Also at ]INFN-Sezione di Firenze, Sesto Fiorentino, ItalyAlso at ]INFN-Sezione
di Firenze and CNR-INO, Sesto Fiorentino, Italy

# Practical limits for large-momentum-transfer clock atom interferometers

Mauro Chiarotti  Jonathan N. Tinsley  Satvika Bandarupally  Shamaila Manzoor
Michele Sacco  Leonardo Salvi  [  Nicola Poli  [<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia and LENS, Università degli Studi di
Firenze, Via G. Sansone 1, 50019 – Sesto Fiorentino, Italy

###### Abstract

Atom interferometry on optical clock transitions is being pursued for numerous
long-baseline experiments both terrestrially and for future space missions.
Crucial to meeting these experiments’ required sensitivities is the
implementation of large momentum transfer ($>$10${}^{3}\hbar k$). Here we show
that to sequentially apply such a large momentum via $\pi$ pulses places
stringent requirements on the frequency noise of the interferometry laser,
finding that the linewidth is required to be considerably lower than has
previously been suggested. This is due to imperfect pulse fidelity in the
presence of noise and is apparent even for an atom at rest interacting with
resonant light, making this a fundamental constraint on operational fidelity
for a given laser and pulse sequence. Within this framework, we further
present and analyse two high-power, frequency-stabilised laser sources
designed to perform interferometry on the 1S0 \- 3P0 clock transitions of
cadmium and strontium, respectively operating at 332 nm and 698 nm.

##  I Introduction

Atom interferometry utilising single-photon optical transitions represents an
emerging technology with the ability to probe physics in a variety of
previously untested regimes [1, 2, 3]. This includes experiments looking for
tests of the quantum twin paradox [4], gravitational red shift [5], and tests
of the weak equivalence for atoms in quantum superpositions [6]. In
particular, however, multiple experiments based upon the 1S0 \- 3P0 clock
transition of Sr at 698 nm have been proposed to search for a very wide set of
fundamental physics goals, but especially for ultra-light dark matter searches
and gravitational wave detection [7, 8, 9, 10, 11, 12]. These experiments are
predicated on the crucial fact that for interferometers based upon a single-
photon transition, the laser phase is set at the point of emission and is
therefore identical for all regions of the interferometer, regardless of their
spatial separation, and cancels out in the read-out stage for a gradiometric
configuration [7, 8]. This allows for much larger interferometer baselines to
be employed than would be possible with counter-propagating, multi-photon
interferometry beams, as used in Raman or Bragg interferometers, where the
finite speed of light introduces a noise term [13].

The use of large baselines and the enabled very large interferometry times are
not, however, in of themselves sufficient to achieve the sensitivity required
for the physics goals of these experiments. Therefore it is necessary to
enhance the device sensitivity by employing the technique of large momentum
transfer (LMT) [8], with very large enhancements of 103-104 ultimately
proposed for terrestrial experiments currently in the development stage [11,
12]. In practice, this typically means increasing the momentum separation
between the two paths of the wavepacket by applying a series of $\pi$ pulses
whose number is of the same order as the required enhancement [8, 3]. For
these schemes it is therefore crucial that each $\pi$ pulse is as efficient as
possible if interferometry contrast and fringe visibility is to be maintained,
as a loss of fringe visibility leads to a decrease in device sensitivity and
therefore acts in opposition to the metrological gain that would otherwise be
achieved by LMT. From a practical standpoint, this requirement translates as
needing an ultracold, velocity-selected atom source illuminated by an intense
and homogeneous beam. Furthermore, although laser frequency noise cancels out
in the phase readout in the clock-transition gradiometer configuration [1, 2],
laser noise can nevertheless degrade device performance by reducing $\pi$
pulse efficiency and interferometer contrast and visibility [14], as has been
experimentally observed for clock atom interferometers [1, 2].

In this article, we concentrate on the effect of laser noise, both intensity
but especially frequency noise, on pulse fidelity and hence interferometer
fringe visibility. We determine the fidelity for the Mach-Zehnder
interferometry schemes proposed for large-baseline clock-transition
interferometers [8], showing that the induced imperfections of fidelity lead
to practical difficulties in scaling LMT to the desired levels of large-
baseline clock atom interferometers, especially at the proposed laser
linewidth of $\sim$10 Hz [12]. It is further shown that, in general, the
fidelity of the pulse sequence is improved for a given noise level with
increasing Rabi frequency, highlighting the need for a high-intensity
interferometry beam, which is an important technological difference between
using optical clock transitions for atom interferometry and frequency
metrology. As we consider the case of a single two-level atom interacting with
a resonant laser, these results hold regardless of the atom cloud temperature
or spatial spread and represent the maximum achievable fidelity for a given
laser system and pulse sequence.

