Energy bands of a single Josephson junction

In a mesoscopic Josephson junction, the quantum mechanical nature of charge has to be taken into account. According to the quantum prescription, the Coulomb energy Q2/2C leads to a term:

 Q - \frac{e}{i}\frac{\partial}{i\partial\Phi/2}

in the Hamiltonian. Therefore, the quantum behavior of a superconducting junction is described by the Schrödinger equation

 \frac{\partial^2\Psi}{\partial(\phi/2)^2} + \left[\frac{E}{E_C}-\frac{E_J}{E_C}\cos\phi \right] \Psi = 0

where the Josephson coupling energy plays a role of a periodic potential.

Presently, we are working mostly on noise spectroscopy applications to detect higher order moments of noise and counting statistics using Coulomb blockaded junctions in the regime EJ/EC << 1.

Energy level spectroscopy of macroscopic quantum systems is of large current interest. In conjunction with quantum computing, superposition of macroscopic quantum states have recently been observed in potential wells with two adjacent minima. In single Josephson junctions, with a periodic potential for the phase difference across the junction, superposition of states leads to delocalization of the macroscopic phase variable. A consequence of this is the formation of energy bands, in a similar fashion as for electrons in solids.

We have verified the existence of these bands experimentally using a novel spectroscopic tool based on inelastic Cooper pair tunneling in a mesoscopic junction. According to the environmental fluctuation theory, non-coherent Cooper pair tunneling is allowed only if energy is exchanged with the surroundings. Thus, transitions between energy bands are seen as current peaks in the IV-curve of a probe junction.

Energy levels λ = E/EC of the Mathieu equation as a function of EJ/C. Labels an and bn denote the sine- and cosine-type solutions. The functions corresponding to an even value of n are π-periodic and those with odd index are 2π-periodic. The energy bands (shaded regions) of the Schrödinger equation appear between a0- a1, b1-b2, a2-a3, etc...
Contour plot of the measured current in the probe junction in the flux-voltage plane; flux penetrating the SQUID loops is given in reduced units Φ/Φ0 where Φ0 = h/2e is the flux quantum. A sequence of resonance peaks is seen in the subgap-conductance. These peaks are identified as coming from transitions between the excited bands of a Josephson junction.

Related publications

  • Mesoscopic Josephson junction as a noise detector

R.K. Lindell, J. Delahaye, M.A. Sillanpää, T.T. Heikkilä, E.B. Sonin, and P.J. Hakonen

Proc. of SPIE 5472, 19 (2004)