1. PHYSICAL REVIEW A 85, 063809 (2012)
Coherence-population-trapping transients induced by an ac magnetic field
L. Margalit,1
M. Rosenbluh,2
and A. D. Wilson-Gordon1
1
Department of Chemistry, Bar-Ilan University, Ramat Gan 52900, Israel
2
The Jack and Pearl Resnick Institute for Advanced Technology, Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel
(Received 16 February 2012; published 8 June 2012)
Coherent-population-trapping transients induced by an ac magnetic field are investigated theoretically for a
realistic three-level system in the D1 line of 87
Rb. The contributions to the transient probe absorption from
the various subsystems that compose the realistic atomic system are examined and the absorption of each
subsystem is compared to that of a simple system. The population redistribution due to optical pumping
is shown to be the dominant cause of the difference between the contributions of the various subsystems to
the oscillatory character of the probe absorption. We also discuss the series of transients that reappear every
half-cycle time of a modulated magnetic field when the system is in two-photon resonance, and we study the
transient behavior as a function of the probe detuning. The effect of a buffer gas on the amplitude and shape of
the transients is considered.
DOI: 10.1103/PhysRevA.85.063809 PACS number(s): 42.50.Gy, 42.50.Md, 32.80.Xx
I. INTRODUCTION
Atomic systems that exhibit quantum coherence between
coupled atomic states are of great interest due to the variety
of novel properties they exhibit. A convenient description of
such media is via a density matrix formalism with nonzero off-
diagonal coherence terms [1]. These systems have been shown
to manifest interesting phenomena such as coherent population
trapping (CPT) [2], electromagnetically induced transparency
(EIT) [3], and electromagnetically induced absorption (EIA)
[4–6].
Systems exhibiting quantum coherence properties have also
been studied for their transient behavior. These studies can be
divided into two types. In the first type, the time-dependent
dynamics is studied from the time that the laser fields are
switched on until the system achieves CPT [7] or EIT [8].
In the second type, which is the subject of this paper, CPT
or EIT is first established and then a magnetic field is
applied either gradually [9] or suddenly [10] or, equivalently,
one of the laser fields is detuned either gradually [11] or
suddenly [12,13].
The transient response of the absorption of a probe laser
interacting with a system [12,13] or a degenerate two-level
system [10,11,14] that occurs due to a sudden change in the
frequency of one [11,13] or both [12] laser fields, due to the
sudden turning on of a static magnetic field [10], or due to
the sudden turning on of a laser field in the presence of a
magnetic field [14], has been examined both theoretically
and experimentally. An analytic solution of a three-level
system in the case where the Raman detuning is suddenly
changed from zero to R has been developed by Park et al.
[13]. They showed that the Raman coherence and hence the
probe absorption oscillate at a constant frequency R. This
observation was later adopted [15] in order to determine the
standard frequency of a 85
Rb atomic clock. Park et al. [13]
also showed experimentally that, for a triangular modulation
of the Raman detuning, a transient response appears after
the detuning crosses zero and the frequency of the transient
oscillations increases as the detuning increases. Similar results
have also been obtained for a degenerate two-level system that
exhibits EIT [11]. It has also been shown [11] that the rate of
decay of the transient is proportional to the intensity of the
pump (coupling) laser.
Recently, Momeen et al. [9] studied the transient response
of the resonant nonlinear magneto-optic rotation (NMOR)
signal in a paraffin-coated Rb vapor cell as the magnetic field
was swept. They showed that at low sweep rates, the nonlinear
rotation appears as a narrow resonance signal for transitions
starting from the lower ground hyperfine level, which can be
useful for high sensitivity magnetometry at low field strengths.
As expected, when the magnetic field sweep rate is increased
the narrow resonance gradually disappears and undesirable
transient oscillations begin to appear.
