# Foundry manufacturing of octave-spanning microcombs

Jizhao Zang  Time and Frequency Division, National Institute of Standards and
Technology, 325 Broadway, Boulder, Colorado 80305, USA  Department of Physics,
University of Colorado, 440 UCB, Boulder, Colorado 80305, USA  Haixin Liu
Time and Frequency Division, National Institute of Standards and Technology,
325 Broadway, Boulder, Colorado 80305, USA  Department of Physics, University
of Colorado, 440 UCB, Boulder, Colorado 80305, USA  Corresponding author:
<EMAIL_ADDRESS>Travis C. Briles  Time and Frequency Division,
National Institute of Standards and Technology, 325 Broadway, Boulder,
Colorado 80305, USA  Scott B. Papp  Time and Frequency Division, National
Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305,
USA  Department of Physics, University of Colorado, 440 UCB, Boulder, Colorado
80305, USA

###### Abstract

Soliton microcombs provide a chip-based, octave-spanning source for self-
referencing and optical metrology. We explore use of a silicon-nitride
integrated photonics foundry to manufacture octave-spanning microcombs. By
group-velocity dispersion engineering with the waveguide cross-section, we
shape the soliton spectrum for dispersive-wave spectral enhancements at the
frequencies for f-2f self-referencing. With the optimized waveguide geometry,
we control the carrier-envelope offset frequency by adjusting the resonator
radius. Moreover, we demonstrate the other considerations for octave
microcombs, including models for soliton spectrum design, ultra-broadband
resonator external coupling, low-loss edge couplers, and the nonlinear self-
interactions of few-cycle solitons. This design process permits highly
repeatable creation of soliton microcombs optimized for pump operation less
than 100 mW, an electronically detectable offset frequency, and high comb mode
power for f-2f detection. However, these design aspects must also be made
compatible with the foundry fabrication tolerance of octave microcomb devices.
Our experiments highlight the potential to manufacture a single-chip solution
for an octave-spanning microcomb, which is the central component of a compact
microsystem for optical metrology.

††journal: opticajournal

Kerr-microresonator soliton frequency combs, or microcombs, have undergone
rapid development because of the insights they provide to nonlinear dynamics
and their application possibilities [1, 2]. These broadband laser sources are
composed of discrete spectral lines with the relation
$f_{n}=nf_{\text{rep}}+f_{\text{ceo}}$, where $n$ is an integer comb mode
number, $f_{\text{rep}}$ is the repetition frequency and $f_{\text{ceo}}$ is
the carrier-envelope offset frequency. Several applications require phase
stabilization of $f_{\text{rep}}$ and $f_{\text{ceo}}$, including optical
frequency synthesis [3] and measurement with respect to either microwave or
optical clock signals, optical clock metrology, and optical frequency division
[4].

Realizing a phase-stabilized microcomb involves generation of an octave-
spanning spectrum with sufficient optical power for f-2f self-referencing and
phase-locking of the $f_{\text{ceo}}$ and $f_{\text{rep}}$ signals [5].
Management of group-velocity dispersion (GVD or dispersion) is critical for
all of these tasks. The spectrum of a soliton microcomb is mostly determined
by the integrated dispersion,
$D_{\text{int}}=\nu_{\mu}-(\nu_{0}+\text{FSR}\,\mu)$, where $\mu$ is the
relative mode number to the pump mode, $\nu_{\mu}$ is the cold cavity
resonance frequency and FSR is the free spectral range at the pump [6]. While
dark solitons in normal GVD have higher pump-to-comb conversion efficiency [7,
8], bright solitons in anomalous GVD offer the broadest spectra, characterized
by a bandwidth adjustable mostly with the second-order dispersion parameter
and coherent emission of dispersive waves (DWs) [9] that arise due to higher-
order dispersion. We consider dual DW microcombs with a shortwave DW (SWDW)
and a longwave DW (LWDW) at higher and lower frequency than the pump laser,
respectively. These DWs are designed according to the condition
$D_{\text{int}}$ = 0, and they greatly enhance the comb mode power and
bandwidth. We control octave microcombs mostly by adjusting $D_{\text{int}}$
with the resonator geometry. Indeed, the SWDW and LWDW are very sensitive to
fabrication tolerance. Therefore, an approximate target design exists,
however, to date it has only been implemented with carefully controlled, low-
volume fabrication, requiring iterative device selection and fabrication or
device trimming post-fabrication [10]. For frequency metrology, foundry
manufacturing as high-volume fabrication of f-2f microcombs would
revolutionize use of the SI second in the optical domain.

Here, we report a single-chip solution of microcombs for f-2f self-
referencing, which we fabricate at volume with a commercial silicon-nitride
foundry. We present an optimized microcomb design that supports a harmonic,
dual DW spectrum with an electronically accessible $f_{\text{ceo}}$ for the
highest efficiency in f-2f detection. To account for the fabrication tolerance
of the foundry, a chip carries numerous resonators with a programmed variation
in ring width ($RW$) and ring radius ($RR$). We model the target design
according to detailed SiN dispersion engineering, optimized microcomb dynamics
that enhance DW power for f-2f detection, and wavelength-dependent external
resonator coupling across octave span. A new feature of our modeling is an
analytical expression for the complex, octave span microcomb. Post-fabrication
device screening in concert with our models characterizes the fabrication
tolerance to yield optimized microcombs for f-2f detection. We demonstrate
that the two DW frequencies and $f_{\text{ceo}}$ can be independently
controlled in a predictable fashion by use of resonator geometry. Finally, we
discuss post-fabrication manipulation of soliton spectrum through control of
the pump laser. This process reveals interesting DW dynamics that can lead to
step-changes in the DW frequencies and power. Our work demonstrates a reliable
procedure of generating ready-to-use octave-spanning combs through a
commercial foundry, and also expands the theory of the DW dynamics in soliton
microcombs.

Figure 1: Key concepts of wafer-scale search for an octave-spanning comb from
commercial foundry. (a) Our chip testing setup (left) and photos of the wafer,
a chip, and a resonator (clockwise from middle). The bottom spectrum shows an
octave-spanning microcomb with an on-chip pump power of $(151\pm 10)$ mW, $RR$
of 23.475 $\mu$m, and $RW$ of 1690 nm. (b,c) enlargements of DW spectra for
different microresonator geometries. The DW locations are sensitive to $RW$
(panel b; $RW$ values: 1670, 1690 and 1710 nm) but less sensitive to $RR$
(panel c; $RR$ values: 23.475, 23.535, 23.58 $\mu$m). This allows DW control
through $RW$ and $f_{\text{ceo}}$ control through $RR$.

We first illustrate in Fig. 1 the procedure to obtain an octave-spanning
microcomb through wafer-level and chip-level device selection. The three
essential parameters of our devices are layer thickness ($th$), $RW$, and
$RR$. Before fabrication, our analytical model and simulation provided a
suggested set of parameters: $th=760$ nm, $RW=1697$ nm, and $RR=23.45$ $\mu$m.
The devices were fabricated by Ligentec with a programmed variation in $RW$
and $RR$ around the suggested values. The upper inset of Fig. 1 (a) shows
photos of our wafer and individual devices (right side), and our experimental
setup for soliton generation and characterization (left side). The pump from a
C-band continuous-wave (CW) laser is amplified by an EDFA and coupled into a
microresonator chip. The chip layer is made of silicon nitride and covered by
silicon dioxide top cladding, providing low edge coupling loss ($<3$ dB per
edge). We initiate the soliton microcombs by fast sweeping the pump frequency
[11, 12], and then finely adjust the detuning to optimize two DWs, monitored
by an optical spectrum analyzer (OSA). The lower inset of Fig. 1 (a) is a
measured spectrum of an octave-spanning microcomb with two DWs at 145.43 THz
and 290.92 THz, respectively. Figure 1 (b) and (c) show more details about how
we vary $RW$ and $RR$ to tune the DW frequencies and $f_{\text{ceo}}$ for a
workable octave-spanning microcomb. In Fig. 1 (b), we fix $RR$ and sweep $RW$
from 1670 nm to 1710 nm, while in Fig. 1 (c), we fix $RW$ and sweep $RR$ from
23.475 $\mu$m to 23.58 $\mu$m. The measurement results indicate that the DW
frequencies can be tuned by $RW$, and they are less sensitive to $RR$. On the
other hand, $f_{\text{ceo}}$ is sensitive to $RR$, which enables us to
optimize $f_{\text{ceo}}$ without changing the DW frequencies. Tuning $RW$
also changes $f_{\text{ceo}}$ but its influence can be easily counteracted by
a relatively wide scan of $RR$ values. With the programmed variation in
microresonator geometries, we can identify microcombs with an electronically
accessible $f_{\text{ceo}}$ and enhanced DWs at frequencies for f-2f self-
referencing.

