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

1Department of Chemistry, Bar-Ilan University, Ramat Gan 52900, Israel2The 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 transversemagnetic 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 thattransients appear as the system is switched between steady-state situations, and we identify the various levelsystem components of the total probe absorption. We discuss the time-dependent evolution of the probe absorptioncaused by the modulated TMF in the presence and absence of a constant longitudinal magnetic field (LMF). Thedifferences 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 inalkali-metal atomic vapors has been extensively investigateddue, in part, to its impact in various applications such asmagnetometry [1–3] and frequency standards [4]. Recently,it was shown both theoretically and experimentally [5,6] thatone can measure the magnitude and direction of an arbitrarymagnetic field using multiple coherent-population-trapping(CPT) resonances, and this was demonstrated in Rb vapor.In addition, a universal theory based on the technique ofsuperoperators has been used to analyze dark resonances inarbitrary electric and magnetic fields [7].

Recently, we theoretically investigated CPT transientsinduced by an ac longitudinal magnetic field (LMF) for arealistic three-level � system in the D1 line of 87Rb [8]. Weexamined the contributions to the probe absorption from thevarious subsystems that compose the realistic atomic systemand compared the absorption of each subsystem to that of asimple � system. We also discussed the series of transientsthat appear every half-cycle time of the modulated magneticfield when the system is in two-photon resonance and studiedthe changes in the timing of the transients as a function of theprobe detuning. Modulation spectroscopy of dark resonanceshas also been investigated for the case of a degenerate two-levelsystem [9], either by scanning the modulation frequency of thelaser field for a fixed magnetic field or by scanning the strengthof the magnetic field at a fixed modulation frequency.

Here, we examine the effect of an oscillating transversemagnetic field (TMF) on the CPT spectrum in the sameatomic system. As shown in [5,6], the application of aconstant TMF generates new � subsystems and creates newdark states. Consequently, the number of CPT dips can varybetween one and seven, according to the polarization ofthe incident electromagnetic fields and the magnitude anddirection of the magnetic field. In addition to creating newsubsystems, the TMF leads to redistribution of the populationamong the Zeeman sublevels [10–13], and the contributions ofthe various subsystems to the total spectrum can be explainedby considering these two effects, which are not alwayssynergetic. Here, we show that when the TMF is modulated intime, 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 probeabsorption spectra. In order to facilitate the discussion of thetime-dependent spectra, we first discuss the steady-state probeabsorption spectrum in the absence and presence of LMF andTMF magnetic fields. The calculations were performed, usingthe equations given in Sec. II, for the D1 line of 87Rb where thepump is resonant with the Fg = 1 → Fe = 2 transition and theprobe is resonant with the Fg′ = 2 → Fe = 2 transition. Thepump and the probe are both σ+ polarized and have the samegeneral Rabi frequency � = �1,2. In the absence of the TMF[see Fig. 1(a)] [8], the system consists of a single two-levelsystem (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 � subsystemsare 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 pumpand the probe, and �hfs is the hyperfine splitting.

In the presence of a LMF, each � subsystem in Fig. 1(a) ischaracterized 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 withrespect to the Zeeman splitting of their lower sublevels, sothat the dips appear at � = 0,±2δ, where δ = μBgB is thedifference 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 alonga new axis. If we relate to this new axis as the quantization axis,the effective laser polarization will have components of σ− andπ polarizations in addition to the original σ+, leading to thecreation of new � subsystems, resulting in the appearance of atotal 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 onlya TMF is applied, the dominant � subsystems seen in thespectrum [Fig. 2, curve (d)] are those with ±δ and ±3δ energydifference between the lower sublevels.

