect density. The errors are calculated from the standard
deviation across different spectra, the spectrometer resolution ($\sim$1cm-1)
and the uncertainty associated with the different methods to estimate the
doping from full width at half maximum of G-peak, FWHM(G), intensity and area
ratios of 2D and G peaks, I(2D)/I(G), A(2D)/A(G). Table 2 summarises Raman
peaks fit are derived EF, doping type, charge carrier density $n$, strain, and
defects density nD. The Raman spectrum of as grown SLG is in Fig.2b, after Cu
photo-luminescence removal[95]. The 2D peak is a single-Lorentzian with
FWHM(2D)=27$\pm$2cm-1, signature of SLG[96, 97]. Pos(G)=1591$\pm$4cm-1 with
FWHM(G)=16$\pm$2cm-1. Pos(2D)=2712$\pm$9cm-1, I(2D)/I(G)$\sim$4.5$\pm$0.7 and
A(2D)/A(G)7.6$\pm$1.1. No D peak is observed, indicating negligible Raman
active defects[98, 99]. First, SLG1 is transferred on 10nm Al2O3 deposited on
SOI. The representative Raman spectrum of transferred SLG1 before Al2O3
encapsulation is in Fig.2b. The 2D peak retains its single-Lorentzian line
shape with FWHM(2D)=31$\pm$1cm-1, Pos(G)=1592$\pm$3cm-1, FWHM(G)=14$\pm$3cm-1,
Pos(2D)=2692$\pm$3cm-1, I(2D)/I(G)=2.7$\pm$0.7, and A(2D)/A(G)=2.2$\pm$0.4
indicating a p-doping with EF=190$\pm$80meV[84, 100]. I(D)/I(G)=0.07$\pm$0.03
corresponds to a defect density n${}_{D}\sim 2.6\pm$0.4x1010[101] for
excitation energy of 2.41eV. SLG1 is then patterned by electron beam
lithography (EBL) using a Raith EBPG5200, followed by a 60s O2 plasma at 10W
using a Vision 320 reactive ion etcher (RIE). Contacts are fabricated using a
double-layer resist mask of Methyl methacrylate and PMMA[102], followed by
15/50nm Ni/Au deposited by sputter coating (Precision Atomics Metallifier
Sputter Coater) and thermal evaporation (M-Braun PROvap PVD system). A 1nm
seed-layer of Al is then thermally evaporated, before 20nm of Al2O3 is
deposited by ALD at 120oC on SLG1. After Al2O3 encapsulation, the 2D peak in
SLG1 retains its single-Lorentzian line shape with FWHM(2D)=30$\pm$1cm-1,
Pos(G)=1596$\pm$1cm-1, FWHM(G)=12$\pm$2cm-1, Pos(2D)=2691$\pm$1cm-1,
I(2D)/I(G)=2.4$\pm$1, and A(2D)/A(G)=5.6$\pm$1.7 indicating a p-doping with
EF=276$\pm$158meV[84, 100]. I(D)/I(G)=0.03$\pm$0.04 corresponds to
nD=1.6$\pm$0.9x1010[101] for 2.41eV excitation. SLG2 is transferred using the
same process as SLG1, and characterized by Raman spectroscopy (Fig.2b). The 2D
peak retains its single-Lorentzian line shape with FWHM(2D)=29$\pm$3cm-1.
Pos(G)=1596$\pm$1cm-1, FWHM(G)=11$\pm$2cm-1, Pos(2D)=2694$\pm$2cm-1,
I(2D)/I(G)=1.8$\pm$0.1, and A(2D)/A(G)=2.7$\pm$0.1 indicating a p-doping with
EF=292$\pm$87meV[84, 100]. I(D)/I(G)=0.05$\pm$0.04 corresponds to
nD=2.5$\pm$0.6x1010[101] for 2.41eV. SLG2 is then patterned by using O2 plasma
after EBL and contacts are fabricated using a double-layer resist mask for EBL
as SLG1, and subsequent Ni/Au (15/50nm) deposition. Finally, 10nm Al2O3 is
deposited on SLG2 after a 1nm Al seed-layer is thermally evaporated. After
Al2O3 encapsulation, the 2D peak in SLG2 retains its single-Lorentzian line
shape with FWHM(2D)=30$\pm$4cm-1, Pos(G)=1585$\pm$5cm-1, FWHM(G)=17$\pm$2cm-1,
Pos(2D)=2689$\pm$2cm-1, I(2D)/I(G)=3.5$\pm$0.5, and A(2D)/A(G)=6.3$\pm$0.7
indicating a p-doping with EF=180$\pm$130meV[84, 100]. I(D)/I(G)=0.13$\pm$0.13
gives nD=4.2$\pm$2.7x1010[101] for 2.41eV. SLG1 and SLG2 show different doping
and defect density even though they are transferred from the same SLG/Cu
because SLG1 is subject to more fabrication steps than SLG2. Strain is
estimated from Pos(G)[103, 104]. Biaxial strain can be differentiated from
uniaxial by the absence of G-peak splitting with increasing strain, however at
low ($\leq$0.5%) strain the splitting cannot be resolved. For uniaxial
(biaxial) strain, Pos(G) depends on both EF and strain[84, 103], to get the
contribution of strain only, we first derive EF from A(2D)/A(G), I(2D)/I(G)
and FWHM(G), which are independent of strain[84, 100, 63], and then calculate
Pos(G) corresponding to this EF. The strain is then retrieved from the
difference between the experimental and calculated Pos(G), Table 2.

A 4-point-probe measurement using top-gated Hall bar structures (Fig.2c) is
performed to derive SLG resistance and conductivity. Fig.2d plots the measured
voltage-dependent resistance and the calculated $\sigma_{\text{d.c.}}$ after
normalising the conductance to the channel geometry. We observe the
expected[105] peak in resistance, which corresponds to the SLG Dirac point.
$\mu$ is estimated from the measured conductivity as
$\sigma_{\text{d.c.}}=ne\mu$[83, 105], where the linear region of
$\sigma_{\text{d.c.}}$ in Fig.2d corresponds to a constant $\mu$. The charge
density $n$ in terms of $C_{\text{ox}}=\epsilon\epsilon_{\text{0}}/d$ can be
written as $n=C_{\text{ox}}(V_{\text{G}}-V_{\text{CNP}})/e$[105], hence the
conductivity becomes
$\sigma_{\text{d.c.}}=\epsilon\epsilon_{\text{0}}\mu(V_{\text{G}}-V_{\text{CNP}})/d$.
Using measured dielectric constant and thickness of $Al_{2}O_{3}$
$\epsilon_{\text{Al2O3}}=8$ and d=20nm, the measured $\sigma_{\text{d.c.}}$
can be fitted to estimate $\mu\sim 1000cm^{2}/Vs$. The linear fit to SLG
conductivity as a function of VG is in Fig.2d.

