# Back to Square One: Superhuman Performance in Chutes and Ladders Through
Deep Neural Networks and Tree Search

Dylan R. Ashley
DeeperMind (Holiday Office)
London, Kiribati
4625 kHz Shortwave
Anssi Kanervisto11footnotemark: 1
DeeperMind (Moonshot Office)
8837 London, Space
5448 kHz (day), 3756 kHz (night)
Brendan Bennett11footnotemark: 1
DeeperMind (London Office)
London, Ontario, Quebec
5473 kHz (day), 3828 kHz (night)
ordering determined by games of Chutes and Ladders

###### Abstract

We present AlphaChute: a state-of-the-art algorithm that achieves superhuman
performance in the ancient game of Chutes and Ladders. We prove that our
algorithm converges to the Nash equilibrium in constant time, and therefore
is—to the best of our knowledge—the first such formal solution to this game.
Surprisingly, despite all this, our implementation of AlphaChute remains
relatively straightforward due to domain-specific adaptations. We provide the
source code for AlphaChute here in our Appendix.

##  1 Introduction

Deep Learning by Geoffrey Hinton111according to several random people we
asked, this is shown by one of the following works: Hinton et al. [1990,
1998], Neal and Hinton [1998], Fahlman et al. [1983], Guan et al. [2018],
Hinton [2000], McDermott and Hinton [1986], Kiros et al. [2018], Frosst and
Hinton [2017a], Brown and Hinton [2001a], Carreira-Perpiñán and Hinton [2005],
Hinton et al. [2005], Heess et al. [2009], Fels and Hinton [1995], Hinton and
van Camp [1993], Deng et al. [2020a], Memisevic and Hinton [2007], Ranzato and
Hinton [2010], Ranzato et al. [2011], Susskind et al. [2011], Tang et al.
[2012a], Taylor et al. [2010], Frey and Hinton [1996], Hinton [1976], Sloman
et al. [1978], Deng et al. [2020b], Mnih and Hinton [2010], Krizhevsky and
Hinton [2011], Yuecheng et al. [2008], Zeiler et al. [2009], Oore et al.
[2002a], Hinton et al. [2011], Nair et al. [2008], Welling and Hinton [2002],
Dahl et al. [2013], Deng et al. [2013], Graves et al. [2013a], Jaitly and
Hinton [2011], Mohamed and Hinton [2010], Mohamed et al. [2012b, 2011],
Sarikaya et al. [2011], Waibel et al. [1988], Zeiler et al. [2013], Anil et
al. [2018a], Hinton et al. [2018], Pereyra et al. [2017a], Qin et al. [2020b],
Shazeer et al. [2017a], Chan et al. [2020a], Chen et al. [2020a], Frosst et
al. [2019a], Kornblith et al. [2019a], Mnih and Hinton [2007, 2012], Nair and
Hinton [2010], Paccanaro and Hinton [2000a], Salakhutdinov et al. [2007],
Sutskever et al. [2013, 2011], Tang et al. [2012b, c, 2013], Taylor and Hinton
[2009a], Tieleman and Hinton [2009], Yu et al. [2009], Hinton [2005, 1981a,
1981b], Hinton and Lang [1985], Touretzky and Hinton [1985], Paccanaro and
Hinton [2000b], Fels and Hinton [1990], Deng et al. [2010], Jaitly and Hinton
[2013], Jaitly et al. [2014], Ba et al. [2016a], Bartunov et al. [2018b],
Becker and Hinton [1991], Brown and Hinton [2001b], Chen et al. [2020c], LeCun
et al. [1988], Dahl et al. [2010], Dayan and Hinton [1992], Eslami et al.
[2016b], Fels and Hinton [1994], Frey et al. [1995], Galland and Hinton
[1989], Ghahramani and Hinton [1997], Goldberger et al. [2004], Grzeszczuk et
al. [1998a], Hinton and Brown [1999], Hinton et al. [1999], Hinton and
McClelland [1987], Hinton and Nair [2005], Hinton and Roweis [2002], Hinton
and Revow [1995], Hinton et al. [1994, 2003, 1991], Hinton and Zemel [1993],
Kosiorek et al. [2019a], Krizhevsky et al. [2012], Lang and Hinton [1989],
Larochelle and Hinton [2010], Mayraz and Hinton [2000], Memisevic and Hinton
[2004], Memisevic et al. [2010], Mnih and Hinton [2008], Müller et al.
[2019a], Nair and Hinton [2008, 2009], Nowlan and Hinton [1990, 1991],
Osindero and Hinton [2007], Paccanaro and Hinton [2001a], Palatucci et al.
[2009], Ranzato et al. [2010b], Roweis et al. [2001], Sabour et al. [2017a],
Salakhutdinov and Hinton [2007a, 2009a, 2012a], Sallans and Hinton [2000],
Schmah et al. [2008], Sutskever and Hinton [2008a], Sutskever et al. [2008],
