1. Introduction

Recent advances in cold atom technology have enabled ultra-cold atomic gas setups as one of ideal platforms for simulating many-body physics of exotic quantum matter [1–8]. By confining atoms in periodic optical potentials, optical lattices not only support the band structures for the atomic motion, but also increase the on-site atom density and thus the interaction strength, which is crucial for quantum simulation of interacting particles. To speed up the simulation before the system is decohered, it is important to have optical lattices with a small lattice constant or spacing for strong on-site confinement and quick inter-site tunneling. Unfortunately, this requirement contradicts the other aspect of the quantum simulation: the in situ control and imaging of atoms with high fidelity prefers relatively large lattice constant. So far most popular optical lattices designs use counter-propagating infrared laser beams to form periodic optical potentials with sub-micron lattice constant. Recent progresses in high resolution quantum gas microscopy, with imaging optics numerical aperture (NA) at 0.7∼0.9 level [9–12], have enabled in-situ imaging and manipulation of quantum gases in these lattices. Nevertheless, there are obstacles to overcome in these state-of-art quantum gas imaging techniques, for example, the lattice constants are so small that light-assisted collisions are difficult to control during in situ imaging. On the other hand, to further reduce the lattice constant in these experiments would require super-resolution imaging techniques [13], which are difficult to implement due to the fragile nature of cold atoms.

It is known that optical lattice with its lattice constant dynamically tunable can substantially address the contradictory requirements of having both small lattice constant and precise on-site control in a single cold atom setup. Such a lattice is referred to as an optical accordion [14]. Previous work on dynamic optical lattices includes the schemes based on far-field wavefront modulation with a spatial light modulator, for example those in [15, 16]; and the schemes based on controlling the intersection angles of interfering laser beams [14, 17–19]. In both cases, imaging optics are required for focusing the wavefronts and/or for converting displacements between the interfering laser beams into controllable intersection angles. The resulting smallest achievable lattice constant is thus limited by the finite NA of the imaging optics.

This work intends to develop an optical accordion setup with a wide lattice constant tuning range not limited by the imaging system, and can be conveniently adapted to ultra-cold atomic gas experiments. To vary the periodicity of two beam interference (Beam A and B in Fig. 1), we utilize the flexibility of fiber optics and simply rotate the fiber port for Beam A with a rotation stage. The fiber motion would inevitably change the optical path-length of the laser and introduce phase noise. We suppress the phase noise with a coherent optical fiber link technique [20], by retro-reflecting part of Beam A from a reference cylindrical surface near the experiment, Fig. 1(a). Furthermore, we dynamically adjust the relative phase between Beam A and Beam B to compensate for imperfections of the referencing optical surfaces during the Beam A rotation, by adjusting either the offset of the phase-locked loop or the phase of the laser Beam B. This phase compensation capacity greatly reduces the technical requirements for this accordion lattice scheme to operate at the wavelength limit. The setup can be conveniently adapted to a quantum gas microscope setup, for example, by mounting the hemispheric lens as in [9, 10] to a cylindrical vacuum glass cell and by reflectively coating the side of the hemisphere lens as the cylindrical reference surface for the optical phase lock. We expect the phase-stable optical accordion scheme demonstrated in this work to be a valuable tool for conveniently bridging single-site few atom control in a sparse lattice [21, 22] with the Bloch band many atom control in a wavelength-limited optical lattice.

 

Fig. 1 Experimental setup of the optical accordion. (a) The schematic in the y-z plane (b) The schematic of the Michelson interferometer on the top layer in the x-y plane. PBS: polarization beam splitter; WP: Wave Plate. Notice AOM₂ is vertically positioned so its 0th and 1st order diffractions are displaced along z. (c) The schematic of the rotating fiber port and the accordion lattice (inset) on the lower layer in the x-y plane.

