अन्तरा सकलसुन्दरीयुगलमिन्दिरारमण संचरन् |

मञ्जुलां तदनु रासकेलिमयि कंजनाभ समुपादधाः ||

naaraayaNiiyam of naaraayaNa bhaTTatri

A hallmark of quantum mechanics is the phenomenon of entanglement. It represents an intertwining of two or more entities where each loses its individual character. Perhaps the simplest example occurs in a hydrogen molecule that consists of two hydrogen atoms. Each atom contains an electron that carries a spin. This spin can take two values. In simplistic terms, each electron rotates in the clockwise or counterclockwise direction (about a reference axis that can be fixed arbitrarily). Effectively, each electron carries a simple identity card - a spin label that takes one of two values. In the hydrogen molecule, the two electrons come together so that their spins cancel each other out. However, it is not as if the first electron spins clockwise while the second spins counterclockwise. Nor is it the other way around. Rather, the two electrons lose their individual identities. All we can say is that the two electrons have spins that are opposite to each other.

Entanglement can occur among any number of particles, not just two. When more than two particles get entangled, we can understand their entanglement by partitioning them into pairs in different ways. This is perhaps best illustrated by benzene, an organic chemical that is responsible for the distinctive smell of petrol (gas) stations. In the mid 1800’s, the chemical structure of this compound was a big mystery. The German chemist August Kekulé provided the solution, proposing that benzene is a cyclic ring of carbon atoms. This was a revolutionary idea at the time, incompatible with what was then known about the rules of chemical bonding. The rules dictated that each carbon atom should form a ‘double’ bond with one its neighbours. This would give rise to two possible configurations, with different positioning of double bonds.

For our purposes, we can think of the double bond as an entangled pair of electrons. Each carbon has an extra dangling electron that entangles with a neighbour on the right or on the left. This is represented as a double line in the cartoon pictures. Now, neither of the two configurations above can explain the properties of benzene. For instance, each has three single bonds and three double bonds -- two types of bonds that should have different lengths. However, experiments show that the six carbon-carbon bonds in benzene are of the same length. Kekulé’s solution was that the molecule is simultaneously present in both configurations. In modern parlance, Benzene exists in a ‘superposition’ of the two configurations. In a way, Kekulé’s solution foresaw the discovery of quantum mechanics that was to come about seven decades later! Superposition is an essential quantum trait – two possibilities can exist simultaneously. If we take two consecutive carbon atoms, they form a single bond and a double bond – with both possibilities being realized simultaneously. In other words, the six electrons (of six carbon atoms) enter into a big entangled state with each electron losing its identity. However, it is not as if electrons break up into pairs that entangle. Rather, they break up into entangled pairs in two different ways at the same time. We may represent this as

This is a simple example of a ‘resonating valence bond’ system. The term ‘valence bond’ refers to an entangled pair of electrons. It is said to ‘resonate’ as the double bonds exist in two positions at the same time. Crudely (although inaccurately), the benzene molecule rapidly shuttles between the two configurations shown above, i.e., its double bonds rapidly bounce between two possible configurations. This evokes the image of a resonating needle or the surface of a drum.

Resonating valence bonds, or RVBs, have stoked the interest of chemists for more than a century. They are a centrepiece of organic chemistry and even molecular biology. In physics, they have been in the limelight for the past three decades or so. They are responsible for several interesting phenomena in quantum magnets and possibly in superconductors. More broadly, they serve as a building block to understand entanglement in several contexts. With such wide-ranging applications, it would be wonderful to synthesize resonating valence bonds and to probe them in a laboratory setting. However, this is a tremendous challenge. If you are a chemist, you may immediately point to a long list of organic compounds that contain resonating bonds. However, molecules are not a good setting to study entanglement. For instance, it is not easy to take a benzene molecule and to remove one double bond. Likewise, it is not possible to insert an additional double bond into a given molecule. These are the kind of operations that we require to understand the nature of entanglement. Some previous studies have created small RVBs with only four spins, see here and here. However, these studies follow complicated protocols that cannot be easily extended to make bigger RVBs.