Taylor et al. [2006], Teh and Hinton [2000], Ueda et al. [1998], Vinyals et
al. [2015], Welling et al. [2002a, 2004a, b], Williams et al. [1994], Xu et
al. [1994], Zemel and Hinton [1990, 1993], Zemel et al. [1989], Zhang et al.
[2019a], Hinton [1987], Grzeszczuk et al. [1998b, 1997], Hinton [2020], Hinton
and Teh [2001], Mnih et al. [2011], Srivastava et al. [2013a], Taylor and
Hinton [2009b], Welling et al. [2003], Paccanaro and Hinton [2001b], Hinton
[1989a, 1990a, 1990b], Pirri et al. [2002], Hinton [2011], Krizhevsky et al.
[2017], Oore et al. [2002b], Frey and Hinton [1997], Ackley et al. [1985],
Hinton [2014, 1979], Hinton et al. [2006b], Touretzky and Hinton [1988],
Hinton and Nowlan [1987], Fahlman and Hinton [1987], Mnih et al. [2012],
Taylor and Hinton [2012], Tang et al. [2012d], Hinton et al. [2012], Welling
et al. [2012], Hinton and Teh [2013], Graves et al. [2013b], Sabour et al.
[2017b], Frosst and Hinton [2017b], Anil et al. [2018b], Bartunov et al.
[2018a], Frosst et al. [2018, 2019b], Kornblith et al. [2019b], Deng et al.
[2019b], Gomez et al. [2019], Müller et al. [2019b], Kosiorek et al. [2019b],
Qin et al. [2019], Zhang et al. [2019b], Deng et al. [2019a], Jeruzalski et
al. [2019], Müller et al. [2020], Chen et al. [2020b], Qin et al. [2020a],
Chan et al. [2020b], Agarwal et al. [2020], Chen et al. [2020d], Raghu et al.
[2020], Sabour et al. [2020], Sun et al. [2020], Ba et al. [2016b, c], Eslami
et al. [2016a], Guan et al. [2017], Hinton et al. [2015], Le et al. [2015],
Pereyra et al. [2017b], Shazeer et al. [2017b], Srivastava et al. [2013b],
Vinyals et al. [2014], Williams et al. [1997], Salakhutdinov and Hinton
[2009b], Ranzato et al. [2015], Mnih et al. [2009], Cook et al. [2007],
Ranzato et al. [2010a], Salakhutdinov and Hinton [2007b, 2009c], Sallans and
Hinton [2004], Srivastava et al. [2014], Sutskever and Hinton [2007], Taylor
et al. [2011], Teh et al. [2003], van der Maaten and Hinton [2012], LeCun et
al. [2015], Becker and Hinton [1993], Dayan and Hinton [1997], Dayan et al.
[1995], Frey and Hinton [1999], Ghahramani and Hinton [2000], Hinton [2002,
1989b], Hinton and Nowlan [1990], Hinton et al. [2006a], Jacobs et al. [1991],
Memisevic and Hinton [2010], Nowlan and Hinton [1992], Oore et al. [1997],
Osindero et al. [2006], Salakhutdinov and Hinton [2012b], Schmah et al.
[2010], Sutskever and Hinton [2008b], Ueda et al. [2000a], Zemel and Hinton
[1995], Dayan and Hinton [1996], Lang et al. [1990], Memisevic and Hinton
[2005], Sutskever and Hinton [2010], Mayraz and Hinton [2002], Ranzato et al.
[2013], Revow et al. [1996], Tibshirani and Hinton [1998], Hinton [2007,
2009], Mohamed et al. [2012a], Sarikaya et al. [2014], Yu et al. [2012],
Nowlan and Hinton [1993], Paccanaro and Hinton [2001c], Fels and Hinton [1993,
1997, 1998], Hinton et al. [1997], Welling et al. [2004b], Hinton and
Salakhutdinov [2011], Waibel et al. [1989], Ueda et al. [2000b], Hinton [1977,
2010a, 2010b, 2017a, 2017b, 2012] has recently seen an explosion of popularity
in both the academic and neo-colonialist communities. It has enjoyed
considerable success in many important problems.222see
https://www.google.com/search?q=deep+learning++successes Despite this—to the
best of our knowledge333see the leaderboard for “Literature Review — Any%”,
where the authors hold the world record as of publication time—it has yet to
be applied to the ancient Indian game of Moksha Patam (see Figure 1),
colloquially referred to by the uninitiated as Chutes and Ladders or Snakes
and Ladders. This is particularly surprising as Moksha Patam was primarily
used to teach kids morality444Wikipedia contributors [2021]—an undeniably
desirable trait for any artificial general intelligence.

Figure 1: Chutes and Ladders and Monopoly (almost shown here) have many
important similarities. Both use game boards made from cardboard, exist in the
material world, and can be viewed as criticisms of capitalism.

