Circulating pulse cavity enhancement as a method for extreme momentum transfer atom interferometry

Large scale atom interferometers promise unrivaled strain sensitivity to midband (0.1 - 10 Hz) gravitational waves, and will probe a new parameter space in the search for ultra-light scalar dark matter. These atom interferometers require a momentum separation above 10^4 \hbar k between interferometer arms in order to reach the target sensitivity. Prohibitively high optical intensity and wavefront flatness requirements have thus far limited the maximum achievable momentum splitting. We propose a scheme for optical cavity enhanced atom interferometry, using circulating, spatially resolved pulses, and intracavity frequency modulation to overcome these limitations and reach 10^4 \hbar k momentum separation. We present parameters suitable for the experimental realization of 10^4 \hbar k splitting in a 1 km interferometer using the 698 nm clock transition in 87Sr, and describe performance enhancements in 10 m scale devices operating on the 689 nm intercombination line in 87Sr. Although technically challenging to implement, the laser and cloud requirements are within the reach of upcoming cold-atom based interferometers. Our scheme satisfies the most challenging requirements of these sensors and paves the way for the next generation of high sensitivity, large momentum transfer atom interferometers.

Traditional atom interferometers rely on multiphoton interactions: two-photon Bragg (24) or Raman (12) transitions, or multiphoton Bragg (25) transitions. Such techniques are sensitive to differential laser phase noise, making them challenging to implement in very large baseline systems where propagation delay causes the laser noise to dominate the overall noise budget. Differential single photon interferometry benefits from greatly reduced susceptibility to laser noise and is the most promising technique for future large-scale interferometers (26,27).
Large momentum transfer with sequential single-photon transitions has been demonstrated in 88 Sr (28)(29)(30), achieving interferometry with a momentum separation of 141 k (31). Extending the technique to 10 4 k raises significant challenges. Spontaneous emission must be minimized during the interferometry sequence, motivating the use of transitions to long lived excited states such as those used in atomic clocks (32,33). The weak coupling to these long lived excited states requires high optical powers to match the Rabi frequencies that can be achieved on short lived states.
The 698 nm optical clock transition in 87 Sr is a suitable candidate for single photon interferometry. With an excited state lifetime of 150s (Γ = 2π · 1 mHz), spontaneous emission is a negligible source of decoherence (34,35). With microsecond π-pulses and therefore MHz Rabi frequencies, the 10 4 k pulse sequence is completed in a small fraction of the total interrogation time. The Rabi frequency is Ω = Γ I/2I S , with I S = 0.4 pW/cm 2 , so microsecond pulses require prohibitively large intensities of several kW/cm 2 . Delivering these pulses with uniform phase also demands flat optical wavefronts (36). These constraints have prevented momentum separations of 10 4 k from being achieved thus far.
Optical cavities offer a potential solution to both of these problems (37): resonant power enhancement reduces the required input power, and spatial filtering improves wavefront flatness.
Impressive early demonstrations of cavity enhanced interferometry have shown the potential of these systems (38,39). However, cavity power enhancement is only effective if the pulse duration exceeds the cavity lifetime. For pulses shorter than, or comparable to the cavity lifetime the cavity fails to reach its maximum power enhancement. The intracavity field will persist long after the input light is removed resulting in significant pulse elongation. This bandwidth limit described by Alvarez et al. (40) restricts the maximum length-finesse product that can be achieved without significant pulse elongation. In a kilometer scale cavity, the bandwidth limit restricts the maximum finesse to F ≤ 10, significantly reducing the mode filtering and power enhancement benefits.
