# Single-color two-photon spectroscopy of Rydberg states in electric fields

T. E. Wall  D. B. Cassidy  S. D. Hogan  Department of Physics and Astronomy,
University College London, Gower Street, London WC1E 6BT, U.K.

###### Abstract

Rydberg states of atomic helium with principal quantum numbers ranging from
$n=20$ to $n=100$ have been prepared by non-resonance-enhanced single-color
two-photon excitation from the metastable $2\,^{3}\text{S}_{1}$ state.
Photoexcitation was carried out using linearly and circularly polarized pulsed
laser radiation. In the case of excitation with circularly polarized
radiation, Rydberg states with azimuthal quantum number $|m_{\ell}|=2$ were
prepared in zero electric field, and in homogeneous electric fields oriented
parallel to the propagation axis of the laser radiation. In sufficiently
strong electric fields, individual Rydberg-Stark states were resolved
spectroscopically, highlighting the suitability of non-resonance-enhanced
multiphoton excitation schemes for the preparation of long-lived
high-$|m_{\ell}|$ hydrogenic Rydberg states for deceleration and trapping
experiments. Applications of similar schemes for Doppler-free excitation of
positronium atoms to Rydberg states are also discussed.

###### pacs:

32.70.Jz,32.80 Ee,32.80.Rm

##  I Introduction

Among their many often extreme properties, Rydberg states of atoms and
molecules can possess long radiative lifetimes. In the absence of external
perturbations low-angular-momentum hydrogenic Rydberg states with principal
quantum number $n>30$, and small quantum defects, typically exhibit lifetimes
exceeding $1~{}\mu$s which scale with $n^{3}$. High-angular-momentum
‘circular’ states with similar values of $n$ have lifetimes greater than
$5~{}$ms which scale with $n^{5}$ Gallagher (1994). Because of these long
lifetimes, optical and microwave transitions involving these states exhibit
narrow natural linewidths. They are therefore very well suited for precision
spectroscopic measurements, including for example, the accurate determination
of molecular ionization and dissociation limits Liu  _et al._ (2009), and
studies of the role of nuclear spins in photoionization Paul  _et al._ (2009);
Saßmannshausen  _et al._ (2013). These long lifetimes are also of importance
for applications of Rydberg states in quantum information processing Gleyzes
_et al._ (2007); Saffman  _et al._ (2010), and in experiments with
antihydrogen and positronium atoms directed toward (i) spectroscopic
investigations of matter–antimatter asymmetries Bluhm  _et al._ (1999), and
(ii) measurements of the acceleration of particles composed of antimatter in
the gravitational field of the Earth Mills and Leventhal (2002); Kellerbauer
_et al._ (2008); Cassidy and Hogan (2014).

In each of these areas, the long radiative lifetimes of the Rydberg states
alone are generally not sufficient for measurements at the highest resolution.
It is also necessary to ensure that the period of time during which the
samples are interrogated, and the interaction time with the desired external
fields approach, or exceed, these lifetimes, and that motional effects such as
Doppler broadening are minimized. Typically these requirements can be
fulfilled simultaneously through the preparation of slowly moving samples with
low translational temperatures. However, there are cases where high resolution
excitation to Rydberg states directly from a ground, or other low-lying
electronic, state is necessary before subsequent deceleration or trapping,
because of the challenges associated with the preparation of sufficiently cold
ground state samples using current techniques. These cases include, for
example, precision spectroscopic studies of the ionization and dissociation
limits of H2 and its isotopomers Liu  _et al._ (2009), and the preparation of
long-lived excited states of positronium Cassidy  _et al._ (2012). In these
cases, it is foreseen to minimize effects of Doppler broadening in the
photoexcitation process by using counter-propagating laser beams to drive non-
resonance-enhanced single-color two-photon transitions to the Rydberg states
Sprecher  _et al._ (2011); Hogan (2013). The spectral resolution of the
resulting Doppler-free excitation Hänsch  _et al._ (1975) will then be limited
only by the bandwidth of the laser radiation used to drive the transition, and
the period of time with which it interacts with the sample.

The photoexcitation of Rydberg states using non-resonance-enhanced single-
color two-photon schemes in the presence of homogeneous electric fields offers
the opportunity of exploiting the large electric dipole moments associated
with the resulting states for efficient acceleration, deceleration and
trapping using inhomogeneous electric fields Yamakita  _et al._ (2004);
Vliegen  _et al._ (2004); Hogan and Merkt (2008). The magnitude of these
dipole moments scale with $n^{2}$, and at $n=30$ can exceed 3000 D. In
addition to the importance of the results presented here for Doppler-free
Rydberg state photoexcitation, when driven using circularly polarized laser
radiation such multiphoton excitation schemes represent a general approach to
the preparation of long-lived non-core-penetrating Rydberg states of small
molecules. With their small quantum defects, and long fluorescence lifetimes,
these states are ideally suited for deceleration and electric trapping
experiments Hogan  _et al._ (2009); Seiler  _et al._ (2011), directed toward
studies of excited state decay processes, and low energy scattering with other
gas-phase targets, or surfaces Softley (2004).

Non-resonance-enhanced single-color two-photon excitation schemes have been
employed previously in high resolution spectroscopic studies of atomic Rydberg
states Stoicheff and Weinberger (1979); Penent  _et al._ (1988) and in studies
of (2+1) resonance-enhanced multiphoton ionization Génévriez  _et al._ (2014).
We report here a detailed investigation of the role of ac Stark energy shifts
in single-color two-photon excitation in beams of metastable helium (He) atoms
under experimental conditions which are similar to those expected in future
experiments with positronium (Ps), and report the efficient excitation of
selected Rydberg-Stark states.

In the following, a description of the experimental apparatus used in the work
reported here is first provided. The numerical methods employed in the
calculation of the spectral profiles associated with non-resonance-enhanced
two-photon transitions from the 2 ${}^{3}\text{S}_{1}$ state in He to high
Rydberg states is then outlined, before the results of the experiments carried
out (i) in the absence of externally applied electric fields, and (ii) in the
presence of homogeneous electric fields oriented parallel to the direction of
propagation of the laser radiation, are presented. Finally, comparisons are
drawn between the single-color two-photon spectra reported here and those
expected in the case of Doppler-free two-photon excitation of positronium
using laser radiation with similar temporal and spectral properties.

##  II Experiment

Figure 1: (Color online) Schematic diagram of the experimental setup (not to
scale). The polarization of the laser beam is set by a Glan-Taylor polarizer
(GT) and then controlled with a rotatable quarter-wave plate (WP). The
coordinate system is defined such that the origin is located at the position
where the laser beam axis intersects the atomic beam axis (shown as a black
point)

A beam of metastable He atoms is prepared in an electric discharge within a
supersonic expansion, as described in Halfmann  _et al._ (2000). Helium is
injected into the source vacuum chamber of the experimental apparatus through
a pulsed valve (Series 9 General Valve). At the exit of the valve the
metastable 2 3S1 state is populated in an electric discharge generated between
a metal anode located $\approx 1~{}$mm downstream from the nozzle to which an
electric potential of $+240~{}$V is applied, and the grounded base plate of
the valve. The discharge is seeded with electrons from a heated tungsten
filament located $\approx 15~{}$mm further downstream. After expanding and
cooling the metastable He beam travels with a mean forward speed of
$v_{z}\simeq 1950~{}$ms-1. Atoms in the shorter-lived 2 1S0 metastable state
also populated in the discharge, although present in the beam, do not play a
noticeable role in the studies of the triplet states described here.

