Calculation of g(2)

[Giovanni Amedeo Cirillo]

Dear all,

I am writing to You for having some suggestions for calculating the second-order correlation function g(2) of a lambda system behaving as a single-photon source. You can find the system to be analyzed in the attached figure. In particular, I replicated the results provided in

Hiroo Azuma, Numerical analyses of emission of a single-photon pulse based on single-atom–cavity quantum electrodynamics, Progress of Theoretical and Experimental Physics, Volume 2019, Issue 6, June 2019, 063A01, https://doi.org/10.1093/ptep/ptz052

which is inherited from

Kuhn, A., Hennrich, M., Bondo, T. et al. Controlled generation of single photons from a strongly coupled atom-cavity system. Appl Phys B 69, 373–377 (1999). https://doi.org/10.1007/s003400050822

The Hamiltonian to be simulated is (17) in Azuma paper and it is the interaction picture. I found the expected eigenstates and expectation value evolutions. Now, I would like to compute the second-order correlation function g(2). I tried to exploit the methodology of the tutorial provided by K.A. Fisher from Stanford University https://nbviewer.ipython.org/github/qutip/qutip-notebooks/blob/master/examples/pulse-wise-second-order-coherence-g2.ipynb

When I try to arrange that code to my system under analysis, the second-order optical coherences (see meshes and G2_t_tau_G in the notebook) are always equal to 0, so the g(2)(0) = 0. It is not clear to me if the problem is in the Hamiltonian formulation itself, e.g. because an Hamiltonian contribution for the repumping pulse inducing |g,0> --> |u,0> transition is not present, or the problem is numerical (I am using the Master Equation solver).

I also observed that in the method _correlation_me_2t in correlation.py the initial state of the Master Equation solver is c_op * rho * a_op, which provides a density matrix with trace lower than 1. Is it usual? I apologize for the question, I am an Electrical Engineer and I study Quantum Mechanics on my own, probably I do not have a sufficiently strong background.

I attach the code providing to me problems in the following. From a practical point of view, G2_t_tau_G contains only 0 values.

Thanks in advance.

*** CODE ***

delta = 0 # detuning

ustate = basis(3, 0)

excited = basis(3, 1)

ground = basis(3, 2)

N = 2 # Set where to truncate Fock state for cavity

sigma_ge = tensor(ground * excited.dag(), qeye(N))  # |g><e|

sigma_ue = tensor(ustate * excited.dag(), qeye(N))  # |u><e|

sigma_ee = tensor(excited * excited.dag(), qeye(N))  # |e><e|

a = tensor(qeye(3), destroy(N))

Omega0 = 2*np.pi*0.11*22 # Gaussian pulse peak

g = Omega0/4  # coupling strength

T = 50e0

gamma = 2.5e-2

#gamma = 0

tOffset = 5*T

H0 = (g * a * sigma_ge.dag() ) + (g.conjugate() * a.dag() * sigma_ge) + (delta * sigma_ee) # time-independent term

H1 = 1/2*(sigma_ue.dag() + sigma_ue)  # time-dependent term

H = [H0, [H1, 'Omega0 * (exp(-((t - t_offset) / T) ** 2))']]

args = {'Omega0': Omega0, 'T': T, 't_offset' : tOffset}

t = np.linspace(0, 2*tOffset, 300) # Define time vector

psi0 = tensor(ustate, fock(N,0)) # initial state |u,0>

G2_t_tau_G = correlation_3op_2t(H, psi0, t, t, [np.sqrt(gamma)*a], a.dag(), a.dag()*a, a, args=args, options=options)