We present a scheme based on circulating, spatially resolved pulses to overcome the bandwidth limit and satisfy the challenging pulse requirements for 10 4 k LMT. A circulating pulse drives a π-pulse once per round trip, removing the need to couple a new pulse into the cavity for each beam splitter. This avoids lifetime elongation and overcomes the bandwidth limit, dramatically increasing the possible lengths and cavity finesses that can be exploited. We present parameters for a cavity and laser system capable of 10 4 k momentum separation on the 698 nm transition in 87 Sr in only 200 ms. Figure 1: Circulating pulses in each of the two running wave modes in this 6 km round trip cavity. On each round trip, additional light is coupled into the cavity to compensate for losses. Serrodyne modulation, applied through a Pockels cell, shifts the frequency of each pulse on each pass to compensate for Doppler shifts. The pulse durations are maximized within the constraint that only one pulse may pass through the atoms (blue) or Pockels cell at a time.

Results
Consider a traveling wave cavity into which we periodically couple short, spatially-resolved pulses of light. The cavity has a 1 km baseline and a round trip path length of 6 km (see Fig. 1).
We use a pulse duration τ = 6 µs, which corresponds to a physical length of 1.8 km and satisfies the condition for spatially resolved pulses; see Materials and Methods. Successive pulses coupled into a single circulating mode are separated by the cavity round-trip time (20 µs) such that they constructively interfere. This periodic train of pulses forms a comb in the frequency domain, but in this regard it is identical to free space LMT schemes using periodic short pulses.
Light is coupled in away from atomic resonance, and shifted onto resonance using Serrodyne modulation once a stationary regime has been reached (see Fig. 2), resulting in a high-intensity pulse circulating inside the cavity. The circulating pulse intensity is adjusted to drive a π-pulse on the atoms loaded in the cavity, delivering a momentum kick of k on each round trip. To ensure successive momentum contributions add constructively, we alternate the pulse direction by populating both circulating modes of the cavity. After 100 ms, 5000 round trips have taken place and the target splitting of 10 4 k is achieved.
In this beam splitter sequence, momentum is only imparted to one arm of the interferometer (referred to as the fast-arm), leaving the other (slow-arm) unaffected. Lifting the initial degeneracy between arms requires careful treatment, outlined in Materials and Methods.
Cavity build-up and pulse requirements In the following analysis we consider only one of the two running wave modes of the cavity containing the circulating pulses, but note that the results apply equally to both. The input intensity is fixed by the requirement that the circulating pulse drives a π−transition on the atoms. The response of the intracavity field to the train of phase coherent input pulses is obtained by solving the propagation equation where t is the transmission coefficient of the input mirror, (1 − γ) is the total round trip loss, and i = √ −1.
If we select t = γ/ √ 2 such that the cavity is impedance matched, this is given bỹ The circulating pulse amplitude increases with each successive input pulse before reaching a stationary value where further pulses only compensate for round-trip losses; see Fig. 2. The stationary pulse intensity is cavity enhanced I circ (t) = (F/π)I in (t) where F is the finesse (41).
Cavity enhancement allows this scheme to achieve π−pulses inside the cavity with reduced input laser power. A finesse of 4000 enables 6 µs π−pulses with an input intensity, at the center of the beam, of only 4.4 W/cm 2 , compared to 5.6 kW/cm 2 without cavity enhancement. This  Figure 2: Cavity response to a periodic train of spatially-resolved coherent square pulses as a function of time. The input intensity I in is plotted in red and the intracavity intensity I circ in blue. The main plot shows the only the intensity maxima, with the full time-dependence of the first input pulses shown in the inset. The system parameters match those described in the text, L = 6 km, F = 4000, τ = 6 µs. The intracavity intensity increases as input pulses coherently add, before saturating at its stationary cavity-enhanced value, which is exactly that of a π-pulse. Grey-shaded areas indicate when the light is off-resonant with the atomic transition. It is shifted on-resonance for one round trip at t = 15 ms to drive the initial π/2-pulse, and again after 120 ms to perform the LMT sequence; see Fig. 4.
limiting factor in the proposed scheme.