Figure 4: (a) Optical micrograph of MZM consisting of a DSLG modulator on each
arm of a Si MZI. Scale bars 100 $\mu$m. (b) Voltage-dependent transmission of
a MZM containing one 450$\mu m$ DSLG modulator on each arm, one biased at 10V
and the other swept from 4V (orange) to 10V (red). (c) Voltage-dependent shift
of interference fringe position (blue) and ER (orange) of an MZM with
a$\sim$100$\mu$m DSLG modulator on each arm, with one modulator biased at 10V.
The MZI is unbalanced, with 2 input and output ports, and 620x220 nm2 Si WGs.
(d) Comparison of measured (solid circles) and simulated (open circles)
$\Delta n_{\text{eff}}$ (blue) and $\Delta\alpha$ (orange) for a 450$\mu$m
DSLG MZM. Simulation performed at 1.55$\mu$m with $\tau$=14fs with same
structure as the measured device, 550x220 nm2 Si WG, overlapping SLG
region$\sim$550nm, ungated SLG region$\sim$1$\mu$m ($E_{\text{F}}$=0.2eV),
20nm Al2O3 with $\epsilon_{\text{Al2O3}}$=8.

The EO response of DSLG EAMs and ERMs are then measured using angled single-
mode optical fibres to couple light into the photonic circuits via grating
couplers. A representative EAM optical microscopy image in Fig.3a. The
position of the fibres and the polarisation of the source laser (Agilent 8164B
Lightwave Measurement System) are adjusted to minimise coupling losses and
maximise the power coupled into the confined optical mode. The steady-state
response is measured by applying a DC voltage across both SLGs and measuring
the optical power at the output, $P_{\text{out}}$. The transmitted power,
$P_{\text{t}}=10\log(P_{\text{out}}/P_{\text{in}})$, is expressed in dB.
Fig.3b is the optical transmission of a DSLG EAM comprising a 75$\mu$m
modulator on a straight WG. The EAM is biased between -10 and 17V at
1.55$\mu$m with $P_{\text{in}}$=1mW. To extract IL, $P_{\text{t}}$ needs to be
normalised to account for the additional propagation and coupling losses
introduced from processing. The increase in power loss compared to the loss
before processing is due to the deposited Al2O3 on the grating couplers,
residues from SLG transfer and device fabrication. The additional losses can
be subtracted by measuring the transmission through a similar WG with same
dimensions and grating couplers, that has undergone the same processing steps
as the DSLG modulator. The lowest VG dependent transmission in EAM occurs in
region I in Fig.3b, between -5 and 0V, when $E_{\text{F}}$ is less than the
half of the photon energy, $\hbar\omega_{\text{1550}}/2\sim 0.4$eV. Therefore,
interband transitions in SLG are allowed in region I. The transmitted power is
minimum in this voltage range around $V_{\text{CNP}}$ where EF is closer to
Dirac point[61]. For intrinsic SLG, where $V_{\text{CNP}}$ coincides with
$V_{\text{G}}$=0, the transmission curve would be centred at 0V. In Fig.3b the
$V_{\text{CNP}}$ is$\sim$-2.5V. This corresponds to $E_{\text{F}}\sim$274meV,
which represents the average EF of both SLGs in the DSLG EAM. This is also
consistent with the average SLGs $E_{\text{F}}=275\pm 170$meV, estimated from
Raman spectroscopy of SLG1 and SLG2 after $Al_{2}O_{3}$ encapsulation, Table
2, Fig.2b. As $V_{\text{G}}$ increases in region II, $E_{\text{F}}$ approaches
$\hbar\omega_{\text{1550}}/2$, and transmission increases due to Pauli
blocking of inter-band transitions. For $V_{\text{G}}>$10V, the transmission
plateaus when $E_{\text{F}}>\hbar\omega_{\text{1550}}/2$ and SLG enters the
transparent regime.

In transparency, IL$\sim$5.6dB for the DSLG phase modulator, corresponding to
loss$\sim$746dB/cm when normalised by the modulator length. This is higher
than state-of-the-art Si depletion ($\sim$22dB/cm[18]), III-V
($\sim$19dB/cm[22]), LN ($\sim$0.25dB/cm[24]), and SLG ($\sim$236dB/cm[32])
MZMs, Table 1. We attribute the additional optical losses to scattering from
resist residues and defects generated in SLG during fabrication, degrading
$\tau$. The simulated optical loss of the same device structure
is$\sim$93dB/cm for $\tau$=440fs. Therefore, IL can be further reduced by
improving SLG processing, increasing $\mu$, hence reducing short-range
scattering, and by developing a selective planarization process which isolates
passive sections of the WGs. The size of overlapping SLG regions can be
increased to further reduce IL, as the ungated SLG sections are not in
transparency, hence contributing optical losses. However, any increase in the
overlapping SLG region, will increase $C_{\text{eq}}$, therefore will reduce
$f_{\text{3\text{dB}}}$. Increasing the overlapping SLG region
to$\sim$1$\mu$m, would decrease optical loss to$<$10dB/cm for $\tau$=440fs,
leading to IL$<$1dB for$\sim$3mm devices, matching IL of LN[60] and III-V[22]
MZMs, Table 1.

The EO BW, or speed, is then measured by applying a sinusoidal voltage to the
DSLG modulators, in either EAM or ERM configuration. The voltage is provided
by a signal generator (Agilent E8257D PSG) combined with a DC voltage via a
bias tee. The optical output from the DSLG modulator is then amplified with an
Er doped fibre amplifier (EDFA, Keyopsys CEFA-C-HG) followed by a 1nm narrow-
band optical filter, before going into a InGaAs photodetector (PD) with a
BW$>$40GHz (Newport 1014). The narrow-band filter is used to remove the noise
resulting from the spontaneous emission from the EDFA[106], and to ensure that
the PD input power is below the safe input power=5mW given by the
specifications of the Newport PD[107]. The modulated output signal is recorded
on an electrical spectrum analyser (ESA, Agilent PSX N9030A). By monitoring
the amplitude of the modulated signal with increasing $f$, we get
$f_{\text{3\text{dB}}}$. The setup is calibrated by repeating the measurements
with the same configuration, but with a Thorlabs LN05S-FC AM with a 3dB cut-
off$\sim$40GHz. A final normalisation is then done for the $f$ response of the
Thorlabs LN05S-FC modulator, taken from the supplied data sheet[108]. Fig.3c
is the $f$ response of a 50$\mu$m DSLG EAM for different DC biases.
$f_{\text{3\text{dB}}}$ increases from 13 to 25GHz between 3 and 5V, then
decreases to 12GHz for 7V, and 8GHz for 11V. We attribute the decrease in
$f_{\text{3\text{dB}}}$ above 5V to a reduction in $\mu$ due to increased
short-range scattering of charge carriers as $E_{\text{F}}$ increases. For our
RC limited devices, we expect $f_{\text{3\text{dB}}}$ to increase with $V$,
because $R_{S}$ reduces with increasing $V$[105], up to an optimum point after
which the increase of $C_{q}$ becomes predominant. The optimal bias point for
operating our device at 25 GHz BW is$\sim 5V$, whilst operating it at CMOS-
compatible voltages ($<$2V) allows 13GHz, Fig.3c. We assign the
$V_{G}$-dependent slow-down in Fig.3c to a decrease in $\mu$ above 5V due to
increased short-range scattering as VG increases. This would lead to a
$f_{\text{3\text{dB}}}$ slow-down of the same order of magnitude as that
measured between 5 and 11V, where $f_{\text{3\text{dB}}}$ drops from 25 to
8GHz. This contrasts the increase from 13 to 25GHz between 3 and 5V, where we
are still in the linear region of $\sigma_{\text{d.c.}}$, and benefit from
decreasing $R_{S}$. The transition to sub-linear behaviour can be pushed to
higher VG by decreasing the sources of short-range scattering in SLG, from SLG
processing improvements. Thus, $f_{\text{3\text{dB}}}$ can be increased by
improving SLG growth and transfer, to limit $\mu$ degradation during
fabrication.

