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Polarization dependence of n = 2 positronium transition rates to Stark-split n = 30 levels via crossed-beam spectroscopyTo cite this article: A C L Jones et al 2016 J. Phys. B: At. Mol. Opt. Phys. 49 064006

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https://doi.org/10.1088/0953-4075/49/6/064006

Polarization dependence of n = 2 positronium

transition rates to Stark-split n = 30 levels via

crossed-beam spectroscopy

A. C. L. Jones

Department of Physics and Astronomy, University of California, Riverside, CA 92521

E-mail: [email protected], [email protected]

T. H. Hisakado

H. J. Goldman

H. W. K. Tom

A. P. Mills, Jr.

Abstract. We produce Rydberg Ps by a two-step laser excitation from 13S → 23P

and from 23P to states of principal quantum level n = 30 ± 1 that are Stark split by

a motionally induced electric field. Our measurements are largely free of first-order

Doppler shifts such that we are able to investigate the impact of laser polarization on

the population of the closely spaced Stark levels. We find a variation in the distribution

that is primarily dependent on the IR laser polarization with respect to the direction

of the motionally induced electric field. With the IR light polarized parallel to the

electric field F, the ratio of excitation probability to the levels of maximal Stark

splitting compared to that of excitation to the states of minimal Stark splitting is

found to be 3.37 ± 0.51, whereas with the IR light polarized perpendicular to F, the

excitation ratio is 0.87± 0.64. Our results agree with those of Wall et al. [1] obtained

with n = 11 and will be useful in the preparation of high-n states of Ps for a variety

of experiments, including measuring the interaction of Ps with gravity, in precision

time-of-flight (TOF) energy spectroscopy, and precision optical spectroscopy of Ps.

PACS numbers:

Keywords Submitted to: J. Phys. B: At. Mol. Phys.

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Polarization dependence of n = 2 positronium transition rates to Stark-split n = 30 levels via crossed-beam spectroscopy2

1. Introduction

Positronium is a unique atomic target composed entirely of leptons and is inherently

unstable to self-annihilation. Its spectrum is without the quantum defect offsets that

are caused by the atomic cores of ordinary atoms, and should be precisely defined by

bound-state QED theory. Due to the short lifetime of ground state positronium (Ps),

long-lived Rydberg states are particularly attractive for studies of Ps for which a long

observation time would be advantageous. Collisions of Ps with anti-protons provides a

possible mechanism for the efficient production of antihydrogen [2, 3], with the charge-

exchange cross section growing rapidly as a function of the principal quantum level of the

Ps atoms and growing with decreasing collision energy like ∼ 1/E [4]. In addition, there

is renewed interest in precision measurements of the 1S − 2S intervals [5, 6] and Lamb

shifts [7] of hydrogenic atoms as a means of resolving the proton radius puzzle [8, 9].

Due to the low mass of Ps, its thermal velocities are substantially higher than

those of other atoms (e.g., v ≈ 30 times that of hydrogen at the same temperature).

Unlike studies of hydrogen, precision studies of Ps must account for unusually significant

Doppler and motional Stark effects [10], the latter particularly when utilizing Rydberg

states [11]. Consideration of the motionally induced Stark effect is critical in experiments

involving thermal Ps atoms in Rydberg states in magnetic fields of more than a few

mT. Where Rydberg Ps is employed in strong magnetic fields, field ionization due to

the motionally induced electric field [12] also puts significant constraints on the range of

Ps kinetic energies that can be utilized. The first-order Doppler effect can be removed,

or at least significantly constrained, via a number of experimental techniques. The

effect can be explicitly removed by two-photon absorption [13] or saturated absorption

spectroscopy [14, 15], and can be substantially reduced by performing crossed-beam

spectroscopy [16].

