Detuning the BSW Effect with Longitudinal String Spreading

Dayshon Mathis

_Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA
94306, USA_

Abstract

Black holes are interesting astrophysical objects that have been studied as
systems sensitive to quantum gravitational data. The accelerated geometry in
the exterior of extremal black holes can induce large center-of-mass energies
between particles with particular momenta at the horizon. This is known as the
Bañados-Silk-West (BSW) effect. For point particles, the BSW effect requires
tuning to have the collision coincide with the horizon. However, this tuning
is relaxed for string-theoretic objects. String scattering amplitudes are
large in the Regge limit, occurring at large center-of-mass energies and
shallow scattering angles, parametrically surpassing quantum field theoretic
amplitudes. In this limit, longitudinal string spreading is induced between
strings with a large difference in light-cone momenta, and this spread can be
used to ‘detune’ the BSW effect. With this in mind, quantum gravitational
data, as described by string theory, may play an important role in near
horizon dynamics of extremal Kerr black holes. Further, though it may be hard
to realize astrophysically, this system acts as a natural particle accelerator
for probing the nature of small-scale physics at Planckian energies.

##  1 Introduction

The Kerr metric, a description of the exterior space-time of a spinning black
hole, was discovered in 1963, and the Penrose effect was deduced in 1969. The
Penrose process is the phenomenon in which a particle decays into debris with
particular kinematics in the ergosphere of a spinning black hole, leading to
an ejected decay product with a large amount of energy. However, there is a
$2\rightarrow 2$ collisional version of the Penrose process [29], which
produces a large center of mass energies analogous to particle accelerators.
Extremal Kerr black holes can generate arbitrarily large center of mass
energies [20] [2] if one of the infaller’s trajectories is tuned to be
critical [44][39][42], this is known as the Bañados-Silk-West (BSW) effect. A
critical trajectory has an energy-angular momentum ratio that matches the
angular speed of the space-time at the horizon, $E/L=\omega_{H}$. In the
extremal limit $a\rightarrow M$, critical trajectories with angular momentum,
$L$, have energy equal to the peak of the effective potential,
$V_{\text{eff}}(L)=E$. The peak of the effective potential for relativistic
orbits coincides with the horizon in the extremal limit, and critical
infallers reaching the horizon have vanishing Boyer-Lindquist radial momentum.
However, their local proper velocities are less than the speed of light. This
is contrary to usual trajectories with finite radial momentum and local
velocity equal to the speed of light at the horizon. With redshifting factors,
the momenta of usual trajectories diverge at the horizon. In contrast,
critical momenta are finite, leading to the collision between a critical and a
usual particle having arbitrarily large center-of-mass energy.

Assuming the infallers are colliding point particles implies the space-time
coordinates must coincide upon collision. This requires tuning the timing of
at least one of the trajectories, not only tuning the energy-angular momentum
ratio. However, objects in string theory have a large extent at high energies
(i.e., large $p^{+}$ in the light-cone coordinates) and spread longitudinally
[34] [12] [10] and with higher amplitudes than QFT models

$\displaystyle\langle\Delta
X^{+}\rangle~{}\sim\frac{p^{+}_{D}}{p_{\perp}^{2}+m_{D}^{2}+\frac{1}{\alpha^{\prime}}}$
(1.1)

if we are in the Regge limit $s\rightarrow\infty$ and
$t\rightarrow\text{fixed}$. In [13] [27], these large momenta strings are seen
as detectors sensitive to early infaller data. Further, the spreading is not a
gauge artifact. By analysis of the six-point amplitudes [11], the spreading
still manifests. Analogously, usual particles have large $p^{+}$ and,
consequently, large longitudinal spread, making usual particles sensitive to
critical infallers and relaxing the need for tuning the timing. We could
assume a statistically large amount of particles are accreting into the hole
to get detection and detuning for point particles, but critical trajectories
are still tuned. In the extremal case, the innermost stable circular orbit
(ISCO) coincides with the horizon, possessing a critical energy-angular
momentum ratio. ISCOs have the largest binding energy, so infallers undergoing
radiative processes naturally reach ISCO with critical momenta. Strings, like
point particles, are driven by the Kerr geometry to criticality. Strings zero-
mode undergo similar point particle phenomena; additionally, objects of finite
extent around Kerr holes go through a stringy Penrose process, where strings
gain or lose energy by stretch as opposed to releasing ejecta [21][19].

Since the extremal Kerr exterior geometry induces large COM energy, it acts as
an effective particle accelerator to probe the quantum nature of gravity.
However, extremal Kerr black holes may be non-astrophysical due to the Thorne
limit [37], back-reactions, and other limitations [3]. There is a chance to
overcome these limits [31][36], but regardless of the physical hindrances an
extremal Kerr background will be assumed for its theoretical interest.

Section 2 reviews the BSW effect and determines the near-horizon kinematics.
Next, in the second section, the string theoretic version is analyzed for the
sake of detuning the BSW effect with the non-local spread of the string form
factor and a Penrose process for an extended object. The third section
qualifies some of the limitations and concludes the paper. Further, if
outgoing strings from the collisions can escape the hole, this could leave a
quantum gravity signature on quasinormal mode.

##  2 Kerr Kinematics

The Kerr metric in Boyer-Lindquist coordinates ($G=c=1$)[4] is

$\displaystyle ds^{2}=-N^{2}dt^{2}+g_{\phi\phi}(d\phi-\omega
dt)^{2}+dl^{2}+g_{\theta\theta}d\theta^{2},$ (2.1)

where $N$ is the lapse function

$N^{2}=\frac{\Sigma\Delta}{(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta}~{},$

$\omega$ is the angular speed on a co-rotating frame at a Boyer-Lindquist
radius, $r$,

$\omega=\frac{2aMr}{(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta},$

an d$l$ is the proper distance $l=\int dr\sqrt{\frac{\Sigma}{\Delta}}$,
$\Sigma=r^{2}+a^{2}\cos^{2}\theta$, and $\Delta=r^{2}+a^{2}-2Mra$. A light-
cone coordinate, the $x^{\pm}$-direction is be defined

$dx^{\pm}=\frac{Ndt\pm dl}{\sqrt{2}}.$ (2.2)

When considering infallers, for simplicity, we will consider equatorial
motion, $\theta=\frac{\pi}{2}$. However, incoming particles are naturally
driven to equatorial motion by the Bardeen-Peterson effect[1] (although non-
equatorial motion exhibits the BSW effect[45]). The equation of motion for a
particle with mass, $\mu$, is

$\displaystyle\sqrt{2}p^{\pm}/\mu$ $\displaystyle=\frac{E-\omega
L}{N}\pm\sqrt{(\frac{E-\omega L}{N})^{2}-(\delta+\frac{L^{2}}{g_{\phi\phi}})}$
(2.3) $\displaystyle(p^{\phi}-\omega p^{t})/\mu$
$\displaystyle=\frac{L}{g_{\phi\phi}}$ (2.4) $\displaystyle p^{\theta}=0$
(2.5)

where $E$ and $L$ are the energy per mass and angular momentum per mass,
respectively, and the value of $\delta$ depends on the geodesic of the
particle being space-like ($\delta=-1$) or time-like ($\delta=1$). The light-
like case ($\delta=0$) requires the momenta per mass to become coordinate-
derivatives of an affine parameter
$p^{\mu}\rightarrow\nu\frac{dx^{\mu}}{d\lambda}$ and $E$ and $L$ represent
energy per frequency and orbital angular momentum per frequency [40]. As
argued in [43] [35] [26][25], the angular frequency in the near horizon
extremal limit

$\omega=\frac{1}{2M}-\frac{N}{M}+\mathcal{O}(N^{2}),$ (2.6)

the non-extremal angular speed deviation from the horizon speed is quadratic
in the lapse function. Seen, the expansion of $\omega$ for the non-extremal
case in terms of $r-r_{+}$ is the same as the extremal case, however in the
extremal case $N^{2}\sim(r-r_{+})^{2}$ while the non-extremal case has
$N^{2}\sim(r-r_{+})(r_{+}-r_{-})$. This expansion means that the momentum at
the horizon is finite for a critical particle ($E-\omega_{H}L=0$) around an
extremal hole.

$\displaystyle\sqrt{2}p^{+}_{C}/m_{C}=2\frac{L_{C}}{M}$ (2.7)
$\displaystyle\sqrt{2}p^{-}_{C}/m_{C}=\frac{M(\delta_{C}+\frac{L^{2}_{C}}{g_{\phi\phi}})}{2L_{C}}$
(2.8)

whereas for horizon momenta of usual particles ($E_{U}-\omega L_{U}\neq 0$),
$p^{+}$ diverges

$\displaystyle\sqrt{2}p^{+}_{U}/m_{U}=2\frac{E_{U}-\omega_{H}L_{U}}{N}$ (2.9)
$\displaystyle\sqrt{2}p^{-}_{U}/m_{U}=\frac{N(\delta_{U}+\frac{L^{2}_{U}}{g_{\phi\phi}})}{2(E_{U}-\omega_{H}L_{U})}$
(2.10)

Calculating the center of mass energy,
$s=-(p_{U}+p_{C})^{2}\sim-p^{+}_{U}p^{-}_{C}$, leads to divergent COM energy,
$s\sim\mathcal{O}(\frac{1}{N})$. This is the essence of the BSW effect. The
large COM energy theoretical can reach Planckian energies, lending credence to
the necessity of quantum gravity data for near horizon extremal black holes
physics.