We further apply this fidelity formalism to two newly developed high-power Hz-
level clock lasers and show the practical LMT limits in principle achievable
with these lasers when performing a standard clock interferometry sequence.
These two systems have been specifically developed for performing atom
intererometry on the 1S0 \- 3P0 clock transitions of Cd and Sr at 332 nm and
698 nm, respectively [15]. For the case of the UV transition of Cd, we build
upon on recently demonstrated technology for a laser source centred at 326 nm
[16], extending that work by demonstrating the capability of obtaining high
coherence via successive frequency-stabilisation stages on optical cavities
with increasing finesse.

The structure of this article is the following: in Section II we discuss the
basics of LMT clock interferometry and visibility loss; in Section III we
present the basic theory of pulse fidelity, with further details in Appendix
A, and show the results of the pulse fidelity calculations applied to an LMT
sequence; Section IV discusses the developed laser systems and analyses them
within this framework; and conclusions are reported in Section V.

##  II LMT Clock Atom Interferometer Sensitivity

The leading-order phase shift of a clock atom interferometer is $\phi\sim
nkgT^{2}$, where $n$ is the LMT order, $k$ is the wavenumber of the
transition, $g$ is the local gravitational acceleration, and $T$ the
interferometry time [1, 8]. Single-shot interferometry sensitivity can be
characterised by the ratio $\Delta\phi/\phi$ where $\Delta\phi$ is the phase
difference between the two interferometer arms that can be experimentally
resolved. It is therefore expected that, in principle, device sensitivity
enhancement should scale linearly with LMT order $n$ and it has consequently
been suggested to improve clock atom interferometers by performing a
succession of $\pi$ pulses propagating in alternating directions [8]. In
brief, an LMT enhancement of $n$ is achieved by applying two additional stages
of $2(n-1)$ $\pi$ pulses independently resonant to the upper and lower
interferometry arms and temporally mirrored about the standard interferometry
mirror pulse sequence (Fig. 1).

However, in experimental conditions $\Delta\phi$ itself is typically found to
be dependent upon the parameters $n$ and $T$. This is because $\Delta\phi$ is
extracted from the relative population difference and is therefore
proportional to the slope and hence the visibility of the interferometry
fringe (see Fig. 1). In a typical atom interferometer, sources of visibility
loss relating to the interferometry pulses can be roughly divided into two
categories: those which lead to the different constituent atoms of the
ensemble becoming out of phase with each other; and those which degrade the
operational fidelity on the single-atom scale [17]. The former is well-studied
and often represents the leading practical constraint on atom interferometers,
with finite atom cloud temperature and size, as well as interferometry beam
imperfections (e.g. wavefront and intensity inhomogeneities), causing
different atoms within the ensemble to interact with effectively different
laser beams during the sequence [14, 18, 19]. The resulting atomic dephasing
causes a loss of contrast and visibility, as the read-out of the
interferometer is effectively the average over all the individual atoms. Not
all such contrast losses, however, can necessarily be considered as due to a
fundamental loss of coherence, as demonstrated by related spin-echo techniques
[20, 21]. Nevertheless, such effects are known to be both crucial and a key
limiting factor for many current atom interferometers [14].

Figure 1: (a) Clock atom interferometers operate on the metastable 1S0 \- 3P0
transition, which is ideally driven by a noiseless laser (blue line). A noisy
laser system (red line) will instead introduce errors when applying
operations, such as for a $\pi$ pulse as shown here in the Bloch sphere
representation of the two-level system. (b) These imperfections lead to
damping in the temporal domain and also to a loss of interferometry visibility
and hence interferometric sensitivity. (c) Proposed LMT scheme for a large-
area, clock atom interferometer [8]. Black lines represent the basic
interferometry scheme, whilst orange and green lines represent the LMT pulses
applied to the upper and lower arms, respectively. The upper and lower arms of
the interferometer receive different pulses, but with a pattern which is
symmetric in time.

