Dynamics of a two-level system in a broadband light field

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by  Tom Savels

Dynamics of a two-level atom in a broadband light field
Tom Savels,∗ Allard P. Mosk, and Ad Lagendijk
Complex Photonic Systems, Dept. Science and Technology
University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands.
(Dated: January 19th, 2004)
5 We provide a fully analytical description of a two-level atom interacting with a broadband light
0 field. The problem we present is a typical example of a physical situation which occurs very com-
0 monly in practice, but is not straightforwardly solvable. However, we show in this paper that,
2 remarkably, only a limited number of elementary calculations are required to treat the problem.
Thepresentedresultsarenotonlyhighlyintuitive,butalsoallow ustostudythenontrivialnonlin-
n
ear response of a two-level atom to incident broadband radiation.
a
J

9
I. INTRODUCTION lap of the incident spectrum and the natural Lorentzian
1
emissionlineoftheatomitselfdeterminestheinteraction

] strength, confirming our intuitive predictions.
n The interaction of a two-level atom and a monochro- In order to simplify the calculations, we will impose a
n
maticfieldhasalwaysbeenaverypopulartopicinquan- very general restriction on the incident field. More pre-
-
s tum physics. The attraction of this system is owe to the cisely, we consider in this paper the class of statistically
di relatively modest mathematical tools needed to describe stationary incident fields. The mathematical simplifica-
. theproblem,combinedwitharichphysicalbehavior,able tionstheyallowfor,andthefactthatmanyfieldsencoun-
t
a to accurately predict many interesting phenomena such teredinpracticearestatisticallystationary,isthereason
m as superradiance [1] or the Mollow triplet [2]. whytheyaretreatedinsomanytextbookson,e.g.,quan-
- Theeffectofthe incidentfieldonthe atom’sdynamics tum optics or magnetic resonance. An important prop-
d isdescribedbytheopticalBlochequations[3,4]. Thisset erty of such fields is that two-time averages hE(t)E(t′)i
n ofequationsallowsonetoobtainexpressionsfortheaver- ofthefieldE(t)onlydependonthetimedifferencet−t′,
o
c age atomic level populations and coherences in the pres- whichimplies thatfieldcomponents atdifferentfrequen-
[ enceofanincidentfield. Iftheincidentlightismonochro- cies are uncorrelated [11].
matic, the optical Bloch equations are easily dealt with. Finally, we will apply a standard approximation to
2
For broadband incident light, however, solving the opti- the evolution of the atom itself: we will average out
v
1 cal Bloch equations is in general a far from trivial but the contribution of highly non-resonating terms. For
6 physically very relevant matter [5, 6, 7, 8, 9, 10]. For a monochromatic incident wave, for example, this ap-
6 example, a common experimental situation we are con- proximationsimply implies that all components oscillat-
0 sidering is an atom or molecule probed by near-resonant ing at twice the incident wave frequency are neglected.
1 light, but at the same time irradiated by a broadband The physicalmeaning of this approximationis that non-
4 light source, such as a trapping laser. energy conserving emission and absorption of photons is
0
In this paper, we show how to transparently describe neglected, or in other words, that “virtual photons” are
/
t the interaction of a two-level system and a broadband not taken into account. For a monochromatic incident
a
m incident wave. An everyday (and practically important) wave, this approximation is referred to in the literature
example of a broadband field is the field of spontaneous as the Rotating Wave Approximation (RWA). Since we
-
d photons emitted by an atom. The spectrum correspond- dealin this paper with fields whichare in generalbroad-
n ing to spontaneous emission has a Lorentzian frequency band, we will extend the standard approximation found
o distribution, and is therefore broadband. Obviously, we in textbooks, and refer to the extension as the General-
c intuitively expect that if the spectrum of the incident ized Rotating Wave Approximation (GRWA).
