## Quantum Heating

Mesoscopic wires exhibit peculiar properties at low temperatures. Their electric conductance can show plateaus at evenly spaced values, which reflects the sequential opening of “quantum transport channels,” each of which can only carry a finite amount of charge or heat. Whereas the step size for the electric conductance depends on the type of the particle carrying the charge, for heat conduction this “quantum” is universal. **Jezouin et al.** (p. 601, published online 3 October; see the Perspective by

**Sothmann and Flindt**) measured the quantum of heat conduction through a single electronic channel by comparing the amount of heat needed to heat a small metal plate to a constant temperature, while varying the number of electronic channels through which the heat was dissipated from the plate. Encouragingly, the measurement was in agreement with the theoretical prediction.

## Abstract

Quantum physics predicts that there is a fundamental maximum heat conductance across a single transport channel and that this thermal conductance quantum, *G _{Q}*, is universal, independent of the type of particles carrying the heat. Such universality, combined with the relationship between heat and information, signals a general limit on information transfer. We report on the quantitative measurement of the quantum-limited heat flow for Fermi particles across a single electronic channel, using noise thermometry. The demonstrated agreement with the predicted

*G*establishes experimentally this basic building block of quantum thermal transport. The achieved accuracy of below 10% opens access to many experiments involving the quantum manipulation of heat.

_{Q}The transport of electricity and heat in reduced dimensions and at low temperatures is subject to the laws of quantum physics. The Landauer formulation of this problem (*1*–*3*) introduces the concept of transport channels: A quantum conductor is described as a particle waveguide, and the channels can be viewed as the quantized transverse modes. Quantum physics sets a fundamental limit to the maximum electrical conduction across a single electronic channel. The electrical conductance quantum *G _{e}* =

*e*

^{2}/

*h*, where

*e*is the unit charge and

*h*is the Planck constant, was initially revealed in ballistic one-dimensional (1D) constrictions (

*4*,

*5*). However, different values of the maximum electrical conductance are observed for different types of charge-carrying particles. In contrast, for heat conduction the equivalent thermal conductance quantum (which sets the maximum thermal conduction across a single transport channel,

*k*

_{B}being the Boltzmann constant and

*T*the temperature) is predicted to be independent of the heat carrier statistics, from bosons to fermions, including the intermediate “anyons” (

*6*–

*16*). In electronic channels, which carry both an electrical and thermal current, the predicted ratio between

*G*and

_{Q}*G*verifies and extends the Wiedemann-Franz relation down to a single channel (

_{e}*8*,

*9*). In general, the universality of

*G*, together with the deep relationship between heat, entropy, and information (

_{Q}*17*), points to a quantum limit on the flow of information through any individual channel (

*6*,

*15*).

The thermal conductance quantum has been measured for bosons, in systems with as few as 16 phonon channels (*18*, *19*), and probed at the single-photon channel level (*20*, *21*). For fermions, heat conduction was shown to be proportional to the number of ballistic electrical channels (*22*, *23*). In (*22*), the data were found compatible, within an order of magnitude estimate, to the predicted thermal conductance quantum, whereas (*23*) demonstrated more clearly the quantization of thermal transport, but *G _{Q}* was not accessible by construction of the experiment.

We have measured the quantum-limited heat flow across a single electronic channel using the conceptually simple approach depicted in Fig. 1A. A micrometer-sized metal plate is electrically connected by an adjustable number *n* of ballistic quantum channels to a cold bath at temperature *T*_{0}. Electrons in the small plate are heated up with a well-known Joule power (*J _{Q}* in Fig. 1A), and the resulting increased electronic temperature

*T*

_{Ω}is measured by direct thermometry based on noise measurements. Heat balance implies that the injected Joule power is compensated by the overall outgoing heat current

where is the electronic heat flow across the *n* ballistic quantum channels. The flow is an additional contribution, here attributed to the transfer of heat from the hot electrons toward the cold phonon bath in the plate (and thus independent of *n*). The heat flow across a single ballistic electronic channel is then directly given by how much *J _{Q}* is increased to keep

*T*

_{Ω}constant when one additional electronic channel is opened. The quantum-limited heat flow for a single electronic channel connecting two heat baths at

*T*

_{Ω}and

*T*

_{0}reads (

*8*,

*9*)

The quadratic temperature dependence reflects the fact that the temperature sets both the average energy of electronic excitations, as well as their number, the latter being proportional to the energy bandwidth.