In this work, we discuss the transients induced by the
modulation of an applied magnetic field in the probe absorption
of a realistic three-level system in the D1 line of 87
Rb
that exhibits CPT. We examine the contributions to the probe
absorption from the various subsystems that compose the
realistic atomic system and compare the absorption of each
subsystem to that of a simple system. We find that for
moderate modulation frequencies the transient behavior is not
dependent on the wave form of the modulation, which can be
of either a sawtooth or a sinusoidal shape. In related work,
Bevilacqua et al. [16] investigated both experimentally and
theoretically the effect of a longitudinal alternating magnetic
field on the CPT resonances of Cs. By applying a static
magnetic field, they were able to investigate a single
subsystem. They showed that the ac magnetic field produces
sidebands to the CPT resonance similar to those obtained using
frequency-modulated (FM) spectroscopy. Here we study the
time dependence of the probe absorption rather than its steady-
state frequency dependence and examine the contributions of
the various subsystems to the total absorption rather than the
behavior of a single subsystem. In addition, we investigate
modulation frequencies much lower than those considered in
Ref. [16].
We distinguish between two processes that occur simulta-
neously in the system, the change in the energy of the sublevels
due to the modulation of the magnetic field and the transfer
of population between the sublevels, and we demonstrate the
effect of each process. The differences between the subsystems
063809-11050-2947/2012/85(6)/063809(6) ©2012 American Physical Society

2. L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 85, 063809 (2012)
that make up the realistic system underline the importance of
optical pumping effects.
Unlike previous studies that dealt with the behavior of a
single transient, here we also study the behavior of a series
of transients as a function of the probe detuning. In addition,
we consider the effect on the behavior of the transients when a
buffer gas is added to the vapor cell. We have observed a series
of transients in preliminary experiments on Cs vapor and these
results will be reported elsewhere.
II. THE BLOCH EQUATIONS
The system consists of two ground hyperfine states, Fg and
Fg , and a single excited hyperfine state, Fe (a configuration).
The Fg → Fe transition interacts with a pump of frequency ω1
and the Fg → Fe transition interacts with a probe of frequency
ω2. We use the equations for the time evolution of the
configuration as given by Boublil et al. [17] for a simple
system and generalize them for a system consisting of Zeeman
sublevels, with the addition of decay from the ground and
excited states to a reservoir, and collisions between the Zeeman
sublevels of the ground states, as was done by Goren et al. [6]
for a degenerate two-level atomic system:
˙ρei ej
= −(iωei ej
+ )ρei ej
− γ ρei ej
− ρeq
ei ei
− (i/¯h)
gk
(ρei gk
Vgkej
− Vei gk
ρgkej
)
− (i/¯h)
gk
(ρei gk
Vgkej
− Vei gk
ρgkej
), (1)
˙ρei gj
= −(iωei gj
+ ei gj
)ρei gj
− (i/¯h)
ek
ρei ek
Vekgj
−
gk
Vei gk
ρgkgj
−
gk
Vei gk
ρgkgj
, (2)
˙ρgi gi
= −(i/¯h)
ek
(ρgi ek
Vekgi
− Vgi ek
ρekgi
) − γ ρgi gi
− ρeq
gi gi
− (2Fg) gi gj
ρgi gi
+
gk,k=i
gkgi
ρgkgk
− (2Fg + 1) gi gj
ρgi gi
+
gk
gkgi
ρgkgk
+ (
·
ρgi gi
)SE
,
(3)
˙ρgi gj
= −(iωgi gj
+ gi gj
)ρgi gj
+ (
·
ρgi gj
)SE
−(i/¯h)
ek
(ρgi ek
Vekgj
− Vgi ek
ρekgj
), (4)
˙ρgi gj
= −(iωgi gj
+ gi gj
)ρgi gj
−(i/¯h)
ek
(ρgi ek
Vekgj
− Vgi ek
ρekgj
), (5)
In Eqs. (2), (4), and (5) one can interchange g and g in order
to obtain the equations for ˙ρei gj
, ˙ρgi gi
, and ˙ρgi gj
, respectively.