:
v field is far detuned comparedto the resonancefrequency In the following, we will start by formulating the op-
Xi of the atom, or very broad compared to the atom’s nat- tical Bloch equations for our configuration. A Fourier
ural linewidth, no significant interaction will take place. transform will lead to a better understanding of the be-
r
a Likewise,iftheincidentfieldhasaverynarrowfrequency havior of our system in frequency space. Application of
distribution,weexpectthattheinteractionwillresemble the GRWA will return steady-state solutions for the op-
the one induced by a monochromatic incident wave. In tical Bloch equations.
this paper, we show that only a few elementary calcula- To conclude our paper, we will show that the results
tions are required to quantify the interaction of such a obtained can be straightforwardly used to express the
broadband field and a two-level atom. We find that not nonlinearresponseofanatomtoabroadbandfield. This
justtheincidentspectrumitself,butinadditiontheover- response,whichcanbequantifiedbytheatom’sdynamic
polarizability,isofprimaryimportanceinstudyingatom-
field interactions, since it allows one to gain full insight
in the internal atomic dynamics.
∗Electronicaddress: [email protected]
Dynamicsofatwo-levelsysteminabroadbandfield
TomSavels,AllardP.MoskandAdLagendijk
II. THE OPTICAL BLOCH EQUATIONS
1 +∞
We consideratwo-levelatomA withlowerlevela and Ω[ω]≡ Ω(t)e−iωtdt, (3a)
upper level c, separated by an energy difference ~ω . 2π Z−∞
ca +∞
Theatominteractswithanincidenttime-dependentreal- Ω(t)= Ω[ω]eiωtdω. (3b)
valued field E(t). The dynamics of the system can be Z−∞
described in terms of the density matrix σˆ(t). The time
evolutionof σˆ(t) is givenby the opticalBlochequations, We will see that the appearance of a delta function in
which can be written, in the electric dipole approxima- (2) will highly facilitate the calculations further on. In
tion, as whatfollows,wewillFouriertransformtheopticalBloch
equations. We therefore define the population difference
. w(t) and its Fourier transform w[ω] as
σ =+iΩ(t) σ −σ −Γσ , (1a)
cc ca ac cc
(cid:16) (cid:17)
.
σaa =−iΩ(t) σca−σac +Γσcc, (1b) +∞
(cid:16) (cid:17) w(t)≡σ (t)−σ (t)≡ w[ω]e−iωtdt, (4)
σ. =−iΩ(t) σ −σ +iω σ − Γσ , (1c) cc aa Z−∞
ac cc aa ca ac ac
(cid:16) (cid:17) 2
. Γ and
σ =+iΩ(t) σ −σ −iω σ − σ , (1d)
ca cc aa ca ca ca
(cid:16) (cid:17) 2
+∞ +∞
. σ (t)≡ σ [ω]eiωtdω ≈ σ [ω]eiωtdω, (5a)
where the notation f(t) ≡ df(t) is used. Equations (1) ac Z ac Z ac
dt −∞ 0
can be found in many elementary books on quantum +∞ 0
optics [12]. The matrix elements σaa and σcc are σca(t)≡Z σca[ω]eiωtdω ≈Z σca[ω]eiωtdω, (5b)
the ensemble-averaged populations of the lower and −∞ −∞
upper atomic level, respectively. They are related by
σ + σ = 1, expressing conservation of population. where the contribution of the non-resonating part of
aa cc
The off-diagonal elements represent coherences; we will the coherences has been omitted. The restriction of the
elaborate on their physical meaning in section IV. The integration interval in expressions (5) has the simple
constant decay rate Γ represents spontaneous emission physical meaning that non-energy conserving terms are
by the system to the surrounding vacuum. The Rabi nottakenintoaccount,asmentionedinthe introduction
frequency Ω(t)≡−1d ·E(t) quantifies the interaction of our paper. This approximation is a straightforward
~ ac
strength between the atom and the incident field, generalization of the Rotating Wave Approximation
with d the c → a transition dipole moment. In the [12] and is therefore referred to as the Generalized
ac
conventional case of a monochromatic incident wave Rotating Wave Approximation. This generalization
E(t)≡E cosω t which is often found in the literature, holds excellently if J[ω] is only appreciably different
0 L
the definition Ω ≡ −1d · E is mostly used, from zero near the atomic resonance.