The simultaneous phase change that accompanies the change of amplitude cannot
be extracted from an electro-absorption configuration, because the
transmission of a straight WG is independent of optical signal phase[12].
Instead, it is measured using an electro-refractive configuration, with a
Mach-Zehnder interferometer (MZI)[12, 109]. The optical microscopy image of a
representative MZI is in Fig.4a. Here, the optical signal is split into two
arms. Depending on the phase difference between these, $\Delta\phi$, the
propagating waves will interfere when recombined. If the propagating waves are
in phase, transmission will not depend on $\lambda$[12]. If they are not in
phase, an interference pattern will appear[12]. This is characterised by the
free spectral range ($FSR=\frac{\lambda_{\text{res}}^{2}n_{g}}{\Delta
L}$[109]), defined as the wavelength difference between each transmission
minima, where $\lambda_{\text{res}}$ is the fringe position, $n_{g}$ is the
group index, and $\Delta L$ is the length difference between the arms of the
MZI. $\lambda_{\text{res}}$ depends on $\Delta\phi$, which can result from a
modulator that induces $\Delta n_{\text{eff}}$, or when the MZI arms are
different lengths, known as an unbalanced MZI[13]. A MZM uses an ERM on one or
both MZI arms to control $\Delta\phi$. $\Delta\phi$ can then be directly
measured by the shift of the output interference pattern[12]. Placing an ERM
on each arm enables the phase to be controlled independently on each arm to
reach the required $\Delta\phi$. Fig.4b shows the V-dependent transmission of
a MZM with a 450$\mu$m DSLG modulator on each arm of an unbalanced MZI. One
device is biased at 10V, so that it is in the transparency regime, and the
other is swept from 4 to 10V. By measuring the fringe shift, $\Delta\lambda$,
for different VG, we determine $\Delta\phi$. The measured $\Delta\lambda$ is
normalised by the FSR, which corresponds to a phase difference of 2$\pi$[12],
giving $\Delta\phi$ in units of $\pi$:
$\Delta\phi[\pi]=\Delta\lambda/(\text{FSR}/2)$[72]. $\Delta\phi$ is related to
the V-induced change in the real component of $n_{\text{eff}}$ along L, by
$\Delta\phi=k_{0}\Delta n_{\text{eff}}L$[12].

The extinction ratio (ER)=$10\log(P_{\text{t,max}}/P_{\text{t,min}})$[13] is
affected by the difference in absorption between the MZI arms, $\Delta\alpha$.
If the propagating wave in one arm is absorbed, there is no interference at
the output, because there will only be one propagating wave remaining[12]. For
losses that do not result in complete absorption, ER will increase when
$\Delta\alpha$ is minimised, and decrease when $\Delta\alpha$ is maximised.
The MZM ER can be related to $\Delta\alpha$ by considering the transmission
through the MZM as the sum of the electric fields propagating down each MZI
arm: $E\propto e^{-\alpha}\sin({\omega t-\phi})$[12, 109]. The ER is given by
the ratio of the maximum and minimum transmission through the MZM, which
occurs when $\Delta\phi=0$ and $\pi$[12]:

$ER=\frac{1+e^{2\Delta\alpha}+2e^{\Delta\alpha}}{1+e^{2\Delta\alpha}-2e^{\Delta\alpha}}$
(5)

Fig.4b shows that, as $V_{G}$ increases, and the SLG on the active arm becomes
transparent, ER increases$>$25dB because $\Delta\alpha$ is reduced. The effect
of $\Delta\phi$ and $\Delta\alpha$ due to the active arm of the MZM is seen by
the simultaneous change in position and ER of the interference fringes with
$V_{G}$. Fig.4c plots the change in position of interference fringe and ER as
a function of $V_{G}$. The MZM has the same behaviour as the EAM in Fig.3a. ER
is minimised near the Dirac point, because the absorption of the SLG on the
active arm is highest, whilst the device on the other arm is transparent. ER
then increases with increasing $V_{G}$, as absorption by the active arm is
reduced, until flattening$>$10V, when the SLGs on both arms are transparent.
This shows that the transparency regime is ideal for pure PM because we have a
quasi-linear change in phase, whilst losses remain constant. A similar
V-dependent change in fringe position and ER is observed on either side of the
Dirac point, where negative VG give a weaker effect than positive ones. The
similarity around the Dirac point is due to the SLG ambipolarity[105], and the
asymmetry can be due to different scattering rates of electrons ($e$) and
holes ($h$), resulting from an uneven distribution of positively or negatively
charged impurities[92, 110, 111, 112]. From $\Delta\lambda$ and $\Delta$ER we
extract $\Delta n_{\text{eff}}$ and $\Delta\alpha$, Fig.4d. The measured and
simulated $\Delta n_{\text{eff}}$ and $\Delta\alpha$ are in Fig.4d. We
attribute the differences in measured and simulated behaviour to asymmetries
between the two SLG as each SLG undergoes different amounts of processing,
since SLG1 is subject to more processing than SLG2. The difference between
measured and simulated $\Delta\alpha$ in transparency is a result of increased
propagation losses outside the DSLG modulator, due to residues remaining on
the WG from SLG processing. In transparency, the measured $\Delta\phi$ gives a
$V_{\pi}L\sim$0.3V$\cdot$cm, matching that of state-of-the-art SLG PMs[32].
However, unlike Ref.32, our devices have pure PM with negligible change in
optical losses, an essential property for IQ modulation[11]. Our DSLG MZMs
have a $V_{\pi}L\sim$ on par with the lowest reported plasmonic LN
MZMs[28],$\sim$5 times better than the lowest reported LN MZMs[48],
and$\sim$2.5 times better than the lowest reported thin film LN MZMs[27] and
Si MZMs[49], Table 1. Due to the high ($\sim$746dB/cm) optical loss, our DSLG
phase modulator has FOM${}_{\text{PM}}>$200VdB, greater than the lowest
reported Si ($\sim$38VdB[18]), LN ($\sim$0.35VdB[24]), and III-V
($\sim$1VdB[22]), Table 1. However, if the optical losses of SLG in
transparency are reduced$<$10dB/cm by increasing the overlapping SLG region
and increasing $\tau$ to$>$300fs, corresponding to $\mu>$6,000cm2V-1s-1, our
low $V_{\pi}L$ would enable FOM${}_{\text{PM}}\sim$3VdB. This is lower than
both Si and LN MZMs, with$\sim$3mm devices instead$\sim$2cm. Even though III-V
MZMs have the lowest FOM${}_{\text{PM}}\sim$1VdB[22], their BW is unsuitable
for Tb/s data transmission because it is limited to the MHz range[22].
Shrinking device dimensions by one order of magnitude results in denser
circuits that benefit from reduced overall power consumption by minimising the
interconnects lengths.

