A pulsed Sisyphus scheme for laser cooling of atomic (anti)hydrogen

1.3 MB·

by  Saijun Wu

A pulsed Sisyphus scheme for laser cooling of atomic (anti)hydrogen
Saijun Wu, Roger C. Brown, William D. Phillips, and J. V. Porto
Joint Quantum Institute, NIST and University of Maryland, Gaithersburg, Maryland 20899
(Dated: January 20, 2011)
Weproposealasercoolingtechniqueinwhichatomsareselectivelyexcitedtoadressedmetastable
state whose light shift and decay rate are spatially correlated for Sisyphus cooling. The case of
coolingmagneticallytrapped(anti)hydrogenwiththe1S-2S-3Ptransitionsusingpulsedultraviolet
andcontinuous-wavevisiblelasersisnumericallysimulated. Wefindanumberofappealingfeatures
includingrapid3-dimensionalcoolingfrom ∼1Ktorecoil-limited, millikelvin temperatures,aswell
1
1 as suppressed spin-flip loss and manageable photoionization loss.
0
2 PACSnumbers: 37.10.De,67.63.Gh

n
a Recent progress [1, 2] in producing antihydrogen (H)
J
improves the prospects for precision spectroscopy and
9
de Broglie wave interferometry of H that may uncover
1
new physics in low-energy experiments [3, 4]. Anti-

] hydrogen atoms are synthesized from anti-protons and
h
positrons [1–5] in such small numbers that trapping and
p
efficient coolingto millikelvin temperatures arelikely re-
-
m quired for precise measurements. In contrast to hydro-
o gen, which may be cooled by collisions with a buffer
t gas[6], or by selecting low-energyatoms froman intense FIG. 1: (a): Level diagram for the proposed cooling scheme
a
. beam, cooling of H will likely rely on laser cooling tech- (seetext). (b): Simplifieddressed-statepictureofthecooling
s
niques [7]. scheme. The pulsed two-photon excitation has bandwidth
c
i The most obvious approach to laser cooling of H [8] 1/τ,isdetunedfromthebottomofthelatticewithdepthU by
s δ0, and provides Doppler cooling by transferring momentum
y or H demands 121.6 nm Ly-α radiation. Apart from the
h difficulty of manipulating VUV light, the generation of ¯hkge to an atom with velocity v. The linewidth Γ(x) of the
dressed excited state |e˜(x)i is indicated by the width of the
p
even 10 nW of CW Ly-α radiation is technically chal- gray curve. As the atom climbs the hill the velocity (green
[ ∼
lenging [9]. In addition, such cooling of H atoms in a arrow)decreaseswhilethedecayprobabilityincreases,leading

1 kelvin-deepmagnetic trapfaces severalinterrelateddiffi- to Sisyphus cooling. (The |e′i → |ei decay is ignored in (b).
v culties. Theneedtoavoidspin-fliplosses,combinedwith Not shown is the other, detuneddressed state |e˜′(x)i.)
2
thelimitedfractionofphasespaceaddressablewithalow
0
6 intensity, single-frequency laser, implies that the cooling
3 willbe slow. Indeed, the only experimentalLy-αcooling 2-photon coupling Ωge. The proposed cooling process
. worksofar[8]usedapulsedlaserwithanaveragepower arises from two effects: 2-photon Doppler cooling [10–
1
0 of160nW(2.5nWatthelocationofatoms)tocoolmag- 12] associated with g e excitation, and Sisyphus
1 netically trapped H, and took more than 15 minutes to cooling [14] associate|diw→ith|thie e e′ lattice.
