Optics and Photonics Journal, 2013, 3, 360-363
Published Online November 2013 (http://www.scirp.org/journal/opj)
http://dx.doi.org/10.4236/opj.2013.37056
Open Access OPJ
The Long-Lived Photon Echo Response Locking Effect in
the Presence of External Non-Resonant Laser Pulses with a
Different Spatial Orientation
Leonid A. Nefed’ev, Elza I. Hakimzyanova, Guzel I. Garnaeva
Department of Educational Technologies in Physics, Kazan Federal University, Kazan, Russia
Received September 9, 2013; revised October 7, 2013; accepted October 28, 2013
Copyright © 2013 Leonid A. Nefed’ev et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The correlation of the inhomogeneous broadening of the resonance transition at different time intervals and the effi-
ciency of long-lived photon echo response locking by the action of standing waves of non-resonant laser pulses are con-
sidered. It is shown that the long-lived photon echo response locking effect may be observed even at the angles of the
relative orientation of the non-resonant standing wave laser pulses of less than one degree, due to the change in the
correlation coefficient of inhomogeneous broadening at time intervals between the first and the second and after the
third resonant laser pulses.
Keywords: Photon Echo; The Information Locking; Spatial Non Uniform Electromagnetic Fields; Relaxation
Processes; Time of The Response; External Space-Non Uniform Non Resonant Laser Pulses
1. Introduction
At present, a problem of erasing information is the greatest
technical challenges in creating specific patterns of opti-
cal storage devices. To solve this problem, various meth-
ods are proposed [1-4]. They are based on the elimina-
tion of the space-frequency modulation of the resonance
levels population by acting on the system with defined
sequence of optical pulses. However, all offered schemes
of removal of the information are difficult enough for
their technical embodiment. In addition, the process of
erasing of the information is energetically unprofitable,
since its implementation requires the energy of the same
order as that for recording information. From this per-
spective, the information locking is more profitable, which
means creating the conditions under which the informa-
tion doesn’t show up in the form of the optical response
of a resonant medium. This can be achieved by breaking
the time-frequency correlation of inhomogeneous broad-
ening of the resonance line at different time intervals [5].
It is known that the formation of photon echo response
consists of two main stages: phase mismatching of the
oscillating dipole moments of optical centers and their
subsequent phase matching. At the second stage, the me-
dium becomes macroscopically polarized, which is ob-
served as a specific response. Thus, an insignificant
violation of the rigid time-frequency correlation of in-
homogeneous broadening should lead to a significant
reduction in the response intensity (the reversible de-
struction of the phase memory of a resonance medium
with a possibility of its reconstruction). This effect can
be obtained by subjecting a resonant medium to the ac-
tion of different spatially inhomogeneous external per-
turbations at different time intervals. Such an action
should lead to random shifts or splitting of the initial
monochromatic components of the inhomogeneously
broadened line.
We should note that the effect of locking of long-lived
photon echo in a La F3Pr3+ crystal is theoretically pre-
dicted and experimentally confirmed in [6].
A similar effect can be achieved through non-resonant
interaction of a quantum system with laser pulses. If the
non-resonant laser fields have spatial inhomogeneity, the
energy shifts δE are functions of the coordinates δЕ


r,
which leads to additional inhomogeneous broadening of
the resonance transition in the sample.
In this study we consider the correlation of the inho-
mogeneous broadening of the resonance transition at
different time intervals and the efficiency of long-lived
photon echo response locking by the action of standing
waves of non-resonant laser pulses.

L. A. NEFED’EV ET AL. 361
2. The Basic Equations
Let us consider the formation of a long-lived photon echo
under the action of sequence of three resonant laser
pulses and state waves of non-resonant laser pulses with
a different spatial orientation (Figure 1).
The equation for the single-particle density matrix in a
rotating coordinate system can be written as


,ti B

 


, (1)
where
0m
BHU A

,
00
ee
iAt iAt
mm
HH


,
ee
iAt iAt
UU


,
where A is the matrix of transition to the rotating coordi-
nate system, U is the operator of the resonant interaction
with the exciting laser pulses, H0m is the Hamiltonian of
an atom in the external spatially non uniform non reso-
nante laser radiation at τm-th time interval, is the ra-
dius vector of the atom location. In the case of two-level
system, A = P22ω12,
r
12
11 22
ee
it iAt PP    ,



0, mm
H  
r,
12

  ,
12 12
12 1221 21
ee
it it
UPU PU
 

,
1212 0
1e
2
iti
UdE
 
kr ,
2121 0
1e
2
iti
UdE
 
kr ,
where Pij are the projective matrices (their ij-th element
is equal to unity and the other elements are zero), ω12 is
the frequency of the resonance transition, dij is the dipole
moment of the resonance transition, E0 is the electric
field strength of resonant laser pulse and its wave
vector, is an additional frequency shift of the
k

,
m
 r

Figure 1. Excitation pulses order in the formation of signals
long-lived photon echo P1, P2 and P3 are the exciting pulses,
τmn is a time interval between the m-th and the n-th pulses,
SW1 and SW2 are a standing waves, τ1 and τ2 are durations
of the non-resonant laser pulses.
resonant transition of an atom in a time interval τm by
external non-resonant spatially inhomogeneous laser ra-
diation.
The dependence ε from the location of the optical cen-
ter in the sample is related to the spatial inhomogeneity
of nonresonant laser radiation. Such inhomogeneity oc-
curs, for example, under the influence of a standing wave.
In this case the energy shift of the n-th state of the atom
can be written as [7,8].

