This document summarizes research on coherent-population-trapping transients induced by a modulated transverse magnetic field. The key points are: 1) Applying a transverse magnetic field causes new subsystems to form, new dark states to be created, and population to be rearranged among Zeeman sublevels. 2) When the transverse magnetic field is modulated over time, transients appear as the system switches between different steady-state solutions. 3) The contributions of the newly formed subsystems to the transient probe absorption spectra are discussed. Both the creation of new subsystems and population redistribution due to the transverse magnetic field influence the spectra.

1. PHYSICAL REVIEW A 88, 023827 (2013)
Coherent-population-trapping transients induced by a modulated transverse 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 6 June 2013; published 14 August 2013)
We present theoretical results of coherent-population-trapping transients induced by a modulated transverse
magnetic field (TMF). The application of a transverse magnetic field causes the appearance of new subsystems,
creation of new dark states, and rearrangement of the population among the Zeeman sublevels. We show that
transients appear as the system is switched between steady-state situations, and we identify the various level
system components of the total probe absorption. We discuss the time-dependent evolution of the probe absorption
caused by the modulated TMF in the presence and absence of a constant longitudinal magnetic field (LMF). The
differences between TMF modulation and LMF modulation are discussed.
DOI: 10.1103/PhysRevA.88.023827 PACS number(s): 42.50.Gy, 42.50.Hz, 32.80.Xx
I. INTRODUCTION
The influence of magnetic fields on coherent spectra in
alkali-metal atomic vapors has been extensively investigated
due, in part, to its impact in various applications such as
magnetometry [1–3] and frequency standards [4]. Recently,
it was shown both theoretically and experimentally [5,6] that
one can measure the magnitude and direction of an arbitrary
magnetic field using multiple coherent-population-trapping
(CPT) resonances, and this was demonstrated in Rb vapor.
In addition, a universal theory based on the technique of
superoperators has been used to analyze dark resonances in
arbitrary electric and magnetic fields [7].
Recently, we theoretically investigated CPT transients
induced by an ac longitudinal magnetic field (LMF) for a
realistic three-level system in the D1 line of 87
Rb [8]. 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 also discussed the series of transients
that appear every half-cycle time of the modulated magnetic
field when the system is in two-photon resonance and studied
the changes in the timing of the transients as a function of the
probe detuning. Modulation spectroscopy of dark resonances
has also been investigated for the case of a degenerate two-level
system [9], either by scanning the modulation frequency of the
laser field for a fixed magnetic field or by scanning the strength
of the magnetic field at a fixed modulation frequency.
Here, we examine the effect of an oscillating transverse
magnetic field (TMF) on the CPT spectrum in the same
atomic system. As shown in [5,6], the application of a
constant TMF generates new subsystems and creates new
dark states. Consequently, the number of CPT dips can vary
between one and seven, according to the polarization of
the incident electromagnetic fields and the magnitude and
direction of the magnetic field. In addition to creating new
subsystems, the TMF leads to redistribution of the population
among the Zeeman sublevels [10–13], and the contributions of
the various subsystems to the total spectrum can be explained
by considering these two effects, which are not always
synergetic. Here, we show that when the TMF is modulated in
time, transients appear as the system is switched between the
various steady-state solutions. We will discuss the contribu-
tions of these newly formed subsystems to the transient probe
absorption spectra. In order to facilitate the discussion of the
time-dependent spectra, we first discuss the steady-state probe
absorption spectrum in the absence and presence of LMF and
TMF magnetic fields. The calculations were performed, using
the equations given in Sec. II, for the D1 line of 87
Rb 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. In the absence of the TMF
[see Fig. 1(a)] [8], the system consists of a single two-level
system (TLS), |Fg ,mg = −2 ↔ |Fe,me = −1 , and three
systems: 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 .
