Zeno Blockade

A quick refresher on the quantum Zeno effect

We have already discussed the quantum Zeno effect here, where a quantum system can be effectively “stopped” by observing it. The key fact to remember about the Zeno effect is that quantum amplitude has to build up, unlike classical probability and anything which disrupts that process can effectively stop a quantum particle. As we hinted at in the Zeno effect explainer, there are many ways to realize the quantum Zeno effect. Anything that removes, (or changes the sign of) quantum probability amplitude can lead to a Zeno effect, and this can actually happen in many ways that are discussed in this paper and this paper by our team. Before we explain how this works, we need to explain the different pieces that we need to put together to make two photons bounce off each other, an effect known as the Zeno blockade. The Zeno blockade is a key ingredient to quantum computing and could be used to construct quantum logic gates. While this isn’t quite how we use it in our technology, it still plays a key role, and could be used that way in future devices.

Optical Cavity

One of the key concepts in implementing a Zeno blockade is an optical cavity. Since light acts as a wave, it can stay in a region of space for a long time if it happens to nicely fit a full number of wavelengths (a condition known as resonance). Resonances are something where we can have classical intuition based on sound waves. For example, a flute plays a certain note because of the wavelengths that fit nicely, or are at resonance. Both light and sound can remain in a cavity for some time. The configurations we are interested in are those where the light can either enter the cavity or continue past. Cavities can also be resonant with multiple different frequencies of light at the same time, a fact which is important later. The process of entering a cavity can be written mathematically in the language of quantum optics, using what is called an “annihilation operator” $\hat{a}$ which has the effect of removing a light particle and a “creation operator”  $\hat{a}^\dagger$ which adds a light particle. If we call the free light particle $f$ and the particle in the cavity $s$, this process can be written as

$\hat{a}_f\hat{a}^\dagger_s$

because quantum mechanics has a property known as being unitary, in which for every process a reverse process also has to be possible. In this case, light can leave the cavity the same way it came in, giving the overall Hamiltonian, an equation that describes what is physically possible in a quantum system. The Hamiltonian of a cavity which light can enter or leave is therefore 

$\hat{H}=c(\hat{a}_f\hat{a}^\dagger_s+\hat{a}^\dagger_f\hat{a}_s)$

where $c$ is a constant describing the rate at which the light enters the cavity. If we wait for the right amount of time, which is proportional to $1/c$, then the light will fully transfer to the cavity.

Sum-frequency Generation

The kind of nonlinear interactions that we consider here is one known as sum frequency generation. In this process, two light particles at specific frequencies combine to make one light particle at a frequency that is the sum of the two. For a Zeno blockade, we consider a nonlinear cavity that can perform sum-frequency generation. In the language of quantum optics, the process of removing light particles $s$ and $i$ and creating light particle $p$ (the letters are those conventionally used in quantum optics, their meaning is not important here) can be written as 

$\hat{a}_s\hat{a}_i\hat{a}^\dagger_p$.

Because quantum mechanics is unitary, the reverse process also has to be possible. The overall Hamiltonian that describes the processes that are allowed to happen is therefore

$\hat{H}=G(\hat{a}_s\hat{a}_i\hat{a}^\dagger_p+\hat{a}^\dagger_s\hat{a}^\dagger_i\hat{a}_p)$,

where $G$ is a constant that relates to the material that is used to create the interactions and light intensity. Concentrating light into a small space makes the interaction stronger. If $G$ is large compared to the rate at which the light can enter the cavity, $\frac{G}{c}\gg 1$, then the light will be prevented from entering.

Simplified Zeno Blockade

With the two ingredients we listed above, we can build a minimal example of a Zeno blockade. The overall Hamiltonian is 

$\hat{H}=c(\hat{a}_f\hat{a}^\dagger_s+\hat{a}^\dagger_f\hat{a}_s)+$

$G(\hat{a}_s\hat{a}_i\hat{a}^\dagger_p+\hat{a}^\dagger_s\hat{a}^\dagger_i\hat{a}_p)$.