| i−| i
1 reach 8 mK, starting from just 80 mK. Furthermore, 3D For magnetically trapped H, g , e and e′ are the
: | i | i | i
v cooling in this approachwas aided by collisional mixing, maximally Zeeman shifted states in the 1S, 2S and 3P
i whichwillbeabsentindilutesamplesofH. Insteadofre- manifoldsrespectively. Weshowthatthecoolingscheme
X
lying on Ly-α radiation,there are severalproposedcool- provides both a large capture velocity (100 m/s) and a
r
a ing schemes using more readily available lasers to drive low final temperature (near the Ly-β single-photon re-
Dopplersensitive2-photontransitions[10–12]. However, coil temperature of 1.8 mK), and allows for large vol-
in addition to limited phase-space addressability similar ume3Dcooling. Advantagesinclude: availabilityofboth
toLy-αcooling,theseschemeshavethedifficultyoflosses nanosecond-pulsedUV2-photon1S-2Sradiation[15]and
due to photoionization. CW 2S-3P radiation at 656 nm; reduction of UV pho-
Motivated by previous work [13], which used transi- toionization losses; and suppressed spin-flip transition
tions between excited states for cooling and trapping, from the 3P level in a high field. In the following we
we propose a 3-level cooling scheme (Fig. 1), applicable first discuss the pulsed Sisyphus cooling scheme in a 1D,
to magnetically trapped H, where a metastable state e semiclassical model. After justifying the model with a
is coupled to a short-lived state e′ by a blue detun|edi 1D quantum simulation [16, 17], we present a 3D semi-
standing wave coupling Ωee′. Ato|msiin the ground state classical simulation for magnetically trapped H.
g are repeatedly excited to the bottom of the dissipa- The proposedcoolingscheme involvesrepeatedpulsed
t|ivie e e′ opticallatticebyapulsed,Dopplersensitive excitations,eachfollowedbyspontaneousdecay[18]. The
| i−| i
2
propagation direction of the 2-photon excitation pulses
alternates between xˆ. The Rabi frequency Ω (x,t) is
ge
(θ/τ)f(t/τ)e±ikgex,±where θ is the pulse area, kge is the
sum of the wavevectors of the two photons, and f(t/τ)
is a normalized pulse-shape function with characteris-
tic duration τ. For θ 1 its Fourier transform F(ωτ)
≪
givesthe excitationspectrum. TheintervalT between
rep
pulses is long enough so that excited atoms, moving in
the e e′ lattice, decay to g with high probability.
| i−| i | i
In the effective 2-levelsystem(Fig.1b), the spatially de-
pendent detuning δ(x) and linewidth Γ(x) (see Eq. (1))
allows spatial selectivity in both pulsed excitation and
subsequentdecay. The e e′ transitionis drivenby a
| i−| i
standing wave coupling Ωee′(x) with a positive detuning
∆,resultingintwodressedstates e˜(x) and e˜′(x) which
arespatiallydependentsuperposi|tionsiof e |and ie′ and
| i | i
connect to those states respectively as Ωee′ 0. The 2-
→ FIG.2: OBEsimulationofcoolingproperties. (a,d): Normal-
photon g e detuning from the unshifted metastable
sdteactaey|erai|t(eids−efcra|oymira|ete′iγ(eFgig≈. 10a))i.sWδ0e, aanssduγmee′g,∆γe≫′e aδr0e,γteh′ge ai(zbge,eded)e:xocvNietorartmxio0anliapznerddobeiasnbeiilrnigtyyupnleoitsvsssopx¯fe0rh¯=ωprku,eglsee′e.xε0/(cvπ,sfa)vn¯:.dRv¯εa=tiisova/ovvfegerε-.
and γe′g γe′e. Atoms are predominantly excited to, to the normalized ionization probability, ε/pi vs v¯. Here
nanotd|e˜a′d(ixa)b≫ia)t[i1c9a]l,lyanfodllow, the dressed state |e˜(x)i (and Ωδ0ee′=(x)−2=5ωΩr,geee′,sτin(=kee′−x2).,5/∆δ0,=f(Ωt)ee′=/2√=1πe−20t20, ωθr,g=e,πa/n8d,
γe′g =8γe′e =2ωr,ge. The red curves in (b,c,e,f) correspond
Γδ((xx))==δ√0Ω−δe0e(′−p(xδ()Ωx2+)ee∆′(2xγ)e2′g+. ∆2−∆)/2, (1) atondΩ5ee.4′ r=es0p.ectTivheelyl.eft and right panels are for kge/kee′=25
In Eq. (1) we have ignored γe′e and γeg, but their inclu-
sion has little influence on the results described below. spent there. Averaged over the position (and velocity)
The depth U of the resulting e e′ optical lattice is dependentdecayprobability,thevelocitydistributionaf-
given by the maximum of δ | iδ−(x|),iand its period is ter decay is centered at zero velocity, with a rms width
0
determined by kee′. We define−kij as the wavevector of less than 21v for v ≫vge. On average this removes more
the i j transition for i,j = g,e,e′, with the associated than 75% of the atomic kinetic energy per 2-photon ex-
recoil−velocities and frequencies defined as v = ¯hk /m citation.
