Table Of ContentElectromagnetically Induced Transparency with Superradiant and Subradiant states
Wei Feng,1,2 Da-Wei Wang,2,∗ Han Cai,2 Shi-Yao Zhu,1,3 and Marlan O. Scully2,4,5
1Beijing Computational Science Research Center, Beijing 100193, China
2Texas A&M University, College Station, TX 77843, USA
3Department of Physics, Zhejiang University, Hangzhou 310027, China
4Baylor University, Waco, Texas 76706, USA
5Xi’an Jiaotong University, Xi’an, Shaanxi 710048, China
(Dated: January 31, 2017)
7
We construct the electromagnetically induced transparency (EIT) by dynamically coupling a
1
superradiant state with a subradiant state. The superradiant and subradiant states with enhanced
0
2 andinhibiteddecayratesactastheexcitedandmetastablestatesinEIT,respectively. Theirenergy
differencedeterminedbythedistancebetweentheatomscanbemeasuredbytheEITspectra,which
n renders this method useful in subwavelength metrology. The scheme can also be applied to many
a
atomsinnuclearquantumoptics,wherethetransparencypointduetocounter-rotatingwaveterms
J
can be observed.
7
2 PACSnumbers: 42.50.Nn,42.50.Ct
]
h
Introduction.–Electromagnetically induced trans- the effective coupling fieldbetween the superradiantand
p
parency (EIT) [1, 2] is a quantum optical mechanism subradiantstatesbringanadditionaltransparencypoint,
-
m that is responsible for important phenomena such as whichisusuallynotachievableintraditionalEITsystems
o slow light [3–5], quantum memory [6–8] and enhanced with three-level atoms.
t nonlinearity [9, 10]. A probe field that resonantly Mechanism.–Two two-level atoms have four quantum
a
. couples the transition from the ground state g to an states, a ground state gg , two first excited states ge
s | i | i | i
c excited state e of an atom, experiences a transparency and eg , and a double excited state ee . Considering
| i | i | i
i point at the original Lorentzian absorption peak, if the the interaction between the two atoms, the eigen basis
s
y excited state is coherently and resonantly coupled to of the first excited states is composed by the symmetric
h a metastable state m . EIT involves at least three and anti-symmetric states,
p levels and naturally|thriee-level atoms are used in most
[
cases. However, proper three-level structures are not
1
1 available in some optical systems, such as in atomic + = [eg + ge ],
v nuclei [11–13] and biological fluorescent molecules [14], | i √2 | i | i
(1)
5 in which EIT can have important applications once 1
7 = [eg ge ],
realized. Interestingly, it has been shown that even with |−i √2 | i−| i
1
8 only two-levelsystems, EIT-like spectra can be achieved
0 by locally addressing the atomic ensembles [15–17]. with decay rates γ± = γ0 γc and energy shifts ∆± =
±
1. However, strict EIT scheme with a dynamic coupling ±∆c. Here γ0 is the single atom decay rate, γc and
0 field is still absent in two-level optical systems. ∆c are the collective decay rate and energy shift (see
7 Supplementary Material [31]). When the distance be-
1 Superradiance and subradiance are the enhanced and tween the two atoms r λ where λ is the transition
v: inhibited collective radiation of many atoms [18–20], as- wavelength, we have γc ≪ γ0 and thus γ+ 2γ0 and
→ →
i sociated with the collective Lamb shifts [21–23]. The γ− 0. The collective energy shift ∆c is divergentwith
X →
superradiance and subradiance of two interacting atoms 1/r3. A weak probe field can only resonantly excite +
ar has attracted much interest both theoretically [24, 25] from |ggi since the collective energy shift ∆c moves|thei
andexperimentally[26–30]. Inthis Letter,weusesuper- transition between + and ee out of resonant with the
| i | i
radiance and subradiance to construct EIT and inves- probe field [29]. We can neglect the two-photon absorp-
tigate the new feature in the EIT absorption spectrum tion for a weak probe field [28, 32]. The states gg , +
| i | i
involving with the cooperative effect and the counter- and form a three-level system, as shown in Fig. 1
|−i
rotatingwaveterms. Foronly twoatoms,the symmetric (a). The symmetric and the anti-symmetric states sat-
(superradiant)state hasmuchlargerdecayratethanthe isfy the requirement on the decoherence rates for EIT,
anti-symmetric (subradiant) state when the distance be- i.e., γ+ γ− when r λ. The eigenenergies of
≫ ≪ |±i
tweenthe two atoms is muchsmaller than the transition states are split by the collective energy shift.