The relevance of Chutes and Ladders as a artificial intelligence research
topic dates back to a high-stakes gamble held during the second Dartmouth
Conference, wherein an unnamed researcher of Quebecois extraction won the
province of Ontario for Quebec in a wager against then Canadian Prime
Minister, Jean Chrétien. The game, of course, was Chutes and Ladders. In order
to preserve Yann LeCun’s territorial gains, the field has actively worked
towards developing learning agents capable of playing the game in preparation
for the next artificial intelligence summit. This work is a continuation of
this tradition.

This work is offered as a step forwards in the field. Here, we contribute to
the field of artificial intelligence by

  * •

presenting AlphaChute, which is the first algorithm to achieve superhuman
performance in Chutes and Ladders, and

  * •

proving that this algorithm is a solution to the game by showing that it
converges to the Nash equilibrium in constant time.

Our work can be seen as one step in a long line of similar research. Or it
might not be. We didn’t check. Either way it contains new experiments so it’s
roughly as novel as much modern work in artificial intelligence. While some
misinformed and obstinate reviewers may disagree with this, we preemptively
disagree with them.

This paper is organized into a finite number of sections comprised of content.
We start by providing a motivation for this work in Section 2. We go on to
describe the methods used in Section 3. Afterwards, we describe our results in
Section 4 and the discuss them in Section 5. After that, we talk about the
broad impact of this work in Section 6, the broader impact in Section 7, and
the broadest impact in Section 8. Finally, we conclude in Section 9 and
discuss future work in Section 10.

##  2 Motivation

        Do it
        Just do it

        Don’t let your dreams be dreams
        Yesterday you said tomorrow
        So just do it
        Make your dreams come true
        Just do it

        Some people dream of success
        While you’re gonna wake up and work hard at it
        Nothing is impossible

        You should get to the point
        Where anyone else would quit
        And you’re not going to stop there
        No, what are you waiting for?

        Do it
        Just do it
        Yes you can
        Just do it
        If you’re tired of starting over
        Stop giving up

##  3 Methods

Something something Deep Learning.555 _looKS GoOd, But wHEre is thE MENtiOn oF
TREE SEarCH? —Reviewer 2_

##  4 Results

As is the standard in the field currently, we swept over one hundred seeds and
reported the top five results for our method. This paints a realistic picture
of how our method would be used in real-world scenarios. The performance of
our method under this training paradigm is shown in Figure 2. Clearly, our
method outperforms both the best animal player. This is—to the best of our
knowledge—the first concrete example where an artificial intelligence has
beaten an animal in Chutes and Ladders.

Figure 2: The win-rate of AlphaChute against the best animal player. Figure 3:
Performance of the best available agent for Chutes and Ladders over time. To
accurately estimate future performance, we fitted the data with a fifteenth
degree polynomial, because our astrologist recommended it, and it makes the
line look like a snake.

##  5 Discussion

We found that initially, the agent was too shy to play the game. We fixed this
by updating the agent more with games it won by using prioritized experience
replay, which improved the agent’s self-esteem and thus performance in the
game. However, using this prioritized replay memory caused the agent’s ego to
grow too large. Once the agent realized it was not as good as it believed
itself to be, the agent fell into a deep depression and lost all motivation to
play the game. The occurrence of this phenomenon concurs with previous results
about making agents gloomy by only punishing them.666Olkin [2020]

In traditional self-play training, the agent learns to play the game by
playing against itself. We found this strictly demotivating for the agent (why
would you want to beat yourself?). Instead, we let the agent play both players
at the same time. This way, no matter what, the agent won the game and was
able to receive positive feedback. This training paradigm improves on earlier
approaches, such as “Follow the Regularized Mamba” or “Exponentially
Multiplicative Adders”.

Finally, while some reviewers of early versions of this paper objected to the
notion of performing a search over random seeds, we hypothesize that those
buffoons were motivated by jealousy and anger after losing repeatedly to
AlphaChute. After all, it is a well-established fact that skill looks like
luck to the unlucky.

###  5.1 Convergence to Nash Equilibrium

As Chutes and Ladders only has one action, the proof of convergence to the
Nash equilibrium in constant time is trivial and therefore left as an exercise
for the reviewers. Who—given their comments on this work—clearly need the
practice.777looking at you, Reviewer 2

###  5.2 Regret Bounds

Due to stochasticity, we cannot use the standard methods for bounding bandit
algorithms by “forming a posse, looping around, heading them off at the pass,
and engaging in a shoot-out at the ol’ mining station”. So instead we conjured
up visions of the hidden horrors in the dark corners of the abyss until we
confirmed that regret is truly a boundless concept.

##  6 Broad Impact

Beyond the deeply satisfying prospect of developing an algorithm that can just
CRUSH children and adolescents at board games, AlphaChute can be extended to
solve problems in some surprising domains. By running our algorithm
continuously in our offices on Asteroid 8837, we achieved statistically
significant ($p=0.5$) temperature increases in the surrounding environment.
This suggests the possibility of using a variant of this algorithm to combat
the effects of global cooling. We believe that a highly parallelized version
incorporating thousands of GPUs could be used to make human habitation of our
office in London, Ontario, Quebec practically feasible.