Figure 2: Analytic model and simulation of comb spectrum. (a) The simulated
$D_{\textrm{int}}$ curves with $th$ = 760 nm, $RR$ = 23.45 $\mu$m and $RW$ =
1670 nm (blue), 1697 nm (black), and 1730 nm (green). The inset shows a
typical mode profile. (b) Simulated (line) and measured (squares)
$Q_{\textrm{c}}$ vs frequency for $RR$ = 23.535 $\mu$m, $WGW$ = 500 nm,
$g_{c}$ = 950 nm, and $L$ = 17 $\mu$m. (c) A comparison among measured comb
spectra (blue), prediction from our analytic approximation (black) and LLE
simulation (red) with $Q_{\textrm{i}}$ = 1.4M, $F^{2}$ = 37.2, $\alpha$ = 33.
(d) SWDW power vs $th$, predicted by our analytic model with $F^{2}$ = 37.2
($P_{\textrm{in}}$ = 150 mW) and $\alpha$ = 33. For each $th$ value, we adjust
$RW$ to ensure that the two DWs span an octave.

Next, we will discuss the steps to generate a pre-fabrication design. We first
regard $th$ as a variable. To ensure an octave span, for each $th$ value, we
find the proper $RW$ by using COMSOL software to calculate $D_{\textrm{int}}$
since DWs appear at $D_{\textrm{int}}\approx 0$. Figure 2 (a) shows simulated
$D_{\textrm{int}}$ with varied $RW$. The image on the left side shows a
typical mode field distribution. The blue, black, and green curves plot the
$D_{\textrm{int}}$ with $RW$ of $1670$nm, $1697$nm, and $1730$nm,
respectively. For the black curve, two solutions of $D_{\textrm{int}}=0$ span
an octave, indicating that $1697$ nm is the proper $RW$ for this $th$ value.

After $RW$ is determined, we design an optimal coupler that maximizes the
coupling rate, denoted by $\kappa_{\textrm{c}\mu}$, at the SWDW, since
$\kappa_{\textrm{c}\mu}$ is usually smaller at short wavelength. The coupling
is determined by three parameters: the bus waveguide width ($WGW$), the gap
between the bus waveguide and the ring, $g_{c}$, and the pulley length, $L$.
We use Lumerical FDTD solver to simulate coupling quality factor, defined by
$Q_{\textrm{c}}=2\pi\nu_{\mu}/\kappa_{\textrm{c}\mu}$, and find the best
values of $WGW$, $g_{\text{c}}$, and $L$. The curve in Fig. 2 (b) shows the
simulated $Q_{\textrm{c}}$ with $th$ = 751 nm, $RR$ = 23.535 $\mu$m, $RW$ =
1659 nm, $WGW$ = 500 nm, $g_{c}$ = 950 nm, and $L$ = 17 $\mu$m. The black
squares are measured $Q_{\textrm{c}}$, which agrees well with the simulation
considering the fabrication uncertainty.

We now present a quasi-analytic model for solitons generated in resonators
with arbitrary $D_{\text{int}}$ profiles and use it to quantitatively predict
the spectral mode distribution of optical power, $P_{\mu}$, given the device
parameters and some estimation about the pump laser. This model is critical
for obtaining sufficient DW power for f-2f self-referencing. Here, we define a
mode-dependent conversion efficiency
$\textrm{CE}_{\mu}=P_{\mu}/P_{\textrm{in}}$, where $P_{\textrm{in}}$ is the
input pump power ($\approx 150$ mW in our experiment). $P_{\mu}$ can be
estimated from $\text{CE}_{\mu}$ which is related to both the field inside the
resonator and the coupling. The intracavity dynamics of Kerr microresonator is
governed by the normalized LLE [13].

$\frac{\partial E_{\mu}}{\partial
t}=-\left(l_{\mu}+i\left(\alpha+D_{\mu}\right)\right)E_{\mu}+F\delta_{\mu,0}+i\sum_{\mu_{1},\mu_{2}}E_{\mu_{1}}E_{\mu_{2}}E_{\left(\mu_{1}+\mu_{2}-\mu\right)}^{*}.$
(1)

Here, the intracavity field is decomposed into different modes $E_{\mu}$. We
account for frequency dependent loss in terms of the ratio of loss rate
between mode $\mu$ and the pump ($\mu=0)$,
$l_{\mu}=(\kappa_{\textrm{i}\mu}+\kappa_{\textrm{c}\mu})/(\kappa_{\textrm{i}0}+\kappa_{\textrm{c}0})$,
where $\kappa_{\textrm{i}\mu}$ is the intrinsic loss rate. $\alpha$ is the
detuning of the pump laser, normalized by the halfwidth of pump mode
$(\kappa_{\textrm{i}0}+\kappa_{\textrm{c}0})/(4\pi)$. $D_{\mu}=4\pi
D_{\textrm{int}}/(\kappa_{\textrm{i}0}+\kappa_{\textrm{c}0})$ is the
normalized $D_{\text{int}}$. $F$ is the normalized driving force and related
to $P_{\text{in}}$ and threshold power $P_{\text{thre}}$ by
$F^{2}=P_{\text{in}}/P_{\text{thre}}$.

The relation between coupling and $\text{CE}_{\mu}$ is characterized by
$r_{\mu}=2K_{\mu}/(K_{\mu}+1)$ where
$K_{\mu}=\kappa_{\textrm{c}\mu}/\kappa_{\textrm{i}\mu}$ is the coupling
coefficient of mode $\mu$, and $\textrm{CE}_{\mu}$ has the following form
[13]:

$\textrm{CE}_{\mu}=\frac{|E_{\mu}|^{2}l_{\mu}r_{\mu}}{F^{2}/r_{0}}\times\frac{\omega_{\mu}}{\omega_{0}},$
(2)

where $\omega_{\mu}$ is the output angular frequency of mode $\mu$. As
discussed above, $\kappa_{\textrm{c}\mu}$ can be calculated from
$Q_{\textrm{c}}$. While $Q_{\textrm{c}}$ varies greatly over frequency, the
intrinsic quality factor $Q_{\textrm{i}}=2\pi\nu_{\mu}/\kappa_{\textrm{i}\mu}$
has much smaller variation. We assumed $Q_{\textrm{i}}$ to be a constant
measured in previous fabrication. Then, $l_{\mu}$ and $r_{\mu}$ can be
calculated from $Q_{\textrm{c}}$. Equation 2 suggests that we can calculate
$\text{CE}_{\mu}$ if we know $E_{\mu}$ and $F$. In resonators with low loss
and purely quadratic dispersion, $D_{\mu}=d_{2}\,\mu^{2}/2$, the soliton
spectra is approximated by [14]:

$E_{\mu}^{(0)}=\frac{\sqrt{d_{2}}}{2}\textrm{Sech}\left[\frac{\pi}{2}\sqrt{\frac{d_{2}}{2\alpha}}\mu\right].$
(3)

For non-quadratic dispersion, currently no model can approximate the soliton
spectra. Here, we provide two extra orders of correction for $E_{\mu}$ which
can characterize the spectrum around DWs through perturbation method. To
distinguish Eqn. 3 from our correction, we denote it with a superscript (0).