023827-11050-2947/2013/88(2)/023827(6) ©2013 American Physical Society

L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 88, 023827 (2013)

-1 0 1

-1 0-21 2

-2 - 1 2102F =

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 ofthe D1 line of 87Rb. The Zeeman shifts for the upper hyperfine levelare not shown. (a) The pump and the probe are σ+ polarized. Thesystem can be divided into one two-level system (TLS) and three �

subsystems. (b) All the � subsystems with δ = 0 energy differencebetween 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 pumpof frequency ω1, and the Fg′ → Fe transition interacts witha probe of frequency ω2. We use the equations for thetime evolution of the � configuration as given by Boublilet al. [14] for a simple � system and adapt them to asystem consisting of Zeeman sublevels with the addition ofdecay from the ground and excited states to a reservoir andcollisions between the Zeeman sublevels of the ground statesas done by Goren et al. for a degenerate two-level atomic

−2 −1 0 1 22

4

6

8

10

12

14

16

Δ (MHz)

Pro

be A

bsor

ptio

n (c

m−

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 = 0.1G and Bx = 0, (c)Bz = 0.1 G and Bx = 0.1 G, (d)Bz = 0 andBx = 0.1 G. The other parameters are � = 4π × 106 s−1, =2π × 6.0666 MHz, γ = 0.001, ∗ = ∗

gigj= ∗

g′ig′j

= ∗gig

′j

= 0,

and gigj= g′

ig′j

= gig′j

= g′igj

= 10−5.

system [15].

ρ̇ei ej= −(

iωeiej+

)ρeiej

− γ(ρeiej

− ρeqeiei

)− (i/h̄)

∑gk

(ρeigk

Vgkej− Veigk

ρgkej

)− (i/h̄)

∑g′

k

(ρeig

′kVg′

kej− Veig

′kρg′

kej

), (1)

ρ̇eigj= −(

iωeigj+ ′

eigj

)ρeigj

− (i/h̄)

( ∑ek

ρeiekVekgj

−∑gk

Veigkρgkgj

−∑g′

k

Veig′kρg′

kgj

), (2)

ρ̇gigi= −(i/h̄)

∑ek

(ρgiek

Vekgi− Vgiek

ρekgi

)− γ

(ρgigi

− ρeqgigi

) − (2Fg)gigjρgigi

+∑

gk,k �=i

gkgiρgkgk

− (2Fg′ + 1)gig′jρgigi

+∑g′

k

g′kgi

ρg′kg

′k+ (

·ρgigi

)SE

, (3)

ρ̇gigj= −(

iωgigj+ ′

gigj

)ρgigj

+ ( ·ρgigj

)SE

− (i/h̄)∑ek

(ρgiek

Vekgj− Vgiek

ρekgj

), (4)

ρ̇gig′j

= −(iωgig

′j+ ′

gig′j

)ρgig

′j

− (i/h̄)∑ek

(ρgiek

Vekg′j− Vgiek

ρekg′j

). (5)

In Eqs. (2), (4), and (5) one can interchange g and g′ in orderto obtain the equations for ρ̇eig

′j, ρ̇g′

i g′i, and ρ̇g′

i g′j, respectively.

Here,

(·ρgigj

)SE

= (2Fe + 1)Fe→Fg

∑q=−1,0,1

Fe∑me,m′

e=−Fe

(−1)−me−m′e

×(

Fg 1 Fe

−mgiq me

)ρ

me m′e

(Fe 1 Fg

−m′e 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 eachFeme sublevel, whereas Fe→Fg,g′ is the decay rate fromFe to one of the Fg,g′ states. gigj

and g′i g

′j

are thecollisional decay rate from sublevels gi → gj , and g′

i → g′j .

Because the energy between the ground hyperfine levelscorresponds to frequencies in the microwave range, thecollisions not only damp the coherence but also affect thepopulations of Fg and Fg′ [16]; therefore, we introducephenomenologically the population transfer rate from mg

to mg′ : gig′j

and g′i gj

. γ is the rate of decay due to time

023827-2

COHERENT-POPULATION-TRAPPING TRANSIENTS . . . PHYSICAL REVIEW A 88, 023827 (2013)

of flight through the laser beams. The dephasing ratesof the excited- to ground-state coherences are given by′

eigj= γ + 1

2 [ + (2Fg)gigj+ (2Fg′ + 1)gig

′j] + ∗, and

′eig

′j

= γ + 12 [ + (2Fg′)g′

i g′j

+ (2Fg + 1)g′i gj

] + ∗,where ∗ is the rate of phase-changing collisions. Thedephasing rates of the ground-state coherences are given by′

