# High Repetition-Rate Pulse Shaping of a Spectrally Broadened Yb Femtosecond
Laser

Julia Codere1,a  Michael Belmonte1,a  Brian Kaufman1,2  Michael Wahl1  Eric
Jones1  Martin G Cohen1  Thomas Weinacht1  and Ruaridh Forbes2  1 Department
of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794,
USA
2 Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo
Park, California 94025, USA
A These authors contributed equally to this work<EMAIL_ADDRESS>
###### Abstract

We demonstrate compression and shaping of few cycle pulses from a high average
power Ytterbium laser system. The pulses from a commercial 20 W, 100 kHz Yb
laser system are spectrally broadened in two-stages using gas-filled,
stretched hollow-core fibers and then compressed and shaped in an acousto-
optic modulator-based pulse-shaper. The pulse-shaper allows for compression,
characterization, and shaping all in one system, producing $\sim$10 fs pulses
with 50 $\mu$J of energy.

††journal: osac††articletype: Research Article

##  1 Introduction

High-average-power, ultrafast laser systems are increasingly in demand for
many experiments which benefit from high repetition rates that enable highly
differential measurements, or ones that require a very low yield per laser
shot (e.g. coincidence measurements) [1, 2, 3]. While there are commercial
Ytterbium laser systems available which can provide high average power with
repetition rates of 100 kHz and greater, they typically produce pulses with
durations greater than 100 fs. These long pulses are not suitable for many
time-resolved spectroscopy or strong-field light—matter experiments, which
require few cycle pulses to resolve the fastest molecular dynamics [4, 5, 6].
Here we report on the spectral broadening and subsequent compression of 250 fs
laser pulses from a 100 kHz, 20 W commercial Ytterbium laser system (Light
Conversion Pharos) using a two-stage, stretched hollow-core fiber apparatus in
conjunction with an acousto-optic modulator (AOM) based ultrafast optical
pulse-shaper [7]. Our system is capable of producing arbitrarily shaped pulses
as short as 10 fs, with energies between 30 and 50 $\mu$J, at repetition rates
of up to 50 kHz.

Shortening the duration of an ultrafast laser pulse can be broken into two
stages: (1) broadening the pulse in the frequency domain, and (2) compression
of the pulse in the time domain. There are many techniques to accomplish
spectral broadening, including stretched hollow core fibers, Herriott cells,
thin plates and filamentation [8, 9, 10, 11, 12, 13]. While there are various
advantages and disadvantages associated with each approach [14, 15], they all
make use of self phase modulation to produce the new spectral bandwidth. Pulse
compression is then usually achieved by applying negative group-delay
dispersion (GDD) to the pulses with chirped mirrors [16]. Few cycle pulses
($\sim$10 fs) have been generated at high average power by means of HCF-based
spectral broadening and chirped mirror-based pulse compression [17, 18, 19].
However, using chirped mirrors and third-order dispersion-compensating
material [20] does not allow for programmable pulse shaping or direct
compensation of higher order dispersion, which can be accomplished with an
AOM-based, ultrafast pulse-shaper [21, 22]. A further advantage of the direct
dispersion control using an AOM-based pulse-shaper is that it can be used for
pulse characterization through dispersion scans (D-scan) [23] and collinear
frequency resolved optical gating (CFROG) [24].

Here, we demonstrate the production of an ultrabroadband spectrum with
cascaded fibers and the ability to compress, control, and measure the pulses
using an AOM-based 4f pulse-shaper. This work builds on our previous work of
pulse compression techniques with a Ti:sapphire laser system [22], but now
uses a commercial Yb laser system. This enables us to perform pulse shaping at
high repetition-rates (50 kHz) and high pulse energies ($\sim$100 $\mu$J). We
are able to compress 250 fs pulses from the output of the laser to ones that
are $\sim$10 fs, a compression factor of about 25. The combination of high
average power and fine control of a broadened spectrum allows for highly-
differential, time-resolved spectroscopy measurements.

##  2 Apparatus

Figure 1: Experimental Apparatus. The pulses from the Yb laser are centered at
about 1025 nm, with a 250 fs duration. They are focused into a 3 m long HCF
filled with 3 bar of Ar and compressed with chirped mirrors. They are then
focused into a 2 m long HCF filled with 0.5 bar of Ar and sent into the AOM-
based pulse-shaper (the inset shows the mode of the beam exiting the pulse-
shaper). The compressed output of the pulse-shaper is sent through a Beta
Barium Borate (BBO) crystal and into the spectrometer for characterization.

The schematic in Figure 1 shows the two-stage pulse-compression system and the
subsequent measurement apparatus. In brief, each stage consists of a spectral
broadening component and a pulse compression component. Spectral broadening is
achieved with Kerr-induced self-phase modulation by filling stretched, hollow-
core fibers (S-HCF) with argon [8, 25]. Pulse-compression is done in two ways:
after the first fiber, there are chirped mirrors, and after the second fiber,
there is an AOM-based pulse-shaper.

Pulses from the Yb laser (50 kHz repetition rate, 400 $\mu$J pulse energy, 20
W average power, 250 fs pulse duration, 1025 nm central wavelength) are
focused to a beam waist of 144 $\mu$m and sent into a 3 m S-HCF. The inner
diameter of the fiber is 450 $\mu$m. The beam waist of 144 $\mu$m satisfies
the mode matching condition [26]. The first fiber is glued onto mounts and
stretched within a vacuum-sealable tube. The tube and enclosed fiber are
filled with 3 bar of Ar. The transmission efficiency of the first fiber is 73%
including reflections from uncoated MgF2 windows. With nano-structured or
anti-reflection coated windows, an efficiency of 89% should be possible. The
output of the fiber is collimated, and the spectrally broadened pulses are
compressed in duration with four bounces off -500 fs2 chirped mirrors
(Ultrafast Innovations HD59). We estimate the pulse duration to be
approximately 35 fs after the first pulse-compression system based on
calculations of the light—matter interaction in the fiber, and the Fourier
transform limit of the spectrum (see Figure 2). The exit of the first fiber is
imaged onto the entrance of a second 2 m long S-HCF with the same mode
matching condition as the first. In comparison to the first fiber, the ends of
the second are directly coupled to vacuum and movable to the beam path [22];
it is filled with 0.5 bar of Ar. The Fourier transform limit of the spectrum
exiting the second fiber is 6 fs (Figure 2), and the efficiency is 80%. The
output of the second fiber is collimated and the pulse is compressed using an
AOM-based pulse-shaper as opposed to chirped mirrors.

Figure 2: Initial spectrum out of the laser centered about 1025 nm (light
blue), spectrum out of the first fiber with a bandwidth of 80 nm tail-to-tail
(medium blue), spectrum out of the second fiber with a bandwidth of 400 nm
(dark blue), and spectrum after the pulse-shaper with a bandwidth of 350 nm
(gray). An inset of the transform limit after the second fiber is included.

The pulse-shaper is set up in a 4f configuration as discussed in earlier work
[21, 22], but optimized for a near-IR, high average power laser. Pulses with a
bandwidth of 400 nm tail-to-tail are sent into the pulse-shaper and diffracted
from a gold-coated ruled diffraction grating. The first order diffracted beam
is propagated to a $f$ = 0.5 m, gold-coated curved mirror, which collimates
the dispersed frequency components and focuses them into a TeO2 AOM, which is
placed in the Fourier plane of the zero dispersion stretcher geometry. The
combination of the grating and focusing mirror map the optical frequencies of
the laser pulse to position in the AOM. Given the relatively slow velocity of
the acoustic wave in the crystal, compared to the speed of light, the acoustic
wave is seen as fixed with respect to the orthogonally propagating laser beam.
Thus, the optical frequencies can be mapped to time in the acoustic wave. The
maximum repetition rate of the laser is limited by the transit time of the
acoustic wave through the active portion of the crystal. We utilize the entire
40 mm optical aperture to shape the optical bandwidth; this corresponds to 10
$\mu$s, or a maximum repetition rate of 100 kHz. However, the maximum laser
output pulse energy at 100 kHz is half that at 50 kHz. In order to work with
the maximum pulse energy from the laser, and avoid a 100% duty cycle on the
AOM, we work at 50 kHz. The shaped pulses exit the pulse-shaper in a
symmetrically equivalent way to the input pulses. The efficiency of the 4f
AOM-based pulse-shaper is about 25%.

In order to shape the pulse, we apply a mask with the acoustic wave onto the
electric field of the pulse $E\textquoteright(\omega)=M(\omega)E(\omega)$,
where $E(\omega)$ is the input pulse, $E^{\prime}(\omega)$ is the output
shaped pulse, and $M(\omega)$ is the pulse-shaper applied mask function. The
mask is applied to the pulse by controlling the amplitude and phase of the
acoustic wave with the radio frequency voltage driving the modulator. While
the mapping of acoustic time to optical wavelength is roughly linear with
optical wavelength for small diffraction angles ($\sin(\theta)\sim\theta$),
the mapping of acoustic time to optical frequency is highly nonlinear, given
the inverse relationship between frequency and wavelength. This nonlinear
mapping leads to a nonlinear optical phase as a function of frequency (higher
order dispersion) [21] when programming a constant acoustic wave that
diffracts the laser. However, depending on the orientation of the AOM and the
acoustic diffraction order that one chooses, the sign of this dispersion can
be chosen and utilized to compensate for normal dispersion from optics in the
beam path [22]. Here, the transducer of the AOM is on the red side of the
optical spectrum and the +1 diffraction order is used, which corresponds to a
nonlinear mapping that introduces a negative group delay dispersion.

