# Superradiance-assisted two-color Doppler cooling of molecules

Caleb Heuvel-Horwitz  Harvard Physics Dept, 17 Oxford St, Cambridge, MA 02138
S. F. Yelin  Harvard Physics Dept, 17 Oxford St, Cambridge, MA 02138

(August 27, 2024)

###### Abstract

For experiments that require a quantum system to be in the ultra-cold regime,
laser cooling is an essential tool. While techniques for laser cooling ions
and neutral atoms have been refined and temperatures below the Doppler limit
have been achieved, present-day techniques are limited to a small class of
molecules. This paper proposes a general cooling scheme for molecules based on
vibrational-state transitions. Superradiance is used to speed up the two-
photon transition. Simulations of this scheme achieve temperatures comparable
to those achieved by existing two-level schemes for neutral atoms and ions.

##  I Introduction

Laser cooling is a useful technique for the preparation of ultra-cold atoms
for use in quantum experiments. The original scheme for Doppler cooling
describes an atomic gas driven by a laser that is tuned to be slightly below
an electronic transition in the atoms of the gas [1, 2, 3]. Due to the Doppler
effect, atoms will absorb more photons if they move toward the light source,
so they’ll always absorb more photons that oppose their direction of motion.
The re-emission of these absorbed photons is isotropic with no net
contribution to the atom momentum. The first successful Doppler cooling
experiments were reported soon after for Mg+ and Be+ [4, 5].

Since the inception of Doppler cooling, other types of cooling schemes have
been demonstrated. The Zeeman slower would achieved temperatures near the
Doppler limit using neutral atoms [6]. Later, atoms would be cooled below the
Doppler limit, the possibility of which was attributed to additional atomic
states and laser polarization [7]. Sub-Doppler temperatures would also be
achieved with Sisyphus cooling, and Doppler cooling techniques would later be
further refined using on optical pumping [8, 9].

The logical next step in the refinement of laser cooling is a method that
works for molecules. A technique for cooling using an optical cycling scheme
with only three lasers was demonstrated for the diatomic molecule SrF[10].
Cooling and laser slowing of several oxides and other atomic molecules have
been shown since then [11, 12, 13, 14, 15, 16, 17]. Cooling of potassium-
rubidium molecules to the ultra-cold regime has been demonstrated, and there
is a proposed method for cooling polyatomic molecules of six or greater atoms
using an attached metal atom as a photon cycling site [18, 19]. Proposals
exist for a cooling technique that feature two-level adiabatic transfers, and
these proposals are less reliant on spontaneous emission, making them good
candidates for cooling of particles without cyclic transitions, such as
molecules [20, 21]. However, these schemes don’t achieve the ultra-cold
temperatures that atoms can reach and work almost exclusively for diatomic
molecules. Currently there are no other proposals for laser cooling a general
class of polyatomic molecules.

Figure 1: (a) The energy level structure of a molecule. In red is the
effective three-level scheme used in our proposal. (b) The
$\Lambda$-configuration considered in this study. The
$|1\rangle,|2\rangle,\text{ and }|3\rangle$ states are the $\Lambda$ part of
the scheme, with modification $|4\rangle$ added to simulate decay into some
non-cyclic energy level (we assume
$\gamma_{34}\ll\gamma_{31},\gamma_{32},\gamma_{21}$). Straight arrows
represent coupling between the counter-propagating beams and the two main
electronic transitions. Only the coupling for two beams are shown here; in the
complete model there is a pair of beams for each of six possible directions in
space. Squiggly arrows represent spontaneous decay, except for $\gamma_{21}$,
which is the superradiant decay. $|1\rangle$ and $|2\rangle$ are in the same
electronic ground state, but with different vibrational modes.

The fundamental problem with the extension of existing laser cooling
techniques to more complex structures is that they rely on cyclic (electronic)
transitions. Most cooling schemes for neutral atoms and ions assume a perfect
two-level structure. Ideal structures like this are rare in most molecules.
The molecular energy level structure consists of electronic levels that are
split by different vibrational and rotational levels (see Fig. 1a). Each of
these transitions would require an individual laser to repump the molecule. We
suggest here to use the two lowest vibrational states of the ground electronic
states as viable cyclic transition in order to cool such a complex system to
the ultra-cold regime.

