Recent progress on micro-and nanometer-scale manipulation has opened the possibility to probe systems small enough that thermal fluctuations of energy and coordinate variables can be significant compared with their mean behavior. We present an experimental study of nonequilibrium thermodynamics in a classical two-state system, namely, a metallic single-electron box. We have measured with high statistical accuracy the distribution of dissipated energy as single electrons are transferred between the box electrodes. The obtained distributions obey Jarzynski and Crooks fluctuation relations. A comprehensive microscopic theory exists for the system, enabling the experimental distributions to be reproduced without fitting parameters.

Recently, the fundamental laws of thermodynamics have been reconsidered for small systems. The discovery of the fluctuation relations [1][2][3][4][5] has spurred theoretical [1,6,7,[9][10][11][12][13] and experimental [14-23] studies on thermodynamics of systems with few degrees of freedom. The concept of entropy production has been extended to the microscopic level by considering stochastic trajectories of a system coupled to a heat bath. However, the experimental observation of the microscopic entropy production remains elusive. We measure distributions of the microscopic entropy production in a single-electron box consisting of two islands with a tunnel junction. The islands are coupled to separate heat baths at different temperatures, maintaining a steady thermal nonequilibrium. As Jarzynski equality between work and free energy is not applicable in this case, the entropy production becomes the relevant parameter. We verify experimentally that the integral and detailed fluctuation relations are satisfied. Furthermore, the coarse-grained entropy production [10-12, 23, 24] from trajectories of electronic transitions is related to the bare entropy production by a universal formula. Our results reveal the fundamental roles of irreversible entropy production in non-equilibrium small systems.Entropy production is a hallmark of irreversible thermodynamic processes. The concept of a stochastic microscopic trajectory allows one to define entropy for small systems [1]. However, such trajectories depend on the scale of observation. If one only accesses mesoscopic degrees of freedom, one observes coarse-grained trajectories of mesoscopic states. The corresponding entropy production then differs from the bare entropy production without coarse-graining. In fact, it has been recently shown that coarse-graining of the slow background degrees of freedom for stochastic dynamics may actually lead to a modification of the fluctuation relations for entropy [23]. To clarify the concept of microscopic entropy production in non-equilibrium, accurate measurements are needed for systems, where the concepts of stochastic dynamics and time scale separation between the system and the heat bath are well-defined.A single-electron box (SEB) device at low temperatures is an excellent test bench for thermodynamics in small systems [22,25,26].The SEB employed here is shown in Fig. 1(a). The electrons in the normal-metal copper island (N) can tunnel to the superconducting Al island (S) through the aluminum oxide insulator (I). The sample fabrication [27] methods are similar to those in Ref. [22], but the design is different in that the S side of the junction does not overlap with the normal conductor in order to intentionally weaken the relaxation of energy in S [28]. Moreover, the main results in Ref. [22] were extracted from measurements at the temperature of 220 mK, whereas these measurements are conducted at 140 mK. Lower temperature further weakens the relaxation significantly [28], leading to and elevated temperature in S. We denote by n ...

We will discuss the relevant conditions to observe a critical tunneling current [New J. Phys. 10, 045018 (2008)] in electron-double layer systems at a total filling factor of one and find they are related to the effective layer separation and the temperature. Our studies suggest that the intensity of the critical tunneling behavior is also directly linked to the area of the sample.PACS numbers:

The effect of tunneling on the transport properties of quantum Hall double layers in the regime of the excitonic condensate at total filling factor one is studied in counterflow experiments. If the tunnel current I is smaller than a critical I C , tunneling is large and is effectively shorting the two layers. For I > I C tunneling becomes negligible. Surprisingly, the transition between the two tunneling regimes has only a minor impact on the features of the filling-factor one state as observed in magneto-transport, but at currents exceeding I C the resistance along the layers increases rapidly.

We have observed a few distinct anomalous avoided level crossings and voltage dependent transitions in the excited state spectrum of an Al/AlOx/Al Cooper-pair box (CPB). The device was measured at 40 mK in the 15 - 50 GHz frequency range. We find that a given level crosses the CPB spectrum at two different gate voltages; the frequency and splitting size of the two crossings differ and the splitting size depends on the Josephson energy of the CPB. We show that this behavior is not only consistent with the CPB being coupled to discrete charged "two-level" quantum systems which move atomic distances in the CPB junctions but that the spectra provide new information about the fluctuators, which is not available from phase qubit spectra of anomalous avoided levels. In particular by fitting a model Hamiltonian to our data, we extract microscopic parameters for each fluctuator, including well asymmetry, tunneling amplitude, and the minimum hopping distance for each fluctuator. The tunneling rates range from less than 3.5 to 13 GHz, which represent values between 5% and 150% of the well asymmetry, and the dipole moments give a minimum hopping distance of 0.3 to 0.8 Anstrom. We have also found that these discrete two-level systems have a pronounced effect on the relaxation time (T1) of the quantum states of the CPB and hence can be a source of dissipation for superconducting quantum bits