In addition to beam and cloud temperature imperfections reducing visibility,
however, imperfect application of the desired quantum operation (e.g. $\pi/2$
and $\pi$ pulses) will lead to contrast loss and other systematic errors. One
source of such imperfections is laser noise and crucially LMT exacerbates the
problem as it increases the number of necessarily imperfect pulses that are
applied. In investigating this problem, previous analyses have tended to
account for laser frequency noise by considering the excited population
fraction from a $\pi$ pulse to be $P_{e}\approx 1-\delta^{2}/\Omega_{R}^{2}$
[8, 12], with $\delta$ the effective detuning and $\Omega_{R}$ the Rabi
frequency. In this regime, it is the small changes in $\delta$ caused by the
non-zero laser linewidth that are considered. Such calculations have concluded
that, for $\Omega_{R}\sim$103 rad/s, the laser linewidth needs to be kept to
the order of 10 Hz for contrast loss on the percent level for an
interferometer with LMT order $n\sim$103 [12]. As 10 Hz is readily achievable
with current technologies, other effects such as finite atom temperature and
intensity variations across the atom cloud are considered as the main source
of dephasing and thus contrast and visibility loss.

However, it is also well known that the laser noise in terms of both intensity
[22] and, especially, frequency [23, 24] is an important parameter in
determining pulse fidelity in quantum systems, as has been extensively studied
in the field of quantum computing which targets 104 operations on a single
qubit system [25, 26]. Furthermore, it has recently been shown that to
correctly determine $\pi$-pulse fidelity it is not sufficient to consider only
the linewidth of the interrogating laser, but rather the whole power spectrum
of fluctuations should be considered [27]. Imperfect pulse fidelity arising
from laser noise reduces the visibility as above, but in an arguably more
fundamental way as it is a damping of a single atom rather than a washing-out
across the ensemble.

We note that although this laser noise also affects the fidelity of e.g. Raman
transitions [27], this represents a crucial practical difference between clock
atom interferometers and more standard, multi-photon transition
configurations. In the latter case the relevant noise term is the _relative_
phase noise between the counter-propagating interferometry beams, whilst for
clock interferometers it is the _absolute_ laser phase noise itself which is
the relevant parameter [2]. The phase noise of Raman and Bragg beams can be
easily reduced to orders of magnitude less than the phase noise that is
currently achievable with even the narrowest clock lasers [28, 29].

##  III Pulse Fidelity & Computational Results

Figure 2: (a) Filter function calculated for different LMT values (see Fig.
1(a)). In these calculations, $T_{1}$=1 s or 10 s and $T_{1}\simeq T_{2}$ due
to the high Rabi frequency values ($\Omega_{R}$=103 rad/s) and short pulse
separation ($\tau$=1 $\mu$s). Fidelity of an LMT sequence as a function of (b)
laser linewidth for a fixed Rabi frquency ($\Omega_{R}=2\pi\times 10^{3}$ Hz)
and (c) as a function of Rabi frequency at a fixed laser linewidth
($\Delta\nu$=1 Hz). In both cases, $\tau$=1 $\mu$s and $T_{1}$=1 s. d)
Fidelity of a single $\pi$ pulse as a function of RIN level and Rabi
frequency.

In determining the effect of the laser noise, we consider the fidelity of the
whole interferometry sequence, where the fidelity of a quantum operation is
defined as the overlap of the target operation with the actually applied
operation. This is quantified by the square modulus of the Hilbert-Schmidt
inner product of these operations [17]. Laser noise causes the state evolution
on the Bloch sphere, defined here by the two-level system formed by the 1S0 \-
3P0 clock transition, to deviate from the ideal case and thus for the fidelity
to deviate from unity (Fig. 1). Specifically, in the standard case of applying
a rotation about the x or y axes, frequency and intensity noise respectively
produce undesired rotations about the $z$ axis and the $x$ and $y$ axes [27].
In the presence of weak noise, the fidelity can be written as [17, 23, 27],

$\mathcal{F}\simeq\frac{1}{2}\left[1+e^{-\chi}\right],$ (1)

where $\chi$ is the fidelity decay constant which is related, to first-order,
to the noise power spectral density (PSD) by the filter function
($F\left(\omega\right)$) of the operation being performed:

$\chi=\sum_{j}\frac{1}{\pi}\int^{\infty}_{0}\frac{d\omega}{\omega^{2}}S_{j}\left(\omega\right)F_{j}\left(\omega\right);$
(2)

where $S_{j}\left(\omega\right)$ is linearly related to the PSD in a manner
dependent upon the type of noise source and the summation is over the
different sources of noise. Specifically, for frequency and intensity noise
$S_{j}\left(\omega\right)=\frac{1}{4}S_{\omega_{L}}\left(\omega\right)$ and
$S_{j}\left(\omega\right)=\frac{1}{16}\Omega_{R}^{2}S_{I}\left(\omega\right)$,
respectively, where $S_{\omega_{L}}\left(\omega\right)$ is the laser frequency
noise, $S_{I}\left(\omega\right)$ the relative intensity noise (RIN), and
$\Omega_{R}$ is the Rabi frequency [27]. The problem of determining the
fidelity of a quantum operation is therefore reduced to the knowledge of the
relevant PSDs and calculating the appropriate filter functions.