ij ij
and ω =¯hk2/(2m) where m is the atomic mass. To characterize the cooling, we define the normal-
r,ij ij
Doppler-sensitive g e absorptionleadsto2-photon ized position and velocity dependent excitation proba-
| i−| i
Doppler cooling [10–12]. The Doppler shifted, spatially bilities p = 4P /θ2 and energy loss per 2-photon pulse
e e
dependent detuning, δ(x,v)=δ(x) k v, allows atoms ε = 4∆E/θ2, where P is the 2-photon excitation prob-
ge e
±
at different velocities to be excited to the lattice poten- ability, ∆E is the energy loss per 2-photon pulse and p
e
tial at different locations. Atoms with velocity v are and ε are θ-independent for θ 1. Figure 2(a,b,d,e)
resonantly excited at positions such that δ(x,v)τ < 1. shows p and ε, determined usi≪ng 3-level optical Bloch
e
For τ > 2π/δ , strong excitation only|occurs|in∼the equations(OBE)fora“draggedatom”followingatrajec-
0
| |
velocity range v < v < v , with decoupling velocity toryx(t)=x +vt. Figure2(a,b)andFig.2(d,e)arefor
d c 0
| |
vd δ0 /kge andcapture velocity vc (δ0 +U/¯h)/kge. kge/kee′ =25and5.4respectively. Thelatterratiocorre-
≃| | ≃ | |
Atoms with v vd or v vc are off resonance and sponds to k1S−2S/k2S−3P in hydrogen. Figure 2 suggests
| | ≪ | | ≫
not efficiently excited. thatthesimplepictureofphase-spaceselectiveexcitation
InadditiontoDopplercooling,thecorrelationbetween plusSisyphuscoolingonlyappliesforkee′ kge,(vτ)−1,
≪
thespatiallydependentdetuningδ(x)andthedecayrate as in Fig. 2(a,b). For atoms that move more than 1/kee′
Γ(x) leads to Sisyphus cooling since atoms preferentially during τ, the excitation is complicated by multi-photon
decay from e˜(x) at the tops of the light shift poten- resonances at velocities with kgev + 2nkee′v δ (the
| i ≈
tial [14]. The Sisyphus effect is particularly efficient for peaks of the black curve in Fig. 2e for v¯ > 20) [13]. In
atomswithv closetovd, whichareexcitednearthe bot- addition, for large kee′, Doppleron resonant coupling to
tom of the lattice. If 12mv2 < U, atoms remain within |e˜′(x)i[20]occursatmoderatespeedsvwith2nkee′v ≈∆
one lattice site and typically oscillate before decaying to forintegern(the sharpdips ofthe blackcurveinFig.2e
g . Thedecayisenhancedattheclassicalturningpoint, near v¯ = 70, 80), and leads to heating. Neverthe-
| i
due to both the larger decay rate and the longer time less, efficient Sisyphus cooling is still possible for mod-
3
erate kge/kee′ (see Fig. 2e). Compared to regular 2-
photon cooling (red curves in Fig. 2(b,e)), the peak ex-
citationprobability is decreasedby approximately 2π/U
τ h¯
due to the spatially inhomogeneous broadening of e˜(x)
| i
(Fig.1b). However,duetotheSisyphusenhancedenergy
removalper excitation, the averageenergy loss per pulse
remains comparable to the Doppler-only case, but with
an increased velocity capture range v < v <v .
d c
| | FIG.3: Comparisonof1DSCSWandQSWsimulations. Here
In addition to the increased velocity capture range,
the decreased excitation probability to e˜(x) also helps Ωee′/(4π) = ∆/(2π)=2.7 GHz as in Fig. 2, δ0 = −2π/τ,
| i θ =π/4. (a): Average speed h|v|i vs pulse number for three
mitigatethephotonionizationlossfrom e˜(x) tothecon-
different pulse durations τ. 4 ns represents a compromise
| i
tinuum. For degenerate 2-photon excitation of H to the between 1 ns, where the bandwidth is so large that there is
2S level, the ionization probability per pulse is given by little spatial selectivity, and 7 ns, where the bandwidth is so
Pioni =R dtγioni(t)ρ2S(t)whereρ2S(t)isthe2Sstatepop- small that only a small fraction of atoms are excited. Thick
ulation and γ is the rate of ionization from 2S due to lines are an average of 30 QSW trajectories, while thin lines
ioni
areanaverageof20SCSWtrajectories. (b): Theequilibrium
the UV radiation[21]. As ameasureofcooling efficiency
temperature T vs pulse duration τ. At small velocities the
per2-photonpulse,inFig.2(c,f)wecompareε/p ,thera-
i SCSW method becomes less accurate for τ > 9 ns, roughly
tio between the normalizedenergylossε andnormalized
set by half the oscillation period in a single |ei−|e′i lattice
ionizationprobabilitypi =12Pioni/θ3,with(blackcurve) site.