Obviously, using vibrational states for radiative transitions poses two major
problems. One is that vibrational radiative lifetimes are forbiddingly long
(often of the order of seconds or longer), while electronic excited states of
molecules typically have similar lifetimes as atoms. Another is that the
photons from a vibrational transition have less energy and thus less momentum
than the higher energy photons of an electronic transition. A third, minor,
problem, is that vibrational transitions don’t have dipole selection rules, so
rotational transitions need to be involved.

Superradiance could mitigate the first of these problems. In a superradiant
system, molecules cooperatively emit spontaneous photons much faster than they
would in vacuum, which is due to the dipoles of the molecules becoming locked
in phase [22, 23, 24]. Superradiance has been demonstrated to speed up
vibrational transitions in atoms and polar molecules, and recent studies have
shown that superradiance can occur on some transitions that are useful in
laser cooling schemes [25, 26]. If the vibrational cooling transition, in
addition, is pumped using counter-propagating Raman beams, this two-photon
transition would impart two high-frequency photon momenta to the molecule,
consequently speeding up the cooling [27]. In order to solve the third
problem, the rotational transition selection rules can be use. In total, this
technique is an ideal candidate for Doppler cooling polyatomic molecules.

The aim of this paper is to describe a novel technique that uses superradiance
to laser cool molecules. First the general theory for two- and three-level
systems will be reviewed, followed by a detailed explanation incorporating
this theory into an effective five-level scheme in three dimensions. Finally,
the results of these simulations will be interpreted, with some concluding
discussion regarding the implications and possible expansion of this work.

##  II Model

We start by providing a short review of laser cooling on a two-level
transition. Consider an electronic transition coupled to a beam with Rabi
frequency $\Omega$ and detuning $\Delta=\omega-\omega_{0}$ for driving
frequency $\omega$ and transition frequency $\omega_{0}$, and spontaneous
decay rate $\gamma$. The system moves with velocity $v$ in the positive
direction. Assume the beam points in the opposite direction of $v$.

The Hamiltonian in the rotating frame is

$\frac{H}{\hbar}=\left(\begin{array}[]{cc}0&-\Omega/2\\\
-\Omega/2&\Delta+kv\\\ \end{array}\right).$ (1)

After stimulated absorption of one photon, the molecule’s momentum changes by
$-\hbar k$. After stimulated emission of one photon, the molecule’s momentum
changes by $\hbar k$. Thus the effective cooling experienced by the system is
proportional to the difference of the two rates of stimulation. Assuming
$\dot{\rho_{eg}}=0$ via adiabatic elimination and solving for
$\dot{\rho_{ee}}$, we find

$\dot{\rho_{ee}}=-\gamma\rho_{ee}+\frac{\gamma\Omega^{2}}{\gamma^{2}+4(\Delta+kv)^{2}}(\rho_{gg}-\rho_{ee})$
(2)

The term $-\gamma\rho_{ee}$ describes how the excited state population changes
with spontaneous decay. The remaining two terms are the rate of stimulated
absorption and the rate of stimulated emission, respectively. Each time a
photon is spontaneously emitted, it kicks the molecule via

$\frac{\hbar k}{m}\gamma\rho_{gg}\cos(\theta)$

where $\theta$ is the angle between the velocity of the molecule and the
velocity of the emitted photon. These random momentum kicks do not contribute
to the average velocity:

$\langle\frac{\hbar k}{m}\gamma\rho_{gg}\cos(\theta)\rangle=0$ (3)

This technique can be readily generalized to more levels, e.g., a three-level
$\Lambda$ scheme, with $|1\rangle$ and $|2\rangle$ denoting the two lowest
ground state vibrational states, and a single electronic excited state
$|3\rangle$. One Raman beam is coupled to the
$|1\rangle\leftrightarrow|3\rangle$ transition with with Rabi frequency
$\Omega_{1}$ and detuning $\Delta_{1}$. A second beam is coupled to the
$|1\rangle\leftrightarrow|3\rangle$ transition with Rabi frequency
$\Omega_{2}$ and detuning $\Delta_{2}$. The Hamiltonian in the rotating frame
is

$\frac{H}{\hbar}=\left(\begin{array}[]{ccc}\Delta_{1}+k_{1}v&0&-\Omega_{1}\\\
0&\Delta_{2}+k_{2}v&-\Omega_{2}\\\ -\Omega_{1}&-\Omega_{2}&0\\\
\end{array}\right).$ (4)

Again via adiabatic elimination, we find the contribution of the
$|1\rangle\leftrightarrow|3\rangle$ and $|2\rangle\leftrightarrow|3\rangle$
transitions to the cooling are

