Rydberg atoms with a reduced sensitivity to dc and low-frequency electric fields

0.27 MB·English

by  L. A. Jones

Rydberg atoms with a reduced sensitivity to dc and low-frequency electric fields
L. A. Jones, J. D. Carter, and J. D. D. Martin
Department of Physics and Astronomy and Institute for Quantum Computing,
University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
(Dated: January 18, 2013)
A non-resonant microwave dressing field at 38.465GHz was used to eliminate the static electric
dipole moment difference between the 49s1/2 and 48s1/2 Rydberg states of 87Rb in dc fields of
≈ 1V/cm. The reduced susceptibility to electric field fluctuations was measured using 2-photon
microwave spectroscopy. An anomalous spectral doublet is attributed to polarization ellipticity in
3
1 the dressing field. The demonstrated ability to inhibit static dipole moment differences — while
0 retaining sensitivity to high frequency fields — is applicable to sensors and/or quantum devices
2 using Rydbergatoms.

n
a
Rydberg atoms have a high sensitivity to resonantos- about this field. Rydberg atoms may require a dc elec-
J
cillatingelectricfields,typicallyatGHzfrequencies. This tric field to break degeneracies [1], enhance interatomic
7
property is useful for exploring the quantum nature of interactionsbymakingthemresonant[17,18]orfortrap-
1
radiation-matter interactions [1], and for high sensitiv- ping [13]. Surprisingly, we find that a dressing field can

] ity rf/microwave detection [2, 3]. But Rydberg atoms make the transition energy less sensitive to dc electric
h
also have a potentially problematic sensitivity to lower- field fluctuations around 1V/cm than at zero dc field
p
- frequency non-resonant fields, which can destroy coher- with no dressing field. This is despite the fact that low
m ence. One source of spurious fields are nearby surfaces, angular momentum Rydberg states are ordinarily more
o as sensed by Rydberg atoms near metals [4], co-planar sensitivetofieldfluctuationsinthepresenceofadcfield.
t resonators [5], and both shielded [6] and unshielded [7] The influence of dressing fields may be considered us-
a
. atom chips. The low-frequency fields near surfaces and ing exact diagonalization of truncated Floquet matrices
s
c thehighsensitivityofRydbergatomstothesefieldsmay [18, 19]. However,perturbationtheory aids the choice of
i limit schemes to couple these atoms to surface devices anappropriatedressingfrequency andpowerfornulling.
s
y [8–10]. We use the methods of Zimmerman et al. [20] to con-
h It is often desirable for an atomic transitionfrequency struct and diagonalize a Hamiltonian in a non-zero dc
p
to have a low sensitivity to electric and/or magnetic field Fdc,0 with no dressing field. The oscillating field
[
fields. For example, the “clock” transitions between al- (of amplitude Fac) and deviations in the static dc field
1 kalihyperfinelevelsareinsensitivetofirst-ordermagnetic ∆Fdc =Fdc−Fdc,0 are then considered as perturbations
v field variations [11]. When two-levelsystems are used to to the ith state, shifting the unperturbed energy Ei,0:
0
implement qubits, a reduced sensitivity to noisy pertur-
17 bations helps preserve qubit coherence, as illustrated by Ei ≈Ei,0−µi∆Fdc− 14αi(ω)Fa2c+βi(ω)Fa2c∆Fdc, (1)
4 Vion et al. [12] for superconducting qubits. where µi is the undressed Fdc,0 induced dipole moment,
1. To reduce the influence of low-frequency electric fields αi(ω)istheacpolarizability,and−βi(ω)Fa2c isanacfield
0 on Rydberg atoms Hyafil et al. [13] have proposed using induced contribution to the dipole moment (calculated
3 non-resonantdressingmicrowavefieldstomodifyelectric using 3rd order perturbation theory). Deviations in the
1
field susceptibilities. A dressing field may be chosen to dc field ∆F are considered to be in the same direction
: dc
v minimize the dc electric field sensitivity of a transition asFdc,0asthesecausethefirst-ordershiftsthatwewould
Xi frequency between two high-angular momentum “circu- like to suppress, whereas symmetry dictates that fluctu-
r lar” Rydberg states while retaining their sensitivity to ations transverse to Fdc,0 only result in 2nd order (and
a higher frequency fields. Dressing fields are a useful tool higher) shifts [21]. We consider a dressing field applied
for the modification of atomic properties: they can re- inthesamedirectionasthestaticfield,asboundarycon-
ducetheinfluenceofmagneticfieldsonopticalclocktran- ditions near metal surfaces dictate that this is the most
sitions [14], enhance interatomic Rydberg interactions likely scenario to be encountered in practice.