Interferometric sequence Pulses of light are coupled into the cavity far-detuned from atomic resonance, avoiding unintentional interactions during the build-up phase. Once the target amplitude has been reached (white areas in Fig. 2), intracavity serrodyne modulation is used to shift the circulating pulse onto atomic resonance. Serrodyne modulation is generated by apply- The cavity baseline is fixed at 1 km, and the maximum pulse duration, minimizing power requirements, is selected. This is the limiting value for the pulses to avoid spatial overlap at the atoms (see Materials and Methods). The parameters used throughout are indicated by the red dot.
ing a linear ramp to an intracavity Pockels cell. High modulation efficiencies and low losses are essential to reaching the required pulse amplitude; some experimental steps towards realizing this are discussed in the materials and methods.
At t = 15 ms the circulating amplitude has reached that of a π/2-pulse and one of the pulses is shifted onto resonance for a single round trip, see white vertical line on Fig. 2. This delivers a π/2-pulse and is followed by a further serrodyne shift to move the pulse away from resonance whilst the amplitude is increased further.
We propose two techniques to lift the initial arm degeneracy: imparting momentum to the slow-arm of the interferometer with the 689 nm transition, and bichromatic beam splitter pulses An initial 698 nm (red, wavy) π/2-pulse transfers the atoms into a superposition between the ground state, slow-arm (blue), and the excited state, fast-arm (red). Sequential counterpropagating π-pulses on the 689 nm transition (pink, wavy) deliver 100 k of momentum to the slow-arm only. With the frequency degeneracy lifted, the main LMT sequence begins at 120 ms, delivering 10 4 k of momentum to the fast-arm through counterpropagating π-pulses on the 698 nm transition.
to track the recoil of both arms.
At t = 120 ms, the circulating intensity has reached the value required for a π-pulse and the initial degeneracy has been lifted. Intracavity serrodyne modulation is used to shift both circulating pulses onto resonance with the fast-arm of the interferometer. On each round trip they deliver a pair of sequential counterpropagating π-pulses, imparting 2 k of momentum.
Recoil shifts and gravitationally induced Doppler shifts are compensated every round trip with further serrodyne modulation. After 5 × 10 3 round trips, a momentum separation of 10 4 k between the interferometer arms has been achieved. The circulating pulses are either dumped from the cavity or serrodyne shifted back away from resonance in preparation for the next beam splitter. The total sequence duration is less than 1 s, a duration over which spontaneous emission losses remain negligible. A similarly constructed beam splitter sequence is used to close the interferometer.
Doppler shift and phase compensation In addition to its primary role of shifting pulses on and off the atomic resonance, serrodyne modulation is used to compensate for Doppler shifts in the sequence. Fig. 4 shows the fast-arm recoiling upwards on each successive beam splitter pulse. The Doppler effect will cause the downward-going pulse to appear blueshifted, whilst the upward-going pulse appears redshifted. We compensate for this by applying negative serrodyne T=100nK T= 50nK T= 10nK T= 5nK T= 1nK Figure 5: Fidelity of the whole LMT sequence as a function of momentum separation, for various atomic temperatures. The sequence fidelity is obtained as the cumulative product of individual pulse fidelities, which are obtained by averaging the transition probability over the cloud instantaneous spatial and velocity distributions. The initial cloud radius is 200 µm and the beam radius is 1.5 cm. To obtain a sufficient contrast at the end of the 10 4 k LMT sequence, a cloud temperature less than 10 nK is required. Cloud expansion, Doppler shift compensation and spontaneous emission are included in the model. The pulse bandwidth is neglected, but any resulting errors can be compensated with frequency and intensity adjustments (44,45).

Implementation in smaller cavities
This scheme could be tested on a smaller system operating on the 689 nm 1 S 0 − 3 P 1 transition in 87 Sr, where higher Rabi frequencies enable shorter pulses and correspondingly smaller cavities. A 40 m-round-trip cavity of finesse 1000, with 16 ns input pulses every 133 ns, will require only 1.2 W of laser power to implement π-pulses with a beam waist of w 0 = 15 mm. Due to the large pulse bandwidth, no Doppler shift compensation is required as both arms are addressed with high-fidelity. Whilst spontaneous emission will limit the performance on this transition below the target of 10 4 k momentum separation, it will still provide a useful performance enhancement to these lab-scale systems, and serve to validate the scheme.