There is a large amount of proper distance at the horizon, although it is not
evident in Boyer-Lindquist coordinates. This leaves plenty of space to explore
string spreading in the near horizon region. Calculating the proper distance
between relativistic orbits shows the large proper length in the near horizon
region. Infalling critical particles asymptotically rests on the unstable
turning point in the extremal limit. These turning points for positive energy
critical particles exist between the innermost-stable-circular orbit (ISCO)

$\displaystyle r_{ISCO}$
$\displaystyle=M\\{3+Z_{2}-\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})}\\}$ (2.11)
$\displaystyle Z_{1}$
$\displaystyle=1+(1-\frac{a}{M}^{2})^{1/3}\\{(1+\frac{a}{M})^{1/3}+(1-\frac{a}{M}\\}{M})^{1/3})$
(2.12) $\displaystyle Z_{2}$ $\displaystyle=\sqrt{3\frac{a}{M}^{2}+Z_{1}^{2}}$
(2.13)

and the circular photon orbit (CPO)

$\displaystyle r_{CPO}=2M\\{1+\cos(\frac{2}{3}\cos^{-1}(-\frac{a}{M}))\\}.$
(2.14)

The proper distances between ISCO, the marginally bound circular orbit
($E=\mu$) (MBCO),

$\displaystyle r_{MBCO}=M\\{1+\sqrt{1-\frac{a}{M}}~{}\\}^{2}$ (2.15)

CPO and the proper distance from the horizon to all these radii are all non-
zero. Nearing extremality, $a=1-\epsilon$, the behaviour of these radii go as

$\displaystyle r_{+}=M+(2\epsilon)^{1/2}$ (2.16) $\displaystyle
r_{\text{CPO}}=M+(\frac{8}{3}\epsilon)^{1/2}$ (2.17) $\displaystyle
r_{\text{MBCO}}=M+2\epsilon^{1/2}$ (2.18) $\displaystyle
r_{\text{ISCO}}=M+(4\epsilon)^{1/3}$ (2.19)

and with the proper distance at the horizon being

$l_{1,2}\sim r_{+}\log{\frac{r_{2}-M}{r_{1}-M}}$ (2.20)

which is large as extremality is approached. Assuming the extremal Kerr
geometry maintains $a=M$, there is a large amount of space at the horizon,
which leaves string extent analysis unhindered by the background.

It can also be shown the energy-angular momentum ratio required to rest at
this turning point is equivalent to criticality. A turning point placed at
radius, $r$, between ISCO and CPO has energy and momentum

$\displaystyle E=$
$\displaystyle\frac{1-2M/r+aM^{1/2}/r^{3/2}}{\sqrt{1-3M/r+2aM^{1/2}/r^{3/2}}}$
(2.21) $\displaystyle L=$
$\displaystyle\frac{M^{1/2}r^{1/2}-2aM/r+M^{1/2}a^{2}/r^{3/2}}{\sqrt{1-3M/r+2aM^{1/2}/r^{3/2}}}.$
(2.22)

and in the near horizon extremal limit $\frac{E}{L}\rightarrow\omega_{H}$.
Critical particles can be considered to be stuck near, have fallen from, or
are approaching its centrifugal barrier, so the conserved energy and angular
momentum, $E$ and $L$, are appropriate for understanding infallers near the
horizon. A large amount of regular space at the horizon leaves plenty of room
for scattering dynamics and string theoretic phenomena. The space near the
horizon is flat and tidal forces are weak for large mass black holes,
$M>>m_{P},m_{\text{string}}$.

The gist of the BSW effect is the generation of a diverge,
$\mathcal{O}(N^{-1})$, COM energy, induced by the geometry, between colliding
usual and critical trajectories. Let the momentum of the usual particle near
an extremal horizon be denoted by $p_{U}$ and the momentum of the critical
particle $p_{C}$. The center of mass energy is therefore

$\displaystyle s$ $\displaystyle=-(p_{U}+p_{C})^{2}$
$\displaystyle=-p^{+}_{U}p^{-}_{C}-p^{+}_{C}p^{-}_{U}+g_{\phi\phi}(p^{\phi}_{C}-\omega
p^{t}_{C})(p^{\phi}_{U}-\omega p^{t}_{U})+m^{2}_{C}+m^{2}_{U}.$

Since $p^{+}_{U}$ is large near the horizon, the dominant term is

$\displaystyle s\sim
m_{U}m_{C}\frac{M(E_{U}-\omega_{H}L_{U})(\delta_{C}+\frac{L^{2}_{C}}{g_{\phi\phi}})}{NL_{C}}+\mathcal{O}(N^{0}).$
(2.23)

Now, with a COM energy that can access Planckian energies, we can analyze
string amplitudes in this regime. The Regge limit, $s\rightarrow\infty$ with
$t\rightarrow\text{fixed}$, has a large scattering amplitude for interacting
strings. The Regge limit corresponds to very small scattering angles, but the
Boyer-Lindquist coordinates obscures angular separation, so

For $2\rightarrow 2$ scattering, the contraction of momenta in the final state
must also be large by conservation of momentum, so a usual-critical pair in
the final state is a good assumption. In the case where we have usual-critical
$\rightarrow$ usual-critical, the Mandelstam variable
$t=-(p_{C}-p_{\hat{C}})^{2}$ is

$\displaystyle
t=m_{\hat{C}}m_{C}(\frac{\delta_{C}L_{\hat{C}}}{L_{C}}+\frac{\delta_{\hat{C}}L_{C}}{L_{\hat{C}}})+m_{C}^{2}+m_{\hat{C}}^{2},$
(2.24)

and $u=-(p_{C}-p_{\hat{U}})^{2}$ can be solved with $s+t+u=\sum_{i}m_{i}^{2}$
and is also large due to the contraction between usual and critical momenta.
The final state constants, $E_{\hat{C}}$, $E_{\hat{U}}$, $L_{\hat{C}}$, and
$L_{\hat{U}}$ are solved by
$p_{C}^{\mu}+p_{U}^{\mu}=p_{\hat{C}}^{\mu}+p_{\hat{U}}^{\mu}$ and we’ll assume
for simplicity the final state particles are also equatorial. To be sure we
are in the stringy Regge regime we impose

$\displaystyle\text{sgn}(p^{l}_{C})$
$\displaystyle=\text{sgn}(p^{l}_{\hat{C}})=\sigma_{C}$ (2.25) $\displaystyle
p^{l}_{U}$ $\displaystyle<0$ $\displaystyle p^{l}_{\hat{U}}$
$\displaystyle<0.$

This leads to the final state constants being

$\displaystyle
E_{\hat{U}}=E_{U}+3\omega_{H}L_{C}+2\omega_{H}\sigma_{C}\sqrt{3L_{C}^{2}-4\delta_{C}M^{2}}...$
$\displaystyle...\pm\sqrt{21L_{C}^{2}+12\sigma_{C}L_{C}\sqrt{3L_{C}^{2}-4\delta_{C}M^{2}}-16M^{2}\delta_{C}}+\mathcal{O}(N)$
(2.26) $\displaystyle E_{U}+E_{C}=E_{\hat{U}}+E_{\hat{C}}$ (2.27)
$\displaystyle L_{U}+L_{C}=L_{\hat{U}}+L_{\hat{C}}$ (2.28)

and particle-$\hat{C}$ is critical, $E_{\hat{C}}-\omega_{H}L_{\hat{C}}=0$. In
this case, we are looking at the Regge regime since $t=\text{fixed}$ as seen
from (2.24). In the Regge limit, string amplitude is large compared to the QFT
amplitude. Additionally, other stringy objects have an extent (open/closed
string, D-brane, superstrings, etc.) and exhibit the same spread longitudinal
spread in the Regge limit. Hence, this analysis is broadly applicable to
string theory.

It’d be interesting to study brane dynamics deep in the throat and elsewhere
in the context of the BSW effect since brane-brane and string-brane amplitudes
also exhibit the Regge behavior. Branes have been used in realizing the
microscopic of extremal black holes in higher dimensions [33], so the BSW
effect inducing longitudinal string spreading could become relevant with the
growth of an extremal black hole from a usual infaller or interactions with
past infallers.