The mapping of optical frequency to position, as mentioned above, is important
for applying the correct amount of dispersion to the pulse and for accurately
decoupling the higher orders of dispersion. Dispersion is often described in
terms of a Taylor series expansion. Thus, we utilize the same formulation for
dispersion control of the pulse shaper and write the mask:

$M_{\text{phase}}(\omega)=\exp\bigg{(}i\frac{\text{GDD}}{2}(\omega-\omega_{0})^{2}+i\frac{\text{TOD}}{6}(\omega-\omega_{0})^{3}+i\frac{\text{FOD}}{24}(\omega-\omega_{0})^{4}\bigg{)}$
(1)

where $\omega_{0}$ is the central frequency of the expansion, GDD is the
group-delay dispersion, TOD is the third-order dispersion, and FOD is the
fourth-order dispersion. Two techniques were used to measure the pulse
duration: D-scans and CFROGs. The pulses were sent through a 10 $\mu$m thick
BBO crystal and the resulting second harmonic generated (SHG) spectrum was
recorded. For D-scans, GDD was for fixed TOD and FOD by programming these
values onto the pulse-shaper.

Similarly, for CFROGs, the dispersion (GDD, TOD, and FOD) is fixed, and the
delay between pulses is programmed and scanned by the pulse-shaper utilizing
the mask:

$M_{\text{CFROG}}(\omega)=1+R\exp\big{(}i(\omega-\omega_{L})\tau+i\phi_{L}\big{)}$
(2)

where $R$ is the relative pulse amplitude, $\omega_{L}$ is the locking
frequency (frequency at which constructive interference occurs independent of
delay), $\tau$ is the pulse delay, and $\phi_{L}$ is the relative pulse phase.
Since CFROGs use a collinear geometry, they are easier to set-up and remove
the potential for temporal smearing associated with non-collinear FROG
measurements. Moreover, normally the scan step would require greater than
Nyquist sampling of the carrier, but an appropriate choice of $\omega_{L}$
softens this requirement such that one only needs a step size small enough to
sample the envelope.

##  3 Apparatus Discussion

Multiple avenues of pulse compression were attempted to achieve sub-10 fs
pulses. We originally tried to compress the 250 fs pulses out of the laser
with a single S-HCF, filled with molecular gas, and the AOM-based pulse-
shaper. We filled the fiber with molecular gas because spectral broadening can
be enhanced by the rotational Raman process [27]. The most promising gas
seemed to be N2O based on the Fourier transform limit of the spectrum out of
the fiber. However, at average powers of 10 W or greater, the molecular gas
absorbed too much energy from the laser as a result of rotational heating,
dramatically decreasing the throughput efficiency and damaging the entrance to
the optical fiber. Thus, we made use of atomic gases, for which the nonlinear
process is strictly parametric, with no heating of the gas. This, however,
required two fibers in order to produce the desired bandwidth and final pulse
duration.

High average power through the pulse-shaper also needed to be considered. Too
high a peak intensity in the TeO2 AOM can lead to self focusing and self phase
modulation, and too high an average power on the diffraction gratings can lead
to mode distortions. While increasing the mode size into the pulse-shaper
mitigates the mode distortions due to nonuniform thermal expansion of the
gratings, it leads to a tighter focus in the AOM, which can lead to both self
focusing and damage. Therefore, we had to strike a balance between these two
requirements, and used a mode size of 2.5 mm FWHM going into the pulse-shaper.
We explored two different pairs of gratings - one with 300 lines/mm, and
another with 200 lines/mm. Both were gold-coated ruled diffraction gratings
blazed for operation at about 1000 nm. They both consist of a glass substrate
with an epoxy that was used to produce a replica grating from a ruled master,
coated with a thin layer of chromium and then gold. Combined with the 0.5 m
focal length mirrors, the dispersed spectrum with the 300 lines/mm grating was
too large for the aperture of the AOM, limiting the spectrum and resulting in
compression to only 20 fs. At average powers over 10 W, these gratings in the
pulse-shaper became deformed by the thermal load from the laser and this
decreased the efficiency of the pulse-shaper to 20%, but the mode and power
were stable. For the gratings with a groove density of 200 lines/mm, we were
able to compress the pulses to $\sim$10 fs, but these gratings provided lower
efficiency and were damaged at average powers over 10 W. We considered
changing the curved mirrors to ones with a shorter focal length to fit more of
the bandwidth on the AOM, but this would also produce a tighter focus in the
AOM, leading to nonlinear phase accumulation in the TeO2. In the future, we
plan on improving the grating performance with a more appropriate substrate
(copper, to avoid nonuniform thermal expansion) and a lower groove density (in
order to fit the entire optical spectrum through the AOM).

Figure 3:  Overall pulse energy out of the system after the pulse-shaper with
two different gratings: 300 lines/mm (blue) and 200 lines/mm (red).

##  4 Results

With this system, we are able to compress the 250 fs pulses out of the laser
down to 11 fs. One of the strengths of the AOM-based pulse-shaper is that it
allows for different types of pulse characterization. By writing a mask on the
phase of the acoustic wave, we can scan through the GDD and adjust the TOD and
FOD accordingly, i.e. Eq. 1. For CFROGs, we can utilize the same phase mask
and scan the delay between two pulses (product of Eqs. 1 and 2). Figure 4
shows a D-scan and CFROG that reconstruct to 11.0 fs and 11.5 fs,
respectively. We obtained shorter pulses in some cases ($<10$ fs), but with
more structure in time. In order to obtain the D-scan, the GDD was scanned
from 6200 to 8000 fs2, and the applied TOD and FOD were -23700 fs3 and 55000
fs4. With the acquired GDD, TOD, and FOD, we then scanned the delay between a
pair of pulses with $\omega_{L}=1.86$ rad/fs and $\phi_{L}=0$. The
reconstructed duration for both types of pulse characterization agree within
0.5 fs. The Fourier transform limited duration calculated with the bandwidth
measured after the pulse-shaper was about 8 fs (Figure 2), a longer duration
than the spectra out of the fibers provide for due to optical clipping on the
AOM crystal. We have included a table that compares our experimentally applied
values of dispersion to the expected values we calculated. While we expected
to achieve the shortest compressed pulse with an applied phase mask that
cancels the sum of the elements (dispersive elements plus the phase associated
with the nonlinear mapping of optical frequency to acoustic time), this was
not exactly the case. The differences in the last row of the table are small
but not insignificant. (If uncorrected, the residual GDD, TOD and FOD would
individually lead to pulse durations of 120 fs, 23 fs and 21 fs respectively).
We attribute these differences mostly to small errors in the calibration of
the optical frequency to acoustic time, with small contributions from our
imperfect characterization of the phase accumulated in the spectral broadening
and subsequent optical elements.

We note that the compressed second harmonic spectra are not centered at 513
nm, which one might expect given the 1025 nm fundamental. The loss of the
lower frequency components in the second harmonic spectra could be due to a
number of reasons, including limited phase matching in the BBO crystal,
frequency dependent losses in the pulse shaper, spatial chirp, etc. We have
ruled out most of these reasons and suspect that some spatial chirp combined
with imperfect coupling into the spectrometer is the main contribution. We are
planning improvements to the collimation of the beam out of the fibers and
pulse characterization to address this.

Figure 4: Top panel: an AOM-based pulse-shaper D-scan and its corresponding, reconstructed pulse duration (11.0 fs) and phase. Bottom panel: an AOM-based pulse-shaper CFROG and its corresponding, reconstructed pulse duration (11.5 fs) and phase. Dispersive Elements | GDD (fs2) | TOD (fs3) | FOD (fs4)
---|---|---|---
Lens | 68 | 123 | -136
Air | 65 | 9 | -
AOM TeO2 | 3618 | 2583 | 1172
Ar Broadening | 250 | - | -
Nonlinear Mapping | -10624 | 16350 | -24201
Sum of Elements | -6623 | 19065 | -23165
Applied Phase Mask | 7166 | -23700 | 55000
Sum + Mask | 543 | -4635 | 31835
Table 1: Table comparing the sum of the applied dispersion and dispersion
accumulated along the table. To have the shortest compressed pulse, we expect
the sum of the elements and the applied mask to cancel each other out.