Here,
(
·
ρgi gj
)SE
= (2Fe + 1) Fe→Fg
q=−1,0,1
Fe
me,me=−Fe
(−1)−me−me
×
Fg 1 Fe
−mgi
q me
ρme me
Fe 1 Fg
−me q mgj
, (6)
with
Fe→Fg
= (2Fg + 1)(2Je + 1)
Fe 1 Fg
Jg I Je
2
≡ b ,
(7)
where is the total spontaneous emission rate from each Feme
sublevel whereas Fe→Fg,g
is the decay rate from Fe to Fg,g .
gi gj
and gi gj
are the collisional decay rates from sublevels
gi → gj and gi → gj , respectively. The frequencies between
the ground hyperfine levels lie in the microwave range so that
the collisions not only damp the coherences but also affect the
populations of Fg and Fg [18]. We therefore introduce the
phenomenological population transfer rate from mg to mg :
gi gj
and gi gj
. γ is the rate of decay due to time-of-flight
through the laser beams. The dephasing rates of the excited to
the ground state coherences are given by ei gj
= γ + 1
2
[ +
(2Fg) gi gj
+ (2Fg + 1) gi gj
] + ∗
and ei gj
= γ +
1
2
[ + (2Fg ) gi gj
+ (2Fg + 1) gi gj
] + ∗
, where
∗
is the rate of phase-changing collisions.
The dephasing rates of the ground state
coherences are given by gi gj
= γ + (2Fg) gi gj
+
(2Fg + 1) gi gj
+ ∗
gi gj
, gi gj
= γ + (2Fg ) gi gj
+ (2Fg + 1)
gi gj
+ ∗
gi gj
, and gi gj
= γ + 1
2
[(2Fg) gi gj
+ (2Fg +
1) gi gj
+ (2Fg ) gi gj
+ (2Fg + 1) gi gj
] + ∗
gi gj
, where ∗
gi gj
,
∗
gi gj
, and ∗
gi gj
are the rates of phase-changing collisions.
The frequency separation between levels ai and bj , including
Zeeman splitting of the ground and excited levels due to an
applied magnetic field, is given by ωai bj
= (Eai
− Ebj
)/¯h,
with a,b = (g,e), and ρ
eq
ai ai , with a = (g,e) being the
equilibrium population of state ai, in the absence of any
electrical fields. The interaction energy in the rotating-wave
approximation for the transition from level gj to ei is written
as
Vei gj
= −μei gj
(E1e−iω1t
+ E2e−iω2t
) (8)
≡ −¯h[Vei gj
(ω1)e−iω1t
+ Vei gj
(ω2)e−iω2t
],
where 2Vei gj
(ω1,2) are the pump and probe Rabi frequencies
for the Feme → Fgmg transition, given by
2Vei gj
(ω1,2) =
2μei gj
E1,2
¯h
= (−1)Fe−me
Fe 1 Fg
−me q mg
1,2, (9)
where 1,2 = 2 Fe||μ||Fg,g E1,2/¯h are the general pump and
probe Rabi frequencies for the Fe → Fg,g transitions and
q = (−1,0,1) depending on the polarization of the incident
laser. In order to calculate the time-dependent probe absorption
which is proportional to Imρei gj
, Eqs. (1) to (5) were solved
063809-2

3. COHERENCE-POPULATION-TRAPPING TRANSIENTS . . . PHYSICAL REVIEW A 85, 063809 (2012)
FIG. 1. (Color online) Energy-level scheme for the D1 line of
87
Rb interacting with σ+
polarized pump and probe. The system can
be divided into one TLS and three subsystems. The Zeeman shifts
for the upper hyperfine level are not shown.
numerically taking into account the magnetic field modulation
and the consequent shifts in the energy of the sublevels.