Figure 3: Relative interferometer sensitivity, defined as $nW$, as a function
of LMT for different values of (a) laser linewidth $\Delta\nu$ (with
$\Omega_{R}=2\pi\times 10^{3}$ Hz) and (b) Rabi frequency $\Omega_{R}$ (with
$\Delta\nu$=1 Hz). Dashed grey lines show linear scaling with LMT and $W=1$
for comparison. In all cases, $\tau$=1 $\mu$s and $T_{1}$=1 s.

We determine the filter function for laser frequency noise for a sequence of
pulses mimicking the proposed LMT sequence in a clock-transition
interferometer for detecting gravitational waves [8] (Fig. 1(c)). These
calculations follow a known method [17], with the key results for this
application summarised in Appendix A. Figure 2 shows the filter function
calculated for a variety of LMT values and shows a structure with a low-
frequency cut-on, two principle resonances around $\Omega_{R}$, and then a
flatter distribution at higher frequencies with some harmonics on top. As can
be seen, the filter function structure is largely unmodified with increasing
LMT, except that the function resonances about the Rabi frequency are
amplified approximately quadratically with LMT order $n$. Increasing the
interferometry time, increases the sensitivity to low-frequency noise and
determines the position of the function’s nodes at low frequency.

The above framework can be used to compute the effect of the laser frequency
noise and RIN on operational fidelity. We calculate the effects independently,
but in the real case they must be combined through the summation in Eq. 2.
When analysing these effects we in all cases assume a spectrally flat
distribution, i.e. the PSD is a constant value, and integrate Eq. 2 in the
range 10-2–107 rad/s. Although ideally Eq. 2 should be integrated over all
angular frequencies, we find that, in our case, when increasing or decreasing
both the upper and lower integration limits by two decades, the resultant
fidelity is only altered by $\sim$1%.

In the case of the RIN, it is observed that the pulse fidelity from a single
$\pi$ pulse is relatively unaffected by the laser noise (Fig. 2(d)), with
excellent fidelities above 0.999 achievable even with high RIN values of -70
dBc/Hz for a wide range of Rabi frequencies. However, it is important to note
that the $\Omega_{R}^{2}$ dependence in Eq. 2 may make the RIN contribution
experimentally important should alternative schemes to increase the Rabi
frequency be used [3, 30]. Nevertheless, we therefore ignore the contribution
of the RIN in the following analysis.

Conversely, however, the fidelity of an LMT sequence is highly degraded by the
presence of laser noise, unless the laser noise is reduced to experimentally
difficult linewidths (Fig. 2(b)). As mentioned, these calculations assume
white noise which we relate to a Lorentzian linewidth via $\Delta\nu=\pi
S_{\nu}=S_{\omega_{L}}/2$. We have focused on Lorentzian linewidths for
generality, whereas in reality laser linewidths typically take the form of a
central Gaussian region and a Lorentzian component at higher frequencies [31],
with an additional servo bump from the employed feedback circuit. The effect
of these spectral features has been studied in a quantum computing context
[27], and allows for the possibility to potentially reduce the degrading
effect of the laser noise on the interferometry fidelity by suppressing the
noise around peaks of the filter function or by judicious selection of the
Rabi frequency. Therefore, for any real application, the measured noise
spectra of the laser sources should therefore be used for analysis, as we show
later in Fig. 5.

As an example of the difficulties posed, however, the fidelity already
approaches its minimum value of 0.5 for $n=1000$ even with a linewidth of 1 Hz
(Fig. 2(b)). For a given LMT and $\Omega_{R}$, the fidelity is initially
relatively unaffected by increasing $\Delta\nu$, but decays exponentially
following a cut-off linewidth. In this regime, the fidelity decays from
approximately 0.9 to the minimum 0.5 with an approximate factor 10 increase in
$\Delta\nu$, meaning it is important to keep $\Delta\nu$ at or lower than this
cut-off value to achieve the highest fidelities.

