The key equation to drive quantum technologies

As the size of modern technology shrinks to the nanoscale, strange quantum effects—such as quantum tunneling, superposition, and entanglement—become prominent. This opens the door to a new era of quantum technologies, where quantum effects can be harnessed. Many everyday technologies use feedback control routinely; an important example is the pacemaker, which must monitor the user’s heart rate and apply electrical signals to control it, only when necessary. But physicists do not yet have an equivalent understanding of feedback control at the quantum level. Now, physicists have developed a “master equation” that will help engineers understand reactions at the quantum scale. Their results are published in the journal Physical review papers.

“It is vital to investigate how feedback control can be used in quantum technologies in order to develop efficient and fast methods for controlling quantum systems so that they can be driven in real time and with high precision,” says co-author Björn Annby-Andersson, a quantum physicist at Lund University in Sweden.

An example of a crucial feedback control process in quantum computing is quantum error correction. A quantum computer encodes information on physical qubits, which can be photons of light or atoms, for example. But the quantum properties of qubits are fragile, so the encoded information is likely to be lost if the qubits are disturbed by vibrations or fluctuating electromagnetic fields. This means that physicists must be able to detect and correct such errors, for example using feedback control. This error correction can be implemented by measuring the state of the qubits and, if a deviation from what is expected is detected, by applying feedback to correct it.

But controlling reactions at the quantum level presents unique challenges, precisely because of the fragility that physicists are trying to mitigate. This delicate nature means that even the feedback process itself can destroy the system. “It is necessary to only weakly interact with the measured system, preserving the properties we want to exploit,” says Annby-Andersson.

Therefore, it is important to develop a thorough theoretical understanding of quantum feedback control, to establish its fundamental limits. But most existing theoretical models of quantum feedback control require computer simulations, which usually provide only quantitative results for specific systems. “It is difficult to draw general, qualitative conclusions,” says Annby-Andersson. “The few models that can provide qualitative understanding are only applicable to a narrow class of feedback-controlled systems—this type of feedback is commonly referred to as linear feedback.”

‘Pen and paper’

Annby-Andersson and his colleagues have now developed a key equation, called the Fokker-Planck Quantum Equation, that enables physicists to trace the evolution of any quantum system by controlling feedback over time. “The equation can describe scenarios that go beyond linear feedback,” says Annby-Andersson. “In particular, the equation can be solved with pen and paper, rather than relying on computer simulations.”

The team tested their equation by applying it to a simple reaction model. This confirmed that the equation provides physically sensible results and also demonstrated how energy can be obtained in microscopic systems using feedback control. “The equation is a promising starting point for future studies of how energy can be manipulated with the help of information at a microscopic level,” says Annby-Andersson.

The team is now investigating a system that uses feedback to manipulate energy in “quantum dots” – tiny semiconductor crystals only billionths of a meter across. “An important future direction is to use the equation as a tool for inventing new feedback protocols that can be used for quantum technologies,” says Annby-Andersson.

Provided by the Fundamental Questions Institute