$-\hbar
k_{1}\frac{\gamma_{31}\Omega_{1}^{2}}{\gamma_{31}^{2}+4(\Delta_{1}+kv)^{2}}(\rho_{11}-\rho_{33})$

and

$-\hbar
k_{2}\frac{\gamma_{32}\Omega_{2}^{2}}{\gamma_{32}^{2}+4(\Delta_{2}+kv)^{2}}(\rho_{22}-\rho_{33})$

respectively, where $\gamma_{ij}$ represents the decay
$|i\rangle\to|j\rangle$. $k_{1}$ ($k_{2}$) is the wavenumber of the Raman beam
coupled to the $|1\rangle\leftrightarrow|3\rangle$
($|1\rangle\leftrightarrow|3\rangle$) transition. If the Raman transition is
kept close to two-photon resonance with the vibrational transition – thus
creating an electromagnetically induced transparency (EIT) resonance –, the
excited state $|3\rangle$ will not be populated. Thus, in this case, there is
no decay into uncontrolled vibrational states.

In practice, the molecule will occasionally be in its excited state. The main
reasons for this departure from ideal EIT come from decoherence due to finite
linewidths, collisions, and the Doppler effect. (Note that the undesired
Doppler shifts in this setup lead to decoherence only relative to the energy
difference of the two lower vibrational states and thus contribute relatively
weakly.) This excitation leads to decay into some non-cyclic energy level that
is not part of the original $\Lambda$ configuration. We account for this by
expanding our Hamiltonian to include a fourth state, which represents the
population lost during the cooling process. The density matrix expands
accordingly. We assume that decay into this fourth energy level does not
contribute to the system’s cooling via spontaneous emission.

The decay $\gamma_{sup}$ describes a superradiant transition between the two
ground states, and it is the principle advantage of this scheme that enables
it to work for systems with complicated cyclic transitions. $\gamma_{sup}$ is
large enough to ensure rapid transitions from
$|2\rangle\leftrightarrow|1\rangle$, allowing for swift cooling despite the
relatively low energy of photons emitted by this transition. Superradiance
approximately enhances the decay between vibration states by the optical
density for the respective transition [28], assuming population $\rho_{22}$ in
the upper superradiant levels:

$\gamma_{\rm sup}=\gamma_{21}\times OD=\gamma_{21}\times
n\times\lambda^{2}\times r\times\rho_{22}$

for number density $n$, vibrational transition wavelength $\lambda$, and cloud
size $r$, which is typically on the order of millimeters to centimeters [28].
Note that $\gamma_{21}$ is the ordinary spontaneous decay rate for the
vibrational transition. $\lambda$ for vibrational transitions is typically a
few micrometers to a millimeter, and density can reach up to $10^{9}$ cm3.
Assuming a typical vibrational lifetime of about a second, this gives an $OD$
of about $1$ kHz to $10$ MHz. A brief calculation using these figures
simplifies the superradiant decay to

$\gamma_{\rm sup}\approx\gamma_{\rm opt}\rho_{22},$

where $\gamma_{\rm opt}$ is the decay rate on the optical electronic
transitions.

Each Raman beam is assumed to be Gaussian, with width $\Gamma$.

With two pairs of beams oriented along each axis and counter-propagating, it
is possible in principal to cool a system with arbitary velocity in three
dimensions.

If one assumes that the $|1\rangle\leftrightarrow|3\rangle$ and
$|2\rangle\leftrightarrow|3\rangle$ transitions are approximately identical, a
quick calculation reveals that the Doppler cooling limit is the same as in the
two-level case, $k_{B}T_{D}=\hbar\gamma/2$, for
$\gamma=\gamma_{31}=\gamma_{32}$.

It should be noted that the three-level scheme presented here is actually
unrealistic: all three transitions are assumed to be strong electric dipole
allowed which, because of parity, is impossible. This can be remedied,
however, by using a six-level scheme with two subsequent rotational levels
(i.e., with opposite parity) for each vibrational level, like in Fig. 2. This
additional modification to the original scheme does not change the calculation
significantly.

Figure 2: The three-level scheme from Fig. 1, extended to six levels to
account for parity changes in each transition.

##  III Implementation

We imagine a three-level system consisting of a molecule that is coupled to
two pairs of counter propagating Raman beams along each axis, for a total of
12 beams.

