Why a single atom? The Zeno Paradox and Photonic Quantum Computers

Performing a computation, any computation, requires mechanisms that are not linear: the result of logic gates is generally not just a summation of the results of the separate inputs. For example, the result of the AND gate between two classical bits is 0 if only one of the two input bits is 1, yet if both inputs are 1 the result is 1.

This is also true for quantum computation. For example, the quantum controlled-Zgate requires a nonlinear mechanism, as it applies a phase of π if both qubits are 1, but it does not apply any phase at all if only one of the qubits is 1.

This obvious need for nonlinear operations between single qubits is the greatest hurdle in the effort to construct a photonic quantum computer, in which the qubits are encoded in single photons, which do not interact with each other at all.

Seemingly, one trivial way to attack this challenge is to construct an optical device that counts the incoming photons at the input modes, and emits new photons into the output modes according to whatever nonlinear algorithm you wish to implement. However, this "artificial nonlinearity" cannot be used for quantum computation, in which the qubits must not be measured. What can be done, however, is to measure some of the photons, and according to the results of these measurements apply some operations to the other photons that were not measured. This is exactly, in a nutshell, the principle behind linear-optics quantum computation (LOQC) [1]. The price of this approach is extremely low efficiency. The probability that the the right number of photons ended up in the right detectors (letting exactly the right number of photons to be the released unmeasured) is very low, resulting in a tremendous amount of overhead in resources, not to mention the challenge of very fast and complex feed-forward mechanisms.

The more direct approach to achieve nonlinear operations is to harness a natural nonlinear optical effect, for example using a nonlinear crystal. However, it has been known, both theoretically and experimentally, that the nonlinearity provided by crystals and other "macroscopic" systems that are composed of many atoms, is not enough for quantum computations [2]. Deterministic quantum logic gates with single photons have been demonstrated so far only by using single atoms [3,4], and accordingly such gates form the basic building blocks of our platform at Quantum Source.

The fact that single atoms are indeed required for deterministic single-photon nonlinearity may not be a surprise - after all, for a single atom two photons are already too much. However, a rigorous scientific description of how the involvement of more atoms leads to a decrease in the single-photon nonlinearity (rather than, say, an increase) has been missing, in particular for the specific interactions that form the basis for photon-atom quantum gates. 

This recent study, published in October 2024 in Physical Review Letters [5], reveals the physical mechanism that causes the reduction in the effectivity of the system as more and more atoms are involved. Surprisingly, this is no other than the Quantum Zeno effect (QZE). 

The Greek philosopher Zeno of Elia (490-430 BCE) wondered how an arrow can actually fly, when at every given instant in time it can be conceived as standing still. 

Zeno's problem was not knowing Calculus (which is understandable, given that Calculus was invented only 2000 years later by Newton and Leibniz). Calculus can easily describe how an infinite series of instants in time can sum to a finite duration (thereby solving Zeno's other paradox of Achilles and the tortoise), and how the derivative can define a specific velocity value to the arrow at every instant in time, thereby solving the Arrow's paradox.

However, quite surprisingly, the Quantum Mechanical description of nature does include situations where Zeno's paradox leads to a valid prediction. As it happens, the evolution of a system can be slowed down, up to a halt, if the system is measured repeatedly at short time periods. The reason is that after a measurement the state of the system "collapses" to the result of the measurement, in some sense" resetting" the time of the evolution back to zero. 

Think about a little kid repeatedly asking his tired father "are you awake?". Obviously, the father's falling asleep process will be slowed down quite significantly by this measurement process. Even if we assume that once the father falls asleep he is not awakened by his kid's questions, the kid can keep his father awake forever if the questions come at time intervals that are much shorter than the times it takes the father to start falling asleep. This painful example shows that what is necessary for this effect to take place in nature is for the evolution process to be not exponential, namely to start slow and only then speed up. 

What is shown in this paper [5] is that single-photon nonlinearity, in which the system's optical response needs to change after one photon, is suppressed because of the Quantum Zeno effect when multiple atoms are involved. In other words, since only one atom can change its state following an interaction with a single photon, the collective state is changed less if there are more atoms, and so when the next photon comes and asks the system "did you change?", there is a higher likelihood to get the answer "no".

So if your kids keep you awake, you may read this paper to them - the story about the connection between a philosopher from 2500 years ago and modern photonic quantum computation. If they fall asleep - all the better. If they stay awake - well, please have them send Quantum Source their CVs!

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[1] E. Knill, R. Laflamme, G. J. Milburn, Nature. 409 ,46 (2016)

[2] J. H. Shapiro, Phys. Rev. A 73, 062305 (2006).

[3] A. Reiserer, N. Kalb, G. Rempe, S. Ritter ,Nature 508, 237 (2014)

[4] O. Bechler, A. Borne, S. Rosenblum, G. Guendelman, B. Dayan, Nature Physics 14, 996 (2018)

[5] How Single-Photon Switching is Quenched with Multiple Λ–LevelAtoms, A. N. Poddubny, S. Rosenblum, B. Dayan, Physical Review Letters,133, 113601 (2024)

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