wavelength. Thesetwostatesserveastheexcitedandthe The challengeis how to resonantlycouple + and
| i |−i
metastable states and their splitting, depending on the states. The key result of this Letter is that + and
| i
distancebetweentheatoms,canbemeasuredbytheEIT states can be coupled by two off-resonant counter-
|−i
spectra. In addition, the counter-rotating wave terms in propagatingplanewaveswithdifferentfrequenciesν and
1
2
ν . If the frequency difference ν = ν ν matches the
2 1 2
splitting between + and states 2−∆c, we obtain on (a)
| i |−i
resonance coupling via two Raman transitions as shown
(cid:1876)(cid:2869) (cid:1876)(cid:2870) (cid:513)(cid:1857)(cid:1857)(cid:1732)
in Fig. 1 (b). The resulting Hamiltonian is (assuming (cid:1876)(cid:3548)
~=1) [31], (cid:1870) (cid:513)(cid:3397)(cid:3)(cid:1732)
H =ω+|+ih+|+ω−|−ih−|+Ωc(t)(|+ih−|+|−ih+|) (cid:513)(cid:513)(cid:1859)(cid:1857)(cid:1732)(cid:1732) (cid:2012)(cid:3043)(cid:2007)(cid:3043) (cid:2011)(cid:2878)(cid:2007)(cid:3030) (cid:884)(cid:1986)(cid:3030)(cid:2011) (cid:2879)(cid:513) (cid:3398)(cid:1732)
−Ωp(e−iνpt|+ihgg|+h.c.), (b) (cid:513)(cid:1859)(cid:1859)(cid:1732)
(2)
where Ωc(t)=Ω0sin(kr)sin(νt φ) with k =νs/c, νs = (cid:1822) (cid:513)(cid:1857) (cid:1857)(cid:1732)
−
(ν +ν )/2,r=x x andφ=k(x +x )withx being
th1ecoo2rdinatesof1t−he2twoatomsalo1ngth2epropag1,a2tionof (cid:2021)(cid:2869) (cid:2021)(cid:2870) (cid:2021)(cid:2870) (cid:2021)(cid:2869)
the planewaves. ThecouplingstrengthΩ =E2d2/(ω (cid:513)(cid:3398)(cid:3)(cid:1732)
0 − (cid:513)(cid:3397)(cid:3)(cid:1732)
νs)withE beingtheamplitudeoftheelectricfieldofthe Probe Laser (cid:2021)(cid:2869) (cid:2021)(cid:2870)
planewaves,dbeingthetransitionmatrixelementofthe
(cid:513)(cid:1859)(cid:1859)(cid:1732)
atoms and ω being the single atom transition frequency.
The transition frequencies of states are ω± = ω
|±i ± FIG. 1. (Color online) (a) Two two-level atoms form an EIT
∆ +δ (t) with δ (t)=Ω [1+cos(kr)cos(νt φ)] being
c u u 0 − systemwiththesymmetric(superradiant)statebeingtheex-
a universal Stark shift induced by the two plane waves.
cited state and the anti-symmetric (subradiant) state being
The absorptionspectracanbe calculatedby the Liou- themetastablestate. (b)Thesymmetricandanti-symmetric
ville equation, statesareresonantlycoupledbytheRamantransitionsoftwo
counter-propagating plane waves. We can also understand
∂ρ γj thiscouplingasinducedbythetime-dependentdifferencebe-
= i[H,ρ]+ [2gg j ρj gg
∂t − 2 | ih | | ih | tween the dynamic Stark shifts of the two atoms induced by
j=X+,− (3) a moving standingwave with velocity v=νxˆ/2k.
j j ρ ρj j ].