III. RESULTS AND DISCUSSION
Our calculations are performed for the D1 line of 87
Rb
(see Fig. 1) where the pump is resonant with the Fg =
1 → Fe = 2 transition and the probe is resonant with the
Fg = 2 → Fe = 2 transition. The pump and the probe are
both σ+
polarized and have the same general Rabi frequency
= 1,2. The system consists of a single two-level transition
|Fg ,mg = −2 ↔ |Fe,me = −1 (TLS) and three sys-
tems: 1, |Fg,mg = −1 ↔ |Fe,me = 0 ↔ |Fg ,mg = −1 ;
2, |Fg,mg = 0 ↔ |Fe,me = 1 ↔ |Fg ,mg = 0 ; and 3,
|Fg,mg = 1 ↔ |Fe,me = 2 ↔ |Fg ,mg = 1 (see Fig. 1).
It should be noted that the 1 and 3 subsystems are
antisymmetric with respect to the Zeeman splittings of their
lower sublevels.
First, the pump and the probe are turned on in the presence
of a constant magnetic field B0 = 4 G parallel to the direction
of propagation of the laser beam, and the system is given
sufficient time for 2 to stabilize in a dark state. Then, at time
t0 = 0.3 ms, we begin to modulate the magnetic field according
to B = B0 sin[π/2 + ω(t − t0)], where ω is the magnetic field
modulation frequency (see Fig. 2).
The magnetic field removes the degeneracy of the Zeeman
sublevels and separates the system into subsystems. In Fig. 2,
we see that the total probe absorption exhibits transient
behavior that is repeated every half-cycle of the magnetic field
modulation.
A. Transient components
The contributions to the probe absorption from each
subsystem are shown in Fig. 3. The absorption of the TLS
stabilizes at a constant value as the magnetic field decreases
toward zero causing the transition to become resonant. 2, the
“clock transition,” is almost uninfluenced by the magnetic field
because the Zeeman sublevels |Fg,mg = 0 and |Fg ,mg = 0
are not shifted to first order by the magnetic field so that
this transition is always two-photon resonant and thus in a
dark state. When the magnetic field passes through zero, the
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
10
20
30
40
50
60
70
80
t (ms)
ProbeAbsorption(cm−1
)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
−4
0
4
MF(G)
probe abs.
MF modulation
T
T/2
FIG. 2. (Color online) Probe absorption for resonant pump and
probe as a function of time in the presence of a magnetic field (MF).
First, the system is stabilized in the presence of a constant MF: B0 =
4 G. Then, starting at t0 = 0.3 ms, the MF is modulated sinusoidally at
a frequency ω = 1 kHz. The transient behavior reappears every half-
cycle (T/2). In practice, the time between the minima was measured.
The parameters used in the calculation are = 4π × 106
s−1
, =
2π × 6.0666 MHz, γ = 0.001 , ∗
= ∗
gi gj
= ∗
gi gj
= ∗
gi gj
=0, and
gi gj = gi gj
= gi gj
= gi gj
= 10−5
.
1 and 3 subsystems enter a CPT state. A short time after
the total magnetic field becomes equal to zero, the transients
reach their minimum values. It should be noted that, as the
magnetic field modulation frequency decreases, the minimum
point moves closer to B = 0, eventually reaching a situation
resembling the Hanle effect in a degenerate TLS where the
minimum occurs at B = 0 [19]. For example, in Figs. 4 and 8
of Ref. [20], a series of EIA and EIT resonances each centered
0.53 0.54 0.55 0.56 0.57
−10
0
10
20
30
40
50
t (ms)
ProbeAbsorption(cm−1
)
total probe
TLS
Λ
1
Λ2
Λ3
0.53 0.54 0.55 0.56 0.57
−4
−3
−2
−1
0
1
2
3
4
MF MF(G)
B=0
FIG. 3. (Color online) Contributions of the subsystems to the total
transient probe absorption in the presence of a modulated MF (black
line), for the same parameters as in Fig. 2. The dashed line represents
the time where the MF crosses zero. 1 and 3 contribute to the
transient behavior, the clock transition 2 is almost uninfluenced
by the magnetic field, and the absorption of the TLS stabilizes at a
constant value. The 3 oscillations are rapidly damped compared to
those of 1.