A problem of this nature involving coupling of two beams to a system can be
solved by working in the rotating frame of the beams. In this scenario there
are six pairs of beams interacting with the system simultaneously. Working in
the rotating frame of one pair of beams yields time-dependent terms in the
Hamiltonian that result from the other pairs. To avoid the unpleasant
computational difficulty of this approach, we describe the system with six
distinct Hamiltonians (and density matrices), one in the rotating frame of
each pair of beams. We then stitch the solutions together at the end to obtain
an approximate picture of the system. We justify this approximation by
pointing out that in each direction, the beams interact with the molecule
independently.

We describe a three-level system with two electronic ground states that differ
by vibrational energy, and a third electronic excited state.

In accordance with our approximation, the Hamiltonian that describes
interaction in the $\pm$ x-direction is

$\frac{H^{(\pm x)}}{\hbar}=\left(\begin{array}[]{cccc}\Delta_{1}\pm
k_{1}v_{x}&0&-\Omega_{1}&0\\\ 0&\Delta_{2}\pm k_{2}v_{x}&-\Omega_{2}&0\\\
-\Omega_{1}&-\Omega_{2}&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ (5)

where $v_{x}$ is the $x-$component of the velocity. We define $v_{x}$ to be in
the positive direction. The Hamiltonians for the y- and z-directions are
identical except with $x$ swapped for $y$ and $z$, respectively.

Just as $H^{(\pm x)}$ is the Hamiltonian for interaction in the $\pm$
x-direction, $\rho^{(\pm x)}$ represents the density matrix for interaction in
the $\pm$ x-direction.

For a transition $|i\rangle\leftrightarrow|3\rangle$, the average deceleration
experienced by the molecule due to stimulated absorption and stimulated
emission in the $\pm$ x-direction is

$\pm\frac{1}{m}\hbar
k_{i}\frac{\gamma_{3i}\Omega_{i}^{2}}{\gamma_{3i}^{2}+4(\Delta_{i}\pm
k_{i}v_{x})^{2}}(\rho^{(\pm x)}_{ii}-\rho^{(\pm x)}_{33})$

Finally, we discuss spontaneous decay. For each transition
$|3\rangle\to|2\rangle$ and $|3\rangle\to|1\rangle$ there is spontaneous
emission. Each time a photon is absorbed by the molecule via one of these
transitions, either in the x-, y-, or z-directions, it must be emitted, and
the emitted photon will have momentum with magnitude equal to that of the
absorbed photon. The deceleration experienced by the molecule from spontaneous
emission in all six directions is

$\sum_{i=1,2}\sum_{j=\pm}\sum_{k=x,y,z}\frac{1}{m}\hbar
k_{i}\gamma_{3i}\rho_{ii}^{(j,k)}\cos(\theta_{k})$

The above system was simulated numerically. For each axis, two Hamiltonians
and two density matrices are created, and two equations of motion are written.
All six equations of motion are then solved numerically for range
$[t_{i},t_{i+1}]$ in a recursive loop for $t_{i+1}=t_{i}+\Delta t$. The loop
solves all six equations simultaneously and iterates over $[0,\Delta t,2\Delta
t,...,N\Delta t]$, where $N\Delta t$ is the total duration of the simulation.
Our simulations had a total running time of $40\tau$. A slowing force $f_{i}$
is computed for $t_{i}$. The contribution of spontaneous emission to this
force is assigned pseudorandomly to each of six directions in space. A new
velocity is computed $v_{i+1}=v_{i}-t_{i}(f_{i}/m)$.

The two transitions in the three-level system were approximated to have the
same wavelength, $\lambda=2\pi/=100$ nm. Each molecule was assumed to have a
mass $m=10^{-27}$ kg. The molecule’s initial velocity is given in terms of the
recoil velocity, $v_{r}=\hbar k/m$. The initial velocity was $v=15v_{r}\approx
100$ m/s. All other parameters are given in terms of $\gamma=1/\tau$. For all
simulations, $\tau=100$ ns. The various decays in this scheme are as follows:

$\gamma_{31}=\gamma_{32}=\gamma_{opt}=\gamma,\text{ }\gamma_{34}=10\gamma.$

Simulations were evaluated a number of times with Rabi frequencies
$\Omega_{i}$ varying from $10\gamma$ to $100\gamma$. Detunings $\Delta_{i}$
varied from $0$ to $-10\gamma$. The system was allowed to be two-photon
resonant and $\Delta_{1}=\Delta_{2}$ and $\Omega_{1}=\Omega_{2}$ for all
simulations. For most of the remainder of this paper, both detunings will be
referred to as $\Delta$, and the Rabi frequencies as $\Omega$.