In the absence of any magnetic field, all the subsystems
are degenerate, resulting in a single CPT resonance at = 0
[Fig. 2, curve (a)], where = ω1 − ω2 − hfs = 0 is the two-
photon detuning, ω1 and ω2 are the frequencies of the pump
and the probe, and hfs is the hyperfine splitting.
In the presence of a LMF, each subsystem in Fig. 1(a) is
characterized by the difference in energy between its lower lev-
els, so that the central CPT resonance splits into three [Fig. 2,
curve (b)]. The 1 and 3 subsystems are antisymmetric with
respect to the Zeeman splitting of their lower sublevels, so
that the dips appear at = 0, ±2δ, where δ = μBgB is the
difference in frequency between adjacent Zeeman sublevels,
g is the gyromagnetic ratio, and μB is the Bohr magneton.
When, in addition to the LMF, a TMF is applied (Bx or By
yields similar results), the total magnetic field is aligned along
a new axis. If we relate to this new axis as the quantization axis,
the effective laser polarization will have components of σ− and
π polarizations in addition to the original σ+, leading to the
creation of new subsystems, resulting in the appearance of a
total of seven CPT dips [Fig. 2, curve (c)]. In addition, the dis-
tance between the dips, which appear at = 0, ±δ, ±2δ, ±3δ,
increases when the total magnetic field increases. When only
a TMF is applied, the dominant subsystems seen in the
spectrum [Fig. 2, curve (d)] are those with ±δ and ±3δ energy
difference between the lower sublevels.
023827-11050-2947/2013/88(2)/023827(6) ©2013 American Physical Society

2. L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 88, 023827 (2013)
-1 0 1
-1 0
-2
1 2
-2 - 1 210
2F =
1F =
2F =
Λ Λ ΛTLS
-1 0 1
-1 0 1
-1 10
ΛΛ
Λ Λ
1
1 2
21
ΛΛ Λ
(a) (b) (c)
FIG. 1. (Color online) Energy-level scheme for the interaction of
the D1 line of 87
Rb. The Zeeman shifts for the upper hyperfine level
are not shown. (a) The pump and the probe are σ+
polarized. The
system can be divided into one two-level system (TLS) and three
subsystems. (b) All the subsystems with δ = 0 energy difference
between the lower sublevels in the presence of TMF. (c) 3 and
subsystems with 3δ energy difference between the lower sublevels
( 7 and 8) that are created by the TMF.
II. THE BLOCH EQUATIONS
The system consists of two ground hyperfine states Fg
and Fg and a single excited hyperfine state Fe (a con-
figuration). 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. [14] for a simple system and adapt them to 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 done by Goren et al. for a degenerate two-level atomic
−2 −1 0 1 2
2
4
6
8
10
12
14
16
Δ (MHz)
ProbeAbsorption(cm−1
)
a
b
c
d
2
3
1
FIG. 2. (Color online) Steady-state probe absorption in the pres-
ence of different magnetic fields: (a)Bz = 0 and Bx = 0, (b)Bz = 0.1
G and Bx = 0, (c)Bz = 0.1 G and Bx = 0.1 G, (d)Bz = 0 and
Bx = 0.1 G. The other parameters 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
.
system [15].
˙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)
is the total spontaneous emission rate from each
Feme sublevel, whereas Fe→Fg,g
is the decay rate from
Fe to one of the Fg,g states. gi gj
and gi gj
are the
collisional decay rate from sublevels gi → gj , and gi → gj .
Because the energy between the ground hyperfine levels
corresponds to frequencies in the microwave range, the
collisions not only damp the coherence but also affect the
populations of Fg and Fg [16]; therefore, we introduce
phenomenologically the population transfer rate from mg
to mg : gi gj
and gi gj
. γ is the rate of decay due to time
023827-2

3. COHERENT-POPULATION-TRAPPING TRANSIENTS . . . PHYSICAL REVIEW A 88, 023827 (2013)
of flight through the laser beams. The dephasing rates
of the excited- to 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), is 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
)
≡ −¯h[Vei gj
(ω1)e−iω1t
+ Vei gj
(ω2)e−iω2t
], (8)
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 transition and q =
(−1,0,1) depending on the polarization of the incident laser.