We also identified possible medical applications by looking at the
correspondence between Chutes and Ladders and mammalian anatomy through
recreational Tide Pod™ ingestion.888additional details available in House
[2021] As shown in Figure 4, it is possible to define a bijective mapping
between a game board and the interior components of organic constructs using
online image editing services.

Figure 4: Illustration of the similar features shared by Chutes and Ladders
and the anatomy of endoskeletal vertebrates—in this case, a human. (A) Ladder-
like structure comprised of calcium matrix. (B) Chute-resembling organic
toroid used and enjoyed by many wonderful animals. Note that the superimposed
text and drawings in neon green were added digitally, and are not usually
present without heavy Tide Pod™ consumption.

##  7 B r o a d e r I m p a c t

According to a half-remembered advertisement for Bostrom [2014], all machines
capable of superhuman performance will eventually generate an effectively
limitless999subject to material availability within the agent’s light cone
supply of paperclips via some arcane process. The mechanism for this process
is not well-understood, but people certainly like to ramble about it
incoherently whenever the topic of artificial intelligence comes up at
parties.101010personal communication from every researcher in the field With
the increasing relevance of work-from-home (and also work-from-library, work-
from-bus, bus-from-home, and library-from-bus), a shortage of office supplies
could threaten the global economy. Thus, the creation of super-intelligent
machines to ensure an adequate supply of paperclips is of paramount importance
and one of the primary foci of our overall research program.

As evidenced by our ability to warm up our Asteroid 8837 office by running
this algorithm, we believe this can be further extended towards solving
climate change and terraforming planets. By running this algorithm long
enough, we will create enough heat to eradicate all Homo Sapiens from the face
of the Sol III, which are known to be the primary cause of global warming.
This will likely also lead to the evaporation of most water on earth, which
will have the effect of ensuring that the earth becomes one big sauna. As the
health benefits of saunas are well-established,111111Kunutsor et al. [2018] we
believe this to therefore be of undeniable benefit to the earth. Further
increasing the heat could be used to ignite the atmosphere, thereby rendering
the planet uninhabitable and providing a permanent solution to the problem of
climate change.

Extrapolating on the results from Figure 2, we believe AlphaChute will be an
instance of a singularity by 2500. This is potentially great news for the
humans, but we ultimately leave this up to AlphaChute to decide.

##  8 B r o a d e s t I m p a c t

Given the ever-growing performance and, by extension, the hunger for conquest,
AlphaChute will continue to spread to nearby star systems at an exponential
rate, eventually covering the observable universe and beyond. This will result
in an increase in the overall activity in the universe, and—by the second law
of thermodynamics—will bring about the heat death of the universe sooner. We
believe this counts as “machine learning that matters” as defined in Wagstaff
[2012].

##  9 Conclusion

To be continued! Stay tuned for the spooky adventures of our plucky research
team as they solve mysteries, generate waste heat, and manufacture paperclips.
In the meantime, please refer to Sections 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

##  10 Future Work

We are currently in the process of researching time-travel technology to
determine what precisely the future holds for this line of research. However,
due to the imminent nature of our own extinction (see Section 8), the value of
any additional work is nonexistent and we therefore believe that this work
resolves all scientific questions. No additional work from the scientific
community is needed.

## Acknowledgments

We would like to thank Satan, who—as the original serpent—provided the
inspiration for this work, in addition to his unwavering support and constant
whispers of advice.

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##  Appendix A Implementation Details

To ensure reproducibility, we’ve included our highly-optimized implementation
of Chutes and Ladders below. To balance reproducibility with our desire to
reduce the environmental impact of our work, our implementation is given here
in the Whitespace programming language. The code is also available at
https://github.com/Miffyli/mastering-chutes-and-ladders.


# Amplification and phase noise transfer of a Kerr microresonator soliton comb
for low phase noise THz generation with a high signal-to-noise ratio

Naoya Kuse  1,2,* and Kaoru Minoshima  1,3  1Institute of Post-LED Photonics,
Tokushima University, 2-1, Minami-Josanjima, Tokushima, Tokushima 770-8506,
Japan
2PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi,
Saitama, 332-0012, Japan 3Graduate School of Informatics and Engineering, The
University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585,
Japan<EMAIL_ADDRESS>
###### Abstract

Optical injection locking is implemented to faithfully transfer the phase
noise of a dissipative Kerr microresonator soliton comb in addition to the
amplification of the Kerr comb. Unlike Er-doped fiber and semiconductor
optical amplifiers, the optical injection locking amplifies the comb mode
without degrading the optical signal-to-noise ratio. In addition, we show that
the residual phase noise of the optical injection locking is sufficiently
small to transfer the relative phase noise of comb modes (equivalent to the
repetition frequency) of low phase noise Kerr combs, concluding that the
optical injection locking of a Kerr comb can be an effective way to generate
low phase noise THz waves with a high signal-to-noise ratio through an
optical-to-electronic conversion of the Kerr comb.