With these parameters, we calculate a Doppler cooling limit of $T_{D}\approx
70$ $\mu$K.

##  IV Results and Discussion

Figure 3: (a) Cooling for $\Omega=30\gamma$ for various detunings. Here, and
for the majority of the remaining results, $\Delta_{1}=\Delta_{2}=\Delta$. As
$\Delta$ approaches $-15$ from either side, the molecule experiences faster
cooling and colder final temperatures. As $\Delta$ strays away from $-15$ the
molecule experiences more stifled cooling, or even heating. In the case of
$\Delta=0$, no heating or cooling occurs, because the driving force on the
molecule is the same in all directions. (b) Fraction of molecules lost in the
cooling process due to noncyclic decay. In the best cases, adequate cooling
occurs in under $10\tau$, corresponding to a fraction of lost population of
about 40 percent on the lower end. This fraction is generally acceptable for
most experiments.

An ideal cooling scheme wants to see fast cooling with low loss of population
to non-cyclic energy levels. Our results show that for proper choice of
$\Omega$, cooling of 99 percent or greater can be achieved within a time
$0<t<5\tau$. Within the parameter regime that was examined, the choice of
$\Omega$ that gives the most options for choice of $\Delta$ that will result
in satisfactory cooling is in the $20\gamma$ to $40\gamma$ range. As shown in
Fig. 3a, for $\Omega=5\gamma$, various $\Delta$s can be chosen that result in
quick cooling, with greater absolute $\Delta$ generally resulting in faster
cooling. The amount of lost population in these results does not exceed 60
percent, and for the fastest cooling (small absolute $\Delta$), lost
population does not exceed 40 percent, (see Fig. 3b). Final velocity, in units
of $v_{r}$, for these simulations ranges from the order of $10^{-2}\text{ to
}10^{-4}$ for the best cooling (see Fig. 4). For a complete comparison of
$\Omega$ to $\Delta$, refer to Fig. 5. There is a nearly parabola-shaped
region inside which cooling of 1 percent or smaller is achieved, with final
velocities on the order of $10^{-4}v_{r}\approx 6\times 10^{-4}$ m/s in the
best cases. As $\Omega$ increases, the range of ideal choices of $\Delta$
increases. Figures 6 and 7 examine the time (in units of $\tau$) and fraction
of molecules lost to noncyclical decay, respectively, at the moment of cooling
of 1 percent or smaller. Since lost population is roughly exponential with
time, the fastest cooling scenarios result in the least population lost (a
time of about $5\tau\approx 500$ ns and population loss of $<40$ percent in
the best cases). Figure 8 shows the population lost at the end of cooling. The
proportion of population lost depends more strongly on the choice of $\Omega$
compared to $\Delta$.

Figure 4: Log plot of cooling for $\Omega=30\gamma$. As $\Delta$ approaches
$-15$ from either side, the molecules experiences faster cooling and colder
final temperatures. In the best cases, the slowest final velocities are
achieved in under $10\tau$ (and as early as $5\tau$) and are as low as
$10^{-4}v_{r}$, corresponding to a temperature of about $10^{-10}$K, well
below our calculated doppler limit of $70$ $\mu$K. Figure 5: (a) Log of
molecule velocity after $4\tau$ of cooling. Velocity is in units of $v_{r}$.
There is a sideways parabola-shaped region inside of which the best cooling
occurs. this region is roughly centered on, and split in half by,
$\Delta=-15$. The region also grows wider with larger $\Omega$, showing that
larger Rabi frequency allows for greater variety of choice for detuning to
still achieve good cooling. (b) Zoomed in section of (a). Figure 6: Fraction
of molecules lost after $4\tau$ of cooling. This figure depends little on the
choice of parameters, but is more affected by different $\Omega$ than
$\Delta$.  Figure 7: Time (in units of $\tau$) to 1 percent cooling. Blank
space indicates that 1 percent cooling did not occur for that pair of Rabi
frequency and detuning. This figure shows that within the region of good
cooling, the molecule can be cooled completely in as few as $5\tau=500$ ns.
Figure 8: Fraction of molecules lost at the instant of 1 percent cooling. In
the regions corresponding to complete cooling in under $5\tau$, the fraction
of population lost can be as low as 40 percent in the best cases.