##  3 Detuning the BSW effect with Longitudinal String Spreading

The amplitudes for interacting strings contain the convolution of vertex
operators, $V(z,\bar{z})$. An open/closed string tachyon has a vertex operator
of the form ($\hbar=1$)

$\displaystyle V\sim\frac{g_{o,c}}{\alpha^{\prime}}\int d^{2}z~{}e^{ip\cdot
X(z,\bar{z})}.$ (3.1)

Higher excited states terms have a more involved vector/tensor mode
construction, but the required Regge behavior manifestation can be deduced
from the simplified tachyon vertex operator. The four-point tree-level
amplitude is a convolution of these vertex operators

$\displaystyle A=\frac{g^{-\chi}}{\text{Vol}}\int
d^{2}z_{U}d^{2}z_{C}d^{2}z_{\hat{C}}d^{2}z_{\hat{U}}\langle
V_{C}(z_{C})V_{U}(z_{U})V_{\hat{C}}(z_{\hat{C}})V_{\hat{U}}(z_{\hat{U}})\rangle.$
(3.2)

The $\text{SL}(2,\mathbb{C})$ group reduces the integrand so that we consider,
$V_{U}(z_{U})$ and $V_{C}(0)$. Regardless of the type of stringy objects
involved, the integrand possesses a term

$\displaystyle e^{ip_{U}\cdot X(z_{U},\bar{z}_{U})}e^{ip_{C}\cdot X(0)},$
(3.3)

which elicits the Regge behavior when evaluated through the operator product
expansion,

$\displaystyle|z_{U}|^{-\frac{\alpha^{\prime}t}{2}-4}e^{i(p_{U}+p_{C})\cdot
X(0)+p_{U}\cdot(z_{U}\partial+\bar{z}_{U}\bar{\partial})X(0)}.$ (3.4)

Integrating over the world sheet leads to the Pomeron vertex amplitude as seen
in [6]

$\displaystyle\sim 2\pi
e^{-i\pi(\alpha^{\prime}t/4+1)}\frac{\Gamma(-1-\alpha^{\prime}t/4)}{\Gamma(2+\alpha^{\prime}t/4)}e^{i(p_{U}+p_{C})\cdot
X(0)}(p_{U}\cdot\partial
X(0)~{}p_{U}\cdot\bar{\partial}X(0))^{1+\alpha^{\prime}t/4}.$ (3.5)

where $\Pi(\alpha^{\prime}t)=2\pi
e^{-i\pi(\alpha^{\prime}t/4+1)}\frac{\Gamma(-1-\alpha^{\prime}t/4)}{\Gamma(2+\alpha^{\prime}t/4)}$,
and putting equation (3.5) back into equation (3.2) give the $s$-dependence
which exhibits the large form factor

$\displaystyle\mathcal{A}\sim\Pi(\alpha^{\prime}t)(e^{-i\pi/2}\alpha^{\prime}s/4)^{2+\alpha^{\prime}t/2}$
(3.6)

In the Regge limit $s\rightarrow\infty$ $t\rightarrow\text{fixed}$, the result
is

$\displaystyle\mathcal{A}\sim(\frac{\alpha^{\prime}s}{4})^{2}\frac{\Gamma(-1-\alpha^{\prime}t/4)}{\Gamma(2+\alpha^{\prime}t/4)}(e^{-i\pi/2}\alpha^{\prime}s/4)^{\alpha^{\prime}t/2}$
(3.7)

which has the modified form factor, $s^{\alpha^{\prime}t/2}$. Replacing the
Mandelstam variable, $t$, with the transverse momentum-squared,
$q_{\perp}^{2}=-t$, shows the form factor becomes

$\displaystyle e^{-\frac{\alpha^{\prime}q^{2}_{\perp}}{2}\mathop{\rm
ln}\nolimits(\alpha^{\prime}s)}$ (3.8)

with the logarithmic piece in the exponent contributing to the effective
growth of the impact parameter. More complicated string interactions also have
a similar amplitude in the Regge limit, exhibiting the logarithmic transverse
spread and, as seen below, the large longitudinal spread.

With the Kerr near-horizon region being flat compared to the string scale, we
can consider large transverse separation $\Delta
X_{\perp}^{2}\sim\alpha^{\prime}\log(s)$ where QFT amplitudes are suppressed,
but string amplitudes are relatively large. The transverse spread

$\langle(\Delta
X_{\perp})^{2}\rangle\sim\alpha^{\prime}\log(s/t)\sim\mathcal{O}(\log(N))$

is also large compared to QFT fields interacting on tails of their wave
function. The hard scattering counterpart with finite angle difference between
$U$ and $\hat{U}$, $s\rightarrow\infty$ and $\frac{t}{s}=\text{fixed}$, has an
amplitude that goes like $e^{-\alpha^{\prime}s}$ which has a softer falloff–
suggesting an object of size $\sqrt{\alpha^{\prime}}$, while the Regge
amplitude is large in $x_{\perp}^{2}\sim\alpha^{\prime}\log(s)$. The large
amplitude is one reason to use the Regge limit to probe information in a
quantum gravitation context, along with the gravitational information in the
$t\rightarrow 0$ limit.

The BSW effect implies string physics in a certain kinematic regime, at least,
with fixed Mandelstam variable, $t\rightarrow\text{fixed}$. The longitudinal
string spreading is the primary effect that relaxes the need for tuning the
initial conditions to guarantee a near-horizon collision. We now turn to the
longitudinal string spreading effect that detunes the timing necessity. As
seen in [12] [11],

$\displaystyle\langle\Delta
X^{+}\rangle~{}\sim\frac{p^{+}_{U}}{p_{U\perp}^{2}+m_{U}^{2}+\frac{1}{\alpha^{\prime}}}$
(3.9)

This comes directly from the expectation value of the $X^{+}-x_{cm}^{+}$ for
an open string in the light-cone gauge, $X^{-}=x^{-}+2\alpha^{-}\tau$,

$\displaystyle X^{+}(\tau=0)=x^{+}_{cm}+i\sqrt{2\alpha}\sum_{n\neq
0}\frac{1}{n}\alpha^{+}_{n}\cos(n\sigma)$ (3.10) $\displaystyle
X^{i}(\tau=0)=x^{i}_{cm}+i\sqrt{2\alpha}\sum_{n\neq
0}\frac{1}{n}\alpha^{i}_{n}\cos(n\sigma)$ (3.11)

calculating the variance for both, $\braket{(X^{\mu}-x^{\mu})^{2}}$ and
implementing the Virasora algebra,

$\displaystyle[\alpha^{+}_{n},\alpha_{m}^{+}]=\sqrt{\frac{2}{\alpha}}\frac{(m-n)}{p^{-}}\alpha^{+}_{m+n}+\frac{4m^{3}}{\alpha^{\prime}(p^{-})^{2}}$
(3.12)
$\displaystyle[\alpha^{i}_{m},\alpha^{j}_{n}]=m\delta_{m+n,0}\delta^{ij}$
(3.13)

leading to the variances,

$\displaystyle\braket{(\Delta
X^{+})^{2}}\sim\frac{1}{(p^{-}_{C})^{2}}\sum_{n>0}^{n_{\text{max}}}n\cos^{2}n\sigma$
(3.14) $\displaystyle\braket{(\Delta
X^{i})^{2}}\sim\alpha^{\prime}\sum_{n>0}^{n_{\text{max}}}\frac{1}{n}\cos^{2}n\sigma.$
(3.15)

The maximum mode, $n_{\text{max}}$, to which the detector, $U$, is sensitive,
in the Regge limit, is given by [13]

$\displaystyle
n_{\text{max}}\sim\frac{p^{-}_{C}p^{+}_{U}}{p_{\perp}^{2}+m^{2}_{U}+\frac{1}{\alpha^{\prime}}}.$
(3.16)

The result is equation (3.9) and a logarithmic dependence for the transverse
mode

$\displaystyle\braket{\Delta
X_{\perp}^{2}}\sim\alpha^{\prime}\log(\frac{s}{t}).$ (3.17)

Longitudinal spreading is proportional to the light-cone momenta of string
$U$. With large $p^{+}_{U}$, longitudinal spreading is also very large,
spreading in the radial direction, relaxing the tuning conditions.

Although the string spreading is large, the interaction involves small regions
on the worldsheet. The interaction on the world sheet has a large contribution
around a point, $z\sim\frac{t}{t+s}$. This saddle point is small in the Regge
limit, since $s\rightarrow\infty$ and $t\rightarrow\text{fixed}$. This is
related to the small time resolution of string $U$; since $p^{+}_{U}$ is
large, the internal clock of $U$ can resolve string excitations on string $C$.

##  4 Discussion of Limitations

###  4.1 COM energy, timing, and redshift

The divergent COM energy generated by the Kerr geometry can be bounded in a
string theory [8]. Since string theory generates a non-commutative geometry,
the Kerr black hole develops a mass distribution. The mass is an incomplete
gamma function

$\displaystyle
M(r)=\frac{2M_{0}}{\sqrt{\pi}}\int^{\lambda\frac{r^{2}}{\alpha^{\prime}}}_{0}dtt^{1/2}e^{-t}$
(4.1)

computing the BSW effect with this addition to the metric bounds the COM
energy to of the order of the black hole mass,
$\sqrt{s}\leq\mathcal{O}(M_{0})$. This new bound is reasonable and implies the
Regge limit intermediate string can theoretically have a large portion of the
black hole’s mass. This is promising for the strings involved in a Blanford-
Znajek-like process.

The presence of the BSW effect leads to implications on complementary and UV-
completion. Understanding where the BSW effect can arise and the limitations
suppressing large COM energies is essential.

Many limitations in the literature haven’t been considered for stringy
processes. The detuning of parameters inducing the BSW effect also relaxes
some suppressive effects. The critical particle may take a long time to reach
the horizon (disallowing multiple scattering) and fall on a floating orbit
where the radiation to infinity matches and changes the extremality of the
hole along the way. With longitudinal spreading, the ample proper time to
reach BSW can also be relaxed by string spreading.

It may take a large amount of time for critical particles from infinity to
reach the collision zone. It is not apparent how string spreading can resolve
this since spreading occurs in the near-horizon region, far from
$r=r^{+}+\delta$. The breakdown of EFT here requires a long waiting time, like
the Schwarzschild rapidity-though-time-displacement calculation.