Having passed through a skimmer with a $2~{}$mm-diameter aperture into a
second vacuum chamber, the atoms pass between two parallel planar electrodes
(labelled E1 and E2 in Fig. 1) which each have a length of $100~{}$mm in the
direction of propagation of the atomic beam and are separated by $18~{}$mm.
Each electrode contains a $3~{}$mm aperture in the center. A single laser beam
($\sim 10$ mJ per pulse at $\approx 522~{}$nm with a pulse length of 6 ns and
a bandwidth of 5 GHz, produced by a Nd:YAG-pumped pulsed dye laser) passes
through these apertures. The laser beam converges as it propagates through the
chamber, reaching a focus $\sim 20~{}$mm beyond the atomic beam axis. Where
the laser intersects the atomic beam axis it has a Gaussian transverse
intensity profile, with a standard deviation $\sigma_{y}\simeq\sigma_{z}\simeq
70~{}\mu$m (see Fig. 1 for the definition of the coordinate system), resulting
in peak intensities on the axis of the atomic beam of $\sim 5\times
10^{13}~{}$Wm-2. By applying a potential difference between E1 and E2, uniform
electric fields of up to $F=155~{}$Vcm-1 can be produced in the excitation
region, allowing the selective preparation of individual Rydberg-Stark states.

Single-color two-photon transitions driven from the 2 ${}^{3}\text{S}_{1}$
state in He to Rydberg states with $n\geq 20$ converging on the
$1\text{s}\,^{2}\text{S}_{1/2}$ ionization limit require radiation with
wavelengths in the range from 520 nm to 524 nm (Fig. 2). With the intermediate
state far from resonance with any bound state, these transitions are not
resonance-enhanced.

Figure 2: (Color online) Energy level diagram showing possible single-color
two-photon excitation pathways from the 1s2s ${}^{3}\text{S}_{1}$ state to
high Rydberg states of He using circularly ($\circlearrowright$) and linearly
($\uparrow$) polarized laser radiation.

In the experiment the laser polarization is set using a Glan-Taylor polarizer
and then controlled using a $\lambda/4$-plate, permitting the production of
circularly polarized radiation with a purity of $>90\%$. In the excitation
region in the apparatus the electrode planes are oriented parallel to the axis
of the atomic beam and perpendicular to the direction of propagation of the
laser beam, such that the circularly polarized radiation drives transitions
for which $\Delta$$M_{J}=+1$ or $\Delta$$M_{J}=-1$, where $M_{J}$ is the
azimuthal quantum number associated with the total angular momentum $\vec{J}$,
of the atom. In the case of the triplet states of He treated here only the
interaction of the optically active electron, with total angular momentum
$\vec{j}$, and azimuthal quantum number $m_{j}$, initially in the 2s orbital
need be considered. Therefore excitation with circularly polarized radiation
results in transitions for which $\Delta$$m_{j}=+1$ or $\Delta$$m_{j}=-1$.
Since individual spin-orbit components are not resolved in the experiments
described, in general the azimuthal quantum number, $m_{\ell}$, associated
with the orbital angular momentum of the single optically active electron is
referred to when discussing the effects of laser polarization in the observed
spectra. With circularly polarized radiation driving single-photon transitions
for which $\Delta$$m_{\ell}=+1$ or $\Delta$$m_{\ell}=-1$, two-photon
transitions from the 1s2s ${}^{3}\text{S}_{1}$ state can result only in the
excitation of states with $|m_{\ell}|=2$. However, linearly polarized
radiation can drive transitions for which $\Delta$$m_{\ell}=\pm$1\. Thus, in
the configuration presented, linearly polarized radiation can drive two-photon
transitions to states with $m_{\ell}=0$ and with $|m_{\ell}|=2$.

Following photoexcitation He Rydberg atoms are detected by pulsed electric
field ionization 118 mm downstream from the excitation region between
electrodes E3 and E4. The resulting ions are then collected on a micro-channel
plate (MCP) detector. With E3 and E4 separated by 50 mm, the pulsed potential
of $+4.0~{}$kV applied to E4 for electric field ionization results in a field
on the atomic beam axis of $\sim 800$ Vcm-1. This field is sufficient for
complete diabatic ionization of atoms in states with principal quantum number
$n\geq 35$ Gallagher (1994). The signal from the He Rydberg atoms is recorded
by integrating the gated output from the MCP, with each data-point the average
of 30 pulses. The experiment is run at a repetition rate of 10 Hz to match
that of the Nd:YAG laser pumping the dye laser.

##  III AC Stark shifts - calculation and lineshapes

The strong laser fields required to drive the non-resonance enhanced two-
photon transitions reported here, give rise to ac Stark shifts (light shifts)
of the associated energy levels. These energy shifts lead to broadenings and
asymmetries of the corresponding spectral features. The magnitudes of these
effects depend upon the frequency-dependent dynamic polarizability,
$\alpha(\omega)$, of each of the states between which the two-photon
transitions are driven. In a laser field with an intensity $I_{\text{laser}}$,
and an angular frequency $\omega_{\text{laser}}$, the ac Stark energy shift,
$E_{\text{ac}}$, experienced by an atom in an electronic state with principal
quantum number $n$, total angular momentum $\vec{J}=\vec{L}+\vec{S}$ (where
$\vec{L}$ is the total electronic orbital angular momentum, and $\vec{S}$ is
the total electron spin) and azimuthal quantum number $M_{J}$, is Deissler
_et al._ (2008)

$\displaystyle E_{\mathrm{ac}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha_{|nJM_{J}\rangle}(\omega)\langle
F_{\text{laser}}^{2}\rangle$ (1)

where $\langle F_{\text{laser}}^{2}\rangle=I_{\text{laser}}/(\epsilon_{0}c)$
is the root-mean-square electric field strength of the laser field, with
$\epsilon_{0}$ and $c$ the permittivity of free space, and the speed of light
in vacuum, respectively.