Figure 5 is a testament to the ability of the pulse-shaper. It shows a CFROG
characterization of two pulses that were delayed by $t=50$ fs, but the two
pulses were $\pi$ out-of-phase with each other on the top panel and in-phase
on the bottom panel. The effect of the different phases can be seen in the
intensity plots to the right of the CFROGs. At 50 fs in the CFROG delay, the
pulses destructively interfere when they are out-of-phase and constructively
interfere when they are in-phase. For the double pulse, the mask is slightly
different than the CFROG mask from above (Eq. 2). The CFROG mask is multiplied
by a double pulse mask:

$M_{\text{Double
Pulse}}(\omega)=\underbrace{\big{[}1+R\exp\big{(}i(\omega-\omega_{L})\tau+i\phi_{L}\big{)}\big{]}}_{\text{CFROG
Delay}}\times\underbrace{\big{[}1+\exp\big{(}i(\omega-\omega_{L})t+i\phi\big{)}\big{]}}_{\text{Double
Pulse Delay}}$ (3)

where $t$ is the delay between the double pulses and $\phi$ is the relative
pulse phase between the double pulses. This mask (Eq. 3) is also multiplied by
the phase mask (Eq. 1) to compress the pulses. Independent control over the
delay and phase can be very important for nonlinear spectroscopies, such as 2D
electronic spectroscopy [28, 29].

Figure 5: CFROG characterizations of double pulses where the pulses in the top
panel were $\pi$ out-of-phase and the pulses in the bottom panel were in-
phase. The plots to the right of the CFROGs are the integrated intensity of
the CFROGs over wavelength.

##  5 Conclusion

We have demonstrated the ability to achieve a high compression ratio of pulses
with high pulse energy and repetition rate. The pulses can be shaped in phase
and amplitude with the AOM-based pulse-shaper, and pulse pairs can be created
with independent control over delay and phase. This allows us to both create
ultrafast pulses, on the order of 10 fs, and characterize them with the same
device. Currently, we are limited by the efficiency and groove density of our
diffraction gratings, but have shown the capability of ruled diffraction
gratings to handle the high average power and large bandwidths. The generation
of intense, ultrafast pulses with arbitrary delays between pulse pairs will
allow for pump-probe measurements of molecular dynamics [30].

## Funding

National Science Foundation (2110376).

## Acknowledgments

We thank Joseph Robinson (SLAC) and John Travers (Heriot Watt) for valuable
discussion as well as Light Conversion for supporting this work. This work was
funded by the Linac Coherent Light Source, SLAC National Accelerator
Laboratory, which is supported by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-76SF00515
and the National Science Foundation under award number 2110376.

## 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.

## References

  * [1] J. Ullrich, R. Moshammer, A. Dorn, R. Dörner, L. P. H. Schmidt, and H. Schmidt-Böcking, “Recoil-ion and electron momentum spectroscopy: reaction-microscopes,” Reports on progress in physics 66, 1463 (2003).
  * [2] A. Zhao, M. van Beuzekom, B. Bouwens, D. Byelov, I. Chakaberia, C. Cheng, E. Maddox, A. Nomerotski, P. Svihra, J. Visser _et al._ , “Coincidence velocity map imaging using tpx3cam, a time stamping optical camera with 1.5 ns timing resolution,” Review of Scientific Instruments 88 (2017).
  * [3] A. E. Boguslavskiy, J. Mikosch, A. Gijsbertsen, M. Spanner, S. Patchkovskii, N. Gador, M. J. Vrakking, and A. Stolow, “The multielectron ionization dynamics underlying attosecond strong-field spectroscopies,” Science 335, 1336–1340 (2012).
  * [4] E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. Gullikson, D. T. Attwood, R. Kienberger _et al._ , “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
  * [5] P. Sándor, V. Tagliamonti, A. Zhao, T. Rozgonyi, M. Ruckenbauer, P. Marquetand, and T. Weinacht, “Strong field molecular ionization in the impulsive limit: Freezing vibrations with short pulses,” Physical review letters 116, 063002 (2016).
  * [6] S. Zherebtsov, T. Fennel, J. Plenge, E. Antonsson, I. Znakovskaya, A. Wirth, O. Herrwerth, F. Süßmann, C. Peltz, I. Ahmad _et al._ , “Controlled near-field enhanced electron acceleration from dielectric nanospheres with intense few-cycle laser fields,” Nature Physics 7, 656–662 (2011).
  * [7] M. R. Fetterman, D. Goswami, D. Keusters, W. Yang, J.-K. Rhee, and W. S. Warren, “Ultrafast pulse shaping: amplification and characterization,” Optics Express 3, 366–375 (1998).
  * [8] M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Applied Physics Letters 68, 2793–2795 (1996).
  * [9] T. Nagy, M. Kretschmar, M. J. Vrakking, and A. Rouzée, “Generation of above-terawatt 1.5-cycle visible pulses at 1 khz by post-compression in a hollow fiber,” Optics Letters 45, 3313–3316 (2020).
  * [10] L. Silletti, A. Bin Wahid, E. Escoto, P. Balla, S. Rajhans, K. Horn, L. Winkelmann, V. Wanie, A. Trabattoni, C. M. Heyl _et al._ , “Dispersion-engineered multi-pass cell for single-stage post-compression of an ytterbium laser,” Optics Letters 48, 1842–1845 (2023).
  * [11] C.-H. Lu, T. Witting, A. Husakou, M. J. Vrakking, A. Kung, and F. J. Furch, “Sub-4 fs laser pulses at high average power and high repetition rate from an all-solid-state setup,” Optics express 26, 8941–8956 (2018).
  * [12] P. He, Y. Liu, K. Zhao, H. Teng, X. He, P. Huang, H. Huang, S. Zhong, Y. Jiang, S. Fang, X. Hou, and Z. Wei, “High-efficiency supercontinuum generation in solid thin plates at 0.1 TW level,” Opt. Lett. 42, 474–477 (2017).
  * [13] C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
  * [14] T. Nagy, P. Simon, and L. Veisz, “High-energy few-cycle pulses: post-compression techniques,” Adv. Phys. 6, 1845795 (2021).
  * [15] L. Gallmann, T. Pfeifer, P. Nagel, M. Abel, D. Neumark, and S. Leone, “Comparison of the filamentation and the hollow-core fiber characteristics for pulse compression into the few-cycle regime,” Appl. Phys. B 86, 561–566 (2007).
  * [16] E. Mayer, J. Möbius, A. Euteneuer, W. Rühle, and R. Szipőcs, “Ultrabroadband chirped mirrors for femtosecond lasers,” Optics letters 22, 528–530 (1997).
  * [17] T. Nagy, S. Hädrich, P. Simon, A. Blumenstein, N. Walther, R. Klas, J. Buldt, H. Stark, S. Breitkopf, P. Jójárt _et al._ , “Generation of three-cycle multi-millijoule laser pulses at 318 w average power,” Optica 6, 1423–1424 (2019).
  * [18] V. Cardin, N. Thiré, S. Beaulieu, V. Wanie, F. Légaré, and B. E. Schmidt, “0.42 tw 2-cycle pulses at 1.8 $\mu$ m via hollow-core fiber compression,” Applied Physics Letters 107, 181101 (2015).
  * [19] M. Gebhardt, C. Gaida, T. Heuermann, F. Stutzki, C. Jauregui, J. Antonio-Lopez, A. Schulzgen, R. Amezcua-Correa, J. Limpert, and A. Tünnermann, “Nonlinear pulse compression to 43 w gw-class few-cycle pulses at 2 $\mu$m wavelength,” Optics letters 42, 4179–4182 (2017).
  * [20] H. Timmers, Y. Kobayashi, K. F. Chang, M. Reduzzi, D. M. Neumark, and S. R. Leone, “Generating high-contrast, near single-cycle waveforms with third-order dispersion compensation,” Optics letters 42, 811–814 (2017).
  * [21] A. C. Jones, M. B. Kunz, I. Tigges-Green, and M. T. Zanni, “Dual spectral phase and diffraction angle compensation of a broadband aom 4-f pulse-shaper for ultrafast spectroscopy,” Optics express 27, 37236–37247 (2019).
  * [22] A. Catanese, B. Kaufman, C. Cheng, E. Jones, M. G. Cohen, and T. Weinacht, “Acousto-optic modulator pulse-shaper compression of octave-spanning pulses from a stretched hollow-core fiber,” OSA Continuum 4, 3176–3183 (2021).
  * [23] M. Miranda, C. L. Arnold, T. Fordell, F. Silva, B. Alonso, R. Weigand, A. L’Huillier, and H. Crespo, “Characterization of broadband few-cycle laser pulses with the d-scan technique,” Optics express 20, 18732–18743 (2012).
  * [24] I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, E. J. Gualda, and D. Artigas, “Ultrashort pulse characterisation with shg collinear-frog,” Optics express 12, 1169–1178 (2004).
  * [25] T. Nagy, P. Simon, and L. Veisz, “High-energy few-cycle pulses: post-compression techniques,” Advances in Physics: X 6, 1845795 (2021).
  * [26] R. Abrams, “Coupling losses in hollow waveguide laser resonators,” IEEE Journal of Quantum Electronics 8, 838–843 (1972).
  * [27] G. Fan, R. Safaei, O. Kwon, V. Schuster, K. Légaré, P. Lassonde, A. Ehteshami, L. Arias, A. Laramée, J. Beaudoin-Bertrand _et al._ , “High energy redshifted and enhanced spectral broadening by molecular alignment,” Optics Letters 45, 3013–3016 (2020).
  * [28] D. Agathangelou, A. Javed, F. Sessa, X. Solinas, M. Joffre, and J. P. Ogilvie, “Phase-modulated rapid-scanning fluorescence-detected two-dimensional electronic spectroscopy,” The Journal of Chemical Physics 155 (2021).
  * [29] E. Collini, C. Y. Wong, K. E. Wilk, P. M. Curmi, P. Brumer, and G. D. Scholes, “Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature,” Nature 463, 644–647 (2010).
  * [30] V. Tagliamonti, B. Kaufman, A. Zhao, T. Rozgonyi, P. Marquetand, and T. Weinacht, “Time-resolved measurement of internal conversion dynamics in strong-field molecular ionization,” Physical Review A 96, 021401 (2017).


