Coupling to Multiple Excited Hyperfine Levels

We now present the code used to calculate atom-cavity interactions that involve multiple hyperfine levels as in Section 6.1. Again, it will be useful to define index functions to help keep track of the atomic states:

function n = index_g_Cs(F,m) if ((m <= F) & (m >= -F))

switch (F)

case 3, n = 4 + m; case 4, n = 12 + m;

otherwise, error(’F out of range in index_g_Cs’); end

else

error(’m out of range in index_g_Cs’); end function n = index_e_Cs(F,m) if ((m <= F) & (m >= -F)) switch (F) case 2, n = 19 + m; case 3, n = 25 + m;

case 4, n = 33 + m; case 5, n = 43 + m;

otherwise, error(’F out of range in index_e_Cs’); end

else

error(’m out of range in index_e_Cs’); end

With the atomic basis thus defined, we present the code which calculates a single atom coupled to a one-mode resonator, appropriate when g is comparable to the excited state hyperfine splittings, but still much less than the ground state splitting:

function [rhoss, a, H, g4_m4, g4_m3, g4_m2, g4_m1, g4_p0, g4_p1, g4_p2,... g4_p3, g4_p4] = kmb_Cs133_scott(gamma, kappa, g, Na, cavity_freq,... probe_freq, eps_p, repump_freq, Wrp, p_xi, p_psi, r_xi, r_psi) % One Cesium atom (D2 transition) coupled to cavity with one mode. % Atom’s 4->N’ transitions are all coupled to cavity.

% A classical beam repumps from 3 ground state.

% This routine calculates the steady state density matrix of the system. % All 4-> N’ transitions (of the same polarization) couple to the same % environmental reservoir. All 3->N’ also couple to one reservoir, % but this is NOT the same reservoir as 4->N’. Thanks to Scott for % working out this rather subtle point in the derivation of the master

% equation.

% gamma is atomic HWHM % kappa is cavity HWHM

% g is half the single photon Rabi frequency for 4,4->5’5’ % Na is number of Fock states for cavity mode

% cavity_freq is cavity frequency relative to F’=5->F=4 transition % probe_freq is probe frequency relative to F’=5->F=4 transition % eps_p is strength of probe. Empty cavity on resonance photon number % is eps_p^2/kappa^2. eps_p must be real

% repump_freq is relative to F’=4->F=3

% Wrp is half of the Rabi freq for repump (Wrp=gamma at saturation) % Wrp must be real

% r_xi, r_psi are angles that set the repump polarization %xi = 0 for 0 polarization, pi/2 for p or m polarization %psi = 0 for p polarization , pi/2 for m polarization

f_5 = 0; % excited state frequencies

f_4 = -251; f_3 = -251-201; f_2 = -251-201-151;

dr = f_4 - probe_freq + repump_freq; % (repump freq) - (probe freq) - % (groundstate splitting)

d2 = f_2 - probe_freq; % (F=2’->F=4 freq) - (probe freq)

d3 = f_3 - probe_freq; % (F=3’->F=4 freq) - (probe freq)

d4 = f_4 - probe_freq; % (F=4’->F=4 freq) - (probe freq)

d5 = f_5 - probe_freq; % (F=5’->F=4 freq) - (probe freq)

dc = cavity_freq - probe_freq; % (cavity freq) - (probe freq)

phi = 0; % phase of repump relative to probe

Ng3 = 2*3 + 1; % number of states in g3 manifold

Ng4 = 2*4 + 1; % number of states in g4 manifold

Ne2 = 2*2 + 1; % number of states in e2 manifold

Ne3 = 2*3 + 1; % number of states in e3 manifold

Ne4 = 2*4 + 1; % number of states in e4 manifold

Ne5 = 2*5 + 1; % number of states in e5 manifold

Natomic = Ng3 + Ng4 + Ne2 + Ne3 + Ne4 + Ne5; % number of atomic states

%--- Atomic lowering operators --- % Change minus sign for 0-polarized transitions to be consistent with % Steck notes.

% lowering operators for 5’ --> 4 Fg=4; Fe=5;