In order to include the effect of an additional transverse
magnetic field, for instance, Bx, we add the following
additional terms [10] to the Bloch equations [Eqs. (1)–(5)]:
˙ρei ej
|Bx
= −i
μBBx
2¯h
ge c+
ei
ρei+1ej
+ c−
ei
ρei−1ej
− c+
ej
ρei ej+1
− c−
ej
ρei ej−1
, (10)
˙ρei gj
|Bx
= −i
μBBx
2¯h
ge c+
ei
ρei+1gj
+ c−
ei
ρei−1gj
− gg c+
gj
ρei gj+1
+ c−
gj
ρei gj−1
, (11)
˙ρgi gj
|Bx
= −i
μBBx
2¯h
gg c+
gi
ρgi+1gj
+ c−
gi
ρgi−1gj
− c+
gj
ρgi gj+1
− c−
gj
ρgi gj−1
, (12)
˙ρgi gj
|Bx
= −i
μBBx
2¯h
gg c+
gi
ρgi+1gj
+ c−
gi
ρgi−1gj
− gg c+
gj
ρgi gj+1
+ c−
gj
ρgi gj−1
, (13)
where c±
FmF
≡
√
(F ∓ mF )(F ± mF + 1). In Eqs. (11) and
(12) one can interchange g and g in order to obtain the
equations for ˙ρei gj
and ˙ρgi gj
. The solution ρ(t) of Eqs. (1)–(5)
and Eqs. (10)–(12) is calculated numerically as a function of
time.
0 1 2 3 4 5 6 7
0
2
4
6
8
10
12
t (ms)
ProbeAbsorption(cm
−1
)
Bx
=0 Bx
=0
total probe
Λ
2
Λ
3
0 1 2 3 4 5 6 7
−0.1
0
0.1
TMF
TMF(G)
FIG. 3. (Color online) Probe absorption and dominant contribu-
tions from the subsystems 2 and 3 to the total transient probe
absorption in the case of = 0 in the presence of a constant Bz = 0.1
G and sinusoidally modulated TMF (ω = 200 Hz; black line) for the
same parameters as in Fig. 2. The dashed lines represent the times
where the TMF crosses zero.
III. RESULTS AND DISCUSSION
We will discuss the time-dependent spectra at various
two-photon detunings (labeled 1, 2, 3 in Fig. 2) for a zero
or constant value of Bz and for a modulated Bx according
to Bx = Bx0
sin[π/2 + ω(t − t0)], where ω is the modulation
frequency of the transverse magnetic field. In each case,
Bx = Bx0
and Bz are first applied, the system is allowed
to establish its steady-state properties, and then the TMF
is modulated (at time t0 = 0.9 ms). We will see that the
time-dependent spectra are easily understood by considering
transitions between the various steady-state solutions shown
in Fig. 2.
A. Modulation in the presence of Bz = 0
In the first case we consider, = 0 (point 1 in Fig. 2) and
Bz = Bx0
= 0.1 G. Thus, when the modulation begins, the
system is in the steady state whose spectrum is given by curve
(c) in Fig. 2, whereas when Bx = 0, the spectrum is given by
curve (b). It should be noted that in the presence of the TMF,
both the depth and the contrast are smaller than in its absence.
The resulting time-dependent spectrum is given in Fig 3.
Although the change seems adiabatic, the minimum ab-
sorption occurs a short time after the TMF reaches zero [8].
This adiabatic evolution is seen even when the modulation
frequency is increased, unlike the case of LMF modulation [8]
and the cases where = 0, which will be discussed below. The
contributions to the total probe absorption are shown in Fig. 3.
