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<?xml version="1.0"?>
<simulation xmds-version="2">
<name>Genas_system</name>
<author>Eugeniy Mikhailov and Simon Rochester</author>
<description>
License GPL.
Solving 4 level atom in 0->1 configuration,
with field propagation along spatial axis Z
no Doppler broadening
Fields:
light field, default circularly polarized
rf field along x
static B-field along z
M=0 upper state sublevel can be shifted independently
|
| ______ |2> m=1
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| _______|3> m=0
|...................................... magnetically unshifted energy
|
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|_______|4> m=-1
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| ______________|1> m=0
We are solving
dE/dz+(1/c)*dE/dt=i*eta*rho_ji, where j level is higher then i.
Note that E is actually a Rabi frequency of electromagnetic field not the EM field
in xmds terms it looks like
dE_dz = i*eta*rhoji - 1/c*L[E], here we moved t dependence to Fourier space
VERY IMPORTANT: all Rabi frequency should be given in [1/s], if you want to
normalize it to something else look drho/dt equation.
No need to renormalizes eta as long as its express through i
the upper level decay rate in the same units as Rabi frequency.
</description>
<features>
<globals>
<![CDATA[
const double pi = M_PI;
const double c=3.e8;
const double lambda=794.7e-9; //wavelength in m
const double N=6e10*(1e6); //number of particles per cubic m i.e. density
const double Gamma=1*(2*M_PI*1e6); // characteristic decay rate of upper level used for eta calculations expressed in [1/s]
const double eta = 3*lambda*lambda*N*Gamma/16.0/M_PI; // eta constant in the wave equation for Rabi frequency. Units are [1/(m s)]
// repopulation rate (atoms flying in/out the laser beam) in [1/s]
const double gt=0.01*(2*M_PI*1e6);
// Natural linewidth of upper state in [1/s]
const double g0=1.*(2*M_PI*1e6);
const double zmax=7e-2;
complex Exc, Eyc; // Complex-conjugated Rabi frequency
complex r21, r31, r41, r32, r42, r43, r44; // density matrix elements
double delta0z(double z) {
// calculates detuning/shift of exited m=0 sublevel as a function of z
double delta;
delta = ( atan(100*(1.0-2.*z/zmax))/pi*2 )*delta0;
return delta;
}
]]>
</globals>
<benchmark />
<arguments>
<!-- Real and imaginary parts of complex Rabi frequency in [1/s] -->
<argument name="ExReo" type="real" default_value="(9.+0.00000001)*(2*M_PI*1.e4)" />
<argument name="ExImo" type="real" default_value="0." />
<argument name="EyReo" type="real" default_value="0." />
<argument name="EyImo" type="real" default_value="(9.-0.00000001)*(2*M_PI*1.e4)" />
<!-- light detuning in [1/s] from unshifted m=0 level-->
<argument name="delta" type="real" default_value="0.0*(2*M_PI*1e6)" />
<!--shift of upper-state M=0 sublevel-->
<argument name="delta0" type="real" default_value="14.*(2*M_PI*1e6)" />
<!--Static B-field Larmor frequency-->
<argument name="deltaL" type="real" default_value="15.*(2*M_PI*1e6)" />
<!--normalized rf Rabi frequency:
Orf = (Larmor frequency due to peak rf field)/sqrt(2)-->
<argument name="Orf" type="real" default_value="0.1*(2*M_PI*1e6)" />
<!--rf frequency-->
<argument name="orf" type="real" default_value="1.*(2*M_PI*1e6)" />
</arguments>
<bing />
<fftw plan="patient" />
<openmp />
<auto_vectorise />
<validation kind="run-time"/>
</features>
<!-- 'z' and 't' to have dimensions [m] and [s] -->
<geometry>
<propagation_dimension> z </propagation_dimension>
<transverse_dimensions>
<dimension name="t" lattice="1000" domain="(-1.0e-6, 1.0e-6)" />
</transverse_dimensions>
</geometry>
<!-- Rabi frequency -->
<vector name="E_field" type="complex" initial_space="t">
<components>Ex Ey</components>
<initialisation>
<![CDATA[
// Initial (at starting 'z' position) electromagnetic field does not depend on detuning
