2.2 Interaction of Light with Atomic Medium
2.2.1 Electromagnetically Induced Transparency
In this chapter we will mostly follow the consideration from [85,86] to explain the physics behind EIT. We will consider three level system consisting of a ground state level|gÍ, an intermediate state |eÍand a long-lived Rydberg state
|rÍin ladder type configuration (see Fig. 2.4). The transition |gÍ ≠|rÍ is dipole forbidden. The population decay due to spontaneous emission is described by
r (for decay of |rÍ to |gÍ) and e (for decay of |eÍ to |gÍ), and additional dephasing due to collisions, stray electric fields, thermal motion of atoms and other dephasing mechanisms are included as “r and “e. Two laser fields are used to couple the ground state |gÍ to the Rydberg state |rÍ: a weak probe field p couples the ground state |gÍ and the intermediate state |eÍ with the detuning from atomic transition p = Êeg ≠Êp; and a strong control field c
couples the intermediate state|eÍ and the Rydberg state |rÍ with the detuning from the atomic transition c = Êre ≠Êc (Êp and Êc are the probe and the control laser frequencies, correspondingly). Here the coupling of the light fields to the atoms is expressed in terms of the Rabi coupling (or Rabi frequency)
= µ·E0/~, with E0 being the amplitude of the electric field E, and µ the transition electronic dipole moment.
Therefore, the system is described by the Hamiltonian H = Hatom +Hint, where Hatom is a Hamiltonian of a bare atom and Hint comes from interac-tion with the light fields. Using the rotating frame approximainterac-tion, interacinterac-tion Hamiltonian in rotating frame takes a form [85]
Hint =≠~ 2
Q ca
0 p 0
p 2 p c
0 c 2( c≠ p)
R
db. (2.26)
In the case of a two photon resonance ( p = c), one of the eigenstates of the system is the state
|a0Í= úc|gÍ ≠ p|rÍ
| c|2+| p|2 , (2.27)
2.2. Interaction of Light with Atomic Medium 23 which has no contribution from the intermediate state|eÍand therefore called a dark state. If we start with the system being in the ground state|gÍ, switching on the control light c does not redistribute the population and it remains in the |gÍ. In the case of the weak probe field p π c applied, the ground state becomes identical to the dark state |a0Í from which excitation cannot occur.
Therefore when the weak probe pulse p arrives to the system (under slow varying amplitude condition), the population adiabatically follows the state
|a0Í, which is the dark state, and thus absorption from the probe pulse is not possible. This effect is called electromagnetically induced transparency (EIT).
To take more insight into EIT the semi-classical analysis can be applied. The coupling of the atom and the light field can be described by the time-dependent interaction Hamiltonian [85, 87]
Hint =≠~ 2
Ë
p(t)ˆ‡egei pt+ c(t)ˆ‡reei ct+h.c.È, (2.28) where ‡ˆij = |iÍ Èj| is the atomic projection operator (i, j = g, e, r). The time evolution of the system in the language of the density matrix is governed by the master equation in the Lindblad form
dfl dt = 1
i~[Hint,fl] + e
2 [2ˆ‡geflˆ‡eg≠‡ˆeefl≠flˆ‡ee] + r
2 [2ˆ‡erflˆ‡re≠‡ˆeefl≠flˆ‡ee] +“e
2 [2ˆ‡eeflˆ‡ee≠‡ˆeefl≠flˆ‡ee] +“r
2 [2ˆ‡rrflˆ‡rr≠‡ˆrrfl≠flˆ‡rr],
(2.29)
where the second and the third terms on the right-hand side describe sponta-neous emission from state |eÍ to |gÍ and from |rÍ to |gÍ with rates e and r. And last two terms describe energy-conserving dephasing processes with rates
“e and “r.