Unlike in the case of the RIN, however, the situation is improved with
increasing Rabi frequency (Fig. 2(c)), as this increases the frequency of the
resonances (Fig. 1(b)). Combined with the $1/\omega^{2}$ dependence of Eq. 2,
this leads to a relative suppression of the laser noise. This is in agreement
with the simple picture of considering the ratio of the effective detuning and
the Rabi frequency. Increasing $\Omega_{R}$ has additional benefits for real
systems with finite temperature [30], although the RIN will become
increasingly important, as discussed above.

To quantify the reduction in interferometer performance, we consider the
parameter $W=\text{exp}\left[{-\chi}\right]$ which varies from 0–1 and is
closely related to the fidelity (Eqs. 1 and 2). For a standard Mach-Zehnder
sequence, this $W$ parameter scales linearly with the interferometer
visibility. Therefore assuming that the interferometer sensitivity is linearly
dependent upon both this value and LMT order (see Sec. II), this allows for
the sensitivity relative to an interferometer with $W=1$ and $n=1$ to be
determined. In the ideal case of perfect pulse fidelity, this relative
sensitivity enhancement would scale linearly with $n$, but this is seen to not
be the case in the presence of laser frequency noise. As shown in Fig. 3,
there is in fact an optimal value of $n$ for a given $\Delta\nu$ and
$\Omega_{R}$ in terms of maximising single-shot sensitivity. For example, for
a laser with $\Delta\nu$=1 Hz and $\Omega_{R}=2\pi\times 10^{3}$ Hz this
occurs for $n\sim 500$ and at $n$=104 the sensitivity is lower than at $n$=1
(Fig. 3(b)). For a 10-Hz-linewidth laser at the same Rabi frequency, however,
this optimum is at $n<100$ and, moreover, the relative sensitivity is
significantly less than 1 in all cases (Fig. 3(a)), highlighting the necessity
of low-noise lasers if the potential sensitivity gains from LMT are to be
realised.

##  IV Clock Interferometry Laser Sources

Figure 4: Setup of the clock atom interferometry laser sources for Sr and Cd
at 332 nm and 698 nm, respectively. The laser for Sr is derived from a
Ti:sapphire laser with an intra-cavity EOM. For Cd, the laser is a frequency-
quadrupled ECDL which is amplified by a Raman fibre amplifier. The lasers can
both be locked to the same super-high-finesse ULE clock cavity. See text for
details of the locking procedures. Figure 5: (a) Measured in-loop
$S_{\nu}\left(f\right)$ for the two clock interferometry lasers at 698 nm and
664 nm for Sr and Cd, respectively. (b) Free-running RIN of the two clock
interferometry lasers. (c) Relative interferometer sensitivity as a function
of LMT for for the developed laser sources with $\tau$=1 $\mu$s and $T$=100 ms
and $\Omega_{\text{Sr}}=2\pi\times$1.7 kHz and
$\Omega_{\text{Cd}}=2\pi\times$740 Hz. In all plots dashed red lines are for
Sr and solid purple lines are for Cd.

As can be deduced from the above results, the principle requirements for a
laser system for performing LMT clock-transition interferometry are high
optical power and high coherence (Fig. 4). For the Sr system the 1S0 \- 3P0 is
at 698 nm meaning that commercial systems can provide the basis of this, in
addition to providing good spatial mode quality and low RIN. We use a
commercial Ti:sapphire system (Matisse TS, Sirah), pumped with up to 25 W at
532 nm, which is locked to a super-high-finesse ultra low expansion (ULE)
optical Fabry-Pérot cavity with a finesse $\mathcal{F}_{698}=4.1\times 10^{5}$
[32]. A high feedback bandwidth ($\sim$300 kHz) is achieved by utilising an
intra-cavity electro-optical modulator (EOM), at the cost of a slight
reduction in output power ($\sim$20$\%$). Nevertheless, a total optical power
of 4 W is achieved at 698 nm in this configuration. Stabilisation is performed
by pre-stabilising the laser to a medium-finesse Fabry-Pérot cavity, using the
EOM and the piezoelectric transducers (PZTs) on two of the laser mirrors. This
cavity is in turn locked to the super-high-finesse cavity by controlling its
length with a PZT on one of the cavity mirrors. For both cavities $\sim$1 mW
of light is sent via polarisation-maintaining fibre and modulated at $\sim$10
MHz by external EOMs for Pound-Drever-Hall locking (PDH) [33]. The PSD of the
in-loop measurement of the error signal is shown in Fig. 5 and yields a
linewidth of 10 Hz.

