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Lambda memory equations

Slightly different equations for the following protocols will be derived.





In [1]:



    


from fast import *
from fast.symbolic import hamiltonian
from matplotlib import pyplot as plt
from sympy import solve
from sympy import exp, I
%matplotlib inline
init_printing()



    











In [2]:



    


fig, axes = plt.subplots(2, 2, figsize=(12.5, 15))
om21 = 1
om31 = 5
om32 = 4
p1 = [0, 0]
p2 = [4, om21]
p3 = [1.8, om31]

framexmax = 5; frameymax = 7; frameymax2 = 6

for i in range(2):
    for j in range(2):
        axij = axes[i, j]; axij.set_aspect("equal")
        if (i,j) in [(0, 0), (1, 1)]:
            draw_state(axij, p1, text=r"$|1\rangle$", l=1.0, alignment='right',
                       label_displacement=0.05, fontsize=25, linewidth=4.0)
            draw_state(axij, p2, text=r"$|2\rangle$", l=1.0, alignment='right',
                       label_displacement=0.05, fontsize=25, linewidth=4.0, atoms=7)
        else:
            draw_state(axij, p1, text=r"$|1\rangle$", l=1.0, alignment='right',
                       label_displacement=0.05, fontsize=25, linewidth=4.0, atoms=7)
            draw_state(axij, p2, text=r"$|2\rangle$", l=1.0, alignment='right',
                       label_displacement=0.05, fontsize=25, linewidth=4.0)
        
        draw_state(axij, p3, text=r"$|3\rangle$", l=1.0, alignment='right',
                   label_displacement=0.05, fontsize=25, linewidth=4.0)
        if i == 0:
            axij.plot([0, 0, framexmax],[0,frameymax2, frameymax2], "k-", alpha=0.0)
        else:
            axij.plot([0, 0, framexmax],[0,frameymax, frameymax], "k-", alpha=0.0)
        axes[i, j].axis("off")

fs = 30
#axes[0, 0].set_title(r"$\star\mathrm{Magic}\star$", fontsize=fs, color=(255/255.,223/255.,0/255.))
#axes[0, 1].set_title(r"$\mathfrak{Ye \ Olde \ Raman}$", fontsize=fs)
#axes[0, 0].set_ylabel(r"$\mathrm{Red \ detuned}$", fontsize=fs, color="r")
#axes[1, 0].set_ylabel(r"$\mathrm{Blue \ detuned}$", fontsize=fs, color="b")
axes[0, 0].text(1, 6, r"$\star\mathrm{Magic}\star$", color=(255/255.,223/255.,0/255.), fontsize=fs)
axes[0, 1].text(0, 6, r"$\mathfrak{Ye \ Olde \ Raman}$", fontsize=fs)
axes[0, 0].text(-1.5, 5, r"$\mathrm{Red \ detuned}$", fontsize=fs, rotation=90, color="r")
axes[1, 0].text(-1.5, 5, r"$\mathrm{Blue \ detuned}$", fontsize=fs, rotation=90, color="b")

l = 0.5
###################################################################################
# Red Magic
delta = -2*om21; disp = 0.25; disp2 = -0.5
excitation(axes[0, 0], [p1[0]+0.25-disp,p1[1]],[p3[0]+0.10-disp,p3[1]+delta],
           fc="r", ec="r",linewidth=10.0, head_width=0.2, head_length=0.2)
excitation(axes[0, 0], [p2[0]+0.25-disp,p2[1]],[p3[0]+0.10-disp,p3[1]+delta],
           fc="b", ec="b",linewidth=3.0, head_width=0.2, head_length=0.2)

excitation(axes[0, 0], [p2[0]+p1[0]+0.25+disp-0.3,p2[1]+p1[1]],[p2[0]-p3[0]-0.10+disp,p2[1]+p3[1]+delta],
           fc="w", ec="r",linewidth=10.0, head_width=0.2, head_length=0.2, alpha=0.5)

decay(axes[0,0], [p2[0]-p3[0]-0.10+disp+disp2,p2[1]+p3[1]+delta],
              [ p1[0]+disp2, p1[1]], 0.05,20.0,color="g",linewidth=1.0)
decay(axes[0,0],[p2[0]+0.5, p2[1]],
                [p2[0]-0.8, p3[1]], 0.05,20.0,color="g",linewidth=1.0)

axes[0, 0].plot( [p3[0]-l, p3[0]+l], [p3[1]+delta, p3[1]+delta], "k--")
axes[0, 0].plot( [p3[0]-l, p3[0]+l], [p3[1]+delta*0.5, p3[1]+delta*0.5], "k--")
axes[0, 0].text(2.4,1.5,r"$\Omega_s$", fontsize=fs)
axes[0, 0].text(1.0,1.1,r"$\Omega_c$", fontsize=fs)
axes[0, 0].text(-0.2,2.1,r"$\mho_a$", fontsize=fs)
axes[0, 0].text(2.8,3.4,r"$\mho_c$", fontsize=fs)
axes[0, 0].text(3.5,4.4,r"$\Omega_a$", fontsize=fs)

###################################################################################
# Red Normal
delta = -om21; disp = -0; disp2 = -0.2
excitation(axes[0, 1], [p2[0]+0.25-disp,p2[1]],[p3[0]+0.10-disp,p3[1]+delta],
           fc="r", ec="r",linewidth=10.0, head_width=0.2, head_length=0.2)
excitation(axes[0, 1], [p1[0]+0.25-disp,p1[1]],[p3[0]+0.10-disp,p3[1]+delta],
           fc="b", ec="b",linewidth=3.0, head_width=0.2, head_length=0.2)
disp = 0.75
excitation(axes[0, 1], [p1[0]+0.25-disp,p1[1]],[p3[0]+0.10-disp-0.4,p3[1]+delta-om21],
           fc="w", ec="r",linewidth=10.0, head_width=0.2, head_length=0.2, alpha=0.5)

decay(axes[0, 1], [p3[0]+0.10-disp-0.2,p3[1]+delta-om21],
              [ p2[0]+disp2, p2[1]], 0.05,20.0,color="g",linewidth=1.0)

