Direct entropy measurement in a mesoscopic quantum system


The entropy of an electronic system offers important insights into the nature of its quantum mechanical ground state. This is particularly valuable in cases where the state is difficult to identify by conventional experimental probes, such as conductance. Traditionally, entropy measurements are based on bulk properties, such as heat capacity, that are easily observed in macroscopic samples but are unmeasurably small in systems that consist of only a few particles1,2. Here, we develop a mesoscopic circuit to directly measure the entropy of just a few electrons, and demonstrate its efficacy using the well-understood spin statistics of the first, second and third electron ground states in a GaAs quantum dot3,4,5,6,7,8. The precision of this technique, quantifying the entropy of a single spin-1/2 to within 5% of the expected value of kB ln 2, shows its potential for probing more exotic systems. For example, entangled states or those with non-Abelian statistics could be clearly distinguished by their low-temperature entropy9,10,11,12,13.

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Fig. 1: Measurement protocol.
Fig. 2: Entropy measurement for a single spin-\(1/2\).
Fig. 3: Magnetic field dependence.
Fig. 4: Entropic signature of a singlet–triplet crossing.


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The authors acknowledge J. Martinis for a helpful discussion on the interpretation of our measurements. N.H., C.O., S.L., M.S. and J.F. were supported by the Canada Foundation for Innovation, the National Science and Engineering Research Council, CIFAR and SBQMI. S.F., G.C.G. and M.M. were supported by the US DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering award no. DE-SC0006671, with additional support from Nokia Bell Laboratories for the MBE facility gratefully acknowledged.

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N.H. and C.O. fabricated the mesoscopic device. GaAs heterostructures and their characterization were provided by S.F., G.C.G. and M.M. S.L. and M.S. worked on early versions of the experiment and provided helpful discussion. N.H. performed measurements and analysed data. The manuscript was written by N.H. and J.F. with additional feedback from all authors.

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Correspondence to Nikolaus Hartman or Joshua Folk.

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Hartman, N., Olsen, C., Lüscher, S. et al. Direct entropy measurement in a mesoscopic quantum system. Nature Phys 14, 1083–1086 (2018).

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