Our model assumes that $D_{\mu}\approx d_{2}\,\mu^{2}/2$ for small $\mu$ but
diverges from quadratic shape at large $\mu$. For a steady comb, $\partial
E_{\mu}/\partial t=i\lambda E_{\mu}$, where $\lambda$ is the time rate of
change of the phase and equals the difference between $f_{\text{rep}}$ and
FSR. For a $D_{\text{int}}$ with even parity, $\lambda$ = 0. We assume
$\lambda$ = 0 in our model but will discuss later that nonzero $\lambda$ will
result in DW switching phenomenon. As a result, for nonzero $\mu$, we rewrite
Eqn. 1 into:

$E_{\mu}=\frac{\sum_{\mu_{1},\mu_{2}}E_{\mu_{1}}E_{\mu_{2}}E_{\left(\mu_{1}+\mu_{2}-\mu\right)}^{*}}{-il_{\mu}+\alpha+D_{\mu}}.$
(4)

With this equation, we can calculate our first order solution $E_{\mu}^{(1)}$
by replacing $E_{\mu}$ on the right side with $E_{\mu}^{(0)}$. Instead of
directly doing the summation, a more clever way is to notice that
$E_{\mu}^{(0)}$ is a good approximation when $D_{\mu}$ is exactly
$d_{2}\,\mu^{2}/2$. So, according to Eqn. 4,
$\sum_{\mu_{1},\mu_{2}}E_{\mu_{1}}^{(0)}E_{\mu_{2}}^{(0)}E_{\left(\mu_{1}+\mu_{2}-\mu\right)}^{(0)*}\approx(-il_{\mu}+\alpha+d_{2}\,\mu^{2}/2)E_{\mu}^{(0)}$,
and $E_{\mu}^{(1)}$ has the following expression:

$E_{\mu}^{(1)}=c_{\mu}^{(1)}E_{\mu}^{(0)},\quad
c_{\mu}^{(1)}=\frac{-il_{\mu}+\alpha+d_{2}\,\mu^{2}/2}{-il_{\mu}+\alpha+D_{\mu}},$
(5)

where $c_{\mu}^{(1)}$ is the first order correction. This expression already
gives insight into comb power distribution across a dispersion profile. A
striking feature of DWs is that $P_{\mu}$ has local maximum at modes where
$D_{int}\approx 0$. This is explained by Eqn. 5: for most modes,
$D_{\mu}\gg\alpha$, $l_{\mu}$, and $|c_{\mu}^{(1)}|^{2}$ is relatively small,
but when $D_{\mu}$ decreases to $-\alpha$, the lineshape of
$|c_{\mu}^{(1)}|^{2}$ becomes a Lorentzian with peak value
$\approx\left(d_{2}\,\mu^{2}/(2l_{\mu})\right)^{2}$.

A more accurate spectrum needs the second order correction:

$E_{\mu}^{(2)}=E_{\mu}^{0}\frac{\sum_{\mu_{1},\mu_{3}}c_{\mu_{1}}^{(1)}c_{\mu_{3}-\mu_{1}}^{(1)}c_{\left(\mu_{3}-\mu\right)}^{(1)*}E_{\mu_{1}}^{(0)}E_{\mu_{3}-\mu_{1}}^{(0)}E_{\left(\mu_{3}-\mu\right)}^{(0)*}/E_{\mu}^{(0)}}{-il_{\mu}+\alpha+D_{\mu}}.$
(6)

Note that $\textrm{Sech}[x]\leqslant 2\,\textrm{Exp}[-|x|]$ and
$|\mu_{1}|+|\mu_{3}-\mu_{1}|+|\mu_{3}-\mu|\geqslant|\mu|$.
$E_{\mu_{1}}^{(0)}E_{\mu_{3}-\mu_{1}}^{(0)}E_{\left(\mu_{3}-\mu\right)}^{(0)*}/E_{\mu}^{(0)}$
vanishes exponentially except for the case where
$\mu\leqslant\mu_{3}\leqslant\mu_{1}\leqslant 0$ or
$\mu\geqslant\mu_{3}\geqslant\mu_{1}\geqslant 0$, and
$E_{\mu_{1}}^{(0)}E_{\mu_{3}-\mu_{1}}^{(0)}E_{\left(\mu_{3}-\mu\right)}^{(0)*}/E_{\mu}^{(0)}\approx
d_{2}$. Then $E_{\mu}^{(2)}$ can be expressed as:

$E_{\mu}^{(2)}=c_{\mu}^{(2)}c_{\mu}^{(1)}E_{\mu}^{(0)},c_{\mu}^{(2)}=\frac{-il_{\mu}+\alpha+d_{2}\sum_{\mu_{1},\mu_{3}}c_{\mu_{1}}^{(1)}c_{\mu_{3}-\mu_{1}}^{(1)}c_{\left(\mu_{3}-\mu\right)}^{(1)*}}{-il_{\mu}+\alpha+d_{2}\,(|\mu|+1)(|\mu|+2)/2},$
(7)

where the summation is done over $\mu\leqslant\mu_{3}\leqslant\mu_{1}\leqslant
0$ or $\mu\geqslant\mu_{3}\geqslant\mu_{1}\geqslant 0$. In $c_{\mu}^{(2)}$, we
retain $-il_{\mu}+\alpha$ in the numerator, and replace $\mu^{2}/2$ with
$(|\mu|+1)(|\mu|+2)/2$. This modification makes $c_{\mu}^{(2)}$ converge to 1
when $c_{\mu}^{(1)}=1$.

Using Eqn. 2 and 7, we can calculate the spectrum of output comb with given
values of $D_{\mu}$, $l_{\mu}$, $r_{\mu}$, $F$, $\alpha$ and
$P_{\textrm{in}}$. Equation 3 indicates that larger $\alpha$ leads to more
power, as observed in experiment as well. While absent from Eqn. 7, $F^{2}$
affects $P_{\mu}$ indirectly by altering the soliton existence range, for
which the upper bound of $\alpha$ is $\pi^{2}F^{2}/8$ [14]. During the design,
we set $F$ based on our laser power and the measured $P_{\text{thre}}$, and
perform numerical simulation with Matlab, based on Eqn. 1 directly, to
estimate the maximum $\alpha$ and the comb spectrum more accurately. Figure 2
(c) is a comparison among our analytic approximation (black curve), LLE
simualtion (red curve), and measured spectrum (blue trace) with nominal $RW$
and $th$ to be 1690 nm and 760 nm, respectively. For the analytic
approximation and numerical simulation, we use the same device parameters as
in Fig. 2 (b) and $Q_{\textrm{i}}$ is 1.4M. Considering the fabrication
uncertainty, the analytic, simulated and experimental spectra agree well.

The last parameter to optimize is $th$. We calculate the SWDW power versus
$th$ using our analytic model, shown in Fig. 2(d). Here, we set $RR$ = 23.45
$\mu$m, $P_{\textrm{in}}$ = 150 mW, $F^{2}$ = 37.2, and $\alpha$ = 33. $RW$ is
adjusted to keep an octave span of DWs. It shows that $th$ = 760 nm will yield
the highest SWDW power.

Figure 3:  Tuning of DW frequencies and $f_{\text{ceo}}$ (a)
$2\nu_{L}-\nu_{S}$ vs $RW$ for $th$ = 754 nm (green), 760 nm (black) and 766
nm (green). The blue squares show the measured data for $th$ = 760 nm. (b)
Simulated (black curve with diamonds) and measured (blue squares)
$f_{\text{ceo}}$ as a function of $RR$.

With the suggested device parameters, we next determine their sweep ranges
based on the sensitivity of DWs and $f_{\text{ceo}}$. To characterize the
harmonic mismatch between two DWs, we define the quantity
$2\nu_{\text{L}}-\nu_{\text{S}}$, where $\nu_{\text{L}}$ and $\nu_{\text{S}}$
are LWDW and SWDW frequencies, and plot it as a function of $RW$ and $th$ with
a fixed $RR=23.5$ $\mu m$ (Fig. 3 (a)). The green, black, and red curves with
diamonds show the theoretical values with $th$ = 754 nm, 760 nm and 766 nm,
respectively. The blue squares are calculated from measured $\nu_{\text{L}}$
and $\nu_{\text{S}}$. The plot shows that 1 nm uncertainty in $th$ requires 5
nm sweep in $RW$ to compensate.

Figure 3 (b) plots the theorectical (black curve with diamonds) and measured
(black squares) $f_{\text{ceo}}$ over $RR$ with $RW=1690$ nm and $th=760$ nm.
The lowest measured $f_{\text{ceo}}$ is 5.1 GHz with $RR=23.535$ $\mu$m and
the corresponding spectrum is shown in Fig. 2 (c). Despite an overall shift
between prediction and experimental result due to fabrication tolerance, the
similar linear dependence and slope over $RR$ highlight the accuracy of our
prediction. Since $f_{\text{ceo}}$ is always less than FSR ($\sim$ 1 THz ), a
sweep range of 0.8 $\mu$m in $RR$ will always compensate the fabrication
uncertainties and enable an electrically accessible $f_{\text{ceo}}$.

