Studying the Mollow Triplet

[athanj...@gmail.com]

Good day!

I'm a relatively new user of QuTiP (which I'm loving) and I'm wondering if anyone could help me out to write some simple code to visualize the Mollow triplet.

The actual situation in which I want to apply this is a bit complicated. However, I believe that the following should be a minimal working sample situation:

We take a qubit with a resonance frequency omega_l, and we drive this with a classical drive A sin(omega_d t). Following the example of http://qutip.org/docs/2.2.0/examples/me/ex-25.html we can easily write down the Hamiltonian. Now, my initial approach was just continuing with the RWA variant and then using the following code

def calc_spectrum1(w1, kappa, gamma, E, wd,wlist):

    # Hamiltonian
    a1 = destroy(2)
    H = (w1-wd)* a1.dag() * a1 + 0.5*E*(a1.dag() + a1)

     
    # collapse operators
    c_op_list = []

    n_th_a = 0.5
    rate = kappa * (1 + n_th_a)
    if rate > 0.0:
        c_op_list.append(sqrt(rate) * a1)

    rate = kappa * n_th_a
    if rate > 0.0:
        c_op_list.append(sqrt(rate) * a1.dag())

           
    A1 = a1.dag() + a1
    B1 = A1

    # calculate the power spectrum
    spec1 = spectrum(H, wlist, c_op_list, A1, B1)
   
    return spec1

which I then run with

w1 = 4 * 2 * pi
wd = (4-0.01) * 2 * pi

kappa = 1*10**-3*2*pi        
E = 0.03*2*pi;

wlist = linspace(-0.5* pi * 2, 0.5 * pi * 2, 2000)

spec1 = calc_spectrum1(w1, kappa, gamma, E, wd,wlist)

# plot results side-by-side
figure(figsize=(20, 10))
subplot(2, 2, 1)
plot(wlist / (2 * pi), abs(spec1), 'b', lw=2)
xlabel('Frequency')
ylabel('Power spectrum')
title('Spectrum of Qubit 1')
show()

Now, this shows a spectrum which does have the features of the mollow triplet (you can see it here https://i.imgsafe.org/6945adf.png), except that it is not symmetric. The cause of this is not looking at the system exactly at resonance, but at 10 MHz below (in the above example). However, by looking at the system at resonance the qubit term drops out of the Hamiltonian. So I was wondering, what should I change in the above? Switching to the time dependent hamiltonian would greatly complicate everything, would it not?

[Kevin Fischer]

Hi,

If you're interested in calculating the power spectrum, in which the Mollow triplet resides, you need to change the operators in your correlator. The optical Wiener-Kinchin theorem states the power spectrum arises from correlations of the negative and positive frequency operators, therefore:

A1 = a1.dag()

B1 = a1

Rerunning your code with this change yields the standard Mollow triplet.

With n_th_a=0 then the triplet is symmetric, however with n_th_a>0 and wl-wd!=0, it is asymmetric due to repopulation of the qubit via a thermal process. This repopulation preferentially excites the qubit's dressed state |e> over |g> when the laser is detuned from resonance. 

Kevin

[Kevin Fischer]

Sorry, correction: dressed state with greater proportion |e> over |g>