gigj= γ + (2Fg)gigj

+ (2Fg′ + 1)gig′j+ ∗

gigj,′

g′i g

′j=

γ + (2Fg′)g′i g

′j+ (2Fg + 1)g′

i gj+ ∗

g′i g

′j, and ′

gig′j=

γ + 12 [(2Fg)gigj

+ (2Fg′ + 1)gig′j+ (2Fg′)g′

i g′j+ (2Fg +

1)g′i gj

] + ∗gig

′j, where ∗

gigj,∗

g′i g

′j, and ∗

gig′j

are the rates of

phase-changing collisions. The frequency separation betweenlevels ai and bj , including Zeeman splitting of the groundand excited levels due to an applied magnetic field, is givenby ωaibj

= (Eai− Ebj

)/h̄, with a,b = (g,e), and ρeqaiai

, witha = (g,e), is the equilibrium population of state ai in theabsence of any electrical fields. The interaction energy in therotating-wave approximation for the transition from level gj

to ei is written as

Veigj= −μ

eigj(E1e

−iω1t + E2e−iω2t )

≡ −h̄[Veigj(ω1)e−iω1t + Veigj

(ω2)e−iω2t ], (8)

where 2Veigj(ω1,2) are the pump and probe Rabi frequencies

for the Feme → Fgmg transition, given by

2Veigj(ω1,2) =

2μeigj

E1,2

h̄

= (−1)Fe−me

(Fe 1 Fg

−me q mg

)�1,2, (9)

where �1,2 = 2〈Fe||μ||Fg,g′ 〉E1,2/h̄ are the general pump andprobe 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 transversemagnetic field, for instance, Bx , we add the followingadditional terms [10] to the Bloch equations [Eqs. (1)–(5)]:

ρ̇eiej|Bx

= −iμBBx

2h̄ge

{c+eiρei+1ej

+ c−eiρei−1ej

− c+ej

ρeiej+1 − c−ej

ρeiej−1

}, (10)

ρ̇eigj|Bx

= −iμBBx

2h̄

{ge

[c+eiρei+1gj

+ c−eiρei−1gj

]− gg

[c+gj

ρeigj+1 + c−gj

ρeigj−1

]}, (11)

ρ̇gigj|Bx

= −iμBBx

2h̄gg

{c+giρgi+1gj

+ c−giρgi−1gj

− c+gj

ρgigj+1 − c−gj

ρgigj−1

}, (12)

ρ̇gig′j|Bx

= −iμBBx

2h̄

{gg

[c+giρgi+1g

′j+ c−

giρgi−1g

′j

]− gg′

[c+g′

jρgig

′j+1

+ c−g′

jρgig

′j−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 theequations for ρ̇eig

′j

and ρ̇g′i g

′j. The solution ρ(t) of Eqs. (1)–(5)

and Eqs. (10)–(12) is calculated numerically as a function oftime.

0 1 2 3 4 5 6 7

0

2

4

6

8

10

12

t (ms)

Pro

be A

bsor

ptio

n (c

m−

1 )

Bx=0 B

x=0

total probe

Λ2

Λ3

0 1 2 3 4 5 6 7

−0.1

0

0.1TMF

TM

F (

G)

FIG. 3. (Color online) Probe absorption and dominant contribu-tions from the subsystems �2 and �3 to the total transient probeabsorption in the case of � = 0 in the presence of a constant Bz = 0.1G and sinusoidally modulated TMF (ω = 200 Hz; black line) for thesame parameters as in Fig. 2. The dashed lines represent the timeswhere the TMF crosses zero.

III. RESULTS AND DISCUSSION

We will discuss the time-dependent spectra at varioustwo-photon detunings (labeled 1, 2, 3 in Fig. 2) for a zeroor constant value of Bz and for a modulated Bx accordingto Bx = Bx0 sin[π/2 + ω(t − t0)], where ω is the modulationfrequency of the transverse magnetic field. In each case,Bx = Bx0 and Bz are first applied, the system is allowedto establish its steady-state properties, and then the TMFis modulated (at time t0 = 0.9 ms). We will see that thetime-dependent spectra are easily understood by consideringtransitions between the various steady-state solutions shownin Fig. 2.

