Dear Denise, Slowly decaying bound states can happen whenever the bound state energy is close to the opening of a channel (e.g. close to the gap in your case). To move it away from the gap, you can put a phase difference across the superconductors.
As a sanity check you can take a finite system and perform an exact diagonalisation (for several sizes that include a finite portion of the leads). This is imprecise when the bound states are slowly decaying, but it should allow you to spot obvious problems. Note that the algorithm is still experimental. As it requires some root finding for a function f(E)=0, it is always a good idea to plot the function f(E) to see if the algorithm has missed a root. Hope this helps. Best regards, Xavier > Le 9 déc. 2019 à 19:27, Denise Puglia <dpuglia....@gmail.com> a écrit : > > Dear All, > > Has anyone tried to calculate bound states in a JJ? I used the package by > Benoit Rossignol available at: > https://gitlab.kwant-project.org/kwant/boundstate > <https://gitlab.kwant-project.org/kwant/boundstate> . > I am sending attached the code I used to simulate it but I am having some > difficulties interpreting the results. First, the algorithm could not find > bound states, i.e., psi_tot returned by the solver is an empty array, for > N<5. Second, it returns a total wavefunction for larger N, however the > wavefunction oscillates between each site and decays very slowly in the > leads. In the original proposal of this method, Istas et al > (arXiv:1711.08250) solves the wavefunction of quantum billiard (Fig 5) in > which this does not seem to happen. Has anyone found something similar? I do > know if it a mistake in the code, an attribute of discretization or something > else. > > Scanning over N=[1,2,3,5,10,20] and ,u=[2, Delta, 0, -2] the algorithm > returns the following non-zero wavefunctions: > SNS junction with N=10, S=20 and mu=2 > <N_10_S_20_mu_2.png> > SNS junction with N=20, S=20 and mu=2 > <N_20_S_20_mu_2.png> > SNS junction with N=5, S=20 and mu=-2 (in this case the amplitude of the > bound states decays to zero in the lead) > <N_5_S_20_mu_-2.png> > > SNSNS junction with 20/10/10/10/20 sites and mu=2. The wavefunction in the N > region is similar to the SNS junction but there seems to be no decay in the > middle S region. > <N_10_S_20_mu_2.png> > and of course it's mirrored version: > <NSN_10_mu_2.png> > > I appreciate any comments on this subject. > > Best regards, > Denise > > <BS_calc.py><fast.py><_common.py>