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2. Setup

2.1. Overview

Our optics layout is composed of two layers as in Fig. 1: A Michelson interferometer on the top and an 1D optical accordion setup on the bottom. The -1st order diffractions of acousto-optic modulators AOM₁ and AOM₂, transmitted through the beamspliter PBS₁, are referred to as Beam A and Beam B respectively. Beam A is delivered from the interferometer layer to the lattice layer by a single-mode polarization maintaining Fiber₁. Beam B is delivered directly by mirror M₃ and M₆. They interfere at the lattice location O beneath a cylindrical mirror C₁, Fig. 1(c). To cancel the optical phase variation along the Fiber₁ path, we follow the coherent optical fiber link technique [20] by retro-reflecting part of Beam A, diffracted by AOM₃ to its +1st order, with the cylindrical mirror C₁. The retro-reflected Beam A interferes with the reference beam R, the PBS₁-reflected -1st order AOM₁ diffraction that is retro-reflected by a stable mirror M₄. The heterodyne beat signal with frequency 2ω (twice the AOM₃ frequency) is detected by a fast photodiode D₁ that contains twice the phase shift variation along the PBS₁-Fiber₁-AOM₃-C₁ path Δϕ(t). We retrieve the phase shift by comparing the amplified beat signal with the RF-signal from a voltage-controlled oscillator VCO₁ using a phase-frequency detector D₂. By feeding the loop-filtered error signal to VCO₁ to drive AOM₁, the AOM₁ diffraction properly creates pre-compensating phase shift for Beam A to cancel the single-pass phase variation Δϕ(t)/2 at the lattice location O.

The Beam A fiber port is mounted on a rotation stage with its rotation axis aligned to overlap with the symmetry axis of the cylindrical mirror C₁. The beams A, B cross at the lattice location O, which is also on the C₁ symmetry axis. (Here ’O ’ is at a distance d ≈ 1mm beneath C₁ lower surface. This location should be in ultra-high vacuum in a quantum gas experiment. The distance is adjustable by shifting the location of AOM₃ along z.) The optical accordion is realized by rotating Beam A relative to the stationary Beam B. With a laser wavelength λ, The lattice constant $d=\frac{\lambda }{2\sin \theta /2}$ is thus controlled by the relative angle θ between Beam A,B. With rigidly mounted C₁ and other mirrors/splitters, the Beam A,B relative phase ϕ is stabilized by the phase-locked loop composed of the detector D₂, a low-pass filter, and VCO₁.

2.2. Opto-mechanical design

We use a commercial rotation stage driven by a stepper motor (Zolix RAK100) to achieve 20 µrad angular tuning resolution. To facilitate prealignment of the rotation axis to the C₁ axis, the rotation stage is mounted on a motorized 2D translation stage with computer control.

For the Michelson interferometer in Fig. 1(a) and 1(b) to faithfully detect the phase variation Δϕ(t) along the Fiber₁ path, it is important to minimize the relative motion among the two end mirrors (M₄ and C₁) and the beam splitter (PBS₁). Vibration between these optics and M₃, M₆ that deliver Beam B to the lattice location should also be minimized to finally achieve a quiet optical lattice. For this purpose, we mount C₁ on the opposite side of a thick breadboard together with the other optics elements to within a 20 cm footprint.

2.3. Cylinder/stage alignment

A stable phase lock of Beam A during its rotation requires normal incidence of its +1st diffraction order from AOM₃ to the reference cylindrical mirror C₁ during the rotation, with incidence angle θ_in = 0, so as to maintain a constant retro-reflection efficiency f back to AOM₃ and Fiber₁. For high quality retro-reflection from the cylindrical mirror C₁ with radius of curvature R, the +1st order diffraction also prefers to be focused onto the C₁ surface with a focal spot size w ≪ R. The tolerance of the normal incident angle is determined by its focal spot size w with δθ_in < λ/w. The tolerance to the focal point shift from the C₁ surface δd is determined by the Rayleigh length, δd < w²/λ. Deviation of δθ_in and δd beyond such tolerances would substantially reduce the retro-reflection efficiency f (δθ_in, δd), leading to reduced interference signal at D₁ as well as an offset to the optical phase δϕ(δθ_in, δd). Fortunately, the interference signal at D₁ is proportional to the amplitude of the retro-reflected Beam A and thus $\sqrt{f}$, and the phase-locked loop can typically operate with f ∼ 1% or even less. Furthermore, the phase offset δϕ can be compensated for electronically by AOM₁. Practically we find it is fairly simple to achieve the alignment of δθ_in and δd for stable phase lock over a large tuning range for θ. In this work with w ≈ 60 µm and R = 2mm, we briefly outline the alignment procedure in this work, Fig. 1.