The 1S0 \- 3P0 of Cd, however, is at 332 nm and a UV light source capable of 1
W of optical power and coherence at the Hz level is not currently commercially
available. It was therefore necessary to develop a system internally,
optimising its performance ad hoc for this application. We follow the same
basic scheme as was recently reported for a laser for the 1S0 \- 3P1 Cd
transition at 326 nm [16], where intense UV radiation is produced by two
successive stages of second-harmonic generation (SHG), i.e., it is quadrupled
from the infrared emission of a high-power semiconductor diode and fibre
amplifier. In brief, a Littrow-configuration extended-cavity diode master
laser (ECDL) based on a gain chip is used to inject a commercial Raman fibre
amplifier, which amplifies the 35-mW input to the 10-W level. SHG of this
light is performed in a periodically-poled lithium niobate (PPLN) crystal,
producing around 3 W at 664 nm, before being doubled again in a resonant bow-
tie cavity using a Brewster-cut beta-barium borate ($\beta$-BaB2O4, BBO)
crystal as the non-linear, doubling medium [34]. This cavity is locked via the
Hänsch-Couillaud technique [35] for total produced UV powers up to $\sim$1 W.

The key difference with the previous scheme is the requirement of greater
coherence than was necessary for the 1S0 \- 3P1 Cd transition
($\Gamma$=$2\pi\times$67 kHz). This is achieved first by optimising the pre-
stabilisation lock of the ECDL onto a medium-finesse optical Fabry-Pérot
cavity ($\mathcal{F}_{pre}=5\times 10^{3}$). The stabilisation loop is closed
by sending in feedback the error signal thus generated, after engineering
through the use of optimised PIDs and filters, to two transduction channels:
the PZT which in the ECDL defines the grating orientation (low frequency – 1
kHz) and the modulation input superimposed on the current supply (high
frequency – 700 kHz). The in-loop noise analysis conducted on the described
stage, which is capable of reducing the source noise to equivalence to an
estimated linewidth of $\sim$Hz in the 100 kHz band [36].

To really reduce the linewidth further to the Hz level the laser must be
locked to the ULE cavity, for which a second feedback stage is applied in the
visible region following SHG. The observation of a 20 Hz beat between the pre-
stabilised and the amplified radiation allows us to conclude that the
coherence length of the source is so far sufficiently large to neglect the
frequency noise introduced by the Raman fibre amplification stage. We also
observe the presence of low-frequency noise (10 dB at 100 Hz, 30 dB at 5 kHz
with respect to the central emission peak), to which this second stabilisation
stage is directed. We use the same cavity and setup as for the Sr laser, with
the only difference being a reduction in cavity finesse to
$\mathcal{F}_{664}=1.3\times 10^{5}$, as measured by the ring-down method. The
PDH error signal is split and the low-frequency component is sent to the PZT
of the pre-stabilisation cavity, to compensate drifts. An acousto-optical
modulator (AOM), in single-pass on the visible SHG, is used for higher
frequencies up to around 50 kHz. The AOM is also used to additionally
stabilise the intensity and reduce the amplitude noise added by the
amplification stage [16]. With this method, the in-loop measured linewidth is
2 Hz at 664 nm, which will correspond to around 8 Hz after the doubling stage
to the UV (Fig. 5(a)). This value represents the lower limit for the
linewidth, although the main noise contribution of the doubling stage should
be frequency-to-amplitude conversion. Direct experimental validation of the
linewidth or stablisation at 332 nm itself would require an evacuated high-
finesse cavity in the UV regime, for which the production of the necessary
substrates and coatings remains an active area of research [37].

Finally, the radiation at 664 nm is coupled (through a T=1.5$\%$ input mirror)
with a commercial bow-tie optical cavity, specifically designed for the
application [34]. A production of UV power up to 1.2 W is achieved, which
matches well with the expected values for the measured finesse
$\mathcal{F}_{BBO}=323\pm 3$ and the nonlinear conversion coefficient
$E_{nl}=1.21\times 10^{-4}$ calculated from the nominal system parameters
[38]. The RIN of the UV (Fig. 5b) is kept low ($\sim$-90 dBc/Hz for 10-105 Hz)
by the AOM stabilisation stage and by passively shielding the BBO cavity to
reduce acoustic resonances and no further active stabilisation is required.