The dynamic polarizability of an individual $|nJM_{J}\rangle$ state can be
calculated by summing the off-resonant single-photon dipole-allowed couplings
induced by the laser field, of this state to all other non-degenerate bound
and continuum states. For a state $|nJM_{J}\rangle$ lying at an energy
$E_{|nJM_{J}\rangle}$ Deissler  _et al._ (2008),

$\displaystyle\alpha_{|nJM_{J}\rangle}(\omega_{\text{laser}})$
$\displaystyle=$
$\displaystyle\frac{2}{\hbar}\sum_{|n^{\prime}J^{\prime}M^{\prime}_{J}\rangle\neq|nJM_{J}\rangle}\Bigg{[}\frac{\Delta\omega}{\Delta\omega^{\,2}-\omega_{\text{laser}}^{2}}$
(2) $\displaystyle\hskip 22.76228pt\times\,|\left\langle
n^{\prime}J^{\prime}M^{\prime}_{J}|\hat{\mu}|nJM_{J}\right\rangle|^{2}\Bigg{]},$

where $\hbar=h/(2\pi)$ ($h$ is Planck’s constant),
$\Delta\omega=(E_{|n^{\prime}J^{\prime}M^{\prime}_{J}\rangle}-E_{|nJM_{J}\rangle})/\hbar$,
and $\hat{\mu}$ is the electric dipole operator.

In electronic states with low values of $n$, such as the initial $2\,^{3}$S1
state in the experiments reported here, the electron is localized close to the
positively charged ion core and high intensity far-off-resonant laser
radiation in the visible region of the electromagnetic spectrum can readily
polarize the atom, giving rise to energy shifts that are proportional to the
laser intensity, $I_{\text{laser}}$. Strong ac Stark shifts observed for
states with low values of $n$ result from a combination of the low density of
electronic states in the surrounding energetic regions, and the general
propensity for bound and continuum states to lie toward higher energies.

In Rydberg states with high values of $n$, the density of states in the
surrounding energetic regions are high, and far-off-resonant visible laser
radiation couples states lying higher in energy, and lower in energy, with
similar coupling strengths. As a result the ac Stark energy shifts of high
Rydberg states in strong high frequency (visible) laser fields are weak and,
as demonstrated in the data presented here, can, to a reasonable
approximation, be neglected when performing experiments with pulsed laser
radiation with bandwidths exceeding 1 GHz.

The determination of the ac Stark shift of a state $|nJM_{J}\rangle$ using
Equation (2) requires the calculation of the dipole matrix elements
$\left\langle
n^{\prime}J^{\prime}M^{\prime}_{J}|\hat{\mu}|nJM_{J}\right\rangle$. These
matrix elements can be calculated using the Wigner-Eckart theorem to express
them in terms of a reduced radial matrix element $\langle
n^{\prime}L^{\prime}||e\hat{r}||nL\rangle$ such that Zare (1988)

$\displaystyle\left\langle
n^{\prime}J^{\prime}M^{\prime}_{J}|\hat{\mu}|nJM_{J}\right\rangle$
$\displaystyle=$
$\displaystyle(-1)^{J+J^{\prime}-M^{\prime}_{J}+L^{\prime}+S+1}\sqrt{(2J^{\prime}+1)(2J+1)}$
(7) $\displaystyle\hskip 42.67912pt\left(\begin{array}[]{ccc}J^{\prime}&1&J\\\
-M^{\prime}_{J}&\Delta
M_{J}&M_{J}\end{array}\right)\left\\{\begin{array}[]{ccc}L^{\prime}&J^{\prime}&S\\\
J&L&1\end{array}\right\\}\langle n^{\prime}L^{\prime}||e\hat{r}||nL\rangle,$

where the curved and curly brackets represent the Wigner 3J and 6J symbols,
respectively. In the particular case of the triplet states of the He atom of
relevance here, the single electron in the 1s orbital has zero orbital angular
momentum. Since this orbital lies 19.8 eV below the next nearest excited level
to which it can be coupled by the laser field, in our treatment of the ac
Stark shift of the 2 3S1 state we consider only the single optically active
electron with orbital angular momentum $\ell$, which is initially in the 2s
orbital. Therefore $L\equiv\ell$ and the reduced radial matrix element
$\langle n^{\prime}L^{\prime}||e\hat{r}||nL\rangle\equiv\langle
n^{\prime}\ell^{\prime}||e\hat{r}||n\ell\rangle$.

From the expression in Equation 7 and using the Numerov method Zimmerman  _et
al._ (1979) to calculate the reduced radial matrix elements, together with
previously published values for the $n$-dependent quantum defects,
$\delta_{n\ell}$, of the triplet states of He ($\delta_{30\mathrm{s}}=0.2967$,
$\delta_{30\mathrm{p}}=0.0683$, $\delta_{30\mathrm{d}}=0.0029$) Martin (1987),
the polarizability of the 2 3S1 state, in an electromagnetic field with a
frequency $\omega_{\text{laser}}/(2\pi)=5.738\times 10^{14}$ Hz ($\equiv
522.48$ nm) – half of the transition frequency associated with the energy
difference between the 2s state and the 25d state – was determined. In doing
this the off-resonant couplings to all $n$p states with values of $n$ in the
range from 2 to 20 were included. For this range of states, the calculated
polarizability of
$\alpha_{2\,^{3}\mathrm{S}_{1}}(\omega_{\mathrm{laser}})=-3.15\times 10^{-39}$
Cm2V-1 (equivalent to a frequency shift of $-8.95~{}$GHz in a laser field of
intensity $10^{13}$ Wm-2) converged to better than one part per thousand.

Using this value for the dynamic polarizability of the 2 3S1 state, the
spectral lineshapes associated with the two-photon transitions studied
experimentally were calculated. This was done by determining the ac Stark
shift of the 2 3S1 state of a spatially distributed ensemble of atoms
interacting with the laser beam. The spatial dimensions of the three-
dimensional ensemble matched the known shape of the supersonic beam (uniform
distributions in $y$\- and $z$-dimensions with width $400~{}\mu$m, much
greater than the size of the laser beam in these dimensions, and a Gaussian
distribution in the $x$-dimension with width $\sigma_{x}=1.5~{}$mm). The
measured laser beam profile and convergence at the intersection region between
it and the atomic beam (a Gaussian transverse laser profile with
$\sigma_{y}\simeq\sigma_{z}\simeq 70\mu$m and gradient
$\text{d}\sigma_{y,z}/\text{d}x=-11.5\mu$m(mm)-1) were then used to generate a
three-dimensional intensity distribution for each laser pulse energy, from
which the ac Stark shifts could be calculated. The resulting ac Stark shifted
transition frequency of each atom in the ensemble was then convolved with the
measured bandwidth of the laser ($\sigma_{L}=\sqrt{2}~{}5.0~{}$GHz at the two-
photon level), and the contributions from all atoms were then summed to
determine the overall lineshape. Further details of the numerical procedure
used to calculate the spectral lineshapes observed experimentally are provided
in the Appendix.

##  IV Results

###  IV.1 Photoexcitation to high Rydberg states

Figure 3: Spectrum of triplet $n$d Rydberg states of He with $|m_{\ell}|=2$
recorded following non-resonance-enhanced single-color two-photon excitation.
The calculated line positions are indicated by the vertical bars at the top of
the figure. The red dashed line represents a function the amplitude of which
depends on $n^{-3}$ that has been fitted to the amplitudes of the spectral
features with $32\leq n\leq 70$. The offset of the zero intensity level close
to the ionization limit is a result of electrical fluctuations during the
measurement. Careful measurements made with and without the laser beam present
confirm that there is no $\text{He}^{+}$ signal recorded at the ionization
limit.