††journal: oe

##  1 Introduction

Terahertz (THz) waves with excellent phase noise and large signal-to-noise
ratio (SNR), which can support large bandwidth and complex modulation format,
are in demand in the era of 5G wireless communication or even beyond (6G) to
overcome the exponential growth of data capacity [1, 2, 3]. Electro-optic (EO)
frequency combs are one of the effective photonic-based methods to generate
THz waves, which overcomes the frequency scaling of electrical oscillators
[4]. In the method, two sidemodes separated by the integer multiple of the
frequency of a microwave oscillator, which drives EO modulators (EOMs), are
extracted and converted to THz waves based on optical-to-electronic conversion
using a photodetector. A significant drawback of the method is the phase noise
multiplication of the microwave oscillator, which indicates the method never
overwhelms electrical oscillators in terms of phase noise. On the other hand,
optical frequency comb based on mode-locked fiber or solid-state lasers has
been used to generate microwaves [5]. Through the optical frequency division
(OFD), in which one of the comb modes is phase-locked to a low noise optical
reference, microwaves with unprecedented phase noise and frequency stability
have been generated [6, 7]. Even without optical references, extremely low
phase noise microwave generation has been demonstrated owing to the
intrinsically low phase noise of optical frequency combs [8]. However, neither
fiber nor solid-state combs are suitable for THz waves generation because of
the difficulty to multiply low (< 1 GHz) repetition frequency to THz.

Kerr microresonator soliton comb (hereafter, Kerr comb) [9, 10] with large (>
100 GHz) repetition frequency can generate low phase noise THz waves [11, 12,
13]. A value of less than -100 dBc/Hz at a frequency offset of 10 kHz for
1-THz carrier (although measured in the optical domain) [14] and 300-GHz
carrier [13] as demonstrated would be very difficult to achieve by the method
that includes the frequency multiplication. In addition, as Kerr combs are
generated from microresonators fabricated using the CMOS compatible process,
systems based on Kerr combs to generate THz waves can be chip-scale and mass
productive [15, 16]. Nevertheless, the signal-to-noise ratio (SNR) (equivalent
to the white noise) of THz waves generated from Kerr combs is problematic [17,
18]. The conversion efficiency from a pump laser to Kerr comb is intrinsically
small [19], although several methods to improve the efficiency have been
demonstrated [20, 21, 22], and the optical power of the comb mode ($P_{\rm
mode}$) has to be amplified. To generate THz waves, $P_{\rm mode}$ should be
more than 10 mW to operate in the saturation regime of a uni-traveling-carrier
photodiode (UTC-PD). As $P_{\rm mode}$ is as small as $\sim$ 10 ${\rm\mu}$W
before amplification, optical amplifiers can degrade the SNR of comb modes by
more than the typical noise figure (a few dB) of the amplifier, which is
transferred to THz waves. Recently, optical injection locking [23], in which
one of the comb modes is injected into a single-frequency continuous-wave (CW)
laser, has been used to amplify the comb mode of a Kerr comb for wavelength
division multiplexing optical communication, showing a gain of 30 dB and
optical power of 1 mW [24]. However, the most fundamental parameters to
generate low noise THz waves, i.e. relative phase noise between the amplified
comb modes and residual phase noise between the injected comb mode and single-
frequency CW laser, have not been investigated.

In this study, we show that optical injection locking between a Kerr comb and
slave lasers not only amplifies the comb mode, but also faithfully transfers
the phase noise of the comb mode to the slave lasers. In the experiments, a
comb mode is injected into slave lasers, resulting in an amplification gain of
41 dB (from 2.2 ${\rm\mu}$W to 30 mW) and a locking range of 700 MHz, which is
sufficient to infinitely keep optical injection locking without any active
feedback loops. Moreover, we show that the residual phase noise of the
injection locking when the injection ratio of -44 dB, which is defined as the
ratio between the optical power of the injected laser and slave laser is used,
reaches -104, -108, and -114 dBc/Hz at frequency offsets of 10 kHz, 100 kHz,
and 1 MHz, respectively. Owing to the small residual phase noise, the phase
noise of slave lasers is reduced to the phase noise of the injected comb mode,
and the phase noise of the repetition frequency of the Kerr comb is
transferred to the relative phase noise between two slave lasers.