First, the C₁ axis is aligned perpendicular to the x-y surface plane of the rotation stage. We find to adjust the relative angle to within 0.5° angle is sufficient, which can be done fairly easily on an optical table with hand. Next, both the 0th and the -1st order diffractions of Beam A from AOM₃ are focused with a convex lens L₁. With the 0th order delivered to the optical lattice location O to interfere with Beam B, the +1st order is focused to the C₁ surface. By adjusting the L₁ position, the retro-reflection back to AOM₃ and Fiber₁ can be optimized to achieve ~10% back-coupling efficiency or better. The focusing Beam A with w ~ 60 µm allows the retro-reflection angle to deviate by δθ_in ∼ 10 mrad for the 780 nm laser. Finally, the stage rotation axis is translated to overlap with the C₁ axis. To achieve this, we mount the rotation stage on a computer controlled 2D translation stage. We repeatedly scan the incident angle θ of Beam A with the rotation stage, keep monitoring the back-coupled Beam A signal with the detector D₁, and optimize the θ scan range by translating the rotation stage with the translation stage, interleaved with Beam A translation with L₁ for back-coupling efficiency. With the alignment procedure, we achieved a Beam A scanning angle of up to 70° with sufficient back-coupling efficiency for phase-locking, limited only by mechanical obstruction in our setup. With a different mechanical design, a phase-locked 180° θ-scan should be straightforwardly achieved.

2.4. Imaging the optical accordion

With the phase-locked Beam A (0th order from AOM₃) and Beam B beneath the cylinder C₁, and with aligned rotation, we are ready to observe the interference by the Beams A and B at location O, Fig. 1(c). The interference is best observed with a high-NA microscope. In this work, we use a home-made NA=0.3 objective [23] to imagine the fringes with θ varying between 3° and 20°. The effective pixel size of the camera is 0.7 µm.

The alignment of the imaging system itself turns out to be important for faithfully observing the fringe phase stability. Indeed, the fringe we see on the camera is due to the interference of the refracted Beam A and B through the imaging optics. Thus any imaging aberration is added to the relative phase between the two beams on the camera.

2.5. Phase-lock loop

With AOM₃ operated at ω = 2π×130 MHz, the 260 MHz beat signal between the retro-reflected Beam A and Beam R is detected by D₁ (DET02AFC from Thorlabs), amplified by an RF amplifier (ZDC-20-3 from Minicircuits), before being sent to D₂ which is ADF4002 from Analog Devices. The VCO₁ is zx95−148s+ with 10 MHz tuning bandwidth from Minicircuits. The ADF4002 is set with prescaler parameters R=4 and N=2 to compare the 260 MHz beat signal with the reference VCO₁ frequency at 130 MHz, with a phase detector frequency of 65 MHz. The digital charge pump signal from ADF4002 is filtered by a low-pass filter before being sent to control VCO₁. In this work, the charge pump signal is conditioned by a Newfocus LB1005 servo controller with a PI corner frequency of 300 kHz. The closed loop has a phase-correction bandwidth of approximately 300 kHz, as verified by a ∼0.5 µs response time of the lattice interference signal measured by a fast photodiode, after a step-function offset voltage is applied to the controller. Since the high frequency spurious noise of the digital charge pump signal would generate too fast phase noise to affect the slow motion of atoms, it should be relatively simple to achieve a larger phase-correction bandwidth with a careful designed loop filter, thereby achieving an even faster phase-lock offset control.

3. Results

We rotate Beam A with θ from a small value up to 20° so as to compress the optical accordion, and compare the interference fringes recorded on the camera during the compression with and without the phase-locked loop.

We first record the lattice pattern without the phase-locked loop. The interference fringe recorded over a period of 40 seconds is complied in Fig. 2(a). Without the feedback control, the interference fringe vibrates rapidly and substantially over the 40 s period of the rotating stage motion. The phase noise is mainly induced by mechanical motion of the fiber port that substantially changes the optical path length of the light propagation along the single mode fiber.

 

Fig. 2 Compression of the 1D lattice with the two-beam angle scan from 8° to 20°. The CCD camera is set with 30 frames per second. The image are generated by compiling a same line of pixels from image frames of a video record. The scale of intensity is individually adjusted for each frame. (a) Complied image without phase lock. (b) Complied image with phase lock loop closed. Here the drift of lattice phase is about 70 rad. The occasional sudden jump is likely due to the mechanical vibrations of M₆, M₄, M₃ and C₁ as in Fig. 1. The stage stops rotating after the 1130 frames in this measurement.

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Next, we repeat the angular tuning in the same period of time, with a closed phase-locked loop. As demonstrated in Fig. 2(b), the phase noise in the interference fringe is substantially suppressed. However, we observe a slow drift of the fringe position, which is due to imperfect alignment of the cylindrical reflector and also the imaging aberrations. We ignore the different origins of the phase drifts, and compensate for the slow drift of the interference on the camera, by applying the same amount of phase shift to the RF-signal to drive AOM₂ and thus Beam B. (We could also do the phase compensation by adjusting the offset of the phase-locked loop in Fig. 1(b).)