The actual sample, displayed in Fig. 1B, was cooled down to mK in our experiment. The noise thermometry was performed with ultrasensitive cryogenic electronics based on a homegrown high–electron-mobility transistor (*24*).

The ballistic quantum channels are formed in a high-mobility Ga(Al)As 2D electron gas by the field-effect tuning of two quantum point contacts (QPCs), labeled QPC_{1} and QPC_{2} in Fig. 1B. The potentials applied to the metal split gates (colored yellow in Fig. 1B) are set to fully transmit *n* electronic channels, in which case the QPC electrical conductances display clear plateaus [see fig. S5 in (*25*)]. The measured conductances *n*_{1}*e*^{2}/*h* and *n*_{2}*e*^{2}/*h* of, QPC_{1} and QPC_{2}, respectively, correspond to *n* = *n*_{1} + *n*_{2}. For fully transmitted channels, the electron-photon coupling observed in the same sample for the case of partially transmitted channels (*26*) vanishes.

The heated-up metal plate, colored brown in Fig. 1B, is a micrometer-sized ohmic contact (*25*), which is electrically connected to cold electrodes located further away exclusively through the two QPCs. To approach the quantum limit of heat flow per channel, the electrical connection between the plate and the 2D electron gas located 94 nm below the surface must have a negligibly low resistance compared to *h*/*e*^{2}. Moreover, the heated-up electrons must dwell in the plate for a time longer than the electron-electron energy exchange time, in order to relax toward a quasi-equilibrium situation characterized by a hot Fermi distribution at *T*_{Ω}. These two conditions set the minimum size of the ohmic contact (*25*); however, the ohmic contact must be small enough to minimize the heat transfer toward phonons, which is proportional to volume. The sample was optimized to fulfill these antagonistic requirements, achieving a negligibly small contact resistance, a typical dwell time in the 10 μs range, and a dominating electronic heat flow for mK.

The sample was subjected to a strong perpendicular magnetic field in order to enter the integer quantum Hall effect (QHE) regime, at filling factors ν = 3 or ν = 4. In this regime, the current flows along the sample edges in so-called edge channels with a unique propagation direction (continuous red lines with arrows in Fig. 1B). One motivation for performing the experiment in this regime is the spatial separation between incoming and outgoing edge channels away from the QPC, which enables use of the large metal electrodes located further away as ideal cold reservoirs. Furthermore, it is easier to tune the QPCs to a discrete set of fully open channels (*25*), and the spin degeneracy is broken so that the electronic channels can be opened one at a time. Finally, the QHE regime allows for a simple implementation of the noise thermometry.

The injected Joule power *J _{Q}* was generated with an applied DC current (Fig. 1B) partly transmitted across the

*n*

_{1}ballistic channels of QPC

_{1}into the plate (

*25*).

The resulting increase *T*_{Ω }– *T*_{0} of the electronic temperature in the plate was determined from the increase Δ*S*_{I} in the measured spectral density of the current noise along the outgoing edge channels (*25*, *27*, *28*)

The raw measurements of excess current noise Δ*S*_{I} versus applied DC current for open electronic channels at ν = 3 are shown as symbols in the inset of Fig. 2. Here *n* = 2 corresponds to (*n*_{1}, *n*_{2}) = (1, 1), *n* = 3 is the average over the two equivalent configurations (1, 2) and (2, 1) [see fig. S6 in (*25*)], and *n* = 4 corresponds to (2, 2). The displayed data are measured on the top left electrode, behind QPC_{2}; the same excess noise, within 2%, was measured with another amplification chain on the bottom right electrode, behind QPC_{1} (*25*).

The main panel of Fig. 2 shows these data recast as the measured electronic temperature in the micrometer-sized plate *T*_{Ω} versus the injected Joule power *J _{Q}*. The base temperature

*T*

_{0}= 24 mK was obtained separately, from a noise thermometry performed at ν = 3 during the same experimental run (

*25*). At fixed

*T*

_{Ω}, the distinct increase in

*J*as the number

_{Q}*n*of ballistic electronic channels is incremented by one directly corresponds to the heat flow across an individual electronic channel. We focus here on the low injected power regime fW, where the electronic heat flow is the most important contribution at

*n*= 4 (

*25*).