and without (red curve) the Sisyphus cooling. Here we
have set γ = 1.6Ω [21]. We see that the Sisyphus
ioni ge
effect enhances the cooling efficiency by approximately nal and external degrees of freedom of a 3-level atom,
U/¯hδ near v = v where the Doppler cooling has the to confirm that the SCSW method correctly predicts
0 d
| |
best ε/p . the cooling dynamics and the final temperature. In
i
We simulate the cooling process with a semiclassical Fig. 3 typical results for smoothed square pulses [24]
stochastic wavefunction (SCSW) method [16, 17]. The are compared, for the appropriate hydrogen 1S-2S-
simulation of a cooling cycle is divided into two stages: 3P parameters, γe′g/(2π) = 26.6 MHz, γe′e/(2π) =
excitation (0 < t < τ) and decay (τ < t < Trep) (we 3.6 MHz, {vge,vee′,vge′} = {3.3,0.6,3.9} m/s, and
ignore quantum jumps during excitation). The exter- ωr,ge,ωr,ee′,ωr,ge′ /(2π) = 13.4,0.46,18.8 MHz,
{ } { }
nal motion of the atom is described by a classical tra- which are also used in Fig. 2(d-f). We find good agree-
jectory x(t). The internal dynamics are described by ment between the SCSW and QSW methods as long as
a stochastic wavefunction ψ(t) , which, after the g e thedraggedatompictureisvalidduringthepulse,i.e.,if
| i −
pulse, is probabilistically projected to either g or the theopticalforceduringtheshortexcitationdoesnotsig-
e , e′ manifold (typically almost all in |e˜(ix) ) as nificantly displace the trajectory compared to the wave-
{| i | i} | i
|ψpi. Due to this post-selection, the optical force in length (kee′ δvpulse τ ≪ 1)(Fig. 3b). The attainable 1D
the excitation stage cannot be evaluated in the usual temperature predicted by the quantum simulation de-
way as ψ(t)Fˆ ψ(t) where Fˆ is the force operator. In- creases with decreasing bandwidth 1/τ, and is remark-
h | | i
stead, the force is estimated as the real part of a “weak ably low ( 3 mK) even with τ =5 ns.
∼
value” [22], ψp(t)Fˆ ψ(t) / ψp(t)ψ(t) , where ψ(t) is Having verified the semiclassical approach for our pa-
h | | i h | i | i
found by forward-propagating the pre-determined state rameters, we use SCSW to simulate 3D cooling of mag-
ψ(0) = g and ψp(t) is found by back-propagating netically trapped H. To describe the 3D light-atom in-
| i | i h |
the post-determined state ψp(τ), both for a dragged teraction, we included all ten electronic levels in the
h |
atom. This estimationmethod reproducesthe quantum- 1S-2S-3P manifold (ignoring hyperfine structure). In a
mechanically expected velocity change during the pulse, high magnetic field, the cooling process is dominated by
δv ,duetoboththerecoileffectandtheexcited-state the three maximally Zeeman-shifted states of the 1S,
pulse
dipole force. During the second stage, the stochastic 2S and 3P levels (corresponding to g , e and e′ ),
wavefunction ψp(t) e , e′ manifold and x(t) are which wouldform a closedsystem und|eri1S|-2iS2-ph|otion
| i ∈ {| i | i}
propagated in small time-steps, until a quantum jump coupling Ωge and perfect σ+ 2S-3P coupling Ωee′. We
occurs[16,17]. Ifthequantumjumpisan e′ e tran- consider a magnetic trap with B = B ,B ,B =
| i→| i { x y z}
se˜it′(ioxn), pwreobparobijleicsttic|ψalpl(yt)[2i3t]o,wthheiledfroerssaend est′ates |ge˜(xju)imopr, B{B1=y−0.B752zTx,/2B,B=10x.−8 TB/2zcmy/2a,nBd0B+B=2(122z2m−Tx/2c−my22,)s/i4m}-,
| i | i→| i 0 1 2
we propagatex(t) freely untilthe nextpulse. Uponeach ilar to those for an existing antihydrogen apparatus [5].