063809-3

4. L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 85, 063809 (2012)
0.5 0.52 0.54 0.56 0.58
−0.01
0
0.01
0.02
0.03
t (ms)
Coherence(Arb.Units)
Λ1
Λ3
FIG. 4. (Color online) Evolution of the coherence between the
lower sublevels of the 1 and 3 subsystems for the same parameters
as in Fig. 2.
at B = 0 is produced as the frequency of the linearly polarized
laser interacting with degenerate two-level systems in Cs is
slowly modulated in the presence of a magnetic field that is
modulated at a faster rate. Another example is that shown
in Fig. 4 of Ref. [21], where a series of CPT resonances each
centered at B = 0 is produced by a magnetic field that is slowly
modulated by a sawtooth waveform.
In Fig. 3, we show that both 1 and 3 contribute to the
transient behavior of the total probe absorption, with 3 giving
the greater contribution. This is due to the fact that the extreme
sublevels |Fg ,mg = 1,2 are the most populated ones since
the σ+
polarized fields optically pump the population to higher
values of mg . We also see that the 3 oscillations are rapidly
damped compared to those of 1. This behavior is also seen
in the time dependence of the coherences between the lower
levels of the 1 and 3 subsystems (Fig. 4). We now discuss
the origin of this different behavior.
There are two processes that occur simultaneously in the
system. The first is the change in the energy of the sublevels
due to the modulation of the magnetic field and the second
is the transfer of population between the sublevels. In order
to distinguish between the effect of each process on the total
absorption, we tested each one separately.
First, we studied the behavior of a single system. We
numerically solved the Bloch equations for a three-level
system for various initial populations of the lower states, while
applying the same time-dependent magnetic field as in Fig. 2.
The transient is created as the system enters the dark state at
two-photon resonance. Oscillations appear in the transient tail
of the absorption, in the coherence between the lower levels
and in the population. We found that the oscillations differ
only in their amplitude when the initial population distribution
between the lower levels is varied. Thus the difference between
the 1 and 3 population distributions is not the cause of the
difference in the behavior of the oscillations.
Returning to the realistic atomic system, we plot the
populations of the Zeeman sublevels of Fg (Fig. 5) as a
function of time. The populations of the Fg Zeeman sublevels
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
t (ms)
PopulationFg’
"trap"
Λ2
Λ3
Λ1
0 0.2 0.4 0.6 0.8 1
−4
−3
−2
−1
0
1
2
3
4
MF
MF(G)
FIG. 5. (Color online) Population evolution of the Fg = 2
Zeeman sublevels for a fixed pump and probe with the same
parameters as in Fig. 2. The MF is modulated sinusoidally at
frequency ω = 1 kHz. The dashed line indicates the time where the
MF equals zero. The TLS behavior is not shown in the figure.
behave similarly to those of the Fg sublevels in the same
subsystem. We show that the difference in the behavior of the
coherences and probe absorption of the two subsystems can
be explained by the transfer of population that occurs during
the transient. Initially, the population is equally distributed
among the eight sublevels in Fg = 1 and Fg = 2. After the
constant magnetic field is switched on, it can be seen in Fig. 5
that most of the population is trapped in the |Fg ,mg = 2
sublevel (the “trap” state) due to lack of pumping from this
sublevel. Simultaneously 2 enters into a dark state and CPT
is created, causing a decrease of population in the trap state.
As the magnetic field approaches zero, the other subsystems
( 1, 3, and the TLS) also become nearly resonant and the
pumping from them is more efficient. This leads to an increase
in the population of the trap state. At B = 0, 1 and 3 exhibit
CPT and a short time afterward population flows into these
subsystems at the expense of the trap state. 1 and 3 exit
CPT as the magnetic field passes B = 0, and some of the
population then returns to the trap state. It should be noted that
oscillations also occur in the population but are very weak.