A. Modulation in the presence of Bz �= 0

In the first case we consider, � = 0 (point 1 in Fig. 2) andBz = Bx0 = 0.1 G. Thus, when the modulation begins, thesystem is in the steady state whose spectrum is given by curve(c) in Fig. 2, whereas when Bx = 0, the spectrum is given bycurve (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 modulationfrequency is increased, unlike the case of LMF modulation [8]and the cases where � �= 0, which will be discussed below. Thecontributions to the total probe absorption are shown in Fig. 3.(The contributions of the TLS and �1 are not shown sincethey are more or less constant.) As can be seen, the subsystemthat controls the total probe absorption is �2, which is in adark state in the absence of TMF, as evidenced by the increasein the lower-level coherence shown in Fig. 4(c) and changesto partial CPT in the presence of the TMF. The TMF alsotransfers population between neighboring Zeeman sublevels[see Eqs. (10)–(13)], as can be seen in Figs. 4(a) and 4(b).

023827-3

L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 88, 023827 (2013)

1 5

0.1

0.15

Pop

Fg’

1 5

0.1

0.15

Pop

Fg

0 1 2 3 4 5 6 70

0.05

0.1

Coh

eren

ce

t (ms)

(a)

(b)

(c)

FIG. 4. (Color online) Population and coherence evolution in thepresence of TMF modulation in the case of � = 0. (a) Population ofthe Fg sublevels: |m = −1〉 (red dotted line), |m = 0〉 (green solidline), 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 tofirst order in Bz, the TMF affects the |m = 0〉 Zeeman sublevelsby mixing them with the |m = ±1〉 Zeeman sublevels so thatnow �2 determines the behavior of the total absorption.

In the presence of Bx , new � subsystems that contributeto 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 shownin Fig. 4(c). In the presence of the TMF, both �2 and �4

contribute to the same lower-level coherence. However, thedominant effect seems to be the transfer of population betweenthe sublevels rather than the addition of �4. As shown inFig. 4(c), the coherence between the lower levels of �5 and�6 is nonzero in the presence of TMF due to the creation ofdark states.

We now consider the case where � = −3δ0, where δ0 =μBg(B2

x0 + B2z )1/2 (point 2 in Fig. 2). As in the previous case,

the system is first prepared by applying constant LMF andTMF with Bz = Bx0 = 0.1 G so that the steady-state spectrumis given by curve (c) in Fig. 2. When the TMF passes throughzero 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 betweencases (c) and (b). In Fig. 5, we see that applying the constantmagnetic fields causes the system to be in a state of CPT. Whenthe TMF modulation is applied, there is an immediate sharpchange in the probe absorption followed by weak oscillationsas the position of the CPT dip moves to a smaller value of�. The changes in the absorption as the absolute value ofthe TMF approaches zero reflect the difference between thebaselines of cases (b) and (c) in Fig. 2. Only when the TMF

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

t (ms)

Pro

be A

bsor

ptio

n (c

m−

1 )

Bx=0 B

x=0

totalprobe

Λ2

Λ3

0 1 2 3 4 5 6 7

−0.1

0

0.1TMF

TM

F (

G)

FIG. 5. (Color online) Probe absorption and dominant contribu-tions from the subsystems �2 and �3 to the total transient probeabsorption in the case of � = −3δ0 in the presence of a constantBz = 0.1 G and TMF modulation (black line) for the same parametersas in Fig. 2. The dashed lines represent the times where the TMFcrosses zero.

again reaches its maximum absolute value and CPT is attainedis there another sharp change in the absorption.

In the presence of the TMF, there are two new �

subsystems with 3δ energy difference between their lowersublevels [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) issignificant 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), hascontributions from both �7 and �8; thus, this coherenceis created in the presence of the TMF and vanishes whenthe TMF vanishes. In order to distinguish between thecontributions of �7 and �8, we plot the time dependenceof the imaginary part of the density matrix (proportional to theabsorption) for the transitions |Fe,me = 1〉 ↔ |Fg′,mg′ = 2〉and |Fe,me = 2〉 ↔ |Fg′,mg′ = 2〉 in Fig. 6(b) and for thetransitions |Fe,me = 1〉 ↔ |Fg,mg = 1〉 and |Fe,me = 2〉 ↔|Fg,mg = 1〉 in Fig. 6(c). We can see that �8, which has acommon transition with �3, gives the greater contribution.

