# Three-Level System(1)¶

This is a series of lectures on some subjects in Quantum Optics.

Author's Infomation:

• Jia-Qi Cai @ School of Physics,Huazhong Univ. of Sci. and Techn.,Wuhan
• Email:caidish@sina.com

A direct extension of the two level problem is the three level system interacting with two light fields. Many nontrivial quantum algorithms and quantum control techniques can be demonstrated by this model.

There are some different transition configurations of three-level atom due to the transitional selection rules in the natural atoms. But in circuitQED there truely exists $\Delta$-type atom thanks to the optical selection rules for circuits. See: PRL,95,087001 for more infomation about selection rules in SQCs.

## Basic Theory 1: Three-Level System with Single Classical Fields¶

Before diving into the complex cases discuss above, an intuitive model is to think the case below:

$$H = {\omega _a}\left| e \right\rangle \langle e| + {\varepsilon _{g1}}\left| {g1} \right\rangle \langle g1| + {\varepsilon _{g2}}\left| {g2} \right\rangle \langle g2| + \frac{\Omega }{2}\left[ {{e^{i\omega t}}\left( {\left| {g1} \right\rangle + \left| {g2} \right\rangle } \right)\langle e| + h.c} \right]$$

In the interacting picture(i.e. rotating picture) with rotating wave approximation, the Hamiltonian reads:

$$H = \Delta \left| e \right\rangle \left\langle e \right| + \left( {\Delta - \Delta '} \right)\left| {{g_2}} \right\rangle \left\langle {{g_2}} \right| + \frac{\Omega }{2}\left[ {\left( {\left| {{g_1}} \right\rangle + \left| {{g_2}} \right\rangle } \right)\left\langle e \right| + h.c.} \right]$$

As the picture suggests, people with field theory background cannot help tracing the up energy level $\left| e \right \rangle$ out to obtain an effective theory. They will write several lines bravely: $$Z = \int {D\left[ {{{\bar \phi }_i},{\phi _i}} \right]} {e^{ - S\left[ {{{\bar \phi }_i},{\phi _i}} \right]}},i = e,g1,g2$$ $$S\left[ {{{\bar \phi }_i},{\phi _i}} \right] = \smallint d\tau \sum\limits_i {{{\bar \phi }_i}{\partial _\tau }{\phi _i}} + \Delta {{\bar \phi }_e}{\phi _e} + \left( {\Delta - \Delta '} \right){{\bar \phi }_{g2}}{\phi _{g2}}{\text{ }} + \frac{\Omega }{2}\left( {{{\bar \phi }_{g1}} + {{\bar \phi }_{g2}}} \right){\phi _e} + \frac{\Omega }{2}{{\bar \phi }_e}\left( {{\phi _{g1}} + {\phi _{g2}}} \right){\text{ }}$$ Then, they can obtain the effective theory:

$$S\left[ {\bar v,v} \right] = {e^{ - \int {d\tau \left( {\bar vMv + \bar Jv + \bar vJ} \right)} }} = {e^{Tr\ln M - \bar J{M^{ - 1}}J}}$$$${S_{eff}}\left[ {{{\bar \phi }_{g1}},{{\bar \phi }_{g2}},{\phi _{g1}},{\phi _{g2}}} \right]{\text{ }} = \smallint {\text{ }}d\tau \sum\limits_{i = g1,g2} {{{\bar \phi }_i}{\partial _\tau }{\phi _i}} + \Delta {{\bar \phi }_e}{\phi _e} + \left( {\Delta - \Delta '} \right){{\bar \phi }_{g2}}{\phi _{g2}}{\text{ }} - \frac{{{\Omega ^2}}}{4}\left( {{{\bar \phi }_{g1}} + {{\bar \phi }_{g2}}} \right){\Gamma ^{ - 1}}\left( {{\phi _{g1}} + {\phi _{g2}}} \right) + Tr\ln \Gamma$$$$\Gamma = {\partial _\tau } + \Delta ,{\Gamma ^{ - 1}} = \frac{1}{{{\partial _\tau } + \Delta }} = \frac{1}{{i{\partial _t} + \Delta }} \cong {\Delta ^{ - 1}}$$

With the large detuning approximation where $\Delta \gg i\partial t$, and for the finity volume the cavity, the $Trln$ term serving as a bias term due to finite volume of the cavity.