Discussion
We have presented a novel scheme for cavity enhanced atom interferometry to enable extremely large momentum transfer beam splitters in large scale atom interferometers. Intracavity serrodyne modulation allows circulating, spatially resolved pulses to be recycled, overcoming cavity lifetime elongation and the pulse bandwidth limit. Serrodyne modulation is used to shift the pulse on and off resonance and compensate for photon recoils and gravitational Doppler shifts.
We analyze the case of a kilometer scale atom interferometer for gravitational wave detection, and find that a cavity with 6 km round trip path length and a finesse of F = 4000 can generate a circulating pulse intensity of 5.6 kW/cm 2 with only 4.4 W/cm 2 at the input. When applied to the 87 Sr 698 nm clock transition this enables 6 µs π-pulses in a mode of 1.5 cm radius with an input laser power of only 16 W. The overall sequence fidelity is found to be limited by the temperature of the atomic cloud. A beam splitter with 10 4 k momentum separation between the arms and combined fidelity > 0.25 requires a vertical velocity spread of 1.2 mm/s, corresponding to a temperature selectivity of 5 nK. Further reductions in cloud temperature increase fidelity and hence the maximum total momentum separation. This is the first practical, albeit challenging, approach to the generation of short, high-fidelity pulses on this transition.
Circulating pulse interferometry can also be applied to 10 m atom interferometers operating on the intercombination line in 87 Sr at 689 nm. Resonant power enhancement and spatial mode filtering will allow for an increase in fidelity and therefore the possible momentum separation in these systems, improving sensitivity.

Materials and Methods
Initial arm degeneracy Our scheme relies on velocity selective pulses such that only one arm of the interferometer is addressed, leaving the other unaffected. However, the large Rabi frequency ( 80 kHz) and small initial frequency separation from the first π/2-pulse (9.4 kHz) results in the two arms being initially degenerate. The two arms cannot be discriminated in frequency with high-fidelity until the frequency separation exceeds the pulse bandwidth. Since one recoil of k produces a Doppler shift of k 2 /m = 2π · 9.4 kHz, 10 − 100 beam splitters are required to lift this degeneracy.
One method to open the interferometer is to create the initial momentum separation with πpulses on the 1 S 0 − 3 P 1 689 nm transition. The initial π/2-pulse on 698 nm splits the atoms into a superposition of the long lived excited state and the ground state, referred to as the fast and slow-arm respectively. We apply one hundred sequentially counterpropagating π-pulses on the 689nm transition, imparting 100 k of momentum to the slow-arm on the interferometer whilst leaving the fast-arm in the excited state unaffected. The resulting Doppler shift is 100 k 2 /m 2π · 1 MHz, greatly exceeding the Rabi frequency of the clock pulses, ensuring that the fast-arm of the interferometer can be uniquely addressed.
The 689nm transition allows for much higher Rabi frequencies ∼ 100 MHz, enabling 10 ns π-pulses without cavity enhancement. The 100 pulse sequence is completed within 5 µs: 3.3 µs for the pulses to propagate 1 km between the clouds and 2 µs for 100 pulses. The pulse bandwidth is large compared to the total Doppler shift so we achieve high-fidelity beam splitters without Doppler compensation (31). The 3 P 1 state has a lifetime of 21.6 µs requiring a rapid pulse sequence ending in the ground state to minimise losses due to spontaneous emission. This beam splitter sequence can be accommodated within the general scheme depicted on Fig. 2  This geometry results in a residual birefringence from each crystal, which we compensate by arranging the two crystals orthogonally. Phase compensation can be achieved with the same methods discussed in the results, but must now be applied to both polarizations simultaneously.