Events near the horizon are red-shifted before escaping, which may spoil the
chance for a direct string signature in the overtones of QNM. The COM energy
is controlled by the near extremal spin parameter $a=1-\epsilon$. For a BSW
event occurring on $r_{+}$, ISCO, MBCO, and CPO have the following dependence
on $\epsilon$ [30],

$\displaystyle s_{\text{ISCO}}\sim\frac{1}{\epsilon^{1/3}}$ (4.2)
$\displaystyle s_{\text{MBCO}}\sim\frac{1}{\epsilon^{1/2}}$ (4.3)
$\displaystyle s_{\text{CPO}}\sim\frac{1}{\epsilon^{1/2}}$ (4.4)
$\displaystyle s_{r^{+}}\sim\frac{1}{\epsilon^{1/2}}.$ (4.5)

In [32], it is shown there are multiple scattering events that can enhance the
energy escaping the near horizon geometry at the cost of spinning down the
black hole. However, the spin-down of an extremal hole to sub-Planckian BSW
event takes over around the Hubble time for supermassive,
$M\sim\mathcal{O}(10^{8}M_{\text{Sun}})$. The Hawking temperature is less than
the CMB time, so the loss of extremality by Hawking radiation, especially for
large holes, is negligible.

In [19], it is shown the induced metric from Nambu-Goto strings has similar
(though not identical) evolution to magnetic ‘field sheet’ around the rotating
black hole. Strings like magnetic fields are slowed when
$\frac{d\phi}{dt}>\omega$ and accelerated by the geometry
$\frac{d\phi}{dt}<\omega$. This is another effect, facilitated by a Penrose-
like process, that is useful for extended objects. The fine-tuning of critical
particles is unnecessary for stringy particles, the geometry drives the string
to have $\frac{d\phi}{dt}=\omega$, which matches the angular speed of the
horizon. String zero modes with deviation from the co-rotating frame have non-
zero energy flux scaled by the string tension. With a modest string coupling
parameter, the geometry can induce string decay. Critical strings are not
stretched nor drained by the geometry and fall unabated decay.

There are several future directions to take using the possible Planckian
energy of a BSW event in the fiducial frame. This effect occurs for other
extremal holes with exotic charges as well. Since the Regge amplitude is large
for extended objects, branes-string or brane-brane scattering is large. In the
Regge limit, gravitational data is probed, and an interesting direction may
involve considering black holes source by string charges.

##  5 Comments on the Implication of the BSW effect for extremal black holes

###  5.1 Signal out?: Comments on Quasinormal Modes and Superradiant modes

Given the kinematics of the Regge limit described in (2.25), where
$\sigma_{C}=1$, particles do not easily escape the horizon regime. The final
state critical particle has a chance to escape but would classically take a
divergent amount of time to overcome its centrifugal barrier. The possibility
of quantum tunneling past the barrier or multiple perturbing scatter processes
could lead to the critical particle leaving a string signature in outgoing
modes. The critical condition, $E_{\hat{C}}-\omega_{H}L_{\hat{C}}=0$, is
reminiscent of the superradiant bound for quasinormal modes,
$\nu-m\omega_{H}\leq 0$, so one might expect a stringy signature in SR mode
111The is quasi-classical argument. There is a large amount of work involving
SR mode and the NHEK/CFT correspondence[16] [5][15]. Analyzing the BSW effect
and detuning from longitudinal string spreading in this context would be
interesting to explore.

Our Regge setup involves an infalling usual string, whose final state
$\hat{U}$ is also plunging; however, outgoing usual strings can similarly
exhibit Regge behavior. The usual trajectories may originate from deeper in
the throat, $p^{l}_{U}>0$ and $p^{l}_{\hat{U}}$, which can possibly leave some
signature in some outgoing quasinormal modes.

There is an additional issue, even if the data can escape the Kerr throat– the
redshift. The large gravitational redshift reduces the asymptotic local final
state energy heavily. There is some tension regarding the strength of the
redshift [23] [5] [41] [24]. Regardless of the level of redshift, multiple
scattering with particles or the barrier can induce Penrose processes
(sometimes called the super-Penrose process for several collisions) that can
increase the energy severalfold per interaction, and string data could escape
with modest energy, assuming the quantum gravitation data survive the
interactions. The efficiency of each collision,
$\eta_{i}=\frac{E_{\hat{C},i}}{E_{D,i}+E_{C,i}}$, is large for extremal black
holes and is divergent at CPO, $\eta_{\text{CPO}}\sim\frac{1}{\sqrt{1-a^{2}}}$
[32]. Near the horizon, the efficiency is $\eta=(2+\sqrt{3})^{2}>10$ and the
COM energy diverges. The hole can donate energy and angular momentum to
$\hat{C}$ but there is a finite number of scattering possible before the
effect fails, and suggests this is

$\displaystyle N_{\text{scatter}}\sim\sqrt{\log_{10}(M/m_{\hat{C}})}.$ (5.1)

For astrophysical black holes and initial electron-like masses, Planckian
energies can extracted before the hole contracts [32]. For point particles,
tuning is required in these processes, but strings whose zero mode follows
point particles (when tidal forces are weak) can reach these energies without
such tuning. Multiple scattering requirements are related to an n-point
scattering amplitude, $n>4$, studied in [27][9], and Regge behavior and
longitudinal spreading are still present.

If $\hat{C}$ is a graviton, which is a non-coherent state,

$V\sim\frac{1}{\alpha^{\prime}}\int d^{2}z\sqrt{g}(h_{\mu\nu}\partial
X^{\mu}\bar{\partial}X^{\nu}),$ (5.2)

while a coherent state is constructed from $e^{V}$. [28] shows gravitational
wave interferometry can detect quantum gravity signatures. Stringy physics may
be found in high-energy processes that produce QNM modes or late-time tails.

It is well-known that near the horizon of an extremal Kerr black hole is a CFT
dual [16]. The only scalar modes that survive are the superradiant ones with
$\nu\hat{t}=m\hat{phi}$ (hatted variable are in NHEK coordinates) and
$\nu\sim(E/m)_{\hat{C}}$. This is the SR bound. We expect the string’s
effective field decomposition to be

$\Psi=\int
d\nu\sum_{l,m}R(\hat{r})S_{sml}(\theta,a\nu)\exp(i\nu\hat{t}-im\hat{\phi})$
(5.3)

but the NHEK limit seems to reject all modes except the superradiant ones. We
therefore assume

$\Psi_{\hat{C}}=R(\hat{r})S_{sml}(\theta,a\nu)\exp(-i\nu\hat{t}+im\hat{\phi})+\mathcal{O}(m_{\hat{C}}/E_{\hat{C}})$
(5.4)

where we ignore the effect of the mass. According to the EK geometry, this
mode is exponentially excited from the hole and rejected, extracting energy.
For $s=\pm 2$ and $a\nu\rightarrow\infty$, we have a peak at the south pole
and north pole, respectively [7]. This may mean there is a stringy signature
in jets. The local tidal force measurable at large $r$ by an interferometer is

$\mathcal{E}_{\hat{\theta}\hat{\theta}}-i\mathcal{E}_{\hat{\phi}\hat{\theta}}\sim\frac{\sqrt{\nu^{3}M}}{r}S_{sml}(\theta,a\nu)\exp(-i\nu\hat{t}+im\hat{\phi}+i\nu
r_{*})$ (5.5)

where the hatted coordinates are local. Deep horizon physics may be detected
at large distances. The vertex operators of string theory are analogous to
creation operators in QFT.

The Kerr background is known to be dual to a CFT representing information near
the superradiant limit on near the extremal horizon. The implications of
longitudinal spread translates to non-local distributions in the CFT.
Quasinormal modes near the SR-limit corresponds to left-moving fields in CFT.
It would be interesting to consider large longitudinal string spreading
parameterise in terms of an outgoing signal.

The ignored mass, $m_{\hat{C}}$, may play an interesting role in the late-time
tail of Kerr holes. The late-time function for massless perturbations is [22]

$\displaystyle|\Psi|\sim t^{-(2l+3)}$ (5.6)

whereas for massive perturbations,

$\displaystyle|\Psi|\sim t^{-(l+3/2)}\sin(m_{\hat{C},\hat{U}}t)$ (5.7)

There may be high-frequency undulations in the late-time tail function.

###  5.2 Large Strings

The dynamics of large cosmic strings in a Kerr background have been of
interest [14] [38]. These large stable strings, captured by a Kerr black hole,
interact by sapping and imparting energy with the hole in a fashion similar to
the Blanford-Znajeck process, but scales by the string tension,
$\mathcal{O}(1/g\alpha^{\prime})$ [21][19]. These studies analyze the
particular case of rigid strings, i.e., time-independent string configuration.
Still, the intersection of the BSW effect, non-local longitudinal spreading,
and large string may be an exciting avenue to explore in future works.

The regime of interest for perturbative string scattering scenarios is still
at weak coupling, $g^{2}_{o}\sim g_{c}\ll 1$, as needed for the tree-level
analysis above, and we can see from the Nambu-Goto action of an on-shell
string the nature of the energy extraction/deposition. For stationary
($r=\text{cons}$) rigid strings with angular velocity $\Omega=\frac{E}{L}$,
the induce metric with the static gauge $t=\tau$ is [19]

$\displaystyle h_{tt}=-N^{2}+g_{\phi\phi}(\Omega-\omega)^{2}$ (5.8)

for critical infallers $\Omega=\omega_{H}$. When $h_{tt}=0$ near the horizon,
rigid data on the worldsheet hits a limit, where the time-translating killing
vector is null– the Killing horizon. The conserved momentum flux onto the
world-sheet goes as

$\mathcal{P}_{\text{flux}}=\frac{1}{4\pi g\alpha^{\prime}}\frac{d\sqrt{-\det
h}}{d(d\phi/d\sigma)}$

which is $\mathcal{O}(N)$ and exhibiting the tension scaling. A rigid critical
string gains no energy from the black holes, while the usual rigid strings do
the bulk of the energy extraction outside the throat where the lapse function
is finite, $N^{2}=g_{\phi\phi}(\Omega-\omega)^{2}$. As seen in [21], rigid
string around a hole is analogous to the Blanford-Znajek effect, which
supplies an astrophysical jet. Therefore, an astrophysical jet may be another
avenue for such quantum gravitational data. So, energy is pumped into the
string if it is too slow, $E/L<\omega_{H}$, and drained is $E/L>\omega_{H}$.
Only critical strings are exempt from this energy exchange. Critical strings
may naturally populate an equilibrium state, with the hole requiring even less
tuning to attain critical momenta.