As in Fig. 3(a), with the compensation, the drift of the lattice phase during its compression is also reduced, leading to constrained variation of the lattice fringe position to within 2µm over the whole compression process. The irregular and varying beam profiles in Fig. 3(b) and 3(c) are likely due to imperfect alignment between the Beam A/B’s cross point O and the axis of the stage rotation. Such misalignment should be easy to suppress in a real cold atom experiment when the Beam A/B cross are aligned to the atomic sample at various intersection angles.

 

Fig. 3 (a) Compiled image for compression of the 1D lattice from d = 11 µm to 2 µm with phase locked and phase drift compensated. Here θ is scanned from 3° to 20°. (b),(c) Original images of the 1D lattice with lattice constant d ≈ 8 µm and 2 µm respectively.

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With Fourier analysis of the fringe distribution, the lattice constant is retrieved and plotted versus the rotating angle θ in Fig. 4(c). The result is in good agreement with formula λ/d = 2 sin θ/2.

 

Fig. 4 (a) The cosine function of the phase ϕ at location O during a 35 seconds θ scan. The green and red curves are respectively extracted from the video image for Fig. 2(a)(green) and 3(a)(red) via 2D Fourier transform. (b) The phase noise spectrum of the data in (a) and of the data with the stage kept stationary. In the red plot a few sharp peaks are likely due to mechanical resonances. (c) Plot of the lattice constant as a function of the stage readout θ according to λ/d = 2 sin θ/2. Blue: line plot of the lattice constant extracted from the video images for Fig. 3; Green: the theoretical curve of λ/d = 2 sin θ/2 with a fitted θ offset. Some experimental data are plotted with error bars to reflect the uncertainty of fringe periods due to digital noise and finite extension of the images.

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4. Discussions

4.1. Vibration noise and rotation speed

The vibrations that locally disturb Beam A along the PBS₁-Fiber₁-AOM₃-C₁ path are detected, with noise efficiently compensated for by our phase-locked loop. The fluctuations of fringe positions in Fig. 2(b) and 3(a) are likely caused by vibrations of stepper-motorized rotating stage that propagate to the interferometer layer of our setup and disturb position of PBS₁, M₃,M₄ and M₆ in Fig. 1. The compact mechanical design of the interferometer layer helps to suppress such vibration responses, leading to phase noise amplitude of less than 1 rad and with frequency below 10 Hz as in Fig. 4(a) and 4(b). The performance may already be useful for applications in cold atom research with strong optical lattice confinement, or if some heating is allowed during the atom transportation.

An immediate step to substantially reduce the vibration induced phase noise is to vibrationally isolate the interferometer layer in Fig. 1 from the rotation stage. In practice, in a real cold atom experiment where the cold atoms are trapped near C₁ attached to the interferometer layer, the rotation stage and the mounted optics (the accordion layer in Fig. 1) should be vibrationally isolated from the rest of the apparatus. Such vibration isolation may introduce drifts of stage position/orientation, which can be detected optically and be compensated for with e.g. the offset of the phase-locked loop. In addition, the amount of Beam R/B optics can be reduced and better designed to suppress vibration sensitivity. For example, C₁ can be shared as the end mirror for both Beam A and Beam R; Beam B and Beam R can then propagate toward C₁ along a common optical path (also see Section 4.2 below). It should also be helpful to choose a quieter rotation stage, e.g., those driven by brushless dc or ultrasonic motors. These rotation stages are not only quieter but can also support a higher rotation speed (e.g., 720 degree/sec for the RGV100BL-S rotation stage from Newport), compared with the stage in this work. The combined improvement should bring the accordion operation into an interesting regime of coherently manipulating atoms with the lattice constant varying over an order of magnitude.

4.2. Optical accordion with broadband light

Sometime it is advantageous to use broadband lasers to form the lattice interference. The setup in this work can be modified so that the two lattice beams A and B travel nearly identical optical path lengths, thereby rendering both the lattice position and interference contrast immune to laser frequency broadening. In the following we briefly outline one example of such modifications as Fig. 5.

 

Fig. 5 A more symmetric scheme in which Beam A and B are treated on equal footing and can support optical accordion with a broadband laser.

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In the modified scheme, the PBS₁ reflected AOM₁ diffraction is delivered by a 2nd fiber (Fiber₂) with the same length as Fiber₁. The AOM₂ is relocated after the fiber port, which, together with an additional AOM₄, split the fiber-delivered laser into the Beam B (+1st order of −1st order) and Beam R (0th order of 0th order) paths without changing the frequency. The reference mirror M₃ is conveniently replaced by C₁ with a beam incident angle different from that of the Beam A path by θ. Finally, to close the phase-lock loop, the phases of both Beams A/B are corrected with AOM₁ first, and the correction is removed in Beam B by the AOM₄ diffraction.