The measured intensity and frequency noise of these laser sources can be used
to analyse their expected performance in an LMT clock atom interferometer. We
input these measured values into Eq. 2 and integrate between the measurement
range of 1–105 Hz. Inputting the measured RIN values for our two laser sources
(Fig. 5(b)) gives single-pulse fidelities of $\mathcal{F_{\pi}}>$0.9999 for
$\Omega_{R}<10^{6}$ in both cases, suggesting the RIN will pose negligible
problems as expected. To instead estimate the effect of the frequency noise,
however, we first fix the expected Rabi frequency. For a Gaussian beam radius
of 5 mm, which is well below the maximum permissible size to keep the
intensity ripple at the 1% level for diffraction from the standard DN100CF
viewports which are to be employed [39], we estimate achievable $\Omega_{R}$
values of $2\pi\times$1.7 kHz and $2\pi\times$740 Hz for the Sr and Cd
sources, respectively. These Rabi frequencies can be used to estimate optimal
LMT parameters of $n_{Sr}\sim$100 and $n_{Cd}\sim$10 for Sr and Cd,
respectively (Fig. 5(c)), based on the measured frequency noise (Fig. 5(a))
and $T$=100 ms, which is a reasonable interferometry time for our proposed
experiments [15]. In the case of Sr, such an LMT-based enhancement in
performance represents an in principle improvement upon what has previously
been demonstrated with clock atom interferometry schemes [3].

##  V Conclusion & Outlook

We have provided a general analysis for the noise requirements for a system to
perform LMT clock atom interferometry with a standard pulse sequence by
investigating the operational fidelity in the single-atom case. These analyses
have shown the challenging nature of proposed long-baseline experiments for
e.g. gravitational wave detection and the difficulties in performing clock
atom interferometer with LMT parameters up to $10^{4}\hbar k$. In order to
fully understand the effect on interferometer performance, especially in a
gradiometer configuration where some common-mode noise cancellation may
persist, it would be necessary to extend the above calculations to a full
quantum simulation incorporating parameters such as atom cloud temperature and
spatial spread, loss mechanisms, and the external degrees-of-freedom of the
system. Nevertheless, the outlined formalism will be useful for determining
the interferometry laser requirements for the future experiments both on Earth
[11, 12] and in space [9, 10] which seek to use clock atom interferometry to
perform fundamental physics experiments. The maintenance of contrast and
visibility is also crucial for achieving metrological gain in experiments
which seek to use squeezing to go below the quantum projection noise limit
[40, 41].

The analysis presented above has further focused solely on square pulses and
we note that it is possible to enhance $\pi$-pulse population transfer
efficiency by techniques such as adiabatic rapid passage [42, 43] or composite
pulses [44, 45], which can further be optimised by pulse shaping [46, 47, 48].
Alternative schemes, such as using the 1S0 \- 3P1 transition for the LMT
portion of the interferometer [3] or using a cavity to enhance the optical
power and increase the Rabi frequency [30], have also been previously been
discussed and may help to circumnavigate some of these problems. Trapped-atom
configurations which use Bloch oscillations to enhance the interferometry
sensitivity may also offer a route towards a solution [49, 50, 51]. In all
cases, however, our analysis suggests that the noise of the interferometry
laser will play an important role and should not be discounted, but rather
integrated into these approaches.

Within this operational fidelity framework, we have presented and analysed two
laser systems suitable for performing clock atom interferometry – one for the
698 nm transition of Sr and one for the 332 nm transition of Cd – possessing
the required characteristics of high power, low RIN, and $\sim$Hz-level
linewidth. In the case of the 332-nm laser we have established the capability
to produce high-coherence and high-power light in the UV by frequency
quadrupling and amplifying an ECDL at 1328 nm. These lasers will be
implemented in a dual-species atom interferometer, with potential applications
to tests of the weak equivalence principle and quantum time dilation [15].

###### Acknowledgements.

We thank Grzegorz D. Pekala for assistance in establishing the Cd clock laser
source. We thank Guglielmo M. Tino and Hidetoshi Katori for useful discussions
and Michael Holynski for a critical reading of the manuscript. This work has
been supported by the European Research Council, Grant No.772126 (TICTOCGRAV).