(The contributions of the TLS and 1 are not shown since
they are more or less constant.) As can be seen, the subsystem
that controls the total probe absorption is 2, which is in a
dark state in the absence of TMF, as evidenced by the increase
in the lower-level coherence shown in Fig. 4(c) and changes
to partial CPT in the presence of the TMF. The TMF also
transfers population between neighboring Zeeman sublevels
[see Eqs. (10)–(13)], as can be seen in Figs. 4(a) and 4(b).
023827-3

4. L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 88, 023827 (2013)
1 5
0.1
0.15
PopFg’
1 5
0.1
0.15
PopFg
0 1 2 3 4 5 6 7
0
0.05
0.1
Coherence
t (ms)
(a)
(b)
(c)
FIG. 4. (Color online) Population and coherence evolution in the
presence of TMF modulation in the case of = 0. (a) Population of
the Fg sublevels: |m = −1 (red dotted line), |m = 0 (green solid
line), and |m = 1 (cyan dashed line). (b) Population of the Fg
sublevels : |m = −1 (red dotted), |m = 0 (green solid line), and
|m = 1 (cyan dashed line). (c) Lower-level coherence of 2 and 4
(green solid line), 5 (dotted blue line), and 6 (black dashed line).
Thus, whereas the LMF has no effect on the clock transition
2 because the Zeeman sublevels |m = 0 are not shifted to
first order in Bz, the TMF affects the |m = 0 Zeeman sublevels
by mixing them with the |m = ±1 Zeeman sublevels so that
now 2 determines the behavior of the total absorption.
In the presence of Bx, new subsystems that contribute
to the spectrum at = 0 are introduced: 4, |Fg,mg =
0 ↔ |Fe,me = −1 ↔ |Fg ,mg = 0 ; 5, |Fg,mg = −1 ↔
|Fe,me = 0 ↔ |Fg ,mg = 1 ; and 6, |Fg,mg = 1 ↔
|Fe,me = 0 ↔ |Fg ,mg = −1 . These are shown in Fig. 1(b).
The lower-level coherences of these subsystems are shown
in Fig. 4(c). In the presence of the TMF, both 2 and 4
contribute to the same lower-level coherence. However, the
dominant effect seems to be the transfer of population between
the sublevels rather than the addition of 4. As shown in
Fig. 4(c), the coherence between the lower levels of 5 and
6 is nonzero in the presence of TMF due to the creation of
dark states.
We now consider the case where = −3δ0, where δ0 =
μBg(B2
x0 + B2
z )1/2
(point 2 in Fig. 2). As in the previous case,
the system is first prepared by applying constant LMF and
TMF with Bz = Bx0
= 0.1 G so that the steady-state spectrum
is given by curve (c) in Fig. 2. When the TMF passes through
zero on modulation, the steady-state spectrum is given by curve
(b) in which the CPT dip at = −3δ no longer exists. Thus,
modulating the TMF in time switches the system between
cases (c) and (b). In Fig. 5, we see that applying the constant
magnetic fields causes the system to be in a state of CPT. When
the TMF modulation is applied, there is an immediate sharp
change in the probe absorption followed by weak oscillations
as the position of the CPT dip moves to a smaller value of
. The changes in the absorption as the absolute value of
the TMF approaches zero reflect the difference between the
baselines of cases (b) and (c) in Fig. 2. Only when the TMF
0 1 2 3 4 5 6 7
0
2
4
6
8
10
12
14
t (ms)
ProbeAbsorption(cm
−1
)
Bx
=0 Bx
=0
total
probe
Λ
2
Λ3
0 1 2 3 4 5 6 7
−0.1
0
0.1
TMF
TMF(G)
FIG. 5. (Color online) Probe absorption and dominant contribu-
tions from the subsystems 2 and 3 to the total transient probe
absorption in the case of = −3δ0 in the presence of a constant
Bz = 0.1 G and TMF modulation (black line) for the same parameters
as in Fig. 2. The dashed lines represent the times where the TMF
crosses zero.
again reaches its maximum absolute value and CPT is attained
is there another sharp change in the absorption.
