Added analytic rabi oscillation

examples/Proposal/parameter_tuning_example.jl

@@ -1,10 +1,12 @@
 
 using QuantumGateDesign
+using LinearAlgebra
 using Plots
 # Single qubit in the rotating frame
 system_sym = zeros(2,2)
-system_sym[1,1] = 0.84
-system_sym[2,2] = 1.78
+# These entries make the problem more challenging (more like lab frame)
+#system_sym[1,1] = 0.84 
+#system_sym[2,2] = 1.78
 system_asym = zeros(2,2)
 
 a = [0.0 1; 0 0]
@@ -15,7 +17,7 @@ u0 = [1.0 0; 0 1]
 v0 = [0.0 0; 0 0]
 
 tf = 3.5 # Change this
-nsteps = 500
+nsteps = 100
 N_ess_levels = 2
 
 prob = SchrodingerProb(
@@ -39,13 +41,55 @@ target = [0.0 1.0
 
 control = QuantumGateDesign.GRAPEControl(1, prob.tf+0.1) # For now, it doesn't matter since amplitudes last for whole duration
 
-N_samples = 31
+N_samples = 11
 var_range = LinRange(-2, 2, N_samples)
-var1s = zeros(N_samples, N_samples)
-var2s = zeros(N_samples, N_samples)
 infidelities = zeros(N_samples, N_samples)
 
+for (j, var1) in enumerate(var_range)
+    println("j=", j)
+    for (k, var2) in enumerate(var_range)
+        pcof = [var1, var2]
+        infidelities[j,k] = infidelity(prob, control, pcof, target, order=4)
+        #infidelities[j,k] = 1
+    end
+end
 
 
-pl = heatmap(var_range, var_range, infidelities, color=:thermal,
+pl1 = heatmap(var_range, var_range, infidelities, color=:thermal,
             xlabel="θ₁", ylabel="θ₂", colorbar_title="Infidelity")
+
+"""
+Analytic result for rabi oscillation in rotating frame
+(assuming control is constant, could asl)
+"""
+function analytic_infidelity(complex_Ω, T)
+    Ω = abs(complex_Ω)
+    θ = angle(complex_Ω)
+
+    U_evo = [cos(Ω*T)     (sin(θ)-im*cos(θ))*sin(Ω*T)
+             -(sin(θ)+im*cos(θ))*sin(Ω*T)    cos(Ω*T)]
+
+    U_init = [1 0; 0 1]
+    U_final = U_evo*U_init
+    U_targ = [0 1; 1 0]
+
+    this_infidelity = 1 - 0.25*abs( dot(U_final, U_targ) )^2
+
+    return this_infidelity
+end
+
+analytic_N_samples = 20001
+analytic_var_range = LinRange(-2,2,analytic_N_samples)
+analytic_infidelities = zeros(analytic_N_samples, analytic_N_samples)
+
+for (j, var1) in enumerate(analytic_var_range)
+    println("j=", j)
+    for (k, var2) in enumerate(analytic_var_range)
+        complex_Ω = var1+im*var2
+        analytic_infidelities[j,k] = analytic_infidelity(complex_Ω, prob.tf)
+        #infidelities[j,k] = 1
+    end
+end
+
+pl2 = heatmap(analytic_var_range, analytic_var_range, analytic_infidelities, color=:thermal,
+              xlabel="Re(Ω)", ylabel="Im(Ω)", colorbar_title="Infidelity")