axes[0, 1].plot( [p3[0]-l, p3[0]+l], [p3[1]+delta, p3[1]+delta], "k--")
axes[0, 1].plot( [p3[0]-l-1, p3[0]+l-1], [p3[1]+delta-om21, p3[1]+delta-om21], "k--")
axes[0, 1].text(0.8,0.8,r"$\Omega_s$", fontsize=fs)
axes[0, 1].text(3.2,2.5,r"$\Omega_c$", fontsize=fs)
axes[0, 1].text(-0.7,1.8,r"$\mho_c$", fontsize=fs)
axes[0, 1].text(2.3,1.2,r"$\mho_a$", fontsize=fs)
###################################################################################
# Blue Magic
delta = 2*om21; disp = -0; disp2 = -0.2
excitation(axes[1, 0], [p2[0]+0.25-disp,p2[1]],[p3[0]+0.10-disp,p3[1]+delta],
           fc="r", ec="r",linewidth=10.0, head_width=0.2, head_length=0.2)
excitation(axes[1, 0], [p1[0]+0.25-disp -0.4,p1[1]],[p3[0]+0.10-disp,p3[1]+delta],
           fc="b", ec="b",linewidth=3.0, head_width=0.2, head_length=0.2)
disp = 0.75
pvir = [0.2, p3[1]+delta*0.5]

excitation(axes[1, 0], [p1[0]+0.25-disp,p1[1]],[pvir[0], pvir[1]],
           fc="w", ec="r",linewidth=10.0, head_width=0.2, head_length=0.2, alpha=0.5)

decay(axes[1, 0], [pvir[0], pvir[1]],
              [ p2[0]+disp2, p2[1]], 0.05,20.0,color="g",linewidth=1.0)
decay(axes[1, 0], [p1[0]+0.2, p1[1]],
                  [p3[0], p3[1]], 0.05,20.0,color="g",linewidth=1.0)

axes[1, 0].plot( [p3[0]-l, p3[0]+l], [p3[1]+delta, p3[1]+delta], "k--")
axes[1, 0].plot( [p3[0]-l-1, p3[0]+l-1], [p3[1]+delta*0.5, p3[1]+delta*0.5], "k--")

axes[1, 0].text(0.2,3.8,r"$\Omega_s$", fontsize=fs)
axes[1, 0].text(3.5,3.5,r"$\Omega_c$", fontsize=fs)
axes[1, 0].text(-0.8,5.0,r"$\mho_c$", fontsize=fs)
axes[1, 0].text(2.0,2.2,r"$\mho_a$", fontsize=fs)
axes[1, 0].text(0.8,1.2,r"$\Omega_a$", fontsize=fs)

###################################################################################
# Blue Normal
delta = 0.7; disp = 0.5; disp2 = -0.0
excitation(axes[1, 1], [p1[0]+0.25-disp,p1[1]],[p3[0]+0.10-disp,p3[1]+delta],
           fc="r", ec="r",linewidth=10.0, head_width=0.2, head_length=0.2)
excitation(axes[1, 1], [p2[0]+0.25-disp,p2[1]],[p3[0]+0.10-disp,p3[1]+delta],
           fc="b", ec="b",linewidth=3.0, head_width=0.2, head_length=0.2)

excitation(axes[1, 1], [p2[0]+p1[0]+0.25+disp2,p2[1]+p1[1]],[p2[0]-p3[0]-0.10+disp2,p2[1]+p3[1]+delta],
           fc="w", ec="r",linewidth=10.0, head_width=0.2, head_length=0.2, alpha=0.5)

decay(axes[1,1], [p2[0]-p3[0]-0.10+disp2,p2[1]+p3[1]+delta],
              [ p1[0]+disp2+0.25, p1[1]], 0.05,20.0,color="g",linewidth=1.0)


axes[1, 1].plot( [p3[0]-l, p3[0]+l], [p3[1]+delta, p3[1]+delta], "k--")
axes[1, 1].plot( [p3[0]-l, p3[0]+l], [p3[1]+delta+om21, p3[1]+delta+om21], "k--")
axes[1, 1].text(0.8,0.8,r"$\mho_a$", fontsize=fs)
axes[1, 1].text(3.5,3.5,r"$\mho_c$", fontsize=fs)
axes[1, 1].text(-0.3,2.9,r"$\Omega_c$", fontsize=fs)
axes[1, 1].text(2.0,2.2,r"$\Omega_s$", fontsize=fs)

plt.savefig("lambda_diagrams.png", bbox_inches="tight")
plt.savefig("lambda_diagrams.pdf", bbox_inches="tight")



    


















    
























In [3]:



    


from sympy import zeros, pi, pprint, symbols, I, diff
Ne = 3
Nl = 3
Ep, omega_laser = define_laser_variables(Nl)
epsilonp = [polarization_vector(0, -pi/2, 0, 0, 1) for l in range(Nl)]
delta1, delta2 = symbols("delta1 delta2", real=True)
detuning_knob = [delta1, delta2]

xi = [zeros(Ne, Ne) for l in range(Nl)]
coup = [[(2, 0), (2, 1)], [(2, 0), (2, 1)], [(2, 0), (2, 1)]]
# coup = [[(2, 0), (2, 1)], [(2, 0), (2, 1)], [(2, 0)]]
# coup = [[(2, 0)], [(2, 1)], [(2, 1)]]


for l in range(Nl):
    for pair in coup[l]:
        xi[l][pair[0], pair[1]] = 1
        xi[l][pair[1], pair[0]] = 1

xi



    


















    
Out[3]:







$$\left [ \left[\begin{matrix}0 & 0 & 1\\0 & 0 & 1\\1 & 1 & 0\end{matrix}\right], \quad \left[\begin{matrix}0 & 0 & 1\\0 & 0 & 1\\1 & 1 & 0\end{matrix}\right], \quad \left[\begin{matrix}0 & 0 & 1\\0 & 0 & 1\\1 & 1 & 0\end{matrix}\right]\right ]$$
















In [4]:



    


rm = define_r_components(Ne, xi, explicitly_hermitian=True,
                         helicity=True, p=-1)
rm = helicity_to_cartesian(rm)
rm[2]



    


















    
Out[4]:







$$\left[\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\r_{0;31} & r_{0;32} & 0\end{matrix}\right]$$
















In [5]:



    


t = symbols("t", real=True)
c, ct, theta = define_psi_coefficients(Ne)
thetat = [theta[i]*t for i in range(Ne)]
thetat



    


















    
Out[5]:







$$\left [ t \theta_{1}, \quad t \theta_{2}, \quad t \theta_{3}\right ]$$
















In [6]:



    


omega_level, omega, gamma = define_frequencies(Ne, True)

H = hamiltonian(Ep, epsilonp, detuning_knob, rm, omega_level, omega_laser,
                xi, RWA=True, RF=thetat)
H



    


















    
Out[6]:







$$\left[\begin{matrix}\hbar \left(\omega_{1} + \theta_{1}\right) & 0 & - \frac{e r_{0;31}}{2} e^{- i \left(- t \theta_{1} + t \theta_{3} + t \varpi_{3}\right)} \overline{E^{3}_{0}} - \frac{e r_{0;31}}{2} e^{- i \left(- t \theta_{1} + t \theta_{3} + t \varpi_{2}\right)} \overline{E^{2}_{0}} - \frac{e r_{0;31}}{2} e^{- i \left(- t \theta_{1} + t \theta_{3} + t \varpi_{1}\right)} \overline{E^{1}_{0}}\\0 & \hbar \left(\omega_{2} + \theta_{2}\right) & - \frac{e r_{0;32}}{2} e^{- i \left(- t \theta_{2} + t \theta_{3} + t \varpi_{3}\right)} \overline{E^{3}_{0}} - \frac{e r_{0;32}}{2} e^{- i \left(- t \theta_{2} + t \theta_{3} + t \varpi_{2}\right)} \overline{E^{2}_{0}} - \frac{e r_{0;32}}{2} e^{- i \left(- t \theta_{2} + t \theta_{3} + t \varpi_{1}\right)} \overline{E^{1}_{0}}\\- \frac{e r_{0;31}}{2} E^{1}_{0} e^{- i \left(t \theta_{1} - t \theta_{3} - t \varpi_{1}\right)} - \frac{e r_{0;31}}{2} E^{2}_{0} e^{- i \left(t \theta_{1} - t \theta_{3} - t \varpi_{2}\right)} - \frac{e r_{0;31}}{2} E^{3}_{0} e^{- i \left(t \theta_{1} - t \theta_{3} - t \varpi_{3}\right)} & - \frac{e r_{0;32}}{2} E^{1}_{0} e^{- i \left(t \theta_{2} - t \theta_{3} - t \varpi_{1}\right)} - \frac{e r_{0;32}}{2} E^{2}_{0} e^{- i \left(t \theta_{2} - t \theta_{3} - t \varpi_{2}\right)} - \frac{e r_{0;32}}{2} E^{3}_{0} e^{- i \left(t \theta_{2} - t \theta_{3} - t \varpi_{3}\right)} & \hbar \left(\omega_{3} + \theta_{3}\right)\end{matrix}\right]$$
















In [7]:



    


omega_level, omega, gamma = define_frequencies(Ne, True)

H = hamiltonian(Ep, epsilonp, detuning_knob, rm, omega_level, omega_laser,
                xi, RWA=True, RF=thetat)

H



    


















    
Out[7]:







$$\left[\begin{matrix}\hbar \left(\omega_{1} + \theta_{1}\right) & 0 & - \frac{e r_{0;31}}{2} e^{- i \left(- t \theta_{1} + t \theta_{3} + t \varpi_{3}\right)} \overline{E^{3}_{0}} - \frac{e r_{0;31}}{2} e^{- i \left(- t \theta_{1} + t \theta_{3} + t \varpi_{2}\right)} \overline{E^{2}_{0}} - \frac{e r_{0;31}}{2} e^{- i \left(- t \theta_{1} + t \theta_{3} + t \varpi_{1}\right)} \overline{E^{1}_{0}}\\0 & \hbar \left(\omega_{2} + \theta_{2}\right) & - \frac{e r_{0;32}}{2} e^{- i \left(- t \theta_{2} + t \theta_{3} + t \varpi_{3}\right)} \overline{E^{3}_{0}} - \frac{e r_{0;32}}{2} e^{- i \left(- t \theta_{2} + t \theta_{3} + t \varpi_{2}\right)} \overline{E^{2}_{0}} - \frac{e r_{0;32}}{2} e^{- i \left(- t \theta_{2} + t \theta_{3} + t \varpi_{1}\right)} \overline{E^{1}_{0}}\\- \frac{e r_{0;31}}{2} E^{1}_{0} e^{- i \left(t \theta_{1} - t \theta_{3} - t \varpi_{1}\right)} - \frac{e r_{0;31}}{2} E^{2}_{0} e^{- i \left(t \theta_{1} - t \theta_{3} - t \varpi_{2}\right)} - \frac{e r_{0;31}}{2} E^{3}_{0} e^{- i \left(t \theta_{1} - t \theta_{3} - t \varpi_{3}\right)} & - \frac{e r_{0;32}}{2} E^{1}_{0} e^{- i \left(t \theta_{2} - t \theta_{3} - t \varpi_{1}\right)} - \frac{e r_{0;32}}{2} E^{2}_{0} e^{- i \left(t \theta_{2} - t \theta_{3} - t \varpi_{2}\right)} - \frac{e r_{0;32}}{2} E^{3}_{0} e^{- i \left(t \theta_{2} - t \theta_{3} - t \varpi_{3}\right)} & \hbar \left(\omega_{3} + \theta_{3}\right)\end{matrix}\right]$$
















In [8]:



    


hbar, e = symbols("hbar, e", positive=True)
t, Z, vZ, delta = symbols("t, Z, vZ, delta", real=True)
Oms = symbols(r"\Omega_s")
Omc = symbols(r"\Omega_c")
Oma = symbols(r"\Omega_a")
Omsp = symbols(r"\mho_s")
Omcp = symbols(r"\mho_c")
Omap = symbols(r"\mho_a")
rho = define_density_matrix(Ne, explicitly_hermitian=True, variables=[t, Z, vZ])
rho