−| ih | − | ih |
Since H is time-dependent with frequency ν, the coher- δ∆ /δr 1/r4. Compared with the existing propos-
c
ence can be expanded +ρgg = ρ[n] einνt. Eq.(3) als for su∝bwavelength imaging of two interacting atoms
h | | i n +gg
can be solved with the Floquet thePorem [33, 34] and the with fluorescences [35], a natural preference for this EIT
absorptionisproportionaltoImρ[0] ,theimaginarypart metrology is that both the dressing field and the probe
+gg
of the zero frequency coherence. fields are weak. This is in particular useful for the bio-
The counter-rotating wave terms of Ω (t) can be ne- logical samples that cannot sustain strong laser fields.
c
glected for small distance between the two atoms and Theabovemechanismcanalsobe understoodasa dy-
weak coupling field when Ω sin(kr) ∆ . We ob- namic modulation of the transition frequency difference
0 c
≪
tain typical EIT absorption spectra with two absorp- between the two atoms [31]. We notice that the differ-
tion peaks and one transparency point, as shown in the ence between + and states is a relative π phase
| i |−i
black curve of Fig. 2 (a). Here the probe detuning factor between eg and ge states. If we can control
| i | i
δ = ω +∆ +Ω ν has taken into account all the the transitionfrequenciesofthe twoatomssuchthatthe
p c 0 p
−
static energy shifts of |+i state, including Ω0, the static states|egiand|geihaveenergyshiftsΩcand−Ωcrespec-
part of the universal Stark shift δ (t). The effect of the tively,aninitialstateofthesymmetricstate ψ(0) = +
counter-rotatingwavetermsandthueuniversalshiftδ (t) evolves with time ψ(t) = (e−iΩct eg +eiΩ|ct gei)/√| 2i.
u | i | i | i
emergeeither whenwe increasethe distance(reduce∆c) At t = π/2Ωc, we obtain |ψ(t)i =−i|−i. Therefore, the
between the two atoms or increase the dynamic Stark states + and are coupled by an energy difference
| i |−i
shift Ω (proportional to the intensity of the standing between the two atoms. In our scheme, the two counter
0
wave),whicharedemonstratedbythemultiplesidepeaks propagating plane waves create a moving standing wave
in Fig. 2 (a). that induces a time-dependent dynamic Stark shift dif-
We can use the following procedure for the subwave- ference between the two atoms, Ωc(t), which serves as
length metrology, as shown in Fig. 2 (b). We first re- the coupling field. This picture enables us to generalize
duce the intensity of the standing wave to only allow the mechanism to many atoms, as shown later.
two peaks to appear in the spectra. Then we tune the The single atom EIT [36] and the superradiance and
frequencydifferenceν untilthe twoabsorptionpeaksbe- subradiance of two ions [27] have been observed in ex-
come symmetric, which yields the collective energy shift periments. The coupling between the symmetric and
∆ = ν/2. The distance between the two atoms can be anti-symmetric states has also been realized with two
c
obtainedby the relationbetween∆ (r) andr [31]. Since atoms trapped in an optical lattice [37]. In particular,
c
∆ (r) 1/r3 for small distance r λ, the sensitivity the cryogenic fluorescence of two interacting terrylene
c
∝ ≪
3
moleculeshasbeenusedforspectroscopywithnanometer
1
(a) resolution [28]. Due to different local electric fields, the
0.8 twomoleculeshavedifferenttransitionfrequencies,which
unit) correspondstoastatic couplingfieldΩc. By introducing
b. 0.6 anoscillatingelectricfieldgradientoramovingstanding
ar
n ( wave,suchasystemcanbeexploitedforthecurrentEIT
o
orpti0.4 experiment of superradiance and subradiance. Very re-
Abs cently,superradiancewasalsoobservedfromtwosilicon-
0.2
vacancy centers embedded in diamond photonic crystal
cavities [38], which provide another platform to realize
0
−8 −6 −4 −2 0 2 4 6 8
δ/γ this mechanism.