Figure 4: (Color online) The single-color two-photon $|m_{\ell}|=2$ spectrum
encompassing transitions to states from $n=65$ up to the ionization limit. The
spectral features are asymmetric, broadened to higher frequency by the ac
Stark shift of the 2 ${}^{3}{\mathrm{S}}_{1}$ state (see Sec. IV.3). The
decrease in signal strength above $n=85$ is a result of electric field
ionization in the fringe fields of the MCP between the photoexcitation region
and detection region.

A single-color two-photon spectrum of $n$d Rydberg states of helium with
$|m_{\ell}|=2$ is presented in Fig. 3. This spectrum was recorded by scanning
the wavelength of the pulsed dye laser from 520 nm to 524 nm. This range
covers two-photon transitions to Rydberg states ranging from $n=20$ up to the
ionization limit. The highest Rydberg states observed are those with $n\simeq
100$, while states up to approximately $n=75$ are clearly resolved (see Fig.
4). In Figure 3 the calculated positions of the Rydberg states are indicated
by the vertical bars above the spectrum and are in very good agreement with
the positions of the experimentally recorded peaks. Careful inspection of Fig.
3 reveals that each spectral feature is asymmetric, broadened to the high
frequency side. This broadening is a result of the ac Stark shift of the 2
${}^{3}{\mathrm{S}}_{1}$ state (see Section III), and is discussed in more
detail in Section IV.3.

The overall shape of the spectrum presented in Fig. 3 results from several
competing effects. The reduction in the intensity of the spectral features
from $n=30$ up to $n=85$ scales with $n^{-3}$ (see dashed curve), and is a
result of the reduction in the transition dipole moments from the virtual
intermediate state at the single photon level with increasing $n$. The cut-off
in signal above $n=110$, below the ionization limit, is a result of electric-
field ionization of the excited atoms in the fringe fields of the MCP detector
before reaching the position in the detection region where they can be
efficiently collected to contribute to the measured signal. The gradual
increase in the intensity of the spectral features for values of $n$ from 20
to 30 arises from a combination of effects of increasing radiative lifetimes,
transitions driven by blackbody radiation and the cut-off of electric field
ionization at low $n$ in the detection region of the apparatus. In the absence
of blackbody radiation, the $\approx 800~{}$Vcm-1 pulsed electric field used
for Rydberg atom ionization/detection is not strong enough to ionize atoms in
states below $n=(F_{0}/9F)^{1/4}=29$, where
$F_{0}=2R_{\text{He}}hc/ea_{0}=5.14\times 10^{9}~{}$Vcm-1, with
$R_{\text{He}}$ the Rydberg constant for He. However, in the 60 $\mu$s flight-
time of the atoms from the position of photoexcitation to the detection
region, transitions from states with values of $n$ below 29 to those above
permit detection of these lower $n$ states Farley and Wing (1981).

An expanded view of the portion of the two-photon spectrum above $n=65$ is
displayed in Fig. 4. From this data individual features up to approximately
$n=75$ can be clearly identified. States with $n>75$ are not well resolved for
two reasons. Firstly, at high $n$ the states are very closely spaced in
energy, and for $n>94$ the separation of adjacent states becomes smaller than
the experimental linewidth. This width, $\simeq 20~{}$GHz (FHWM) as a result
of ac Stark shifts (see Sec. IV.3), is comparable to the level spacing for
$n\simeq 74$. Secondly, these high-$n$ states possess very large electric
dipole moments and weak stray electric fields in the excitation region will
broaden the corresponding spectral features. For example, a field of only 0.1
Vcm-1 will cause the $n=76$ state to broaden to $\sim 2.5$ GHz. Finally, for
states with $n\geq 85$ the measured signal begins to decrease, reaching zero
intensity at $n\simeq 110$. This cut-off is a result of electric field
ionization in weak fringe electric fields of the MCP, between the excitation
and detection regions. A stray electric field of $\sim 10~{}$Vcm-1 is
sufficient to begin to ionize states with $n\approx 85$ and strong enough to
fully ionize states with $n>110$, where $F_{\text{ion}}=2F_{0}/(9n^{4})~{}$.
Since the range of values of $n$ corresponds well to the extent of the cut-off
region in Fig. 4, we conclude that between the photoionization region and
detection region atoms encounter fringe electric fields from the MCP of $\sim
10~{}$Vcm-1, precluding detection of states above $n\simeq 115$. This drop in
signal at high $n$ also indicates that in the apparatus used here,
$\text{He}^{+}$ ions formed by multiphoton ionization in the excitation
region, or blackbody photoionization are not detected. This confirms that the
recorded spectral features arise from atoms in long-lived Rydberg states and
not from free ions.

###  IV.2 Effects of laser polarization

Figure 5: (Color online) The spectral region in the vicinity of the transition
to the 29d state measured with circularly (blue, dashed curve) and linearly
(red, continuous curve) polarized radiation. With circular polarized radiation
only the transition to the 29d state is observed while with linear polarized
radiation the transition to the 29s state is also seen.

The spectrum in Fig. 3 contains transitions to $n$d Rydberg states with
$|m_{\ell}|=2$ for atoms excited using circularly polarized laser radiation.
From the selection rules for electric dipole transitions, excitation with
linearly polarized light leads to transitions for which $\Delta m_{\ell}=\pm
1$. From the virtual intermediate state with $m_{\ell}=\pm 1$ the second
photon can therefore drive transitions either to Rydberg states with
$m_{\ell}=0$, or with $|m_{\ell}|=2$. The latter transitions will populate
$n$d-states with $|m_{\ell}|=2$, whereas the former will populate $n$s- and
$n$d-states with $m_{\ell}=0$ [Fig. 1(b)]. To confirm the
$|m_{\ell}|$-selective character of the excited states, spectra are presented
in Fig. 5, that were recorded close to $n=29$ using linearly and circularly
polarized laser radiation. In the case of circularly polarized excitation,
only the transition to the 29d state is observed, while in the case of
linearly polarized excitation transitions to the short-lived 29s state, and
the 29d state can be seen.

The transition to the 29s state occurs at a frequency $\sim 80~{}$GHz below
the transition to the 29d state because of the larger value of the 29s quantum
defect ($\delta_{29s}=0.2967$, $\delta_{29d}=0.0029$ Martin (1987)). The
reduced intensity of the transition to the 29d state in the case of excitation
with linear polarized radiation is a result of the smaller angular integral
associated with the transition moments to excited $n$d-states with
$m_{\ell}=0$ than to $n$d-states with $|m_{\ell}|=2$. In the case here, the
spectral intensity associated with the transition to the 29d states with
$|m_{\ell}|=0$ and 2 deiven using linearly polarized radiation is expected to
be 0.61 times that associated with the excitation of the 29d state with
$|m_{\ell}|=2$ using circularly polarized radiation of the same intensity. As
can be seen from the data in Fig. 5, this intensity ratio is consistent with
that observed experimentally. The signal associated with the transition to the
29s state is weaker because of a smaller transition dipole moment than that to
the 29d state and because of its shorter fluorescence lifetime. This reduced
lifetime makes it difficult to compare the observed relative intensity of this
transition to that expected theoretically.