Simultaneous modulation of LMF and TMF at the samefrequency, in the absence of constant magnetic fields, can bedescribed by switching between cases (c) and (a) of the steady-state spectrum shown in Fig. 2. When � = 0, the evolutionof the probe absorption will be similar to that obtained inthe presence of only LMF modulation (Fig. 2 of [8]) or onlyTMF modulation (see Fig. 7), namely, a sharp transient thatappears a short time after the MFs cross zero. For a detunedprobe there are two kinds of transients. The first type occursimmediately after the total MF deviates from the value thatoffsets the probe detuning in each individual � subsystem,so that several transients are obtained. The second kind oftransient occurs immediately after the total MF crosses zero

023827-4

COHERENT-POPULATION-TRAPPING TRANSIENTS . . . PHYSICAL REVIEW A 88, 023827 (2013)

1 5−0.1

0

0.1

Coh

eren

ce

1 5

−3

0

3

Imρ eg

’

0 1 2 3 4 5 6 7

0

5

10

t (ms)

Imρ eg

(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 dottedline) and |Fe,me = 2〉 ↔ |Fg′ ,mg′ = 2〉 (blue solid line) and (c) forthe 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 bymodulation of the TMF in the presence of a constant LMF.Now, we will examine the CPT transients induced in theabsence of the LMF. Thus, when the modulation begins, thesystem 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)

Pro

be A

bsor

ptio

n (c

m−

1 )

Bx=0 B

x=0

total probe

0 1 2 3 4 5 6 7

−0.1

0

0.1TMF

TM

F (

G)

FIG. 7. (Color online) Probe absorption for resonant pump andprobe � = 0 as a function of time in the presence of a TMFmodulation (black line) and in the absence of Bz for the sameparameters as in Fig. 2. The dashed lines represent the times wherethe TMF crosses zero.

0 1 2 3 4 5 6 7

12

t (ms)

Pro

be A

bsor

ptio

n (c

m−

1 )

Bx=0 B

x=0

total probe

0 1 2 3 4 5 6 7

−0.1

0

0.1TMF

TM

F (

G)

FIG. 8. (Color online) Probe absorption when � = −3δ0 as afunction 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. Thedashed lines represent the times where the TMF crosses zero.

by curve (a). In the presence of the TMF, both the depth andthe contrast are much smaller than in its absence. In Fig. 7,the total probe absorption in the presence of the modulatedTMF at � = 0 (point 1 in Fig. 2) is shown. The time evolutionin this case is similar to that obtained in the presence of amodulated LMF [8]: namely, a sharp transient that appearsa 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 isgiven by curve (d) in Fig. 2, whereas when Bx = 0, thespectrum is given by curve (a). We see in Fig. 8 that thereare two sets of transients: the transients that occur just afterthe TMF reaches its maximal absolute value that also occurat this value of � in the presence of Bz (see Fig. 5) and thetransients that occur immediately after the TMF passes throughzero (see Fig. 7). The transients that occur just after the TMFreaches its maximal absolute value are caused by the shift ofCPT to a lower value of �. The transients that happen afterthe TMF passes through zero reflect the difference betweenthe baselines in cases (d) and (a).

A three-dimensional figure that summarizes the effect ofthe TMF modulation on the probe absorption spectrum as afunction of detuning in the absence of the LMF is shown inFig. 9. The plot begins at time t0 and ends after almost ahalf cycle of TMF modulation. At time t0, we see the sevenresonance dips clearly. As the absolute value of the TMFdecreases, the dips at � = 0,±2δ become weaker, and thedistance between the dips decreases. When the TMF crosseszero, the Zeeman sublevels become degenerate, and all thedips appear at � = 0. For any other value of �, the transientthat appears after the TMF crosses zero reflects the differencebetween the baselines in cases (d) and (a) in Fig 2. A similarpicture in the presence of a constant LMF (not shown here)also displays the seven CPT resonances changing their positionand intensity as the value of δ changes. When the TMF crosseszero, only the original � subsystems at � = 0,±2δ remain.