Rabi ~ ac 0
explicitly removing the oscillatory time dependence of If we now split the Fourier transformΩ[ω] of the Rabi
the field from the Rabi frequency (obviously, the optical frequency Ω(t) into a positive- and a negative-frequency
Bloch equations then contain extra factors e±iωLt). part
However, we will see that for a more general time
dependence, as the one we deal with here, it is beneficial
+∞ 0
to use our definition and consider a time-dependent Ω(t)= Ω[ω]eiωtdω+ Ω[ω]eiωtdω
Rabi frequency. Z0 Z−∞
Our aim in this section of the paper is to derive ≡Ω+(t)+Ω−(t), (6)
steady-state solutions for equations (1). The statistical
properties of the field are especially advantageous in
the frequency domain, since if the two-time average we see that neglecting all highly non-resonant terms in
hΩ(t)Ω(t+τ)i of the phase of Ω only depends on τ, one (1) results in slightly altered optical Bloch equations:
can deduce (see, e.g., [11]) that in the frequency domain
.
σcc =+iΩ+(t)σca−iΩ−(t)σac−Γσcc, (7a)
hΩ[ω]Ω[ω′]i=J[ω]δ(ω+ω′), (2) σ.aa =−iΩ+(t)σca+iΩ−(t)σac+Γσcc, (7b)
. Γ
σ =−iΩ (t) σ −σ +iω σ − σ , (7c)
ac + cc aa ca ac ac
(cid:16) (cid:17) 2
whereJ[ω]isthespectraldensityfunctionoftheincident . Γ
radiation and the Fourier transform of Ω(t) is defined as σca =+iΩ−(t)(cid:16)σcc−σaa(cid:17)−iωcaσca− 2σca. (7d)
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It is clear that equations (7) are far simpler to deal Wecanconcludethatinsteady-state,wefindthedesired
with than the original Bloch equations (1). Fourier solution
transforming equation (7a) and (7b) gives
X
σst ≡1−σst = , (14a)
+∞ cc aa 2X+1
(Γ+iω)w[ω]=+2i Ω[ω′]σ [ω−ω′]dω′
Z ca 1 0 iΩ[ω]
0 σst ≡(σst)∗ = eiωtdω,
−2i 0 Ω[ω′]σ [ω−ω′]dω′−Γδ(ω). ca ac 2X +1Z−∞ −iωca−iω− Γ2
Z ac (14b)
−∞
(8)
with
Fourier transforming (7c) and (7d), on the other hand,
gives +∞ 1
X ≡ dωJ[ω] . (15)
Z (ω −ω)2+(Γ)2
0 ca 2
Γ +∞
(iω−iω + )σ [ω]=−i Ω[ω′′]w[ω−ω′′]dω′′, Expressions (14) are the key result of our paper. The
ca 2 ac Z
0 influence of the spectral properties of the incident field
(9a)
enters the dynamics of the density matrix througha sin-
Γ 0 gle interaction parameter X. The expression (15) for X
(iω+iω + )σ [ω]=+i Ω[ω′′]w[ω−ω′′]dω′′.
ca 2 ca Z−∞ is surprisingly simple andappealing: it is the overlapin-
tegral of the spectral density of the incident field, and
(9b)
the naturalLorentzianemissionline ofthe two-levelsys-
tem itself. The structure of the interaction parameter
If we now substitute the previous expressions in (8), we
confirmswhatweintuitivelyexpect: resonantfields with
find
a narrow distribution interact strongly with the atom,
while the interaction with broad, far off-resonance fields
(Γ+iω)w[ω]+Γδ(ω)= is far less pronounced. As examples, we will in the next
sectionfocusontwospecificandinterestingvaluesforX.
+∞ 0
=−2 dω′ dω′′Ω[ω′]Ω[ω′′]w[ω−ω′−ω′′]×
Z Z III. EXAMPLES
0 −∞
1 1
+ ,
(cid:16)iω−iω′+iωca+ Γ2 iω−iω′′−iωca+ Γ2(cid:17) verAifsyathfiartstthdeemwoenlls-ktrnaotwionneoxfpreeqsusaiotniosnsfor(1a4n), iwnceidweniltl
(10)
monochromatic field [12] can be retrieved from our
results. We consider a real-valued monochromatic
which is a self-consistent equation in the population
incident field E(t) ≡ E cos(ω t) with ω > 0. The
difference. We use the following trial solution for w[ω]: 0 L L
corresponding Rabi frequency is
w[ω]=w[0]δ(ω), (11) Ω
Ω[ω]≡ 0 δ(ω−ω )+δ(ω+ω ) , (16)
L L
2 (cid:16) (cid:17)
which is appropriate since we are interested in the
therefore
regime for which the populations are time-independent
(which is also referred to as the “steady-state” regime).