Recently Wall et al. [1] demonstrated Stark-state selective production of n = 11

Rydberg Ps in a static electric field, and investigated the effect of the excitation laser

polarizations. A set of similar measurements are performed here in the n = 29, 30,

and 31 manifolds for which the level splittings are ∼ 20 times smaller. We elucidate

the impact of motional Stark effects in the production, use, and study of these higher n

Rydberg Ps atoms. In our experiment the detection of atoms over a small angular range

of emission normal to the target surface greatly reduces the observed Doppler spread,

yielding measurements that are largely free of the first-order Doppler shift. With this

approach it is possible to resolve neighboring n states up to ∼ 35 using thermal Ps

produced in a heated target.

2. Experimental Details

The present apparatus, illustrated in Fig. 1, is modified from a similar one described

elsewhere [17]. Moderated positrons [18] from a 22Na source are accumulated in a buffer

gas trap [19] in ∼ 4 s cycles. Collected positrons are dumped from the trap in a

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bunched pulse, and then further accelerated in a high-voltage pulser. Ps is produced by

implanting positrons, with kinetic energies of ∼ 1 keV and guided by an axial magnetic

field of 6.9± 0.2 mT, into a p-Si(100) target heated to ∼ 1000 K [20]. Ps emitted from

the target is optically excited into Stark-split Rydberg states via a two-step process

using a pair of pulsed dye lasers (Quanta Ray PDL-1), pumped with the frequency-

doubled and -tripled light of a pulsed Nd:YAG laser (Continuum Surelite III-10). First

a ∼ 1 mJ UV pulse of duration ∼ 3 ns FWHM, centered about ∼ 243.02 nm with a

∼ 100 GHz bandwidth, excites ground state ortho-Ps to 23P states. A second laser

pulse (∼ 1 mJ/pulse, ∼ 3 ns FWHM, ∼ 30 GHz bandwidth) of IR light in the range

732.0-732.5 nm, fired within < 2 ns of the first pulse, excites atoms from n = 2 to final

states of n = 29 - 31. The two laser beams propagate in directions that are within

±3◦ of perpendicular to the velocity of the Ps atoms that could subsequently hit the

detector.

In our previous work [17], the lasers were directed parallel to the sample surface,

at 45◦ to the velocity vector of the detected Ps atoms. We corrected for the first order

Doppler effect by using the TOF to determine the Ps velocity and hence the magnitude

of the shift. In the present experiment we have made several improvements to the design

of the apparatus: (1) The near-perpendicular orientation of the laser with respect to the

trajectory of the detected atoms largely removes the first-order Doppler effect, resulting

in shifts of 6 4 parts in 105, equivalent to roughly half the IR bandwidth. (2) The

flight path is ∼ 3 times longer, which improves the accuracy in our determination of

the Ps velocities, thus allowing for more accurate measurement of Stark effects. (3) In

contrast to our earlier experiments on Rydberg Ps, which used scintillators coupled to

photomultiplier tubes to detect Rydberg Ps atoms via their annihilation radiation, the

detection scheme employed here involves a new design [21] in which Rydberg atoms are

field-ionized in the 1.5 kV/cm electric field produced by a pair of 85 mm diameter grids

(90 lines/inch, 90% transmitting). The first grid is at ground potential and is located

1.49± 0.01 m from the target, subtending an acceptance cone of half angle 1.64± 0.04

degrees about the target normal. The field-dissociated positrons are accelerated and

focused onto a 4 cm diameter micro-channel plate (MCP) detector (Hamamatsu model

F1217-21S). This detector scheme offers roughly an order of magnitude improvement in

the detection efficiency compared to the scintillators, as well as a much lower background

count rate and improved timing accuracy (< 10 ns). Each event signaling the detection

of a Rydberg Ps atom is recorded as a wavelength and TOF pair (λ, t), with the IR

wavelength λ measured at a Bristol 821 pulsed wavelength meter. Ps flight times t are

recorded with a ∼ 1 ns precision and overall ±5 ns statistical uncertainty.