##  II conclusions

We reported DSLG MZMs showing pure PM in the transparency regime for
$E_{\text{F}}>$0.4eV, with VπL$\sim$0.3Vcm. We reached the transparency regime
by device design and process optimisation, ensuring the dielectric can
withstand the required 10V to reach $E_{\text{F}}>0.4$eV without breakdown.
Our low $V_{\pi}L$=0.3Vcm means we are able to overcome the loss limitations
of Si MZMs, deliver increased circuit densities compared to LN, and match the
performance of III-V (InGaAsP) MZMs, without expensive fabrication
requirements. Reaching transparency is critical for graphene-based
communications and metrology platforms that use complex modulation formats to
maximise the density of transmitted information.

##  III acknowledgments

We acknowledge funding from EU Graphene Flagship, ERC Grants Hetero2D, GIPT,
EU Grants GRAP-X, CHARM, EPSRC Grants EP/K01711X/1, EP/K017144/1,
EP/N010345/1, EP/L016087/1, EP/V000055/1, EP/X015742/1, Nokia Bell Labs, and
MUR - Italian Minister of University and Research under the ”Research projects
of relevant national interest-PRIN 2020-Project No.2020JLZ52N ”Light-matter
interactions and the collective behavior of quantum 2D materials (q-LIMA)”.

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# Covariant formulation of Generalised Uncertainty Principle

Raghvendra Singh<EMAIL_ADDRESS>Institute of Mathematical Sciences,
Homi Bhabha National Institute (HBNI), IV Cross Road, C. I. T. Campus,
Taramani, Chennai 600 113, India  Dawood Kothawala<EMAIL_ADDRESS>Centre
for Strings, Gravitation and Cosmology, Department of Physics, Indian
Institute of Technology Madras, Chennai 600 036

###### Abstract

We present a formulation of the generalised uncertainty principle based on
commutator $\left[{\hat{x}}^{i},{\hat{p}}_{j}\right]$ between position and
momentum operators defined in a covariant manner using normal coordinates. We
show how any such commutator can acquire corrections if the momentum space is
curved. The correction is completely determined by the extrinsic curvature of
the surface $p^{2}=$ constant in the momentum space, and results in non-
commutativity of normal position coordinates
$\left[{\hat{x}}^{i},{\hat{x}}^{j}\right]\neq 0$. We then provide a
construction for the momentum space geometry as a suitable four dimensional
extension of a geometry conformal to the three dimensional relativistic
velocity space - the Lobachevsky space - whose curvature is determined by the
dispersion relation $F(p^{2})=-m^{2}$, with $F(x)=x$ yielding the standard
Heisenberg algebra.

###### pacs:

04.60.-m

##  I Introduction

The Heisenberg algebra satisfied by position and momentum operators,
${\hat{x}}^{i}$ and ${\hat{p}}_{j}$, forms the cornerstone of quantum
mechanics, and in standard treatments, this is characterized by the
commutators:

$\displaystyle\left[{\hat{x}}^{i},{\hat{p}}_{j}\right]$ $\displaystyle=$
$\displaystyle i\hbar\delta^{i}_{\phantom{i}j},$
$\displaystyle\left[{\hat{p}}_{i},{\hat{p}}_{j}\right]$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\left[{\hat{x}}^{i},{\hat{x}}^{j}\right]$
$\displaystyle=$ $\displaystyle 0.$ (1)

However, the generalisation and interpretation of the above commutators in a
fully relativistic theory which is also invariant under general coordinate
transformations is not straightforward. The key issue is already evident from
the explicit appearance of ${\hat{x}}^{i}$ in the commutator relationships,
which spoils manifest general covariance. This issue needs to be addressed
since general covariance is also the first step towards generalization of a
theory to a curved background space, or spacetime. Therefore, a curved
space(-time) generalisation of the Heisenberg algebra requires a covariant
definition of the quantities appearing in the above relations. This, in
itself, is not difficult if one identifies the coordinates appearing in the
commutators via exponential map - the so called normal coordinates. These
coordinates, by definition, depend on the geodesic spray from a given point
${\mathcal{P}}_{0}$, and hence their characterization depends directly on the
curvature of the manifold. We will discuss this in Sec. II.

A second generalization of the Heisenberg algebra, that has gained
considerable interest over the past two decades Kempf  is tied to what we
expect from measurements at very high energies or at very small length scales.
Very general arguments rooted in basic principles of quantum mechanics and
general relativity suggest that Planck scale might provide a fundamental
operational bound on the measurement of observables such as spacetime
intervals paddy . One therefore expects the above commutators to be modified
at large momentum, with Planck scale as the characteristic scale at which the
modification occurs. This has motivated a large amount of research exploring a
modification of the form

$\displaystyle\left[{\hat{x}}^{i},{\hat{p}}_{j}\right]$ $\displaystyle=$
$\displaystyle i\hbar\Theta^{i}_{\phantom{i}j}({\hat{p}}_{k}),$ (2)

with the most popular form of the function $\Theta$ being a quadratic in
${\hat{p}}_{k}$ that reduces to $\delta^{i}_{\phantom{i}j}$ at low momentum
GUP . However, although there exists several arguments based on certain well
motivated requirements that have been used to constrain
$\Theta^{i}_{\phantom{i}j}$ at least in flat spacetime (using, for instance,
translation and rotation invariance, see Kempf2 ), no such generic constraints
are available for a curved spacetime.

Our first aim in this paper will be to provide a geometrically well motivated
argument that gives a curved spacetime generalization of the Heisenberg
algebra, in terms of quantities defined covariantly in a specific region of
spacetime, provided only that it is a geodesically convex neighbourhood of
some spacetime point. Our analysis will also give us a hint towards a
geometrical interpretation of $\Theta^{i}_{\phantom{i}j}({\hat{p}}_{k})$,
constructing which will be our next aim.

As a first step towards explicitly constructing
$\Theta^{i}_{\phantom{i}j}({\hat{p}}_{k})$, we will begin by first analysing
the geometry of the momentum space. To do this, we will start with the well-
known expression for center-of-mass energy of a system of two point particles
with rest masses $m_{1}$ and $m_{2}$, and use it to obtain a metric on the
three dimensional momentum space which turns out to be conformal to the well-
known Lobachevsky metric on the relativistic velocity space. We then extend
this metric to four dimensions by demanding that, for particles with zero
relative velocities, it gives the (squared) difference of rest masses:
$(\Delta m)^{2}=(m_{2}-m_{1})^{2}$. The momentum space so constructed is flat
(${\rm\bf Riemann}=0$) as long as the relation $p^{2}=g^{ab}p_{a}p_{b}=-m^{2}$
holds for a given $\left(p_{a},m\right)$. This also yields
$\Theta^{i}_{\phantom{i}j}({\hat{p}}_{k})=\delta^{i}_{\phantom{i}j}$. For a
general dispersion relation $F(p^{2})=-m^{2}$, ${\rm\bf Riemann}\neq 0$ and we
show that $\Theta^{i}_{\phantom{i}j}({\hat{p}}_{k})$ is now determined by the
extrinsic geometry of $p^{2}=$ constant surface in momentum space.

To summarise, we present in this paper a novel formulation of the generalised
uncertainty principle that incorporates and interconnects three key ideas: (i)
covariant characterization of ${\hat{x}}^{i}$ appearing in the uncertainty
principle, (ii) momentum space geometry based on the relativistic velocity
space, and (iii) modified dispersion relation $F(p^{2})=-m^{2}$. For some
earlier work that comments on similar issues, see Sabine . Note, however, that
our formalism involves only Lorentz invariant modifications of the dispersion
relation, incorporated via a very specific construction of the four
dimensional momentum space geometry. Our geometrical approach is also somewhat
closer in spirit to Ref. Wagner . However, the conceptual set-up and
mathematical implementation here are completely different.