Substitution of (11) in (10) and phase-averagingyields Ω[ω]Ω[−ω′]=
Ω2
Γw[0]+Γ=−2w[0] +∞dω′J[ω′] Γ , 40(cid:16)δ(ω−ω′)δ(ω−ωL)+δ(ω−ω′)δ(ω+ωL)
Z0 (ω′−ωca)2+(Γ2)2 +δ(ω+ω′)δ(ω−ωL)+δ(ω+ω′)δ(ω+ωL) . (17)
(12) (cid:17)
Of the 4 terms appearing in (17), only the first remains
and therefore
in the (G)RWA, yielding
1
w[ω]=−1+2 +∞dω′J[ω′] 1 δ(ω). (13) J[ω]= Ω20δ(ω−ω ), (18)
R0 (ω′−ωca)2+(Γ2)2 4 L
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to incident light, one has to determine the strength with
which transforms expressions (14) into which the atom interacts with the incident light. A key
propertyquantifyingthisstrengthistheatom’sdynamic
polarizability↔α. The associated polarization P induced
Ω 1
σst =( 0)2 , (19a) by the incident field E is then given by P = ε ↔α·E .
cc 2 (ωL−ωca)2+(Γ2)2+ Ω220 In the limit of low-inten0sity incident fields, the r0espons0e
Ω 1 is linear and the polarization is then proportional to the
σst = 0e−iωLt ,
ca 2 (ω −ω )+iΓ + 1 Ω20 incident field. However, at higher incident intensities,
L ca 2 2(ωL−ωca)−iΓ2 the response is no longer linear since the atom will
(19b) start to saturate. The polarizability will then depend
on the incident field and the polarization will no longer
which corresponds exactly to the solutions for incident be proportional to the incident field. We will now
monochromatic fields found in the literature [12], justi- show how the results from the previous sections can be
fying our method. straightforwardly used to quantify the rate at which the
As a second demonstration of equations (14), we atom saturates due to a incident (broadband) field. In
consider the nontrivial case of an atom interacting other words, we will show how expressions (14) allow us
with incident spontaneous emission. The incident field to calculate the nonlinear dynamic polarizability of a
then has a Lorentzian spectrum with a width Γ(1+κ), two-level atom.
κ≥−1, centered around ωL ≫Γ: We take the general case of an incident field E(t)
consisting of a monochromatic cosine component
J Γ(1+κ) with amplitude E0 and frequency ωL, and a non-
J[ω]≡ 0 2 , (20) monochromatic component E (t), with a Lorentzian
π (ω−ωL)2+(Γ2(1+κ))2 frequency distribution. The fiLelds E(t) thus obtained
define a very general class, containing the special cases
where the factor
ofincident monochromaticlight, aswellas pure incident
spontaneousemission. The Lorentziancomponentofthe
∞ incident field has a linewidth Γ(1+κ), κ≥−1, centered
J ≡ J[ω]dω = Ω(t)2 (21)
0 Z−∞ (cid:10) (cid:11) around ωL ≫Γ:
is proportional to the total incident field energy. We 1
find as an explicit solution for X: E[ω]=E0 δ(ω+ωL)+δ(ω−ωL) +EL[ω]. (23)
2(cid:16) (cid:17)
+∞ 1 The corresponding Rabi frequencies are
X = dωJ[ω]
Z (ω−ω )2+(Γ)2
0 ca 2
+∞ 1 1
≈ dω × −~Ω =d ·E , (24a)
Z−∞ (ω−ωL)2+(Γ2(1+κ))2(ω−ωca)2+(Γ2)2 −~Ω [ω0]=dac·E0[ω], (24b)
L ac L
J Γ
0 (1+κ)
π 2
(2+κ) obeying
=J , (22)
0(ω −ω )2+(Γca)2(2+κ)2
L ca 2
where the extension of the integral in (22) from 0 to hΩL[ω]ΩL[ω′]i=JL[ω]δ(ω+ω′), (25)
−∞ is clearly justified by ω ≫ Γ. Equation (22)
L
clearly shows that the atom-field interaction is weak
withJ [ω]thespectraldensityoftheincidentLorentzian
for spectrally broad or far-detuned fields, as mentioned L
field
earlier. To conclude this paper, we will in the next
section use the result (14) with (19) and (22) obtained
here to determine the nonlinear response of an atom to
J(κ) Γ(1+κ)
incident broadband light. J [ω]≡ 2 . (26)
L π (ω−ω )2+(Γ(1+κ))2
L 2
Equations (23)-(26) fully describe the incident field.
