Observation of polarization instabilities
in a two-photon laser
M.D. Stenner, W.J. Brown, O. Pfister, and D.J. Gauthier
Duke University Department of Physics
Supported by the National Science Foundation
This is the text of the talk given at the Opto-Southeast conference in Charlotte, NC on 19 September, 2000. The conference was held by the OSA and
SPIE.
2
Two-Photon Laser
The two-photon laser is based on two-photon stimulated emission, a process
by which an excited atom interacts with two photons simultaneously. These
two photons stimulate the emission of two new photons. Just as in one-photon
stimulated emission, the new photons are clones of the original photons. In
this process, the atom goes from the excited state to the ground state via an
intermediate virtual level.
The only requirement on the frequencies of the photons is that the sum of
the incoming photons’ energies equals the energy difference between the excited
and ground states of the atom. However, all of our experiments are in the
degenerate case where the photons have the same frequency.
The laser based on this process was originally proposed by Prokhorov in
1963 and independently proposed by Sorokin and Braslau in 1964. It was not
until 1987 that Brune et al. successfully constructed a two-photon maser, and
until 1992 that Gauthier et al. built the first two-photon laser.
The two-photon laser has many interesting properties and differs from the
one-photon laser in many ways. In this talk I will focus on the two-photon
laser’s nonlinear behavior.
3
Ingredients of a Laser
I’ll begin by reviewing some laser basics.
All lasers are based on the combination of optical gain (or amplification) and
feed back. There is some gain medium which amplifies light, and laser cavity
which continuously sends the light back into the gain medium, with some light
trickled out at some point.
It is this combination of gain and feedback which allow laser oscillation.
For the the next few pictures, I’ll discuss only one-photon processes.
In our experiments, we use atoms as our gain medium. Although the atoms
have many levels, I’ll discuss only two, for now. In my notation, I’ll refer to
1
the energy difference between the levels as h̄ω, where omega is the frequency of
a photon with that energy. I’ll use N to refer to the population of each level,
where the population is the number of atoms in that level.
4
The Light-Matter Interactions: One-Photon
Processes
There are three main types of light-matter interactions: spontaneous emission,
absorption, and stimulated emission.
In spontaneous emission, and excited atom spontaneously drops to the ground
state, and gives off a photon with random direction and phase, and with energy
that the atom lost. By the process, the population of the upper state decays
exponentially on timescales of the order of tens of nanoseconds.
In absorption, a photon interacts with an atom in the ground state, and is
absorbed. In this process, the atom goes into the excited state. This process
causes exponential decay of the ground state population, at a rate the depends
on the interaction cross section and the photon flux density. Here, I is the
intensity, and h̄ωeg is the energy per photon.
In stimulated emission, an atom in the excited state interacts with a photon,
and the atom emits a second photon. This new photon is a clone of the first one
in every way. This process causes exponential decay of the excited state with
the same rate as absorption. This is the process responsible for laser action.
5
Two-Photon Processes
These processes also come in the two-photon variety, although I won’t discuss
two-photon absorption here.
Two-photon processes connect two states with the same parity. In twophoton spontaneous emission, an atom in the excited state, spontaneously emits
two photons of random phase and direction. In doing this, the atom drops to
the ground state via an intermediate virtual level. The only requirement on the
energies of these photons is that their total energy is the difference between the
energies of the ground and excited states.
This is the dominant decay mechanism for many metastable states, and
happens with typical timescales of about 1s.
In two-photon stimulated emission, an excited atom interacts with two photons simultaneously, and falls to the ground state, emitting two new photons.
As in the one-photon case, the new photons are clones of the original two. The
freedom in photon frequency still holds, but if we consider the degenerate case
when the photons have the same frequency, then we see that the excited state
population decays exponentially, just as in the one-photon case, except that
now the intensity is squared. This causes much of the interesting behavior of
two-photon systems.
2
6
Two-Photon Amplifier
If we build an amplifier based on this process, and look at the increase in the
intensity as light propagates through the amplifier, we find that it looks very
much like the one-photon case, except that everywhere the intensity appears, it
comes in squared.
To see the effects of this, we can look of the limiting case of low intensity
and low gain. In that case, we find that, like the one-photon case, the output
intensity is input intensity times this exponential. As expected, the exponent
contains a gain coefficient and the path length, but it also contains the input
intensity!
The most notable effect of this is that for zero input intensity, there is no
gain! In fact, if we plot the gain vs. the input intensity, we see that the gain
increases linearly with intensity, until the two-photon saturation intensity, when
it begins to drop off.
This behavior is the cause of much of the nonlinear behavior of two-photon
lasers. For comparison, I’ll discuss the one-photon laser first.
7
Nonlinearity in one-photon lasers
Here, I’ve plotted the one-photon laser intensity vs. the laser pumping rate,
normalize to the threshold pump rate. As we first begin pumping, the laser
remains off, with spontaneous emission dominating. When we reach threshold,
gain becomes equal to losses, stimulated emission starts to dominate, and the
laser turns on smoothly. As we continue to increase the pump rate, the intensity
increases linearly.