Conventions: Throughout the paper, Latin indices (serif/non-serif) run from 0
to 3, and the sans-serif font represent frame indices. Boldface quantities
represent four vectors.

##  II Covariant characterizations of commutators

###  II.1 Normal coordinates

Figure 1: Normal coordinates defined in the tangent space ${T}_{{\cal
P}_{0}}(M)$. Their variations are completely determined by the geometry of the
equi-geodesic surfaces anchored at $\mathcal{P}_{0}$ (dashed lines), with
$\lambda$ the length of the geodesic connecting $\mathcal{P}$ to
$\mathcal{P}_{0}$. The passive version of these variations correspond to
shifting the base point $\mathcal{P}_{0}$, in which case the change in normal
coordinates is determined by extrinsic geometry anchored at ${\mathcal{P}}$.

By definition, the normal coordinates are defined by
$\Phi^{\textsf{a}}=-\eta^{\textsf{ab}}e^{i}_{\textsf{b}}\nabla_{i}\Omega$
where $e^{i}_{\textsf{b}}$ is an orthonormal tetrad at ${\cal P}_{0}$:
$\bm{e}_{\textsf{a}}\cdot\bm{e}_{\textsf{b}}=\eta_{\textsf{ab}}$ (lorentzian)
or $\delta_{\textsf{ab}}$ (euclidean), and $\Omega({\cal P}_{0},{\cal
P})=\sigma^{2}({\cal P}_{0},{\cal P})/2$ is the Synge world function book-jls
with $\sigma^{2}({\cal P}_{0},{\cal P})=\pm\lambda^{2}$; see Fig. 1. In terms
of frame components of the tangent vector, this is equivalent to assigning to
${\cal P}$ the coordinates $\Phi^{\textsf{a}}(\mathcal{P})=\lambda
t^{\textsf{a}}\left(\mathcal{P}_{0}\right)$. The variations in these
coordinates are therefore given by Dewitt Book

$\displaystyle\delta\Phi^{\textsf{a}}(\mathcal{P})=t^{\textsf{a}}\left(\mathcal{P}_{0}\right)\left[\delta\lambda\right]_{t^{\textsf{a}}\;{\mathrm{fixed}}}+\lambda\left[\delta
t^{\textsf{a}}\left(\mathcal{P}_{0}\right)\right]_{\lambda\;{\mathrm{fixed}}}.$
(3)

We now wish to compute the variation which arises due to a shift in the origin
${\cal P}_{0}$, which we characterise by a vector $\varepsilon^{\sf k}$ in
${T}_{{\cal P}_{0}}(M)$. The above expression can then be written in a neat
form by observing that the variation appearing in the second term on the RHS
lies purely within the equi-geodesic surface $\lambda=$ constant, and is
therefore given by

$\displaystyle\left[\delta
t^{\textsf{a}}\left(\mathcal{P}_{0}\right)\right]_{\lambda\;{\mathrm{fixed}}}=K^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}\varepsilon^{\textsf{b}},$
(4)

where $K^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}$ represents the frame
components of the extrinsic curvature tensor associated with the equi-geodesic
surfaces, and we have used
$K^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}t^{\textsf{b}}=0$. We can
therefore write

$\displaystyle\delta\Phi^{\textsf{a}}(\mathcal{P})=\left(\lambda
K^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}-t^{\textsf{a}}t_{\textsf{b}}\right)\varepsilon^{\textsf{b}}.$
(5)

The extrinsinc curvature $K^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}$ of
the equi-geodesic surfaces is completely determined by curvature tensor of
$M$, and it’s expansion in normal coordinates can be found in Appendix A of
the second reference in Dawood .

###  II.2 The commutator

The discussion so far has been general and applies to any manifold. For
instance, it can be applied to the case when $M$ represents curved spacetime,
and moving to quantum mechanics by identifying the variation of
$\Phi^{\textsf{a}}\equiv{\sf x}^{\textsf{a}}$ with the commutator of ${\sf
x}^{\textsf{a}}$ with the generator of this variation, which can then be
identified with the momentum operator Kempf3 . Alternatively, when $M$
represents the momentum space, the variation of $\Phi_{\textsf{a}}\equiv{\sf
p}_{\textsf{a}}$ can again be identified, quantum mechanically, with the
commutator ${\sf p}_{\textsf{a}}$ with the generator of this variation, which
we can then identify with the position operator.

When $M$ represents the spacetime manifold, the normal coordinates
$\Phi^{\textsf{a}}(\mathcal{P})\equiv{\sf x}^{\textsf{a}}$, and the RHS of Eq.
(5) suggests the following for the commutator between ${\sf x}^{\textsf{a}}$
and the operator $\widetilde{\sf p}_{\textsf{b}}$ generating the shift of
origin by $\varepsilon^{\textsf{b}}$:

$\displaystyle\left[{\sf x}^{\textsf{a}},\widetilde{\sf
p}_{\textsf{b}}\right]\overset{\rm def}{:=}i\hbar\left(\lambda
K^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}-u^{\textsf{a}}u_{\textsf{b}}\right).$
(6)

where $t^{\textsf{a}}=u^{\textsf{a}}$ is the four velocity. In flat spacetime,
it follows from the geometry of the equi-geodesic surfaces, that $\lambda
K^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}-u^{\textsf{a}}u_{\textsf{b}}=\delta^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}$
poisson ; Dawood  and one recovers the standard Heisenberg commutators. On
curved spacetimes, the above arguments would imply that the Heisenberg
commutators are modified to
$[\textsf{x}^{\textsf{a}},\widetilde{\textsf{p}}_{\textsf{b}}]=i\hbar\Theta^{\textsf{a}}_{\
\textsf{b}}(\boldsymbol{\textsf{x}})$. However, demanding that the candidate
for momentum operator be symmetric adds certain constraints, which we briefly
discuss; a very detailed analysis can be found in the seminal papers by De
Witt Dewitt . To begin with, it is easy to show that the symmetric momentum
operator in position representation with respect to the integration measure
i.e. $\mathrm{d}^{n}x\sqrt{-g}$, is given by

$\displaystyle\widetilde{\textsf{p}}^{\rm{sym}}_{\textsf{a}}=-i\hbar(\frac{1}{2}\Theta^{\textsf{m}}_{\
\textsf{a}}\Gamma^{\textsf{i}}_{\
\textsf{m}\textsf{i}}+\frac{1}{2}\Theta^{\textsf{m}}_{\
\textsf{a},\textsf{m}}+\Theta^{\textsf{m}}_{\
\textsf{a}}\partial_{\textsf{m}}).$ (7)

The standard commutator relations

$\displaystyle\left[{\textsf{x}}^{\textsf{a}},{\textsf{p}}_{\textsf{b}}\right]:=i\hbar\delta^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}},$
(8)

can be obtained even in curved spacetime if we define
$\textsf{p}_{\textsf{a}}=(\Theta^{-1})^{\textsf{b}}_{\
\textsf{a}}\widetilde{\sf p_{b}}$. But the new momentum operator is no more
symmetric with respect to the integration measure. As soon as one tries to
make it symmetric, it takes the following form

$\displaystyle{\sf p}^{\rm{sym}}_{\sf a}=-i\hbar\left(\partial_{\sf
a}+\frac{1}{2}\Gamma^{\sf i}_{\phantom{\sf i}\sf a\sf i}\right).$ (9)

The point we want to emphasize is that, one does not need to modify the
commutator relations even in curved spacetime if ‘$\Theta$’ is a function of
‘x’ only. As stated above, this is consistent with known results about
quantisation in curved spaces; see, for instance, Dewitt .