In Maxwell’s equations polarization plays a role of a source of electromag-netic field, therefore we would like to investigate the polarization generated in the atomic medium by the applied electric fields to study electromagnetic filed dynamics [85]:
P˛(t) =≠ÿÈerjÍ/V = Natom
V
˵geflgee≠iÊegt+µerflere≠iÊret+ c.c.È (2.30) HereNatom amount of atoms are contained in the volumeV, yielding the atomic densityÍ =Natom/V, and been identically coupled to the electromagnetic field.
For the case of the weak probe field p π c it is possible to derive the linear
Figure 2.5: EIT absorption spectra for different values of the control field and
“rg = 0,“eg= 2fi·6.05 MHz: (a) c = 2“eg; (b) c = 0.4“eg.
susceptibility‰ [85]
‰= |µeg|2Í
‘0~
A 4”1| c|2≠4” p
2≠4 p“2rg
-
--| c|2 + (“eg+i2 p) (“rg+i2”)---2
+i 8”2“eg+ 2“rg
1| c|2+“rg“eg2
-
--| c|2+ (“eg+i2 p) (“rg +i2”)---2
B (2.31)
where ” = c ≠ p; “eg = e +“e and “rg = r +“r determine the decay of fleg and flrg respectively. In general it is also possible to account for the finite laser linewidth by changing the effective linewidths “eg æ “eg +“p and
“rg æ “rg +“p +“c [88], where “p and “c are the half linewidths of the probe laser and control laser, respectively (this approach is held only for Lorentzian lineshapes). In the Rydberg experiments with 87Rb one can use the following approximation of the dephasing rates: “eg = e and “rg = “r, which will be implied everywhere later if other is not explicitly used.
The linear susceptibility given by Eq.2.31predicts many important features of the EIT. Beyond the modification of the absorption due to the appearance of dressed atomic states it is seen that for two-photon Raman resonance (” = 0), both the real and imaginary parts of the linear susceptibility vanish in the ideal limit of “rg = 0. It is important to note that this result is independent of the strength of the coherent control field and it is shown in Fig.2.5. For the case when| c|>“eg, the absorption profile carries the signature of an Autler-Townes doublet: at”= 0 the loss vanishes and on the high and low energy sides of the doublet absorption is enhanced (Fig. 2.5(a)). For the case, where | c| π “eg
one can observe a sharp transmission window with a linewidth much narrower than “eg (Fig.2.5(b)).
In our experiments we work with optically thick media therefore we are
2.2. Interaction of Light with Atomic Medium 25 interested in the collective response of the entire medium, which is given by the amplitude transfer function, or in reduced variant as a transmission [85]
T = exp3≠OD “eg
2 Im [‰/‰0]4, (2.32) where OD = 3⁄20ÍL/2fi is an optical depth of the medium (OD), ⁄0 is the wavelength of the probe light in vacuum and L is the length of the medium;
‰0 =|µeg|2Í/‘0~is the prefactor of the linear susceptibility from Eq.2.31. The EIT transmission window near the resonance” = 0 has a Gaussian profile with the width
ÊEIT = 2c
e
Ô1
OD. (2.33)
The two remarkable features are that the width of the spectral window in which EIT medium appears transparent decreases with OD and on the other side can be significantly broadened by a strong control field c. From this, one can see that by decreasing the power of control field and increasing the OD of the medium it is possible to obtain the infinitely narrow transmission peak with a unity transmission. However it is only the case when “rg = 0 and including nonzero value into consideration leads to the modification of the transmission value at the resonance ”= 0:
Tmax= exp
Q
a≠ OD 1 + e“2crg
R
b. (2.34)
This restricts the possible values that OD and c can take in the experiment to observe the EIT transmission window.
The discussed features of the transmission are depicted in Fig.2.6(a). First of all, the classical two-level absorption valley of|gÍ ≠|eÍtransition is shown as a reference (blue dashed line). Adding the control field coupling to the Rydberg state |rÍ results in the appearance of a narrow transmission window in the center of the absorption valley (green line) with the transmission reaching unity exactly at the resonance. Adding a dephasing rate of the Rydberg state “rg
results in decreasing of the transmission on resonance (red line); and increasing of the Rabi frequency of the control light broadens the transmission window (black line).