Note that because we assume light particles $s$, $i$, and $p$ can exist in the cavity without being lost, we are implicitly assuming it is resonant with all three. This is a situation that is difficult to engineer in practice, but mathematically easier to describe. If some were not in resonance, we would have to also mathematically describe loss from the cavity, which would significantly complicate our equation. 

We now consider what happens if we wait the amount of time that it would take a light particle to fully transfer to the cavity if there were no nonlinear effects (proportional to $1/c$). We first consider the case where there are no light particles in the cavity. In this case, the nonlinear effects do nothing because there are no particles $i$ or $p$, so neither of the two nonlinear terms can do anything. The first term involves $\hat{a}_i$ so with no particle $i$ it cannot have any effect (physically, it needs the different energy of the two photons to add), and the second involves $\hat{a}_p$, so with no particle $p$ it also cannot have an effect. Regardless of the size of $G$, the light particle will just enter the cavity unimpeded. 

Now let’s consider what happens if there is a light particle $i$, what happens here depends on the size of $G$ relative to $c$. If $\frac{G}{c}\approx 1$, then what happens will be complicated. Some light will enter the cavity and some will be converted to $p$, and the results will be messy. If $\frac{c}{G} \gg 1$, then the nonlinear effects won’t really do much of anything and the light particle will just transfer. However, if $\frac{G}{c}\gg 1$, then we can have a Zeno blockade. The amplitude will not be able to build up in $i$ so the light particle will have to remain outside the cavity in $f$. This is exciting, because we can literally bounce one light particle off another. The behavior, both with and without a light particle already in the cavity (red), is shown below. The light particle that is potentially able to enter the cavity from a fiber ($f$ or $s$) is shown in green, and the photon that can be created by the nonlinear interaction ($p$) is shown in blue, with saturation indicating probability.

Some more astute readers might notice a slight problem with this argument. Can you see what it is? Because quantum mechanics is unitary, there is also a term $G\hat{a}^\dagger_s\hat{a}^\dagger_i\hat{a}_p$, so the light particle in $p$ will quickly split back into two in $s$ and $i$, because $G$ is large. From this perspective, it looks like the amplitude will be there to build up after all. However, this misses a piece of the mathematics of quantum mechanics that we have not talked about yet. It turns out that because of the structure of the equations of quantum mechanics, every time a light particle is changed by the Hamiltonian, it gets multiplied by a factor of $\sqrt{-1}$. Therefore when the light gets converted into $p$ and comes back, the amplitude has a factor of $-1$. Recall that from our other material unlike classical probabilities which can only add, quantum amplitudes can add or subtract. This is exactly what happens here. The amplitudes coming directly from $f$ cancel with those coming back from $p$, leaving the light outside of the cavity, and indeed causing the light particles to bounce off each other.

One might also notice that if there is no blockade, the light particle will re-enter $f$. Thanks to the term $\hat{a}^\dagger_f\hat{a}_s$,  this might cause you to question what the blockade has accomplished. However, similar to creating a light particle $p$, transferring into the cavity and back will lead to a factor of $-1$, and this can be used to perform quantum logical operations, known as quantum gates.

In a more realistic case where $p$ can be lost, the mathematics are more complicated, but the basic behavior is the same.

Real Zeno Blockade

The description we have given here is highly simplified, intending to give a taste of the mathematics behind the effects. In reality, there are several complications. We won’t discuss all of them, but here are some experimental challenges in implementing a Zeno Blockade

• Real light particles move: our simplified mathematical description hasn’t taken into account that light particle $f$ will actually be moving across a fiber and will not have constant amplitude as it passes by, and we have to get light particle $i$ into the cavity somehow. Issues related to these effects can be addressed by shaping the pulses of light.

• It is hard to make $G$ large in practice since the nonlinear effects themselves are weak. The fields need to be made strong by confining them to small regions of space. Our team has made great progress though and has been able to demonstrate Zeno blockade at the single photon level.

  • Alternatively, the nonlinear effects are stronger if more light particles are involved, but our vision is to compute with few photons, so this is not an option for us.

• The cavities need to be very high quality so light particles aren’t lost. Our team has been able to make excellent cavities with lithium niobate.