Then the effective Hamiltonian reads: $${H_{eff}} = \left( {\Delta - \Delta '} \right)\left| {g2} \right\rangle \left\langle {g2} \right| - \frac{{{\Omega ^2}}}{{4\Delta }}\left( {\left| {g1} \right\rangle \left\langle {g1} \right| + \left| {g2} \right\rangle \left\langle {g2} \right|} \right) - \frac{{{\Omega ^2}}}{{4\Delta }}\left( {\left| {g1} \right\rangle \left\langle {g2} \right| + \left| {g2} \right\rangle \left\langle {g1} \right|} \right)$$

This effective Hamiltonian is vilid when the population in $\left | e \right \rangle$ is much less than $\left | g1 \right \rangle$ and $\left | g2 \right \rangle$ , and $\left| {\Delta {c_e}} \right| \gg \left| {{\partial _t}{c_e}} \right|$.

The effective Hamiltonian can also be obviously obtained by reducing the equation of motion.(Exercise)

In [29]:
from qutip import *
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

g1 = basis(3,0)
g2 = basis(3,1)
e = basis(3,2)

Ee=10*2*np.pi
Eg1 = 0 * 2*np.pi
Eg2 = 0.4 *2*np.pi
omega = 1*2*np.pi
wa = Ee-Eg1
wa_n = Ee-Eg2
Delta = wa-omega
Delta_n = wa_n-omega
Omega = 0.4 * 2*np.pi #Omega should match the gap between Eg1 and Eg2

H = Delta * e * e.dag() + (Delta-Delta_n)*g2*g2.dag()+Omega/2.0*(g1+g2)*e.dag()+Omega/2.0*e*(g1.dag()+g2.dag())
H_eff = (Delta-Delta_n)*g2*g2.dag() - Omega**2.0/(4*Delta)*(g1*g2.dag()+g2*g1.dag())

psi0 = (g1 + g2).unit()
t = np.linspace(0,10,10000)
result = mesolve(H,psi0,t,[],[g1*g1.dag(),g2*g2.dag()])
result_eff = mesolve(H_eff,psi0,t,[],[g1*g1.dag(),g2*g2.dag()])

fig,ax = plt.subplots(1,2,figsize=(16,9))
ax[0].plot(t,result.expect[0],label='Real Result:g1')
ax[0].plot(t,result_eff.expect[0],label = 'Effective Result:g1')
ax[0].legend()
ax[1].plot(t,result.expect[0],label='Real Result:g2')
ax[1].plot(t,result_eff.expect[0],label = 'Effective Result:g2')
ax[1].legend()

Out[29]:
<matplotlib.legend.Legend at 0x1e0eea0abe0>

## Basic Theory 2: Three-Level System with Two Classical Fields¶

The Hamiltonian of this system can be readly written as: $$H = \sum\limits_{i = a,b,c} {{\omega _i}\left| i \right\rangle \left\langle i \right|} + {\Omega _1}\cos \left( {{\nu _1}t + {\phi _1}} \right)\left( {\left| a \right\rangle \left\langle b \right| + \left| b \right\rangle \left\langle a \right|} \right) + {\Omega _2}\cos \left( {{\nu _2}t + {\phi _2}} \right)\left( {\left| a \right\rangle \left\langle c \right| + \left| c \right\rangle \left\langle a \right|} \right)$$ After RWA and in the interaction picutre, we have:

$${H_I} = - \Delta \left( {\left| b \right\rangle \left\langle b \right| + \left| c \right\rangle \left\langle c \right|} \right) + \left( {\frac{{{\Omega _1}}}{2}{e^{ - i{\phi _1}}}\left| a \right\rangle \left\langle b \right| + \frac{{{\Omega _2}}}{2}{e^{ - i{\phi _2}}}\left| a \right\rangle \left\langle c \right| + h.c.} \right)$$

