A theoretical framework for creating practical quantum thermometers that are optimized for taking the temperature of tiny structures as varied as living cells and quantum bits has been created by physicists in Spain and the UK. The work identifies an important trade-off between the precision and operating range of such quantum probes, and also describes how they could be used in the lab.

The temperature of an everyday object can easily be taken by putting it into contact with a thermometer. But this is much trickier when the object is extremely cold, extremely small or both. This is because heat will be exchanged between the thermometer and the object, ultimately affecting the measurement. The growing interest in creating nanometre-sized quantum devices for use in computing, metrology and other applications means that physicists must gain an understanding of the thermodynamics of these systems. This requires the ability to measure the temperature of such devices to nanometre precision – an ability that could also be also useful for taking the temperature of living cells.

Gold and diamond thermometers

In 2013, for example, Mikhail Lukin of Harvard University and colleagues used a tiny diamond (just 100 nm in diameter) to measure the temperature within a living cell at a spatial resolution of 200 nm (see "Nanodiamond thermometer takes temperature of biological cells"). Other researchers have shown that quantum dots – tiny pieces of semiconductor – can be used to measure the temperature of electrons in samples cooled below 1 K.

These tiny probes can be thought of as "quantum thermometers" because they sense the effect of heat energy on the fragile quantum states of an otherwise isolated system. Such systems are very sensitive to external stimuli such as heat, which makes them very good temperature probes. However, their development is in its infancy, and much work must be done to create quantum probes that are optimized for use in specific situations.

Now, Luis Correa, Mohammad Mehboudi and Anna Sanpera of the Universitat Autònoma de Barcelona, along with Gerado Adesso of the University of Nottingham, have done a theoretical investigation of what types of quantum systems make the best temperature probes. They have also done calculations to find out what are the best ways of operating these hypothetical devices.

Fully thermalized

The team looked at "fully thermalized thermometers", which are devices that are in thermal equilibrium with the system they are measuring. This is much like a traditional mercury or alcohol thermometer immersed in a beaker of water.

The team's calculations revealed that the probe's heat capacity – the amount of energy required to raise its temperature by one degree – is an important parameter in the design of a quantum thermometer. The heat capacity is related to the number of different ways that the system can be excited from its ground state – its degrees of freedom – and the calculations show that increasing the number of degrees of freedom boosts the precision of the measurement.

However, the team found that increasing the degrees of freedom also narrows the range of temperatures in which the probe is effective – which means that designing a practical probe would involve a trade-off between precision and range.

Range versus precision

In the case of the Lukin's diamond thermometer, thermal energy can excite the ground state of its atom-like system into one of two degenerate excited states (degrees of freedom). If more degenerate excited states were available, then the diamond thermometer would be more precise, but would function over a narrower range of temperatures. This trade-off suggests that other quantum systems that do not have a large number of degenerate excited states – such as the harmonic oscillator – could be used as practical temperature probes that work over wide temperature ranges.

The team points out that an effective way of probing the temperature of a tiny object would involve first using a low-precision, wide-range probe to make a rough measurement. Then successively more precise probes would be used to reduce the uncertainty in the measurement.

The team also looked at "partially thermalized thermometers" that are able to take the temperature of an object without being at thermal equilibrium with it. Such a probe would be useful if the system to be measured was unstable and existed for a time that is much shorter than the time it would take for probe and object to equilibrate. In this case, they found that the probe should be at its lowest possible temperature when applied. They also found that while the precision of a partially thermalized measurement increases with the number of degrees of freedom, the temperature range of the probe remains constant.

Correa and colleagues hope that their theoretical framework will help to improve experiments that measure heat dissipation in nanometre-sized circuits, and also guide scientists in developing new ways to study the transfer of heat in living cells.

The research is described in Physical Review Letters.