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

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 16 February 2012; published 8 June 2012)

Coherent-population-trapping transients induced by an ac magnetic field are investigated theoretically for arealistic three-level � system in the D1 line of 87Rb. The contributions to the transient probe absorption fromthe 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 pumpingis shown to be the dominant cause of the difference between the contributions of the various subsystems tothe oscillatory character of the probe absorption. We also discuss the series of transients that reappear everyhalf-cycle time of a modulated magnetic field when the system is in two-photon resonance, and we study thetransient behavior as a function of the probe detuning. The effect of a buffer gas on the amplitude and shape ofthe 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 betweencoupled atomic states are of great interest due to the varietyof novel properties they exhibit. A convenient description ofsuch media is via a density matrix formalism with nonzero off-diagonal coherence terms [1]. These systems have been shownto manifest interesting phenomena such as coherent populationtrapping (CPT) [2], electromagnetically induced transparency(EIT) [3], and electromagnetically induced absorption (EIA)[4–6].

Systems exhibiting quantum coherence properties have alsobeen studied for their transient behavior. These studies can bedivided into two types. In the first type, the time-dependentdynamics is studied from the time that the laser fields areswitched on until the system achieves CPT [7] or EIT [8].In the second type, which is the subject of this paper, CPTor EIT is first established and then a magnetic field isapplied either gradually [9] or suddenly [10] or, equivalently,one of the laser fields is detuned either gradually [11] orsuddenly [12,13].

The transient response of the absorption of a probe laserinteracting with a � system [12,13] or a degenerate two-levelsystem [10,11,14] that occurs due to a sudden change in thefrequency of one [11,13] or both [12] laser fields, due to thesudden turning on of a static magnetic field [10], or due tothe sudden turning on of a laser field in the presence of amagnetic field [14], has been examined both theoreticallyand experimentally. An analytic solution of a three-level �

system in the case where the Raman detuning is suddenlychanged from zero to �R has been developed by Park et al.[13]. They showed that the Raman coherence and hence theprobe absorption oscillate at a constant frequency �R . Thisobservation was later adopted [15] in order to determine thestandard frequency of a 85Rb atomic clock. Park et al. [13]also showed experimentally that, for a triangular modulationof the Raman detuning, a transient response appears afterthe detuning crosses zero and the frequency of the transientoscillations increases as the detuning increases. Similar resultshave also been obtained for a degenerate two-level system thatexhibits EIT [11]. It has also been shown [11] that the rate of

decay of the transient is proportional to the intensity of thepump (coupling) laser.

Recently, Momeen et al. [9] studied the transient responseof the resonant nonlinear magneto-optic rotation (NMOR)signal in a paraffin-coated Rb vapor cell as the magnetic fieldwas swept. They showed that at low sweep rates, the nonlinearrotation appears as a narrow resonance signal for transitionsstarting from the lower ground hyperfine level, which can beuseful for high sensitivity magnetometry at low field strengths.As expected, when the magnetic field sweep rate is increasedthe narrow resonance gradually disappears and undesirabletransient oscillations begin to appear.

In this work, we discuss the transients induced by themodulation of an applied magnetic field in the probe absorptionof a realistic three-level � system in the D1 line of 87Rbthat exhibits CPT. We examine the contributions to the probeabsorption from the various subsystems that compose therealistic atomic system and compare the absorption of eachsubsystem to that of a simple � system. We find that formoderate modulation frequencies the transient behavior is notdependent on the wave form of the modulation, which can beof either a sawtooth or a sinusoidal shape. In related work,Bevilacqua et al. [16] investigated both experimentally andtheoretically the effect of a longitudinal alternating magneticfield on the CPT resonances of Cs. By applying a staticmagnetic field, they were able to investigate a single �

subsystem. They showed that the ac magnetic field producessidebands to the CPT resonance similar to those obtained usingfrequency-modulated (FM) spectroscopy. Here we study thetime dependence of the probe absorption rather than its steady-state frequency dependence and examine the contributions ofthe various subsystems to the total absorption rather than thebehavior of a single � subsystem. In addition, we investigatemodulation frequencies much lower than those considered inRef. [16].