3. Results and Analysis

In Fig. 2 plots (a)-(b) we present measurements of the Rydberg Ps count rate as a

function of TOF and IR wavelength sorted by the IR laser polarization, as labeled

in the plots. The motionally induced electric field experienced by the detected atoms


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Figure 1. Schematic of the Rydberg Ps (Ps∗) time-of-flight (TOF) apparatus.

Positron pulses containing ∼ 105 e+ per 5 ns pulse are accelerated from a mean energy

of 180 eV to 1.18 keV in a pulsed high voltage tube, and then implanted into a p-Si(100)

target. Ps atoms are rapidly emitted from these targets (sub-ns). Emission from the

Si target produces an epithermal velocity distribution [22]. A pulsed laser system

is used to excite ground state ortho-Ps atoms to Rydberg states, with 19 ≤ n ≤ 31.

Rydberg atoms entering the 1.5 m flight tube can be detected via a micro-channel plate

(MCP) based detector. The inset shows closer detail of the target (which measures

12.5 mm across) and the region of Rydberg excitation. The angle between the lasers

is exaggerated to better show the method by which laser overlap is achieved. In our

experiments the angle between the lasers is typically 6 5◦. Due to symmetry the laser

propagation angle with respect to the Ps trajectories is roughly half this value.

is oriented vertically in the lab reference frame (the horizontal plane of which is that

projected in the experimental schematic of Fig. 1). As a result the axis of polarization

(or z-axis) for the Stark states is in the vertical direction. The polarizations of the IR

and UV lasers were varied between the lab vertical (↕) and horizontal (↔) orientations

covering the four basic combinations. As the observed distributions appear to depend

primarily on the IR laser polarization, the data have been summed together over the

two UV polarization directions for each IR polarization for improved statistics [23]. The

range of IR wavelengths used covers final states between n = 29 and n = 31. Expected


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0

20

40

60

0

20

40

60

0204060

732.1 732.2 732.3 732.40

20

40

732.1 732.2 732.3 732.4

8

12

16

0.65 kV/m

0.87 kV/m

1.13 kV/m

1.63 kV/m

F

(b4)

(b3)

(b2)

(b1)

(b)

(a1)

TOF

(s)

n = 29n = 30

(a4)

(a3)

(a2)

n = 31

IR

14.5-19 s

6-7 s

8-9 s

10.5 -12.5 s

(a5) (b5)

IR wavelength (nm)

coun

ts (/

1000

sho

ts)

(a)

0.5

1.0

1.5

2.0

2.5

elec

tric

field

(kV/

m)

IR

Figure 2. (color online) Experimental data are presented, sorted as a function of IR

wavelength and Ps time of flight (TOF), for measurements of Rydberg Ps production

in the vicinity of the n = 30 resonance for measurements made with the polarization

of the IR excitation laser (a) perpendicular and (b) parallel to the motionally induced

electric field experienced by detected Rydberg atoms. In plots (a1) and (b1) the data

are rebinned as a function of the motionally induced electric field. Arrows indicate

the expected Stark splitting of the n = 30 states. In these plots the intensity of the

shading is proportional to the number of events detected, with white indicating no

counts and black indicating the maximum count rate. Plots (a2,...,a5)-(b2,...,b5) show

a selection of data produced by summing over narrow bands of TOF, as labelled in

plots (a2,...,a5), corresponding to sets of approximately constant electric field as noted

in plots (b2,...,b5). Also shown in each plot is a fitted model curve, described in detail

in the text.


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Polarization dependence of n = 2 positronium transition rates to Stark-split n = 30 levels via crossed-beam spectroscopy6sign

al

n = 10, m = 1

IR wavelength0

Y0

A1

A1+A2

Figure 3. (color online) Schematic representation of the fitted model of the Stark-

split resonances, illustrated here for n = 10. Each manifold of Stark-split states

is represented by a sum of Gaussian distributions (corresponding to the substates

k), evenly spaced apart about a line center λ0 up to a maximum splitting of ∆λ.

