A tunable CW UV laser with $<$35 kHz absolute frequency instability for
precision spectroscopy of Sr Rydberg states

Elizabeth M. Bridge∗, Niamh C. Keegan, Alistair D. Bounds, Danielle Boddy,
Daniel P. Sadler, and Matthew P. A. Jones

Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham
University, Durham DH1 3LE, UK

<EMAIL_ADDRESS>
###### Abstract

We present a solid-state laser system that generates over 200 mW of
continuous-wave, narrowband light, tunable from 316.3 nm – 317.7 nm and 318.0
nm – 319.3 nm. The laser is based on commercially available fiber amplifiers
and optical frequency doubling technology, along with sum frequency generation
in a periodically poled stoichiometric lithium tantalate crystal. The laser
frequency is stabilized to an atomic-referenced high finesse optical transfer
cavity. Using a GPS-referenced optical frequency comb we measure a long term
frequency instability of $<$ 35 kHz for timescales between $10^{-3}$ s and
$10^{3}$ s. As an application we perform spectroscopy of Sr Rydberg states
from $n\,=\,37$ – 81, demonstrating mode-hop-free scans of 24 GHz. In a cold
atomic sample we measure Doppler-limited linewidths of 350 kHz.

OCIS codes: (020.5780) Rydberg states; (140.0140) Lasers, solid-state;
(140.3600) Lasers, tunable; (140.3610) Lasers, ultraviolet.

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##  1 Introduction

Tunable continuous-wave (CW) lasers are the workhorse for quantum technologies
based on ultracold atoms and ions. While the visible and near-infrared
spectral regions are readily accessible by solid-state laser systems, the
extension of these techniques to the near-ultraviolet region remains an active
area of research. Many ionic and atomic species have their principal cooling
transitions in the near-UV, including species such as Be+ [1] and Hg [2] that
are relevant for optical frequency standards.

In addition to laser cooling, narrowband CW excitation enables precision
spectroscopy of highly excited Rydberg states. Rydberg atoms have high
principal quantum number $n$ and exhibit large atomic polarizabilities, which
scale as $n^{7}$, thus providing tunable long-range interactions and high
sensitivity to electric fields. With sufficiently narrowband excitation the
long-range interactions between pairs of Rydberg atoms can be observed. In
particular, if the interaction shift exceeds the linewidth of the excitation,
then a blockade effect occurs [3] and a single excitation is shared between
all the atoms within a blockade radius
$R_{\mathrm{B}}=\left(C_{6}/\hbar\Omega\right)^{1/6}$, where $C_{6}$ is the
van der Waals interaction strength and $\Omega$ is the excitation Rabi
frequency.

The Rydberg blockade effect has been used to demonstrate quantum effects in
neutral atoms, including cooperativity [4], entanglement [5], a controlled-NOT
gate [6] and coherent many-body Rabi oscillations [7]. Other applications of
precision Rydberg spectroscopy include electrometry [8, 9, 10], production of
single photons [11] and the creation of ultra-long range molecules [12, 13,
14, 15]. By optically “dressing” the ground or a metastable state with a
Rydberg state, further potential applications include demonstration of a
super-solid phase of matter [16, 17] and spin-squeezing in optical lattice
clocks for increased signal-to-noise beyond the standard quantum limit [18].
Keating et al. [19] have recently demonstrated Rydberg dressing in Cs atoms
for studying low-decoherence implementation of a controlled-Z gate.

So far, most of these experiments have relied on a two-step excitation scheme
via a short-lived, low-lying intermediate state. While this has the advantage
that both lasers may be in the visible or near-IR spectral regions, the short
lifetime of the intermediate state introduces additional decoherence.
Electromagnetically induced transparency [20] or off-resonant two-photon
excitation [21] can reduce this effect, but at the expense of a lower Rydberg
excitation efficiency. More recently, experiments in the alkali atoms have
used direct excitation from the ground state, which requires tunable CW light
at 297 nm (Rb) [22] and 319 nm (Cs) [23].