###  IV.3 ac Stark shift

As discussed in Sec III the intense laser radiation employed to drive the non-
resonance-enhanced single-color two-photon transitions presented here causes
the $2\,^{3}\mathrm{S}_{1}$ state to shift to lower energy with increasing
intensity, this shifts the 2s-$n$d transitions to higher frequency. The
spatial spread of the atoms in the supersonic beam,which is much larger than
the laser beam waist, leads to an asymmetric broadening of the observed
transitions, since the atom cloud covers a volume over which there is a
considerable gradient in the laser intensity. Atoms that are in the center of
the laser beam, where the intensity is greatest, experience a large ac Stark
shift, while those at the edge of the beam experience only a small shift.
Since the atoms that are subjected to the greatest intensities are also the
most likely to be excited to a Rydberg state, this would indicate that for a
collimated laser beam the spectral features would be shifted towards the
frequency associated with the greatest ac Stark shift. However, the recorded
spectra show features with maxima close to the unperturbed line-position, with
a long tail towards higher frequency. This line shape is a result of the
intensity gradient of the converging laser beam, which reaches a focus beyond
the atomic beam axis. Atoms that interact with the laser beam at a position
close to the focus can experience strong ac Stark shifts, while atoms located
further from the focus experience weaker shifts. The ensemble of atoms located
further from the focus experiences a smaller range of ac Stark shifts, causing
the overall signal from these atoms to ‘bunch up’ close to the unperturbed
transition frequency, whereas closer to the focus atoms experience a wider
range of intensities, causing the signal from these atoms to be spread out in
frequency.

Figure 6: (Color online) Experimentally measured spectrum of the transition to
the 26d state, recorded with a laser energy of 9.5 mJ per pulse focussed to a
beam width of $\sigma_{x}\simeq\sigma_{y}\simeq 70~{}\mu$m resulting in a
spatially averaged peak laser intensity of $I_{\mathrm{laser}}=2.3\times
10^{13}~{}$Wm-2 (blue, continuous curve), and the calculated line shape (red,
dashed curve)

The cumulative effect of the three-dimensional ensemble of atoms interacting
with the converging laser beam is a spectral line with a maximum close to the
unperturbed transition frequency, and a long tail towards higher frequencies.
An example of this is shown in Fig. 6 where the spectral feature corresponding
to the transition to the $26$d state is shown. In this figure the frequency is
displayed relative to the unperturbed $26$d transition frequency. The
transition was measured with a spatially averaged peak laser intensity on the
atomic beam axis of $2.3\times 10^{13}~{}$Wm-2, corresponding to a maximal ac
Stark shift of $21~{}$GHz. The data show a strong peak located $\sim 12~{}$GHz
above the unperturbed frequency, with a FWHM of 19 GHz and a long tail
extending to $\sim 60~{}$GHz. This long tail results from the contribution of
atoms passing through the beam closer to the focus, where the local peak
intensity is $7.3\times 10^{13}~{}$Wm-2, and the maximum ac Stark shift is 65
GHz. Also shown is the result of a numerical calculation that accounts for the
laser beam profile and the shape of the atomic beam, and is in excellent
agreement with the experimental data.

Figure 7: (Color online) (a) Measured and (b) calculated spectra of the
transition to the 29d state for peak laser intensity
$I_{\text{laser}}=4.6\times 10^{13}~{}$Wm-2 (red, continuous curve),
$3.2\times 10^{13}~{}$Wm-2 (orange, dashed-dotted curve), $1.9\times
10^{13}~{}$Wm-2 (green, long-dashed curve), and $1.1\times 10^{13}~{}$Wm-2
(blue, short-dashed curve).

Figure 7(a) shows spectra close to the transition to the 29d state recorded
with a range of spatially averaged peak laser intensities. With a low
intensity on the atomic beam axis of $I_{\text{laser}}=1.1\times
10^{13}~{}$Wm-2 the spectral profile is slightly asymmetric, and is centered
close to the unperturbed transition frequency. As the intensity is increased
the profile becomes gradually more asymmetric, with the tail shifting out to
higher frequencies. The maximum also moves gradually to higher frequencies,
but remains close to the unperturbed transition frequency. Figure 7(b) shows
calculated line profiles which are in very good agreement with the
experimental data, confirming the accuracy of the numerical procedure used to
model the excitation process. In the spectra recorded at high laser
intensities, the fluctuations in the He+ ion signal exceeds the shot-to-shot
fluctuations in the experiment. The observed fluctuations (most clearly
displayed in Fig. 7(a) in the data recorded with $I_{\text{laser}}=4.6\times
10^{13}~{}$Wm-2) appear to result from interference between different pathways
in the excitation process, and will be investigated in more detail in the
future.

The spectral intensity of the non-resonance-enhanced two-photon transitions
studied here are expected to be proportional to $I_{\mathrm{laser}}^{2}$
because of the intensity-dependence of the two-photon process. However, the
features in the recorded spectra are broadened by the intensity-dependent ac
Stark shift of the $2\,^{3}\mathrm{S}_{1}$ state. Therefore the integral of
the spectral intensity is the quantity dependent upon
$I_{\mathrm{laser}}^{2}$, and not simply the spectral amplitude. This can be
seen in Fig. 8 where the integral of the spectral intensity over the frequency
range encompassed in Fig. 7 is displayed as a function of the spatially
averaged peak laser intensity on the atomic beam axis, together with a
quadratic function which confirms the $I_{\text{laser}}^{2}$ dependence of the
integrated signal.

Figure 8: (Color online) Integrated spectral signal for the measured
transitions to the 29d state shown in Fig. 7 as a function of the peak laser
intensity along the atomic beam axis, $I_{\mathrm{laser}}$. Also shown is a
quadratic function that has been fit to the experimental data (purple
continuous curve).

Because the ac Stark shift is a feature of the interaction of the 2 3S1 state
with the laser field, and does not affect the Rydberg states, similar
asymmetric lineshapes are seen for all recorded two-photon transitions. This
is also the reason why all the spectral features in Fig. 3 are shifted
slightly to higher frequency from the calculated line positions. The peak
widths are derived from the overlap of the atoms with the laser beam (as
described in the Appendix), and are thus independent of the principal quantum
number of the Rydberg states. This is illustrated in Fig. 9 which shows the
$n=23$, 40 and 50 spectral features (scaled to equal amplitude for ease of
comparison) overlaid, clearly demonstrating the $n$-independence of the line
shape.