spontaneous emission, we use random velocity jumps to Both g and e feel a trapping potential V µ B
B
| i | i ≈ | |
account for the recoil effect. (µ is the Bohr magneton), so the g e detuning
B
We use a 1D full quantum stochastic wavefunction δ is nearly free from Zeeman shifts [2|5i].−T|hie e e′
0
| i−| i
(QSW) simulation [16, 17], which includes both inter- detuning ∆(B) ∆ µ B/¯h, on the other hand, is
B
≈ −
4
flip loss is found to be less than 0.1%. As with other
hydrogen cooling proposals [10–12], one must consider
limitations imposed by photoionization losses. We per-
turbatively calculate photoionization from state popula-
tions determined by internal state dynamics that ignore
photoionization. Forareasonabletwo-color2-photonex-
citation scheme where the stronger laser beam cannot
ionize H from the 2S state in a single step [15], we found
FIG. 4: Evolution of atomic velocity (a) and position (b) for ionizationlosseslessthan25%[28]. Evenif243nmradia-
a typical classical trajectory during the simulated cooling of tionisusedforthe1S-2Sexcitation[21],photoionization
magneticallytrappedH.Theinsetsgivethequasi-equilibrium loss wouldstill be less than25%for cooling,by applying
distributions. N =109, θ =2.5 mrad pulses [29] in 2000 s.
total
WehaveproposedandanalyzedapulsedSisyphuslaser
cooling scheme applicable to magnetically trapped H or
field-sensitive and has a position-dependent shift.
H. The approach leads to rapid 3D cooling to <10 mK
We considera2S-3Platticecomposedofthreepairsof
in a magnetic trap over a large volume with small spin-
standing waveGaussianbeams, eachwith1/e2 diameter
flip losses. Approaches to reduce photoionization losses
d, arrangedsymmetrically with equalintersectionangles
to practical levels are proposed. Cooling efficiency may
α to zˆ. The choice of relative phases between standing
be further improved by exploring the spatial-temporal
waves is not critical to the cooling scheme. The beams
control of both the 2-photon excitation and the excited
are circularly polarized to maximize the σ+ components
state lattice. This excited-state Sisyphus method may
(relative to Bˆ). In the Paschen-Back regime considered
open new possibilities for cooling of deuterium, tritium
here, with µ B ¯h∆ (∆ /(2π)=3.25 GHz),
B 3P,fine 3P,fine or other species with metastable states.
≫
the π coupling induces spinflip lossesafter 2Sexcitation
WewouldliketothankAmyCassidy,GretchenCamp-
with a branching ratio of r = 2∆2 /(∆+µ B/¯h)2
sf 9 3P,fine B bell, and Jonathan Wrubel for helpful discussions.
(similar for σ− coupling). Even for 2S-3P light that is
purelyπ orσ− polarized,thespin-flipprobabilityper 2S
excitation is still less than 0.3% in a field of 1 T.
Figure 4 plots a typical classical trajectory of H dur-
ing cooling. The simulation starts with H in g at the [1] G. B. Andresen et al, Nature468, 673 (2010).
| i [2] Y. Enomoto et al,Phys. Rev.Lett. 105, 243401 (2010).
trap bottom (B = 0.75 T), with initial longitudinal (z)
[3] M. Amoretti et al, Nature419, 456 (2002).
and transverse (x,y) kinetic energy of El = 0.5 K and [4] G. Gabrielse et al, Phys.Rev. Lett.89, 213401 (2002).