2
2
0
22
1cos
2
ns ns
nηη

j
sns
ωd
δEE
ωω


kr (2)
and










12
21 0
,c
Dj
EECE

 

rrr kos,

r
(3)
where the СD is the constant of the dynamic Stark effect,
0
E

is the electric field amplitude of the η-th non-reso-
nant laser pulse, ns

is the frequency of transitions
between bound states of n and s. If ns

  expression
(2) becomes the common formula for the quadratic Stark
effect
 

2
2
0
2
0
1cos
2
cos ,
ns
nj
sns
sh j
d
EE
CE


 




rk
kr

r
(4)
where Csh is the constant of the quadratic Stark effect. In
most cases the constants of quadratic Stark effect for
different atomic transitions lie in the range of 10
Hz/(В/см)2 < Csh <1 kHz/(В/см)2 [9], which is compara-
ble with the dynamic Stark effect constants values CD.
This allows us to obtain a sufficiently large frequency
shifts


,

 r compared to the width of the excitation


1kk 

of an inhomogeneously broadened line of the
resonance transition by selecting the appropriate power
non-resonant laser pulses. If the non-resonant frequency
ω of the laser radiation is close to one of the frequencies
ns

(transitions to the perturbing levels), the values of
constant CD can greatly exceed the value of Csh.
The initial distribution of the optical centers over the
frequencies


g

we will assume as Gaussian with
dispersion 2

:

2
1exp
2π2
g




 

 


(5)
Comparing the frequencies shifts of optical centers
on different time intervals


, by interaction with a
different spatially oriented standing waves it is conven-
ient to define the vector

k in the coordinate system


,,
x
yz


,associated with the direction of propaga-
tion of laser radiation:
x
y
kkk
z


 


kijk (6)
Open Access OPJ

L. A. NEFED’EV ET AL.
362
where the

,,



i
j
k

,,
are the unit vectors of the coor-
dinate system

x
yz


. In the laboratory coordinate
system

,,
x
yz

,,
x
x
y
zz
k
k
kA k
kk






 
 

 
 
 
y

(7)
where A is the matrix of rotations. From (7) we find that



cos cos cossin sin
coscos sinsincos
cossin ,
xx
y
z
kk
k
k



 

 


 

 






sincos coscos sin
sincossincos cos
sinsin ,
yx
y
z
kk
k
k



 

 

 

 


 

sin cossin sincos
zxy z
kkk k

.


 
 

here


,


and


are Euler angles defining the
relative orientation by coordinate systems

,,

x
yz and

,,
x
yz


.
3. The Correlation of Inhomogeneous
Broadening and Effectiveness of Locking
a Long-Lived Photon Echo Depending on
the Relative Orientation of Non-Resonant
Laser Pulses
The coefficient of time-frequency correlation of inho-
mogeneous broadening at the time intervals


and


taking into account (1) and (3) has the form [10]:
 

,
,, ,,
1dd
V
R
fufu
g
V
V








 


 rr ,
(8)
where V is the volume of the excited part of sample,

1,, dd
V
ufgV
V




 r, (9)
 
22
1(,, )dd
V
f
ug V
V




 r. (10)
The numerical calculation of the correlation coeffi-
cient (8), depending on the relative orientation of the
non-resonant standing wave of laser pulses is shown in
Figure 2. It implies that small changes of the angle be-
tween the non-resonant standing waves of laser pulses
CW1 and CW2 (within 0.1 degrees) results a change in
the correlation coefficient of inhomogeneous broadening
at different time intervals that means the destruction of
the reversible phase memory of the system.
Figure 3 shows the results of numerical calculation of
the intensity of a long-lived photon echo response as a
function of angle β. Analysis of the resulting angular
dependence shows that there is effect of locking a
long-lived photon echo that takes place if β < 0.1 degrees.
Thus, using the excitation scheme of Figure 1, you can
create a large number of independent channels of re-
cording and reproducing information, with the angle β as
associative key of access.
4. Conclusion
The formation of the stimulated photon echo in the pres-
ence of external non-resonant laser pulses leads to the
effect of a long-lived photon echo locking. The informa-
Figure 2. Angular dependence of the frequency-time corre-
lation coefficient in the presence of non-resonant standing
waves in La F3Pr3+, τ1 = τ2 =20 nc, σ = 5 nc−1, Δt1 = Δt1 = 5 nc.
Figure 3. The angular dependence of the long-lived photon
echo intensity in the presence of non-resonant standing
waves in La F3Pr3+, τ1 = τ2 = 20 nc, σ = 5 nc−1, Δt1 = Δt1 =
5 nc.
Open Access OPJ

L. A. NEFED’EV ET AL.
Open Access OPJ
363
tion locking effect in the presence of non-resonant
standing waves occurs if angles of relative orientation of
the non-resonant standing waves of laser pulses are less
than 0.1 degress.
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