### Coherent Population Trapping¶

There is a special eigenstate of this interaction Hamiltonian that reads as: $$\left| D \right\rangle = {e^{i{\phi _1}}}\cos \theta \left| b \right\rangle - {e^{i{\phi _2}}}\sin \theta \left| c \right\rangle$$ With $\theta = \arctan \Omega_1/\Omega_2$ and the eigen value $-\Delta$ Note that accidentally the above energy level is eliminated. But it is different from what we do above, which is totally on purpose!

If the initial state is this state, then the state will only rotate and won't be pumped into up state. So we call this state 'trapped' and this technique 'coherent population trapping'(CPT).

In [1]:
from qutip import *
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

a = basis(3,0)
b = basis(3,1)
c = basis(3,2)
n_a = a*a.dag()
n_b = b*b.dag()
n_c = c*c.dag()
w_a = 0 * 2*np.pi
w_b = 0.4 * 2*np.pi
w_c = 1* 2*np.pi
Delta = 0.1* 2*np.pi
nu1 = w_a-w_b-Delta
nu2 = w_a-w_c-Delta
O_1 = 1* 2*np.pi
O_2 = 2* 2*np.pi
theta = np.arctan(O_1/O_2)

H0 = w_a*n_a+w_b*n_b+w_c*n_c
def Omega_1(t,args):
return O_1/2.0*np.exp(-1j*nu1*t)
def Omega_1_dag(t,args):
return O_1/2.0*np.exp(1j*nu1*t)
def Omega_2(t,args):
return O_2/2.0*np.exp(-1j*nu2*t)
def Omega_2_dag(t,args):
return O_2/2.0*np.exp(1j*nu2*t)
H=[H0,[a*b.dag(),Omega_1],[b*a.dag(),Omega_1_dag],[a*c.dag(),Omega_2],[c*a.dag(),Omega_2_dag]]
psi0 = np.cos(theta)*b-np.sin(theta)*c

t=np.linspace(0,10,1000)
result = mesolve(H,psi0,t,[],[n_a,n_b,n_c])
fig,ax = plt.subplots(figsize=(16,9))
plt.plot(t,result.expect[0],label='N_a')
plt.plot(t,result.expect[1],label='N_b')
plt.plot(t,result.expect[2],label='N_c')
result_state = mesolve(H,psi0,t)
result_b = [b.overlap(t.dag()) for t in result_state.states]
plt.plot(t,np.real(result_b),label='Real part of Cb')

plt.legend(fontsize='large')

Out[1]:
<matplotlib.legend.Legend at 0x1ffb3cb5ef0>

## Stimulated Raman Process¶

By applying quantum adiabatic theorem, one may find a convenient way to manipulate the system from one trapping state to another.

E.g.:

When $$\begin{gathered} \frac{{{\Omega _1}\left( { - \infty } \right)}}{{{\Omega _2}\left( { - \infty } \right)}} = 0,\frac{{{\Omega _1}\left( { + \infty } \right)}}{{{\Omega _2}\left( { + \infty } \right)}} = \infty \hfill \\ \theta \left( { - \infty } \right) = 0,\theta \left( { + \infty } \right) = \frac{\pi }{2} \hfill \\ \end{gathered}$$

This pulse will pump the atom from $\left|b\right\rangle$ to $\left|c\right\rangle$

For example: $${\Omega _1} = {\Omega _0}\exp \left[ { - \frac{{{{\left( {t - \Delta t} \right)}^2}}}{{{T^2}}}} \right],{\Omega _2} = {\Omega _0}\exp \left[ { - \frac{{{{\left( {t + \Delta t} \right)}^2}}}{{{T^2}}}} \right]$$

