# Probing Andreev States in superconducting atomic contacts

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## Escape driven by an asymmetric noise

The goal of this section is to explain how the escape out of the zero- voltage state of a Josephson junction can be used to probe noise, and in particular the asymmetry of its probability distribution. So far, we only considered thermal fluctuations arising from macroscopic resistors, therefore Gaussian noise. In the following, we introduce predictions for the escape rate when the noise does not arise only from thermal fluctu- ations, but also from a specific noise source, producing for example shot noise which is not Gaussian [21]. Predictions have been derived for a noise presenting a finite second and third moment [11, 31, 32, 33, 65, 68], and neglecting higher order moments.1 The electrical setup considered in this part is shown in Fig. 2.9. A Josephson junction is biased by a current IB flowing through a resistor RB. To this resistor is associated a current source in parallel, producing Gaussian current fluctuations δIB. In addition, a non-Gaussian noise source with impedance RN is present. To this noise source is associated a current fluctuation δIN. The equation describing this circuit is similar to Eq. (2.8): CJϕ0¨γ + ϕ0 R γ˙ + EJ [sin γ − s] = δIB + δIN.

### Frequency scales

The frequency scales of the problem are defined here from the smallest to the largest (see also Fig. 2.10):
• The escape rate: typically, escape rates probed experimentally are in the sub-MHz range, corresponding to a measurement time of 1 μs or more.
• The plasma frequency: the plasma frequency considered in the follow- ing experiment is around 1 GHz.
• The thermal noise cut-off frequency: At the relevant temperature of the experiment, which is of the order or 100mK (from 20mK to 500mK, thermal noise extends to frequencies of the order of kBT h ≃ 2GHz [21]).
• The superconducting gap: In the experiments described in this thesis, all superconductors are aluminum for which the frequency correspond- ing to the superconducting gap is h ≃ 50 GHz.
• Non-Gaussian noise cut-off frequency: In the experimental case of a tunnel junction biased at a voltage VN, the maximal frequency of the shot noise is eVN h [21]. For VN ≃ 400 μV, which is the lowest voltage probed in the experiment presented in the following, it corresponds to frequencies higher than 100GHz. At the scale of the plasma frequency, non-Gaussian noise thus appears completely frequency-independent.

#### Noise statistical properties

In the framework of Full Counting Statistics (FCS), the properties of the current fluctuations are treated through the probability distribution function. Experimentally, this distribution is accessed its the moments or through its cumulants which are two ways to represent the same infor- mation (see [72] for details). However, moments and cumulants are equal up to the third order, as is recalled in Appendix B.3, and since this thesis  only deals with moments up to the third one, it is not necessary here to make a distinction between moments and cumulants. In the following, we deal only with the moments.

Escape rate in presence of an asymmetric noise

We now turn to the theoretical predictions for the effect of an asymmetric noise on the escape rate out of a single well.
The first prediction, obtained by J.T. Peltonen et al. [65] considers the adiabatic response of the junction to the noise. Therefore, it only deals with the effect of noise at frequencies much smaller than the plasma frequency. In our experiments, this corresponds to only a small fraction of the noise spectrum, as shown in Fig. 2.10. The damping in the junction dynamics, which appears in the following to be of central importance, does not enter in this approach.
The second prediction was obtained by E. Sukhorukov and A. Jordan [33] with a stochastic path integral formalism, in the two limits of low and high damping limits only, which were unfortunately out of the relevant experimental range.
The third prediction, obtained by J. Ankerhold [31, 32, 35] is based on a Fokker-Planck equation approach to calculate the escape rate. The resolution of the Fokker-Planck equation relies on the fact that escape is only a small perturbation to the Boltzmann equilibrium in the well. As was already discussed by Kramers, this assumption might not be appropriate for very large values of the quality factor.
The last prediction that we present, obtained by H. Grabert [11], relies on the calculation of the action along the escape trajectory using also a path integral formalism. Its validity range spans over the complete range of quality factor, and recovers the two limits of E. Sukhorukov and A. Jordan [33] calculated with a similar method. All the predictions are compared at the end of the chapter.