With the relative path length of the Michelson interferometer tuned to be identical, the modified scheme should support optical lattices using a broadband laser. Additional benefits include a maximum common-mode vibration noise suppression for Beams B, R, and that both lattice beams are now free to rotate.

4.3. 2D extension

The 1D optical accordion in this demonstration can be extended to 2D by adding another 1D accordion formed by a second pair of laser beams (Beam A′ and B′ in Fig. 6(a)) at a different optical frequency. With large enough frequency difference (e.g., at or beyond MHz level), the rapidly varying Stark shift caused by the interference between the two 1D accordion lattices does not affect the slow motion of ultracold atoms in the lattice, leading to averaged 2D potential with lattice vectors k_A − k_B and $k_{A^{\prime }}-k_{B^{\prime }}$. By mounting Beam A and A′ fiber ports on a same rotation stage, the relative angle of the two 1D accordion can be scanned simultaneously. With phase of each accordion lattice locked, a phase-stable 2D accordion lattice can have tunable lattice constant only limited by light path obstructions from the fixed Beam B and B′ optics. For example, when Beam B and B′ intersect at 120° as in Fig. 6(a), the triangular lattice constant can be tuned from about 10λ down to approximately λ.

 

Fig. 6 Two schemes of 2D optical accordion. (a) Accordion by two laser beam pairs. Beam A and A′ are mounted on the same motorized rotation stage. The relative phase between the pair A, B and the pair A′, B′ are individually locked. The lattice constant is adjusted by scanning the equal angle θ between the A, B beams and A′, B′ beams simultaneously. Bottom: Typical calculated lattice intensity profiles during θ scan for B and B′ intersect at 90° (first line) and 120° (second line). (b) Accordion by three laser beams. The beams share the same linear polarization along z axis. With Beam B fixed, Beam A and A′ rotate oppositely by two rotation stages by angle θ and −θ. Bottom: Calculated lattice intensity profiles with θ = 60, 90, 120, 150°.

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It is worth mentioning that in Fig. 6(a) rotation of lattice axis is accompanied with the accordion operation, which is likely too slow to affect the matterwave tunneling dynamics in our particular accordion lattice scheme. However, at single site level, the conserved lattice symmetry during the accordion operation may open up interesting opportunities to adiabatically compress trapped atoms with a conserved angular momentum, thereby increasing the energy gap between different orbital angular momentum states (an effective increase of effective magnetic field).

The scheme in Fig. 6(b) instead considers a 2D scheme with three beam interference, where Beam A and A′ are both phase-locked to Beam B and are rotated into opposite directions by two rotation stages. Compared with the two-beam-pair scheme in Fig. 6(a), the three-beam scheme requires a more sophisticated dual rotation stage control, but offers two additional advantages: First, the three-beam scheme has the flexibility of creating a 2D accordion of vector light shifts. In particular, with all the three beams linearly polarized in the x-y plane, spatially displaced σ⁺ and σ⁻ lattices are created with adjustable lattice constants. Second, in contrast to the two-beam-pair scheme where the lattice symmetry is largely fixed by the Beam B-B′ angle, the symmetry in the 3-beam scheme can be dynamically adjusted, e.g., from C₃ to C₄ symmetry as θ is scanned from 120° to 90°. Utilizing the fast phase tuning capacity of the accordion scheme, it should be possible to adjust the aspect ratio of the harmonic trap near the lattice site with rapid phase scan, and to rotate the deformed trap minimum [22] so as to create an effective magnetic field. A combination of lattice compression and local rotation would be another interesting scenario offered by our accordion scheme. Finally, Combined with superlattice techniques such as the schemes in [24, 25], an accordion lattice with continuously adjusted 2D symmetry may facilitate preparation and detection of topological orders in quantum gases.

5. Conclusion

In conclusion, we apply the coherent fiber optical link technique [20] in the context of dynamic optical lattices, and realize a phase-stable optical accordion with wavelength-limited tuning capacity. We demonstrate alignment procedures to achieve wavelength-limited phase-stable accordion performance, even with imperfect beam alignment and/or reference optical surfaces. With an improved design for vibration isolation, the scheme should be conveniently applicable to a quantum gas experiment with a section of circular optical access to adiabatically manipulate atoms in a wavelength-limited optical accordion.

Besides applications in ultracold atom physics, the demonstrated technique on phase-stable rotation of a laser beam may also find applications in holographic tomography and other imaging techniques [26, 27] that may benefit from coherent illumination with dynamically adjustable incident angles.