An alternative approach to reducing the decoherence of the two-photon
excitation scheme is provided by two-electron atoms such as Sr. Here,
intercombination transitions give access to long-lived intermediate states.
These provide a coherent two-photon excitation route to the triplet Rydberg
states, including the 5s$n$s 3S1 series which have near-isotropic
wavefunctions and interaction strengths [24]. Isotropic interactions relax the
geometric constraints of the atomic sample for many of the proposed Rydberg
and dressing experiments, and could be key for demonstrating a supersolid
phase of matter [16, 17]. Moving to two-electron atoms also enables
applications, such as spin-squeezing [18] and blackbody thermometry [25], in
optical lattice clocks. In Sr this requires tunable radiation in the 316 nm –
322 nm region [Fig. 1(a)]. Recently, the first experiments using this
intercombination excitation route in Sr have led to the observation of Sr
Rydberg molecules [15] and the study of Autler-Townes spectra in a dense Sr
Rydberg gas [26]. Combined with the ability to optically manipulate the second
valence electron for trapping [27], state detection [28, 29] and imaging [10],
intercombination line Rydberg excitation provides a powerful toolbox for
exploring many of these ideas.

The conventional approach to generating tunable CW light in the near-UV region
of the spectrum has been to frequency-double a dye laser [22, 30, 31]. Solid-
state systems using sum-frequency generation (SFG) have been developed for
particular cooling transitions, offering high power and narrow linewidth [23,
32, 33], but a tuning range of only a few GHz. Recently these techniques have
been extended to lasers for Rydberg spectroscopy [23], where the restricted
tuning range means that only a narrow range of principal quantum numbers are
accessible.

In this paper we describe a solid-state, narrow-linewidth UV laser system
tunable from 316.3 nm – 317.7 nm and 318.0 nm – 319.3 nm, which enables the
two-step excitation of Sr Rydberg states with principal quantum number in the
range $n=35\rightarrow$ $>$ 300 from two different intermediate states that
are separated by 5.6 THz. We achieve a stable output power of $>200\,$mW. A
dual stabilization scheme based on a high-finesse tunable optical transfer
cavity and an atomic reference provides a long-term absolute frequency
instability of $<35\,$kHz, measured relative to a GPS-stabilized optical
frequency comb. The utility of the laser for Rydberg physics is demonstrated
by the spectroscopy of high-lying triplet Rydberg states in Sr.

##  2 Laser construction and characterization

The goal was to develop a laser system suitable for exciting to a large range
of Rydberg states in the triplet series of Sr, via two different intermediate
states [Fig. 1(a)]. A narrow laser linewidth is required in order to achieve a
large Rydberg blockade radius, and high power is necessary for off-resonant
Rydberg dressing since the coupling matrix elements are weak. With these
criteria in mind, we have built a high power, narrow-linewidth, widely-tunable
laser system for excitation to the triplet Rydberg series from both the 5s5p
3P0 and 5s5p 3P1 intermediate states. The full range of wavelengths
achievable, and the corresponding Rydberg states that can be addressed, are
given in Table 1.

Fig. 1: (a) Energy level diagram of relevant transitions in atomic Sr. The
primary cooling transition at 461 nm is used for Zeeman slowing and cooling
the atoms in a magneto-optical trap (MOT). Two-photon excitation with the 461
nm and 413 nm lasers drives atoms up to the singlet Rydberg series, as was
used for our previous work [10, 28, 29]. The second-stage cooling transition
at 689 nm is used to cool the atoms to $\sim\,1$ – $10\,\upmu$K in a MOT. Two-
photon excitation with 689 nm and 319 nm, or 698 nm and 316 nm drives the
atoms up to the triplet Rydberg series. (b) Schematic of the laser system. A
PPSLT crystal is used to sum the frequencies of two infra-red lasers at
wavelengths of $\lambda_{1}$ and $\lambda_{2}$. The resulting light at
$\lambda_{3}$ is frequency doubled to produce $>\,200\,$mW in the UV
($\lambda_{4}$). The laser frequency is locked to an optical transfer cavity
stabilized to the 5s2 1S0 $\rightarrow$ 5s5p 3P1 intercombination line in Sr
at 689 nm. A wideband electro-optic modulator (EOM) is used to bridge the
frequency gap between the cavity mode and the Rydberg transition. The laser
frequency is measured on a GPS-referenced optical frequency comb. (EDFA = Er-
doped fiber amplifier, YDFA = Yb-doped fiber amplifier. $D_{1}-D_{4}$ =
dichroic mirrors).

Our approach, shown in Fig. 1(b), is based on that described in [32], and is
similar to that used in [23, 33]. Two infra-red seed lasers are amplified and
combined using sum frequency generation (SFG) in a periodically-poled
stoichiometric lithium tantalate (PPSLT) crystal to produce red light at
$\sim$ 633 nm – 639 nm. Subsequently, UV light is produced using resonantly-
enhanced second harmonic generation (SHG). A small amount of the red light is
separated off for locking to an atomic-referenced optical cavity, and for
independent absolute frequency measurements using a GPS-referenced optical
frequency comb.