p 0
Generalization tomanyatoms.–Themechanismcanbe
1 extended to large ensembles of two-levelsystems. Let us
(b)
considertwoatomicensembles,onewith e and g ,and
0.5 | i | i
the other with a and b as their excited and ground
b. unit) 01 −4 −3 −2 −1 0 1 2 3 4 states. Each ens|emible ha|siN atoms and both ensembles
ar are spatially mixed together. The transition frequency
on (0.5 difference between the two atomic ensembles is within
pti
or 0 thelinewidthsuchthatasinglephotoncanexcitethetwo
bs 1 −4 −3 −2 −1 0 1 2 3 4
A ensembles to a superposition of two timed Dicke states
0.5 [39, 40],
0
−4 −3 −2 −1 0 1 2 3 4 1
δ/γ +k = (ek + ak ) (4)
p 0 | i √2 | i | i
FIG. 2. (Color online) Absorption spectra of two-atom su- where
perradianceEIT.(a)Theabsorptionspectrafordifferentdis-
tances r and Rabi frequencies Ω0. Black line: r = 0.1λ 1 N
(∆c = 2.60γ0, γc = 0.92γ0), Ω0 = 2γ0; red line: r = 0.2λ ek = eik·rn g1,...,en,...,gN b1,b2,...,bN ,
(∆c =0.38γ0,γc =0.71γ0),Ω0 =2γ0andblueline: r=0.1λ, | i √N nX=1 | i⊗| i
Ω0 =10γ0. The coupling field is on resonance for each case, N
1
ν = 2∆c. (b) The absorption spectra with different stand- ak = g1,g2,...,gN eik·sn b1,...,an,...,bN .
ingwavedetunings. ν =7γ0 (blackline)9γ0 (redline)10.5γ0 | i | i⊗ √N X | i
n=1
(blueline). Ω0 =γ0. Whenν =10.5γ0 =2∆c,theabsorption (5)
spectrumissymmetric. Fromtherelation between∆c andr, Here r and s are the positions of the nth atom in the
n n
we obtain the distance between the two atoms r = 0.08λ0, twoensembles. kisthewavevectorofthesinglephoton.
which agrees with theparameters that we set.
ThetimedDickestates ek and ak areexcitedfromthe
| i | i
same ground state G g ,g ,...,g b ,b ,...,b
1 2 N 1 2 N
| i ≡ | i⊗| i
by a single photon. They have directional emission in
the direction of k, so as their superposition state +k ,
(cid:1865)(cid:3032) | i
(cid:1853)(cid:3042)(cid:3048)(cid:3047) (cid:2012)(cid:3032) (cid:513)(cid:1857)(cid:1732) (cid:3397)(cid:885)(cid:932)(cid:884) associatedwithenhanceddecayrateandcollectiveLamb
(cid:135)(cid:1835) (cid:3404)(cid:885)(cid:932)(cid:884) (cid:3397)(cid:883)(cid:932)(cid:884) shift. On the other hand, the state
(cid:513)(cid:1853)(cid:1732)
(cid:3398)(cid:883)(cid:932)(cid:884)
(cid:3398)(cid:885)(cid:932)(cid:884) 1
(cid:2033)(cid:3404)(cid:883)(cid:886)(cid:484)(cid:886)(cid:141)(cid:135)(cid:25) k = (ek ak ), (6)
(cid:1863)(cid:3552) (cid:1776) (cid:887)(cid:889)(cid:9)(cid:135) (cid:2011)(cid:2868)(cid:3404)(cid:886)(cid:484)(cid:889)(cid:144)(cid:135)(cid:25) |− i √2 | i−| i
(cid:1853)(cid:3036)(cid:3041) (cid:513)(cid:1854)(cid:1732) (cid:3398)(cid:1865)(cid:883)(cid:932)(cid:3034)(cid:884) is a subradiant state in the sense that its decay rate is
(cid:1779) (cid:137)(cid:1835) (cid:3404)(cid:883)(cid:932)(cid:884) (cid:2012)(cid:3034) (cid:513)(cid:1859)(cid:1732) (cid:3397)(cid:883)(cid:932)(cid:884) estimated to be similar to that of a single atom [41].