# Yang-Mills instantons as the end point of black hole evaporation

Pisin<EMAIL_ADDRESS>Xiao Yan
<EMAIL_ADDRESS>Misao<EMAIL_ADDRESS>and Dong-han<EMAIL_ADDRESS>aLeung Center for Cosmology
and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan
bDepartment of Physics, National Taiwan University, Taipei 10617, Taiwan
cGraduate Institute of Astrophysics, National Taiwan University, Taipei 10617,
Taiwan
dKavli Institute for Particle Astrophysics and Cosmology, SLAC National
Accelerator Laboratory, Stanford University, Stanford, California 94305, USA
eSchool of Science, Jiangsu University of Science and Technology, Zhenjiang,
212100 China
fKavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Chiba 277-8583, Japan
gYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502,
Japan
hDepartment of Physics Education, Pusan National University, Busan 46241,
Republic of Korea
iResearch Center for Dielectric and Advanced Matter Physics, Pusan National
University, Busan 46241, Republic of Korea

###### Abstract

Non-perturbative contributions of the Euclidean path integral are important to
understand the information loss paradox. In this paper, we revisit the Yang-
Mills instantons in the Einstein-Yang-Mills theory. There exists a globally
regular solution that is known as the Bartnik-McKinnon solution and a black
hole solution. The regular and the black hole solutions are smoothly connected
in the small horizon limit. Their Euclidean action is solely characterized by
the ADM mass, and the transition probability follows the usual Bekenstein-
Hawking entropy formula. Therefore, the Yang-Mills instantons provide a non-
perturbative channel to the black hole evaporation, which competes effectively
with perturbative processes, and becomes dominant toward the end of
evaporation. We show that these instantons provide a smooth transition
mechanism from a black hole to regular spacetime.

††preprint: YITP-23-70

###### Contents

  1. I Introduction
  2. II Model
    1. II.1 Einstein-Yang-Mills theory
    2. II.2 Solutions
      1. II.2.1 Regular solutions: Bartnik-McKinnon model
      2. II.2.2 Black holes in Einstein-Yang-Mills Theory
  3. III Applications for evaporation
    1. III.1 Euclidean action
    2. III.2 Interpretation
      1. III.2.1 Large black holes
      2. III.2.2 Final stage of black hole evaporation
  4. IV Implications to the information loss problem
  5. V Conclusion

##  I Introduction

The evaporation of black holes is a consequence of general relativity and
quantum mechanics Hawking:1975vcx . It poses, however, a serious challenge
about the consistency between these two fundamental theories since it appears
to imply the violation of unitarity. In other words, a consistent description
of an evaporating black hole demands the reconciliation of the tension between
general relativity and unitary quantum mechanics Yeom:2009zp . Recently, many
important and interesting solutions are reported in the literature
Almheiri:2019hni . The common features of them indicate that the following two
conditions are required to explain the unitarity of black hole evaporation
Chen:2021jzx :

  * –

1\. Multi-history condition: One must take account of the contribution of
different semi-classical histories to the wavefunction of the system, which
include information-preserving histories, i.e., geometries without event
horizons.

  * –

2\. Late-time dominance condition: The contribution of information-preserving
histories should dominate the wave function at the late time of the black hole
evaporation.

If these two conditions are realized, the unitary evolution of the black hole
system is automatically guaranteed, and one can consistently derive the Page
curve Page:1993df  for the entanglement entropy Chen:2021jzx .

Although the above two conditions look conceptually simple, they are
practically highly non-trivial to realize. First, one needs a formulation that
is naturally endowed with the multi-history condition which includes
information-preserving histories. One possibility is to take account of
replica wormholes Almheiri:2019hni . In this case, the two conditions seem
realizable by including replica wormhole configurations in the path integral.
However, since the computations of replica wormholes are based on the density
matrix instead of the wave function, it is unclear whether they can be
embedded in the canonical path integral formalism. An alternative is to
develop a method to compute the contribution of trivial geometries, i.e.,
those with neither horizon nor singularity, in the evolution of the system
Hawking:2005kf . In this case, a crucial issue is to exhibit the late-time
dominance condition. In our recent papers Chen:2021jzx , we showed that the
probability of tunneling toward trivial geometries will dominate as the black
hole approaches the end point of evaporation, and hence the late-time
condition is realized, under the assumption that instantons that mediate
tunneling from the black hole geometry to a trivial geometry exist in general.
A price to pay was the modification of the Page curve in such a way that it no
longer respects the Bekenstein-Hawking entropy bound Chen:2014jwq .

It is fair to say that, at this moment, the final answer is not yet in sight.
It is clear, however, that there is more to be learned about non-perturbative
tunneling channels that may contribute to the black hole evaporation. In this
paper, we take the second, canonical wavefunction approach, and discuss non-
perturbative tunneling channels in the Einstein-Yang-Mills (EYM) theory, with
the attempt to shed some lights on the important issue of information loss
paradox.

An example of a non-perturbative tunneling channel was discussed under the
thin-shell approximation, namely, a matter field that allows for a thin
spherical shell configuration Sasaki:2014spa . This is a nice example of non-
perturbative tunneling channels to trivial geometries, but it is valid only
under the assumption of a highly tuned potential, and hence the generality is
lost. Another interesting example is to consider a black hole plus free scalar
field instantons Chen:2018aij , where the Hawking radiation is described as
tunneling mediated by these instantons in the small back-reaction limit.555The
idea of describing the Hawking radiation as tunneling seems quite old. See,
e.g., Massar:1997ic . We thank Ted Jacobson for letting us know about this
reference. However, while we should be able to consistently describe not only
the perturbative Hawking radiation but also the non-perturbative tunneling to
trivial geometries, explicit construction of these instantons that includes
back-reaction remains a technical challenge, as one has to deal with the
space-time dependent dynamics of non-perturbative complex-valued instantons.

In this context, it is of great interest if there exists a fully nonlinear
instanton solution that can describe the non-perturbative tunneling of a black
hole to a trivial geometry. In this paper, with this point of view in mind, we
revisit the EYM instantons in the light of the black hole evaporation. Since
Yang-Mills (YM) fields surely exist in Nature, such instanton solutions are
expected to make an important contribution to black hole evaporation.

It has been shown that there are no globally regular solutions in the pure YM
field Deser:1976wq ; Coleman:1975qj ; Deser:1983fk . However, Bartnik and
McKinnon discovered that regular and asymptotically flat solutions exist once
gravity is taken into account, that is, in EYM theory Bartnik:1988am .
Subsequently, a black hole solution with a non-trivial YM field configuration
was also constructed Bizon:1990sr ; Volkov:1990sva ; Kuenzle:1990is . These
two solutions are connected smoothly in the small horizon limit. Furthermore,
they possess infinite sequences of solutions, which can be labeled as $k$ to
represent the higher nodes for the excitation of the gauge field and approach
the extremal Reissner-Nordstrom black hole for $k\rightarrow\infty$. It is
important to mention that they are unstable against linear or nonlinear
perturbationsStraumann:1989tf ; Straumann:1990as ; Zhou:1991nu ; Bizon:1991nt
; Bizon:1991hw ; Galtsov:1991du ; Galtsov:1991nk ; Lavrelashvili:1994rp ;
Volkov:1994dq ; Mavromatos:1995kc ; Brodbeck:1994vu . For a more detailed
review, see, e.g., Galtsov and Volkov Volkov:1998cc . Thus they will
eventually decay into particles and radiation.

The fact that they are actually saddle points of the Euclidean action was
pointed out by Moss and Wray Moss:1992sb , which implies that they can be
legitimately regarded as instantons that contribute to quantum tunneling. The
question is then whether they contribute to the black hole evaporation. By
computing the bounce action, we argue that this is indeed the case. In
particular, we conclude that the regular Yang-Mills instantons (and their
eventual decay products) constitute part of the information-preserving
histories. We note that, from the point of view of the Page curve, as these
instantons somewhat enhance the probability of trivial geometries relative to
the black hole geometry, the violation of the Bekenstein-Hawking bound is
slightly eased, though the modification will is significant only toward the
end point of evaporation.