It isn’t until far above threshold, nearing the saturation intensity that nonlinear behavior becomes important. This is because saturation is the dominant
cause of nonlinear behavior.
To understand this, consider the susceptibility. The susceptibility is proportional to the inversion, which is in turn proportional to this quantity. This can
be expanded in terms of the intensity as long as the intensity is less than the
one-photon saturation intensity.
Normally, we keep only the first two terms of the expansion. This is the
basis of third-order laser theory and accurately describes most laser behavior.
8
Nonlinearity in two-photon lasers
The two-photon laser is very different. Here, I’ve plotted the two-photon laser
intensity, normalized to the two-photon saturation intensity, vs. the pump rate,
normalized to the threshold pump rate. Again, we see that as we first begin
pumping, the laser remains off, with spontaneous emission dominating. However, in the case of the two-photon laser, the laser remains off no matter how
hard I pump. This is because, as I mentioned before, there is no gain when the
3
intensity is zero. In order to turn the laser on, one must create some perturbation (such as an injected pulse) to knock the laser onto this other solution. But
look! Even at the minimum pump rate, the laser is already at the saturation
intensity. As a result, nonlinearity is always important.
As in the one-photon laser, the susceptibility is proportional to the inversion
(2)
and to this quantity (with I/Isat squared this time), but in this case, it cannot be
(2)
expanded. I is always greater than Isat and so the expansion does not converge,
and one must keep all orders in the expansion. As a result, the two-photon laser
is highly nonlinear under all operating conditions and cannot be modeled with
a susceptibility expansion.
9
Experimental Setup
In our experimental setup, we have have a high finesse cavity in vacuum chamber. Our cavity is aligned along the y-axis. We have a beam of potassium atoms
propagating in the x direction, and Raman and optical pumping beams along
the z-axis. The cavity and pumping beams are perpendicular to avoid wavemixing effects. We also have a weak magnetic field in z direction to swamp out
stray fields. We inject the starting pulse into one end of the cavity and have a
detector at the other end to measure the pulse and the laser power.
10
Two-Photon Gain via Multi-Photon Scattering
Our gain mechanism is a hyper-Raman transition in optically pumped potassium
39. Two optical pumping beams deposit the atoms in the F=2, m=2 Zeeman
sub-level.
Then, two incoming photons stimulate this transition, whereby two Raman
pump photons of frequency ωd are annihilated and two new laser photons of
frequency ωp are created.
I know this looks different from the diagram I showed you earlier, but it is
fundamentally the same. This is the excited state, this is our first photon, this
is the intermediate state, our second photon, and our final state.
The resonance condition is that ωd and ωp differ by half of the ground state
splitting.
11
Threshold Behavior of the Two-Photon Laser
When we put atoms in the cavity and begin pumping, the laser initially remains
off, as I said before. Then, when we inject a starting pulse, the laser turns on
and runs stably at about .3 µW for several tenths of a second.
4
12
Observation of Polarization Instabilities
If we now place a linear polarizer after the laser, we see that the power in a
single polarization oscillates rapidly on µs timescales. The total intensity is
constant, while the the intensity in a single polarization oscillates rapidly. We
see that while the intensity is stable, the polarization is unstable.
We have seen both periodic and random-like oscillations for varying magnetic
field strengths.
13
Magnetic Field Dependence
We see that for low fields, the polarization oscillates regularly, but as we increase
the field, the oscillations begin to look much more random.
14
How can we have polarization instabilities?
We have seen that the two-photon laser is highly nonlinear, so we might expect
intensity instabilities, but why would we have polarization instabilities?
We can have polarization instabilities because there are multiple final states
for the transition. Each of these final states has multiple quantum pathways
which lead to it. Each of these pathways produces a different polarization. Since
all pathways are frequency degenerate, all of them can happen simultaneously.
15
Other pathways
Here are a few example pathways:
The first is the pathway shown before. Here, two σ− photons are annihilated
and two z polarized photons are emitted.
In this second example, first a σ− transition happens, and then a σ+ . This
transition ends in the same final state as the first example.
This third example first undergoes a σ+ transition and then a z transition.
This leaves the atom in a different final state.
All three of these processes produce different polarizations.
16
What determines the polarization instability
oscillation frequency?
We observed that the polarization oscillates at 9.1 MHz. What determines
this frequency? We have considered the other important frequencies in our
experiment. Several of these frequencies are similar, but none of them are right
on.
We are continuing to explore the gain both experimentally and theoretically
to investigate further.
5
17
Conclusions
The two photon laser has many interesting features which require investigation.
It exhibits highly nonlinear behavior, has demonstrated complex polarization
instabilities, and has multiple degenerate quantum pathways starting from the
same initial state. This last one leads us to believe that it may be a bright
source of entangled photons.
6