The contributions from the k substates have amplitudes which vary with ki as

Ai = (A1 +A2 ∗ (|ki|/kmax)). The standard deviation σ of each Gaussian represents

the bandwidth of the IR laser, broadened by the range of Stark shifts within each TOF

band, and is found to be consistent among the various data sets.

line centers are indicated by dashed vertical lines, shown in each of the subplots for

n = 31 to 29, from left to right. The subplots of Fig. 2 are ordered (in columns) by the

IR laser polarization. In plot (a) the IR light is polarized horizontally (i.e., perpendicular

to the motionally induced electric field), while in (b) it is polarized vertically (parallel

to the electric field). There is a clear distinction between the distribution of intensity

as a function of the IR laser polarization: Where the IR laser is polarized horizontally,

counts are relatively uniform amongst the Stark-split substates, whereas the measured

spectra taken with the IR laser polarized parallel to the electric field are split about each

line center, indicating preferential excitation to the highest and lowest lying states. In

plots (a1) and (b1) the same data are rebinned and plotted versus the motional electric

field associated with each detected event, calculated from the measured flight time and

distance and the known magnetic field (6.9±0.2 mT) in the vicinity of the target. In this

representation the Stark splitting is more easily seen, with each set of states n producing

an envelope of counts expanding linearly outwards as a function of electric field from

the expected line centers, as suggested by the ‘V’ shaped arrows that have been drawn

about the n = 30 line centers (see plot (a1) for n-state labels), which represent the

calculated electric-field dependence of the outermost Stark states (k = ±kmax, where

kmax = n − |m| − 1). As each data set is a sum of measurements made with the UV

both horizontally and vertically polarized, both sets of measurements have a maximal

splitting with |m| = 0.

For further analysis, the data of plots (a) and (b) are subdivided and summed over

narrow bands of flight times (4-5, 5-6, 6-7, 7-8, 8-9, 9-10.5, 10.5-12.5, 12.5-14.5 and

14.5-19 µs), yielding signals as a function of IR wavelength for various narrow ranges of

motionally induced electric field. A selection of these data subsets are shown for each of

the IR polarizations in Fig. 2 plots (a2,...,a5) and (b2,...,b5) as labeled. A basic model is

fitted to each spectrum, yielding a description of the excitation profiles for each subset,


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and providing measures of the Stark splitting ∆λ and n = 30 line centers λ0 of the

resonances as a function of the motionally induced electric field for each data set. The

model illustrated in Fig. 3 comprises a sum of Gaussian curves, corresponding to the

expected available Stark-split levels k. As a result of the resolution of the IR laser used,

it is not possible to resolve distinct k-states, the typical splitting of substates being

on the order of 1/10th of the laser bandwidth. The amplitude of individual k-state

contributions to the spectra are treated as having a simple linear dependence, varying

as Ai = A1 + A2|ki/kmax|. The general form of the fitted models is given as follows,

Y = Y0 +∑i,n

Ai exp

(−(λ− λ0,n +

kikmax

∆λn

)2

/2σ2

), (1)

where Y is the count rate signal as a function of λ, the IR wavelength. Y0 is the

background signal due to ion production in the detector by scattered UV light and to

dark counts. The model is thus obtained by summing over the allowed Stark states

ki. The parameter ∆λn is half of the total Stark splitting and describes the maximum

deviation from the line centers λ0,n for each resonance. To reduce the number of free

parameters in the model, only the n = 30 line center and width are explicitly fit. The

positions of the adjacent resonances are fixed with respect to the n = 30 line center

at the calculated separations, while the widths are taken to be (n/30)2 × ∆λ30, i.e.,

that expected due to Stark splitting. The distribution of signal counts is assumed to

be symmetric about the line center in the model, a simplification which appears to be

reasonable given the quality of the fits. The curves in Fig. 2, being composed of partly

overlapping contributions from adjacent resonant manifolds n, are not symmetric about

the centers of plots (a2)-(b5) because of the different Stark splittings and decreasing

line center spacings going from n = 29 to 31.