We distinguish between two processes that occur simulta-neously in the system, the change in the energy of the sublevelsdue to the modulation of the magnetic field and the transferof population between the sublevels, and we demonstrate theeffect of each process. The differences between the subsystems

063809-11050-2947/2012/85(6)/063809(6) ©2012 American Physical Society

L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 85, 063809 (2012)

that make up the realistic system underline the importance ofoptical pumping effects.

Unlike previous studies that dealt with the behavior of asingle transient, here we also study the behavior of a seriesof transients as a function of the probe detuning. In addition,we consider the effect on the behavior of the transients when abuffer gas is added to the vapor cell. We have observed a seriesof transients in preliminary experiments on Cs vapor and theseresults will be reported elsewhere.

II. THE BLOCH EQUATIONS

The system consists of two ground hyperfine states, Fg andFg′ , 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 frequencyω2. 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 Zeemansublevels, with the addition of decay from the ground andexcited states to a reservoir, and collisions between the Zeemansublevels of the ground states, as was done by Goren et al. [6]for a degenerate two-level atomic system:

ρ̇eiej= −(iωeiej

+ �)ρeiej− γ

(ρeiej

− ρeqeiei

)− (i/h̄)

∑gk

(ρeigkVgkej

− Veigkρgkej

)

− (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

(ρgiekVekgi

− Vgiekρekgi

) − γ(ρgigi

− ρeqgigi

)− (2Fg)�gigj

ρgigi+

∑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

(ρgiekVekgj

− Vgiekρekgj

), (4)

ρ̇gig′j

= −(iωgig′j+ �′

gig′j)ρgig

′j

−(i/h̄)∑ek

(ρgiekVekg

′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)

where � is the total spontaneous emission rate from each Feme

sublevel whereas �Fe→Fg,g′ is the decay rate from Fe to Fg,g′ .�gigj

and �g′i g

′j

are the collisional decay rates from sublevelsgi → gj and g′

i → g′j , respectively. The frequencies between

the ground hyperfine levels lie in the microwave range so thatthe collisions not only damp the coherences but also affect thepopulations of Fg and Fg′ [18]. We therefore introduce thephenomenological population transfer rate from mg to mg′ :�gig

′j

and �g′i gj

. γ is the rate of decay due to time-of-flightthrough the laser beams. The dephasing rates of the excited tothe 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.The dephasing rates of the ground statecoherences 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 + 1

2 [(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 between levels ai and bj , includingZeeman splitting of the ground and excited levels due to anapplied magnetic field, is given by ωaibj

= (Eai− Ebj

)/h̄,

with a,b = (g,e), and ρeqaiai

, with a = (g,e) being theequilibrium population of state ai, in the absence of anyelectrical fields. The interaction energy in the rotating-waveapproximation for the transition from level gj to ei is writtenas

Veigj= −μ

eigj(E1e

−iω1t + E2e−iω2t ) (8)

≡ −h̄[Veigj(ω1)e−iω1t + Veigj

(ω2)e−iω2t ],

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′ transitions andq = (−1,0,1) depending on the polarization of the incidentlaser. In order to calculate the time-dependent probe absorptionwhich is proportional to Imρeigj

, Eqs. (1) to (5) were solved

063809-2

COHERENCE-POPULATION-TRAPPING TRANSIENTS . . . PHYSICAL REVIEW A 85, 063809 (2012)

FIG. 1. (Color online) Energy-level scheme for the D1 line of87Rb interacting with σ+ polarized pump and probe. The system canbe divided into one TLS and three � subsystems. The Zeeman shiftsfor the upper hyperfine level are not shown.

numerically taking into account the magnetic field modulationand the consequent shifts in the energy of the sublevels.

III. RESULTS AND DISCUSSION

Our calculations are performed for the D1 line of 87Rb(see Fig. 1) where the pump is resonant with the Fg =1 → Fe = 2 transition and the probe is resonant with theFg′ = 2 → Fe = 2 transition. The pump and the probe areboth 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 areantisymmetric with respect to the Zeeman splittings of theirlower sublevels.