FIG. 3. Superradiance EIT in nuclear quantum optics. A The directional emissions of ek and ak are canceled
| i | i
thin-film cavity is probed by hard x-rays with grazing angle because of the relative phase factor 1 between them.
incidence. The57Fe nucleiare embedded in thecenterof the The collective Lamb shift of k can−be very different
|− i
cavity. We add an oscillating magnetic field parallel to the from that of the +k state.
electricfieldofthelinearlypolarizedincidentx-ray. Onlythe We can dynam|icalily couple +k and k states in a
two transitions denoted by the dashed arrows between the | i |− i
wellstudiednuclearquantumopticalsystem[11,40,42],
magneticZeemanlevelscanhappen. Theenergydifferenceof
asshowninFig. 3. Thenucleiembedded inawaveguide
these two transitions serve as the effective coupling between
are 57Fe with the transition frequency ω = 14.4keV and
thesuperradiantandsubradiantstates. TheEITspectracan
be detected with thereflected signal. thelinewidthγ0 =4.7neV. Inthepresenceofamagnetic
field, the ground and excited states with I = 1/2 and
g
4
I = 3/2 split into multiplets with Zeeman energy split-
e 25
tingδ (j =e,g). ApplyingamagneticfieldBparallelto (a)
j
theincidentandoutgoingelectricfieldsE andE and
in out 20
perpendiculartok,thelinearlypolarizedinputx-raycan
onlycoupletwotransitions,asshowninFig. 3. Atroom nit)15
u
temperature, the populations on the two magnetic sub- arb.
levels of the ground state are approximately equal [40]. 2R| (10
|
Here we can use a magnetically soft 57FeNi absorber foil
5
with zero magnetostriction [13] to avoid the mechanical
sidebands and other complications in a time-dependent
0
external magnetic field. −40 −30 −20 −10 δ0/γ 10 20 30 40
p 0
TheHamiltonianintheinteractionpicturecanbewrit-
ten as, 25
(b)
H =Ωc(t)(+k k e−iω0t+ k +k eiω0t) 20
| ih− | |− ih | (7)
Ωp(e−iδpt +k G +h.c.), nit)15
− | ih | u
b.
ar
where Ωc(t) = Ω1cos(νt) with Ω1 = (δg+δe)/2 is in- 2R| (10
duced by a magnetic field B = B cosνt. ω is the col- |
0 0
lectiveLambshiftdifferencebetweenthestates +k and 5
| i
k . δp is the probe detuning from the +k state. The
|− i | i
reflectance of the thin film cavity is dominated by the −080 −60 −40 −20 0 20 40 60 80
δ/γ
coherence ρ+G 2 where ρ+G +k ρG (see [31]), p 0
| | ≡h | | i
T FIG. 4. (Color online) The reflectance of x-raywith effective
R2 lim 1 ρ (t)2dt= ρ[n] 2, (8) Rabifrequencies(a)Ω1 =5γ0 and(b)Ω1 =20γ0. Thedecay
| | ∝T→∞T Z | +G | X(cid:12) +G(cid:12) rates of |+ki and |−ki are γ+ = 50γ0 and γ− = γ0. The
0 n (cid:12)(cid:12) (cid:12)(cid:12) collective Lamb shift difference ω0 = 10γ0. The oscillation
frequenciesofthemagneticfieldsareν =15γ0 (bluedashdot
wherewehavemadeaverageinatime intervalT ≫1/ν. line),10γ0 (blacksolid line), 5γ0 (reddash line)and0(green
The coherence ρ has multiple frequency components dot line).