Figure 4: Dependence of comb spectrum on pump laser. (a) Simulated soliton
spectra with $F^{2}=37.2$, $\alpha$ = 19 (blue), 26 (green), and 33 (red)
using resonator parameters from Fig. 2 (c) except $\alpha$. ($\mu_{\text{L}}$,
$\mu_{\text{S}}$) are (-51,102) (blue), (-52,100) (green), and (-53,99) (red).
(b) Simulated comb power as a function of $\alpha$. Each colored diamond
corresponds to the soliton spectrum with the same color in (a). (c) Measured
soliton spectra with the same devices but different $P_{\text{in}}$: 74 mW
(top), 89 mW (middle), and 126 mW (bottom).

Besides device design, we can also use post-fabrication methods to manipulate
soliton spectrum through control of the pump laser. Here, we investigate how
$\alpha$ and $P_{\textrm{in}}$ affect DWs and the DW switching phenomenon.
According to Eqn. 3, $\alpha$ determines the attenuation rate of $P_{\mu}$
versus $\mu$. In addition, we found both in experiment and simulation that DWs
might switch from one mode to another nearby mode during the detuning sweep,
enabling fine tuning of DW frequencies. This behavior comes from the nonlinear
self-interactions between DWs and its parent soliton, causing $f_{\text{rep}}$
to diverge from FSR [15], namely $\lambda\neq 0$, which can be regarded as an
effective shift in $D_{\textrm{int}}$ and switches DW modes. In Fig. 4 (a), we
plot three LLE-simulated spectra with varied $\alpha$: 19 (blue), 26 (green)
and 33 (red) ($F^{2}=37.2$). The simulated comb power $P_{\text{comb}}$ versus
$\alpha$ is shown in Fig. 4 (b), where the corresponding $\alpha$ for each
spectrum in Fig. 4 (a) is indicated by diamonds in the same color. Each comb
is a stable solution of LLE and we have experimentally verified that these
states correspond to low noise combs [5]. The relative mode numbers of DWs
($\mu_{\text{L}}$, $\mu_{\text{S}}$) switch from (-51,102) (blue) to (-52,100)
(green), and then to (-53,99) (red). In contrast to quadratic GVD combs,
$P_{\text{comb}}$ does not increase smoothly in Fig. 4 (b). Instead we observe
that $P_{\text{comb}}$ vs $\alpha$ behavior is interrupted by high noise
regions around $\alpha=25$ and $32$. These regions are related to the
intracavity dynamics of DWs that cause jumps of DW frequency and power [15].

As mention above, $P_{\textrm{in}}$ can also affect CEμ by affecting the range
of $\alpha$ where the soliton can exist. Figure 4 (c) plots three measured
spectra from the same device but with increasng $P_{\textrm{in}}$: 74 mW, 89
mW, 126 mW. For each $P_{\textrm{in}}$, we tuned $\alpha$ to maximize the SWDW
power in experiment. While $P_{\textrm{in}}$ increases by less than two times,
the SWDW power increases by tens of dB, which we attribute to a larger
$\alpha$ able to support the soliton.

In summary, microresonator-based optical frequency combs are promising for
chip-scale integration and low-power operation but suffer from the high
sensitivity to the fabrication process. Benefiting from the consistent
fabrication tolerance of a commercial foundry, we demonstrate a systematical
design process and wafer-scale search method to achieve octave-spanning combs
with an electrically detectable $f_{\textrm{ceo}}$ (5.1 GHz). We also provide
an analytical model to predict the soliton spectrum with two DWs and
investigate its dependence on the device parameters as well as the pump laser.
The experiment results agree well with this analytical model and LLE
simulations. Our work represents an important step towards the practical
applications of the microresonator-based octave-spanning combs for self
referencing.

Acknowledgement. We thank Atasi Dan for taking the microscope photos. This
research has been funded by the AFOSR FA9550-20-1-0004 Project Number
19RT1019, DARPA DODOS, NSF Quantum Leap Challenge Institute Award OMA –
2016244, and NIST. This work is a contribution of the U.S. government and is
not subject to copyright. Tradenames provide information only and not an
endorsement.

Disclosures. The authors declare no conflicts of interest.

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May 3, 2022

Multi-field cold and warm inflation and the de Sitter swampland conjectures

Orfeu Bertolami1,2 and Paulo M. Sá3,4

1 Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do
Porto,

Rua do Campo Alegre s/n, 4169-007 Porto, Portugal

2 Centro de Física das Universidades do Minho e do Porto, Rua do Campo Alegre
s/n, 4169-007 Porto, Portugal

3 Departamento de Física, Faculdade de Ciências e Tecnologia, Universidade do
Algarve,

Campus de Gambelas, 8005-139 Faro, Portugal

4 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

###### Abstract

We discuss under which conditions multi-field cold and warm inflationary
models with canonical kinetic energy terms are compatible with the swampland
conjectures about the emergence of de Sitter solutions in string theory. We
find that under quite general conditions the slow-roll conditions for multi-
field cold inflation are at odds with the swampland conjectures for an
arbitrary number of scalar fields driving inflation. However, slow-roll
conditions can be reconciled with the swampland conjectures in the strong
dissipative regime of warm inflation.

E-mail addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

.Despite its impressive capability to comprise the most desirable features of
a fundamental theory of quantum gravity, string theory has failed, among other
shortcomings, to allow for a natural scenario for inflation. This situation
arises as the fundamental scalar field of the theory, the dilaton, does not
acquire any potential in any order of perturbation theory and non-perturbative
contributions tend to lead to very shallow and phenomenologically untenable
models. In fact, de Sitter solutions seem to be hard to obtain in string
theory, even though possible routes have been proposed [1]. This is sharply
contrasting with the situation of $N=1$ supergravity in the context of which
successful inflationary models can be rather easily achieved [2]. This is
somewhat surprising as, in fact, it is thanks to inflation that, for instance,
some specific issues of superstring models with intermediate scale can be
circumvented [3]. Of course, beyond single-field inflation, proposals for two-
field inflation have been put forward (see e.g. [4, 5]), but to our knowledge
these have not been systematically explored in the context of string theory.

The recent set of conjectures that de Sitter solutions in string theory cannot
be found in its landscape, but lye actually in a “swampland”, i.e. the set of
consistent-looking theories that do not admit a suitable ultraviolet
completion in string theory, has led to an intense discussion whether the
resulting solutions are consistent with the conditions required by inflation.
These set of conjectures comprise quite general criteria: charge symmetries
are supposed to be local gauge symmetries so that at least one particle must
have a mass in Planck units less than the gauge coupling strength so to ensure
that gravity is weak; the sign of some higher-order terms in the effective
action is constrained to warrant the absence of superluminal propagation;
there are a finite number of massless particles (see Ref. [6] for a review).
More specifically, in what concerns inflation, these conjectures can be
expressed in terms of constraints on scalar fields, generically denoted by
$\phi$ in the field space [7, 8]:

${\Delta\phi\over M_{\rm P}}<c_{1},$ (1) $M_{\rm P}{|\partial_{\phi}V|\over
V}>c_{2},$ (2)

where $\Delta\phi$ is the range of variation of the field, $M_{\rm P}\equiv
M_{\rm Pl}/\sqrt{8\pi}$ is the Reduced Planck’s mass, $V(\phi)$ is the scalar
field potential, $\partial_{\phi}V\equiv\partial V/\partial\phi$, and $c_{1}$
and $c_{2}$ are $\mathcal{O}(1)$ constants. It is argued that one should also
consider a more refined condition [9, 10, 11]

$M_{\rm P}^{2}{\partial_{\phi\phi}^{2}V\over V}<-c_{3},$ (3)

where $\partial_{\phi\phi}^{2}V\equiv\partial^{2}V/\partial\phi^{2}$ and,
likewise $c_{1}$ and $c_{2}$, the constant $c_{3}$ is of order one.

