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arXiv:1201.6238 · Physical Review B

Quantum current noise from a Born-Markov master equation

Peter Kirton, Andrew D. Armour, Manuel Houzet, F. Pistolesi

Published Aug 27, 2012• 9 citations
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Abstract

Quantum coherent oscillations in the electric charge passing through a mesoscopic conductor can give rise to a current noise spectrum which is strongly asymmetric in frequency. The asymmetry reveals the fundamental difference between quantum and classical fluctuations in the current. We show how the quantum current noise can be obtained starting from a Born-Markov master equation, an approach which is applicable to a wide class of systems. Our method enables us to analyze the rich behavior of the current noise associated with the double Josephson quasiparticle resonance in a superconducting single-electron transistor (SSET). The asymmetric part of the noise is found to be strongly dependent on the choice of operating point for the SSET and can be either positive or negative. Our results are in good agreement with recent measurements.

DOI
10.1103/physrevb.86.081305

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Quantum coherent oscillations in the electric charge passing through a mesoscopic conductor can give rise to a current noise spectrum which is strongly asymmetric in frequency. The asymmetry reveals the fundamental difference between quantum and classical fluctuations in the current. We show how the quantum current noise can be obtained starting from a Born-Markov master equation, an approach which is applicable to a wide class of systems. Our method enables us to analyze the rich behavior of the current noise associated with the double Josephson quasiparticle resonance in a superconducting single-electron transistor (SSET). The asymmetric part of the noise is found to be strongly dependent on the choice of operating point for the SSET and can be either positive or negative. Our results are in good agreement with recent measurements.
We analyze the charge dynamics of a superconducting single-electron transistor (SSET) in the regime where charge transport occurs via Cooper-pair resonances. Using an approximate description of the system Hamiltonian, in terms of a series of resonant doublets, we derive a Born-Markov master equation describing the dynamics of the SSET. The average current displays sharp peaks at the Cooper-pair resonances and we find that the charge noise spectrum has a characteristic structure which consists of a series of asymmetric triplets of peaks. The strongest feature in the charge noise spectrum is the triplet of peaks centered at zero frequency which has a peak spacing equal to the level separation within the doublets and is similar to the triplet in the spectrum of a driven, damped, two-level system. We also explore the backaction that the SSET charge noise would have on an oscillator coupled to the island charge, measurement of which provides a way of probing the charge noise spectrum.
We analyze the current and zero-frequency current noise properties of a superconducting single-electron transistor (SSET) coupled to a resonator, focusing on the regime where the SSET is operated in the vicinity of the Josephson quasiparticle resonance. We consider a range of coupling strengths and resonator frequencies to reflect the fact that in practice the system can be tuned to quite a high degree with the resonator formed either by a nanomechanical oscillator or a superconducting stripline fabricated in close proximity to the SSET. For very weak couplings the SSET acts on the resonator like an effective thermal bath. In this regime the current characteristics of the SSET are only weakly modified by the resonator. Using a mean-field approach, we show that the current noise is nevertheless very sensitive to the correlations between the resonator and the SSET charge. For stronger couplings, the SSET can drive the resonator into limit-cycle states where self-sustained oscillation occurs and we find that regions of well-defined bistability exist. Dynamical transitions into and out of the limit-cycle state are marked by strong fluctuations in the resonator energy, but these fluctuations are suppressed within the limit-cycle state. We find that the current noise of the SSET is strongly influenced by the fluctuations in the resonator energy and hence should provide a useful indicator of the resonator's dynamics.
The subject of this thesis is the study of a particular open quantum system consisting of a resonator coupled to a superconducting single electron transistor (SSET). The theoretical model we use is applicable to both mechanical and superconducting stripline resonators leading to a large parameter regime that can be explored. The SSET is tuned to the Josephson quasi-particle resonance, in which the transport occurs via Cooper pairs coherently tunnelling across one junction followed by the incoherent tunnelling of quasi-particles across the other. The SSET can be thought of as an artificial atom since it has a similar energy level structure and transitions to an atom. We investigate to what extent the current and current noise through the SSET can be used to infer the state of the resonator. In order to carry out these investigations we describe the system with a Born-Markov master equation, which we solve numerically. The evolution of the density matrix of the system is described by a Liouvillian superoperator. In order to better understand the results we perform an eigenfunction expansion of the Liouvillian, which is useful in connecting the behaviour of the resonator to the current noise. The mixture of coherent and incoherent processes in the SSET leads to interesting back action effects on the resonator. For weak coupling the SSET acts as an effective thermal bath on the resonator. Depending on the operating point the resonator can be either heated or cooled in comparison to its surroundings. In this regime we can use a set of mean field equations to describe the system and also capture certain aspects of the behaviour with some simple models. For sufficient coupling the SSET can drive the resonator into states of self-sustained oscillations. At the transition between stable and oscillating states of the resonator we also find regions of co-existence between oscillating and fixed point states of the resonator. The current noise provides a way to identify these transitions and the state of the resonator. The system also shows analogies with quantum optical systems such as the micromaser. We calculate the linewidth of the resonator and find deviations from the expected behaviour.
The electrical noise of mesoscopic devices can be strongly influenced by the quantum motion of electrons. To probe this effect, we have measured the current fluctuations at high frequency (5 to 90 gigahertz) using a superconductor-insulator-superconductor tunnel junction as an on-chip spectrum analyzer. By coupling this frequency-resolved noise detector to a quantum device, we can measure the high-frequency, nonsymmetrized noise as demonstrated for a Josephson junction. The same scheme is used to detect the current fluctuations arising from coherent charge oscillations in a two-level system, a superconducting charge qubit. A narrow band peak is observed in the spectral noise density at the frequency of the coherent charge oscillations.
We measure current fluctuations of mesoscopic devices in the quantum regime, when the frequency is of the order of or higher than the applied voltage or temperature. Detection is designed to probe separately the absorption and emission contributions of current fluctuations, i.e. the positive and negative frequencies of the Fourier transformed nonsymmetrized noise correlator. It relies on measuring the quasiparticles photon assisted tunneling current across a superconductor-insulator-superconductor junction (the detector junction) caused by the excess current fluctuations generated by quasiparticles tunneling across a Josephson junction (the source junction). We demonstrate unambiguously that the negative and positive frequency parts of the nonsymmetrized noise correlator are separately detected and that the excess current fluctuations of a voltage biased Josephson junction present a strong asymmetry between emission and absorption.
We analyze charge tunneling statistics and current noise in a superconducting single-electron transistor in a regime where the Josephson-quasiparticle cycle is the dominant mechanism of transport. Due to the interplay between Coulomb blockade and Josephson coherence, the probability distribution for tunneling events strongly deviates from a Poissonian and displays a pronounced even-odd asymmetry in the number of transmitted charges. The interplay between charging and coherence is reflected also in the zero-frequency current noise which is significantly quenched when the quasiparticle tunneling rates are comparable to the coherent Cooper pair oscillation frequency. Furthermore, the finite frequency spectrum shows a strong enhancement near the resonant transition frequency for Josephson tunneling.