Although energy and momentum fluxed into the string are proportional to $N$, a
$1/N$ amount of time needs to be spent in the throat to extract a significant
amount of energy. Assuming stabilizing factors like weak coupling to suppress
decay or warped tension, ala the Randall-Sundrum model, to get long-lived
strings allowing for a stringy version of the Blanford-Znajek process. It
seems the large string, in particular, has a chance to extract a large amount
of energy from the hole. Further, considering the intermediate string, between
a usual and critical string, as usual, it will also have divergent energy
extracted from the hole. The Penrose effect is less efficient than its
collisional counterpart, the BSW effect. However, the Penrose process may play
a pivotal role in exciting decaying strings into black hole polar jets,
analogous to the BZ process. As studied in [38], string theory signatures may
uniquely be found in astrophysical jets and probe effective string tension at
high energies.

###  5.3 Tidal forces

In [17], it is shown adding curvature terms to the leading effective
gravitational theory leads to diverging tidal forces consistent with
analyticity, unitary, and causality, and supported by local string theory. The
divergence in tidal force at the extremal Kerr horizon may be related to the
divergence COM energy, a la the BSW effect. The context is unclear but it is a
curiosity to see if there is more than a connection between divergence in
tidal force from curvature terms and longitudinal spreading, as both have
origins in a string theoretical context. This effect could happen in tandem
with the BSW effect since modest acceleration does not spoil the BSW effect
[36].

It is natural for string theoretical black holes to be embedded in an
asymptotic AdS space-time. Extremal BHs in this context may exhibit curvature
singularities at the horizon [18]. This singularity, generated by
backreactions of the field, leads to a curvature singularity that grows with
the size of the horizon, $r_{+}$, as opposed to typical tidal forces, which
increase for smaller black holes. Considering backreactions with the BSW
effect and longitudinal spreading would be interesting. Lastly, a string with
a large $p^{+}$ encounters tidal forces as a shock wave, changing the string’s
transverse modes [9]. The study of shocks seems relevant to this analysis.

###  5.4 Warping and Compactification

The Kerr space-time as described above can be embedded in a higher space-time
to one compact dimension making loop calculations tractable– and AdS radius,
R[27]. With sufficiently large R and small string coupling leads to the
supergravity theory being dual to quark-gluon plasma QCD with vanishing beta
function. The warped metric is then

$\displaystyle
ds^{2}=e^{2A(u)}g_{BL,\mu\nu}dx^{\mu}dx^{\nu}+e^{-2A(u)}du^{2}+ds^{2}_{M^{5}}$
(5.9)

For center-of-mass energy, $s$, sufficiently large, $G_{N}s\sim 1$, we may
need higher loop corrections. In [27], eikonal contributions are added to the
interaction amplitude to calculate higher loop effects. This warping scales
momenta for observers seeing non-compact momenta, so Mandelstam variable are
now $s\rightarrow e^{2A(u_{0})}s$ and $t\rightarrow e^{2A(u_{0})}t$. Further,
this also scales the string tension,

$\displaystyle\frac{1}{4\pi\alpha}\rightarrow\frac{e^{2A(u_{0})}}{4\pi\alpha}$
(5.10)

and create KK excitation with the lightest mode having a mass of
$R_{AdS}e^{A(u_{0})}$, where $u_{0}$ is the position in the compact AdS
direction where the theory is localized. This warping makes the tension a free
parameter and provides the extra dimension that allows for tractability the
eikonal amplitude for string scattering. This warped string tension may allow
for a regime near the black hole horizon where large strings are long-live. An
analysis of analogous to cosmic string in a cosmological model may provide
insight into quantum gravity dynamics. The strings are fairly localized in
around $u=u_{0}$, such that $A(u_{0})<<1$. This can be seen in its dressed
wave function.

$\displaystyle\Phi=e^{ip_{u}\cdot
x_{u}-\frac{1}{2}\sum_{i}\omega_{i}(e^{2A}X_{i}^{\perp}X_{i}^{\perp}-e^{-2A}U_{i}U_{i})}$
(5.11)

the frequency of strings transverse mode, $\omega_{i}$, is inverse
proportional to the the longitudinal momentum

$\omega_{n}^{\perp}\sim\frac{n}{\alpha^{\prime}p^{+}_{U}}.$

At large $p^{+}_{U}$, the string transverse diffusion becomes large, but
interestingly, and spread into the large $A(u_{0})<<A(u)$ region is suppressed
by the larger effective frequency.

Considering the warping in the extra-dimensional throat

$\displaystyle
ds^{2}=\frac{r^{2}}{R^{2}}dx^{\mu}dx_{\mu}+\frac{R^{2}}{r^{2}}dr^{2}+R^{2}d\Omega^{2}_{5}$
(5.12)

and assuming the throat is sufficiently weakly curved

$\displaystyle\frac{R^{2}}{\alpha^{\prime}}>>1$ (5.13)

leads to a modification of the drama condition where longitudinal spreading
dominates. Assuming the position in the throat is $r=r_{B}>r_{+}\rightarrow
r_{B}>\frac{R^{2}}{r_{+}}$

$\displaystyle m$
$\displaystyle>\frac{r_{+}}{\alpha^{\prime}_{B}}=\frac{r_{+}r_{B}^{2}}{\alpha^{\prime}R^{2}}\implies$
(5.14) $\displaystyle m$ $\displaystyle>\frac{R^{2}}{\alpha^{\prime}r_{+}}$
(5.15)

This modification relaxes the drama condition, which becomes more accessible
for larger black holes.

###  5.5 Comparison to Schwarzschild longitudinal spread

For longitudinal spreading around a Schwarzschild black hole, a large amount
of time separation is required to induce a significant boost. This leads to
stringy interaction between late and early infallers. In an extremal Kerr near
horizon background, the large boost needs only a critical and usual infaller
involved. Without multiple scattering or infallers origin from infinity,
critical infallers take a large amount of time to approach the centrifugal
barrier at the horizon, so it seems that infallers with asymptotic origins may
need a large time separation to generate a large COM energy even for EK
geometry. Still, generically, with multiple scattering or origins from the
horizon, like EK superradiant modes, implies drama. Just as early/late hawking
modes can interact with late/early infallers, critical modes in the Kerr
geometry may be sensitive to usual infallers. The connection between horizon
modes and infallers through longitudinal spreading seems intrinsic to black
hole physics and the understanding of microstates in a quantum gravitational
context. It seems inevitable that extremal black holes induced stringy center-
of-mass energies between usual and critical trajectories. Extremal black
branes endowed with superstring charges reproduce the entropy of black holes
[33][16]. So, analyzing the fate of the BSW event in the D-brane description
at small $gN$ or the black brane description at large $gN$ is interesting.
Fundamentally, the information probes in each setup should be the same as long
as the interaction survives the $gN$ scaling. What implications does this have
for complimentary? Given that D-brane interaction can probe below the string
scale, the BSW effect may be an avenue to probe the quantum nature of gravity
at sub-string scales.

##  6 Future direction and Conclusion

The BSW effect is relevant for string theoretic scenarios in an extremal Kerr
background, making Planckian energies accessible. String spreading, especially
the longitudinal direction, detunes the kinematic requirements, leading to the
relevance of string theoretic interaction for the EK geometry. Effective field
theory, in the context of the BSW effect, breaks down in this geometry. So,
the near-horizon EK region is an interesting theoretical laboratory for
studying quantum gravitation physics.

Kerr black holes have been studied in several contexts relating to dualities,
magnetic fields, cosmic string interaction, prediction of microstates,
quasinormal modes, the addition of curvature terms, and more. It would be
interesting to analyze if the application of the BSW to string theoretic data
is consistent with other phenomena developed for spinning black holes. Lastly,
since the BSW effect occurs for non-equatorial physics, generalizing the above
conclusions away from the equator would be an interesting future direction.

##  7 Acknowledgement

We are grateful to Eva Silverstein, Matthew Dodelson, Byungwoo Kang, and Alex
Mousatov for the extensive discussion. Also, thanks to Giuseppe Bruno De Luca
for the support.

