The Arrhenius equation, which has accurately described rates of chemical reactions for more than a century, may have to be tweaked for the quantum realm.

A double potential energy well with quantum wave functions on a ladder of different energy levels N. E. Frattini et al. |

A 135-year-old chemistry law is getting a quantum makeover. An experiment with a single quantum bit, or a qubit, has uncovered that the Arrhenius equation, which describes how reaction rates relate to temperature, must be modified to apply in the quantum realm.

Rodrigo Cortiñas at Yale University says he never doubted that the Arrhenius equation would translate directly to a quantum experiment – until he and his team found otherwise. “This was really an accident,” he says.

Cortiñas and his colleagues built a qubit – the basic building block of quantum computers – from a small superconducting circuit, which they could control with microwaves. They made the current in the circuit oscillate within a so-called potential energy well. It’s a bit like a cherry tomato rolling back and forth at the bottom of a deep salad bowl, except that in their setup, there were two potential wells. Because of the quantum phenomenon of tunnelling, in which quantum objects can breach normally impassable energy barriers, disturbances in the system could make the qubit unexpectedly move from one of the wells to the next.

Nick Frattini at Nord Quantique in Canada, who worked on the experiment, says that the team wanted to learn how to stop this, and the Arrhenius equation should have been helpful. According to this law, increasing the energy difference between the two wells should exponentially decrease how often the qubit tunnels between them. This means that plotting this energy difference and the rate of tunnelling should produce a smooth curve. Instead, the researchers’ data showed a jagged line resembling a staircase.

Eventually, they realised they were seeing a consequence of the fundamental quantum constraints on which energies the qubit could have. In the macroscopic world, the Arrhenius equation assumes that any energy level is allowed, but quantum theory only allows a set of special values – a ladder of discrete energy levels. For quantum objects like a qubit, Arrhenius’s law must reflect this structure.

So, each step in the researchers’ data corresponded to a specific qubit energy level. Cortiñas says it is technically difficult to experiment with a qubit that can be at any rung of the energy ladder, and experiments typically only explore the first two, which has prevented the full effect from showing up in previous work.

Jonathan Friedman at Amherst College in Massachusetts studied a similar stepwise behaviour in how particles change their quantum states in magnets. He says that using the conventional, smooth version of the Arrhenius equation in systems susceptible to quantum effects can lead to large errors. For objects like magnets, superconductors and qubits, you may expect that you can create a big change in their quantum states by making a small change – for example in temperature or magnetic field – but if you hit a step on the Arrhenius staircase, next to no change will actually occur, he says.

For Cortiñas and Frattini, the accidental discovery also helped them improve their qubit as they intended. They can now make the qubit stay in one well 10 times longer than before, a kind of reliability that is necessary before it can be connected to other qubits to make a working quantum computer.

Journal reference

Physical Review X DOI: 10.1103/PhysRevX.14.031040