First, the pump and the probe are turned on in the presenceof a constant magnetic field B0 = 4 G parallel to the directionof propagation of the laser beam, and the system is givensufficient time for �2 to stabilize in a dark state. Then, at timet0 = 0.3 ms, we begin to modulate the magnetic field accordingto B = B0 sin[π/2 + ω(t − t0)], where ω is the magnetic fieldmodulation frequency (see Fig. 2).

The magnetic field removes the degeneracy of the Zeemansublevels and separates the system into subsystems. In Fig. 2,we see that the total probe absorption exhibits transientbehavior that is repeated every half-cycle of the magnetic fieldmodulation.

A. Transient components

The contributions to the probe absorption from eachsubsystem are shown in Fig. 3. The absorption of the TLSstabilizes at a constant value as the magnetic field decreasestoward zero causing the transition to become resonant. �2, the“clock transition,” is almost uninfluenced by the magnetic fieldbecause the Zeeman sublevels |Fg,mg = 0〉 and |Fg′ ,mg′ = 0〉are not shifted to first order by the magnetic field so thatthis transition is always two-photon resonant and thus in adark state. When the magnetic field passes through zero, the

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

10

20

30

40

50

60

70

80

t (ms)

Pro

be A

bsor

ptio

n (c

m−

1 )

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−404

MF

(G

)

probe abs. MF modulation

TT/2

FIG. 2. (Color online) Probe absorption for resonant pump andprobe 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 ata 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�, �∗ = �∗

gigj= �∗

g′ig′j

= �∗gig

′j=0, and

�gigj= �g′

ig′j

= �gig′j

= �g′igj

= 10−5�.

�1 and �3 subsystems enter a CPT state. A short time afterthe total magnetic field becomes equal to zero, the transientsreach their minimum values. It should be noted that, as themagnetic field modulation frequency decreases, the minimumpoint moves closer to B = 0, eventually reaching a situationresembling the Hanle effect in a degenerate TLS where theminimum occurs at B = 0 [19]. For example, in Figs. 4 and 8of 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)

Pro

be A

bsor

ptio

n (c

m−

1 ) total probe

TLS

Λ1

Λ2

Λ3

0.53 0.54 0.55 0.56 0.57−4

−3

−2

−1

0

1

2

3

4

MFM

F (

G)

B=0

FIG. 3. (Color online) Contributions of the subsystems to the totaltransient probe absorption in the presence of a modulated MF (blackline), for the same parameters as in Fig. 2. The dashed line representsthe time where the MF crosses zero. �1 and �3 contribute to thetransient behavior, the clock transition �2 is almost uninfluencedby the magnetic field, and the absorption of the TLS stabilizes at aconstant value. The �3 oscillations are rapidly damped compared tothose of �1.

063809-3

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)

Coh

eren

ce (

Arb

. Uni

ts)

Λ1

Λ3

FIG. 4. (Color online) Evolution of the coherence between thelower sublevels of the �1 and �3 subsystems for the same parametersas in Fig. 2.

at B = 0 is produced as the frequency of the linearly polarizedlaser interacting with degenerate two-level systems in Cs isslowly modulated in the presence of a magnetic field that ismodulated at a faster rate. Another example is that shownin Fig. 4 of Ref. [21], where a series of CPT resonances eachcentered at B = 0 is produced by a magnetic field that is slowlymodulated by a sawtooth waveform.

In Fig. 3, we show that both �1 and �3 contribute to thetransient behavior of the total probe absorption, with �3 givingthe greater contribution. This is due to the fact that the extremesublevels |Fg′,mg′ = 1,2〉 are the most populated ones sincethe σ+ polarized fields optically pump the population to highervalues of mg′ . We also see that the �3 oscillations are rapidlydamped compared to those of �1. This behavior is also seenin the time dependence of the coherences between the lowerlevels of the �1 and �3 subsystems (Fig. 4). We now discussthe origin of this different behavior.