023827-5

L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 88, 023827 (2013)

FIG. 9. (Color online) Evolution of probe absorption spectrum asa function of detuning in the presence of a TMF modulation and inthe absence of Bz for the same parameters as in Fig. 2.

IV. CONCLUSIONS

We have examined the probe absorption transients inducedby a modulated TMF in the absence and presence of a LMF.We showed the transients that appear as the system is switchedbetween the various steady-state situations whose spectra are

shown in Fig. 2. We discussed the contributions of the varioussubsystems to the transient probe absorption spectra in eachcase.

We chose two different probe detunings, � = 0 and � =−3δ0, which are indicative of the time-dependent behavior asa function of probe detuning, in order to describe the evolutionof the probe absorption in time during the TMF modulation.For any probe detuning chosen, one can see that the probeabsorption changes due to the creation and disappearance of� subsystems.

We showed that the evolution in the presence of a constantLMF seems adiabatic and that transients appear only after theCPT dip moves from the chosen value of � to a different value.In the absence of a constant LMF, the evolution is sharper, andin addition to the transients that appear after the CPT dipchanges its position, there are also transients that appear as theTMF crosses zero. This can provide a means for recognizingthe presence of a LMF in the system.

We also noticed differences between the TMF modulationand the LMF modulation, studied previously [8]. Whereasmodulating the LMF leads to variation in the contributions tothe absorption that derive from the original � subsystems dueto their entry and exit from CPT, the TMF modulation leads tothe creation and destruction of new � subsystems.

[1] P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, andJ. Kitching, Appl. Phys. Lett. 85, 6409 (2004).

[2] A. Huss, R. Lammegger, L. Windholz, E. Alipieva, S. Gateva,L. Petrov, E. Taskova, and G. Todorov, Opt. Soc. Am. B 23,1729 (2006).

[3] D. Budker and M. Romalis, Nat. Phys. 3, 227 (2007).[4] S. A. Zibrov, I. Novikova, D. F. Phillips, R. L. Walsworth, A. S.

Zibrov, V. L. Velichansky, A. V. Taichenachev, and V. I. Yudin,Phys. Rev. A 81, 013833 (2010).

[5] V. I. Yudin, A. V. Taichenachev, Y. O. Dudin, V. L. Velichansky,A. S. Zibrov, and S. A. Zibrov, Phys. Rev. A 82, 033807(2010).

[6] K. Cox, V. I. Yudin, A. V. Taichenachev, I. Novikova, and E. E.Mikhailov, Phys. Rev. A 83, 015801 (2011).

[7] Y. V. Vladimirova, B. A. Grishanin, V. N. Zadkov, N. N.Kolachevsky, A. V. Akimov, N. A. Kisilev, and S. I. Kanorsky,J. Exp. Theor. Phys. 96, 629 (2003).

[8] L. Margalit, M. Rosenbluh, and A. D. Wilson-Gordon, Phys.Rev. A 85, 063809 (2012).

[9] Y. V. Vladimirova, V. N. Zadkov, A. V. Akimov, A. Y. Samokotin,A. V. Sokolov, V. N. Sorokin, and N. N. Kolachevsky, Appl.Phys. B 97, 35 (2009).

[10] F. Renzoni, S. Cartaleva, G. Alzetta, and E. Arimondo, Phys.Rev. A 63, 065401 (2001).

[11] K. Nasyrov, S. Cartaleva, N. Petrov, V. Biancalana, Y. Dancheva,E. Mariotti, and L. Moi, Phys. Rev. A 74, 013811 (2006).

[12] Y. J. Yu, H. J. Lee, I. H. Bae, H. R. Noh, and H. S. Moon, Phys.Rev. A 81, 023416 (2010).

[13] L. Margalit, M. Rosenbluh, and A. D. Wilson-Gordon, Phys.Rev. A 87, 033808 (2013).

[14] S. Boublil, A. D. Wilson-Gordon, and H. Friedmann, J. Mod.Opt. 38, 1739 (1991).

[15] C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, andH. Friedmann, Phys. Rev. A 67, 033807 (2003).

023827-6