Effect of a low-frequency cutoff

In the experiment, the dc part of the noise is cut by an RC filter [35, 38] to ensure that the Poisson noise added to the bias current of the junction has zero mean value. To probe the effect of this cut-off on the rate asym- metry, we simulated the dynamics of the circuit shown in Fig. 3.13. In this circuit, both the Poisson noise δIN and the Johnson-Nyquist noise δIB2 are filtered. Only the components of the noise at large enough fre- quencies reach the Josephson junction, while the low frequency part flows through the resistor R2. Beware that two resistors are present, each of them producing a Johnson-Nyquist noise. For simplicity, we consider in this section that they have different resistances but are at the same tem- perature.

Probing shot noise with a Josephson junction

At the beginning of my work, a simplified version of this strategy, aiming at measuring only the third moment and not the FCS, had been explored experimentally by two groups: the group of J. Pekola at Helsinki University of Technology (HUT) [29, 37, 85] and the Quantronics group in Saclay [36, 35]. In both cases, the shot noise of a tunnel junction was added to the bias current of the Josephson junction (JJ), in order to extract the third moment of the noise.
To do so, one option is to have the full current in the noise source IN(t) flowing through the JJ. In this case however, the escape rate is modified not only by the noise, but also trivially by the dc current hINi. To get rid of this contribution, the dc part hINi of the current was either compensated for or filtered out.

1 Introduction
1.1 Josephson effect and mesoscopic physics
1.2 Detecting asymmetric noise with a Josephson junction
1.3 Towards Andreev states spectroscopy
I Detecting noise asymmetry using a Josephson junction
2 Escape of a Josephson junction out of the metastable state
2.1 The Josephson junction in an electromagnetic environment
2.2 Escape rate out of the zero-voltage state
2.3 Escape driven by an asymmetric noise
2.4 Conclusion
3 Numerical simulation of the escape
3.1 Simulation algorithm
3.2 Rate estimation
3.3 Results on the escape rate
3.4 Effect of a low-frequency cutoff
3.5 Conclusion
4 Experimental detection of an asymmetric noise with a Josephson junction
4.1 Introduction
4.2 Experimental setup
4.3 Circuit characterization & measurement techniques
4.4 Results on Sample JJD1
4.5 Results on Sample JJD2
4.6 Perspectives
4.7 Conclusion
Article reporting the measurement of asymmetric noise using a Josephson junction
II Probing Andreev States in superconducting atomic contacts
5 Josephson effect and Andreev states
5.1 Andreev Bound States
5.2 An experimental test-bed: superconducting atomic contacts
5.3 Supercurrent in atomic contacts
5.4 Current-phase relation of well-characterized contacts
5.5 Out-of-equilibrium effects
5.6 Conclusions
6 Towards Andreev states spectroscopy
6.1 Predictions for the Andreev transition
6.2 Design of an experimental setup
6.3 Probing the new on-chip environment with a standard SQUID circuit
III Sample Fabrication and Measurement Techniques
7 Sample Fabrication
7.1 Samples JJD for noise detection experiments
7.2 Samples AC1 and AC2 for atomic contacts experiments
7.3 SQUID sample
7.4 Lithography recipes
8 Low-Noise Measurement Techniques
8.1 Sample Holder & Bending Mechanism.
8.2 Cryostat wiring
8.3 Room temperature connections and instruments
IV Appendix
A.1 Back-Bending in the I(V ) characteristics of Josephson junctions .
A.2 Heating effects in switching measurements
B Miscellaneous
B.1 Approximations for the tilted washboard potential
B.2 Resonant activation through the modulation of the critical current
B.3 Moments and Cumulants, and Poisson Process.
B.4 Details on the simulations
B.5 The Andreev Levels Qubit
B.6 Critical current of a Josephson junction with electrodes having different gaps
B.7 Attenuators
B.8 Correspondence between names
Bibliography

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