This paper is organized as follows. In Sec. II, we discuss our model and
solutions of EYM theory. In Sec. III, we evaluate the Euclidean action of the
solutions; based on this computation, we conclude that they indeed play the
role of instantons mediating the evaporation process, and they contribute not
only near the endpoint of the evaporation but also in a large black hole
background. Finally, in Sec. IV and Sec. V, we discuss in light of the
information loss paradox and discuss possible future research directions.

##  II Model

In this section, we review the Einstein-Yang-Mills instantons and discuss
their physical properties.

###  II.1 Einstein-Yang-Mills theory

We consider the Einstein-Yang-Mills (EYM) theory with the SU(2) non-Abelian
gauge group:

$\displaystyle S=\int\left[\frac{R}{16\pi
G}-\frac{1}{2e^{2}}\text{Tr}\left(F_{\mu\nu}F^{\mu\nu}\right)\right]\sqrt{-g}dx^{4},$
(1)

where $R$ is the Ricci scalar, $e$ is the coupling constant of the YM theory,
and ${F_{\mu\nu}}$ is the field strength for the non-Abelian gauge field
${A_{\mu}}$ which is defined as

$\displaystyle{F_{\mu\nu}}$ $\displaystyle=$
$\displaystyle\nabla_{\mu}{A_{\nu}}-\nabla_{\nu}{A_{\mu}}+i[A_{\mu},A_{\nu}],$
(2) $\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\tau^{a}A^{a}_{\mu}\,,$ (3)

where $\tau_{a}$ is the Pauli matrices.

Taking variations with respect to the metric $g_{\mu\nu}$ and the non-Abelian
gauge field ${A_{\mu}}$ yields the Einstein equation and the Yang-Mills
equation, respectively,

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle\frac{\kappa}{e^{2}}\,\text{Tr}\left({F_{\mu\alpha}}{F_{\nu}^{\,\alpha}}-\frac{1}{4}g_{\mu\nu}{F_{\alpha\beta}}{F^{\alpha\beta}}\right),$
(4) $\displaystyle\nabla_{\mu}F^{\mu\nu}+i\left[{A_{\mu}},F^{\mu\nu}\right]$
$\displaystyle=$ $\displaystyle 0\,.$ (5)

where $\kappa=8\pi G$. In the following, we use the Planck units, $G=1$, and
set $e=1$. Note that the $e\rightarrow 0$ limit corresponds to the decoupling
limit of the Yang-Mills field; $e\gg 1$ corresponds to the strong coupling
limit and $e\ll 1$ corresponds to the weak coupling limit.

We employ a spherically symmetric ansatz for the metric,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2m(r)}{r}\right)\sigma(r)dt^{2}+\frac{dr^{2}}{1-\dfrac{2m(r)}{r}}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\,,$ (6)

and a purely magnetic gauge field $(A_{t}=0)$ with the spherically symmetric
property to construct the solutions in the EYM theory,

$\displaystyle
A_{\mu}dx^{\mu}=\frac{1-w(r)}{2}\left(\tau_{\varphi}d\theta-\tau_{\theta}\sin\theta
d\varphi\right)\,,$ (7)

where $m$, $\sigma$, and $w$ are the functions of $r$. The function $m(r)$ is
known as the Misner-Sharp mass Misner:1964je , where it provides the ADM mass
$M$ at the spatial infinity, i.e., $m(\infty)=M$.

The substitution of Eqs. (6) and (7) into the Einstein and Yang-Mills
equations yields the following ordinary differential equations (ODEs) for $m$,
$\sigma$, and $w$:

$\displaystyle m^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2}\left(1-\frac{2m}{r}\right)w^{\prime
2}+\kappa\frac{\left(w^{2}-1\right)^{2}}{4r^{2}},$ (8)
$\displaystyle\sigma^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2\kappa\sigma}{r}w^{\prime 2},$ (9) $\displaystyle
w^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{w^{\prime}}{2r^{3}\left(1-\dfrac{2m}{r}\right)}\left(\kappa\left(w^{2}-1\right)^{2}-4rm\right)+\frac{w\left(w^{2}-1\right)}{r^{2}\left(1-\dfrac{2m}{r}\right)}.$
(10)

###  II.2 Solutions

There exist two regular and asymptotically flat solutions in the EYM theory.
The first solution is known as the Bartnik-Mckinnon (BM) solution that is
entirely regular and does not have a horizon at the center; the second
solution is a black hole with a non-trivial Yang-Mills hair outside the event
horizon. We need to solve the ODEs with two different sets of boundary
conditions in order to construct them numerically.

To construct the BM solution numerically, we integrate the ODEs from the
origin $(r=0)$ to infinity. All functions have to be regular at the center
$(r=0)$. The requirement that all functions have to be regular at the center
($r=0$) is attained by requiring the functions to be finite and their first-
order derivatives vanish at $r=0$; thus a series expansion of a regular
solution at $r=0$ takes the form:

$\displaystyle m(r)$ $\displaystyle=$
$\displaystyle\kappa\overline{W}^{2}_{2}r^{3}+\frac{4}{5}\kappa\overline{W}^{3}_{2}r^{5}+\mathcal{O}\left(r^{7}\right)\,,$
(11) $\displaystyle\sigma(r)$ $\displaystyle=$
$\displaystyle\sigma_{0}-2\kappa\overline{W}^{2}_{2}r^{2}+\mathcal{O}\left(r^{4}\right)\,,$
(12) $\displaystyle w(r)$ $\displaystyle=$ $\displaystyle
1+\overline{W}_{2}r^{2}+\frac{1}{10}\overline{W}^{2}_{2}\left(3+4\kappa\overline{W}_{2}\right)r^{4}+\mathcal{O}\left(r^{4}\right)\,,$
(13)

where $\sigma_{0}$ and $\overline{W}_{2}$ are the values at the origin.

For the black hole solution, which we call the Yang-Mills black hole (YMBH)
solution, we integrate the ODEs from the horizon $r_{h}$ to infinity. Like
most BM solutions, all the functions and their derivatives have to be regular
at the horizon $r_{h}$. Imposing these regularity conditions, we also obtain a
series expansion of a regular solution at $r_{h}$ in the form:

$\displaystyle m(r)$ $\displaystyle=$ $\displaystyle
m_{h}+\frac{\kappa}{4r^{2}_{h}}\left(1-W^{2}_{h}\right)^{2}\left(r-r_{h}\right)+\mathcal{O}\left(\left(r-r_{h}\right)^{2}\right)\,,$
(14) $\displaystyle\sigma(r)$ $\displaystyle=$
$\displaystyle\sigma_{h}-\kappa\frac{\widetilde{W}^{2}_{1}}{r_{h}}\left(r-r_{h}\right)+\mathcal{O}\left(\left(r-r_{h}\right)^{2}\right)\,,$
(15) $\displaystyle w(r)$ $\displaystyle=$ $\displaystyle
W_{h}+\widetilde{W}_{1}\left(r-r_{h}\right)+\mathcal{O}\left(\left(r-r_{h}\right)^{2}\right)\,,$
(16)

where $m_{h}=r_{h}/2$ is the Misner-Sharp mass at the horizon (if there is no
horizon, $r_{h}=m_{h}=0$), $\sigma_{h}$ and $W_{h}$ are the values at the
horizon, and $\widetilde{W}_{1}$ is given by

$\widetilde{W}_{1}=\frac{2r_{h}W_{h}\left(1-W^{2}_{h}\right)}{\kappa\left(1-W^{2}_{h}\right)^{2}-2r^{2}_{h}}\,.$
(17)

At infinity, both the YMBH solution and the BM solution share the same
asymptotic behavior by requiring the metric functions to be asymptotically
flat and the gauge field is bounded to unity. Namely, both solutions have the
asymptotic behavior given by

$\displaystyle m(r)$ $\displaystyle=$ $\displaystyle M-\frac{\kappa
w^{2}_{1}}{2r^{3}}\mp\frac{\kappa w^{2}_{1}\left(4w_{1}\pm
5M\right)}{4r^{4}}+\mathcal{O}\left(\frac{1}{r^{5}}\right)\,,$ (18)
$\displaystyle\sigma(r)$ $\displaystyle=$ $\displaystyle 1-\frac{\kappa
w^{2}_{1}}{2r^{4}}\mp\frac{6\kappa w_{1}\left(w_{1}\pm
2M\right)}{5r^{5}}+\mathcal{O}\left(\frac{1}{r^{6}}\right)\,,$ (19)
$\displaystyle w(r)$ $\displaystyle=$ $\displaystyle\pm
1+\frac{w_{1}}{r}\pm\frac{3w_{1}\left(w_{1}\pm
2M\right)}{4r^{2}}+\mathcal{O}\left(\frac{1}{r^{3}}\right)\,,$ (20)

where $w_{1}$ is a constant determined by each solution.