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## Appendix

###  7.1 Local Kinematics

The BL coordinates obscure the scattering dynamics, so we use proper distance
and three-velocities

$v^{i}=\frac{p^{\mu}e_{\mu}^{i}}{-p^{\mu}e_{\mu}^{0}},$ (7.1)

to show the difference between critical trajectories, and usual trajectories
near the horizon. Asymptotically infalling critical particles have only enough
energy to reach the turning point with vanishing radial speed,
$\dot{r}=\sqrt{E^{2}-V_{\text{eff}}^{2}}$, while the proper velocity is
finite, $\dot{l}=\sqrt{\frac{3L_{C}^{2}}{4M^{2}}-\delta_{C}}$. We can analyze
the angle separation in terms of three velocities

$\displaystyle v^{l}$
$\displaystyle=-\sqrt{1-\frac{M^{2}}{L_{C}^{2}}(\delta_{C}+\frac{L_{C}^{2}}{g_{\phi\phi}})}$
(7.2) $\displaystyle v^{\phi}$ $\displaystyle=\frac{M}{\sqrt{g_{\phi\phi}}}.$
(7.3)

The lowest momentum critical velocity vector is

$(v^{l},v^{\phi})=(0,\frac{1}{2})$ (7.4)

with its turning point at ISCO. Its CPO counterpart has three-velocity

$(v^{l},v^{\phi})=(\pm\frac{\sqrt{3}}{2},\frac{1}{2}).$ (7.5)

Plunging usual orbits have vanishing $v^{\phi}$ at the horizon regardless of
energy or angular momentum, so for equatorial usual orbits it is

$\displaystyle v^{l}$
$\displaystyle=\pm\sqrt{1-\frac{N^{2}}{(E_{U}-\omega_{H}L_{U})^{2}}(\delta_{U}+\frac{L_{U}^{2}}{g_{\phi\phi}})}$
(7.6) $\displaystyle v^{\phi}$
$\displaystyle=\frac{L_{U}N}{\sqrt{g_{\phi\phi}}(E_{U}-\omega_{H}L_{U})},$
(7.7)

which is $(\pm 1,0)$ at the horizon. The longitudinal string spreading spreads
in the direction of large light cone momentum, which follows the usual
trajectory in this frame.


# Neural Network Enhanced Single-Photon Fock State Tomography

Hsien-Yi Hsieh,1, Yi-Ru Chen,1, Jingyu Ning,1 Hsun-Chung Wu,1 Hua Li Chen,2
Zi-Hao Shi,1 Po-Han Wang,3 Ole Steuernagel,1 Chien-Ming Wu,1 and Ray-Kuang
Lee1,2,3,4,5 1Institute of Photonics Technologies, National Tsing Hua
University, Hsinchu 30013, Taiwan
2Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
3Department of Electrical Engineering, National Tsing Hua University, Hsinchu
30013, Taiwan
4Center for Theory and Computation, National Tsing Hua University, Hsinchu
30013, Taiwan
5Center for Quantum Science and Technology, Hsinchu 30013, Taiwan
<EMAIL_ADDRESS>
###### Abstract

Even though heralded single-photon sources have been generated routinely
through the spontaneous parametric down conversion, vacuum and multiple photon
states are unavoidably involved. With machine-learning, we report the
experimental implementation of single-photon quantum state tomography by
directly estimating target parameters. Compared to the Hanbury Brown and Twiss
(HBT) measurements only with clicked events recorded, our neural network
enhanced quantum state tomography characterizes the photon number distribution
for all possible photon number states from the balanced homodyne detectors. By
using the histogram-based architecture, a direct parameter estimation on the
negativity in Wigner’s quasi-probability phase space is demonstrated. Such a
fast, robust, and precise quantum state tomography provides us a crucial
diagnostic toolbox for the applications with single-photon Fock states and
other non-Gaussisan quantum states.

##  I Introduction

Quantum state tomography (QST) refers to the methodology in reconstructing the
unknown quantum state with the acquired experimental data QST-book ; RMP-CV .
The maximum likelihood estimation (MLE) for QST finds the best-fitted
probability distribution by treating the whole density matrix as the target of
estimation Banaszek ; Hradil ; Lvovsky . As long as a sufficient computational
effort is applied, MLE consistently yields a robust estimation, with the
effectiveness in estimation strongly depending on the quantity of available
data. Nowadays, QST has been successfully implemented as a diagnostic toolbox
both for many qubits (or qudits) systems in higher dimensions and for
continuous variables in infinite dimensions QST-Furusawa ; QST-atom1 ; QST-
atom2 ; QST-ion-1 ; QST-ion-2 ; QST-SC .

However, MLE suffers from the overestimation problem as the required amount of
measurements to reconstruct the quantum state exponentially increases with the
number of involved modes. To overcome the overestimation problem in MLE,
several modified algorithms are proposed, such as permutationally invariant
tomography permu , quantum compressed sensing compress , tensor networks
tensor-1 ; tensor-2 , generative models generative , and restricted Boltzmann
machine RBM , by assuming some physical restrictions imposed upon the state in
question. Moreover, unavoidable coupling from the noisy environment makes the
reconstructions on the density matrix with state degradation embedded,
resulting in dealing with a non-sparse matrix in a larger Hilbert space.

With the power to find the best fit to arbitrarily complicated solutions,
machine-learning (ML) enhanced QST has demonstrated its advantages in
extracting complete information about the quantum states RBM ; QML-1 ; QML-2 ;
PRL-22 ; GAN . Furthure more, instead of using the reconstruction model in
training a truncated density matrix, with ML one may directly generate the
target parameters with a supervised characteristic model Symmetry . Such a
characteristic model-based ML-QST can be easily installed on edge devices such
as FPGA, serving as an in-line diagnostic toolbox for all the possible
applications. As an example, this ML-QST has also been applied to the
reconstruction of Wigner current current , demonstrating experimentally
quantum dynamics in phase space in great detail. Compared to the time-
consuming MLE, ML-QST paves the road toward a real-time and online QST current
; online .

With the benefits from the good properties of the Gaussian states, including
vacuum and squeezed states, a neural network can directly analyze the raw data
to obtain the first and second moments of probability density function. By
applying the well-developed methods in pattern recognitions generalizability ;
Pattern-1 ; Pattern-2 ; Pattern-3 , one can easily deal with various Gaussian
states, producing a single scan QST in speeding up data acquisition and data
processing PRL-22 . Nevertheless, difficulties arise for such a relatively
simple prediction map when non-Gaussian states are attacked. One may increase
the number of neurons in dealing with non-Gaussian states, however the
training process tends to cause overfitting problem.

In the family of non-Gaussian states, single-photon Fock states play the core
role as photonic qubits to carry quantum information encoded single-photon .
Although the request for an on-demand source of single photons has led to
intense research into developing truly deterministic single-photon states,
heralded single-photon sources can be easily generated through correlated
pairs of photons, by detecting one photon (the heralding photon). After the
first experimental observation in 1970 SPDC , nowadays, creating correlated
photon pairs from spontaneous parametric downconversion (SPDC) has been
routinely demonstrated with $\chi^{(2)}$ nonlinear crystals typeII .

To characterize a single-photon Fock state, the common method is based on the
second-order correlation function, $g^{(2)}(\tau)$, i.e., Hanbury Brown and
Twiss (HBT) interferometer HBT ; Glauber ; Sudarshan . The standard test for
single-photon sources is a value of the second-order correlation function of
the emitted field below $1/2$ at zero time delay, i.e., $g^{(2)}(0)<1/2$.
However, this criterion alone provides no information regarding the amplitude
of the single-photon contribution for general quantum states. In particular, a
low-intensity light source always has a vacuum contribution in the quantum
state of light, cloaking actual single-photon projection g2 ; low .

In addition to HBT measurements, nonclassical effects in the single-photon
Fock states can be demonstrated in phase space sq-thermal-3 , such as a
negative value in the Wigner function 9 . Homodyne detection of the rotated
quadratures provides an experimental implementation for the reconstruction on
the Wigner function in phase space, through the inverse Radon problem 10 ; 15
. Tomographic reconstruction of the single-photon states has been
experimentally realized first with phase-randomized pulsed optical homodyne
tomography single ; instant ; high-freq , then with continuous temporal-mode
matching 11 , toward having real-time and complete temporal characterization
of a single photon complete . The development on homodyne tomography also
provides a bridge between the single-photon and squeezed-vacuum states bridge
; antibunching ; antibunching-loss , as well as a methodology for various non-
Gaussian states such as two-photon Fock states and optical cat states two-
photon ; cat .

As the neural network predictor is often trained from some specific and
limited amount of data, in this work, we develop the machine-learning enhanced
single-photon Fock state tomography with the histogram-based architecture.
Histogram-based approaches are often used to reduce the computational cost 3 .
With an appropriately chosen bin-width for histogram, we demonstrate that the
resulting quantum state reconstruction can still keep fidelity high. Further
more, with the capability of hybrid quantum-classical neural networks or
quantum neural network, the improvement in increase accuracy while reducing
computational resources is also possible with quantum machine-learning QML .

The paper is organized as follows: in Section II, we introduce our
experimental setup to perform the homodyne detections on heralded single-
photon Fock states, generated from SPDC process inside a bow-tie cavity. Then,
the implementations of the histogram-based neural network are illustrated in
Section III. The comparisons on the predicted photon-number distributions from
MLE and neural network, also as a function of the SPDC pumping power, are
given in Section IV. Moreover, a direct parameter prediction on the negativity
is demonstrated, validating the feature extraction from our direct parameter
estimations. Finally, we summarize this work with some perspectives in Section
V.