We address this problem with a new approach that uses another essential aspect of quantum mechanics. Essentially, we trick a collection of particles into entangling with one another. They form RVBs in precise arrangements that we can pre-select. Our method is based on ‘wavefunction collapse’. This involves making a measurement on a quantum system. Let us say the system can be into two states, each of which gives a distinct outcome. The rules of quantum mechanics allow the system to be in both states at the same time. However, when we make a measurement with a large classical apparatus, we cannot find both outcomes at the same time. To bridge these worlds, the quantum system ‘collapses’. It falls into one of the two states at random. The measurement then gives the corresponding outcome. If the experiment is repeated by recreating the original ‘superposition’ state, it may now choose the other outcome. At each instance of the experiment, the system collapses into one of the two states. The outcomes are truly random and cannot be predicted beforehand.

In our scheme, we propose to take a collection of spins that are initially unentangled. Each spin is an object that can be in one of two states. We label the lower energy state as ‘down‘ and the higher energy state as ‘up‘. In technical terms, the spin refers to a two-level atom that can be made in the laboratory in a several ways. We start with several spins - some up and some down. We envision this ‘classical’ state as a superposition of several RVB states. Although each RVB component is entangled, their superposition is a simple, unentangled, classical state. We then make a measurement that has several possible outcomes. The measurement is designed such that each outcome comes from one of the RVB states. As a result, the measurement collapses the classical state of spins onto an RVB state.

To explain this, we start with the simplest example. We take two spins in a cavity. A cavity is a carefully designed box that can carry one 'mode' of light. It is akin to a guitar string that is tuned to a particular note. The spins interact with light as follows. Each spin can be in one of two states – up or down. An up-spin can emit a photon (a packet of light) and switch to the down state. A down-spin can absorb a photon and switch to the up state. A loose photon will leak out from within the cavity where it will be captured by a detector. There are several ways to realize this setup in experiments. Now, consider an initial state of the two spins, one up and one down. Will this pair collectively emit a photon? Surprisingly, there are two possibilities here. In the first, the up-spin will emit a photon that leaves the system to reach the detector. In this case, both spins will be left in the down state. The second possibility is much more interesting. The two spins can form a valence bond so that no photon is emitted! This is precisely analogous to the chemical bond in the Hydrogen molecule as we discussed above. This state cannot emit a photon as the two spins have ‘cancelled’ each other out. To sum up, we perform a measurement with two possible outcomes: the detector either sees a photon or not. If a photon is seen, both spins are left in the down state. However, if no photon is seen (after waiting for a sufficiently long period), the spins have collapsed onto an entangled chemical-bond-like state.

This example can be easily generalized. Consider an initial state with one up-spin and three down-spins. This state can either emit a photon or not. If it emits a photon, it must necessarily be emitted by the the up-spin as it transitions into a down-spin. As a result, the system is left with all spins pointing down. However, there is an alternative where no photon is emitted. In this case, the up-spin must form a valence bond with a down spin so that the two spins cancel each other out. However, there are three choices here. The up-spin can form a bond with any of the three spins that were initially pointing down. Remarkably, all of these possibilities are realized simultaneously with a three-fold superposition! This can be visualized as a valence bond that resonates (or rapidly shuttles) among three positions.

The strength of our proposal is that it can be generalized to complex RVBs with several valence bonds. For instance, we can initially take three up spins and three down spins. We can then make a measurement to see if photons are emitted. One possible outcome is that no photon will be emitted. In this case, the spins are left in a complex RVB state with three valence bonds that resonate.

We have suggested a reliable route to creating entangled states in the laboratory. There are several interesting consequences that come out from our work. For instance, we can increase or decrease the entanglement at will, e.g., by creating states with more valence bonds or fewer. This can be further used to study deep problems that emerge in the context of superconductivity or in quantum information storage. We hope experimental groups will take up our proposal and conjure up resonance!

Our papers on this topic:

• Generating resonating-valence-bond states through Dicke subradiance

Physical Review A 96, 033829 (2017); arXiv version here.

• Resonating valence bonds and spinon pairing in the Dicke model

Scipost Physics 4, 044 (2018)