Since one expects quantum gravitational effects to play some role in modifying
the conventional Heisenberg algebra and hence the uncertainty principle, it is
reasonable to expect the deformations to occur at large momenta (suitably
identified). We will therefore now turn to the case when $M$ represents the
momentum space, and look for the kind of deformations that the geometry of the
momentum space can produce in the commutators. To do so, we will now identify
the the normal coordinates $(-{\sf x}^{\sf a})$ as generating shifts in ${\sf
p}_{\sf a}$ generators-flows . To compute the computers, we need information
about the equi-geodesic surface and its extrinsic curvature in the momentum
space, and for this, we need to first characterise the geometry of the
momentum space. We will do so in such a manner that the standard dispersion
relation $p^{2}=-m^{2}$ will represent.

##  III Geometry of momentum space

Figure 2: The behaviour of functions $G_{1}(p^{2})$ and $G_{2}(p^{2})$ for
three different dispersion relations. These functions are defined by:
$\left[{\sf x}^{a},{\sf
p}_{b}\right]=i\hbar\left(G_{1}(p^{2})\delta^{a}_{b}+G_{2}(p^{2}){\sf
p}^{a}{\sf p}_{b}\right)$. The three curves correspond to $F(x)$: (i)
$x+x\left[\exp{(x/\Lambda)^{2}}-1\right]$ (Red), (ii) $x\exp{({x}/{\Lambda})}$
(Blue), (iii) $x\left[1+({x}/{\Lambda})^{2}\right]$ (Orange). Here, $\Lambda$
is a momentum scale and $\Lambda\to\infty$ corresponds to $F(x)=x$.

We will now characterise the geometry of the momentum space by starting with
the following spacetime picture: two particles of rest energies (masses)
$m_{1}$ and $m_{2}$ and four momenta $p^{i}_{1}$ and $p^{i}_{2}$ respectively,
intersect at an spacetime event. The energy of this system in it’s center-of-
momentum frame has the invariant form

$\displaystyle E_{\rm com}^{2}$ $\displaystyle=$
$\displaystyle-(p_{1}+p_{2})^{2}$ (10) $\displaystyle=$
$\displaystyle\mu_{1}^{2}+\mu_{2}^{2}-2\mu_{1}\mu_{2}{\bm{u}_{1}}\cdot{\bm{u}_{2}}$
$\displaystyle=$
$\displaystyle\left(\mu_{1}+\mu_{2}\right)^{2}+2\mu_{1}\mu_{2}\left(\gamma_{\rm
rel}-1\right),$

where we have defined $\bm{p}_{1,2}=\mu_{1,2}\bm{u}_{1,2}$ (with
${\bm{u}}_{1,2}^{2}=-1$), so that $\bm{p}^{2}=-\mu^{2}$ and $\gamma_{\rm
rel}=-\bm{u}_{1}\cdot\bm{u}_{2}$ is the relative gamma factor corresponding to
the relative velocity $v_{\rm rel}$.

Note that, conventionally, $\mu$ would be identified with the rest mass $m$ of
the particle. However, we postpone this replacement at this stage since we
will very soon be interested in generalisation $\mu=f(m)$ which will
correspond to the modified dispersion relation $F(p^{2})=-m^{2}$. We will
therefore keep the discussion general, but will refer to the case $f(m)=m$ to
provide motivation for certain definitions and interpretation.

Consider now the the quantity $\ell^{2}>0$ defined by:

$\displaystyle{\ell}^{2}$ $\displaystyle=$ $\displaystyle
2\mu_{1}\mu_{2}\left(\gamma_{\rm rel}-1\right)$ $\displaystyle=$
$\displaystyle
2\mu_{1}\mu_{2}\left(\cosh\chi_{1}\cosh\chi_{2}-\sinh\chi_{1}\sinh\chi_{2}\cos\Omega-1\right),$

where the second equality obtains by parameterizing $u^{a}$ in terms of
standard Lorentz transformations:
$u^{a}(\chi,\Omega^{A})=\left(\cosh{\chi}\right)T^{a}+\left(\sinh{\chi}\right)N^{a}$,
where $T^{a},N^{a}$ are arbitrary unit timelike, spacelike vectors in the
tangent space ${T}_{{\cal P}_{0}}(M)$, with $T^{a}N_{a}=0$, $\chi$ is rapidity
and $\Omega^{A}=(\theta,\phi)$. Note that, in the non-relativistic limit
$v_{\rm rel}\ll c$, and with $\mu_{1,2}=m_{1,2}$, the above expression yields

$\frac{{\ell}^{2}}{2M}\approx\frac{1}{2}\mu_{\rm red}v_{\rm rel}^{2},$

with $M=m_{1}+m_{2}$ and $\mu_{\rm red}=m_{1}m_{2}/M$ (reduced mass). This is
therefore the relative kinetic energy of the system.

We will now take ${\ell}^{2}$ as the measure of (squared) “three momentum
distance” between the two particles. 111Note that one can write
$\ell^{2}=E_{\rm com}^{2}-\left[E_{\rm com}^{2}\right]_{v_{\rm rel}=0}$, which
has a nice physical interpretation. For a composite system, the 2nd term on
the RHS represents the “rest energy” in the center-of-momentum frame since
$v_{\rm rel}=0$. The above relation therefore has the structure
“$|\vec{p}|^{2}=E^{2}-m^{2}$” valid for a point particle. Defining
$\xi^{a}=(\mu,\chi,\Omega^{A})$ The corresponding metric
$g^{3{\rm{-mom}}}_{ab}$ may be obtained by

$\displaystyle g^{3-\rm mom}_{ab}$ $\displaystyle=$
$\displaystyle\lim\limits_{\xi_{2}^{a}\to\xi_{1}^{a}}\frac{\partial^{2}}{\partial\xi_{1}^{a}\xi_{2}^{b}}\left(\frac{{\ell}^{2}}{2}\right)$
(15) $\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}0&0&0\\\
0&\mu^{2}&0\\\ 0&0&\mu^{2}\sinh^{2}(\chi)\sigma_{AB}\\\ \end{array}\right)$

where $\sigma_{AB}$ is the canonical metric on the $2$-sphere. We note that,
apart from a zero eigenvalue, the above metric is conformal to the Lobachevsky
metric of the relativistic velocity space, $\mathrm{d}l_{\rm
rel}^{2}=\mathrm{d}\chi^{2}+\sinh^{2}\chi\mathrm{d}\Omega^{2}$ Landau :
$\mu^{2}\mathrm{d}l_{\rm rel}^{2}=-p^{2}\mathrm{d}l_{\rm rel}^{2}$. This
therefore gives a rigorous justification for our definition of distance
measure: it correctly gives a locally Lorentz invariant measure of relative
momentum on the space of three momenta. To extend it to a four dimensional
metric $\mathrm{d}\ell^{2}$, we demand that (i) the four momentum geometry is
Lorentzian, and (ii) for points in momentum space that have zero relative
velocity, that is, $\mathrm{d}l_{\rm rel}=0$, the metric gives the difference
in rest masses (or rest energies) associated with the corresponding momenta.
That is, we require that $\mathrm{d}\ell^{2}=-\mathrm{d}m^{2}$ when
$\mathrm{d}l_{\rm rel}^{2}=0$. This then gives the full metric as:

$\displaystyle\mathrm{d}\ell^{2}$ $\displaystyle=$
$\displaystyle-\mathrm{d}{m}^{2}-p^{2}\mathrm{d}l_{\rm rel}^{2}$ (16)
$\displaystyle=$ $\displaystyle-\frac{p^{2}F^{\prime
2}}{F}\mathrm{d}\mu^{2}+\mu^{2}\mathrm{d}l_{\rm rel}^{2},$

where $F^{\prime}=\mathrm{d}F/\mathrm{d}p^{2}$.

When $F(p^{2})=p^{2}$, we have the standard dispersion relation $p^{2}=-m^{2}$
and the above metric is easily recognised as flat Minkowski metric in
hyperbolic coordinates, and hence ${\rm\bf Riemann}=0$.

Modified dispersion relations which are Lorentz invariant, conventionally
given in the form $F(p^{2})=-m^{2}$, correspond to
$f(x)=\sqrt{-F^{-1}(-x^{2})}$. These would generically yield ${\rm\bf
Riemann}\propto f^{\prime\prime}/f\neq 0$. In terms of the dispersion relation
$F(p^{2})=-m^{2}$, the commutators are given by,

$\displaystyle\left[{\sf x}^{\textsf{a}},{\sf
p}_{\textsf{b}}\right]:=i\hbar\left\\{\delta^{\textsf{a}}_{\textsf{b}}+\left(\frac{F(p^{2})}{p^{2}F^{\prime}(p^{2})}-1\right)h^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}\right\\},$
(17)

where $h^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}$ is the projector
orthogonal to $\partial/\partial\mu$. One can write the above expression in a
form in which it is easily compared with the existing works on generalised
uncertainty principle (for instance, the now familiar version that was
introduced in Kempf ; Kempf2 ). To do this, one uses the basic definition of
normal coordinates, given in Sec. II, applied to the momentum space geometry,
and recognise that $m$ represents the geodesic length (from the origin) in the
momentum space metric. The above expression then becomes:

$\displaystyle\left[{\sf x}^{\textsf{a}},{\sf p}_{\textsf{b}}\right]$
$\displaystyle=$ $\displaystyle
i\hbar\left(\frac{F}{p^{2}F^{\prime}}\delta^{\textsf{a}}_{\textsf{b}}-\frac{1}{F}\left(\frac{F}{p^{2}F^{\prime}}-1\right){\sf
p}^{\textsf{a}}{\sf p}_{\sf b}\right).$ (18)

The remaining commutators then follow from a straightforward application of
the Jacobi identity. The final form for the modified Heisenberg algebra is
then given by:

$\displaystyle\left[{\sf x}^{\textsf{a}},{\sf p}_{\textsf{b}}\right]$
$\displaystyle=$ $\displaystyle
i\hbar\Biggl{(}G_{1}(p^{2})\delta^{\textsf{a}}_{\textsf{b}}+G_{2}(p^{2}){\sf
p}^{\textsf{a}}{\sf p}_{\sf b}\Biggl{)},$ (19) $\displaystyle\left[{\sf
p}_{\textsf{a}},{\sf p}_{\textsf{b}}\right]$ $\displaystyle=$ $\displaystyle
0,$ (20) $\displaystyle\left[{\sf x}^{\textsf{a}},{\sf x}^{\textsf{b}}\right]$
$\displaystyle=$ $\displaystyle
2i\hbar\left(G_{2}-2G_{1}^{\prime}-\frac{2G_{1}^{\prime}G_{2}p^{2}}{G_{1}}\right)\sf
x^{[a}{\sf p}^{b]},$ (21)

where $G_{1}={F}/(p^{2}F^{\prime})$ and $G_{2}=-\left(G_{1}-1\right)/F$.
(Note, however, that the last commutator above holds for arbitrary
$G_{1},G_{2}$.) Evidently, the modified dispersion relation introduces a non-
commutativity in normal coordinates. It will be interesting to study the
implications of this non-commutativity vis-a-vis the its connection with
modified dispersion Dispersion relation . The behaviour of $\left[\sf
x^{a},p_{b}\right]$ for some model dispersion relations are shown in the Fig.
2.

Some useful limits: To obtain some useful limits, let us write
$F(p^{2})=p^{2}(1+q(p^{2}))$ which defines $q(p^{2})$. If we demand $q(p^{2})$
to be analytic at $p^{2}=0$, and $q(0)=0$, then $F\sim p^{2}$ as $p^{2}\to 0$
and $F^{\prime}(0)=1$. In this case, we have

$\displaystyle G_{1}(p^{2})$ $\displaystyle=$ $\displaystyle
1-q^{\prime}(0)p^{2}+O(p^{4})$ $\displaystyle G_{2}(p^{2})$ $\displaystyle=$
$\displaystyle q^{\prime}(0)+O(p^{2})$ (22)

which induces a non-commutativity $\left[{\sf x}^{\textsf{a}},{\sf
x}^{\textsf{b}}\right]\sim 6i\hbar q^{\prime}(0)\sf x^{[a}{\sf p}^{b]}$ to the
lowest order.

The opposite limit, $p^{2}\to\infty$, depends very much on the fall-off
behavior of $G_{1},G_{2}$, which in turn would be determined by the theory
that yields the modified dispersion. For instance, an exponential fall-off in,
say, $G_{2}$, of the form $\exp[-\alpha(p^{2})^{n}]$ ($\alpha$ a real positive
constant, $n\in$ positive integers) would give a divergent contribution for
timelike momenta $p^{2}<0$ provided $n$ is odd. As a concrete example that
does not involve exponentials, consider a dispersion relation of the form
$F(p^{2})={p^{2}}({1+\alpha p^{2n}})^{-1}$ ($\alpha$ a real constant, $n\in$
positive integers). For such cases, in the high energy limit,
$G_{1}\sim{1}/{(1-n)}$, while $G_{2}\sim\alpha(n/(n-1))p^{2(n-1)}$. We
therefore see that $G_{2}$ would blow up in the high energy limit for $n>1$.
One can do a similar analysis for any dispersion relation; such analysis would
be particularly interesting since it might connect analytic properties of
$F(p^{2})$ with the nature of commutators.

##  IV Discussion

We have presented a geometric formalism for the generalised uncertainty
principle which is covariant and connects features of the underlying geometry
with the deformation of canonical commutator relations. This deformation is
tied to extrinsic geometry of the equi-geodesic surfaces of the manifold. When
the manifold in question is the momentum space, we characterised its geometry
in terms of a four dimensional extension of the relative velocity
(Lobachevsky) space, whose Riemann curvature is determined by the modified
dispersion relation $F(p^{2})=-m^{2}$. Our work therefore interconnects
generalised uncertainty principle, momentum space geometry, and modified
dispersion relations in a covariant setting, whose only free function is the
one that yields the dispersion relation.