The key challenge compared to previous work was to extend the SFG technique to
meet our tuning requirements. Firstly, we exploit the wide gain bandwidth of
Er- and Yb-doped fiber amplifiers to achieve high fundamental power over a
wide range of wavelengths. We employ two commercial fiber amplifier systems;
an Er-doped amplifier (Manlight) that produces up to 5 W over the wavelength
range $\lambda_{1}$ = 1549 nm to 1565 nm, and a Yb-doped amplifier (Nufern)
that produces up to 10 W over $\lambda_{2}$ = 1064 nm to 1083 nm. Table 1
summarizes how we exploit these wavelength ranges to address the Rydberg
series of interest.

Table 1: Wavelengths achievable with our laser system and the corresponding principal quantum numbers $n$ of the Sr Rydberg states we can access. Wavelengths $\lambda_{1}$ and $\lambda_{2}$ are combined using SFG to produce $\lambda_{3}$, which is frequency doubled to give the desired wavelength, $\lambda_{4}$.  | $\lambda_{1}$
---
(nm)
| $\lambda_{2}$
---
(nm)
| $\lambda_{3}$
---
(nm)
| $\lambda_{4}$
---
(nm)
| Intermediate
---
state
| Rydberg
---
$n$
1549.0 – 1565.0 | 1069.5 | 632.7 – 635.3 | 316.3 – 317.7 |  3P0 | 32 – 320
1549.0 – 1565.0 | 1079.0 | 636.0 – 638.7 | 318.0 – 319.3 |  3P1 | 35 – $\infty$

The 5.6 THz frequency shift required to change between the two intermediate
states is achieved by switching between two different seed lasers centered at
$\lambda_{\mathrm{2a}}\,=$ 1069.5 nm, or $\lambda_{\mathrm{2b}}\,=$ 1079.0 nm.
Both seed lasers are narrow-linewidth ($\sim$ 10 kHz), high stability fiber
lasers (NP Photonics) with polarization-maintaining single-mode optical fiber
outputs. The advantage of switching the seed laser in this way is that it
requires no significant optical realignment. Tunability within each Rydberg
series is achieved by using a highly tunable external cavity diode laser
(ECDL) (Toptica Photonics) as the other seed laser ($\lambda_{1}$). The laser
linewidth is somewhat broader than available fiber laser technology ($\sim$
100 kHz), but the frequency can be tuned over the whole usable gain bandwidth
of the fiber amplifier.

The final step is to ensure that the SFG process is quasi-phase-matched across
these wavelength ranges. The phase mismatch $\Delta k(T)$ can be calculated
[34] by considering the extraordinary refractive index $n_{e,\lambda_{i}}(T)$
of the crystal [35] for a particular temperature $T$ and vacuum wavelength
$\lambda_{i}$ (where $i=1,2,3$ corresponds to the two input, and one output,
beams respectively), and the poling period of the crystal $\Lambda_{c}(T)$:

$\frac{\Delta
k\left(T\right)}{2\pi}=\frac{n_{e,\lambda_{3}}\left(T\right)}{\lambda_{3}}-\frac{n_{e,\lambda_{2}}\left(T\right)}{\lambda_{2}}-\frac{n_{e,\lambda_{1}}\left(T\right)}{\lambda_{1}}-\frac{1}{\Lambda_{c}(T)}\mbox{.}$
(1)

We considered both PPSLT and periodically-poled lithium niobate (PPLN)
crystals. The latter offers a higher nonlinear coefficient
($d_{33\mathrm{LT}}/d_{33\mathrm{LN}}\approx 0.67$) [36], at the expense of
higher light-induced infra-red absorption and a lower photo-refractive damage
threshold [36, 37], and therefore we chose to work with PPSLT. Inserting the
relevant material parameters [35, 38] in to Eq. (1) we find that five equally
spaced poling periods between 13.05 $\upmu$m and 13.45 $\upmu$m will provide
quasi-phase-matching across the full Rydberg series for crystal temperatures
in the range 100 ∘C – 140 ∘C.