    


















    
Out[8]:







$$\left[\begin{matrix}\rho_{11}{\left (t,Z,vZ \right )} & \overline{\rho_{21}{\left (t,Z,vZ \right )}} & \overline{\rho_{31}{\left (t,Z,vZ \right )}}\\\rho_{21}{\left (t,Z,vZ \right )} & \rho_{22}{\left (t,Z,vZ \right )} & \overline{\rho_{32}{\left (t,Z,vZ \right )}}\\\rho_{31}{\left (t,Z,vZ \right )} & \rho_{32}{\left (t,Z,vZ \right )} & \rho_{33}{\left (t,Z,vZ \right )}\end{matrix}\right]$$

We build the interpretation of this general naming scheme into whichever memory protocol we are interested in.





In [9]:



    


magic = True
# magic = False
red = True
# red = False
if (magic and red) or (not magic and not red):
    print "Naming scheme 1"
    ss_lin = {rho[0, 0]: 0, rho[1, 1]:1, rho[2, 2]: 0}
    ss_om = {rm[2][2, 0]*Ep[0]: hbar*Omc/e,
             rm[2][2, 1]*Ep[0]: hbar*Omcp/e,
             rm[2][2, 1]*Ep[1]: hbar*Oms/e,
             rm[2][2, 0]*Ep[1]: hbar*Omsp/e,
             rm[2][2, 1]*Ep[2]: hbar*Oma/e,
             rm[2][2, 0]*Ep[2]: hbar*Omap/e}

    ss_delta = {omega_laser[0]: delta1 + omega_level[2]-omega_level[0],
                omega_laser[1]: delta2 + omega_level[2]-omega_level[1]}
    ss_delta.update({omega_laser[2]: ss_delta[omega_laser[0]] + omega_level[1]-omega_level[0]})

else:
    print "Naming scheme 2"
    ss_lin = {rho[0, 0]: 1, rho[1, 1]:0, rho[2, 2]: 0}
    ss_om = {rm[2][2, 0]*Ep[0]: hbar*Oms/e,
             rm[2][2, 1]*Ep[0]: hbar*Omsp/e,
             rm[2][2, 1]*Ep[1]: hbar*Omc/e,
             rm[2][2, 0]*Ep[1]: hbar*Omcp/e,
             rm[2][2, 0]*Ep[2]: hbar*Oma/e,
             rm[2][2, 1]*Ep[2]: hbar*Omap/e}

    ss_delta = {omega_laser[0]: delta1 + omega_level[2]-omega_level[0],
                omega_laser[1]: delta2 + omega_level[2]-omega_level[1]}
    ss_delta.update({omega_laser[2]: ss_delta[omega_laser[1]] - omega_level[1]+omega_level[0]})

ss_delta0 = {delta1: delta, delta2: delta}
ss_om_con = {key.conjugate() : ss_om[key].conjugate() for key in ss_om}
ss_om.update(ss_om_con)



    


















    






Naming scheme 1

We rename the couplings in the Hamiltonian





In [10]:



    


H = H.subs(ss_om)
H



    


















    
Out[10]:







$$\left[\begin{matrix}\hbar \left(\omega_{1} + \theta_{1}\right) & 0 & - \frac{\hbar \overline{\mho_a}}{2} e^{- i \left(- t \theta_{1} + t \theta_{3} + t \varpi_{3}\right)} - \frac{\hbar \overline{\mho_s}}{2} e^{- i \left(- t \theta_{1} + t \theta_{3} + t \varpi_{2}\right)} - \frac{\hbar \overline{\Omega_c}}{2} e^{- i \left(- t \theta_{1} + t \theta_{3} + t \varpi_{1}\right)}\\0 & \hbar \left(\omega_{2} + \theta_{2}\right) & - \frac{\hbar \overline{\Omega_a}}{2} e^{- i \left(- t \theta_{2} + t \theta_{3} + t \varpi_{3}\right)} - \frac{\hbar \overline{\Omega_s}}{2} e^{- i \left(- t \theta_{2} + t \theta_{3} + t \varpi_{2}\right)} - \frac{\hbar \overline{\mho_c}}{2} e^{- i \left(- t \theta_{2} + t \theta_{3} + t \varpi_{1}\right)}\\- \frac{\Omega_c \hbar}{2} e^{- i \left(t \theta_{1} - t \theta_{3} - t \varpi_{1}\right)} - \frac{\mho_a \hbar}{2} e^{- i \left(t \theta_{1} - t \theta_{3} - t \varpi_{3}\right)} - \frac{\mho_s \hbar}{2} e^{- i \left(t \theta_{1} - t \theta_{3} - t \varpi_{2}\right)} & - \frac{\Omega_a \hbar}{2} e^{- i \left(t \theta_{2} - t \theta_{3} - t \varpi_{3}\right)} - \frac{\Omega_s \hbar}{2} e^{- i \left(t \theta_{2} - t \theta_{3} - t \varpi_{2}\right)} - \frac{\mho_c \hbar}{2} e^{- i \left(t \theta_{2} - t \theta_{3} - t \varpi_{1}\right)} & \hbar \left(\omega_{3} + \theta_{3}\right)\end{matrix}\right]$$

We calculate the full equations.