Conditions given by Eqs. (2) and (3) are somewhat at odds with the onset
conditions of single-field inflation which require that the parameters for the
inflaton field [12],

$\epsilon_{\phi}={M_{\rm P}^{2}\over 2}\left({\partial_{\phi}V\over
V}\right)^{2}$ (4)

and

$\eta_{\phi}=M_{\rm P}^{2}{\partial_{\phi\phi}^{2}V\over V},$ (5)

satisfy the slow-roll requirements, $\epsilon_{\phi}\ll 1$ and
$|\eta_{\phi}|\ll 1$ at the onset of inflation, so that at the end of
inflation $\epsilon_{\phi}\sim|\eta_{\phi}|\sim 1$. In order to successfully
solve the initial conditions problems of standard cosmology, the number of
e-foldings of inflation must satisfy the condition $N_{e}>65$, where

$N_{e}\equiv\ln\left({a_{e}\over a_{i}}\right)\simeq-{1\over M_{\rm
P}^{2}}\int_{\phi_{i}}^{\phi_{e}}{V\over\partial_{\phi}V}d\phi=-{1\over\sqrt{2}M_{\rm
P}}\int_{\phi_{i}}^{\phi_{e}}{1\over\sqrt{\epsilon_{\phi}}}d\phi,$ (6)

with $\phi_{i}$ and $\phi_{e}$ corresponding to the initial and final values
of the inflaton field. Since during inflation the slow-roll parameters are
approximately constant,
$d\epsilon_{\phi}/dN\simeq\mathcal{O}(\epsilon_{\phi}^{2})$, hence

$N_{e}\simeq{1\over\sqrt{2\epsilon_{\phi}}}{\Delta\phi\over M_{\rm P}},$ (7)

and thus for $\epsilon_{\phi}\lesssim 10^{-4}$ one obtains $\Delta\phi\sim
M_{\rm P}$, which is consistent with the constraint given by Eq. (1).

Further contact with observations can be established through the amplitude of
the inflaton quantum fluctuations and its imprint of the Cosmic Microwave
Background Radiation (CMB) through curvature and density fluctuations,

$\Delta_{R}^{2}=A_{s}\left({k\over k_{*}}\right)^{n_{s}-1},$ (8)

where $k_{*}$ is a chosen scale measured on the CMB, the scalar spectral index
is

$n_{s}\simeq 1-6\epsilon_{\phi}+2\eta_{\phi},$ (9)

and the amplitude of scalar fluctuations is given by

$A_{s}\simeq{1\over 24\pi^{2}}\left({V(\phi_{H})\over M_{\rm
P}^{4}}\right){1\over\epsilon_{\phi}(k_{H})},$ (10)

with $\phi_{H}$ and $\epsilon_{\phi}(k_{H})$ corresponding to a scale when the
fluctuating modes cross the horizon, that is between 50 and 60 e-foldings
before the end of inflation.

Furthermore, from the spectrum of tensor modes generated by the inflaton
quantum fluctuations,

$\Delta_{T}^{2}=A_{t}\left({k\over k_{*}}\right)^{n_{t}},$ (11)

where

$A_{t}\simeq{2\over 3\pi^{2}}\left({V(\phi_{H})\over M_{\rm P}^{4}}\right)$
(12)

and

$n_{t}\simeq-2\epsilon_{\phi},$ (13)

it is found that the tensor-to-scalar ratio is given by

$r={\Delta_{T}^{2}\over\Delta_{R}^{2}}\simeq 16\epsilon_{\phi}.$ (14)

Planck 2018 temperature and polarisation data indicate that for
$0.008\,h^{-1}{\rm Mpc}^{-1}\leq k\leq 0.1\,h^{-1}{\rm Mpc}^{-1}$, the
expansion rate $H=(67.4\pm 0.5)\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$, and absence
of running, the scalar spectral index is

$n_{s}=0.9649\pm 0.0042,$ (15)

while BICEP2/Keck Array, Planck and other data place an upper bound on the
tensor-to-scalar ratio at $k=0.002\,{\rm Mpc}^{-1}$, namely,

$r<0.06.$ (16)

From which follows, dropping higher-order slow-roll parameters, that

$\epsilon_{\phi}<0.0044$ (17)

and

$\eta_{\phi}=-0.015\pm 0.006,$ (18)

which do not match $c_{2}$ and $c_{3}$.

In what follows we shall consider the situation in the context of multi-field
cold inflationary models with canonical kinetic energy terms, initially for
two and then for an arbitrary number of fields (section 2.1). We shall then
address in section 2.2 the problem in the context of warm inflationary models
which exhibit dissipation. We shall see that the swampland conjectures and the
slow-roll conditions can be reconciled in the strong dissipative regime of
warm inflation for one, two, and multi-field models. Conclusions will be drawn
in section 3.

##  2 Beyond Single Field inflation

.As seen above, the de Sitter swampland conjectures and the slow-roll
conditions cannot be matched for single-field cold inflation. As discussed,
the purpose of the swampland conjectures is to ensure that a suitable
effective field theory in a de Sitter background arises from string theory. It
is further assumed that these conjectures also allow for the emergence of a
classical theory of gravity, which then drives inflation. The connection
between curvature and matter-energy as established by General Relativity (GR)
is well supported experimentally and observationally (see e.g. [13, 14] for
discussions). Indeed, theoretical and experimental evidence suggest that GR
reflects a provisional stage, although highly relevant, towards the ultimate
description of gravity. Well-known theoretical difficulties concern the
existence of spacetime singularities and the cosmological constant problem,
both related to the key issue of making quantum mechanics and GR compatible
with each other, string theory being, of course, quite relevant in this
respect.

It is quite logical that de Sitter swampland conjectures have attracted great
interest of the cosmological community given that they impose conditions on
the field spacing and on the first two derivatives of the potential of the
background effective field theory (see, for instance, Ref. [11] for a partial
list), which can be accessed in single-field inflation with features of the
CMB as discussed above.

Clearly, the conjectures (2) and (3) are at odds with constraints arising from
the CMB data, Eqs. (17) and (18). This conflict has been pointed out, for
instance, in Ref. [15], even though it has been argued that definite
conclusions about an actual tension depend on the knowledge about the origin
of the adiabatic curvature perturbations, within the slow-roll single-field
models of inflation [16]. In fact, the incompatibility of the swampland
conjectures with the observations has been an object of critique from the
authors of Ref. [17]. In any case, the swampland conjectures have given origin
to many ideas and sparked interesting proposals [18, 19, 20, 21, 22].

Even though the conflict is still depending on the nature of the
perturbations, a natural way to avoid this tension is to consider multi-field
inflationary models. Most of these models are known to show no contradiction
with the CMB features [23]. In fact, multi-field models open interesting
perspectives, for instance, for unification with dark matter and dark energy
[24, 25]. Two-field inflationary models were first considered in the context
of $N=1$ supergravity [26] and their dynamics was scrutinized in Refs. [27,
28] for a broader class of models. In a broad context and in string theory,
two-field inflationary with different mass scales and an interaction term were
considered in Refs. [4, 5, 29]. In the context of the swampland conjectures,
two-field inflationary models were discussed in Refs. [30, 31], where in Ref.
[30] non-canonical kinetic energy terms have been considered. However, in what
follows, we shall present quite general arguments which seem to rule out a
putative conciliation using multi-field cold inflation with canonical kinetic
energy terms (see section 2.1). We shall consider first a two-scalar cold
inflationary model and then generalize the argument for multi-field
inflationary models. In section 2.2 we turn to the case of warm inflation, for
which, in the strong dissipative regime, slow-roll conditions can be
reconciled with the swampland conjectures.

###  2.1 Cold inflation

.Quite generically, inflationary models driven by two scalar fields $\phi$ and
$\chi$ can be described by the Lagrangian density

${\cal L}=-{1\over 2}\partial_{\mu}\phi\partial^{\mu}\phi-{1\over
2}\partial_{\mu}\chi\partial^{\mu}\chi-U(\phi,\chi),$ (19)

where the potential $U(\phi,\chi)$ contains the potentials of the fields
$\phi$ and $\chi$ and an interaction term between these fields. This potential
is assumed to be designed to yield a healthy period of inflation. Let us also
assume that it satisfies the conjecture (1), even though it might be trickier
to achieve this condition for many fields. But, as we shall see, if conjecture
(1) can, at least in principle, be satisfied, the conjecture (2) cannot be met
in the context of quasi-exponential cold inflation.