The PPSLT crystal we use (Laser 2000) has a length of $L$ = 50 mm and a
thickness of 1 mm. Distributed across its 15 mm width are the five independent
periodically-poled channels. The crystal is mounted in a temperature
stabilized oven (Laser 2000), which allows for fine tuning of the refractive
index and poling period. The IR beams are independently mode-matched using
lenses of focal length $f_{\mathrm{1a}}=-50$ mm and $f_{\mathrm{1b}}=+50$ mm
in the $\lambda_{1}$ beam, and $f_{\mathrm{2a}}=-100$ mm and
$f_{\mathrm{2b}}=+100$ mm in the $\lambda_{2}$ beam (as shown in Fig. 1(b)).
The beams are subsequently overlapped on a dichroic optic $D_{1}$, and make a
single pass of the PPSLT crystal. Optimum frequency conversion should occur
when the confocal parameter $b$ of the input beams are related to the length
of the nonlinear crystal $L$ by $\xi=L/b=2.84$ [39]. However, the conversion
efficiency is only weakly related to $\xi$, and focusing the beams too tightly
(high $\xi$) puts a tighter constraint on the overlap of the beam waists in
the axial direction, which is technically challenging to achieve. Instead we
opt for a more relaxed $b$ = 32 mm ($\xi$ = 1.56) for both input beams. Note
that $b$ is the confocal parameter of the beam inside the crystal, which is
quite different to that of the beam in air due to the high refractive index of
the material. Our lens set-up allows for some adjustment of our chosen $\xi$
value, but we observe the conversion efficiency to be relatively insensitive
to this parameter. After the crystal, dichroic mirrors ($D_{2,3}$ in Fig.
1(b)) are used to separate the red and IR beams, and the unused IR power is
dumped.

The SFG relative conversion efficiency as a function of temperature
$\eta^{\prime}\left(T\right)$ is predicted to follow a sinc2 function:

$\eta^{\prime}\left(T\right)\propto\frac{\sin^{2}\left(\Delta
k\left(T\right)L/2\right)}{\left(\Delta
k\left(T\right)L/2\right)^{2}}\mbox{.}$ (2)

The measured temperature dependence of the conversion efficiency is plotted in
Fig. 2(a) along with the expected curve from Eq. (1) and Eq. (2). We observe a
feature with FWHM $\approx$ 1 ∘C, so the 0.1 ∘C temperature resolution of our
temperature controller is only just sufficient to stabilize the temperature to
the top of the peak.

The power dependence of the SFG process is shown in Fig. 2(b). The output
power $P_{3}$ is broadly linear with the product of the input powers $P_{1}$
and $P_{2}$. We obtain a maximum power of $P_{3}$ = 1.6 W of $\lambda_{3}$
light from the SFG process. A least-squares linear fit to the data in Fig.
2(b) yields the normalized conversion efficiency $\eta$,

$\eta=\frac{P_{3}}{P_{1}P_{2}L}=\left(0.53\pm
0.02\right)\%\mbox{W}^{-1}\mbox{cm}^{-1}\mbox{,}$ (3)

where $P_{1,2,3}$ are the beam powers of $\lambda_{1,2,3}$ and $L$ is the
length of the PPSLT crystal. This measured value is favourable in comparison
to other reported CW conversion efficiencies for PPSLT, which are typically
$\sim$ 0.3$\,\%$ to 0.4$\,\%$ W-1 cm-1 [36, 40, 41]. However, it is noticeably
worse than the normalized SFG conversion efficiencies achieved with the more
widely used PPLN, which are typically $\sim$ 2.8$\,\%$ W-1 cm-1 [32, 42]. The
lower nonlinear coefficient $d_{33}$ of LT does not account for the total
difference in conversion efficiency, which could be due to the greater
difficulty of growing and poling stoichiometric lithium tantalate [36]. In the
future, MgO doped stoichiometric LN could be a good compromise if the higher
conversion efficiency is required, as the doping offers some level of
protection against the damage mechanisms discussed above [37, 43].

Fig. 2: (a) Dependence of the SFG efficiency on PPSLT crystal temperature $T$.
Measured data are shown in red (a temperature independent background due to
non-phase-matched SHG has been removed), with the prediction from Eq. (1) and
Eq. (2) shown in black dashes. A small temperature offset (0.55 ∘C) has been
added to the prediction plot to match the experimental data. (b) Output power
($P_{3}$) of the SFG process for a range of input powers ($P_{1}\times
P_{2}$). Data points are shown as red circles, and the blue line shows the
least-squares linear fit to the data.