[15], and can increase Rydberg susceptibility to electric In the absence of the dressing field, the difference in
fields [16]. the dipole moments between two states ∆µ = µ1 −µ2
In this work we experimentally demonstrate dipole dictates the first-order dependence of the transition fre-
nulling–applicationofamicrowavedressingfieldtodra- quency on fluctuations in the dc field. The dressing field
matically reduce Rydberg atom susceptibility to varying inducedcontributiontothedipolemomentcanbeusedto
dc electric fields – using low-angular momentum states. counteractthissensitivity: setting−∆µ+∆β(ω)F2 =0
ac
Specificallywestudythe2-photon49s →48s tran- eliminates the first-order dependence of the transition
1/2 1/2
sition of 87Rb in a field of 1V/cm and find that we frequency to field variations ∆Fdc about Fdc,0.
can eliminate the first-order dependence on fluctuations To demonstrate dipole nulling we have chosen the
2
(a)
10 (a) probe (b) dressed
2m)) 5 dressing non-dressed
z / (V/c -50 -1.5 5500ss11/2/2 494p93p/23/2 49s1/2
(b3cm)) (GH)Da -11.00 energy (THz)Energy (THz)-1.6 4499ss11/2/2 444988444ppp988131ppp///222113///222 energy 48s1/2
z/(V/ 0.0 4488ss11/2/2 47p 47p3/2
H 3/2 632 MHz
G -1.0
(Db -1.7 4477ss11/2/2 47p1/2 365 MHz 0 Fdc,0
47p
30 32 34 36 38 40 1/2 dc electric field
dressing frequency (GHz)
FIG.2. (a)Zerofieldatomicenergylevelsandthemicrowave
dressing and probe frequencies. (b) The dipole nulling effect
FIG. 1. a) ac polarizability, and b) dressing field induced
underidealconditions: withtheapplicationofadressingfield
dipolemomentdifferences(seeEq.1)betweenthe49s1/2 and thereisnofirst-orderdependenceof thetransition energy on
48s1/2 states of 87Rb in a 1V/cm dc field. thedcelectricfieldandnodifferentialacStarkshiftatFdc,0.
49s1/2 → 48s1/2 transition of 87Rb in a dc field of (seeFig.3b). Rydbergstatepopulationsasafunctionof
1V/cm. This choice is based on practical consider- probe frequency were measured using selective field ion-
ations: given a (n + 1)s1/2 → ns1/2 system in Rb, ization (SFI) [18]. The strength of the probe was chosen
this is the lowest n for which the involved frequencies to avoid excessive power broadening of the spectral line.
(probe and dressing) [22] are below the 40GHz upper The dressing field was applied through a retro-
frequency limit of our available synthesizers [23]. Figure reflectingmirrorandvacuumwindow(see Fig.3a). This
1b illustrates the results of a perturbative calculation of configuration was chosen to reduce reflections from sur-
∆β(ω) = β49s1/2(ω) − β48s1/2(ω) for these two states. faces within the vacuum chamber, as these reflections
As ∆µ=µ49s1/2 −µ48s1/2 is positive (6.0MHz/(V/cm)) cause standing waveswhich decreasethe homogeneityof
we are confined to dressing frequencies ω where ∆β(ω) thedressingfield. LikewisetheRydbergexcitationbeam
is positive, to satisfy −∆µ+∆β(ω)Fa2c = 0. Figure 1 was orthogonal to the microwave propagation direction
shows two frequency ranges where ∆β(ω)>0. to reduce the influence of partial standing waves along
The remaining flexibility in the choice of ω may be the direction of microwave propagation.