Note that there are three parameters at the center ($m_{h}=0$, $\sigma_{0}$,
and $\overline{W}_{2}$ for the BM solution; $m_{h}$, $\sigma_{h}$, and $W_{h}$
for the black hole solution). The choice of $m_{h}$ determines $M$, while the
other two must be tuned so that $\sigma\to 1$ and $w\to\pm 1$ at infinity.
Technically, as clear from Eq. (9), $\sigma$ has a constant rescaling degree
of freedom. This is fixed by the asymptotic value $\sigma(\infty)=1$, which
consequently fixes the value of $\sigma_{0}$. Therefore, the only non-trivial
parameter is $\overline{W}_{2}$ for the BM solution or $W_{h}$ for the YMBH
solution, which is to be determined by the condition $w(\infty)=\pm 1$. An
interesting feature of the BM and the YMBH solutions is that there exists a
sequence of solutions, where each solution has a different number of zeros of
$w(r)$. We call the solution with $k$ nodes the $k$-th node solution, where
$k=1\,,2\,,3\,,\cdots$.

(a)  (b)  (c)

Figure 1: The profiles of the functions (a) $m(r)$, (b) $\ln(\sigma)$, and (c)
$w(r)$, describing the BM solutions with $k$ nodes ($k=1$, 2, 3, 4, 5) in the
logarithmic scale of radial coordinate $r$.

(a)  (b)

Figure 2: Properties of the BM solution: (a) the ADM mass $M_{k}$ and (b) the
location $r^{\text{max}}_{k}$ of the outermost zero as a function of $k$.

####  II.2.1 Regular solutions: Bartnik-McKinnon model

(a)  (b)  (c)

Figure 3: Profiles of the YMBH solution with $r_{h}=1$ for $k$ nodes in the
logarithmic scale of radial coordinate $r$: (a) $m(r)$, (b) $\ln(\sigma(r))$,
(c) $w(r)$.

(a)  (b)  (c)  (d)

Figure 4: Properties of the YMBH solutions with: (a) The mass $M_{k}$ and (b)
the locations of the outermost zero, $\text{log}_{10}(r^{\text{max}}_{k})$, as
a function of $k$, for $r_{h}=1$. (c) The mass $M$ versus the horizon radius
$r_{h}$ for $k=1,\,2,\,3,\,4$, and (d) the same but in a log-log plot.

Let us first review the profiles of the BM solutions for $m(r)$, $w(r)$ and
$\sigma(r)$ with nodes $k=1$, 2, 3, 4, 5, in the logarithmic scale of the
radial coordinate $r$ in Fig. 1. Figs. 1(a) and (b) show that the functions
$m(r)$ and $\sigma(r)$ behave monotonically regardless of $k$. Since the BM
solution possesses infinite sequences of solutions, then the number of zeros
for $w(r)$ increases when $k$ increases as shown in Fig. 1(c). Note that
$w(r)$ is bounded in between $\pm 1$. The ADM mass of BM solutions can be read
directly from Fig. 2(a). In the limit of $k\to\infty$, $w\to 0$ in the region
$1<r<r_{k}$, and the geometry there approaches the extremal Reissner-Nordstrom
black hole solution, with the asymptotic ADM mass approaching a limiting value
of the order of $\sim M_{\mathrm{Pl}}$ as shown in Fig. 2(a). Fig. 2(b) shows
the location of the outermost zero for each BM solution. If we define the
outermost zero of the BM solution as its effective size, then the size of the
back hole increases exponentially when $k$ increases linearly, as shown in
Fig. 2(c).

Note that, from the Einstein equations, Eq. (4), one readily sees that the
effect of $e\neq 1$ is to replace $\kappa$ by $\kappa/e^{2}$. Then an
inspection of Eqs. (8), (9) and (10) tells us that the solution can be
obtained by the rescaling Bartnik:1988am ,

$\displaystyle m_{e}(r)=\frac{1}{e}m_{1}(er),$ (21)

where $m_{e}$ is the solution for an arbitrary $e$ and $m_{1}$ is the solution
for $e=1$. Since $m_{1}(\infty)\simeq M_{\mathrm{Pl}}$, the ADM mass of the
solution increases as $e$ decreases. One can easily notice that, as $e$ goes
to zero, the size of the solution increases, and accordingly the mass
approaches infinity. Oppositely, if $e$ goes to infinity, the size and the
mass of the solution become zero.

####  II.2.2 Black holes in Einstein-Yang-Mills Theory

Similar to the case of the BM solutions, the black holes also possess an
infinite sequence of solutions. Fig. 3 shows the profiles of the YMBH
solutions with nodes $k=1,\,2,\,3,\,4,\,5$, respectively, for $r_{h}=1$ in the
logarithmic scale of the radial coordinate $r$. The similarity also holds for
the behaviors of the functions $m(r)$, $\sigma(r)$ and $w(r)$. Fig. 3(a) and
(b) show that $m(r)$ and $\sigma(r)$ behave monotonically regardless of $k$,
and Fig. 3(c) shows that $w(r)$ is bounded in between $\pm 1$ and the number
of zeros for $w(r)$ increases as $k$ increases. The ADM mass of the black hole
solutions can be read off from Fig. 3(a).

In Fig. 4, we show the properties of these infinite sequences of solutions.
Fig. 4(a) shows the ADM mass as a function of $k$ in the case of $r_{h}=1$.
Fig. 4(b) shows the locations of the outermost zero of $w$, which increases
exponentially with $k$. Fig. 4(c) exhibits the ADM mass by varying the radius
of horizon $r_{h}$ for the solutions with nodes $k=1,\,2,\,3,\,4$. In the
limit $k\to\infty$, the solution also approaches the extremal Reissner-
Nordstrom black hole. The difference is that we may have an arbitrary large
black hole by choosing an arbitrarily large Misner-Sharp mass at the horizon
$m_{h}$. Hence, the ADM mass approaches the Misner-Sharp mass in the large
mass limit, $M\sim m_{h}$ as shown in Figs. 4(c) and (d).

Figure 5: Schematic diagram of the quantum transition from a black hole to a
regular Yang-Mills solution. The red dotted line in the lower figure is the
hypersurface through which the Lorentzian black hole is analytically continued
to the Euclidean solution. The red dotted lines of the upper figure is the
hypersurface through which the Euclidean BM solution is analytically continued
to its Lorentzian counterpart. The hypersurfaces $\tau=0$ and $\tau=\tau_{0}$
are to be identified, with $\tau_{0}$ being equal to the inverse black hole
temperature $\beta$.

##  III Applications for evaporation

In this section, regarding the BM and YMBH solutions as instantons, we compute
their Euclidean actions. As they are static, the Euclidean solutions are
easily obtained by the Wick rotation of the time coordinate. The periodicity
in the Euclidean time is arbitrary in the case of the BM solutions, while it
is fixed by the Misner-Sharp mass in the case of the black hole solutions.

To describe the process of the initial black hole tunneling to a BM or YMBH
solution, we consider the analytic continuations from Lorentzian to Euclidean
geometries and vice versa, as illustrated in Fig. 5. The lower part of Fig. 5
describes the initial Lorentzian Schwarzschild black hole and its analytic
continuation to a Euclidean Schwarzschild spacetime. It is joined to the
instantons in the Euclidean spacetime, and analytically continued back to the
Lorentzian geometry as illustrated in the upper part of Fig. 5. We note that
since there is neither horizon nor singularity in the BM solution, which is
topologically equivalent to that of Minkowski, there is only a single
asymptotic infinity. In order to compute the tunneling probability, following
the standard procedure, we evaluate the Euclidean actions of the two Euclidean
geometries and subtract each other.

###  III.1 Euclidean action

The Euclidean action is

$\displaystyle
S_{\mathrm{E}}=\int_{\mathcal{M}}\left[-\frac{R}{16\pi}+\frac{1}{2}\text{Tr}\left({F_{\mu\nu}}{F^{\mu\nu}}\right)\right]\sqrt{+g}dx^{4}-\frac{1}{8\pi}\int_{\partial\mathcal{M}}\left(K-K_{0}\right)\sqrt{+h}d^{3}x,$
(22)

where $K$ and $K_{0}$ in the second term are the Gibbons-Hawking boundary term
of the solution and the Minkowski background, respectively.

The Ricci scalar term is, after the integration by parts,

$\displaystyle\int\left[-\frac{R}{16\pi}\right]\sqrt{+g}dx^{4}$
$\displaystyle=$ $\displaystyle
4\pi\beta\int_{r_{h}}^{\infty}dr\sqrt{\sigma}\left(-\frac{2}{\kappa}m^{\prime}\right)+\frac{\beta}{4}\left.\left(\left(2m-2rm^{\prime}\right)\sqrt{\sigma}+\left(r-2m\right)r\frac{\sigma^{\prime}}{\sqrt{\sigma}}\right)\right|_{r_{h}}^{\infty}$
(23) $\displaystyle=$ $\displaystyle
4\pi\beta\int_{r_{h}}^{\infty}dr\sqrt{\sigma}\left(-\frac{2}{\kappa}m^{\prime}\right)+\frac{\beta}{2}\left(M-m_{h}\right),$

where $\beta$ is the Euclidean time period. Recall that $r_{h}=m_{h}=0$ in the
absence of a black hole. Note that the asymptotic behavior of the solution,
Eqs. (18) and (19), requires $\sigma\to 1$, $\sigma^{\prime}=O(r^{-5})$ and
$m^{\prime}=O(r^{-4})$ at infinity.