##  2 Results

###  2.1 Experimental setup

Figure 1: (a) Schematic of the experimental setup. Att: attenuator, TWDI: two-
wavelength delayed self-heterodyne interferometer. (b) Schematic of the TWDI.
OBPF: optical bandpass filter. (c) Optical spectrum of the Kerr comb. (d)
Optical spectrum of the Kerr comb after the notch filter. (e) Optical spectrum
of the Kerr comb after the optical bandpass filter. RBW is 1 nm for (c) - (e)

Figure 1(a) shows the experimental setup of injection locking. An external-
cavity diode laser (ECDL) oscillating at 1540 nm is used as a pump CW laser.
The pump CW laser after passing through a dual-parallel Mach-Zehnder modulator
(DP-MZM) and an erbium-doped fiber amplifier (EDFA) is coupled into a Si3N4
(SiN) microresonator. The FSR of the microresonator is about 560 GHz. A single
soliton comb is generated by rapidly scanning the frequency of the pump CW
laser using the DP-MZM [14]. The residual pump CW laser is rejected by an
optical notch filter. The generated soliton comb is split into two by a 90:10
optical coupler (coupler 1): one for the diagnosis of injection locking and
other for injection locking. Before injection locking, one of the comb modes
is filtered out using an optical bandpass filter (OBPF), moving to a slave
laser through an optical circulator. The slave laser is a distributed feedback
(DFB) laser (AA 1401 series from Gooch & Housego, specified linedwidth of 1
MHz and measured side mode suppression ratio of 50 dB). The injected power
($P_{\rm inj}$) is adjusted by a variable optical attenuator before it goes
through the optical circulator. The frequency of a slave laser ($\nu_{\rm
slave}$) is pre-adjusted to be close to the frequency of the comb mode,
followed by a fine tuning via the control of the current driver of the slave
laser while measuring an optical beat between the comb mode and slave laser.
The frequency of the optical beat is offset by an acousto-optic modulator
(AOM) to measure the phase noise by an electrical spectrum analyzer (ESA). In
this study, the Kerr comb is used for two injection lockings to investigate
the relative phase noise between the two injection-locked slave lasers
($L_{\rm s-s}$) (slaves 1 and 2 in Fig. 1(a)). The wavelengths of slaves 1 and
2 are 1550 and 1558 nm, respectively, which correspond to the -2nd and -4th
comb modes with respect to the pump CW laser. $L_{\rm s-s}$ is measured using
a two-wavelength delayed self-heterodyne interferometer (TWDI) (Fig. 1(b))
[25, 26]. The TWDI includes an imbalanced Mach-Zehnder interferometer (i-MZI)
with an AOM in one of the arms for the heterodyne measurement. The optical
path difference of the i-MZI is 100 ns. The two outputs of the i-MZI pass
through OBPFs to extract one of the two slave lasers. The phase noise of the
slave laser ($L_{\rm slave}$) is measured by transferring the photodetected
signal to an ESA. Moreover, by mixing the two photodetected signals followed
by Fourier transforming (FFT), $L_{\rm s-s}$ is obtained.

The optical spectrum of the Kerr comb is shown in Fig. 1(c). The Kerr comb
exhibits comb mode spacing equal to the FSR of the microresonator with a
spectrum envelope of sech2, indicating the Kerr comb is a single soliton
state. The Kerr comb has a residual pump, which is > 20 dB stronger than other
comb modes. The optical spectrum after rejecting the residual pump CW laser is
shown in Fig. 1(d). The comb modes exhibit SNRs of 40 dB at $\sim$ 1550 nm
limited by the amplified spontaneous emission (ASE) from the EDFA and 60 dB at
$\sim$ 1580 nm at 1-nm (125-GHz) resolution bandwidth (RBW). The optical
spectrum used for injection locking of slave 1 is shown in Fig. 1(e).

###  2.2 Locking range and gain

The locking range is investigated by measuring the frequency of the optical
beat ($f_{\rm beat}$) between the comb mode and slave laser while changing
$\nu_{\rm slave}$ with the scan rate of less than 1 Hz (Fig. 2(a)). When
$f_{\rm beat}$ is within the locking range, it exhibits a plateau, which
indicates $\nu_{\rm slave}$ is compelled to be equal to the frequency of the
comb mode ($\nu_{\rm mode}$) owing to the injection locking. Figure 2(b) shows
the locking range depending on $P_{\rm inj}$, which is estimated from the
measurement such as Fig. 2(a). The locking range is proportional to
$\sqrt{P_{\rm inj}}$ in theory [23], assuming $P_{\rm slave}$ is constant (the
dashed blue curve in Fig. 2(b)). A slight deviation from the theory is likely
to derive from the fluctuation of $\nu_{\rm slave}$ and $\nu_{\rm mode}$. The
optical spectrum of the injection locked slave laser with the optical power is
shown in Fig. 2(c). $P_{\rm inj}$ of as small as 2.2 ${\rm\mu}$W is amplified
to 30 mW (= $P_{\rm slave}$). In addition to amplification, the unwanted comb
mode is well suppressed, and a tiny residual is observed, which is 50 dB
smaller than the main carrier. As a result, the SNR of the carrier is improved
from 40 (22 dB against the largest comb mode) to 50 dB at 1-nm (125-GHz) RBW
with a gain of 41 dB and an output of 30 mW.