In general, cooling should occur more rapidly when the transitions are red-
detuned. When both single-photon transitions are red-detuned but the two-
photon transition is blue-detuned, it should still be possible to achieve
reasonable cooling if the two-photon transition is near resonance (small
absolute two-photon detuning). In all of our simulations the system had two-
photon resonance.

Figure 9 shows the results of different sign combinations for the two
detunings. When both transitions are blue-detuned, heating occurs, and when
both are red-detuned, cooling occurs. When the detunings alternate red and
blue, whether heating or cooling occurs is idenified by the sign of
$\Delta_{1}$. The pace of the heating/cooling, whether fast or slow, is
identified by the sign of $\Delta_{2}$. This is likely because of the fact
that the $|3\rangle\to|1\rangle$ transition is faster than the
$|3\rangle\to|2\rangle\to|1\rangle$, which contains an extra step. So when the
$|3\rangle\to|1\rangle$ is red-detuned, the system primarily absorbs photons
that oppose its motion and cooling occurs. When it is blue detuned, the system
primarily absorsb photons that are parallel to its motion, and heating occurs.

Consider one of the two transitions. In one dimension, the net force
contributing to the cooling is the difference of the cooling forces of the
opposing lasers,

$\hbar
k\gamma_{opt}\Omega^{2}(\rho_{11}-\rho_{33})(\frac{1}{\gamma_{opt}^{2}+4(\Delta+kv)^{2}}-\frac{1}{\gamma_{opt}^{2}+4(\Delta-
kv)^{2}})$

The value of $\Delta$ that maximizes this expression at $t=0$, or
$\rho_{11}=1$, does not depend on $\Omega$. Inserting our parameter values
yields a maximizing value for the detuning of $\Delta\approx-15\gamma$. The
cooling process should be faster the closer the detunings are to this value,
which our simulations support.

In the best cooling cases, $\Omega=20\gamma$ to $30\gamma$, the simulated
final velocity of the molecules is $\approx 6\times 10^{-3}$, which
corresponds to a temperature of $\approx 10^{-10}$K, 5 orders of magnitude
below our calculated Doppler limit. We attribute this to the significant
acceleration achieved by the superradiant transition $|2\rangle\to|1\rangle$.

Although our scheme is similar to that of velocity selective coherent
population trapping (VSCPT), the key difference is that in VSCPT there is no
driving force to cool the system. Instead, the system is allowed to relax into
the cooling process. Similarly to VSCPT, the molecule does experience some
electromagnetically induced transparency (EIT), as shown in Fig. 10. At the
the end of the cooling process, most of the population that is not lost is in
on of the two ground vibrational states, primarily $|2\rangle$.

Figure 9: Cooling scenarios for the different sign combinations of detunings. For all four scenarios $\Omega=30\gamma$. The sign of $\Delta_{1}$ dictates whether the molecule will cool or heat up. This is because the first electronic transition scatters more photons than the second electronic transition, which is due to the fact that the upper vibrational state will sometimes decay into the lower vibrational state via superradiant decay. The sign of $\Delta_{2}$ dictates whether the cooling/heating experienced by the molecule will be fast or slow. All simulations were run for a total duration of $40\tau=4\mu$s. Temperature data was recorded in terms of the molecular recoil velocity $v_{r}$. An initial velocity of $15v_{r}$ corresponds to about 100 m/s. | $\Delta_{1}<0$ | $\Delta_{1}>0$
---|---|---
$\Delta_{2}<0$ | rapid cooling | gradual heating
$\Delta_{2}>0$ | gradual cooling | rapid heating
Table 1: Different cooling scenarios for the three-level scheme by sign
combination of the detunings. Figure 10: The molecule experiences partial EIT
during the cooling process. When cooling is finished, the population in the
electronic excited state is close to zero. Most of the population that isn’t
lost is in one of the two vibrational states, mostly the upper state
$|2\rangle$. For $\Omega=30\gamma$ and $\Delta=0$ to $\Delta=-20$: (a)
Population of the lowermost vibrational state $\rho_{11}$. (b) population of
the intermediate vibrational state $\rho_{22}$. (c) Population of the excited
state $\rho_{33}$. (d) Fraction of population lost.