There are two processes that occur simultaneously in thesystem. The first is the change in the energy of the sublevelsdue to the modulation of the magnetic field and the secondis the transfer of population between the sublevels. In orderto distinguish between the effect of each process on the totalabsorption, we tested each one separately.

First, we studied the behavior of a single � system. Wenumerically solved the Bloch equations for a three-level �

system for various initial populations of the lower states, whileapplying the same time-dependent magnetic field as in Fig. 2.The transient is created as the system enters the dark state attwo-photon resonance. Oscillations appear in the transient tailof the absorption, in the coherence between the lower levelsand in the population. We found that the oscillations differonly in their amplitude when the initial population distributionbetween the lower levels is varied. Thus the difference betweenthe �1 and �3 population distributions is not the cause of thedifference in the behavior of the oscillations.

Returning to the realistic atomic system, we plot thepopulations of the Zeeman sublevels of Fg′ (Fig. 5) as afunction of time. The populations of the Fg Zeeman sublevels

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

t (ms)

Pop

ulat

ion

F g’

"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′ = 2Zeeman sublevels for a fixed pump and probe with the sameparameters as in Fig. 2. The MF is modulated sinusoidally atfrequency ω = 1 kHz. The dashed line indicates the time where theMF 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 thecoherences and probe absorption of the two � subsystems canbe explained by the transfer of population that occurs duringthe transient. Initially, the population is equally distributedamong the eight sublevels in Fg = 1 and Fg′ = 2. After theconstant magnetic field is switched on, it can be seen in Fig. 5that most of the population is trapped in the |Fg′,mg′ = 2〉sublevel (the “trap” state) due to lack of pumping from thissublevel. Simultaneously �2 enters into a dark state and CPTis 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 thepumping from them is more efficient. This leads to an increasein the population of the trap state. At B = 0, �1 and �3 exhibitCPT and a short time afterward population flows into thesesubsystems at the expense of the trap state. �1 and �3 exitCPT as the magnetic field passes B = 0, and some of thepopulation then returns to the trap state. It should be noted thatoscillations also occur in the population but are very weak.

The transfer of population in our system occurs via decayto a reservoir (γ ) and collisional decay rate (�gigj

and �g′i g

′j),

which are the same for �1 and �3. Population which istransferred from the trap state goes mainly to �3 since theoptical pumping to that subsystem is stronger than that to�1. The addition of population to �3 that occurs during theCPT leads to damping of the oscillations in the populationand in the lower-level coherence, and eventually to dampingof the oscillations in the probe absorption. In order to confirmthat the optical pumping is the cause of the difference in thebehavior of the transients, we returned to the single � systemand found that increasing the Rabi frequency and thus theoptical pumping indeed dampened the oscillations.

This feature is changed when a buffer gas is added to thesystem. When velocity-changing collisions can be neglected[22,23], the presence of the buffer gas can be simulatedby increasing the time the atoms spend in the laser beams

063809-4

COHERENCE-POPULATION-TRAPPING TRANSIENTS . . . PHYSICAL REVIEW A 85, 063809 (2012)

(decreasing γ ) and increasing the rate of phase-changingcollisions �∗ [24]. The probe absorption in a single � systemis proportional to Imρeg′ (t). The analytical solution for ρeg′(t)when the two-photon (Raman) detuning is suddenly changedfrom zero to �, assuming that the pump and probe Rabifrequencies Veg are equal, the initial population is equallydivided between the lower levels, and the pump and probedetunings are antisymmetric such that �eg′ = −�eg = −�/2,is given by

ρeg′ (t) = iV

�′eg′ − i 1

2�

(1

2+ A

BeBt − 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 addedto the system (�′

eg′ increases), the decay rate of the transientis decreased while the oscillation frequency is unchanged,leading to decreased damping of the oscillations. Thiswas confirmed numerically for the modulating MF case.However, in the realistic system (the following parameterswere changed: γ = 10−5� and �∗ = 10�), we find that, inaddition to the decrease in the decay rate of the transient,less population is trapped in the trap state due to a decreasein the optical pumping effect (introducing buffer gas is insome 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 differencebetween the �1 and �3 oscillations does not occur when abuffer gas is added to the system. Increasing � in order toincrease optical pumping leads to damping of the oscillationsof �3 as in the absence of a buffer gas.