This dual polarization, dual frequency scheme requires careful control of the light intensities on a pulse-by-pulse basis to ensure this bichromatic light field delivers high-fidelity π-pulses to both arms on each round trip. The details of the specific powers and frequencies required to achieve this are beyond the scope of this paper.
Maximum pulse duration The scaling of input power with pulse duration motivates the use of the longest pulses possible. However, the interferometry scheme described in this work requires successive pulses to alternate in direction and to be temporally distinct when they pass through the atoms and the Pockels cell. This constrains the maximum pulse duration.
We define an exclusive length L ex as the region within the cavity where the pulses must not overlap. Two pulses, of equal limiting duration τ max , propagating in opposite directions, will fully overlap when one pulse leaves the exclusive length and the other is about to enter.
This condition is repeated on both sides of the exclusive length. Symmetry considerations show that there is an equal exclusive length on the opposite side of the cavity. Fig. 6 illustrates the locations of the pulses and exclusive length in a generalized ring cavity. Summing the exclusive length and pulse length the condition becomes clear: For pulses with τ ≤ τ max and correct input timings, singular occupation of the exclusive lengths will be achieved. Pulse durations approaching τ max reduce performance requirements for the cavity and input lasers. The cavity presented in this paper has L RT = 6 km and L ex = 1 km yielding τ max = 6.66 µs, just exceeding the selected pulse duration of τ = 6 µs. is achieved by applying a single, linear ramp for the duration of the pulse (47). This is a χ (2) effect, so the phase shift φ, is proportional to applied voltage:

Control of frequency and phase
Frequency is defined as dφ/dt so a linear phase chirp produces a frequency shift. By adjusting the value of k the frequency can be shifted by up to 1/τ Hz for a standard length Pockels cell, (ramping from φ = −π → π over the pulse duration τ ) and up to N/τ Hz for an extended length N π-Pockels cell. The performance of this scheme is limited by the modulation efficiency and any losses associated with the modulator itself. Both circulating modes use the same po-larization of light, so reflection losses can be reduced by using a Pockels cell with Brewster's angle cut crystal facets (48).
Phase compensation can be achieved in two ways. DC voltages can be applied to the intracavity Pockels cell to ensure that the phase on the input mirrors is constant for every pulse in both circulating modes. The voltages required will differ every round trip, and be different for each circulating mode. Longer Pockels cells will increase flexibility to introduce a phase shift whilst leaving enough headroom for modulation. A simpler approach experimentally, is to adjust the phase of the input pulses on each round trip to match that of the circulating field.
A Pockels cell or acousto-optic modulator on each of the input beams could achieve this in an agile and controllable way. Regardless of which phase compensation technique is adopted, the frequency of the input beams must also be tuned to track the circulating fields.
Laser system This scheme places demanding constraints on the input lasers, requiring narrow linewidth, rapid tunability, and high output power. It will require at least two separate lasers to inject light into both circulating modes of the cavity. The coherence length of the lasers must exceed the average distance traveled by a photon within the cavity. This ensures that successive input pulses continue to constructively interfere with the intracavity circulating pulse. With a cavity photon lifetime of t c = LF/πc 30 ms, a laser linewidth of ∼ 1 Hz will ensure the limit is comfortably met. The optical paths on the input to the cavity must also be phase stable at ∼ 1 Hz level. These performance levels are demanding, but regularly achieved in optical clock lasers operating on the 698 nm transition (49,50).
Clock lasers do not typically produce the output powers required for this scheme, so we propose the use of a clock laser as the master to injection-lock a high power slave (51). Commercially available high power Ti:sapphire lasers generate ≥ 5W, so the required output of 16 W would require the coherent combination of four of these. The pulsed nature of the output may allow the use of a single Q-switched slave laser, reducing the system complexity. Injection locking also simplifies frequency and phase compensation which may be accomplished by appropriate modulation of the master laser, prior to injection seeding. This retains the stability properties of the master laser and allows modulation to occur at low power levels.