used to satisfy a second constraint (analogous to the Figure 4 shows probe spectra of the 49s −48s
1/2 1/2
magic wavelengths for optical Stark shifts [24]). By transition with different dc fields and varying dressing
choosingthedifferenceintheacpolarizabilities∆α(ω)to powers (the powers quoted are the synthesizer level set-
be zero, we can reduce the influence of inhomogeneities tings). As the dressing power is increased the probe line
in the dressing field over the sample [25]. The condi- splits into a doublet, which we attribute to a polariza-
tion ∆α(ω) = 0 occurs at a frequency of 38.44GHz (see tion imperfection in the dressing field (discussed further
Fig. 1), where ∆β = 0.37GHz/(V/cm)3 and thus for below). Nulling occurs for both lines of the doublet. In
nulling Fac = p∆µ/∆β = 0.13V/cm. This is a rela- particular, we can compare the non-dressed to dressed
tivelyweakmicrowavefieldamplitude –anenhancement Rydberg state transition frequencies at several dc fields
cavity is not necessary. Figure 2 illustrates the relevant as shown in Fig. 4b. There is a distinct dipole mo-
energy levels for the particular n we have studied. ment at 1V/cm in the absence of dressing, whereas the
Rubidium atoms were gathered in a 87Rb magneto- first-orderdependence ondc field is eliminatedwhen the
optical trap (MOT) and optically excited from the dressing microwave field is present. The nine measured
5p ,F = 3 state to the 49s ,F = 2 Rydberg level peak centers of the dipole-nulled lower sideband have a
3/2 1/2
using a frequency doubled Ti:sapphire laser, stabilized standard deviation of 0.06MHz over the dc field range
using a transfer cavity [26]. DC electric fields were ap- 0 to 1.5V/cm compared with the systematic deviation
plied using two field plates positioned above and below of 7MHz over the same range when no dressing field is
the MOT center (see Fig. 3a). The dressing microwave applied. The strength of the two-photon probe coupling
fieldwasoncontinuouslyduringopticalexcitation,while remains unchanged, with slight line broadening due to
theenergydifferencebetweenRydbergstateswasprobed dressing field inhomogeneity.
using a second microwave field after optical excitation Both the observations and the full Floquet diagonal-
3
(a)
(a)
dielectric mirror
Rb+ 0.30 RF OFF
1.0 V/cm
0.20 1.1 V/cm
1.2 V/cm
0.10
(embx)iccirtoawtioanve horn probe 8.1cm 3.6cm s population1/2 000...100284 5dBm
48 0.10 8dBm
dressing SFI malized 000...000864
MOT beams or 0.02
0 10 20 30 40 50 n 0.08 11dBm
time (ms) Rydberg excitation beam
0.06
0.04
0.02
FIG.3. (a)Experimentalapparatus. (b)Thesequenceofap-
plied fields. Prior to Rydberg excitation, the anti-Helmholtz
-10 -5 0 5
coil current is ramped down and the residual magnetic field
(b) probe frequency - zero field frequency (MHz)
nulled using compensation coils; the coil current is switched
on again once selective field ionization (SFI) is complete. 6
no dressing
This sequence is repeated at 10Hz as the probe frequency is
z) 4 dipole nulled doublet
stepped, and theRydbergpopulations measured using SFI. MH dipole enhanced doublet
y ( 2 Floquet calculation
c
n
e
u 0
ization show surprisingly small variations in the dressed eq
energies with dc field – even down to low fields. In fact, eld fr -2
∆β is very nearly proportional to Fdc,0 together with ero fi -4
∆µ. This suggests an alternative theoretical formula- y - z -6
tion where we consider dressing field induced changes in nc
e
the polarizabilities about zero dc field (rather than in- qu -8
e
duced dipole moments about non-zero fields). The per- fr -10
turbations to achieve “polarizability nulling” would be
4thorder(2ndorderin eachofthe fields). However,this 0.0 0.5 1.0 1.5 2.0
alternative approach will break down when considering dc electric field (V/cm)
couplingtostatesintheStarkmanifoldwhosedipolemo-
ments are independent of dc field. In this case only first FIG. 4. (a) The 49s1/2 →48s1/2 probetransition for differ-
ordersuppressionispossible. The dipolenullingdescrip- entmicrowavedressingpowersanddcelectricfields(labeled).