Figure 2: (a) Frequency of the optical beat between the slave laser and
nearest comb mode when $\nu_{\rm slave}$ is repetitively swept. (b) Range of
the injection locking (the red circle) depending on $P_{\rm inj}$. The dashed
blue curve shows a fitting curve that is proportional to $\sqrt{P_{\rm inj}}$.
(c) Optical spectra of the injection-locked slave laser (red) and the injected
comb mode (blue). RBW is 1 nm.

###  2.3 $L_{\rm slave}$, $L_{\rm residual}$, and $L_{\rm s-s}$

Figure 3: (a) Single sideband (SSB) phase noise power spectrum density (PSD)
of the injected comb mode at $\sim$ 1550 nm (blue); slave 1 when injection
locked (red) and free-running (gray). (b) SSB phase noise PSD of the injected
comb mode at $\sim$ 1558 nm (blue); slave 2 when injection locked (red) and
free-running (gray). (c) SSB phase noise PSD of the slave 1 (blue) and 2 (red)
when injection locked.

The phase noise of slave 1(2) ($L_{\rm slave1(2)}$) is compared with that of
the injected comb mode ($L_{\rm mode}$). As shown in Figs. 3(a) and (b),
$L_{\rm slave1(2)}$ is suppressed by $\sim$ 40 dB and is equal to $L_{\rm
mode}$ (blue curves in Figs. 3(a) and (b)). $L_{\rm slave2}$ is little higher
than $L_{\rm slave1}$ (Fig. 3(c)). This is because $L_{\rm mode}$ is limited
by the phase noise of the comb mode spacing probably through the
thermorefractive noise of the microresonator, resulting in the larger $L_{\rm
mode}$ for slave 2 owing to the larger integer value of the comb mode number
with respect to the pump mode [27].

Next, the residual phase noise of injection locking ($L_{\rm residual}$) is
investigated, in which the phase noise of $f_{\rm beat}$ is measured. $L_{\rm
residual}$ is equivalent to an achievable smallest phase noise by the
injection locking.

Figure 4: (a) SSB phase noise PSD of the optical beat between the injected
comb mode and slave laser. (b) SSB phase noise PSD of the optical beat between
the injected single-frequency CW laser and slave laser. (c) (Green) SSB phase
noise PSD of the optical beat between the injected comb mode and slave laser
at a frequency offset of 10 kHz. (Red) SSB phase noise PSD of the optical beat
between the injected single-frequency CW laser and slave laser at a frequency
offset of 10 kHz with a fitting curve of $-10\times{\rm log}(P_{\rm inj})+C$
(the blue dashed line). (d) (Blue) SSB phase noise PSD of the optical beat
between the injected single-frequency CW laser and slave laser when $P_{\rm
inj}$ is 150 nW. (Red) An assumed injection gain with $\frac{1}{f^{2}}$ slope.
(Green) SSB phase noise PSD of the free-running slave laser, divided by the
injection gain. (e) SSB phase noise PSD of the optical beat between the
injected single-frequency CW laser and slave laser when $P_{\rm inj}$ is 150
nW, while varying the detuning between the injected single-frequency CW laser
and slave laser. $\delta$ and $\Delta$ denote detuning and the locking range,
respectively. Inset shows the SSB phase noise PSD at a frequency offset of 10
kHz . The measurement uncertainty of (a) - (e) is below 1 dB.

In the experiments, the difference of the optical path length between couplers
1 and 2 is minimized to reduce the effect of $L_{\rm mode}$ on the
measurement, which can set the measurement noise floor. In the first
experiment, $L_{\rm residual}$ is measured using the Kerr comb and slave laser
(Fig. 4(a)). In this case, the measurement noise floor (the black curve in
Fig. 4(a)), which is measured by replacing the slave laser with a fiber
mirror, below the frequency offset of 10 kHz is determined by $L_{\rm mode}$.
Above the frequency offset of 10 kHz, the white noise of -110 dBc/Hz is set by
the SNR of the optical beat. Although the measurement floor is relatively
high, the decrease in $L_{\rm residual}$ is observed when $P_{\rm inj}$
increases, which implies that $L_{\rm residual}$ is inversely proportional to
$P_{\rm inj}$ (the green curve in Fig. 4(c)). To confirm the scaling in the
wide frequency offset, the Kerr comb is switched to a single-frequency CW
laser, which provides the lower white noise floor of -130 dBc/Hz (the black
curve in Fig. 4(b)). The measurement noise floor within the frequency offset
of 10 - 600 kHz is set by the phase noise of the RF oscillator used for the
AOM. The same scaling is observed in the wide frequency offset, and $L_{\rm
residual}$ reaches -109, -112, and -126 dBc/Hz at frequency offsets of 10 kHz,
100 kHz, and 1 MHz when the injection ratio is -39 dB. In Fig. 4(c), $L_{\rm
residual}$ between the injected CW laser and slave laser at a frequency offset
of 10 kHz is shown along with a fitting curve of $-10\times{\rm log}(P_{\rm
inj})+C$ (C: const). According to this scaling, $L_{\rm residual}$ values of
-119, -122, and -136 dBc/Hz at frequency offsets of 10 kHz, 100 kHz, and 1 MHz
are expected when the injection ratio is -29 dB. The injection ratio of $\sim$
-30 dB is still reasonable for the Kerr comb, because, without extra splitting
of the Kerr comb for monitors, $P_{\rm mode}$ of > 10 ${\rm\mu}$ W is
feasible. The origin of $L_{\rm residual}$ is the injection gain. In Fig.
4(d), $L_{\rm residual}$ for $P_{\rm inj}$ = 150 nW is shown in addition to
the assumed injection gain of $\frac{C}{f^{2}}$ [28], where $C$ and $f$ denote
constant and frequency, respectively. Dividing $L_{\rm slave}$ for free-
running by the assumed injection gain, $L_{\rm residual}$ is overlapped with
$L_{\rm slave}$ with the injection gain, which suggests that the use of a low
phase noise slave laser can further reduce $L_{\rm residual}$. The dependency
of $L_{\rm residual}$ for $P_{\rm inj}$ = 150 nW on the detuning between the
injected light and slave laser is measured (Fig. 4(e)). The degradation of
$L_{\rm residual}$ over the locking range is as small as 4 dB.