Figure 9: (Color online) Measured spectral features associated with the $n=23$
(red, continuous curve), 40 (blue, dashed-dotted curve) and 50 (green, dashed
curve) Rydberg states, scaled to equal amplitude for ease of comparison. The
spectral features were recorded with the same laser intensity, and clearly
demonstrate the $n$-independence of the spectral line profile.

###  IV.4 Electric field

The application of a homogeneous dc electric field in the excitation region
mixes Rydberg states with the same value of $m_{\ell}$ but with values of $l$
differing by $\pm 1$.

Figure 10: (a) Spectra of $|m_{\ell}|=2$ Rydberg states of He with $n=25$–27,
recorded with a spatially averaged peak laser intensity of
$I_{\text{laser}}=2.3\times 10^{13}$Wm-2 in the presence of dc electric fields
of (i) 0 Vcm-1, (ii) 28 Vcm-1, (iii) 56 Vcm-1, (iv) 111 Vcm-1 and (v) 154
Vcm-1.(b) Calculated spectra for the same conditions as those of the
experimental data in (a).

As the applied electric field is increased the resulting Stark states in a
given $n$-manifold separate in energy. Figure 10(a) shows measured spectra of
the transitions to the $n=25$, 26 and 27 states in a range of dc electric
fields applied in the excitation region. In zero field the lines are
asymmetric, with widths of $\sim 20$ GHz (FWHM) resulting from the ac Stark
shift (Sec. IV.3). As the field is increased to 28 Vcm-1 the lines start to
broaden symmetrically as the Stark states begin to separate in energy.
Increasing the field further to 56 Vcm-1 the energy splittings of the Stark
states further increase, with the features becoming symmetric as the dc Stark
shifts in the Rydberg states dominate the ac Stark shifts of the 2 3S1 state.
With this applied field strength the Stark states begin to be resolved.

As the electric field is increased further the dc Stark shifts continue to
increase and the spectral features become more symmetric. In a field of 111
Vcm-1, the Stark state resolution improves and the structure is clearer. With
the maximum applied field of 154 Vcm-1 the Stark states are clearly separated
with each individual state identifiable. Note that in this field the $n=26$
and $n=27$ Stark manifolds are sufficiently broad that they overlap. This
field is greater than the Inglis-Teller field of $\sim 144~{}$Vcm-1 for $n=26$
Gallagher (1994). Fig. 10(b) shows calculations corresponding to the
experimental data in Fig. 10(a). These spectra were calculated by convolving
the calculated line profile arising from the ac Stark shift of the
$2\,^{3}\mathrm{S}_{1}$ state with the spectral intensities of the transitions
to each individual Stark state determined by diagonalization of the
Hamiltonian matrix Zimmerman  _et al._ (1979). Fig. 11 shows the spectrum at
$n=26$ in a field of 154 Vcm-1 in more detail. In this spectrum the 24 Stark
states can be clearly identified. Also shown is the calculated spectrum which
agrees well with the experimental data. The outer Stark states are not
completely resolved because they are more susceptible to electric field
inhomogeneities in the excitation region. Since the transition strengths to
these states is weakest their spectral appearance is also more susceptible to
shot-to-shot laser intensity variations, which slightly wash out their
structure.

Figure 11: (Color online) Spectrum of the $n=26$ Stark manifold of He in a dc
electric field of 154 Vcm-1 (blue continuous curve), shown with the calculated
spectrum (red, dashed curve). Each of the 24 Stark states are labelled with
the index $k$ which ranges from $-(n-|m_{\ell}|-1)$ to $+(n-|m_{\ell}|-1)$ in
steps of 2. The lowest-energy Stark states of the $n=27$ manifold are visible
toward the high-frequency end of the spectrum.

These measurements show that by applying a dc electric field in the excitation
region it is possible to split the Stark states sufficiently to allow
selective excitation using this non-resonance-enhanced two-photon scheme. The
limiting factor in the resolution is the ac Stark shift of the initial state.

##  V Application to Doppler-free photoexcitation of Rydberg states of
positronium

In addition to being of importance for precision spectroscopic studies and the
preparation of long-lived high-angular-momentum Rydberg states, non-resonance
enhanced Doppler-free two-photon excitation schemes are expected to be
particularly well suited to the photoexcitation of Rydberg states of
positronium (Ps) with high spectral resolution.

Positronium atoms are most efficiently created by implanting positrons into
solid state materials. Various mechanisms for Ps formation exist, which are
either thermal in nature Mills (1979), or rely on other energy transfer
processes Cassidy  _et al._ (2010), and typically produce atoms with
velocities on the order of $10^{5}$ ms-1. This leads to very large Doppler
broadenings of single photon transitions (e.g., increasing the 50 MHz natural
width of the 1s–2p transition to $\sim 500~{}$GHz Cassidy  _et al._ (2012)).
Such large Doppler widths preclude the selective excitation of individual Ps
Rydberg-Stark states using previously demonstrated two-color two-photon
excitation schemes Jones  _et al._ (2014). However, in order to overcome some
of the challenges associated with performing gravitational Mills and Leventhal
(2002) or spectroscopic Fee  _et al._ (1993) measurements on Ps it is
desirable to photoexcite them to well-defined Rydberg-Stark states with long
radiative lifetimes, and large electric dipole moments, so that they can be
guided, decelerated and manipulated using inhomogeneous electric fields
Vliegen  _et al._ (2004); Vliegen and Merkt (2006); Hogan and Merkt (2008);
Hogan  _et al._ (2011). By implementing two-photon excitation schemes of the
kind demonstrated here, but using two counter-propagating laser beams in a
Doppler-free configuration Hänsch  _et al._ (1975), a spectral resolution
limited by the combination of the laser bandwidth and the ground state ac
Stark shift will be achievable. This approach would provide sufficient
resolution to selectively prepare individual Rydberg-Stark states of
positronium (as demonstrated for He in Fig. 11).

In the experiments reported here, using circularly polarized laser radiation
at a wavelength of $\sim 520~{}$nm, with a spatially averaged peak intensity
of $\sim 5\times 10^{13}~{}$Wm-2, approximately 10% (determined by comparison
with the measured He+ ion signal recorded by single photon excitation at
saturation) of the metastable helium atoms that interact with the laser beam
were excited from the 2 ${}^{3}\text{S}_{1}$ state to $n$d Rydberg states,
close to $n=25$. In these non-resonance-enhanced single-color two-photon
excitation processes the transition rates depend on the sum of the off-
resonant (electric dipole allowed) couplings at the single-photon level,
between the initial state and all available non-degenerate bound and continuum
intermediate states, and the further off-resonant couplings from these
intermediate states to the excited Rydberg state Haas  _et al._ (2006). By
summing the contributions of all far off-resonant intermediate states up to
$n=50$ we have determined the approximate transition rate from the 2
${}^{3}\text{S}_{1}$ state to the triplet 25d state in He to be $2\times
10^{-5}~{}$Hz/(Wm-2).