The transfer of population in our system occurs via decay
to a reservoir (γ ) and collisional decay rate ( gi gj
and gi gj
),
which are the same for 1 and 3. Population which is
transferred from the trap state goes mainly to 3 since the
optical pumping to that subsystem is stronger than that to
1. The addition of population to 3 that occurs during the
CPT leads to damping of the oscillations in the population
and in the lower-level coherence, and eventually to damping
of the oscillations in the probe absorption. In order to confirm
that the optical pumping is the cause of the difference in the
behavior of the transients, we returned to the single system
and found that increasing the Rabi frequency and thus the
optical pumping indeed dampened the oscillations.
This feature is changed when a buffer gas is added to the
system. When velocity-changing collisions can be neglected
[22,23], the presence of the buffer gas can be simulated
by increasing the time the atoms spend in the laser beams
063809-4

5. COHERENCE-POPULATION-TRAPPING TRANSIENTS . . . PHYSICAL REVIEW A 85, 063809 (2012)
(decreasing γ ) and increasing the rate of phase-changing
collisions ∗
[24]. The probe absorption in a single system
is proportional to Imρeg (t). The analytical solution for ρeg (t)
when the two-photon (Raman) detuning is suddenly changed
from zero to , assuming that the pump and probe Rabi
frequencies Veg are equal, the initial population is equally
divided between the lower levels, and the pump and probe
detunings are antisymmetric such that eg = − eg = − /2,
is given by
ρeg (t) =
iV
eg − i 1
2
1
2
+
A
B
eBt
−
A
B
, (10)
where
A = −
V 2
eg − i 1
2
, (11)
B = −
2V 2
eg − i 1
2
+ g g + i , (12)
eg = 1
2
( + gg ) + γ + ∗
, and gg = gg + γ .
Equation (10) shows that, when a buffer gas is added
to the system ( eg increases), the decay rate of the transient
is decreased while the oscillation frequency is unchanged,
leading to decreased damping of the oscillations. This
was confirmed numerically for the modulating MF case.
However, in the realistic system (the following parameters
were changed: γ = 10−5
and ∗
= 10 ), we find that, in
addition to the decrease in the decay rate of the transient,
less population is trapped in the trap state due to a decrease
in the optical pumping effect (introducing buffer gas is in
some ways equivalent to reducing the Rabi frequency [25]),
which is the cause of the difference between the 1 and 3
oscillations in the absence of a buffer gas. Thus the difference
between the 1 and 3 oscillations does not occur when a
buffer gas is added to the system. Increasing in order to
increase optical pumping leads to damping of the oscillations
of 3 as in the absence of a buffer gas.
B. Sequence of transients
So far, we have dealt with a single transient where both the
pump and the probe are on resonance. In this section we discuss
a sequence of transients as a function of the probe detuning,
where the value of ω is chosen to be sufficiently large so that
transient oscillations appear in the probe absorption spectrum
and sufficiently small so that successive transients are identical
and well-separated in time. Unlike the case of a resonant probe
where the 1 and 3 subsystems enter a dark state at the same
time, for a detuned probe, each subsystem enters a dark state at
the appropriate value of the modulating magnetic field which
offsets its detuning. As the detuning increases, the total probe
absorption transient begins to widen and then breaks up due to
the fact that the 1 and 3 transients appear at different times,
as shown in Fig. 6. Due to the antisymmetry of the 1 and
3 subsystems, the transient contributed by each subsystem
occurs at the same absolute value of the magnetic field but
opposite signs. When the probe is detuned, the time between
two consecutive transients is no longer equal to the half-cycle
time T /2 of the magnetic field frequency (as in the case of
a resonant probe, see Fig. 2). However, the time between the
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0
20
40
60
80
t (ms)
ProbeAbsorption(cm
−1
)
total probe
Λ3
Λ1
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
−4
0
4
MF(G)
MF
FIG. 6. (Color online) Probe absorption for resonant pump and
detuned probe as a function of time in the presence of a magnetic field
(MF). The transients in 1 and 3 occur at different times so that the
total probe absorption splits into components. eg = 2 MHz, ω =
1 kHz, and the other parameters are the same parameters as in Fig. 2.