The Yang-Mills term is

$\displaystyle\int\left[\frac{1}{2}\text{Tr}\left({F_{\mu\nu}}{F^{\mu\nu}}\right)\right]\sqrt{+g}dx^{4}$
$\displaystyle=$ $\displaystyle
4\pi\beta\int_{r_{h}}^{\infty}dr\sqrt{\sigma}\left(\frac{(1-w^{2})^{2}}{2r^{2}}+\left(1-\frac{2m}{r}\right)w^{\prime
2}\right)$ (24) $\displaystyle=$ $\displaystyle
4\pi\beta\int_{r_{h}}^{\infty}dr\sqrt{\sigma}\left(\frac{2}{\kappa}m^{\prime}\right),$

where we have used Eq. (9) for $m^{\prime}$ to obtain the second line.

The contribution from the Gibbons-Hawking boundary term, which is the same as
that in the Schwarzschild black hole case, is given by

$\displaystyle-\frac{1}{8\pi}\int_{\partial\mathcal{M}}\left(K-K_{0}\right)\sqrt{+h}d^{3}x=\beta\frac{M}{2}.$
(25)

Adding Eqs. (24) and (25), we obtain

$\displaystyle S_{\mathrm{E}}=\beta M-\frac{\beta}{2}m_{h}\,.$ (26)

We note that regarding $\beta$ as the inverse of the temperature determined by
the central black hole, $\beta=1/T=8\pi m_{h}$, this has a natural
thermodynamic interpretation. Namely, since the second term may be expressed
as $S=4\pi m_{h}^{2}$, which is the entropy of the black hole at the center,
we have $TS_{\mathrm{E}}=M-TS$, which is the free energy of the black hole.

###  III.2 Interpretation

####  III.2.1 Large black holes

If one considers the Hawking radiation from a Schwarzschild black hole as
tunneling mediated by the above instanton, we should take $\beta=8\pi M$,
where $M$ is the total energy of the system that remains the same before and
after the tunneling. In this case, since the horizon mass is $m_{h}=M-\omega$,
where $\omega$ is the energy carried out as Hawking radiation, there appears a
cusp at the horizon. However, by using the regularization method
Gregory:2013hja , one can compute the correction due to the cusp, and the
difference between the final and initial Euclidean actions is found to be
given exactly by the difference between the initial and final horizon surface
areas Moss:1992sb ,

$\displaystyle B\equiv
S_{E}(M-\omega)-S_{E}(M)=\frac{\Delta\mathcal{A}}{4}=4\pi\left(M^{2}-\left(M-\omega\right)^{2}\right)\simeq
8\pi M\omega=\beta\omega,$ (27)

where we assumed $\omega\ll M$.

Here a comment is in order. Since there exist infinitely many solutions for a
given $M$ with increasing size $r_{k}$ ($k=1,2,3.\cdots$), one should sum over
all the contributions. Let us define $\omega_{k}=M-m_{h,k}$. As $\omega_{k}$
approaches a constant in the limit $k\to\infty$, as discussed in Section II,
one might worry that these infinitely many solutions would lead to a divergent
contribution to the probability, whereas we have only considered the leading
order WKB action so far. The contribution from the next WKB order is commonly
interpreted as the prefactor $A$ of the tunneling rate, that is,

$\displaystyle\Gamma\simeq Ae^{-B}.$ (28)

It is known in general that $A$ has the volume dependence Dunne:2005rt , which
is proportional to the energy divided by the volume (see also Coleman:1978ae ,
where $A$ is inversely proportional to the fourth power of the instanton
size). Therefore, although we do not explicitly compute the prefactor here, as
it is beyond the scope of the present paper, it is reasonable to expect that
the prefactor $A_{k}$ for the $k$-node solution will provide a large
additional suppression factor $\sim V_{k}^{-1}=r_{k}^{-3}$, since the volume
is exponentially large, $V_{k}\propto e^{C\,k}$, where $C=O(10)$ (see Fig.
2(b)).

To summarize, the dominant contribution comes only from small $k$’s, and hence
we obtain

$\displaystyle\Gamma_{\mathrm{bh}\rightarrow\mathrm{sol}}\simeq
e^{-\beta\omega}$ (29)

as we expected.

####  III.2.2 Final stage of black hole evaporation

The contribution of the regular BM solutions to the evaporation of a black
hole of mass $M$ should be important only when $M\simeq M_{k}$, based on mass
conservation. Since $M_{k}=\mathcal{O}(M_{\mathrm{Pl}})$ for any $k$, they
play an important role only near the end of the evaporation, that is, only for
Planck-mass scale black holes. Note that since the mass of the BM solution
scales as $1/e$, where $e$ is the gauge coupling constant as given in Eq.
(21), it could be much heavier than the Planck mass if $e\ll 1$. However, we
expect $e\sim 0.1$ as a typical value in the standard model. Thus, it is
expected that the mass is somewhat larger than the Planck mass, but not
extremely larger.

The tunneling rate due to the $k$-node solution is then approximately

$\displaystyle\Gamma_{k}\simeq e^{-\beta M_{k}+4\pi M^{2}}=e^{-4\pi
M_{k}^{2}},$ (30)

when $M=M_{k}$. If all $k$’s would contribute equally, this would mean a
catastrophic end of evaporation when the black hole mass becomes equal to that
of the BM solution with asymptotically large $k$. However, repeating the same
argument as the one in the previous subsection, we do not expect this to
happen. Still, we expect that the regular BM solutions may have a mild
influence on the evaporation at the final stage, by contributing a few
additional evaporating channels. In any case, as the black hole mass
approaches the Planck scale, the semi-classical approximation would break
down. It is therefore premature to draw any definitive conclusion.

The final question is, what happens after the tunneling? The instability of
the Yang-Mills instantons is known in the literature Straumann:1989tf . On the
other hand, it is important to note that the Bartnik-McKinnon solution is a
natural generalization of the sphalerons Galtsov:1991du . It is therefore
reasonable to expect that after they are formed, they would decay into
particles. The decay of these instantons may contribute to the baryon number
violation.

##  IV Implications to the information loss problem

In order to explain the unitary of the Page curve, let us recall the following
two ansatz Chen:2021jzx :

  * –

1\. Multi-history condition: For the entire path integral, there must be
contributions from various semi-classical histories, some of which are
information-preserving.

  * –

2\. Late-time dominance condition: The contribution from the information-
preserving histories should dominate at the late time of the black hole
evaporation.

For simplicity, let us assume that there are only two semi-classical
histories; one is the information-losing history, in which the black hole
evaporates and vanishes in the end, and the information-preserving history, in
which the black hole tunnels to a trivial geometry mediated by an instanton.
Let $p_{1}(t)$ be the probability of the information-losing history, which is
a function of time, and $p_{2}(t)$ be the probability of the information-
preserving history, where $p_{1}+p_{2}=1$ at each instant of time.

Now consider the entanglement entropy by separating the system into two: the
regions inside and outside the black hole. It is natural to expect that
$S_{1}$ monotonically increases. Namely, if $S_{0}=4\pi M_{0}^{2}$ is the
initial Bekenstein-Hawking entropy of the black hole and $S=4\pi M^{2}$ is the
Bekenstein-Hawking entropy at some later time, we may assume

$\displaystyle S_{1}(S)=S_{0}-S\,,$ (31)

as long as the Hawking radiation produces maximally entangled pairs
Hwang:2017yxp . On the other hand, for the information-preserving history, in
which there is neither horizon nor singularity, we have

$\displaystyle S_{2}(S)=0.$ (32)

Then the expectation value of the entanglement entropy of the entire system is
given by

$\displaystyle S=p_{1}S_{1}+p_{2}S_{2}=p_{1}(S_{0}-S)\,.$ (33)

It is clear that the entanglement entropy increases at the early stage of the
evaporation when $p_{1}(t)\approx 1$, while it decreases to zero toward the
end of evaporation as $p_{1}(t)=1-p_{2}(t)\rightarrow 0$ by assumption. This
explains the unitary Page curve.

Then the next question is: what happens if we include additional non-
perturbative contributions like those discussed in this paper? Note that
$p_{2}$ will dominate near the Planck scale. If multiple instantons contribute
to the tunneling toward trivial geometries, it is quite reasonable to expect
that the probability for $p_{2}$ would be enhanced. This would then render the
dominance of information-preserving histories to shift toward an earlier time.
Namely, for a given black hole mass, the probability of tunneling to a trivial
geometry should be higher with multiple instantons than without. Consequently,
the Page time, at which the expectation value of the entanglement entropy
starts to decrease, may appear earlier than the case without such non-
perturbative channels.

