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#Simple example of transport properties of a coherent double quantum dot. The counting statistics functions used here
#return the current, shot noise and noise spectrum (if required) for left and right junctions, and cross-correlations
#between junctions
#For the current and zero frequency shot noise comparisons are made to known analytical results.
#I believe the frequency dependance is also analytical, but I cannot for the life of me find the formula in my thesis.
#Author: Neill Lambert
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# setup the matplotlib graphics library and configure it to show
# figures inline in the notebook
%pylab inline
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# make qutip available in the rest of the notebook
from qutip import *
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#DDOT example, frequency dependance
#number of data points:
data = 500
wmax = 2
#list of frequencies
w_vec = linspace(0., wmax, data)
#Parameters
eps = 0.5 # detuning
Tc = 0.1 # tunnelling strength
GammaR = 0.025
GammaL = 0.025
Lstate = basis(3,0)
Rstate = basis(3,1)
Empty = basis(3,2)
H = 0.5*eps * (Lstate*Lstate.dag() - Rstate*Rstate.dag()) + Tc*(Lstate*Rstate.dag() + Rstate*Lstate.dag())
c_ops = []
rate = GammaL
if rate > 0.0:
c_ops.append(sqrt(rate)*Lstate*Empty.dag())
rate = GammaR
if rate > 0.0:
c_ops.append(sqrt(rate)*Empty*Rstate.dag())
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rho_ss = steadystate(H, c_ops)
L = liouvillian(H, c_ops)
I, S = countstat_current_noise(L, [], wlist=w_vec, rhoss=rho_ss,
J_ops=[GammaR*sprepost(Empty*Rstate.dag(), Rstate*Empty.dag()),
GammaL*sprepost(Lstate*Empty.dag(), Empty*Lstate.dag())], sparse=False)
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fig, axes = subplots(1, 1, sharex=True, figsize=(8,8))
axes.plot(w_vec, S[0,0,:]/I[0], 'g', linewidth=2, label="Numerical noise spectrum in right junction")
axes.legend(loc=0, prop={"size":14})
axes.set_xlabel(r'$\omega$', fontsize=18)
axes.set_ylabel(r'$F(\omega)$', fontsize=18)
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#DDOT example, Detuning dependance
#number of data points:
data = 200
eps_max = 1
#list of frequencies
eps_vec = linspace(-eps_max, eps_max,data)
#Parameters
Tc = 0.1 # tunnelling strength
GammaR = 0.0025
GammaL = 0.1
Lstate = basis(3,0)
Rstate = basis(3,1)
Empty = basis(3,2)
c_ops = []
rate = GammaL
if rate > 0.0:
c_ops.append(sqrt(rate) * Lstate*Empty.dag())
rate = GammaR
if rate > 0.0:
c_ops.append(sqrt(rate) * Empty*Rstate.dag())
Num_of_J_ops=2
I_vec=[[] for N_ops in range(Num_of_J_ops)]
F_vec=[[[] for N_ops in range(Num_of_J_ops)] for N_ops in range(Num_of_J_ops)]
I_ana=[]
F_ana=[]
for eps in eps_vec:
I_ana.append(4.*GammaR*GammaL*Tc**2/(4.*Tc**2*(2.*GammaL+GammaR)+GammaL*GammaR**2+4.*eps**2*GammaL)) #Analytical Current
F_ana.append((1-8.*GammaL*Tc**2*(4*eps**2*(GammaR-GammaL) + GammaR*(3*GammaL*GammaR + GammaR**2 +
8*Tc**2))/(4.*Tc**2*(2.*GammaL+GammaR)+GammaL*GammaR**2+
4.*eps**2*GammaL)**2)) #Analytical Zero frequency Fano factor (noise normalized by current)
H = 0.5*eps * (Lstate*Lstate.dag() - Rstate*Rstate.dag()) +Tc*(Lstate*Rstate.dag() + Rstate*Lstate.dag())
rho_ss = steadystate(H, c_ops)
L = liouvillian(H, c_ops)
I, S = countstat_current_noise(L, [], rhoss=rho_ss,
J_ops=[GammaR*sprepost(Empty*Rstate.dag(), Rstate*Empty.dag()),
GammaL*sprepost(Lstate*Empty.dag(), Empty*Lstate.dag())],sparse=False)
for i in range(Num_of_J_ops):
I_vec[i].append(I[i])
for j in range(Num_of_J_ops):
F_vec[i][j].append(S[i,j]/sqrt(I[i]*I[j]))
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fig, axes = subplots(1, 1, sharex=True, figsize=(8,8))
axes.plot(eps_vec, I_vec[0], 'g', linewidth=2, label="Numerical Current in right junction")
axes.plot(eps_vec, I_ana, 'r--', linewidth=2, label="Analytical Current in right junction")
axes.legend(loc=0,prop={"size":14})
axes.set_xlabel(r'$\epsilon$', fontsize=18)
axes.set_ylabel(r'I', fontsize=18)
fig1, axes1 = subplots(1, 1, sharex=True, figsize=(8,8))
axes1.plot(eps_vec, F_vec[0][0], 'g', linewidth=2, label="Numerical Fano Factor in right junction")
axes1.plot(eps_vec, F_ana, 'r--', linewidth=2, label="Analytical Fano Factor in right junction")
axes1.legend(loc=0,prop={"size":14})
axes1.set_xlabel(r'$\epsilon$', fontsize=18)
axes1.set_ylabel(r'F(0)', fontsize=18)
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from qutip.ipynbtools import version_table
version_table()
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