Figure 1: (a) Our experimental setup to generate single-photon Fock state and
its quantum state tomography. (b) A simple schematic for generating heralded
single photon precess by spontaneous parametric down conversion (SPDC)
process. Here, SHG refers to the second harmonic generator; BHD: balanced
homodyne detector; OSC: oscilloscope; SNSPD: superconducting nanowire single
photon detector; FC: filter cavities; BS: beam splitter; PBS: polarizing beam
splitter. Figure 2: Measured data on the second-order correlation function,
$g^{(2)}(0)$, as a function of the recored heralding rate in kHz, with the
corresponding pump power labeled (on top) in mW. Figure 3: (a) The time
sequence of recorded BHD raw quadrature data measured from the oscilloscope.
Here, the SPDC pump power is 3 mW. (b) The histogram of the corresponding
probability density distribution, $p_{i}$ ($i=1\dots N$). (c) With $N=50$
inputs, a shallow neural network is applied to generate directly the predicted
probability for different photon numbers $w_{n}$, with $n=0,1,2$. Figure 4:
Measured noise level in dBm of our balanced homodyne detector (BHD). Here, the
spectrum for dark noise is depicted in Red color, up to 10 MHz. When the local
oscillator is operated at 30 mW, the spectrum of noise level is depicted in
Blue color, illustrating a maximum clearance of 18.6 dB, along with a 3 dB
bandwidth up to $7.45$ MHz.

##  II Experimental setup of our single photon quantum state tomography

The experimental setup for our heralded single-photon source and the quantum
state tomography is illustrated in Fig. 1. Here, the main laser source is a
continuous-wave (CW) Nd:YAG laser at the wavelength of 1064 nm. This laser is
split into two parts via a beam splitter (BS): one serves as the local
oscillator (LO) beam for balanced homodyne detector (BHD); while the other one
serves as the pump field for the second harmonic generator (SHG). The SHG
provides the frequency doubling at 532 nm, through a nonlinear crystal, i.e.,
periodically poled Lithium Niobate (PPLN), inside a bow-tie cavity. Then, the
green light of SHG signal is injected into another bow-tie cavity with a type-
II PPLN crystal inside, in order to perform the spontaneous parametric down
conversion (SPDC) process, which generates photon-pairs in two orthogonal
polarizations. The full width at half maximum (FWHM) of our SPDC cavity is
31.8 MHz and the free spectral range (FSR) is 1.052 GHz. The output of
orthogonal polarization beams, denoted as signal $|1\rangle_{s}$ and idler
$|1\rangle_{i}$ photons in Fig. 1(b), are separated by a polarization beam
splitter (PBS).

To ensure the mode-matching in degenerate modes, the idler photons from SPDC
cavity is injected into a filter cavity (FC) system with 6.5 MHz in bandwidth.
This FC system is composed of a triangle cavity and two Fabry–Pérot cavities.
The detection on idler photons is performed by a superconducting nanowire
single photon detector (SNSPD). Finally, the SNSPD triggers the BHD to record
the signal photons. As the Fock states are phase independent, we do not
perform the measurement on rotated quadratures. The extracted quadrature data
are obtained by integrating the temporal mode function on the experimental
data, i.e.,

$\hat{X}_{i}=\int_{\infty}f(t)\hat{x}_{i}(t)\,dt,$

with $\hat{x}_{i}$ being the $i$-th measurement data from the BHD. Here, the
temporal mode function $f(t)=\sqrt{\pi\gamma}e^{-\pi\gamma|t-t_{c}|}$ is
described by the center time for a trigger event $t_{c}$, with the decay rate
of SPDC, denoted as $\gamma$.

In Fig. 2, we report our measured data on the second-order correlation
function at zero time delay, $g^{(2)}(0)$, as a function of the recored
heralding rate from the detection on the heralding idler photon into SNSPD. At
the same time, the corresponding pump power (in mW) into the SPDC cavity is
also depicted. One can see clearly that our single-photon source demonstrates
$g^{(2)}(0)<1/2$ when the pump power is smaller than 10 mW, or with the
heralding rate about 4,000 kHz (4 MHz).

In addition to the second-order correlation function, we also perform the
quantum state tomography for the heralded single-photon state with homodyne
detection scheme. In Fig. 3(a), a typical time sequence of our noisy single-
photon source is demonstrated from our experimental raw data measured from the
oscilloscope. Our BHD output is then integrated after convolution with a
double-decayed (two time-constants) temporal mode function, along with a
detection correction treatment that assumes a detection efficiency of 92$\%$,
which is composed of the quantum efficiency of photodiodes ($99\%$), homodyne
visibility efficiency ($96\%$), and the circuit noise of homodyne detection
($97\%$).

To make sure the measured noise level is not contaminated, our homodyne
detectors are designed with a high common mode rejection ratio of more than
$80$ dB CMRR . As shown in Fig. 4, when the local oscillator is operated at 30
mW the spectrum of noise level depicted in Blue color demonstrates a clearance
of higher than 15 dB (with a maximum value of 18.6 dB). However, our BHD only
supports a 3 dB bandwidth up to $7.45$ MHz.

In the following, by considering the limit in our BHD bandwidth, we analyze
the SPDC pump power up to 3 mW (or the heralding rate lower than 2 MHz) on our
single-photon Fock state tomography with machine-learning. This operation
condition also reflects the scenario when the influence of vacuum
significantly enters into the actual single-photon projection, denoted as the
low-intensity limit.

##  III Histogram-based machine-learning enhanced quantum state tomography

Before introducing the histogram-based machine-learning architecture, we
conduct several tests on single-photon Fock state tomography by applying our
previously developed convolution neural network for quantum state tomography
with simulated raw quadrature data PRL-22 . Here, more than 10,000 mock data
set on noisy single-photon experiments are prepared, including different
percentages of single-photon and vacuum states 11 ; single ; high-freq ;
instant ; complete . However, the inferred fidelity from these testing data
set is limited to $0.96$ even the input dimensions are up to $10,000$. One
possible solution to improve the fidelity is to apply advanced machine
learning architectures or to perform complex hyper parameter adjustments.

As our goal is focusing on building a lightweight inference system which can
be embedded into quantum optics experiments, working on raw quadrature data
needs much more computation efforts in dealing with complicated data sets.
Instead, we construct histograms to reduce the input data size, as well as the
required computational cost.

To reconstruct the quantum state in our SPDC experiment, the corresponding
tomographic data is the recorded event from our homodyne measurement, i.e.,

$\displaystyle
p(X)=\sum_{n=0}^{\infty}w_{n}\frac{1}{\pi^{1/2}2^{n}n!}H^{2}_{n}(X)e^{-X^{2}},$
(1)

with the Hermit polynomial $H_{n}(X)$. Here, we already expand the probability
probability distribution in Fock (photon number) basis, with $X$ being the
value of rotated quadrature and $w_{n}$ being the photon number probability
(weighting factor). As the Fock states are independent to the quadrature
phase, we also apply the phase-average measurement 10 ; 11 ; 15 ; single ;
high-freq ; instant ; complete  to our homodyne data. Tomographic
reconstruction here is to estimate the photon number distributions $w_{n}$
from the measured quadrature data $X$.

By checking all the experimental data, we first set the quadrature value
between -3.2 and +3.2 as our range. Then, we divide this closed interval
$[-3.2,3.2]$ in the quadrature into 50 sub-intervals (bins), see Fig. 3(b).
The relative frequency of the $i$-th bin is calculated by
$f_{i}=\frac{N_{i}}{N}$, with $N_{i}$ denoting the counts in the $i$-th bin
and $N$ being the total counts in the quadrature axis. We want to remark that
in our numerical experiments $N=50$ bins (sub-intervals) are enough to achieve
good results.

The relative frequency of $i$-th bin, $f_{i}$, is used to estimate the value
of probability density defined on sampled quadrature values $\hat{X}_{i}$.

$\displaystyle\frac{f_{i}}{\Delta X}\simeq p_{i}(\hat{X}_{i}),\quad i=1\dots
N.$ (2)

Here, $\Delta X$ is the length of each bin and $\hat{X}_{i}$ is a specific
point in $i$-th bin. With a uniform gird on $X$, the estimated value of
probability density, $p_{i}$ is illustrated in Fig. 3(b). After this binning
process, our tomographic problem now is transferred to predict $w_{n}$ from
the estimated value $p_{i}$ defined on the discrete grid $\hat{X}_{i}$. When
the number of input quadratures is large enough, the relative frequency
converges to the probability, which enables a good approximated value of
probability density $p_{i}$. Otherwise, errors occur in the binning process.

In Fig. 3(c), the schematic of this histogram-based neural network for single-
photon Fock state QST is illustrated. Here, we apply a shallow neural network
for 50 inputs from the histogram-based inputs, i.e., the estimated values of
quadrature probability density $p_{i}$. Our learning task is to build a map
supporting multiple instance setting:

$\displaystyle p_{i}\rightarrow w_{n}.$ (3)

Then, the outputs can generate directly the predicted probability for
different photon numbers $w_{n}$.

To train our prediction map inferring different quantum states from various
tomographic data, we feed machine with well prepared training data set
$\left\\{p^{k}_{i},w^{k}_{n}\right\\}$. Here the index $k$ counts for
different instances which describe specific quantum states. In this learning
task, we use 10,000 training data set ($k=10,000$) and another 10,000 testing
data with different weighting values of $w_{0}$, $w_{1}$, $w_{2}$, i.e.,
vacuum, single-photon, and two-photon Fock states, respectively. A uniform
distribution in $[0,1]$ is sampled. Further more, by considering the low-
intensity condition in our SPDC experiments, we let $w_{0}+w_{1}+w_{2}=1$
without other multi-photon events. With the simulated data as the ground
truth, our histogram-based neural network can ensure the average fidelity
higher than 0.999 with 10,000 instances in testing data sets.