In [4]:
from qutip import *
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

a = basis(3,0)
b = basis(3,1)
c = basis(3,2)
n_a = a*a.dag()
n_b = b*b.dag()
n_c = c*c.dag()
w_a = 0 * 2*np.pi
w_b = 0.4 * 2*np.pi
w_c = 1* 2*np.pi
Delta = 0.1* 2*np.pi
nu1 = w_a-w_b-Delta
nu2 = w_a-w_c-Delta
O_1 = 10* 2*np.pi
O_2 = 10* 2*np.pi
theta = np.arctan(O_1/O_2)
Delta_t = 1.1
T=2

H0 = w_a*n_a+w_b*n_b+w_c*n_c
def Gas_1(t):
return np.exp(-(t-Delta_t)**2/T**2)
def Gas_2(t):
return np.exp(-(t+Delta_t)**2/T**2)
def Omega_1(t,args):
return Gas_1(t)*O_1/2.0*np.exp(-1j*nu1*t)
def Omega_1_dag(t,args):
return Gas_1(t)*O_1/2.0*np.exp(1j*nu1*t)
def Omega_2(t,args):
return Gas_2(t)* O_2/2.0*np.exp(-1j*nu2*t)
def Omega_2_dag(t,args):
return Gas_2(t)*O_2/2.0*np.exp(1j*nu2*t)
H=[H0,[a*b.dag(),Omega_1],[b*a.dag(),Omega_1_dag],[a*c.dag(),Omega_2],[c*a.dag(),Omega_2_dag]]
psi0 = b

t=np.linspace(-20,20,1000)
result = mesolve(H,psi0,t,[],[n_a,n_b,n_c])
fig,ax = plt.subplots(figsize=(16,9))
plt.plot(t,result.expect[0],label='N_a',linewidth=4.0)
plt.plot(t,result.expect[1],label='N_b',linewidth=4.0)
plt.plot(t,result.expect[2],label='N_c',linewidth=4.0)
plt.legend(fontsize='large')

Out[4]:
<matplotlib.legend.Legend at 0x1ffb445e978>

## Electromagnetically Induced Transparency¶

(Removed)EIT will be introduced in NEXT chapter

In [1]:
from qutip import *
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

In [2]:
u = basis(3,0)
g = basis(3,1)
e = basis(3,2)

sigma_ee = e*e.dag()
sigma_uu = u*u.dag()
sigma_gg = g*g.dag()
sigma_ue = u*e.dag()
sigma_ge = g*e.dag()

Delta_min = -3.0
Delta_max = 3.0
step = 300
Delta_list = np.linspace(Delta_min,Delta_max,step)

gamma_eu = 5
gamma_eg = 0.1
c_o = [np.sqrt(gamma_eg)*sigma_ge,np.sqrt(gamma_eu)*sigma_ue]
phi = 0
Omega1 = 0.1
Omega2 = 0.5

result = []
for Delta in Delta_list:
H = -Delta*(sigma_ee-sigma_gg) + phi*(sigma_ee-sigma_uu)+ Omega1*(sigma_ge+sigma_ge.dag())+Omega2*(sigma_ue+sigma_ue.dag())
result.append(expect(sigma_ge.dag(),rhoss))
fig,ax = plt.subplots()
plt.plot(Delta_list,[i.real for i in result],label = 'Real Part')
plt.plot(Delta_list,[i.imag for i in result],'-.',label = 'Imaginary Part')
plt.xlabel(r'\Delta')
plt.ylabel(r'\sigma_{ge}')
plt.legend()

Out[2]:
<matplotlib.legend.Legend at 0x283428ccef0>

## General Quantum Theory: Can three level system be reduced to two level system?¶

See PRA,54,1586 for more detail.