first and third transients of the individual subsytems remains
equal to the cycle time. In Fig. 7, we plot the deviation
from the half-cycle time (DHCT) as a function of the probe
detuning and show that it increases as the probe detuning
increases. In order for this phenomenon to be useful, the
DHCT should change strongly as a function of the absolute
value of the detuning. It is shown in Fig. 7 that the slope
becomes steeper as ω decreases (or alternatively B0 decreases).
However, it should be noted that the slope is independent of ω
if the DHCT is plotted as a function of eg T .
Surprisingly, the addition of a buffer gas does not cause
the DHCT versus the probe detuning to becomes narrower.
−6 −4 −2 0 2 4 6
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Probe detuning (MHz)
DHCT(ms)
200Hz
1kHz
200Hz &
Bdc=0.2G
FIG. 7. (Color online) Deviation from half-cycle time as a
function of probe detuning. The slope of the DHCT is steeper in the
case of ω = 200 Hz (blue line) than for ω = 1 kHz (red dashed line).
DHCT for ω = 200 Hz and in the presence of a constant magnetic
field (0.2 G) in the z direction (blue dotted line). Parameters are the
same as in Fig. 2.
063809-5

6. L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 85, 063809 (2012)
Although the width of the transients becomes narrower, the
transients occur exactly at the same times as in the absence of
the buffer gas, so that the DHCT is unchanged.
The sequence of transients can be applied to magnetometry.
The magnetic field shielding along the z axis is problematic in
experimental configurations and not perfect due to the entrance
of the laser fields through the z direction. Based on the transient
behavior, we propose to use the system as a magnetometer
for dc magnetic fields along the z axis, thereby simplifying
the atomic clock configuration and making it more accurate.
When in addition to the varying magnetic field there exists a
constant magnetic field in the z direction the total magnetic
field in the system will be equal to zero at different times and
this will be reflected in the appearance of the transient. In this
case the DHCT (the time between consecutive transients) will
change. This time can be measured and one can calculate the
unknown dc magnetic field as shown by the blue dotted line
in Fig. 7. If a transverse dc magnetic field exists in addition
to the longitudinal dc magnetic field, we find the DHCT to be
unchanged so that a measurement of the DHCT is indicative
solely of the longitudinal dc magnetic field. The individual
transients that appear in the probe absorption, however, change
their shapes depending on the transverse magnetic field and
hold out the possibility of providing some information for use
in vector magnetometry.
IV. CONCLUSIONS
In this work, we discussed the transient behavior caused
by the modulation of an applied magnetic field in the probe
absorption of a realistic three-level system in the D1 line of
87
Rb that exhibits CPT. We examined the contributions to the
probe absorption from the various subsystems that compose
the realistic atomic system and compared the absorption of
each subsystem to that of a simple system. We showed that
the population redistribution due to optical pumping is the
dominant cause of the difference between the contributions of
the various subsystems to the oscillatory character of the probe
absorption.
We also discussed the interesting behavior of a series of
transients. For a resonant pump and probe, the time from the
appearance of one transient to the consecutive one is equal
to the half-cycle time of the modulation frequency. However,
when the probe laser frequency is detuned, this time deviates
from the half-cycle time. This asymmetry in the periodicity of
the consecutive transients could be useful in applications such
as frequency standards and magnetometry.
ACKNOWLEDGMENTS
We are grateful to P. Phoonthong, F. Renzoni, and Y. Rubin
for stimulating discussions.
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