In the cases of IR light polarized parallel to the motionally induced electric field

(column (b) in Fig. 2), the observed favoring of transitions to the outer k-states is

qualitatively explained by their having increasingly large dipole moments with increasing

k. This behavior is in qualitative agreement with that seen in a comparable experiment

involving the production of Rydberg hydrogen [24], and more explicitly with comparable

measurements made in Rydberg Ps of n = 11, split by a large DC field such that the

individual substates k are resolved [1], and was the motivating reason for choosing a

linear dependence of the model’s amplitudes Ai on ki. It should be noted that two other

fit models were also tested, one with a purely quadratic dependence on ki and another

with a simple step function of intensities (Ai = A1 for ki ≤ k′ and Ai = A2 for ki > k′).

Both of these models produce similar quality fits and results, and have been left out of

the discussion for the sake of clarity.

The first step in fitting each of the data sets involves a fit of equation (1) to each

set with all parameters treated as free variables. From these fits, the weighted mean σ of

the Gaussian standard deviation σ is calculated, and is taken as a constant σ = σ in all

subsequent fits. The parameter σ is principally associated with the IR laser bandwidth,

and is shown in Fig. 4 for the fits to each TOF subset of the two IR laser polarization


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0.4 0.6 0.8 1.0 1.2 1.4 1.60

10

20

30

40

50 IR IR

electric field (kV/m)

fitte

d w

idth

(p

m)

Figure 4. (color online) Fitted widths σ (principally representative of the IR laser

bandwidth) are presented for each of the data sets, presented as a function of the mean

electric field. Results are presented for each of the two data sets, as indicated in the

figure legend. The weighted average σ of the fits is shown by the horizontal line, with

the uncertainty indicated by the shaded bar.

0.0 0.5 1.0732.25

732.26

732.27

0.0 0.5 1.0

IR


(nm

)

(a) (b)IR

Figure 5. (color online) Fitted line centers are presented, plotted as a function of

the motionally induced electric field, for the IR laser polarized (a) horizontally and

(b) vertically. Linear fits to the data sets yield a zero-field line center and a slope,

∆λ0/∆F which is attributed to a residual first-order Doppler shift, corresponding to

an angle of ∼ 1.4◦ between the IR laser and the detected Ps beam.

sets presented in Fig. 2. The weighted mean and its uncertainty calculated from these

fits is σ = 18.4±0.6 pm indicating a bandwidth of (24±1) GHz FWHM ‡, and is shown

by the dashed horizontal line centered in a filled band in Fig. 4.

In Fig. 5 the n = 30 line centers λ0 of each fit are shown as a function of the

expected motionally-induced electric field for (a) IR horizontal and (b) IR vertical,

up to the Inglis-Teller limit [25] (where the Stark manifolds of neighboring principle

quantum numbers n overlap, occurring here at > 1.1 kV/m), above which the fit results

become unreliable. Linear fits to the fitted centers are plotted as dashed lines, with

results summarized in Table 1. The y-intercept yields an estimate of the field-free line

center, while the slope of the lines δλ0/δF can be attributed to a residual Doppler shift.

In both sets of measurements, the zero velocity line center is offset from the expected

position by comparable margins, with a mean offset of ∼ 0.0073 ± 0.0009 nm, in good

accord with the expected level of accuracy of the wavemeter (0.01 nm) [26]. The mean

slope is found to be (−0.0122±0.0012) nm/(kV/m), indicating an angle between the IR

‡ The bandwidth measured here is approximately a factor of two better than that previously reported

(cf. [17]). This is attributed to improvements in the operation of the dye lasers and the lack of residual

broadening from the Doppler correction previously implemented.