IV. APPLICATION We will now focus on the response of the atom to this
incident field. Since the dynamic polarizability and the
Let us now focus on a practical application of the density matrix both describe the response of the atom
previous sections. If one studies the response ofan atom to incident light, both quantities should be connected.
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TomSavels,AllardP.MoskandAdLagendijk
Indeed, they are related by [13] dynamic polarizability of a two-level atom irradiated by
a monochromatic field is recovered
ε ↔α(ω˜)·E[ω˜]≡d σ [−ω˜]. ω >0, (27)
0 ac ca
↔α(ω)=−↔α 1 ωca . (32)
02ω−ω +iΓ + Ω20/2
Our general result (14) now allows us to deduce that ca 2 ω−ωca−iΓ2
(27) leads to
Secondly, for Ω → 0, we find the saturation due to
0
pure spontaneous emission
ε ↔α(ω)· 1E δ(ω+ω )+E [ω]
0 0 L L
(cid:16)2 (cid:17)
=−2Xda+c 1(cid:16)21ωΩc0aδ−(ωω+−ωiLΓ2) + ωcaΩ−Lω[ω−] iΓ2(cid:17) ↔α(ωL)=−↔α02(2X1+1)ωL−ωωccaa+iΓ2, (33)
d 1d ·E δ(ω−ω ) d ·E [ω]
= ac · ac 0 L + ac L , with X given by expression (15).
2X +1 (cid:16)2 ωca−ω−iΓ2 ωca−ω−iΓ2(cid:17) Finally, we note that expressions (32) and (33) imply
(28)
that for small incident fields, the same expression
where ⊗ represents the tensor product of two vectors.
The nonlinear polarizability is thus expressed as ↔α(ω )=−↔α 1 ωca , (34)
L 02ω −ω +iΓ
L ca 2
↔α(ω)= dacε⊗0~dac2X1+1ωca−1ω−iΓ2 fitorshthouelldinbeea.r dynamic polarizability [14] is obtained, as
=−↔α 1 ωca , (29)
02ω−ω +iΓ + 2S
ca 2 ω−ωca−iΓ2
V. SUMMARY
where the static polarizability is given by
In this paper, we have solved the optical Bloch equa-
tions for a two-level system interacting with a statisti-
↔α ≡ 2 d ⊗d , (30) cally stationary broadband field. The resulting steady-
0 ωca~ε0 ac ac state density matrix is similar to the result found for
a monochromatic incident field; the difference between
and where the saturation parameter can be written in a both results can be intuitively understood. Finally, we
surprisingly simple way as haveappliedthe obtainedresults to calculatethe full re-
sponseofatwo-levelatomtoabroadbandfield,expressed
by the atom’s dynamic polarizability.
Ω2 +∞ (ω −ω )2+(Γ)2
S ≡ 0 + dωJ [ω] L ca 2 . (31)
4 Z L (ω−ω )2+(Γ)2
0 ca 2
Acknowledgments
Expressions (29)-(31) fully describe the response of a
two-level atom to an incident field of the general class Thisworkispartoftheresearchprogramofthe‘Sticht-
(23). ing voor Fundamenteel Onderzoek der Materie’ (FOM),
Two limits for the incident field are in particular which is financially supported by the ‘Nederlandse Or-
interesting. Firstly, forJ [ω]→0,the expressionfor the ganisatie voor Wetenschappelijk Onderzoek’ (NWO).
L
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