In the presence of the TMF, there are two new
subsystems with 3δ energy difference between their lower
sublevels [Fig. 1(c)] that contribute to the CPT dip.
These are 7, |Fg,mg = 1 ↔ |Fe,me = 1 ↔ |Fg ,mg = 2 ,
and 8, |Fg,mg = 1 ↔ |Fe,me = 2 ↔ |Fg ,mg = 2 . The
|Fg,mg = 1 ↔ |Fe,me = 2 transition of 8 is also part of
3, so that the contribution from 3 (as shown in Fig. 5) is
significant even though its energy difference is only 2δ.
The lower-level coherence between sublevels |Fg,mg =
1 and |Fg ,mg = 2 , which is shown in Fig. 6(a), has
contributions from both 7 and 8; thus, this coherence
is created in the presence of the TMF and vanishes when
the TMF vanishes. In order to distinguish between the
contributions of 7 and 8, we plot the time dependence
of the imaginary part of the density matrix (proportional to the
absorption) for the transitions |Fe,me = 1 ↔ |Fg ,mg = 2
and |Fe,me = 2 ↔ |Fg ,mg = 2 in Fig. 6(b) and for the
transitions |Fe,me = 1 ↔ |Fg,mg = 1 and |Fe,me = 2 ↔
|Fg,mg = 1 in Fig. 6(c). We can see that 8, which has a
common transition with 3, gives the greater contribution.
Simultaneous modulation of LMF and TMF at the same
frequency, in the absence of constant magnetic fields, can be
described by switching between cases (c) and (a) of the steady-
state spectrum shown in Fig. 2. When = 0, the evolution
of the probe absorption will be similar to that obtained in
the presence of only LMF modulation (Fig. 2 of [8]) or only
TMF modulation (see Fig. 7), namely, a sharp transient that
appears a short time after the MFs cross zero. For a detuned
probe there are two kinds of transients. The first type occurs
immediately after the total MF deviates from the value that
offsets the probe detuning in each individual subsystem,
so that several transients are obtained. The second kind of
transient occurs immediately after the total MF crosses zero
023827-4

5. COHERENT-POPULATION-TRAPPING TRANSIENTS . . . PHYSICAL REVIEW A 88, 023827 (2013)
1 5
−0.1
0
0.1
Coherence
1 5
−3
0
3
Imρeg’
0 1 2 3 4 5 6 7
0
5
10
t (ms)
Imρeg
(a)
(b)
(c)
FIG. 6. (Color online) (a) The lower-level coherence of 7
and 8 in the presence of TMF modulation when = −3δ0.
Time dependence of the imaginary part of the density matrix
(b) for the transitions |Fe,me = 1 ↔ |Fg ,mg = 2 (red dotted
line) and |Fe,me = 2 ↔ |Fg ,mg = 2 (blue solid line) and (c) for
the transitions |Fe,me = 1 ↔ |Fg,mg = 1 (red dotted line) and
|Fe,me = 2 ↔ |Fg,mg = 1 (blue solid line).
and reflects the difference between the baselines in cases (c)
and (a).
B. Modulation in the absence of Bz
So far, we have discussed the CPT transients induced by
modulation of the TMF in the presence of a constant LMF.
Now, we will examine the CPT transients induced in the
absence of the LMF. Thus, when the modulation begins, the
system is in the steady state whose spectrum is given by curve
(d) in Fig. 2, whereas when Bx = 0, the spectrum is given
0 1 2 3 4 5 6 7
4
8
12
16
t (ms)
ProbeAbsorption(cm−1
)
B
x
=0 B
x
=0
total probe
0 1 2 3 4 5 6 7
−0.1
0
0.1
TMF
TMF(G)
FIG. 7. (Color online) Probe absorption for resonant pump and
probe = 0 as a function of time in the presence of a TMF
modulation (black line) and in the absence of Bz for the same
parameters as in Fig. 2. The dashed lines represent the times where
the TMF crosses zero.