In our single-photon Fock state QST, we also remark that the prediction map
can be builded only with a shallow neural network, see Fig. 3(c). Additional
hidden layers are not needed here. We also perform the numerical test
confirming that the neural network can maintain a good performance without any
introduction of nonlinear activation functions. In the training process, we
train 10 epochs such that the mean squared loss of both training and testing
data decreases to $10^{-7}$. The optimization process employs the Adam
optimizer with default hyper-parameter settings, including a learning rate of
0.01.

Figure 5: (a) The measured probability density in the quadrature ($X$) from
homodyne tomography data, with SPDC pump power at 3 mW. (b) The corresponding
Wigner distribution function in the phase space. Here, the fitting curve for
the probability density (in Yellow color) is fitted by Eq. (4) with
$\eta=0.631$. Figure 6: The predicted probability distribution generated from
our histogram-based QST from the same measured homodyne tomography data shown
in Fig. 5(a), with SPDC pump power at 3 mW. Here, we have $w_{0}=0.363$,
$w_{1}=0.606$, and $w_{2}=0.027$, corresponding to vacuum, single-photon, and
two-photon Fock states, respectively.

##  IV Results and Discussions

To verify the validity of our histogram-based QST, first we compare directly
with the measured probability density in the quadrature ($X$) from homodyne
tomography data. As shown in Fig. 5(a), with the SPDC pump power at 3 mW, the
recorded homodyne data illustrates clearly a non-Gaussian probability
distribution. By assuming the light field has the form:

$|\psi\rangle=(1-\eta)|0\rangle+\eta|1\rangle,$

as a noisy single-photon state $|1\rangle$ coupled to the vacuum $|0\rangle$,
with the weighting factor $\eta$ single . The corresponding probability
distribution function has the form

$\displaystyle P(X;\eta)=\sqrt{\frac{2}{\pi}}[1-\eta(1-4X^{2})]e^{-2X^{2}},$
(4)

which gives the best fitting curve depicted in Yellow color, see Fig. 5(a),
with $\eta=0.631$.

With the Wigner-transform, ${\cal
W}[\hat{O}](x,p)=\int_{-\infty}^{\infty}dy\;O(x-\frac{y}{2},x+\frac{y}{2})\;{\rm
e}^{\frac{{\rm i}}{\hbar}py}$ for a single-mode operator given in coordinate
representation $\langle x-y|\hat{O}|x+y\rangle=O(x-y,x+y)$ Hancock_EJP04 ;
Cohen_LectureNotes18 , in Fig. 5(b) we show the corresponding Wigner
distribution function in the phase space. A dip in the origin can be clearly
seen, representing the negativity in the Wigner’s quasi-probability
distribution as a signature of single-photon Fock states.

In Fig. 6, with the same measured homodyne tomography data shown in Fig. 5(a),
i.e., the SPDC pump power at 3 mW, we show the predicted probability
distribution generated from our histogram-based QST. Here, in addition to
vacuum state $|0\rangle$ and single-photon Fock state $|1\rangle$, we also
take possible two-photon Fock state $|n=2\rangle$ into consideration. The
resulting photon number distribution gives $w_{0}=0.363$, $w_{1}=0.606$, and
$w_{2}=0.027$, corresponding to vacuum, single-photon, and two-photon Fock
states, respectively. The discrepancy between the direct fitting and our
histogram-based QST, see Fig. 5(a) and Fig.6, comes from the small portion in
the two-photon Fock states.

As a benchmark, in Fig. 7 we also apply MLE method to verify the experimental
data at different SPDC pump power. Here, both MLE and neural network generate
a tiny value for the three-photon Fock state, i.e., $w_{3}<10^{-13}$,
confirming that at most only up to $w_{2}$ (corresponding to two-photon Fock
states) is non-negligible. With an increment in the SPDC pump power, the
coefficient $w_{1}$ for single-photon Fock states increases; while the
coefficient $w_{0}$ for vacuum states decreases. As shown in Fig. 7, both two
approaches exhibit an almost the same curve, thereby indicating the
equivalence and accuracy of these two estimations. To our surprise, at the
same time our SPDC process inside a cavity also produces a small portion of
two-photon Fock states, i.e., the average value of $w_{2}=0.044$ single .

To avoid the overfitting problem in applying machine learning, we start with
the simplest single-layer shallow neural network (only with 50 neurons). As we
do not apply any complicated structures, the only factor changes the shallow
neural network is the input size, which depends on how many discrete points
are taken for the quadrature probability density. For the tests, we have
increased the input size to $75$ and $100$, but the resulting fidelity both
generate $0.999$ without showing significant improvements.

We want to remark that using a finer discretization also requests the
increment in the number of input quadrature points. In other words, more data
needs to be obtained in the experiment, which reduces the overall tomographic
reconstruction efficiency due to the speed of data collection. In our single-
photon experiment, even though we only collected $8,000$ quadrature points,
our current setting can achieve the target, which is also verified with the
maximum likelihood estimation. As a comparison, we also apply the CNN
architecture developed for squeezed states PRL-22 , to our single-photon
experiments. However, the resulting fidelity can only achieve $0.95$ due to
the intrinsic overfitting problem by applying Gaussian states to map non-
Gaussian Fock states.

Figure 7: Photon number distributions ($w_{0}$, $w_{1}$, and $w_{2}$) versus
different SPDC pump power, compared with two different methods: maximum
likelihood estimation (MLE) in ‘circles’ and our histogram-based neural
network in ‘squares’. For the histogram-based neural network, the input bin
number is $N=50$. Figure 8: Negativity in the Wigner function, $W(0,0)$,
versus different SPDC pump power, compared with two different methods: maximum
likelihood estimation (MLE) in ‘squares’ and and our histogram-based neural
network in ‘squares’. For MLE, $W(0,0)$ is calculated after the reconstruction
on the quantum state; while the neural network directly estimates $W(0,0)$ by
Eq. (5) form the quadrature histogram.

In addition to the photon number probability estimation, our histogram-based
neural network can also predict directly the target parameters, without the
reconstruction on the full quantum state. Here, we focus on the negativity in
the Wigner’s quasi-probability distribution, which manifests the most non-
classical signature of single-photon Fock states. Analytically, the value in
the origin of Wigner function $W(0,0)$ has the form

$\displaystyle W(0,0)=\frac{1}{\pi}\sum_{n=0}^{2}(-1)^{n}w_{n}L_{n}(0),$ (5)

with the Laguerre polynomial $L_{n}(x)$ 9 . In Fig. 8, we illustrate the
powerful feature in our histogram-based neural network by directly infer the
negativity in the Wigner function, $W(0,0)$, versus different SPDC pump power.
Here, we also compare the results generated from two different methods: MLE-
QST and our histogram-based neural network. It is noted that in MLE, $W(0,0)$
is calculated after the reconstruction on the quantum state. Nevertheless, our
neural network directly estimates $W(0,0)$ directly by using Eq. (5) form the
quadrature histogram. As one can see, again, our ML parameter estimation gives
almost the same results as that from MLE.

By considering SPDC experiments in the low-intensity limit, the condition to
have a negative value in $W(0,0)$ happens at $w_{1}=0.5$, corresponding to our
SPDC pump power at 0.8 mW. As shown in Fig. 8, our histogram-based neural
network, also confirmed by MLE, precisely estimate the negativity happens when
the SPDC pump power exceeds 0.8 mW.

Last but not least, due to the perfect agreement between the results from MLE
method and our histogram-based QST, we choose 50 bins as a good estimation.
Unlike MLE method relying on the iteration algorithm, we can have a reusable
prediction map from our neural network. This lightweight feature makes it
easier to install such an inference function on edge devices like FPGA. Most
of time-consuming task in our approach is the pre-processing, i.e., the
histogram binning process, which takes bout 0.01 seconds. However, it only
takes about 3 msec to subsequently go through such a tiny $50\times 3$ network
for inference. The total time consumed is about 0.01 + 0.003 seconds.

##  V Conclusion

In summary, we develop a neural network enhanced single-photon Fock state
tomography and apply it to the heralding single-photon source from spontaneous
parametric down conversion (SPDC) process experimentally. Instead of tackling
on the raw quadrature data, which needs a lot of computational cost but infers
a limited fidelity up to $0.96$, our histogram-based quantum state tomography
(QST) keeps the fidelity as high as $0.999$. Moreover, target parameters, such
as the photon number distribution and the negativity in Wigner function, can
be directly predicted, without dealing with the density matrix in a higher
dimensional Hilbert space.

Through the validation with the experimentally measured data acquired from the
balanced homodyne detectors, perfect agreement to the results obtained by
maximum likelihood estimation (MLE) is also clearly demonstrated. Our machine-
learning enhanced QST can be easily installed on edge devices such as FPGA as
an in-line diagnostic toolbox for all the possible applications with single
photons. Moreover, this fast and easy-to-install methodology helps us with a
better understanding on quantum optics experiments with non-Gaussian states,
such as two-photon Fock state tomography two-photon , photon-added squeezed
states cat , tomographic test of Bell’s inequality Bell , and the
reconstruction of non-classicality decoherence .

## Acknowledgements

This work is partially supported by the Ministry of Science and Technology of
Taiwan (Nos 112-2123-M-007-001, 112-2119-M-008-007, 112-2119-M-007-006),
Office of Naval Research Global, the International Technology Center Indo-
Pacific (ITC IPAC) and Army Research Office, under Contract No.
FA5209-21-P-0158, and the collaborative research program of the Institute for
Cosmic Ray Research (ICRR) at the University of Tokyo.

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