##  V Conclusion

This paper presents a novel scheme for doppler cooling where the cycling
transition happens along a vibrational transition instead of an electronic
one. Our scheme’s configuration contains two vibrational states in the
electronic ground state. The level scheme presented here is realistic for most
molecules if one can choose the lowest two vibrational states. The vibrational
transitions’ characteristic slow scattering time is alleviated by
superradiance, the low photon momentum by a Raman transition in the visible
part of the spectrum. This lends this protocol a unique advantage that allows
for even relatively complicated molecules to be cooled with relative
efficiency. The six-level modification to the original scheme accounts for the
fact that vibrational transitions don’t have selection rules without affecting
the calculation presented here in any meaningful way.

Our results indicate that with the proper choice of parameters rapid cooling
can be achieved on a system with arbitrary initial state within the studied
manifold. Further study of this proposal may benefit from the incorporation of
chirping into the Doppler cooling scheme, which was not investigated here.
This scheme should be generalized to include more levels and possibly
different types of level schemes that describe more realistic molecules. The
concept of superradiance-assisted cooling may also be applied to different
types of laser cooling altogether.

##  VI Acknowledgments

We acknowledge funding from the NSF for this work.

## Appendix

The decays in the three-level scheme are

$\begin{split}\Gamma_{11}&=\gamma_{sup}\rho_{22}+\gamma_{31}\rho_{33},\\\
\Gamma_{12}&=-\frac{1}{2}\gamma_{sup}\rho_{12},\\\
\Gamma_{13}&=-\frac{1}{2}(\gamma_{31}+\gamma_{32}+\gamma_{34})\rho_{13},\\\
\Gamma_{21}&=-\frac{1}{2}\gamma_{sup}\rho_{21},\\\
\Gamma_{22}&=\gamma_{32}\rho_{33}-\gamma_{sup}\rho_{22},\\\
\Gamma_{23}&=-\frac{1}{2}\rho_{23}(\gamma_{sup}+\gamma_{31}+\gamma_{32}+\gamma_{34}),\\\
\Gamma_{31}&=-\frac{1}{2}(\gamma_{31}+\gamma_{32}+\gamma_{34})\rho_{31},\\\
\Gamma_{32}&=-\frac{1}{2}\rho_{32}(\gamma_{sup}+\gamma_{31}+\gamma_{32}+\gamma_{34}),\\\
\Gamma_{33}&=(\gamma_{31}+\gamma_{32}+\gamma_{34})(-\rho_{33}),\\\
\Gamma_{44}&=\gamma_{34}\rho_{33},\\\ \Gamma_{4j}&=0,\text{ }j\neq 4,\\\
\Gamma_{i4}&=0,\text{ }i\neq 4.\end{split}$