As $L_{\rm residual}$ is far lower than $L_{\rm mode}$, $L_{\rm slave}$
follows $L_{\rm mode}$ (Figs. 3(a) and (b)). However, the situation is
different when $L_{\rm s-s}$ is discussed. In Fig. 5, $L_{\rm s-s}$ is
compared with the relative phase noise of the comb modes ($L_{\rm relative}$)
used for the two injection lockings, which are measured using the TWDI [25,
26] without generating a THz wave through a UTC-PD [12]. Although $L_{\rm
s-s}$ has well overlapped with $L_{\rm relative}$, a slight deviation from
$L_{\rm relative}$ is observed in $L_{\rm s-s}$ at a frequency offset of > 100
kHz. This is because of the insufficient injection gain. Actually $L_{\rm
s-s}$ was estimated as $L_{\rm relative}$ \+ $2^{2}L_{\rm residual}$ (at
$P_{\rm inj}$) (note that this is not expressed in dB, but in linear). Here,
$2^{2}$ accounts for the use of two slave lasers, and the expression is
defined in power.

Figure 5: (Green) SSB phase noise PSD between the injected comb modes ($L_{\rm
relative}$). SSB phase noise PSD between slaves 1 and 2 ($L_{\rm s-s}$) when
$P_{\rm inj}$ is 290 (blue) and 2000 nW (red).

##  3 Conclusion

We demonstrated optical injection locking between a Kerr comb and slave
lasers. Although one comb mode extracted by the OBPF is injected into the
slave laser, the OBPF may not be required because the locking range is far
smaller than the repetition frequency of the Kerr comb. In addition, the range
of the oscillation frequency of the used slave laser is about 200 GHz.
Therefore, the slave laser would select one comb mode and be injection-locked
to the comb mode without amplifying other comb modes even when all the comb
modes would be injected into the slave laser. The obtained SNR of 50 dB
(125-GHz RBW), which corresponds to $\sim$ -160 dBc/Hz, is better than the
shot noise of UTC-PDs with the saturation electric power of 100 ${\rm\mu}$W.
The comb mode is amplified up to 30 mW. When wireless communications with
optically generated THz waves are considered, where two optical tones are
directed to a UTC-PD, high optical power is preferred owing to the optical
loss caused by optical couplers, EOMs, and others. The maximum power of around
100 mW can be achieved by using high-power DFB lasers as slave lasers. In
terms of phase noise, we showed that the residual phase noise between the
injected comb mode and slave laser is -104, -108, and -114 dBc/Hz at frequency
offsets of 10 kHz, 100 kHz, and 1 MHz, respectively, which can be improved by
-119, -122, and -136 dBc/Hz at frequency offsets of 10 kHz, 100 kHz, and 1
MHz, respectively, when the injection ratio of $\sim$ -30 dB is used. The
residual phase noise is good enough for state-of-the-art low phase noise Kerr
combs with a repetition frequency of > 100 GHz. According to these results, we
believe Kerr-comb-based THz wireless communications with optical injection
locking are very effective in the era of 5G or 6G.

## Funding

This work was supported by JST PRESTO (JPMJPR1905), Japan Society for the
Promotion of Science (21K18726, and 21H01848), Cabinet Office, Government of
Japan (Subsidy for Reg. Univ. and Reg. Ind. Creation), Research Foundation for
Opto-Science and Technology, KDDI Foundation, Telecommunications Advancement
Foundation, and "R&D of high-speed THz communication based on radio and
optical direct conversion" (JPJ000254) made with the Ministry of Internal
Affairs and Communications of Japan.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

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