tion — making use of perturbation expansions around At 8dBm dressing power the spectral lines for the three dif-
non-zero dc fields — is a more general approach. ferent dc fields coincide, illustrating the dipole nulling effect.
Basonet al. [16]havedemonstratedthatnon-resonant Thedressingfrequencyis38.465GHz,chosensothatthedou-
dressing fields can enhance Rydberg susceptibility to dc blethaszeroaverageacStarkshiftat1.1V/cm. (b)Observed
electric fields. In our case it is possible to increase frequenciesofthe49s1/2−48s1/2 probetransitionsasafunc-
tion of dc electric field under different dressing field condi-
the dipole moment difference between the 49s and
1/2 tions,illustratingthedipolenullingandenhancementeffects,
48s1/2 statesbyusingadressingfrequencywhere∆β(ω) comparedwiththenon-dressedcase. Estimatedstatisticaler-
is the opposite sign of ∆µ. Figure 4b shows that the rorsare on theorderof thesymbolsizes. Alsoshown arethe
49s −48s transitionfrequencybecomesmoresensi- results of a Floquet diagonalization with a linearly polarized
1/2 1/2
tive to dc field fluctuations when a dressing frequency dressing field (Fac = 0.13V/cm and ω/(2π) = 38.465GHz).
Since the probe is a two-photon transition, “frequency” in
of 36.225GHz is applied, consistent with the calcula-
both (a) and (b) refers to twice the applied frequency, for
tion of ∆β(ω) in Fig. 1; i.e. ∆β(ω) < 0 for ω/(2π) =
correspondence with energy level differences.
36.225GHz. However, application of a non-zero dc bias
electric field — in situations where this is possible — is
a more straightforwardway to increase Rydberg suscep-
tibility to electric fields. zeroelectric fieldanddoes notdepend onRydberg atom
We now discuss the origin of the doublet in the probe density. As we show below, by varying the microwave
spectra. This is not an Autler-Townes doublet, as the polarizationwehaveestablishedthatthissplittingisdue
dressingfrequencyisdetunedfromthenearesttransition to a slight polarization ellipticity in the dressing field.
by 365 MHz (see inset to Fig. 2), which is much larger Althoughthemicrowavesemergingfromthehornshould
than the observedsplitting. This splitting also occurs at haveahighdegreeofpolarizationpurity,reflectionsfrom
4
thevacuumchamberwalls,fieldplates,andwindowscan
intentionally applied dressing
introduce ellipticity in the field at the location of atoms. (a)
dressing component causing ellipticity
Thesplittingissimilartothemagneticfield-likevectorac
Stark shift observedfor ground-state atoms in circularly
polarized fields [27].