In [11]:



    


gamma[1, 0] = 0
gamma[0, 1] = 0

rhs = I/hbar*(rho*H-H*rho) + lindblad_terms(gamma, rho, Ne)

We select only the equations for the coherences





In [12]:



    


rhs = Matrix(sum([sum([[rhs[i, j]] for j in range(i)],[]) for i in range(Ne)], []))
lhs = Matrix(sum([sum([[diff(rho[i, j],t)] for j in range(i)],[]) for i in range(Ne)], []))

eqsign = symbols("=")
eqs = Matrix([[lhs[i], eqsign, rhs[i]] for i in range((Ne**2-Ne)/2)])
eqs = eqs.expand()
eqs



    


















    
Out[12]:







$$\left[\begin{matrix}\frac{\partial}{\partial t} \rho_{21}{\left (t,Z,vZ \right )} & = & - \frac{i \Omega_c}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{1}} \overline{\rho_{32}{\left (t,Z,vZ \right )}} - \frac{i \mho_a}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{3}} \overline{\rho_{32}{\left (t,Z,vZ \right )}} - \frac{i \mho_s}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{2}} \overline{\rho_{32}{\left (t,Z,vZ \right )}} + i \omega_{1} \rho_{21}{\left (t,Z,vZ \right )} - i \omega_{2} \rho_{21}{\left (t,Z,vZ \right )} + i \theta_{1} \rho_{21}{\left (t,Z,vZ \right )} - i \theta_{2} \rho_{21}{\left (t,Z,vZ \right )} + \frac{i \overline{\Omega_a}}{2} \rho_{31}{\left (t,Z,vZ \right )} e^{i t \theta_{2}} e^{- i t \theta_{3}} e^{- i t \varpi_{3}} + \frac{i \overline{\Omega_s}}{2} \rho_{31}{\left (t,Z,vZ \right )} e^{i t \theta_{2}} e^{- i t \theta_{3}} e^{- i t \varpi_{2}} + \frac{i \overline{\mho_c}}{2} \rho_{31}{\left (t,Z,vZ \right )} e^{i t \theta_{2}} e^{- i t \theta_{3}} e^{- i t \varpi_{1}}\\\frac{\partial}{\partial t} \rho_{31}{\left (t,Z,vZ \right )} & = & \frac{i \Omega_a}{2} \rho_{21}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{3}} + \frac{i \Omega_c}{2} \rho_{11}{\left (t,Z,vZ \right )} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{1}} - \frac{i \Omega_c}{2} \rho_{33}{\left (t,Z,vZ \right )} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{1}} + \frac{i \Omega_s}{2} \rho_{21}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{2}} + \frac{i \mho_a}{2} \rho_{11}{\left (t,Z,vZ \right )} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{3}} - \frac{i \mho_a}{2} \rho_{33}{\left (t,Z,vZ \right )} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{3}} + \frac{i \mho_c}{2} \rho_{21}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{1}} + \frac{i \mho_s}{2} \rho_{11}{\left (t,Z,vZ \right )} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{2}} - \frac{i \mho_s}{2} \rho_{33}{\left (t,Z,vZ \right )} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{2}} - \frac{\gamma_{31}}{2} \rho_{31}{\left (t,Z,vZ \right )} - \frac{\gamma_{32}}{2} \rho_{31}{\left (t,Z,vZ \right )} + i \omega_{1} \rho_{31}{\left (t,Z,vZ \right )} - i \omega_{3} \rho_{31}{\left (t,Z,vZ \right )} + i \theta_{1} \rho_{31}{\left (t,Z,vZ \right )} - i \theta_{3} \rho_{31}{\left (t,Z,vZ \right )}\\\frac{\partial}{\partial t} \rho_{32}{\left (t,Z,vZ \right )} & = & \frac{i \Omega_a}{2} \rho_{22}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{3}} - \frac{i \Omega_a}{2} \rho_{33}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{3}} + \frac{i \Omega_c}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{1}} \overline{\rho_{21}{\left (t,Z,vZ \right )}} + \frac{i \Omega_s}{2} \rho_{22}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{2}} - \frac{i \Omega_s}{2} \rho_{33}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{2}} + \frac{i \mho_a}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{3}} \overline{\rho_{21}{\left (t,Z,vZ \right )}} + \frac{i \mho_c}{2} \rho_{22}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{1}} - \frac{i \mho_c}{2} \rho_{33}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{1}} + \frac{i \mho_s}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{2}} \overline{\rho_{21}{\left (t,Z,vZ \right )}} - \frac{\gamma_{31}}{2} \rho_{32}{\left (t,Z,vZ \right )} - \frac{\gamma_{32}}{2} \rho_{32}{\left (t,Z,vZ \right )} + i \omega_{2} \rho_{32}{\left (t,Z,vZ \right )} - i \omega_{3} \rho_{32}{\left (t,Z,vZ \right )} + i \theta_{2} \rho_{32}{\left (t,Z,vZ \right )} - i \theta_{3} \rho_{32}{\left (t,Z,vZ \right )}\end{matrix}\right]$$
















In [13]:



    


epsilon = symbols("epsilon", positive=True)
ss_lin2 = {rho[1, 0]: rho[1, 0]*epsilon,
           rho[2, 0]: rho[2, 0]*epsilon,
           rho[2, 1]: rho[2, 1]*epsilon,
           Oms: Oms*epsilon}
ss_lin.update(ss_lin2)
ss_lin



    


















    
Out[13]:







$$\left \{ \Omega_s : \Omega_s \epsilon, \quad \rho_{11}{\left (t,Z,vZ \right )} : 0, \quad \rho_{21}{\left (t,Z,vZ \right )} : \epsilon \rho_{21}{\left (t,Z,vZ \right )}, \quad \rho_{22}{\left (t,Z,vZ \right )} : 1, \quad \rho_{31}{\left (t,Z,vZ \right )} : \epsilon \rho_{31}{\left (t,Z,vZ \right )}, \quad \rho_{32}{\left (t,Z,vZ \right )} : \epsilon \rho_{32}{\left (t,Z,vZ \right )}, \quad \rho_{33}{\left (t,Z,vZ \right )} : 0\right \}$$
















In [14]:



    


eqs_lin = eqs.subs(ss_lin)
eqs_lin = eqs_lin.subs({epsilon**2: 0})
eqs_lin = eqs_lin.subs({epsilon: 1})
eqs_lin = Matrix([eqs_lin[0, :], eqs_lin[1, :], eqs_lin[2, :]])
eqs_lin



    


















