# 2.3. Second example: Fabry-Perot interferometer¶

We consider an infinite one-dimensional chain with nearest-neighbor hopping, in which two potential barriers A and B form a Fabry-Perot cavity. A sketch of the system is

At time $$t = 0$$, the electric potential $$V(t)$$ of the left electrode is suddenly raised from zero to a finite value, which is taken into account by a time-dependent coupling element (shown in red) between the left electrode and the central system. We want to study the transient regime of the current $$I(t)$$ before it eventually reaches its stationary value. This system has been studied in Ref. [1]. The actual simulation script in this tutorial is taken from Ref. [2], but simulation time and accuracy are both reduced in this tutorial in order to speed up the calculation. The entire simulation script is:

from math import sin, pi
import matplotlib.pyplot as plt

import tkwant
import kwant

def am_master():
"""Return true for the MPI master rank"""
return tkwant.mpi.get_communicator().rank == 0

def make_fabry_perot_system():

# Define an empty tight-binding system on a square lattice.
lat = kwant.lattice.square(norbs=1)
syst = kwant.Builder()

# Central scattering region.
syst[(lat(x, 0) for x in range(80))] = 0
syst[lat.neighbors()] = -1
# Backgate potential.
syst[(lat(x, 0) for x in range(5, 75))] = -0.0956
# Barrier potential.
syst[[lat(4, 0), lat(75, 0)]] = 5.19615

# Attach lead on the left- and on the right-hand side.
sym = kwant.TranslationalSymmetry((-1, 0))
lead[(lat(0, 0))] = 0

return syst, lat

# Phase from the time integrated voltage V(t).
def phi(time):
vb, tau = 0.6, 30.
if time > tau:
return vb * (time - tau / 2.)
return vb / 2. * (time - tau / pi * sin(pi * time / tau))

def main():

times = range(220)

# Make the system and add voltage V(t) to the left lead (index 0).
syst, lat = make_fabry_perot_system()
syst = syst.finalized()

# Define an operator to measure the current after the barrier.
hoppings = [(lat(78, 0), lat(77, 0))]
current_operator = kwant.operator.Current(syst, where=hoppings)

# Set occupation T = 0 and mu = -1 for both leads.

# Initialize the time-dependent manybody state. Use a lower
# accuracy for adaptive refinement to speed up the calculation.
state = tkwant.manybody.State(syst, tmax=max(times), occupations=occup,
refine=False, combine=True)
state.refine_intervals(rtol=0.3, atol=0.3)

# Loop over timesteps and evaluate the current.
currents = []
for time in times:
state.evolve(time)
current = state.evaluate(current_operator)
currents.append(current)

# Plot the normalized current vs. time.
if am_master():
plt.plot(times, currents / currents[-1])
plt.xlabel(r'time $t$')
plt.ylabel(r'current $I$')
plt.show()

if __name__ == '__main__':
main()


The complete source code of this example can be found in fabry_perot.py.

The result of the simulation shows the current increases through plateaus that correspond to the different trajectories of the cavity. The first plateau corrsponds to a direct transmission, whereas the second one is due to reflection at B followed by reflection at A then transmission. For longer simulation times, this series continues until a stationary current value is reached, see Refs. [1, 2]. Another detail is that on each plateau, the current oscillates with a frequency $$e V_b / h$$, where $$V_b$$ is the stationary value of the electric potential $$V(t)$$.

Warning

The examples in this section take several minutes on a single core desktop computer. To speed up the computation the script can be run in parallel, see section Parallelization with MPI.

## 2.3.1. References¶

[1] B. Gaury, J. Weston, X. Waintal, The a.c. Josephson effect without superconductivity, Nat. Commun. 6, 6524 (2015). [arXiv]

[2] T. Kloss, J. Weston, B. Gaury, B. Rossignol, C. Groth and X. Waintal, Tkwant: a software package for time-dependent quantum transport, arXiv:2009.03132 [cond-mat.mes-hall].