About obtaining Mollow Triplet

[Alperen Tüğen]

Hi all,

Recently, I have tried to verify the simulation results of the article "Phonon mediated off-resonant quantum dot–cavity coupling under resonant excitation of the

quantum dot"(Majumdar et al. PHYSICAL REVIEW B 84, 085309 (2011)) however I could not obtain the desired results in Figure 1 (c,d). My code for blue detuning case is the following :

-------------------------------------------------------------------------------------------------------------------- 

import numpy as np

from qutip import *

import pylab as plt

N =4                         # number of cavity fock states

wc =1.0 * 2 * np.pi     # cavity and atom frequency

g  =20 * 2 * np.pi       # coupling strength

kappa =20*2*np.pi     # cavity dissipation rate

gamma =2*np.pi*1    # atom dissipation rate

gamma_r=2*np.pi*0.5

gamma_d =2*np.pi*0.5

delta =6*kappa

wa = wc+delta

omega =2*np.pi*6

wl= wa                     # resonantly driven QD with Laser

deltawc =wc-wl

deltawa =wa-wl

# Jaynes-Cummings Hamiltonian

a  = tensor(qeye(N), destroy(2))

sm = tensor(destroy(N), qeye(2))

H = deltawc * a.dag() * a + deltawa * sm.dag() * sm + 1j * g * (a.dag() * sm - a * sm.dag()) + omega * (sm + sm.dag())

# collapse operators

n=0 #limit approximation

c_ops =[np.sqrt(2*kappa)*a,np.sqrt(2*gamma)*sm,np.sqrt(2*gamma_r*n)*spre(sm.dag())*spost(a),np.sqrt(2*gamma_r*(n+1))*spre(sm)*spost(a.dag()),np.sqrt(2*gamma_d)*spre(sm.dag())*spost(sm)]

# calculate the correlation function using the mesolve solver, and then fft to

# obtain the spectrum. Here we need to make sure to evaluate the correlation

# function for a sufficient long time and sufficiently high sampling rate so

# that the discrete Fourier transform (FFT) captures all the features in the

# resulting spectrum.

tlist = np.linspace(0.25, 50, 50000)

corr = correlation_2op_1t(H,None,tlist, c_ops,a.dag(),a)

wlist1,spec1 = spectrum_correlation_fft(tlist, corr)

# calculate the power spectrum using spectrum, which internally uses essolve 

# to solve for the dynamics (by default)

wlist2 = np.linspace(0.25, 1.75, 200000) * 2 * np.pi

spec2 = spectrum(H, wlist2, c_ops, a.dag(), a)

# plot the spectra

fig, ax = plt.subplots(1, 1)

ax.plot((wlist1 ) / (kappa*(2 * np.pi)), spec1, 'b', lw=2, label='eseries method')

ax.plot((wlist2 ) / ((2 * np.pi)*kappa), spec2, 'r--', lw=2, label='me+fft method')

ax.legend()

ax.set_xlabel('Frequency')

ax.set_ylabel('Power spectrum')

ax.set_title('Vacuum Mollow Triplet')

ax.set_xlim(-wlist2[0]/(2*np.pi), wlist2[-1]/(2*np.pi))

plt.show()

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For those of you could not reach the article, corresponding master equation of the system is $

\frac{\mathrm{d} }{\mathrm{d} t}{\rho}=-i[H,\rho]+2\kappa L(a)+2\gamma L(\sigma) + 2\gamma_rn L(\sigma^{\dagger}a)+2\gamma_r(n+1) L(\sigma a^{\dagger})

$

and the corresponding Hamiltonian is $ H = \Delta w_ca^{\dagger}a+\Delta w_a\sigma^{\dagger}\sigma+ig(a^{\dagger}\sigma-a\sigma^{\dagger})+\Omega(\sigma+\sigma^{\dagger})\\ where\\ \Delta w_c = w_c-w_l\ \\ \Delta w_a = w_a-w_l \\ w_a = QD \ Frequency \\ w_c = Cavity \ Frequency \\ w_l = Laser \ Driving \ Frequency \\ g = coherent \ interaction \ strength \ between \ QD\ and \ Cavity \\

a = annihilation \ operator \ for \ Cavity \ Mode \\

\sigma= lowering \ operator \ for \ QD \ Transitions \\ \Omega = Rabi \ Frequency \ of \ driving \ the \ laser $ 

 Expected graph is attached. 

Can anyone help me about whether my code has structural problem(s) cause(s) modeling the system in the paper  to fail ? 

Best Regards,

Alperen

[Andrew M.C. Dawes]

Alperen,

I am a big fan of repeating analyses from published papers as a way to demonstrate the use QuTip so I’m always up for trying these things out. I haven’t had much time to tinker with this but in my early try, I found the following:

First, I think you swapped the labels in the plot, the wlist1 data comes from the mesolver and fft but is labeled with the eseries label.

Second, the eseries data appears to at least be close to what is expected. The huge scale of the other data makes it hard to see the small features of the eseries data set. Looking at only that set (and changing the axis scale a bit) you can see some features from the paper. I’ve pushed a notebook with my changes to gist:

https://gist.github.com/amcdawes/c852d7a3feb6ad02b334c5aeb8ea5683

I’ll keep tinkering as I don’t have an immediate answer about why the mesolve approach breaks.

Andy

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<Screenshot_2.png>

[Kevin Fischer]

Hi Alperen,

The effective phonon transfer processes are dissipative, so the correct collapse operators are (changes in red):

c_ops =[np.sqrt(2*kappa)*a,

             np.sqrt(2*gamma)*sm,

             np.sqrt(2*gamma_r*n)*sm.dag()*a,

             np.sqrt(2*gamma_r*(n+1))*sm*a.dag(),

             np.sqrt(2*gamma_d)*sm.dag()*sm]

Using the spre/post super-operators is only necessary if you're trying to model some explicitly coherent effect in the dissipator term, ex. simulations with some non-Hermitian Hamiltonian.

Kevin