###  2.1 Second harmonic generation to the UV

The majority of the light generated from the SFG process ($\lambda_{3}$) is
coupled into a 10 m long high power, polarization maintaining, single-mode
optical fiber, with $>$ 65 % efficiency to a commercial frequency doubling
unit (Toptica Photonics). A nonlinear crystal optimized for second harmonic
generation using type-I phase matching is housed in a hermetically sealed bow-
tie cavity. The unit contains a resonant electro-optic modulator (EOM) to
apply sidebands to the $\lambda_{3}$ light for Pound-Drever-Hall (PDH) locking
[44] of the SHG cavity length by feedback to a piezo-mounted mirror. For a
well-optimized system we have measured conversion efficiencies of $\sim$ 30 %,
producing 230 mW of UV light from 780 mW of red light. During typical
operation we obtain $>$ 100 mW of UV power across a tuning range of 318.0 nm
to 319.3 nm. The alignment of the SHG cavity requires re-optimization when the
UV frequency is changed by more than $\sim$ 100 GHz; this is a relatively
quick and simple procedure.

###  2.2 Laser frequency stabilization

A tunable optical cavity is used to transfer the frequency stability of the
7.5 kHz wide 5s2 1S0 $\rightarrow$ 5s5p 3P1 689 nm transition in 88Sr [Fig.
1(a)] to the UV laser system. An ECDL operating at 689 nm is locked to a
resonant mode of the optical cavity, and then the cavity length is stabilized
to the 689 nm sub-Doppler resonance feature in a thermal beam of Sr. The
stable reference cavity can then be exploited to lock the frequency of the
light at $\lambda_{3}$ [Fig. 1(b)].

Full details of the optical cavity design and construction are given in [45].
Briefly, the optical transfer cavity is 10.5 cm long and consists of a ZERODUR
® spacer with a ring-shaped piezo between the spacer and each high reflectance
mirror. One of the cavity piezos is a large-stroke piezo for long term drift
correction, whist the other is a short-stroke piezo for faster length
corrections [46]. The cavity is housed in a temperature stabilized vacuum
chamber at $\sim$ 27 ∘C and $\sim$ 10-7 mbar. The finesse is
$\left(41.3\,\pm\,0.6\right)$ $\times$ $10^{3}$ (FWHM = 35 kHz) at 689 nm and
$\left(109.9\,\pm\,0.4\right)$ $\times$ $10^{3}$ (FWHM = 13 kHz) at 638 nm, as
measured by cavity-ring-down spectroscopy. The 689 nm laser is locked to the
cavity using the PDH technique, and a portion of its output is used to
generate an atomic error signal using sub-Doppler fluorescence spectroscopy in
a Sr atomic beam. For this, the light is frequency modulated using a double-
passed acousto-optic modulator and retro-reflected through the atomic beam.
The resulting fluorescence is measured on a high gain photodiode, followed by
demodulation using a lock-in amplifier to derive the error signal. Feedback is
provided to both cavity piezos with different bandwidths, stabilizing the
cavity length to the atomic reference signal such that the 689 nm laser is
locked on resonance. To lock the UV laser frequency, a small portion of the
$\lambda_{3}$ light is picked off from the main beam path and overlapped with
the 689 nm beam on a dichroic filter ($D_{4}$ in Fig. 1(b)) providing
alignment into the same optical cavity. The wavelength of the $\lambda_{3}$
light is locked to a resonant mode of the atom-referenced cavity using the PDH
technique. A fast servo-loop controller (Toptica Photonics) provides feedback
to both the diode current and piezo voltage of the $\lambda_{1}$ seed laser,
with an overall loop bandwidth of $\sim$ 200 kHz. The transfer cavity lock
therefore both narrows the linewidth of the UV light as well as providing
absolute frequency stability. Different modulation frequencies are used for
the two PDH locks in order to prevent cross-talk between them.

Since the atom-referenced cavity mode is not necessarily at a frequency that
is useful for driving the Rydberg transitions, we use a fiber-coupled wideband
EOM (JENOPTIK, PM635) to offset-lock the laser frequency from that of the
cavity mode. We use the “electronic sideband” technique [47], where a tunable
sideband is locked to the cavity at a detuning $\nu_{\mathrm{c}}$ from the
main laser frequency. The carrier frequency of the EOM ($\nu_{\mathrm{c}}$ =
0.1 MHz – 5 GHz) is phase modulated at a frequency $\nu_{\mathrm{sb}}$ = 8 MHz
to produce the modulation sidebands required for PDH locking [Fig. 1(b)].
Demodulation at $\nu_{\mathrm{sb}}$ yields a standard PDH error signal,
locking the laser at a frequency $\nu_{\mathrm{c}}$ away from the cavity mode.
The UV laser frequency can be stepped or scanned while locked to the cavity by
varying $\nu_{\mathrm{c}}$ [47]. This allows us to manipulate the laser
frequency with high precision during an experimental sequence, such as for the
spectrum shown in Fig. 4(b). The laser remains locked (or relocks very
quickly) for step sizes within the capture range of the PDH lock ($\sim$ 1 MHz
in our case).