B. Sequence of transients

So far, we have dealt with a single transient where both thepump and the probe are on resonance. In this section we discussa sequence of transients as a function of the probe detuning,where the value of ω is chosen to be sufficiently large so thattransient oscillations appear in the probe absorption spectrumand sufficiently small so that successive transients are identicaland well-separated in time. Unlike the case of a resonant probewhere the �1 and �3 subsystems enter a dark state at the sametime, for a detuned probe, each subsystem enters a dark state atthe appropriate value of the modulating magnetic field whichoffsets its detuning. As the detuning increases, the total probeabsorption transient begins to widen and then breaks up due tothe 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 subsystemoccurs at the same absolute value of the magnetic field butopposite signs. When the probe is detuned, the time betweentwo consecutive transients is no longer equal to the half-cycletime T/2 of the magnetic field frequency (as in the case ofa 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)

Pro

be A

bsor

ptio

n (c

m−

1 )

total probe

Λ3

Λ1

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1−404

MF

(G

)

MF

FIG. 6. (Color online) Probe absorption for resonant pump anddetuned 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 thetotal 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 remainsequal to the cycle time. In Fig. 7, we plot the deviationfrom the half-cycle time (DHCT) as a function of the probedetuning and show that it increases as the probe detuningincreases. In order for this phenomenon to be useful, theDHCT should change strongly as a function of the absolutevalue of the detuning. It is shown in Fig. 7 that the slopebecomes 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 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Probe detuning (MHz)

DH

CT

(m

s)

200Hz1kHz200Hz &Bdc=0.2G

FIG. 7. (Color online) Deviation from half-cycle time as afunction of probe detuning. The slope of the DHCT is steeper in thecase 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 magneticfield (0.2 G) in the z direction (blue dotted line). Parameters are thesame as in Fig. 2.

063809-5

L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 85, 063809 (2012)

Although the width of the transients becomes narrower, thetransients occur exactly at the same times as in the absence ofthe 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 inexperimental configurations and not perfect due to the entranceof the laser fields through the z direction. Based on the transientbehavior, we propose to use the system as a magnetometerfor dc magnetic fields along the z axis, thereby simplifyingthe atomic clock configuration and making it more accurate.When in addition to the varying magnetic field there exists aconstant magnetic field in the z direction the total magneticfield in the system will be equal to zero at different times andthis will be reflected in the appearance of the transient. In thiscase the DHCT (the time between consecutive transients) willchange. This time can be measured and one can calculate theunknown dc magnetic field as shown by the blue dotted linein Fig. 7. If a transverse dc magnetic field exists in additionto the longitudinal dc magnetic field, we find the DHCT to beunchanged so that a measurement of the DHCT is indicativesolely of the longitudinal dc magnetic field. The individualtransients that appear in the probe absorption, however, changetheir shapes depending on the transverse magnetic field andhold out the possibility of providing some information for usein vector magnetometry.

IV. CONCLUSIONS

In this work, we discussed the transient behavior causedby the modulation of an applied magnetic field in the probeabsorption of a realistic three-level � system in the D1 line of87Rb that exhibits CPT. We examined the contributions to theprobe absorption from the various subsystems that composethe realistic atomic system and compared the absorption ofeach subsystem to that of a simple � system. We showed thatthe population redistribution due to optical pumping is thedominant cause of the difference between the contributions ofthe various subsystems to the oscillatory character of the probeabsorption.

We also discussed the interesting behavior of a series oftransients. For a resonant pump and probe, the time from theappearance of one transient to the consecutive one is equalto the half-cycle time of the modulation frequency. However,when the probe laser frequency is detuned, this time deviatesfrom the half-cycle time. This asymmetry in the periodicity ofthe consecutive transients could be useful in applications suchas frequency standards and magnetometry.

ACKNOWLEDGMENTS

We are grateful to P. Phoonthong, F. Renzoni, and Y. Rubinfor stimulating discussions.

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