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Table 1. Fitted line centers, found from the weighted means of fits to the data of

Fig. 5, performed with a fixed slope (attributable to the residual first-order Doppler

shift) as described in the text. The mean offset between the expected line centers and

those found from the fits is consistent with the accuracy of the wavemeter used to

measure the IR wavelength.

λ0,30 (nm) Expected (nm) Offset (nm)

IR ↔ 732.2698± 0.0017 732.2617 0.0081± 0.0017

IR ↕ 732.2687± 0.0011 0.0070± 0.0011

mean offset: 0.0073± 0.0009δλ0

δF (nm / kV/m) θ (deg.)

IR ↔ −0.0131± 0.0024 1.50± 0.27

IR ↕ −0.0119± 0.0014 1.36± 0.16

mean angle: 1.40± 0.14

0.0 0.5 1.0 1.5

0

2

4

6 (b)


1 +

A 2/A1

(a) IRIR

0.0 0.5 1.0 1.5

Figure 6. (color online) Ratio of the fitted amplitude A1 + A2, describing the

outermost Stark-split states of |k| = kmax, to that of the innermost states, A1, is

displayed for each of the narrow TOF subsets of Fig. 2 (a) and (b). Weighted means

are indicated by dashed horizontal lines, with the associated uncertainty shown by the

filled band. Sets are sorted by the IR laser polarization, noted in each plot. In (a) fits

to the data taken with horizontal IR polarization (i.e., where the IR light is polarized

perpendicular to the motionally induced electric field) are displayed, while plot (b)

represents fits to the data with vertical IR polarization (parallel to the electric field).

laser and detected Ps beam of 1.40± 0.14◦, consistent in both magnitude and direction

with that anticipated from the geometry of the experiment (cf. inset of Fig. 1).

In Fig. 6 we plot the ratios of the maximum and minimum amplitudes (given by

(1+A2/A1)) from the model fits to the data as a function of the motional electric field,

grouped by IR polarization. Plot (a) shows the fit results for the data of Fig. 2 column

(a) with the IR laser polarized horizontally, while plot (b) shows the results for fits to

the data of Fig. 2 column (b). In both plots a dashed line indicates the weighted mean

of the fit results, with the uncertainty (±1 standard deviation) in the mean represented

by the extent of the filled band. Where the IR light is polarized horizontally, the

outermost states (i.e., |k| = kmax) have an observed signal rate that is 0.87 ± 0.64

times that of the states about k = 0, consistent with a relatively uniform distribution.

On the other hand, in the measurements taken with the IR light polarized vertically,

the outermost states are observed with 3.37 ± 0.51 times the intensity of states about


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0.0 0.5 1.00

25

50

75

100


(pm

)

IRIR

(a) (b)

0.0 0.5 1.0 1.5

Figure 7. (color online) Fitted Stark splittings for data taken with the IR laser

polarized (a) horizontally and (b) vertically. The Stark splitting is represented here

by the half-width ∆λ about the line centre λ0 of the n = 30 resonance, for the two

polarization data sets (a) IR horizontal and (b) IR vertical. In each case the observed

splitting exceeds that predicted (shown by the solid lines), based on the magnitude

of the motional electric field, though the slope of each line is consistent with that

expected. This indicates the presence of a DC electric field, oriented vertically, i.e.

parallel to the motionally induced field in the vicinity of the target. Fits to the data

(dashed lines), assuming a fixed slope equal to that expected from the motionally

induced electric field, indicate a splitting at zero velocity of 51.0± 0.8 pm, consistent

with a DC electric field with a vertical component of 391± 19 V/m.

k = 0. The maximum signal rates of the two data sets are similar (compare rates across

rows of Fig. 2 plots (a2)-(b5)), thus suggesting that the cross sections as a function of

k are strongly dependent on the polarization of the IR light, with the excitation being

suppressed near k = 0 by a factor of 3.4± 0.5 compared to states of |k| ≈ kmax in the

measurements taken with the IR light polarized vertically (i.e., parallel to the electric

field).