###  2.3 Laser frequency characterization

We characterize the long-term frequency instability of the 638 nm and 689 nm
lasers using heterodyne beat measurements with a GPS-referenced optical
frequency comb (Toptica Photonics). The fiber laser-based frequency comb has a
difference frequency generation design that eliminates the carrier-envelope
offset frequency. The tenth harmonic of the 80 MHz repetition rate is locked
to an ultra-low noise oven-controlled oscillator (OXCO), which in turn is
locked to a 10 MHz GPS-disciplined quartz oscillator (GPSDO) (Jackson Labs
Fury) for long-term ($\gtrsim$ 0.1 s) stability. Independent measurements
carried out by Toptica Photonics indicate that the performance of the
frequency comb is limited by the RF-reference for timescales larger than the
inverse of the locking bandwidth ($\sim$ few kilohertz) [48]. Each CW laser is
overlapped with the corresponding section of the frequency comb and the beat
note between the laser and the comb is measured on a fast photodiode. Repeated
measurements of the beat frequency ($f$) are made over a period of a few hours
using a zero-deadtime counter (Tektronix FCA3003). From these measurements we
calculate the overlapping Allan deviation [49] of the fractional frequency
error $(f-\bar{f})/f_{0}$, where $\bar{f}$ is the mean beat frequency and
$f_{0}$ is the laser frequency (e.g. $\sim$ 434.829 THz for the 689 nm laser).

The results are shown in Fig. 3, along with the manufacturer’s data for the
GPSDO. The Allan deviations for the 638 nm and 689 nm lasers are in reasonable
agreement, which is expected because they are locked to the same optical
reference cavity. A significant difference between these two measurements
would indicate one or both of the lasers were not well locked to the transfer
cavity. For $\tau\,\lesssim\,0.1\,$s, the beat frequency has less noise than
the GPSDO, reflecting the much lower noise of the OXCO. Above $\tau\,\approx$
0.1 s the instability of the optical frequency comb should follow that of the
GPSDO [48]. Instead, the measurements reveal excess frequency noise on both
the CW lasers. We attribute this noise to the performance of the transfer
cavity length servo to the atomic reference [Fig. 1(b)]. The low atomic
fluorescence rate ($2\pi\,\times\,7.5$ kHz) imposes a low bandwidth on the
cavity length servo ($\sim$ 10 Hz), which combined with a relatively high
cavity drift rate due to piezo relaxation (up to $\sim$ 1 MHz/s), leads to
significant fluctuations in the lock. Nevertheless, Fig. 3 shows that the
fractional frequency instability of both lasers systems is $<4\times 10^{-11}$
for all measurement times between $10^{-3}$ s and $\gtrsim$ $10^{3}$ s,
corresponding to frequency deviations of just $\sim$ 35 kHz for the UV laser
and $\sim$ 15 kHz for the 689 nm laser.

Fig. 3: Allan deviation plots showing the fractional frequency instability of
the 638 nm (red squares) and 689 nm (blue diamonds) lasers as measured by the
optical frequency comb. Measurements were made with either a 1 ms or 1 s
frequency counter gate time, as indicated by filled and empty symbols
respectively. The black line shows the specified instability of the GPSDO,
reproduced with permission from Jackson Labs.

For measurement times $\tau\lesssim\,10^{-3}\,$s, the frequency deviation
measurements eventually become limited by the short-term jitter of the
frequency comb, so this method cannot be used to infer the short-term
linewidth. Instead, we estimate the short-term frequency noise of the lasers
relative to the optical transfer cavity by looking at the spectrum of the in-
loop PDH error signal. Note that this measurement gives a relative frequency
instability, rather than the absolute instability provided by the optical
frequency comb, since the resonance frequency of the cavity may fluctuate due
to acoustic vibrations and electrical noise on the mirror piezos. However,
these cavity fluctuations will be relatively small at the timescales of
interest for these measurements ($<$ 1 ms). The in-loop error signal of the
638 nm laser is converted into frequency deviations from the cavity mode
center by dividing through by the PDH discriminator gradient
$\mathcal{D}=V_{\mathrm{pp}}/\delta\nu_{\mathrm{cav}}$, where
$V_{\mathrm{pp}}$ is the peak to peak voltage of the unlocked error signal and
$\delta\nu_{\mathrm{cav}}$ is the cavity mode FWHM. The overlapping Allan
deviation yields a relative fractional frequency instability of $<4\times
10^{-12}$ for all timescales between 2 $\upmu$s and 1 ms, which corresponds to
frequency deviations of $<$ 4 kHz in the UV. Similar analysis of the 689 nm
laser in-loop error signal is detailed in [45], where we estimate the short-
term linewidth to be $\sim$ 1.5 kHz.