Coupling these fields to gravity in an homogeneous, isotropic and flat
cosmological spacetime background and assuming that the scalar fields can
decay to radiation leads to the field equations

$\ddot{\phi}+3H\dot{\phi}+\partial_{\phi}U=-\Gamma_{\phi}\dot{\phi},$ (20)
$\ddot{\chi}+3H\dot{\chi}+\partial_{\chi}U=-\Gamma_{\chi}\dot{\chi},$ (21)
$\dot{\rho}_{\rm R}+4H\rho_{\rm
R}=\Gamma_{\phi}{\dot{\phi}}^{2}+\Gamma_{\chi}{\dot{\chi}}^{2},$ (22)
$H^{2}={1\over 3M_{\rm P}^{2}}\left({1\over 2}{\dot{\phi}^{2}}+{1\over
2}\dot{\chi}^{2}+U+\rho_{\rm R}\right),$ (23) $\dot{H}=-{1\over 2M_{\rm
P}^{2}}\left(\dot{\phi}^{2}+\dot{\chi}^{2}+{4\over 3}\rho_{\rm R}\right),$
(24)

where, in general terms, $\Gamma_{\phi}=\Gamma_{\phi}(\phi)$ and
$\Gamma_{\chi}=\Gamma_{\chi}(\chi)$ denote the decay widths of these fields to
radiation, $\rho_{\rm R}$ is the energy density of the radiation fluid,
$H=\dot{a}/a$ is the expansion rate, and $a$ the scale factor of the
Friedman–Lemaître–Robertson–Walker metric. In the above equations, an overdot
denotes a derivative with respect to the cosmic time $t$.

In this subsection we shall consider that $\Gamma_{\phi}=\Gamma_{\chi}=0$ and
neglect $\rho_{\rm R}$, meaning that inflation will supercool the Universe and
a reheating process must be considered separately. Non-vanishing dissipative
terms and a non-negligible energy density of radiation will be considered in
section 2.2.

Let us assume that inflation is almost exponential, namely,

${|\dot{H}|\over H^{2}}\ll 1.$ (25)

From this expression, using Eqs. (23) and (24), follows that
$\dot{\phi}^{2}+\dot{\chi}^{2}\ll U$, which implies that Eq. (23) can be
written as

$U\simeq 3M_{\rm P}^{2}H^{2}.$ (26)

In the above expression and in what follows, the symbol $\simeq$ means “equal
within the slow-roll approximation”.

Taking the time derivative of Eq. (26) and using Eq. (24) we obtain

$\dot{\phi}\partial_{\phi}U+\dot{\chi}\partial_{\chi}U\simeq-3H(\dot{\phi}^{2}+\dot{\chi}^{2}),$
(27)

which can be written as

$\partial_{\phi}U+3H\dot{\phi}\left(1+{\dot{\chi}^{2}\over\dot{\phi}^{2}}\right)\left(1+{\dot{\chi}\partial_{\chi}U\over\dot{\phi}\partial_{\phi}U}\right)^{-1}\simeq
0.$ (28)

Imposing the condition

$\left(1+{\dot{\chi}^{2}\over\dot{\phi}^{2}}\right)\left(1+{\dot{\chi}\partial_{\chi}U\over\dot{\phi}\partial_{\phi}U}\right)^{-1}\simeq
1,$ (29)

we obtain

$\partial_{\phi}U+3H\dot{\phi}\simeq 0,$ (30)

which is equivalent to neglecting $\ddot{\phi}$ in Eq. (20).

Now, using Eq. (30), the condition given by Eq. (29) can be simplified to
yield

$\partial_{\chi}U+3H\dot{\chi}\simeq 0,$ (31)

which is equivalent to neglecting $\ddot{\chi}$ in Eq. (21).

Following Ref. [32], the slow-roll parameter $\epsilon$ is defined as

$\epsilon={1\over 2}M_{\rm P}^{2}{|\nabla U|^{2}\over U^{2}},$ (32)

which is a straightforward generalization of the case with a single scalar
field [see Eq. (4)]. Using Eqs. (24)–(26), (30), and (31), we obtain

$\epsilon\simeq{|\dot{H}|\over H^{2}}\ll 1.\\\ $ (33)

The de Sitter swampland conjecture [see Eq. (2)]

$M_{\rm P}{|\partial_{\phi}U|\over U}>c_{2}$ (34)

can be written as

$c_{2}^{2}\lesssim 2{|\dot{H}|\over H^{2}}f,\\\ $ (35)

where

$f(\dot{\phi},\dot{\chi})\equiv{\dot{\phi}^{2}\over\dot{\phi}^{2}+\dot{\chi}^{2}}.$
(36)

Since the above function satisfies the condition
$0<f(\dot{\phi},\dot{\chi})\leq 1$, hence we conclude that $c_{2}^{2}\ll 1$.

The main issue is that, if we assume that inflation is quasi-exponential,
i.e., $|\dot{H}|/H^{2}\ll 1$, then both $\epsilon$ and $c_{2}$ are much
smaller than one.

These results can be straightforwardly generalized to the case of an arbitrary
number of scalar fields. Indeed, let us consider the scalar fields $\phi$ and
$\chi_{i}$ ($i=1,\dots,N$) with potential $U=U(\phi,\chi_{1},\dots,\chi_{N})$.

Once again, assuming quasi-exponential inflation and using the evolution
equations — generalized to the case of the above $N+1$ scalar fields — we
obtain

$\sum_{i=1}^{N}\left(\dot{\chi_{i}}\partial_{\chi_{i}}U+3H\dot{\chi}_{i}^{2}\right)\simeq
0.$ (37)

Since this equation does not imply

$\partial_{\chi_{i}}U+3H\dot{\chi_{i}}\simeq 0,\quad(i=1,\dots,N),$ (38)

these latter expressions must be assumed (as in Ref. [32]), instead of being
derived from Eq. (26) as in the two-scalar field case. The parameter
$\epsilon$ satisfies the condition given by Eq. (33), i.e., it is much smaller
than unity during the inflationary period, and $c_{2}^{2}$ is given by Eq.
(35), but now

$f(\dot{\phi},\dot{\chi_{1}},\dots,\dot{\chi_{N}})\equiv{\dot{\phi}^{2}\over\dot{\phi}^{2}+\sum_{i=1}^{N}\dot{\chi_{i}}^{2}}.$
(39)

Since the above function satisfies the condition
$0<f(\dot{\phi},\dot{\chi_{1}},\dots,\dot{\chi_{N}})\leq 1$, it then follows
that $c_{2}^{2}\ll 1$, as in the two-scalar-field case.

Hence, it has been shown that, under the quite general conditions of cold
exponential inflation, the swampland conjectures cannot be met for any number
of scalar fields.

###  2.2 Warm Inflation

.In the warm inflation scenario, dissipation plays a crucial role in slowing
down the inflaton, $\phi$, as it rolls down the potential $V(\phi)$ [33].
Within a strong dissipative regime, it is known that the swampland conjectures
can be made compatible with single-field warm inflation [34, 35]. In what
follows, we show that this result extends to multi-field warm inflationary
models.

To characterize the slow-roll regime for a single field scenario, in addition
to the parameters $\epsilon_{\phi}$ and $\eta_{\phi}$, defined by Eqs. (4) and
(5), an additional parameter $\beta_{\phi}$ should be introduced,

$\beta_{\phi}=M_{\rm
P}^{2}\,{\partial_{\phi}\Gamma_{\phi}\over\Gamma_{\phi}}\,{\partial_{\phi}V\over
V},$ (40)

where $\Gamma_{\phi}=\Gamma_{\phi}(\phi)$ is the so-called dissipation
coefficient [cf. Eq. (20)], responsible for a continuous energy transfer from
the inflaton field $\phi$ to a radiation bath with energy density $\rho_{\rm
R}$.