Using a Doppler-free single-color two-photon excitation scheme to drive
transitions from the 1s ground state of ortho-Ps to $n$d Rydberg states will
require circularly polarized laser radiation at $\sim 364~{}$nm. From the
exact calculations for two-photon transitions in hydrogenic systems by Haas et
al. Haas  _et al._ (2006) the corresponding transition rate for the
1s$\rightarrow$25d transition in Ps is also $\sim 2\times
10^{-5}~{}$Hz/(Wm-2). The similarity of this number to the corresponding rate
for the two-photon transitions in He reported here indicates that similar
excitation efficiencies can be expected in Ps. In addition, the frequency-
dependent ground state polarizability of Ps, calculated using the methods
outlined in Sec. III for laser radiation with a photon energy appropriate for
driving the Doppler-free two-photon 1s$\rightarrow$25d transition is
$+1.4\times 10^{-39}~{}$Cm2V-1 (equivalent to a frequency shift of +4 GHz in a
laser field of intensity $10^{13}~{}$Wm-2), with a magnitude of approximately
half that of the $2\,^{3}\mathrm{S}_{1}$ state in He. It is expected that this
lower dipole polarizability and hence reduced ac Stark energy shift at a given
laser intensity will also enhance the excitation efficiency in Ps.

The laser intensities used to drive single-color two-photon transitions in He
were achieved by focussing the output of a pulsed dye laser to a width of
$\sigma\sim 70~{}\mu$m. While it is possible to employ greater laser powers
using commercially-available dye laser systems, a practical upper limit to the
dimension of the beam size required to obtain the appropriate intensity is
expected to be $\sigma\sim 200~{}\mu$m, corresponding to a FWHM of $\sim
0.5~{}$mm. The production of a $\sim 1~{}$mm diameter positron (and hence also
Ps) pulse from an accumulator can be achieved with moderate magnetic fields of
500 G Cassidy  _et al._ (2006). For experiments in which such fields are
acceptable this means that approximately half of the available atoms could be
addressed, and $\sim 5\%$ excited. However, because Ps velocities are
generally high, ionization of Rydberg states by motionally induced electric
fields can lead to significant losses Cassidy  _et al._ (2012); Hogan (2013).
For magnetic fields of 50 G, such losses are largely mitigated, but then the
positron beam size will be $\sim 3~{}$mm, and the overlap with a 0.5 mm FWHM
laser beam would be about 15%, leading to an overall excitation efficiency of
1.5%. A typical Ps pulse in an apparatus operating at 1 Hz would contain $\sim
10^{6}$ atoms, so that more than $10^{4}$ Rydberg atoms could be produced per
second.

An alternative approach is to remoderate the positron beam Mills (1980). In
this arrangement the positrons would be extracted from the trap magnetic field
completely, and focussed into a solid material (the remoderator). Non-
conservative interactions in the remoderator make it possible to circumvent
phase-space limitations, and the re-emitted positron beam brightness may be
considerably enhanced. For typical trap parameters this would result in a loss
of $\sim 90\%$ of the particles, but with a spot size of less than
$100~{}\mu$m. In this case the initial $10^{6}$ positrons would again result
in $\sim 10^{4}$ Rydberg atoms, but this time in zero magnetic field, and with
a spatial spread smaller than the laser beam size, so that significantly
smaller laser pulse energies ($\sim$1 mJ) would be required.

A further reduction in the required laser energy can be achieved by using a
laser with a reduced bandwidth. Seeding a pulsed dye laser amplifier, with,
for example a narrow-band continuous wave (cw) diode laser tuned to the
desired fundamental wavelength, will result in a reduction in the laser
bandwidth by a factor of $\sim 20$. Using such a transform-limited pulse will
reduce the energy required for a similar excitation efficiency by an order of
magnitude. The use of transform-limited laser radiation and a brightness-
enhanced re-moderated positron source could reduce the energy required to
drive two-photon transitions in Ps to $\sim 100~{}\mu$J per pulse.

##  VI Conclusions

Non-resonance-enhanced single-color two-photon excitation to long-lived
Rydberg states in He has been demonstrated by driving transitions from the
metastable $2\,^{3}\text{S}_{1}$ state to Rydberg states in the range from
$n=20$ up to $n\simeq 100$. The effect of the intensity and polarization of
the laser radiation on the excitation efficiency and spectral profiles has
been investigated in detail. The spectral features in the recorded spectra
were observed to be broader than the bandwidth of the laser, and asymmetric
with a long tail to high frequency. It has been shown by comparison of the
experimental data with the results of numerical calculations that this is a
result of the ac Stark shift of the $2\,^{3}\text{S}_{1}$ state in the laser
field, which non-resonantly couples the $2\,^{3}\text{S}_{1}$ state to other
bound states. The polarizability of the $2\,^{3}\text{S}_{1}$ state was
calculated by summing all off-resonant couplings to $n$p states in the range
from $n=2$ up to 20. With detailed measurements of the laser beam profile this
allowed the calculation of the spectral lineshapes, which were found to be in
excellent agreement with those recorded experimentally.

Photoexcitation in dc electric fields was also performed, and with a field
strength of 154 Vcm-1 individual Rydberg-Stark states were clearly identified.
This selective population of individual Rydberg-Stark states in a non-
resonance-enhanced multi-photon photoexcitation scheme as demonstrated is
important for the preparation of atoms and molecules in long-lived high-
angular-momentum Rydberg states for deceleration and trapping experiments
using inhomogeneous electric fields.

In particular, the application of similar schemes to Ps has been discussed.
Based on the measured excitation rates in He under experimental conditions
similar to those expected in experiments with Ps, we conclude that it is
feasible to excite ground state Ps atoms to Rydberg states with $n\simeq 25$
in Doppler-free two-photon transitions with sufficient resolution to address
individual Stark states. This will enable new experiments designed to control
Rydberg Ps atoms with inhomogeneous electric fields via their large electric
dipole moments.

###### Acknowledgements.

This work has been supported financially by the Department of Physics and
Astronomy, and the Faculty of Mathematical and Physical Sciences at University
College London, and by the Engineering and Physical Sciences Research Council
under grant number EP/K028774/1. We are grateful to the Engineering and
Physical Sciences Research Council Laser Loan Pool (loan number 13250005) for
the loan of the laser system used in this research.

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*

##  Appendix A Lineshape simulation

The calculation of the lineshapes associated with the spectral features
observed in the experiments reported here was particularly important in
understanding the role of ac Stark shifts, and the effects of the properties
of the laser and atomic beams in the experiments. The features in the spectra
presented are broadened as a result of ac Stark shifts of the
$2\,^{3}\text{S}_{1}$ state, leading to asymmetric lineshapes with widths
greater than those expected from the bandwidth of the laser. The precise shape
of these features was determined by the overlap of the spatial distribution of
atoms in the supersonic beam with the intensity distribution of the focussed
laser beam.