// as well as time
Ex=(ExReo+i*ExImo)*exp(-pow( ((t-0.0)/1e-7),2) );
Ey=(EyReo+i*EyImo)*exp(-pow( ((t-0.0)/1e-7),2) );
]]>
</initialisation>
</vector>
<vector name="density_matrix" type="complex" initial_space="t">
<components>r11 r22 r33 r12 r13 r14 r23 r24 r34 r44</components>
<initialisation>
<![CDATA[
r11 = 1; r22 = 0; r33 = 0; r44 = 0;
r12 = 0; r13 = 0; r14 = 0;
r23 = 0; r24 = 0;
r34 = 0;
]]>
</initialisation>
</vector>
<vector name="rfField" type="real" initial_space="t">
<components> rf </components>
<initialisation>
<![CDATA[
rf = Orf*sin(orf*t);
]]>
</initialisation>
</vector>
<sequence>
<integrate algorithm="ARK45" tolerance="0.05e-7" interval="zmax">
<samples>200 200</samples>
<operators>
<operator kind="cross_propagation" algorithm="SI" propagation_dimension="t">
<integration_vectors>density_matrix</integration_vectors>
<dependencies>E_field rfField</dependencies>
<boundary_condition kind="left">
<![CDATA[
r11 = 1; r22 = 0; r33 = 0; r44 = 0;
r12 = 0; r13 = 0; r14 = 0;
r23 = 0; r24 = 0;
r34 = 0;
]]>
</boundary_condition>
<![CDATA[
Exc = conj(Ex);
Eyc = conj(Ey);
r21=conj(r12);
r31=conj(r13);
r41=conj(r14);
r32=conj(r23);
r42=conj(r24);
r43=conj(r34);
// Equations of motions according to Simon's mathematica code
dr11_dt = gt - gt*r11 + (-Ey - Ex*i)*r12 + (-Ey + Ex*i)*r14 + (-Eyc + Exc*i)*r21 + g0*r22 + g0*r33 + (-Eyc - Exc*i)*r41 + g0*r44;
dr12_dt = (Eyc - Exc*i)*r11 + (-g0/2. - gt - delta*i + deltaL*i)*r12 + i*rf*r13 + (-Eyc + Exc*i)*r22 + (-Eyc - Exc*i)*r42;
dr13_dt = i*rf*r12 + (-g0/2. - gt - delta*i + i*delta0z(z))*r13 + i*rf*r14 + (-Eyc + Exc*i)*r23 + (-Eyc - Exc*i)*r43;
dr14_dt = (Eyc + Exc*i)*r11 + i*rf*r13 + (-g0/2. - gt - delta*i - deltaL*i)*r14 + (-Eyc + Exc*i)*r24 + (-Eyc - Exc*i)*r44;
dr22_dt = (Ey + Ex*i)*r12 + (Eyc - Exc*i)*r21 + (-g0 - gt)*r22 + i*rf*r23 - i*rf*r32;
dr23_dt = (Ey + Ex*i)*r13 + i*rf*r22 + (-g0 - gt - deltaL*i + i*delta0z(z))*r23 + i*rf*r24 - i*rf*r33;
dr24_dt = (Ey + Ex*i)*r14 + (Eyc + Exc*i)*r21 + i*rf*r23 + (-g0 - gt - 2*deltaL*i)*r24 - i*rf*r34;
dr33_dt = -(i*rf*r23) + i*rf*r32 + (-g0 - gt)*r33 + i*rf*r34 - i*rf*r43;
dr34_dt = -(i*rf*r24) + (Eyc + Exc*i)*r31 + i*rf*r33 - i*(deltaL - (g0 + gt)*i + delta0z(z))*r34 - i*rf*r44;
dr44_dt = (Ey - Ex*i)*r14 - i*rf*r34 + (Eyc + Exc*i)*r41 + i*rf*r43 + (-g0 - gt)*r44;
]]>
</operator>
<operator kind="ex" constant="yes">
<operator_names>Lt</operator_names>
<![CDATA[
Lt = i*1./c*kt;
]]>
</operator>
<integration_vectors>E_field</integration_vectors>
<dependencies>density_matrix</dependencies>
<![CDATA[
dEx_dz = i*eta*conj(r12-r14) - Lt[Ex] ;
dEy_dz = -eta*conj(r12+r14) - Lt[Ey] ;
]]>
</operators>
</integrate>
</sequence>
<!-- The output to generate -->
<output format="binary" filename="Genas_system.xsil">
<group>
<sampling basis="t(100)" initial_sample="yes">
<dependencies>E_field</dependencies>
<moments>Ex_re_out Ex_im_out Ey_re_out Ey_im_out</moments>
<![CDATA[
Ex_re_out = Ex.Re();
Ex_im_out = Ex.Im();
Ey_re_out = Ey.Re();
Ey_im_out = Ey.Im();
]]>
</sampling>
</group>
<group>
<sampling basis="t(100)" initial_sample="yes">
<dependencies>density_matrix</dependencies>
<moments>
r11_out r22_out r33_out r44_out
r12_re_out r12_im_out r13_re_out r13_im_out r14_re_out r14_im_out
r23_re_out r23_im_out r24_re_out r24_im_out
r34_re_out r34_im_out
</moments>
<![CDATA[
// populations output
r11_out = r11.Re();
r22_out = r22.Re();
r33_out = r33.Re();
r44_out = r44.Re();
// coherences output
r12_re_out = r12.Re();
r12_im_out = r12.Im();
r13_re_out = r13.Re();
r13_im_out = r13.Im();
r14_re_out = r14.Re();
r14_im_out = r14.Im();
r23_re_out = r23.Re();
r23_im_out = r23.Im();
r24_re_out = r24.Re();
r24_im_out = r24.Im();
r34_re_out = r34.Re();
r34_im_out = r34.Im();
]]>
</sampling>
</group>
</output>
</simulation>
<!--
vim: ts=2 sw=2 foldmethod=indent:
-->
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