##  V Conclusion

In this paper, we presented an alternative interpretation of the known static
Einstein-Yang-Mills solutions as instantons that may contribute to the black
hole evaporation. There are two classes of solutions: One is the globally
regular Bartnik-Mckinnon (BM) solutions, and the other is the Yang-Mills black
holes (YMBHs) that have black holes at the center. The YMBH solutions smoothly
reduce to the BM solutions in the limit of vanishing horizon radius. It is
known that there exists an infinite number of solutions, characterized by the
number of nodes in the gauge field. However, we gave a reasonable argument
that only the solutions with zero node would make a dominant contribution to
the evaporation.

Both BM and YMBH solutions have no gauge charges. They are characterized
solely by their ADM mass at spatial infinity. Hence their Euclidean actions
agree with the usual entropy formula without conserved charges. This allows us
to regard these solutions as instantons that provide a non-perturbative
channel to the black hole evaporation, that can compete with perturbative
processes at late time. Furthermore, when the mass of an evaporating black
hole becomes equal to that of the BM solution, it will provide a smooth
transition mechanism from the black hole geometry to a regular spacetime.

To conclude, let us emphasize again the importance of non-perturbative
instantons. To demonstrate our point, we focused on the $SU(2)$ gauge theory
in this paper, knowing that there should exist more diverse solutions from
larger fundamental gauge symmetries. Even though we still retain the
conclusion that the Bekenstein-Hawking entropy bound should be violated in our
description, the violation is tamed when non-perturbative solutions are
considered as the black hole mass approaches the Planck scale. We will
continue to investigate such non-perturbative instantons.

Finally, let us mention that there may be instantons that violate global
symmetries, such as the baryon number. The implications of the global charge
violation, not only in terms of Euclidean quantum gravity but also in terms of
particle physics and cosmology, must be an interesting future research topic.

## Acknowledgment

This work was supported in part by JSPS KAKENHI Grant No. 20H05853. PC is
supported by Taiwan’s National Science and Technology Council (NSTC) under
project number 110-2112-M-002-031, and by the Leung Center for Cosmology and
Particle Astrophysics (LeCosPA), National Taiwan University. DY was supported
by the National Research Foundation of Korea (Grant No. : 2021R1C1C1008622,
2021R1A4A5031460). XYC is supported by the starting grant of Jiangsu
University of Science and Technology (JUST).

## References

  * (1) S. W. Hawking, Commun. Math. Phys. 43 (1975), 199-220 [erratum: Commun. Math. Phys. 46 (1976), 206].
  * (2) D. Yeom and H. Zoe, Int. J. Mod. Phys. A 26 (2011), 3287-3314 [arXiv:0907.0677 [hep-th]];
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, JHEP 02 (2013), 062
[arXiv:1207.3123 [hep-th]].

  * (3) A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, JHEP 03 (2020), 149 [arXiv:1908.10996 [hep-th]];
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, JHEP 05
(2020), 013 [arXiv:1911.12333 [hep-th]].

  * (4) P. Chen, M. Sasaki, D. Yeom and J. Yoon, [arXiv:2111.01005 [hep-th]];
P. Chen, M. Sasaki, D. Yeom and J. Yoon, Int. J. Mod. Phys. D 31 (2022) no.14,
2242001 [arXiv:2205.08320 [gr-qc]];
P. Chen, M. Sasaki, D. Yeom and J. Yoon, [arXiv:2206.10251 [gr-qc]].

  * (5) D. N. Page, Phys. Rev. Lett. 71 (1993), 1291-1294 [arXiv:gr-qc/9305007 [gr-qc]];
D. N. Page, Phys. Rev. Lett. 71 (1993), 3743-3746 [arXiv:hep-th/9306083 [hep-
th]].

  * (6) S. W. Hawking, Phys. Rev. D 72 (2005), 084013 [arXiv:hep-th/0507171 [hep-th]].
  * (7) P. Chen, Y. C. Ong and D. Yeom, Phys. Rept. 603 (2015), 1-45 [arXiv:1412.8366 [gr-qc]];
L. Buoninfante, F. Di Filippo and S. Mukohyama, JHEP 10 (2021), 081
[arXiv:2107.05662 [hep-th]].

  * (8) M. Sasaki and D. Yeom, JHEP 12 (2014), 155 [arXiv:1404.1565 [hep-th]];
B. H. Lee, W. Lee and D. Yeom, Phys. Rev. D 92 (2015) no.2, 024027
[arXiv:1502.07471 [hep-th]];
P. Chen, G. Domènech, M. Sasaki and D. Yeom, JCAP 04 (2016), 013
[arXiv:1512.00565 [hep-th]];
P. Chen, G. Domènech, M. Sasaki and D. Yeom, JHEP 07 (2017), 134
[arXiv:1704.04020 [gr-qc]];
P. Chen, M. Sasaki and D. Yeom, Int. J. Mod. Phys. D 29 (2020) no.12, 2050086
[arXiv:2005.07011 [gr-qc]].

  * (9) P. Chen, M. Sasaki and D. Yeom, Eur. Phys. J. C 79 (2019) no.7, 627 [arXiv:1806.03766 [hep-th]].
  * (10) S. Massar and R. Parentani, Phys. Rev. Lett. 78, 3810-3813 (1997) [arXiv:gr-qc/9701015 [gr-qc]].
  * (11) S. Deser, Phys. Lett. B 64 (1976), 463-464.
  * (12) S. R. Coleman, Subnucl. Ser. 13 (1977), 297.
  * (13) S. Deser, Class. Quant. Grav. 1 (1984), L1.
  * (14) R. Bartnik and J. Mckinnon, Phys. Rev. Lett. 61 (1988), 141-144.
  * (15) P. Bizon, Phys. Rev. Lett. 64 (1990), 2844-2847.
  * (16) M. S. Volkov and D. V. Galtsov, Sov. J. Nucl. Phys. 51 (1990), 747-753.
  * (17) H. P. Kuenzle and A. K. M. Masoodul Alam, J. Math. Phys. 31 (1990), 928-935.
  * (18) N. Straumann and Z. H. Zhou, Phys. Lett. B 237 (1990), 353-356.
  * (19) N. Straumann and Z. H. Zhou, Phys. Lett. B 243 (1990), 33-35.
  * (20) Z. h. Zhou and N. Straumann, Nucl. Phys. B 360 (1991), 180-196.
  * (21) P. Bizon and R. M. Wald, Phys. Lett. B 267 (1991), 173-174.
  * (22) P. Bizon, Phys. Lett. B 259 (1991), 53-57.
  * (23) D. V. Galtsov and M. S. Volkov, Phys. Lett. B 273 (1991), 255-259.
  * (24) D. V. Galtsov and M. S. Volkov, Phys. Lett. A 162 (1992), 144-148.
  * (25) G. V. Lavrelashvili and D. Maison, Phys. Lett. B 343 (1995), 214-217 [arXiv:hep-th/9409185 [hep-th]].
  * (26) M. S. Volkov and D. V. Galtsov, Phys. Lett. B 341 (1995), 279-285 [arXiv:hep-th/9409041 [hep-th]].
  * (27) N. E. Mavromatos and E. Winstanley, Phys. Rev. D 53 (1996), 3190-3214 [arXiv:hep-th/9510007 [hep-th]].
  * (28) O. Brodbeck and N. Straumann, J. Math. Phys. 37 (1996), 1414-1433 [arXiv:gr-qc/9411058 [gr-qc]].
  * (29) M. S. Volkov and D. V. Gal’tsov, Phys. Rept. 319 (1999), 1-83 [arXiv:hep-th/9810070 [hep-th]].
  * (30) I. Moss and A. Wray, Phys. Rev. D 46 (1992), R1215-R1218.
  * (31) C. W. Misner and D. H. Sharp, Phys. Rev. 136 (1964), B571-B576.
  * (32) R. Gregory, I. G. Moss and B. Withers, JHEP 03 (2014), 081 [arXiv:1401.0017 [hep-th]].
  * (33) G. V. Dunne and H. Min, Phys. Rev. D 72 (2005), 125004 [arXiv:hep-th/0511156 [hep-th]];
O. J. C. Dias and J. P. S. Lemos, J. Math. Phys. 42 (2001), 3292-3298
[arXiv:hep-ph/0103193 [hep-ph]].

  * (34) S. R. Coleman, Subnucl. Ser. 15 (1979), 805
  * (35) J. Hwang, H. Park, D. Yeom and H. Zoe, J. Korean Phys. Soc. 73 (2018) no.10, 1420-1430 [arXiv:1712.00347 [hep-th]];
J. M. Bae, D. J. Lee, D. Yeom and H. Zoe, Symmetry 14 (2022) no.8, 1649
[arXiv:2002.03543 [hep-th]].