Therefore if the short-term fluctuations of the transfer cavity length are
small, then the in-loop error signal analysis of the locked lasers shows
short-term laser frequency fluctuations to be less than the natural linewidths
of the 689 nm and Rydberg transitions ($\approx$ 10 kHz). The long-term
analysis with the GPS referenced frequency comb shows frequency deviations
outside of the natural linewidths, and so in future we plan to improve the
long-term performance by locking to an ultra-stable optical reference cavity
of the type used in optical atomic clocks. Despite this, due to Doppler and
power broadening effects, the measured frequency instability is still
sufficient to perform high resolution spectroscopy of Rydberg states, as we
show in the next section.

##  3 Atomic spectroscopy

To demonstrate the versatility of this laser system, including the high
tunability and narrow linewidth, we performed spectroscopy of a range of
triplet Rydberg states in 88Sr. We use a two-step excitation scheme at 689 nm
and 319 nm, as shown in Fig. 1(a). Rydberg spectroscopy was carried out in a
laser cooled sample of $\sim$ $10^{6}$ 88Sr atoms. A detailed description of
the apparatus can be found in [45]. Briefly, atoms from a thermal beam of Sr
are slowed, cooled and trapped in a “blue” MOT using the 5s2 1S0 $\rightarrow$
5s5p 1P1 transition at 461 nm, where they have a temperature of $\sim$ 5 mK.
They are subsequently loaded into a “broadband red” MOT using the 689 nm
intercombination line. In the broadband MOT, the 689 nm light is frequency
modulated to broaden its spectrum by $\sim$ 4 MHz, which ensures efficient
transfer from the relatively hot blue MOT. At the end of the broadband cooling
stage the atoms typically have a temperature of $\sim\,10\,\upmu$K and a
density of $\sim\,10^{11}$ cm-3.

To locate individual Rydberg states, we perform continuous single-mode 24 GHz
scans of the UV laser frequency while the broadband MOT is operating. The UV
laser is unlocked from the reference cavity, and scanned by piezo tuning of
the $\lambda_{1}$ ECDL cavity length. The single-mode tuning range is
ultimately limited by the $\sim 40$ GHz bandwidth of the SFG process, although
in practice the mode-hop free tuning range of the ECDL imposes a slightly
lower limit. When the UV frequency matches a transition frequency, atoms are
resonantly excited from the 5s5p 3P1 state to a Rydberg state. Some of these
Rydberg atoms spontaneously ionize, and we count the resulting ions using a
multi-channel plate (MCP) and a fast oscilloscope. Results of such scans in
the region $n\,\approx\,50$ and $n\,\approx\,80$ are shown in Fig. 4(a), where
the Rydberg states appear as sharp peaks in the ion signal. We identify the
Rydberg states by using the Rydberg-Ritz parameters given in [24] to
extrapolate from previous measurements at lower principal quantum numbers $n$
[30, 50, 51]. A small electric field ($\sim$ 1 V cm-1) applied during the
excitation sequence, broadens and Stark splits the 3S1\- and 3DJ-states as
well as allowing excitation to the 3PJ-states. Both effects are clearly
visible in the $n\approx 80$ data.

Fig. 4: (a) Ion signal obtained from continuous scans of the UV detuning
(measured relative to the start of the scan) in the regions around
$n\,\approx\,50$ and $n\,\approx\,80$. The intensities used for the excitation
are $\sim$ 20 mW cm-2 and $\sim$ 1 W cm-2 for the 689 nm and UV beams
respectively, and each beam is larger than the size of the atom cloud. The
variation in signal height across the scan is largely due to depletion of
atoms from the MOT and is not an indication of transition strength. The
frequency axis of the scans is calibrated on a 10 MHz resolution wavemeter
(High Finesse). The offset frequencies are are $\sim$ 940.649 THz
($n\,\approx\,50$) and $\sim$ 941.616 THz ($n\,\approx\,80$). (b) High
resolution scan of the
$\mathrm{5s5p}\,^{\mathrm{3}}\mathrm{P}_{1},\,m_{J}=-1\rightarrow\mathrm{5s37s}\,^{\mathrm{3}}\mathrm{S}_{1},\,m_{J}=0$
transition. The solid blue line shows the Voigt profile fit. The intensities
used for the excitation are $\sim$ 0.1 mW cm-2 and $\sim$ 250 mW cm-2 for the
689 nm and UV beams respectively, and each beam is larger than the size of the
atom cloud. The central frequency of this feature is at $\sim$ 939.274 THz.