Et =0.25 K respectively. The 2-photon excitation beam [5] G. Gabrielse et al, Phys.Rev. Lett.100, 113001 (2008).
overlaps with the 2S-3P beams in a 12 cm long and 1.8 [6] R. deCarvalho et al, Can. J. Phys. 83, 293 (2005).
cm wide cooling zone, approximately covering the trap [7] Laser Cooling and Trapping, H. Metcalf and P. van
up to the 0.2 K equipotential surface [26]. We choose der Straten (Springer-Verlag, 1999).
d = 3 cm and α = 0.1 for the 2S-3P beams, with peak [8] I. Setija et al, Phys.Rev.Lett. 70, 2257 (1993).
intensity of 0.46 kW/cm2 per beam corresponding to [9] M. Scheid et al, OpticsExpress 17, 11274 (2009).
[10] M. Allegrini et al,Phys. Lett. A 172, 271 (1993).
Ωee′/(2π) =1.3 GHz. ∆(0.75T)/(2π) = 5.3 GHz is cho- [11] V. Zehnleet al,Phys. Rev.A 63, 021402 (2001).
sen so that ∆(B) γe′g within the cooling zone. In [12] D. Kielpinski, Phys. Rev.A 73, 063407 (2006).
≫
a flatter octopole trap [1] the detuning constraint is re- [13] S. Wu et al, Phys.Rev.Lett. 103, 173003 (2009).
duced,allowingforareducedΩee′ andless656nmpower. [14] A. Aspect et al,Phys. Rev.Lett. 57, 1688 (1986).
The total power requirements can also be lessened with [15] Atwo-colorexcitationschemeisproposedbyL.Yatsenko
a moderate finesse optical cavity. We choose θ = π/8, etalinPhys.Rev.A60,4237(1999).AlsoseeT.Kanaiet
al,Opt.Exp.17,8696(2009)forwavelengthconversions.
τ = 4 ns and T = 2 µs [27]. The 2-photon detuning
rep [16] J. Dalibard et al, Phys.Rev. Lett.68, 580 (1992).
δ = π/τ is chosen to improve the scattering rate for
0 − [17] H. J. Carmichael, Lecture Notes in Physics, Vol. m18
longitudinally cold atoms that are transversely hot.
(Springer, Berlin, 1993).
The rapid cooling trajectory shown in Fig. 4 is typi- [18] T. Breeden et al, Phys.Rev.Lett. 47, 1726 (1981).
cal for atoms with E < 0.5 K and E < 0.25 K, which [19] Foratommovingwithvelocityv,theadiabaticfollowing
l t
are cooled to quasi-equilibrium within Ntotal = 4 104 requires he˜(x)|v∂x|e˜(x)i ≈ kee′vΩ∆ee′ ≪ ∆. The excita-
pulses during a cooling time of only 80 ms. While×some tion predominantly to |e˜(x)i requires kgev≪∆.
[20] J. J. Tollett et al, Phys.Rev.Lett. 65, 559 (1990).
atoms with E significantly larger than 0.2 K may orbit
t [21] M. Haas et al, Phys. Rev.A 73, 052501 (2006).
aroundthecoolingzoneandnotbeefficientlycooled,the
[22] Y. Aharonov et al, Phys. Rev.Lett. 60, 1351 (1988).
final velocity distribution for most atoms is remarkably [23] J. Chen et al,Phys. Rev.A 47, 2128 (1993).
isotropic with 5 m/s width (Fig. 4a inset), which can be [24] Rising/falling edges are smoothed bysine functionsover
furtherreducedbyincreasingτ (Fig.3b). Thetotalspin- 20% of the pulseduration.
5
[25] C. L. Cesar et al, Phys.Rev. Lett.77, 255 (1996). [28] The 2S-3Pradiation also photoionizes Hfrom 3P, which
[26] To avoid excitation outside the cooling zone, one can inthesimulationleadstolessthan4%totallossforcool-
overlap two excitation beams with distinct frequencies ing from El+Et at 0.75 K. Also see P. Antoineet al,J.
of ω1 and ω2 = ω2S−ω1 respectively, where ω2S is the Phys. B 29, 5367 (1996).
frequency of the1S-2S transition. Also see ref. [15]. [29] Reducing the pulse area while increasing the total pulse
[27] To allow slowest atoms confined by the |ei−|e′i lattice number suppresses the total photoionization loss, as the
todecay,the|ei→|gidecayrate(γeg)isincreasedfrom ionization loss ∝θ3 for θ≪1.
0to10µs−1 fort>1µs.Thiscanbeeffectivelyrealized
with a broadband |ei−|e′i coupling pulse.