+G
ρ (t)= ρ[n] ei2νt due to the counter-rotating wave
+G n +G
terms. OnPly when ν =0, no time average is needed.
The typical collective Lamb shift of 57Fe nuclear en- Compared with the result in [40] where +k and k
| i |− i
semble is 5 10γ [11]. The internal magnetic field in have the same energy and the magnetic field is static,
0
the57Fesam∼plecanbetensofTeslainanexternalradio- here the two peaks are not symmetric for ν =0 due to a
frequency field [13, 43]. The effective coupling field Rabi finiteLambshiftdifference. Therefore,theresultscanbe
frequencyΩ canbe easilytuned fromzeroto20γ . The comparedwithexperimentaldatatoobtainthecollective
1 0
magnetic field amplitudes corresponding to the effective Lamb shifts.
coupling strengths Ω = 5γ and Ω = 20γ taken in Inconclusion,weconstructanEITscheme by dynam-
1 0 1 0
Fig. 4 (a) and (b) are B = 5.3T and B = 21.3T, re- ically coupling the superradiant state with the subradi-
0 0
spectively. ant state. The interaction between atoms can be mea-
The reflectance spectra can be used to investigate the sured by the EIT spectra. Compared with the EIT-
effect of the counter-rotatingwave terms of the coupling like schemes with a static coupling in atomic ensembles
fieldandtodeterminethecollectiveLambshift. Forarel- [15,17,40,42,44],thelocaldynamicalmodulationofthe
ativelysmallΩ ,therearetwodipsinasingleLorentzian transition frequencies of the atoms introduces a tunable
1
peak,asshowninFig. 4(a). Theleftandrightonescor- detuning for the coupling field. Therefore, our scheme
respondto the rotatingandcounter-rotatingwaveterms contains all the ingredients of EIT. In particular, for the
of the coupling field, respectively. The distance between systemswherethesplittingbetweenthesuperradiantand
the two dips is approximately 2ν. When ν = 0, these subradiantstates is larger than the decay rate of the su-
two dips merge and the spectrum is the same as the one perradiant state, the dynamic modulation can bring the
of the previous EIT experiments with a static coupling EIT dip to the Lorentzian absorptionpeak of the super-
between two ensembles mediated by a cavity [15]. For radiantstate,asshowninFig. 2(b). Thedynamicmod-
a larger Ω = 20γ in Fig. 4 (b), we still have the two ulation enables a precise measurement of the distance
1 0
dips since Ω < γ and the vacuum induced coherence between two atoms and brings new physics of the EIT
1 +
stillexists [42],but wealsohavetwopeaksbasicallycor- point due to counter-rotating wave terms.
responding to the two magnetic transitions in Fig. 3. The authors thank G. Agarwal, A. Akimov, A.
5
Belyanin, J. Evers, O. Kocharovskaya, R. Ro¨hlsberger taneous Emission and Their Relation to Other Ap-
and A. Sokolov for insightful discussions. We ac- proaches,editedbyG.H¨ohler,SpringerTractsinModern
knowledge the support of National Science Foundation Physics Vol. 70 (Springer,Berlin, 1974).
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Grant EEC-0540832 (MIRTHE ERC), Office of Naval
[22] M. Scully,Laser Phys. 17, 635 (2007) .
Research (Award No. N00014-16-1-3054) and Robert
[23] D. W. Wang, Z. H. Li, H. Zheng, and S.-Y. Zhu, Phys.
A. Welch Foundation (Grant No. A-1261). Wei Feng Rev. A 81, 043819 (2010).
wassupported by China ScholarshipCouncil(Grant No. [24] D.PetrosyanandG.Kurizki,Phys.Rev.Lett.89,207902
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