There is a fundamental quantum limit to heat flow, just as there is to electric current. This limit is independent of what carries the heat, and could also have a role in an unexpected quarter: information theory.
In the past 20 years, physicists have learnt a tremendous amount about the transport of matter and energy through devices small enough for quantum effects to come into play. One surprising fact that has emerged is that the rates of transport in such devices, expressed for example by their electronic or thermal conductance, have simple quantum-mechanical limits. On page 187 of this issue, Meschke et al.1 extend this principle to heat conduction by photons. Although the result will certainly have practical ramifications for the engineering of ultra-sensitive detectors, sensors and microelectronic refrigerators, the physics behind it hints at more fundamental truths.
The experimental trail leading to this point starts in 1988, when two groups independently demonstrated2,3 the quantization of electron transport through a single 'ballistic' channel in which the electron's movement is impeded only negligibly by scattering. The conductance of such a channel (the inverse of its resistance) was found to vary in discrete steps of size 2e2/h, where e is the electron charge and h is Planck's constant. This quantum of electrical conductance has a value of around 7.8210−5 Ω−1 in conventional units. In small systems, such as a single channel, that have only one way for electrons to propagate (one 'mode of conduction'), it also represents the maximum value of the conductance.
This first demonstration of quantized electronic transport was a tour de force of cryogenics, microelectronics and materials science. Since then, the phenomenon has been observed in thousands of situations: in larger, mesoscopic devices, in atomic point contacts and break junctions, in single molecules, nanotubes and so on. It has become an example of what physicists love, and strive for, most: a prediction of complete generality. Without specifying any of the material details, such as the structure of electronic bands or the density, one can estimate the electronic conductance of any quantum-scale device.
Experimental evidence that the transport of thermal energy is also quantized dates back to 1999. Then, Michael Roukes and I demonstrated4 that the movement of heat through discrete, freely suspended mechanically vibrating channels approached a previously predicted maximum rate5 for each independent vibrational mode — longitudinal, transverse and torsional — of the structure. This universal rate is the quantum of thermal conductance, GQ, and comes in units of π2kB2T/3h. Here, kB is Boltzmann's constant and T is the prevailing temperature; the exact maximum rate of heat transport through a quantum device therefore increases linearly with temperature.
The mechanical vibrations that transported the heat in this case are called phonons, and are analogous to the quantized vibrations of the electromagnetic field — better known as photons. With photons, the channels of conduction are the propagating electromagnetic modes of a transmission line or of an optical waveguide such as a single-mode fibre-optic cable.
The advance made by Meschke and colleagues1 is to demonstrate for the first time the quantized conduction of heat by photons. In their experiment, two microscopic electronic resistors exchanged heat through random thermal voltage fluctuations transmitted through two superconducting wires. With some clever use of further superconducting circuitry, the authors could switch the electrical conduction channel on and off, and thus expose the thermal connection between the two resistors brought about by the photons emitted and absorbed by them both. The authors show that the rate of heat exchange between the two resistors in this case is given simply by GQ.
Thus, like the quantum of electrical conductance, GQ is very general, and independent of the nature of the material connection between two heat reservoirs. Nor does it depend on the type of particle that carries the heat: the same GQ is the limit for the conductance of heat by electrons, phonons, photons, gravitons, you-name-it-ons. Thus, GQ is universal in a much deeper sense6,7,8 than the quantum of electrical conductance, which depends on the quantum statistics of the particles; that is, whether they are bosons, fermions or something in between, such as the 'anyons' that crop up in certain two-dimensional systems.
A further connection that might seem somewhat surprising at first glance is that the quantum limit for heat transport is intimately related to the maximum classical information capacity of a single quantum channel (Fig. 1). This connection rests on a deep relationship between information and entropy, which was established by Claude Shannon in 1948 (ref. 9) and has been investigated by many authors over four decades10. One of the more interesting treatments5 made explicit the connection between maximum heating and cooling rates through both fermionic and bosonic channels on the one hand, and maximum data rates on the other. Although the maximum data rates of fermionic and bosonic channels differ, there has been speculation11 that a deeper underlying principle might exist that could be used to extract a universal statement about information capacity from the truly universal nature of thermal transport in one dimension. This connection reinforces the emerging view of the basic nature of information in fundamental physics12.
Cast in this light, the work of Meschke and colleagues1 is of more fundamental importance than just investigating the behaviour of microscopic resistors exchanging heat. What they have demonstrated is two resistors babbling to each other using thermal voltage noise. As these resistors are near-perfect 'black-body' radiators, they emit and absorb radiation fields of maximum entropy, and so — according to Shannon's work9 — maximum information content. The authors have proved that this information can be carried by particles of very different natures: photons do just as well as phonons. In this spirit, I look forward to the day when a measurement of the thermal conductance through a channel of anyons in a quantum Hall fluid demonstrates the extraordinary, universal nature of GQ.
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Nature Nanotechnology (2008)