    
Out[14]:







$$\left[\begin{matrix}\frac{\partial}{\partial t} \rho_{21}{\left (t,Z,vZ \right )} & = & - \frac{i \Omega_c}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{1}} \overline{\rho_{32}{\left (t,Z,vZ \right )}} - \frac{i \mho_a}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{3}} \overline{\rho_{32}{\left (t,Z,vZ \right )}} - \frac{i \mho_s}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{2}} \overline{\rho_{32}{\left (t,Z,vZ \right )}} + i \omega_{1} \rho_{21}{\left (t,Z,vZ \right )} - i \omega_{2} \rho_{21}{\left (t,Z,vZ \right )} + i \theta_{1} \rho_{21}{\left (t,Z,vZ \right )} - i \theta_{2} \rho_{21}{\left (t,Z,vZ \right )} + \frac{i \overline{\Omega_a}}{2} \rho_{31}{\left (t,Z,vZ \right )} e^{i t \theta_{2}} e^{- i t \theta_{3}} e^{- i t \varpi_{3}} + \frac{i \overline{\mho_c}}{2} \rho_{31}{\left (t,Z,vZ \right )} e^{i t \theta_{2}} e^{- i t \theta_{3}} e^{- i t \varpi_{1}}\\\frac{\partial}{\partial t} \rho_{31}{\left (t,Z,vZ \right )} & = & \frac{i \Omega_a}{2} \rho_{21}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{3}} + \frac{i \mho_c}{2} \rho_{21}{\left (t,Z,vZ \right )} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{1}} - \frac{\gamma_{31}}{2} \rho_{31}{\left (t,Z,vZ \right )} - \frac{\gamma_{32}}{2} \rho_{31}{\left (t,Z,vZ \right )} + i \omega_{1} \rho_{31}{\left (t,Z,vZ \right )} - i \omega_{3} \rho_{31}{\left (t,Z,vZ \right )} + i \theta_{1} \rho_{31}{\left (t,Z,vZ \right )} - i \theta_{3} \rho_{31}{\left (t,Z,vZ \right )}\\\frac{\partial}{\partial t} \rho_{32}{\left (t,Z,vZ \right )} & = & \frac{i \Omega_a}{2} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{3}} + \frac{i \Omega_c}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{1}} \overline{\rho_{21}{\left (t,Z,vZ \right )}} + \frac{i \Omega_s}{2} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{2}} + \frac{i \mho_a}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{3}} \overline{\rho_{21}{\left (t,Z,vZ \right )}} + \frac{i \mho_c}{2} e^{- i t \theta_{2}} e^{i t \theta_{3}} e^{i t \varpi_{1}} + \frac{i \mho_s}{2} e^{- i t \theta_{1}} e^{i t \theta_{3}} e^{i t \varpi_{2}} \overline{\rho_{21}{\left (t,Z,vZ \right )}} - \frac{\gamma_{31}}{2} \rho_{32}{\left (t,Z,vZ \right )} - \frac{\gamma_{32}}{2} \rho_{32}{\left (t,Z,vZ \right )} + i \omega_{2} \rho_{32}{\left (t,Z,vZ \right )} - i \omega_{3} \rho_{32}{\left (t,Z,vZ \right )} + i \theta_{2} \rho_{32}{\left (t,Z,vZ \right )} - i \theta_{3} \rho_{32}{\left (t,Z,vZ \right )}\end{matrix}\right]$$
















In [15]:



    


eqs_phase = [-theta[0]+theta[2]+omega_laser[0],
             -theta[1]+theta[2]+omega_laser[1],]

theta_sol = solve(eqs_phase, theta)
theta_sol, ss_delta



    


















    
Out[15]:







$$\left ( \left \{ \theta_{1} : \theta_{3} + \varpi_{1}, \quad \theta_{2} : \theta_{3} + \varpi_{2}\right \}, \quad \left \{ \varpi_{1} : \delta_{1} - \omega_{1} + \omega_{3}, \quad \varpi_{2} : \delta_{2} - \omega_{2} + \omega_{3}, \quad \varpi_{3} : \delta_{1} - 2 \omega_{1} + \omega_{2} + \omega_{3}\right \}\right )$$
















In [16]:



    


eqs_lin_new = eqs_lin.subs(theta_sol).subs(ss_delta).subs(ss_delta0).expand()
eqs_lin_new



    


















    
Out[16]:







$$\left[\begin{matrix}\frac{\partial}{\partial t} \rho_{21}{\left (t,Z,vZ \right )} & = & - \frac{i \Omega_c}{2} \overline{\rho_{32}{\left (t,Z,vZ \right )}} - \frac{i \mho_a}{2} e^{- i \omega_{1} t} e^{i \omega_{2} t} \overline{\rho_{32}{\left (t,Z,vZ \right )}} - \frac{i \mho_s}{2} e^{i \omega_{1} t} e^{- i \omega_{2} t} \overline{\rho_{32}{\left (t,Z,vZ \right )}} + \frac{i \overline{\Omega_a}}{2} \rho_{31}{\left (t,Z,vZ \right )} e^{2 i \omega_{1} t} e^{- 2 i \omega_{2} t} + \frac{i \overline{\mho_c}}{2} \rho_{31}{\left (t,Z,vZ \right )} e^{i \omega_{1} t} e^{- i \omega_{2} t}\\\frac{\partial}{\partial t} \rho_{31}{\left (t,Z,vZ \right )} & = & \frac{i \Omega_a}{2} \rho_{21}{\left (t,Z,vZ \right )} e^{- 2 i \omega_{1} t} e^{2 i \omega_{2} t} + \frac{i \mho_c}{2} \rho_{21}{\left (t,Z,vZ \right )} e^{- i \omega_{1} t} e^{i \omega_{2} t} + i \delta \rho_{31}{\left (t,Z,vZ \right )} - \frac{\gamma_{31}}{2} \rho_{31}{\left (t,Z,vZ \right )} - \frac{\gamma_{32}}{2} \rho_{31}{\left (t,Z,vZ \right )}\\\frac{\partial}{\partial t} \rho_{32}{\left (t,Z,vZ \right )} & = & \frac{i \Omega_a}{2} e^{- 2 i \omega_{1} t} e^{2 i \omega_{2} t} + \frac{i \Omega_c}{2} \overline{\rho_{21}{\left (t,Z,vZ \right )}} + \frac{i \Omega_s}{2} + \frac{i \mho_a}{2} e^{- i \omega_{1} t} e^{i \omega_{2} t} \overline{\rho_{21}{\left (t,Z,vZ \right )}} + \frac{i \mho_c}{2} e^{- i \omega_{1} t} e^{i \omega_{2} t} + \frac{i \mho_s}{2} e^{i \omega_{1} t} e^{- i \omega_{2} t} \overline{\rho_{21}{\left (t,Z,vZ \right )}} + i \delta \rho_{32}{\left (t,Z,vZ \right )} - \frac{\gamma_{31}}{2} \rho_{32}{\left (t,Z,vZ \right )} - \frac{\gamma_{32}}{2} \rho_{32}{\left (t,Z,vZ \right )}\end{matrix}\right]$$
