Let us highlight some key features inherent in the setup we have described:

As already stated in the Introduction and described in Sec. II, our choice of
variables $({\sf x}^{\textsf{a}},{\sf p}_{\textsf{b}})$ are intrinsically
determined by geodesic flows in spacetime and momentum space. These flows are
nicely described in terms of Synge’s World function and the van Vleck matrix.
Therefore, the construction is completely covariant and in terms of physically
well-defined variables.

It is also non-local, since one must choose a base point of the geodesic flow,
in both position and momentum space. The non-locality in position space is
evident from the corresponding commutators in Eq. (21).

In our description of the momentum space, we have chosen a coordinatization
which is based on manifestly covariantly variables, viz the rest mass $m$ and
the relative velocity space metric. It will therefore not make much physical
sense to talk about general coordinate transformations in the momentum space,
although nothing in the formalism forbids it. The equations remain covariant
under arbitrary coordinate reparametrizations of the momentum space.

Figure 3: The choice of reference point in the momentum space can be obtained
from the vector $u^{\sf a}$ giving normal coordinates in position space.

More interesting is the issue of choice of origin in the momentum space. The
astute reader would have noticed that we have taken $m=0$ as the origin.
However, we can keep the reference point arbitrary, in which case, the
expression for extrinsic curvature
$K^{\textsf{a}}_{\phantom{\textsf{a}}\textsf{b}}$ will be messy.
Mathematically, one can still compute it, but conceptually, the interpretation
is subtle since when choice of origin corresponds to $m\neq 0$, the
coordinates in the relative velocity space $(\chi,\theta,\phi)$ must refer to
some specific timelike vector. However, note that one does not have to
introduce such a vector as an additional object. If ${\sf x}\in I^{+}({\sf
x}_{0})$, then the tangent to the geodesic connecting ${\sf x}_{0}$ to ${\sf
x}$ already provides such a vector; see Fig. 3. This nicely intertwines the
issue of reference point in momentum space with the normal coordinates in the
position space, but also makes the former “position” dependent. It will be
worth exploring the physical implications of this. Finally, we wish to
highlight that our construction of the momentum space geometry has an elegant
interpretation for the standard dispersion relation $p^{2}=-m^{2}$: it
describes flat spacetime in the Milne coordinates, with rest mass $m$ giving
the measure of geodesic length from origin.

##  V Acknowledgements

DK acknowledges the support from the Institute of Eminence scheme of IIT
Madras funded by the Ministry of Education of India.

*

## Appendix A

Here, we briefly discuss the commutators between position and momentum
operators, their phase-space representations, and constraints arising from the
Jacobi identity.

$[\textsf{x}^{\textsf{a}},\textsf{p}_{\textsf{b}}]_{\textbf{{p}}_{0}}=i\hbar\Theta^{\textsf{a}}_{\
\textsf{b}}(\textbf{{p}}_{0},\textbf{{p}}).$ (23)

One can show that the Jacobi identities yields,

$\left[\textsf{p}_{\textsf{c}},[\textsf{x}^{\textsf{a}},\textsf{x}^{\textsf{b}}]\right]=2i\hbar(\Theta^{[\textsf{b}}_{\
\textsf{c}}\textsf{x}^{\textsf{a}]}-\textsf{x}^{[\textsf{a}}\Theta^{\textsf{b}]}_{\
\textsf{c}})=2i\hbar\left[\Theta^{[\textsf{b}}_{\
\textsf{c}},\textsf{x}^{\textsf{a}]}\right].$ (24)

This clearly indicate that $\left[\sf x^{a},\sf x^{b}\right]$ is not going to
commute. Let us try to write the possible representation of the position
operator in the momentum basis, then we will give the $[\sf x^{a},\sf x^{b}]$
relation in terms of $\Theta(\textbf{\sf p})$.

$\textsf{x}^{\textsf{a}}=i\hbar\Theta^{\textsf{a}}_{\
\textsf{m}}\partial^{\textsf{m}}.$ (25)

But for being the physical observable the operator must be symmetric with
respect to integration measure of the momentum space i.e.
$d^{4}\textsf{p}\sqrt{-g}$. Using the definition of the symmetric operator,

$\int d^{4}\textsf{p}\sqrt{-g}\phi^{*}\hat{\textsf{x}}\psi=\int
d^{4}\textsf{p}\sqrt{-g}(\hat{\textsf{x}}\phi)^{*}\psi.$ (26)

We write momentum representation of the position operator as,

$\textsf{x}^{\textsf{a}}_{\rm
sym}=i\hbar(\frac{1}{2}\Theta^{\textsf{m}\textsf{a}}\Gamma^{\textsf{i}}_{\
\textsf{m}\textsf{i}}+\frac{1}{2}\Theta_{\textsf{m}}^{\
\textsf{a},\textsf{m}}+\Theta_{\textsf{m}}^{\
\textsf{a}}\partial^{\textsf{m}}).$ (27)

where $f^{,\textsf{m}}={\partial f}/{\partial\textsf{p}_{\textsf{m}}}$. By
using the equations (24),(27), we write the commutation relation between
$\hat{\textsf{x}}_{\textsf{i}}$, as

$\left[\textsf{x}^{\textsf{a}},\textsf{x}^{\textsf{b}}\right]=i\hbar\\{\textsf{x}^{\textsf{l}},(\Theta^{-1})^{\textsf{m}}_{\
\textsf{l}}\Theta_{\textsf{n}}^{\ [\textsf{a}}\Theta_{\textsf{m}}^{\
\textsf{b}],\textsf{n}}\\}.$ (28)

The {,} represent anti-commutators bracket.

## References

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  * (8) These topics are discussed, for instance, in Chapter 16 of: B. S. DeWitt, The global approach to quantum field theory, Vol. 1, Oxford University Press Inc., New York (2003). See also Sec II (A) of the second reference in Dawood .
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  * (11) E. Poisson, A. Pound, I. Vega, Living Rev. Relativity 14, 7 (2011).
  * (12) B. S. DeWitt, Rev. Mod. Phys. 29, 377  (1957).
  * (13) This follows simply from the classical picture of a phase space $({\sf x}^{\textsf{a}},{\sf p}_{\textsf{b}})$. A function $\mathscr{G}({\sf x}^{\textsf{a}},{\sf p}_{\textsf{b}})$ on this space defines the vector field  $X_{\mathscr{G}}=\frac{\partial\mathscr{G}}{\partial\sf p_{k}}\frac{\partial}{\partial\sf x^{k}}-\frac{\partial\mathscr{G}}{\partial\sf x^{k}}\frac{\partial}{\partial\sf p_{k}}$ which implies, for any phase space function $\Phi$, $\delta_{\mathscr{G}}\Phi=X_{\mathscr{G}}[\Phi]=(\Phi,{\mathscr{G}})_{\rm PB}$ where “PB” stands for Poisson brackets. Our identification of the commutators is then the conventional one.
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  * (15) Various examples of such dispersion relations are discussed, for example, in: D. Nesterov, S. N. Solodukhin, Nuclear Physics B 842, 141 (2011); also see- T.Padmanabhan, Physics Letters B 811, 135912 (2020).