## References

  * Schawlow and Hänsch [1975] A. L. Schawlow and T. W. Hänsch, Cooling of gases by laser radiation, Optics Communications 13, 68 (1975).
  * Wineland and Dehmelt [1975] D. J. Wineland and H. G. Dehmelt, Principles of the stored ion calorimeter, Journal of Applied Physics 46, 919 (1975).
  * Ashkin [1978] A. Ashkin, Trapping of atoms by resonance radiation pressure, Physical Review Letters 40, 729 (1978).
  * Wineland  _et al._ [1978] D. J. Wineland, R. E. Drullinger, and F. L. Walls, Radiation-pressure cooling of bound resonant absorbers, Physical Review Letters 40, 1639 (1978).
  * Neuhauser  _et al._ [1978] W. Neuhauser, M. Howenstatt, P. Toschek, and H. G. Dehmelt, Optical-sideband cooling of visible atom cloud confined in parabolic well, Physical Review Letters 41, 233 (1978).
  * Phillips and Metcalf [1982] W. D. Phillips and H. Metcalf, Laser deceleration of an atomic beam, Physical Review Letters 48, 596 (1982).
  * Chu  _et al._ [1985] S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure, Physical Review Letters 55, 48 (1985).
  * Dalibard and Cohen-Tannoudji [1989] J. Dalibard and C. N. Cohen-Tannoudji, Laser cooling below the doppler limit by polarization gradients: Simple theoretical models, Journal of the Optical Society of America B 6, 2023 (1989).
  * Cohen-Tannoudji and Phillips [1990] C. N. Cohen-Tannoudji and W. D. Phillips, New mechanisms for laser cooling, Physics Today 43, 33 (1990).
  * Shuman  _et al._ [2010] E. Shuman, J. Barry, and D. DeMille, Laser cooling of a diatomic molecule, Nature 467, 820 (2010).
  * Yeo  _et al._ [2015] M. Yeo, M. T. Hummon, A. L. Collopy, B. Yan, B. Hemmerling, E. Chae, J. M. Doyle, and J. Ye, Rotational state microwave mixing for laser cooling of complex diatomic molecules, Phys. Rev. Lett. 114, 223003 (2015).
  * Lev  _et al._ [2008] B. L. Lev, A. Vukics, E. R. Hudson, B. C. Sawyer, P. Domokos, H. Ritsch, and J. Ye, Prospects for the cavity-assisted laser cooling of molecules, Phys. Rev. A 77, 023402 (2008).
  * Stuhl  _et al._ [2008] B. K. Stuhl, B. C. Sawyer, D. Wang, and J. Ye, Magneto-optical trap for polar molecules, Phys. Rev. Lett. 101, 243002 (2008).
  * Ding  _et al._ [2020] S. Ding, Y. Wu, I. A. Finneran, J. J. Burau, and J. Ye, Sub-doppler cooling and compressed trapping of yo molecules at $\mu\mathrm{K}$ temperatures, Phys. Rev. X 10, 021049 (2020).
  * Xu  _et al._ [2003] X. Xu, T. H. Loftus, J. L. Hall, A. Gallagher, and J. Ye, Cooling and trapping of atomic strontium, J. Opt. Soc. Am. B 20, 968 (2003).
  * Hummon  _et al._ [2013] M. T. Hummon, M. Yeo, B. K. Stuhl, A. L. Collopy, Y. Xia, and J. Ye, 2d magneto-optical trapping of diatomic molecules, Phys. Rev. Lett. 110, 143001 (2013).
  * Hemmerling  _et al._ [2016] B. Hemmerling, E. Chae, A. Ravi, L. Anderegg, G. K. Drayna, N. R. Hutzler, A. L. Collopy, J. Ye, W. Ketterle, and J. M. Doyle, Laser slowing of CaF molecules to near the capture velocity of a molecular MOT, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 174001 (2016).
  * Marco  _et al._ [2019] L. D. Marco, G. Valtolina, K. Matsuda, W. G. Tobias, and J. Y. Jacob P Covey, A degenerate fermi gas of polar molecules, Science 363, 853 (2019).
  * Kozyryev  _et al._ [2016] I. Kozyryev, L. Baum, K. Matsuda, and J. M. Doyle, Proposal for laser cooling of complex polyatomic molecules, Chemphyschem 17, 3641 (2016).
  * Norcia  _et al._ [2018] M. A. Norcia, J. R. K. Cline, J. P. Bartolotta, M. J. Holland, and J. K. Thompson, Narrow-line laser cooling by adiabatic transfer, New Journal of Physics 20, 023021 (2018).
  * Bartolotta  _et al._ [2018] J. P. Bartolotta, M. A. Norcia, J. R. K. Cline, J. K. Thompson, and M. J. Holland, Laser cooling by sawtooth-wave adiabatic passage, Physical Review A 98, 023404 (2018).
  * Dicke [1954] R. H. Dicke, Coherence in spontaneous radiation processes, Physical Review 93, 99 (1954).
  * Gross and Haroche [1982] M. Gross and S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission, Physics Reports 93, 301 (1982).
  * Lin and Yelin [2012a] G.-D. Lin and S. F. Yelin, Chapter 6 - superradiance: An integrated approach to cooperative effects in various systems, in  _Advances in Atomic, Molecular, and Optical Physics_, Advances In Atomic, Molecular, and Optical Physics, Vol. 61, edited by P. Berman, E. Arimondo, and C. Lin (Academic Press, 2012) pp. 295 – 329\.
  * Lin and Yelin [2011] G.-D. Lin and S. Yelin, Superradiance in spin-j atoms: Effects of multiple atomic levels, Physical Review A 85, https://doi.org/10.1103/PhysRevA.85.033831 (2011).
  * Lin and Yelin [2013] G.-D. Lin and S. Yelin, Vibrational spectroscopy of polar molecules with superradiance, Molecular Physics 111, 1917 (2013).
  * Wang and Yelin [2005] T. Wang and S. F. Yelin, Theory for raman superradiance in atomic gases, Phys. Rev. A 72, 043804 (2005).
  * Lin and Yelin [2012b] G.-D. Lin and S. F. Yelin, Superradiance: an integrated approach to cooperative effects in various systems, Advances in Atomic, Molecular, and Optical Physics 61, 295 (2012b).