We now separate the electronic heat flow from the additional, a priori unidentified, contribution , which is independent of the number *n* of open electronic channels. For this purpose, we use as a reference the *n* = 4 data corresponding to the most robust QPC plateaus *n*_{1} = *n*_{2} = 2.

First, we show that important information can already be extracted at *n* = 4 within a model-dependent approach. The *n* = 4 reference data are fitted with the standard expression for electron-phonon cooling in diffusive metals (*29*–*31*) with ΣΩ a free parameter; for we used the predicted power law (see Eq. 2), with α_{4} a free parameter. The fit is shown as a continuous yellow line in Fig. 2. The electron-phonon coupling parameter extracted from the fit is a very typical value for similar metals (*29*–*31*) given the micrometer-sized ohmic contact volume . The extracted electronic heat flow is found to be , which is within 5% of the theoretically predicted value given by Eq. 2. The same ΣΩ and electronic heat current are obtained by repeating this analysis on the *n* = 4 data at the higher temperature *T*_{0} = 40 mK, whereas a relatively small difference of about 20% is seen at filling factor ν = 4 [see figs. S8 and S9 in (*25*)].

Second, following the model-free approach described earlier, we extract the amount of heat flowing across the additional (*n* – 4) ballistic electronic channels by subtracting from the measured *J _{Q}* with

*n*open channels the

*n*= 4 reference signal. At

*n*> 4 (

*n*< 4), channels are opened (closed) with respect to the reference configuration. Given the extremely accurate fit described above, within experimental error bars, we choose to subtract the fit function, instead of using an arbitrary interpolation function between the measured

*n*= 4 data points. The extracted variations of the electronic heat currents are plotted as symbols in Fig. 3A as a function of the squared temperature , for the data at ν = 3 and

*T*

_{0}= 24 mK. Similar data obtained for

*T*

_{0}= 22.5 mK at the different filling factor ν = 4 and for up to

*n*= 6 ballistic electronic channels (Fig. 3B) show larger scatter simply because of a lower experimental accuracy, due to the less favorable current-voltage conversion at this filling factor and a smaller acquisition time per point. We find that is proportional to , as expected from theory (Eq. 2).

We now compare the extracted electronic heat currents with the quantitative prediction Eq. 2 for the quantum-limited heat flow. The experimentally extracted are in good agreement with the theoretical predictions shown as continuous lines in Fig. 3, A and B. We then extract a quantitative experimental value for the quantum-limited heat flow by fitting the data using the predicted and observed functional , with the only free parameter. Taking altogether the set of normalized slopes obtained within the model-free approach at ν = 3 and ν = 4, we find

(4)in agreement, at our experimental accuracy, with the theoretical prediction for the quantum-limited heat flow across a single channel, and therefore with the predicted value given by the thermal conductance quantum. The displayed 7% uncertainty is the standard error on the mean value obtained from the six values , each weighted by the corresponding number of electronic channels (*25*). This uncertainty ignores systematic sources of error, e.g., on the calibrated gain of the amplification chain (*25*). The accuracy can be improved by including the values of α_{4} obtained at ν = 3 and ν = 4 within the model-dependent approach detailed earlier. Figure 3C displays as symbols the full electronic heat current factors versus *n*, with the corresponding theoretical predictions falling on the continuous line. The same statistical analysis on the eight values of {α* _{n}*/

*n*} yields (

*25*).

The present experiment demonstrates that the quantum-limited heat flow across a single electronic channel, which sets the scale of quantum interference effects, is now attainable at a few percent accuracy level. This opens access to many studies in the emergent field of quantum heat transport (*32*, *33*), such as quantum phase manipulation of heat currents.

## Supplementary Materials

## References and Notes

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**Acknowledgments:**We gratefully acknowledge the contribution of V. Andreani to the noise measurement setup. This work was supported by the European Research Council (ERC-2010-StG-20091028, no. 259033).