In [17]:



    


Gamma_s = symbols("Gamma_s")
sol_ad=solve([eqs_lin_new[1, 2], eqs_lin_new[2, 2]], [rho[2, 0], rho[2, 1]])
sol_ad[rho[2,0]]=sol_ad[rho[2,0]].subs({-2*I*delta+gamma[2,0]+gamma[2,1]:2*Gamma_s}).expand()
sol_ad[rho[2,1]]=sol_ad[rho[2,1]].subs({-2*I*delta+gamma[2,0]+gamma[2,1]:2*Gamma_s}).expand()
sol_ad



    


















    
Out[17]:







$$\left \{ \rho_{31}{\left (t,Z,vZ \right )} : \frac{i \Omega_a}{2 \Gamma_{s}} \rho_{21}{\left (t,Z,vZ \right )} e^{- 2 i \omega_{1} t} e^{2 i \omega_{2} t} + \frac{i \mho_c}{2 \Gamma_{s}} \rho_{21}{\left (t,Z,vZ \right )} e^{- i \omega_{1} t} e^{i \omega_{2} t}, \quad \rho_{32}{\left (t,Z,vZ \right )} : \frac{i \Omega_a}{2 \Gamma_{s}} e^{- 2 i \omega_{1} t} e^{2 i \omega_{2} t} + \frac{i \Omega_c}{2 \Gamma_{s}} \overline{\rho_{21}{\left (t,Z,vZ \right )}} + \frac{i \Omega_s}{2 \Gamma_{s}} + \frac{i \mho_a}{2 \Gamma_{s}} e^{- i \omega_{1} t} e^{i \omega_{2} t} \overline{\rho_{21}{\left (t,Z,vZ \right )}} + \frac{i \mho_c}{2 \Gamma_{s}} e^{- i \omega_{1} t} e^{i \omega_{2} t} + \frac{i \mho_s}{2 \Gamma_{s}} e^{i \omega_{1} t} e^{- i \omega_{2} t} \overline{\rho_{21}{\left (t,Z,vZ \right )}}\right \}$$
















In [18]:



    


eq_s=eqs_lin_new[0, 2].subs(sol_ad).expand()
ss_slow = {exp( I*omega_level[0]*t)*exp(-I*omega_level[1]*t): 0,
           exp(-I*omega_level[0]*t)*exp( I*omega_level[1]*t): 0}
eq_s = eq_s.subs(ss_slow)
eq_s



    


















    
Out[18]:







$$- \frac{\Omega_c \overline{\Omega_c}}{4 \overline{\Gamma_{s}}} \rho_{21}{\left (t,Z,vZ \right )} - \frac{\Omega_c \overline{\Omega_s}}{4 \overline{\Gamma_{s}}} - \frac{\mho_a \overline{\mho_a}}{4 \overline{\Gamma_{s}}} \rho_{21}{\left (t,Z,vZ \right )} - \frac{\mho_a \overline{\mho_c}}{4 \overline{\Gamma_{s}}} - \frac{\mho_s \overline{\mho_s}}{4 \overline{\Gamma_{s}}} \rho_{21}{\left (t,Z,vZ \right )} - \frac{\Omega_a \overline{\Omega_a}}{4 \Gamma_{s}} \rho_{21}{\left (t,Z,vZ \right )} - \frac{\mho_c \overline{\mho_c}}{4 \Gamma_{s}} \rho_{21}{\left (t,Z,vZ \right )}$$
















In [19]:



    


epsilon = symbols("epsilon", real=True)
ss_lin2 = {Oms: Oms*epsilon, Omsp: Omsp*epsilon,
           Oma: Oma*epsilon, Omap: Omap*epsilon}
eq_rho21 = eq_s.subs(ss_lin2).subs({epsilon**2: 0}).subs({epsilon: 1})
eq_rho21



    


















    
Out[19]:







$$- \frac{\Omega_c \overline{\Omega_c}}{4 \overline{\Gamma_{s}}} \rho_{21}{\left (t,Z,vZ \right )} - \frac{\Omega_c \overline{\Omega_s}}{4 \overline{\Gamma_{s}}} - \frac{\mho_a \overline{\mho_c}}{4 \overline{\Gamma_{s}}} - \frac{\mho_c \overline{\mho_c}}{4 \Gamma_{s}} \rho_{21}{\left (t,Z,vZ \right )}$$

Wave equation

From Maxwell's equations in a dielectric medium it can be shown that in the abscence of bound charges, and magnetization currents, the electric field and the polazation of the medium follow the inhomogeneous wave equation.

\begin{equation} \nabla^2 \vec{E} - \frac{1}{c^2} \partial^2_t \vec{E}= \mu_0 \partial_t^2 \vec{P} \end{equation}

We have also taken our fields to be of the form

\begin{equation} \vec{E}(t,\vec{R})=\vec{E}^{+}+\vec{E}^{-}=\frac{1}{2} \sum_l \vec{E}^{l(+)}(t,\vec{R}) e^{i(\vec{k}^l \cdot \vec{R} -\omega^l t)} +c.c. \end{equation}

In the slowly varying envelope approximation we have

\begin{equation} (\vec{k}_l\cdot\nabla+\frac{\varpi_l}{c^2}\partial_t) \vec{\epsilon}_l\cdot\vec{E}_l^{(+)}= -\mu_0 n e\varpi_l^2 Tr( \vec{\epsilon}_l\cdot \vec{\hat{r}} \hat{\rho}) \end{equation}