Assuming quasi-exponential inflation, it follows [36]

${\dot{H}\over H^{2}}\simeq-{\epsilon_{\phi}\over 1+Q},$ (41)
${\ddot{\phi}\over H\dot{\phi}}\simeq-{1\over
1+Q}\left(\eta_{\phi}-\beta_{\phi}+{\beta_{\phi}-\epsilon_{\phi}\over
1+Q}\right),$ (42) ${\dot{\rho}_{\rm R}\over H\rho_{\rm R}}\simeq-{1\over
1+Q}\left(2\eta_{\phi}-\beta_{\phi}-\epsilon_{\phi}+2{\beta_{\phi}-\epsilon_{\phi}\over
1+Q}\right),$ (43)

where the dissipation ratio $Q$ is defined as

$Q\equiv{\Gamma_{\phi}\over 3H}.$ (44)

Taking into account that $|\dot{H}|/H^{2}\ll 1$ and that the slow-roll
approximation requires $|\ddot{\phi}|\ll|H\dot{\phi}|$ and $|\dot{\rho}_{{\rm
R}}|\ll|H\rho_{{\rm R}}|$, from the above equations we conclude that

$\epsilon_{\phi}\ll 1+Q,\quad|\eta_{\phi}|\ll 1+Q,\quad|\beta_{\phi}|\ll 1+Q.$
(45)

This contrasts with the case of cold inflation, for which $\epsilon_{\phi}\ll
1$ and $|\eta_{\phi}|\ll 1$.

The constants $c_{2}$ and $c_{3}$, arising within the de Sitter swampland
conjectures, Eqs. (2) and (3), are related to the slow-roll parameters
$\epsilon_{\phi}$ and $\eta_{\phi}$ as $c_{2}^{2}<2\epsilon_{\phi}$ and
$c_{3}<|\eta_{\phi}|$, respectively, implying that

$c_{2}^{2}\ll 1+Q,\qquad c_{3}\ll 1+Q.$ (46)

Thus, in the strong dissipative regime of warm inflation, for which $Q\gg 1$,
both $c_{2}$ and $c_{3}$ can be of order unity, even if the expansion is
quasi-exponential, $|\dot{H}|/H^{2}\ll 1$. This behavior contrasts with the
situation in cold inflation, for which both $c_{2}$ and $c_{3}$ are much
smaller than unity during a quasi-exponential inflationary period.

We now turn to the case of multi-field warm inflation.

Let us start with two scalar fields $\phi$ and $\chi$ with potential
$U(\phi,\chi)$ and the evolution Eqs. (20)–(24). From the condition that
inflation is almost exponential [see Eq. (25)] and using Eqs. (23) and (24),
follows that $\dot{\phi}^{2}+\dot{\chi}^{2}+\rho_{R}\ll U$, which implies that
Eq. (23) can be written as

$U\simeq 3M_{\rm P}^{2}H^{2}.$ (47)

Taking the time derivative of the latter expression, and using Eq. (24), we
obtain

$\dot{\phi}\partial_{\phi}U+\dot{\chi}\partial_{\chi}U\simeq-3H\left(\dot{\phi}^{2}+\dot{\chi}^{2}+{\textstyle{{4}\over{3}}}\rho_{R}\right),$
(48)

which can be written as

$3H\dot{\phi}\left(1+{4\rho_{R}\over
3\dot{\phi}^{2}}\right)+\partial_{\phi}U\left(1+{\dot{\chi}\over\dot{\phi}}{\partial_{\chi}U\over\partial_{\phi}U}+{3H\dot{\phi}\over\partial_{\phi}U}{\dot{\chi}^{2}\over\dot{\phi}^{2}}\right)\simeq
0.$ (49)

Comparing with Eq. (20), this is equivalent to neglecting $\ddot{\phi}$ and
assuming

${\dot{\chi}\over\dot{\phi}}{\partial_{\chi}U\over\partial_{\phi}U}+{3H\dot{\phi}\over\partial_{\phi}U}{\dot{\chi}^{2}\over\dot{\phi}^{2}}\simeq
0$ (50)

and

${4\rho_{R}\over 3\dot{\phi}^{2}}\simeq{\Gamma_{\phi}\over 3H},$ (51)

yielding

$\partial_{\phi}U\simeq-3H\dot{\phi}\left(1+Q\right).$ (52)

Note that Eq. (50) can be written as

$\partial_{\chi}U\simeq-3H\dot{\chi}\,.$ (53)

which is equivalent to neglecting $\ddot{\chi}$ and $\Gamma_{\chi}\dot{\chi}$
in Eq. (21), while Eq. (51) can be written as

$4H\rho_{R}\simeq\Gamma_{\phi}\dot{\phi}^{2},$ (54)

which is equivalent to neglecting $\dot{\rho_{R}}$ and
$\Gamma_{\chi}\dot{\chi}^{2}$ in Eq. (22).

Now, taking the time derivative of Eqs. (52)–(54), we obtain

$\dot{\phi}\partial_{\phi}\Gamma_{\phi}\simeq\left(3\dot{H}-6H{\ddot{\phi}\over\dot{\phi}}+3H{\dot{\rho}_{\rm
R}\over\rho_{\rm R}}\right)Q,$ (55)
$\partial_{\phi\phi}^{2}U+{\dot{\chi}\over\dot{\phi}}\partial_{\phi\chi}^{2}U\simeq-3\dot{H}(1+Q)-3H{\ddot{\phi}\over\dot{\phi}}(1-Q)-3H{\dot{\rho}_{\rm
R}\over\rho_{\rm R}}Q,$ (56)
$\partial_{\chi\chi}^{2}U+{\dot{\phi}\over\dot{\chi}}\partial_{\phi\chi}^{2}U\simeq-3\dot{H}-3H{\ddot{\chi}\over\dot{\chi}}\,.$
(57)

Following Ref. [32], we define the slow-roll parameters $\epsilon$ and
$\eta_{ij}$ as

$\epsilon={1\over 2}M_{\rm
P}^{2}\left({(\partial_{\phi}U)^{2}+(\partial_{\chi}U)^{2}\over
U^{2}}\right),$ (58) $\eta_{\phi\phi}=M_{\rm
P}^{2}{\partial_{\phi\phi}^{2}U\over
U},\qquad\eta_{\phi\chi}=\eta_{\chi\phi}=M_{\rm
P}^{2}{\partial_{\phi\chi}^{2}U\over U},\qquad\eta_{\chi\chi}=M_{\rm
P}^{2}{\partial_{\chi\chi}^{2}U\over U},$ (59)

while for $\beta$ we adopt the definition [see Eq. (40)]

$\beta=M_{\rm
P}^{2}{\partial_{\phi}\Gamma_{\phi}\over\Gamma_{\phi}}{\partial_{\phi}U\over
U}.$ (60)

Using Eqs. (47), (52)–(57), it is straightforward to obtain

${\dot{H}\over H^{2}}\simeq-{\epsilon\over F},$ (61) ${\ddot{\phi}\over
H\dot{\phi}}\simeq-{1\over
1+Q}\left(\eta_{\phi\phi}+{\dot{\chi}\over\dot{\phi}}\,\eta_{\phi\chi}-\beta+{\beta\over
1+Q}-{\epsilon\over F}\right),$ (62)

${\ddot{\chi}\over H\dot{\chi}}\simeq{\epsilon\over
F}-\eta_{\chi\chi}-{\dot{\phi}\over\dot{\chi}}\eta_{\phi\chi},$ (63)

${\dot{\rho}_{\rm R}\over H\rho_{\rm R}}\simeq-{1\over
1+Q}\left(2\eta_{\phi\phi}+2{\dot{\chi}\over\dot{\phi}}\,\eta_{\phi\chi}-\beta-\epsilon{1+Q\over
F}+{2\beta\over 1+Q}-{2\epsilon\over F}\right),$ (64)

where the following notation was introduced,

$F(Q,\dot{\phi},\dot{\chi})\equiv{(1+Q)^{2}\dot{\phi}^{2}+\dot{\chi}^{2}\over(1+Q)\dot{\phi}^{2}+\dot{\chi}^{2}}.$
(65)

Now, taking into account that $|\dot{H}|/H^{2}\ll 1$ and that the slow-roll
approximation requires $|\ddot{\phi}|\ll|H\dot{\phi}|$,
$|\ddot{\chi}|\ll|H\dot{\chi}|$, and $|\dot{\rho}_{\rm R}|\ll|H\rho_{\rm R}|$,
we obtain

$\epsilon\ll F,\quad|\beta|\ll 1+Q,\quad|\eta_{\phi\phi}|\ll
1+Q,\quad|\eta_{\chi\chi}|\ll
1,\quad\left|{\dot{\chi}\over\dot{\phi}}\eta_{\phi\chi}\right|\ll(1+Q),\quad\left|{\dot{\phi}\over\dot{\chi}}\eta_{\phi\chi}\right|\ll
1.$ (66)

From these expressions we conclude that, if $\dot{\chi}^{2}/\dot{\phi}^{2}\ll
1+Q$, then $F\simeq 1+Q$, and, in the strong dissipative regime of warm
inflation, for which $Q\gg 1$, the slow-roll parameters $\epsilon$, $\beta$,
and $\eta_{\phi\phi}$ can be of order unity.