##  Appendix A Filter Function Calculation

The filter transfer function $F(\omega)$ is calculated according to the
procedure outlined in [17], briefly summarised here. The transfer function
results from the norm of the control matrix, generally introduced to describe
the effect that laser noise sources, such as phase and intensity noise, have
on the fidelity of a quantum operation. The loss of fidelity of a quantum
operation is in fact proportional to the integral of the noise spectrum
describing the radiation that induces the operation, averaged through the
function $F(\omega)/{\omega}^{2}$ (Eq. 2). In the particular case of dephasing
noise, the control matrix is reduced to a three-dimensional row vector
($\vec{R}\left(\omega\right)$), whose squared norm gives $F(\omega)$:

$F(\omega)=\left|\vec{R}\left(\omega\right)\right|^{2}.$ (3)

The control vector $\vec{R}\left(\omega\right)$ for a sequence of pulses is
calculated according to the formula,

$\vec{R}\left(\omega\right)=\sum_{l=1,..N}e^{i\omega
t_{l-1}}\vec{R}^{P_{l}}\left(\omega\right)\Lambda^{(l-1)},$ (4)

where the row vector $\vec{R}^{P_{l}}\left(\omega\right)$ is introduced and
$\Lambda^{\left(l-1\right)}$ indicates a component, which is a matrix, of the
$\vec{\Lambda}$ vector.

In this case of dephasing noise, and for a piecewise defined control sequence,
$R^{P_{l}}_{i}\left(\omega\right)$ is given by,

$\begin{split}R^{P_{l}}_{i}\left(\omega\right)&=\frac{\omega}{{\omega}^{2}-{\Omega_{R}}^{2}}\Big{\\{}\delta_{zi}\left(i\Omega_{R}^{l}g_{l}\left(\omega\right)-\omega
f_{l}\left(\omega\right)\right)+\\\
&+\frac{1}{2}\left(\Omega_{R}^{l}f_{l}\left(\omega\right)-i\omega
g_{l}\left(\omega\right)\right)\mathrm{Tr}\left(\sigma_{\phi_{l}}\sigma_{z}\sigma_{i}\right)\Big{\\}};\end{split}$
(5)

where,

$f_{l}(\omega)=\cos{\left(\Omega_{R}^{l}\left(t_{l}-t_{l-1}\right)\right)}e^{i\omega\left(t_{l}-t_{l-1}\right)}-1,$
(6a)

$g_{l}(\omega)=\sin{\left(\Omega_{R}^{l}\left(t_{l}-t_{l-1}\right)\right)}e^{i\omega\left(t_{l}-t_{l-1}\right)},$
(6b)

$\sigma_{\phi_{l}}=\sigma_{x}\cos{\phi_{l}}+\sigma_{y}\sin{\phi_{l}},$ (6c)

where $\delta_{ij}$ is the Kronecker delta function and $\sigma_{i}$ denotes
the usual Pauli matrices, with $i=x,y,z$.

The coefficients of the matrix $\Lambda^{l}$ are then calculated according to,

$\displaystyle\Lambda^{l}_{ij}$
$\displaystyle=\frac{1}{2}\,\mathrm{Tr}\left(Q_{l-1}^{\dagger}\sigma_{i}Q_{l-1}\sigma_{j}\right),$
(7a) $\displaystyle Q_{l}$ $\displaystyle=P_{l}P_{l-1}..P_{1},$ (7b)
$\displaystyle P_{l}$
$\displaystyle=\exp\left({-\frac{i}{2}\Omega_{R}^{l}\left(t_{l}-t_{l-1}\right)\sigma_{\phi_{l}}}\right),$
(7c)

where the cumulative matrices $Q_{l}$ are introduced and $P_{l}$ is the
propagator of the $l$-th operation of a quantum sequence. Observe that for the
case $l=0$: $Q_{0}=P_{0}=\mathcal{I}$.

The calculation of the transfer function therefore requires the definition of
a chain of $N$ operations ($l=1,..N$) of pseudo-rotation, sequenced through a
vector of operation times $\vec{t}=\\{t_{l}\,|\,l=1,..N\\}$, a vector of Rabi
frequencies $\\{\Omega_{R}^{l}\\}$, describing the rotations’ amplitudes, and
a vector of azimuthal angles $\\{\phi_{l}\\}$ describing the rotations’ axes,
with $\phi=0$ for a rotation about the $x$ axis and $\phi=\pi$ for a counter-
propagating beam. For the case of free propagation, $\Omega_{R}^{l}$=0 and
$P_{l}=\mathcal{I}$.

The required sum on $l$ described in Eq. 4 requires the calculation of N
functions, each obtained by performing the calculation described by Eqs. 5-7
relatively to step $l$, which must be iterated for values $l=1,..N$.

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