0 1 2 3 4 5 6 7
12
t (ms)
ProbeAbsorption(cm
−1
)
Bx
=0 Bx
=0
total
probe
0 1 2 3 4 5 6 7
−0.1
0
0.1
TMF
TMF(G)
FIG. 8. (Color online) Probe absorption when = −3δ0 as a
function of time in the presence of a TMF modulation (black line)
and in the absence of Bz for the same parameters as in Fig. 2. The
dashed lines represent the times where the TMF crosses zero.
by curve (a). In the presence of the TMF, both the depth and
the contrast are much smaller than in its absence. In Fig. 7,
the total probe absorption in the presence of the modulated
TMF at = 0 (point 1 in Fig. 2) is shown. The time evolution
in this case is similar to that obtained in the presence of a
modulated LMF [8]: namely, a sharp transient that appears
a short time after the magnetic field passes through zero,
followed by damped oscillations.
We now detune the probe to = −3δ0 (point 3 in Fig. 2).
When the modulation begins, the steady-state spectrum is
given by curve (d) in Fig. 2, whereas when Bx = 0, the
spectrum is given by curve (a). We see in Fig. 8 that there
are two sets of transients: the transients that occur just after
the TMF reaches its maximal absolute value that also occur
at this value of in the presence of Bz (see Fig. 5) and the
transients that occur immediately after the TMF passes through
zero (see Fig. 7). The transients that occur just after the TMF
reaches its maximal absolute value are caused by the shift of
CPT to a lower value of . The transients that happen after
the TMF passes through zero reflect the difference between
the baselines in cases (d) and (a).
A three-dimensional figure that summarizes the effect of
the TMF modulation on the probe absorption spectrum as a
function of detuning in the absence of the LMF is shown in
Fig. 9. The plot begins at time t0 and ends after almost a
half cycle of TMF modulation. At time t0, we see the seven
resonance dips clearly. As the absolute value of the TMF
decreases, the dips at = 0, ±2δ become weaker, and the
distance between the dips decreases. When the TMF crosses
zero, the Zeeman sublevels become degenerate, and all the
dips appear at = 0. For any other value of , the transient
that appears after the TMF crosses zero reflects the difference
between the baselines in cases (d) and (a) in Fig 2. A similar
picture in the presence of a constant LMF (not shown here)
also displays the seven CPT resonances changing their position
and intensity as the value of δ changes. When the TMF crosses
zero, only the original subsystems at = 0, ±2δ remain.
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6. L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 88, 023827 (2013)
FIG. 9. (Color online) Evolution of probe absorption spectrum as
a function of detuning in the presence of a TMF modulation and in
the absence of Bz for the same parameters as in Fig. 2.
IV. CONCLUSIONS
We have examined the probe absorption transients induced
by a modulated TMF in the absence and presence of a LMF.
We showed the transients that appear as the system is switched
between the various steady-state situations whose spectra are
shown in Fig. 2. We discussed the contributions of the various
subsystems to the transient probe absorption spectra in each
case.
We chose two different probe detunings, = 0 and =
−3δ0, which are indicative of the time-dependent behavior as
a function of probe detuning, in order to describe the evolution
of the probe absorption in time during the TMF modulation.
For any probe detuning chosen, one can see that the probe
absorption changes due to the creation and disappearance of
subsystems.
We showed that the evolution in the presence of a constant
LMF seems adiabatic and that transients appear only after the
CPT dip moves from the chosen value of to a different value.
In the absence of a constant LMF, the evolution is sharper, and
in addition to the transients that appear after the CPT dip
changes its position, there are also transients that appear as the
TMF crosses zero. This can provide a means for recognizing
the presence of a LMF in the system.
We also noticed differences between the TMF modulation
and the LMF modulation, studied previously [8]. Whereas
modulating the LMF leads to variation in the contributions to
the absorption that derive from the original subsystems due
to their entry and exit from CPT, the TMF modulation leads to
the creation and destruction of new subsystems.
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