On the other hand, the de Sitter swampland conjecture given by Eq. (34), can
be written as

$c_{2}^{2}\lesssim 2{|\dot{H}|\over H^{2}}G,$ (67)

where

$G(Q,\dot{\phi},\dot{\chi})\equiv{(1+Q)^{2}\dot{\phi}^{2}\over(1+Q)\dot{\phi}^{2}+\dot{\chi}^{2}}.$
(68)

If $\dot{\chi}^{2}/\dot{\phi}^{2}\ll 1+Q$, then $G\simeq 1+Q$ and,
consequently, $c_{2}^{2}\ll 1+Q$. Therefore, we conclude that, in the strong
dissipative regime of warm inflation, $c_{2}$ can be of order unity. If,
however, $\dot{\chi}^{2}/\dot{\phi}^{2}\simeq(1+Q)^{2}$, then $G\simeq 1$ and,
consequently, $c_{2}^{2}\ll 1$.

The de Sitter swampland conjecture [see Eq. (3)]

$M_{\rm P}^{2}{\partial_{\phi\phi}^{2}U\over U}<-c_{3}$ (69)

can be written in terms of the slow-roll parameter $\eta_{\phi\phi}$ as
$c_{3}<|\eta_{\phi\phi}|$, meaning that $c_{3}\ll 1+Q$. For $Q\gg 1$,
corresponding to the strong dissipative regime of warm inflation, $c_{3}$ can
be of order unity.

As in the case of cold inflation, these results can be generalized for an
arbitrary number of scalar fields, $\phi$ and $\chi_{i}$ ($i=1,\dots,N$), with
potential $U(\phi,\chi_{1},\dots,\chi_{N})$. As before, the evolution
equations yield Eq. (37), which, again, does not imply
$\partial_{\chi_{i}}U+3H\dot{\chi_{i}}\simeq 0$; these latter expressions must
be assumed, instead of being derived from Eq. (47) as in the two-scalar field
case. With this assumption and taking into account that $|\dot{H}|/H^{2}\ll 1$
and that the slow-roll approximation requires $|\ddot{\phi}|\ll|H\dot{\phi}|$,
$|\ddot{\chi}_{i}|\ll|H\dot{\chi}_{i}|$, and $|\dot{\rho}_{\rm
R}|\ll|H\rho_{\rm R}|$, we obtain

$\displaystyle\epsilon\ll F,\quad|\beta|\ll 1+Q,\quad|\eta_{\phi\phi}|\ll
1+Q,\quad\left|{1\over\dot{\chi}_{j}}\sum_{i=1}^{N}\eta_{\chi_{i}\chi_{j}}\dot{\chi}_{i}\right|\ll
1,\quad$
$\displaystyle\left|{1\over\dot{\phi}}\sum_{i=1}^{N}\eta_{\phi\chi_{i}}\dot{\chi}_{i}\right|\ll
1+Q,\quad\left|{\dot{\phi}\over\dot{\chi}_{i}}\eta_{\phi\chi_{i}}\right|\ll
1,$ (70)

where $\eta_{\phi\chi_{i}}$ and $\eta_{\chi_{i}\chi_{j}}$ are slow-roll
parameters defined as

$\eta_{\phi\chi_{i}}=M_{\rm P}^{2}{\partial_{\phi\chi_{i}}^{2}U\over
U},\qquad\eta_{\chi_{i}\chi_{j}}=M_{\rm
P}^{2}{\partial_{\chi_{i}\chi_{j}}^{2}U\over U},$ (71)

and $F$ is given by

$F(Q,\dot{\phi},\dot{\chi_{1}},\dots,\dot{\chi_{N}})\equiv{(1+Q)^{2}\dot{\phi}^{2}+\sum_{i=1}^{N}\dot{\chi}_{i}^{2}\over(1+Q)\dot{\phi}^{2}+\sum_{i=1}^{N}\dot{\chi}_{i}^{2}}.$
(72)

Furthermore, $c_{2}$ is given by Eq. (67) with

$G(Q,\dot{\phi},\dot{\chi_{1}},\dots,\dot{\chi_{N}})\equiv{(1+Q)^{2}\dot{\phi}^{2}\over(1+Q)\dot{\phi}^{2}+\sum_{i=1}^{N}\dot{\chi}_{i}^{2}},$
(73)

while $c_{3}$ relates to the slow-roll parameter $\eta_{\phi\phi}$ as
$c_{3}<|\eta_{\phi\phi}|$. If
$\sum_{i=1}^{N}\dot{\chi}_{i}^{2}/\dot{\phi}^{2}\ll 1+Q$, then $F\simeq 1+Q$
and $G\simeq 1+Q$, implying that, in the strong dissipative regime of warm
inflation ($Q\gg 1$), both $c_{2}^{2}$ and $c_{3}$ can be of order unity.

##  3 Discussion and Conclusions

.In this work we have examined, on quite general grounds, the possibility of
matching the slow-roll conditions of inflation and the de Sitter swampland
conjectures in the context of cold and warm inflationary models driven by more
than one scalar field.

We have shown that irrespective of the number of scalar fields with canonical
kinetic energy terms, quasi-exponential cold inflation and the swampland
conjectures are incompatible up to issues related to the origin of adiabatic
curvature perturbations.

The situation is different once dissipation is introduced. Indeed, in the
context of single-field warm inflation, it is known that the slow-roll
conditions and the swampland conjectures can be matched provided the
dissipation ratio, $Q=\Gamma_{\phi}/(3H)$, satisfies the condition $Q\gg 1$,
corresponding to a strong dissipative regime. Even though the situation of
warm inflation driven by two scalar fields is more complex as it requires more
parameters in order to fully characterize the slow-roll conditions, we have
shown that quasi-exponential inflation and de Sitter swampland conditions can
be reconciled provided a strong dissipative regime is ensured for one of the
scalar fields, say $\phi$, and the other scalar field $\chi$ satisfies the
condition $\dot{\chi}^{2}/\dot{\phi}^{2}\ll 1$. For more than two scalar
fields, as expected, the slow-roll conditions are richer, however, they can be
made compatible with the swampland conditions, likewise in the cases of
single- and two-field inflation, if the strong dissipative regime holds for
one of the fields, say $\phi$, and if the condition
$\sum_{i=1}^{N}\dot{\chi}_{i}^{2}/\dot{\phi}^{2}\ll 1+Q$ is satisfied by the
remaining fields, $\chi_{i}$. However, differently from the two-scalar-field
case, the conditions $\partial_{\chi_{i}}U+3H\dot{\chi}_{i}\simeq 0$ should be
imposed, since they do not follow from the slow-roll expression $U\simeq
3M_{\rm P}^{2}H^{2}$.

To close the discussion we emphasize that our work opens the possibility of
reconciling the swampland conjectures with warm inflation, provided the
single- or multi-field models admit, during the slow-roll, a strong
dissipative regime and satisfy the conditions discussed above. These extra
requirements might be relevant to narrow down the possible class of viable
effective models that, although not in the landscape of string theory, are
suitable from the point of view of inflation. It is intriguing that inflation
with dissipative features is also an important ingredient in a recent proposal
to address the cosmological constant problem [37]. Moreover, it is quite
interesting that dissipation, being a quite generic manifestation of the arrow
of time at the macroscopic level, appears in a fundamental discussion about
the viable emerging solutions of string theory.

Acknowledgments

PMS acknowledges support from Fundação para a Ciência e a Tecnologia
(Portugal) through the research grants UIDB/04434/2020 and UIDP/04434/2020.

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