Where $n$ is the atomic density (this should be shown explicitly later).

\begin{equation} \partial_t (\vec{\epsilon}_l\cdot\vec{E}_l^{(+)})= \frac{c^2}{\varpi_l}\vec{k}_l\cdot\nabla (\vec{\epsilon}_l\cdot\vec{E}_l^{(+)}) -c^2\mu_0 n e\varpi_l Tr( \vec{\epsilon}_l\cdot \vec{\hat{r}} \hat{\rho}) \end{equation}

For forward going beams $\vec{k}_l = \vec{e}_z$

\begin{equation} \partial_t (\vec{\epsilon}_l\cdot\vec{E}_l^{(+)})= c\partial_z (\vec{\epsilon}_l\cdot\vec{E}_l^{(+)}) -c^2\mu_0 n e\varpi_l Tr( \vec{\epsilon}_l\cdot \vec{\hat{r}} \hat{\rho}) \end{equation}

And we will only take the signal and antiStokes electric fields as dynamic variables





In [20]:



    


from fast import polarization_vector
from sympy import pi, trace



    











In [21]:



    


c, epsilon0, mu0, n = symbols("c, epsilon0, mu0, n", positive=True)
es=polarization_vector(0, 0, 0, 0, 1)
ea=polarization_vector(0, 0, pi/4, 0, 1)
es, ea



    


















    
Out[21]:







$$\left ( \left[\begin{matrix}1\\0\\0\end{matrix}\right], \quad \left[\begin{matrix}0\\1\\0\end{matrix}\right]\right )$$
















In [22]:



    


rp = [rm[i].adjoint() for i in range(3)]
rp = [rp[2] for i in range(3)]

We determine which of the numbered fields is the signal field





In [23]:



    


Z = symbols("Z", real=True)
Ep, omega_laser = define_laser_variables(Nl, variables=[t, Z])

if (magic and red) or (not magic and not red):
    Es = Ep[1]
    omega_s = omega_laser[1]
else:
    Es = Ep[0]
    omega_s = omega_laser[0]

We get the signal equation





In [24]:



    


eq_signal = c*diff(Es, Z)- c**2*mu0*n*e*omega_s*trace(cartesian_dot_product(rp, es)*rho)
eq_signal = eq_signal
eq_signal



    


















    
Out[24]:







$$- c^{2} e \mu_{0} n \varpi_{2} \left(r_{0;31} \rho_{31}{\left (t,Z,vZ \right )} + r_{0;32} \rho_{32}{\left (t,Z,vZ \right )}\right) + c \frac{\partial}{\partial Z} \operatorname{E^{2}_{0}}{\left (t,Z \right )}$$
















In [25]:



    


eq_antistokes = c*diff(Ep[2], Z)- c**2*mu0*n*e*omega_laser[2]*trace(cartesian_dot_product(rp, ea)*rho)
eq_antistokes = eq_antistokes
eq_antistokes



    


















    
Out[25]:







$$- c^{2} e \mu_{0} n \varpi_{3} \left(r_{0;31} \rho_{31}{\left (t,Z,vZ \right )} + r_{0;32} \rho_{32}{\left (t,Z,vZ \right )}\right) + c \frac{\partial}{\partial Z} \operatorname{E^{3}_{0}}{\left (t,Z \right )}$$
















In [26]:



    


eq_signal = eq_signal.subs(sol_ad).expand(ss_slow).subs(exp(-I*omega_level[0]*t)*exp( I*omega_level[1]*t), 0)
eq_signal



    


















    
Out[26]:







$$c \frac{\partial}{\partial Z} \operatorname{E^{2}_{0}}{\left (t,Z \right )} - \frac{i \Omega_c e r_{0;32}}{2 \Gamma_{s}} c^{2} \mu_{0} n \varpi_{2} \overline{\rho_{21}{\left (t,Z,vZ \right )}} - \frac{i \Omega_s e r_{0;32}}{2 \Gamma_{s}} c^{2} \mu_{0} n \varpi_{2} - \frac{i \mho_s e r_{0;32}}{2 \Gamma_{s}} c^{2} \mu_{0} n \varpi_{2} e^{i \omega_{1} t} e^{- i \omega_{2} t} \overline{\rho_{21}{\left (t,Z,vZ \right )}}$$
















In [27]:



    


eq_antistokes = eq_antistokes.subs(sol_ad).expand(ss_slow).subs(exp(-I*omega_level[0]*t)*exp( I*omega_level[1]*t), 0)
eq_antistokes



    


















    
Out[27]:







$$c \frac{\partial}{\partial Z} \operatorname{E^{3}_{0}}{\left (t,Z \right )} - \frac{i \Omega_c e r_{0;32}}{2 \Gamma_{s}} c^{2} \mu_{0} n \varpi_{3} \overline{\rho_{21}{\left (t,Z,vZ \right )}} - \frac{i \Omega_s e r_{0;32}}{2 \Gamma_{s}} c^{2} \mu_{0} n \varpi_{3} - \frac{i \mho_s e r_{0;32}}{2 \Gamma_{s}} c^{2} \mu_{0} n \varpi_{3} e^{i \omega_{1} t} e^{- i \omega_{2} t} \overline{\rho_{21}{\left (t,Z,vZ \right )}}$$
















In [28]:



    


eq_rho21



    


















    
Out[28]:







$$- \frac{\Omega_c \overline{\Omega_c}}{4 \overline{\Gamma_{s}}} \rho_{21}{\left (t,Z,vZ \right )} - \frac{\Omega_c \overline{\Omega_s}}{4 \overline{\Gamma_{s}}} - \frac{\mho_a \overline{\mho_c}}{4 \overline{\Gamma_{s}}} - \frac{\mho_c \overline{\mho_c}}{4 \Gamma_{s}} \rho_{21}{\left (t,Z,vZ \right )}$$

These equations have the same form as those in [1], I think.

[1] https://arxiv.org/abs/1601.00157





In [ ]:

Content source: oscarlazoarjona/quantum_memories

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