What is a Simulton?
In December Physical Review Letters published Quasisimultons in Thermal Atomic Vapours, a paper detailing work I did with colleagues at Durham Quantum Light and Matter. It was highlighted as an Editors’ Suggestion ‘due to its particular importance, innovation, and broad appeal’ and featured in APS Physics. We were pleased with this reception of the work, which came out of my PhD research.
I’ll try to answer three questions in this blogpost:
 What is a simulton?
 What is a quasisimulton?
 What is significant about simultons?
From Solitons to Simultons
In an earlier post I introduced the concept of an optical soliton: a pulse of light that can hold its shape without dispersion and travel at fixed speed over large distances through a medium that would otherwise absorb it. If you haven’t read that post yet I would suggest to do that now and come back.
The key idea I introduced in that post was the pulse area
\[\theta(z) = \int^\infty_{\infty} \Omega(z, t) \mathrm{d} t.\]A pulse with an area of $2\pi$ (the units don’t matter for now) will form a sechshaped soliton in a field consisting of a single frequency of light nearresonant with the transition in a twolevel quantised system, such as between two electronic excitation levels in an atom.
In a 1981 paper Konopnicki & Eberly showed theoretically that it would be possible for solitonlike pulses to form analytic solutions in a threelevel system coupled by a field consisting of two nearresonant frequencies, under the condition that the pulses are matched^{1} and the two transitions have the same oscillator strength.
A threelevel system can be coupled in different ways. Here we’ll consider the V configuration, where the atom has two excited states, $\left 1 \right>$ and $\left 2 \right>$, and a single ground state level $\left 0 \right>$ which is coupled to both excited states. The field component resonant with $\left 0 \right> \leftrightarrow \left 1 \right>$ we’ll call $A$ and the field component resonant with $\left 0 \right> \leftrightarrow \left 2 \right>$ we’ll call $B$.
If we synchronise the input field envelopes $\Omega_A(z=0, t)$ and $\Omega_B(z=0, t)$ such that
\[\Omega_B(0, t) = r\Omega_A(0, t)\]for some constant $r$, the threelevel system in superposition acts like a twolevel system and the concept of the area theorem then extends to the pair of field envelopes. If we likewise define the pulse areas of $A$ and $B$ with
\[\theta_{\{A,B\}}(z) = \int^\infty_{\infty} \Omega_{\{A,B\}}(z, t) \mathrm{d} t\]we can derive a matched pulse area
\[\theta'(z) = \sqrt{\theta_A^2(z) + \theta_B^2(z)}\]where, like a soliton, if the area $\theta’(z) = 2\pi$ the pulse can move through the atoms unimpeded. But note here that the area theorem applies to the sum of squares, and looks like the hypotenuse from Pythagora’s theorem. This means we can fix the length of the ‘hypotenuse’ at $2\pi$ and make $\theta_A(z)$ bigger or smaller like the base of the rightangled triangle.
Let’s see what this looks like. In figure 1 a $\theta_A(z) = 1\pi$ pulse in field $A$ approaches from the left and enters a cell of $V$configuration atoms represented by the blue region. There is no matched pulse in field $B$, so the system acts like a twolevel system where the pulse does not have enough area to propagate as a soliton. The pulse is quickly absorbed on entry ($z = 0$) and, apart from some small oscillations, no pulse reaches the back of the medium at the right ($z = 1$).
In figure 2 the same $\theta_A(z) = 1\pi$ pulse in field $A$ approaches from the left, but this time it is matched with a $\theta_B(z) = \sqrt{3}\pi$ pulse in field $B$. We see that both pulses travel through the atoms without absorption and are received at the back of the cell on the right with the same profile.
We can understand this result by looking at the matched pulse area:
\[\theta'(z) = \sqrt{\theta_A^2(z) + \theta_B^2(z)} = \sqrt{1 + 3}\pi = 2\pi.\]The two pulses are able to travel together as one simulton connected by quantum superposition in the threelevel atom. Solitons are observed in classical situations, such as water waves, but simultons are a purely quantal phenomenon.
Observing Simultons
The requirement of matched pulses and equal oscillator strengths, along with some other fairly strict conditions^{2}, made simultons difficult to realise experimentally, and in nearly 40 years since that original prediction they had never been observed in an atomic vapour.
What we were able to show in theory and then in simulations like the ones I’m showing here is that the conditions were too strict. If the pulses aren’t initially matched, the simulton can bring them together. If the couplings have different oscillator strengths, the simulton can hold them together. The different oscillator strengths mean that the propagating pulses don’t match the exact Konopnicki & Eberly definition of a simulton, so in the PRL paper we give the generalised pulses the name quasisimultons to be precise. Here I’ll just call them simultons.
We can see this robustness of simulton propagation in figure 3. Two seperate $2\pi$ pulses, one in field $A$ (green) and one in field $B$ (red), are shown entering the medium. These each have the pulse area to travel individually as solitons. The $A$ field has a stronger interaction strength^{3} and so travels slower than the $B$ soliton, which catches it up. When the pulses meet about half way across the medium, they combine. The matched pulse area is now $4\pi$, so the combined pulse breaks up and forms two $2\pi$ simultons. Note that the sign of field $A$ has flipped in the later simulton.
We also added thermal broadening, spontaneous decay and hyperfine pumping to the simulations, and it was still possible for simultons to form. Simultons are robust!
Simulton Surfers
Finally, we demonstrated that it is not necessary to put in two matched pulses. If one field is weak continuous wave and the other is a $2\pi$ pulse, it will ‘pick up’ a part of the continuous field and carry it along like a surfer on a wave. So we informally call these simulton surfers. An example is shown in figure 4.
What is Significant About Simultons?
Our colleagues in the lab set up an innovative optical experiment with this kind of simulton surfer configuration and were able to observe simulton formation in a thermal atomic vapour for the first time since the prediction in 1981.
Simultons promise to open up an interesting domain of research in quantum technologies. The fact that the simulton area theorem applies to the pulses in combination gives us an extra degree of freedom. For example, we might make one field arbitrarily weak, even just a few photons. If we encode quantum information in the polarisation of a photon, we could use a strong pulse in a simulton to transmit it through a material that would otherwise absorb it. Using Rydberg atom interactions, we might then be able to make simultons interact in a controllable way for use in quantum computation.
References

Ogden et al. (2019), Quasisimultons in Thermal Atomic Vapours. Physical Review Letters 123 (24)  arXiv:1909.07161

M. J. Konopnicki and J. H. Eberly (1981), Simultaneous propagation of short differentwavelength optical pulses. Physical Review A 24 (5)

By matched we mean that they are simultaneous, i.e. have the same time profile. ↩

The analytic model assumed no thermal broadening, which would require the atomic vapour to be close to absolute zero. It also neglected spontaneous decay of the atoms, following the quantum dynamics of a pure state via the Schrödinger equation. ↩

The interaction strength of a field with an atomic transition is proportional to the square of the transition dipole moment. ↩