With the laser system locked to the transfer cavity (as described in section
2.2) we can perform Rydberg spectroscopy with $<$ 10 kHz-level precision.
Here, atoms are released from the broadband MOT, before being excited to the
5s37s 3S1 Rydberg state using resonant 689 nm and 319 nm light. The 689 nm
excitation beam counter-propagates at an angle of 30 ∘ to the UV beam, and
both lasers are pulsed on simultaneously for 300 $\upmu$s. A quantization
magnetic field ($B$ $\approx$ 0.14 mT) applied during the excitation pulse
separates the $m_{J}$ states. Following the Rydberg excitation, a 20 $\upmu$s
pulse of 408 nm laser light is used to autoionize the Rydberg atoms and an
electric field is applied to direct the ions towards the MCP [28]. We repeat
each measurement 10 times in the same atomic sample, after which the MOT is
reloaded and the UV laser frequency stepped to the next detuning value by
changing the EOM offset frequency $\nu_{\mathrm{c}}$. We take the spectrum in
50 kHz steps, tuning the laser frequency up over the feature, then back down
again.

The results are shown in Fig. 4(b). The lineshape is very well described by a
Voigt profile (reduced $\chi^{2}$ = 1.1) with an overall FWHM of
$\left(350\,\pm\,10\right)$ kHz. From the fit we extract the FWHM of the
inhomogeneous (Gaussian) $\left(280\,\pm\,30\right)$ kHz and homogeneous
(Lorentzian) $\left(120\,\pm\,30\right)$ kHz contributions. Both are in
agreement with our expected Doppler broadening and power broadening
respectively. The fit yields the line center with a precision of $\sim$ 5 kHz.
Approximate transition frequencies are given for the observed states in the
caption for Fig. 4. In the future, absolute frequency measurements at the
$10^{-12}$ level will be possible with the use of the GPS-referenced optical
frequency comb. However, this will require a detailed analysis of the electric
and magnetic field induced line shifts, which is beyond the scope of this
paper.

##  4 Conclusion

In conclusion, we have developed a solid-state laser system for generating
high power, narrow-linewidth UV light, which can be tuned across the range
316.3 nm – 317.7 nm and 318.0 nm – 319.3 nm. The Rydberg states shown in Fig.
4 span a frequency range of over $2.3$ THz, providing a clear illustration of
the wide tunability of the laser design. At the same time, we demonstrate
broad single-mode tuning range (24 GHz) suitable for atomic spectroscopy, and
precision control of the absolute frequency, enabling Rydberg spectroscopy
with kilohertz precision. The long-term frequency instability of the laser
system ($<$ 35 kHz for timescales between $10^{-3}$ s and $10^{3}$ s) could be
reduced by using an ultra-low drift optical reference cavity, and either an
improved atomic spectroscopy set-up or the frequency comb as a long-term
absolute reference. With these modifications, an absolute frequency
instability below $1$ kHz should be attainable. The laser output power is
limited by the conversion efficiency of the PPSLT crystal. Although we obtain
one of the highest conversion efficiencies so far reported for this material,
replacement with MgO doped PPLN could provide a substantial increase in UV
power. The precision and versatility of this laser system make it ideal for
future Sr Rydberg experiments, including high precision spectroscopy of the
Rydberg states and their interactions, and experiments involving Rydberg
dressed potentials. The all solid-state design makes this an attractive
replacement for complex frequency-doubled dye laser systems.

## Acknowledgments

The authors would like to thank Thomas Puppe, Russell Kliese and Alexander
Sell from Toptica Photonics for their work on developing and installing the
optical frequency comb, and Chunyong Li at Durham University for maintaining
it. We also thank Ian Hill from the National Physical Laboratory for comments
on the manuscript. Financial support was provided by EPSRC grant EP/J007021/
and EU grants FP7-ICT-2013-612862-HAIRS and H2020-FETPROACT-2014-640378-RYSQ.
The data presented in this paper are available for download [52].