The laser beam was focussed such that where it intersected the atomic beam
axis its intensity profile was Gaussian with a width
$\sigma_{y}(x=0)\simeq\sigma_{z}(x=0)=70~{}\mu$m, where $x$ is the direction
of propagation of the laser beam, and $z$ is the direction of propagation of
the atomic beam (see Fig. 1). Because of the limited precision, $\pm 1$ mm,
with which the crossing point of the atomic beam and the laser beam in the
$x$-dimension could be determined in the experiments, the error on the
experimental value for $\sigma_{y}(x=0)$ and $\sigma_{z}(x=0)$ was $\pm
11~{}\mu$m. The numerical value of $70~{}\mu$m used in the simulation of the
lineshapes represented the best fit to the experimental data within the range
of values encompassed by these error bars.

The spatial distribution of atoms in the supersonic beam was defined by the
geometry of the pulsed valve and the 2 mm-diameter skimmer. At the position
where the atomic beam intersected the laser beam it occupied a significantly
greater volume than the laser beam. Traveling with a mean forward speed of
$v_{z}=1950~{}$ms-1, the distance the atoms travelled during the time that the
laser beam was present was negligible, and it was therefore assumed that the
atoms were stationary during excitation. The calculations presented here
involved the generation of ensembles of atoms with uniform distributions in
the $y$ and $z$ dimensions, perpendicular to the direction of propagation of
the laser beam, and a Gaussian distribution (with a width
$\sigma_{x}=1.5~{}$mm) in the $x$-direction, along which the laser beam
propagated. The width of this Gaussian distribution was determined from the
geometry of the experimental apparatus, in particular the relative positions
of the pulsed valve, the skimmer and the photoexcitation region. As the
transverse temperature of the atomic beam was not precisely known, the
numerical value of $\sigma_{x}$ employed in the lineshape simulations was
adjusted within a range of 15% of the value determined geometrically to
realize the best fit of the calculated lineshapes with the those measured in
the experiments. In the $y$\- and $z$-directions the spatial extent of the
ensemble of atoms was sufficiently large when compared to the size of the
converging laser beam that the assumption of a uniform distribution of atoms
was reasonable. Including the correct distribution of atoms in the
$x$-direction was crucial for the accurate calculation of the experimental
line shapes.

Figure 12: (Color online) A schematic diagram showing the overlap of the
atomic cloud (blue points) with the converging laser beam (green shaded
region). The atomic distribution in the $xz$-plane, where the distribution has
a Gaussian shape along $x$ with width $\sigma_{x}=1.5~{}$mm and centroid
$x_{0}=0~{}$mm. The black lines show the positions of the $yz-$planes
described in the text.

A three-dimensional laser intensity distribution was generated by making
detailed beam profile and energy measurements. Every atom in the calculations
was assigned a laser intensity depending on its position,
$I_{\text{laser}}(x,y,z)$ from which the ac Stark shift was then calculated.
This shift was then convolved with the laser bandwidth, a Gaussian function
with width $\sigma=5.0\sqrt{2}~{}$GHz (where the factor of $\sqrt{2}$ accounts
for the two-photon nature of the transitions). The two-photon transition
strength for each atom was then included by multiplying the amplitude of each
convolved single atom spectrum with $I_{\text{laser}}(x,y,z)^{2}$. The
validity of this approach is based upon the $I_{\text{laser}}(x,y,z)^{2}$
dependence of the integrated signal associated with the transition to each
individual Rydberg state, an example of which is presented in Fig. 8. The
lineshapes for each atom were then summed to generate the overall lineshape.
The calculated lineshapes are in excellent agreement with those in the
experimental spectra, demonstrating that the line profiles resulting from the
ac Stark shift of the $2\,^{3}\text{S}_{1}$ state are therefore, under the
conditions of the experiment, independent of $n$ (see Fig. 9), while the exact
shape of the spectral features was determined by the spatial overlap of the
atomic and laser beams.

Figure 13: (Color online) (a) Line profiles calculated for atoms located in
$yz$-planes with positions in the $x$-dimension as indicated. These planes are
shown as black lines in Fig 12. (b) The line profiles generated by summing the
spectral contributions from a continuous three-dimensional atomic ensemble.

One surprising feature of the recorded lineshapes is the strong contribution
from atoms that experience only a weak ac Stark shift. This is particularly
striking in the data in Fig. 7, where, as the laser energy is increased the
features broaden as the ac Stark shifts increase, while the position of the
maximum of each feature remains close to the unperturbed line position, and
the signal from weakly-shifted atoms is observed to increase with laser
intensity. This spectral behaviour is explained by considering the
contributions to the overall lineshape from atoms at different positions in
the $x$-dimension. In Fig. 12 a schematic diagram highlighting the overlap of
the atoms with the laser beam in the $xz-$plane is presented. The green shaded
region represents the area enclosed by the $2\sigma$ full-width of the laser
beam as it converges in the $x$-direction, and the blue dots represent the
cloud of metastable He atoms in the 2 ${}^{3}\text{S}_{1}$ state at the
instant that the laser beam is present. The spatial contribution to the
spectral profiles are seen by calculating the $29$d lineshape associated with
atoms in a number of $yz$-planes located at selected positions in the
$x$-dimension (black vertical lines in Fig. 12). Fig. 13(a) shows the
lineshapes calculated for atoms in planes located at $x=+2.5$, $+1.5$, $+0.5$,
$-0.5$, $-1.5$ and $-2.5~{}$mm. Each plane contributes an asymmetric
lineshape, with a maximum at a frequency greater than the unperturbed line
position, close to the transition frequency corresponding to the greatest ac
Stark shift in that plane. This shape stems from the fact that in a given
plane the greatest signal always comes from those atoms near the centre of the
laser beam, where the excitation probability is highest. These atoms also
experience the greatest ac Stark shifts, thus generating the asymmetric
lineshapes, shifted to higher frequency. This effect is seen most clearly in
Fig. 13(a) in the line profile from the atoms in the $x=+2.5~{}$mm plane,
which is located closest to the laser focus, where the peak intensity
$I_{\text{laser}}(2.5~{}\text{mm},0,0)=1.5\times 10^{14}~{}$Wm-2 (resulting in
a maximal ac Stark shift of 133 GHz). This lineshape is very asymmetric, with
a maximum $\sim$130 GHz above the unperturbed line position. In planes further
from the laser focus the peak intensities are lower, and the lineshapes shift
towards the unperturbed position, becoming more symmetric as they do so. Since
the peak intensity varies quadratically in the $x$-dimension the contributions
from the planes located further from the laser focus bunch up and overlap
close to the unperturbed line position, creating a large spectral peak near
this frequency, while the atoms that are positioned closer to the laser focus
experience a wide range of intensities, creating a broad spectral signal
spread over a wide range of frequencies. This is clearly seen in the
calculated spectral profile shown in Fig. 13(b), which was calculated with a
continuous three-dimensional atomic ensemble.