Finally, in Fig. 7 we plot the half-width of the Stark splitting ∆λ for the n = 30

resonances as a function of electric field for (a) the IR laser horizontally polarized and (b)

vertically polarized. In both cases, the observed splitting is larger than that anticipated,

as indicated by the solid line, however the slope of the observed fit is in agreement with

that expected. A linear fit is performed to each data set assuming the expected slope

and is plotted as a dashed line. Extrapolating the trend back to zero-field indicates a

mean splitting of 51.0± 0.8 pm in the absence of the motionally induced electric field.

This suggests that there is a small DC electric field in the region of excitation, oriented in

the same direction as the motionally induced field, with a magnitude of ∼ 413±7 V/m.

UV photoemission or secondary electron emission from the target is unlikely to produce

sufficient charging of the target holder to account for the observed offset. Alternatively

the field may result from build up of charge on unshielded insulators above the target

due to operation of the target heater, from patch fields, or transient oscillation of the

nominally grounded target potential due to the few ns rise time of the kV-potential

pulsed positron accelerator, which is situated ∼ 0.12 m from the target. Although

the field thus induced would be expected to lie primarily in the plane of the detected

Ps atoms velocities (i.e., perpendicular to the sample surface), any component of this

field in the same direction as the motionally induced electric field will add linearly,

as necessary to explain the observed excess of splitting. Any component of the non-


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motional electric field in the plane of the Ps velocities does not produce a noticeable

curvature in the splitting at low velocities as would be expected, indicating that any

such component must be smaller than ∼ 500 V/m.

4. Concluding Remarks

We have presented here an experimental investigation into the motional Stark effect in

Rydberg Ps atoms produced in a magnetic field performed by crossed-beam first-order

Doppler-free TOF spectroscopy. We find that the population of maximally Stark-split

k-states can be favorably selected for with the correct combination of laser polarization

with respect to the motionally induced electric field. The effect of laser polarization is

evident when changing the polarization of the IR laser used to excite the final transition

from 23P to triplet Stark-split states of principal quantum number n, in agreement with

the recent results of Wall et al. for n = 11 Ps [1]. When the IR laser is aligned with the

motionally induced electric field, the population of Stark states favors those of extreme

|k|, with a relative population of 3.4 ± 0.5 to 1 as compared with states of |k| ≈ 0.

Experiments performed with the IR laser polarized perpendicular to the motionally

induced electric field, on the other hand, yield a relatively uniform distribution across

the available k levels. This type of effect could be exploited to preferentially excite

states of large |k|, even when the individual Rydberg states are not resolved. One could

also envision repeatedly exciting transitions with one IR polarization and de-exciting

with a perpendicular polarization to produce a population of Rydberg Ps atoms with

a narrow distribution of k-states with either k = kmax or k = 0. By the central limit

theorem, the population of states should approach a narrow Gaussian distribution after

many repeated cycles of low amplitude excitations. These distributions would have

either large dipole moments (|k| = kmax) or a dipole moment of ∼ 0 (k = 0), which

could be used to obtain large Rydberg Ps beam deflections in an electric field gradient

or immunity from stray fields, respectively.

The effects explored here are of direct interest to proposed Ps gravity experiments

which will require the production of long-lived Rydberg Ps states, and have important

implications for the possible use of Rydberg Ps in production of anti-hydrogen. For

example, both the AEGIS [2] and GBAR [3] collaborations intend to utilize excited state

Ps (n ≥ 3) to produce antihydrogen for measurements of the gravitational deflection

of antimatter. Additionally, sufficiently long-lived Rydberg Ps may be used directly

to perform gravitational deflection measurements [27] and selecting the proper k-states

without their being resolved, which could be advantageous where a sufficiently narrow

laser bandwidth is not practical.

Acknowledgements

This work was supported in part by the US National Science Foundation under grant

PHY 1206100 and 1500900.


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