Tounderstandthesplitting,itisusefultoconsidertwo
extreme cases: 1) a linearly polarized dressing field, and 15°
interfering microwave field
2) a circularly polarized dressing field. In the linearly
polarized field, with an axis of quantization along the
ocoscuiplllaetdinagnfidesldhodwirethcteiosna,mteheacmSjta=rk±s1h/i2ftss.taItnesaapreuruenly- (b) y (MHz) 20 cdaaltcaulation
circular field, a natural axis of quantization is normal to nc
e 15
the plane containing the oscillating E field. With this qu
e
choice, the mj = ±1/2 states are also uncoupled, but d fr 10
show different ac Stark shifts, due to their different al- o fiel
lowed couplings (i.e. with σ+: s1/2,mj =1/2 is coupled zer 5
to p3/2,mj = 3/2 whereas s1/2,mj = −1/2 is coupled y -
c
to p ,m = 1/2 and p ,m = 1/2). Elliptical po- n
1/2 j 3/2 j ue 0
larizationsareintermediatebetween these twoextremes, q
e
andthe splitting canbe calculatedusing Floquettheory. fr -π/2 0 π/2 π 3π/2
Although both the 49s1/2 and 48s1/2 states split, the relative phase (radians)
calculations indicate that splitting of the 49s state is
1/2
40 times smaller than that of the 48s state and thus
1/2
unresolved in our probe spectra [28]. FIG. 5. (a) Arrangement of interfering (nearly) orthogonal
microwave polarizations for verifyingsplitting isduetoellip-
Toconfirmthatpolarizationisresponsibleforthesplit-
ting we aimed a second horn towards the atoms (see ticity. (b) The location of two lines of the 48s1/2 −49s1/2
doublet as a function of phasebetween two interfering dress-
Fig. 5a). The microwave dressing signal from a synthe- ing fields at 38.1GHz. The experimental data is shifted hor-
sizer was split and sent to each of the two horns, with izontally by a constant phase for best agreement with the
adjustable relative phase and power. The intensity and calculation. Theshadingaroundthecalculationindicatesun-
ellipticity of the fields due to each horn was determined certainty related to variations in the transmitted amplitude
by observing the splitting of the 49s −48s probe throughthephaseshifter. Uncertaintiesinthemeasuredline
1/2 1/2 positions are on theorder of the symbolsizes.
transition, using a dressing field frequency where the ac
Stark shift is non-zero (38.1GHz – see Fig. 1) and selec-
tively blocking one or the other of the horns. The field
m = ±1/2 degeneracy of the ns states, this degen-
intensitiesweredeterminedfromtheobservedaverageac j 1/2
eracy could also be lifted in a more controllable way by
Starkshiftandthe theoretical∆α. The ellipticities were
application of a dc magnetic field. In this case, an el-
determined by comparison of the ratio of the splitting
liptical dressing field would cause no further splitting,
to the ac Stark shift (both scale linearly with microwave
and the dressing frequency could be tuned to avoid any
power), with a Floquet calculation. We characterize the
differential ac Stark shift for the component of interest.
polarization ellipses by the ratio of the minor to major
axis: −17dB for one of the horns and −11dB for the Our choice of n for dipole-nulling was based on avail-
other. able equipment, but nulling could be achieved for many
Therelativepowerswereadjustedsothattheintensity Rb (n+1)s1/2−ns1/2 pairs. In particular, the dressing
atthelocationoftheatomswasthesameforbothhorns, frequency (for ∆α(ω) = 0) scales like 1/n3, and the ac-
and then both beams were allowed to interfere. Figure field magnitude to obtain zero dc quadratic Stark shifts
5b shows that as the relative phase between the two in- about Fdc =0 scales like Fac ∝1/n6 (determined by di-
terferingfieldsischanged—modulatingthepolarization agonalizationofFloquet Hamiltonians for n=30 to 55).
—thesplittingvariesinmagnitude. Wecancomputethe Nulling with different species and transitions can be ex-
expectedsplittingasafunctionofphasebetweenthetwo amined using the perturbative approach presented here
fieldsusingFloquettheory(seeFig.5b). Therelativeori- (of particular interest are low-angular momentum states
entationofthe twopolarizationellipsesis variedforbest connected by a single-photon transition [5]).
agreement with the data. The discrepancies (< 3MHz) In summary, we have demonstrated that it is possi-
arepossiblyduetovariationoftherelativephaseoverthe ble to selectively inhibit the dipole moment difference
sample, and/or variations in transmitted power through between two Rydberg states using non-resonant dress-
the phase shifter with varying phase shift. ing fields. This technique offers a means to preserve the
Since the observed splitting is associated with the coherence of Rydberg atom superpositions and is com-
5
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