Optical cavity array

Abstract

The optical cavity array includes a plurality of mirrors that form an optical cavity, a first lens system located within the optical cavity, and a second lens system located within the optical cavity. The first lens system has a first output facing a first mirror of the plurality of mirrors and a second output facing a second mirror of the plurality of mirrors. The second lens system has a second input facing the first input and a second output facing the second mirror. The first and second lens systems are configured such that the optical cavity supports a longitudinal mode that is non-degenerate in the lateral direction and forms spatially separated waists that exist along a focal plane located axially between the first input and the second input. When the longitudinal mode is excited, the waists can be used as an array of optical dipole traps.

Description

Background Art

[0001] (Related Application) This application claims priority to U.S. Provisional Patent Application No. 63 / 364,678, filed May 13, 2022, which is incorporated herein by reference in its entirety.

[0002] (Background) Cavity quantum electrodynamics (cQED) is the study of the interaction between light and matter inside a resonant cavity. The matter can be atoms, ions, molecules, or other types of quantum particles or emitters that couple to light. The quantum particles can be trapped near the waist of the cavity's excitation mode, creating optical tweezers inside the cavity.

Summary of the Invention

Means for Solving the Problems

[0003] (Abstract) This embodiment includes an optical cavity array that can be used to generate multiple longitudinal modes within an optical cavity. Advantageously, these longitudinal modes are laterally separated from each other and thus not coupled to each other. Using a lens system, each longitudinal mode forms an individual one of a plurality of foci that coincide with the focal plane existing within the optical cavity. Due to the lateral separation of the modes, these foci are also laterally separated. Longitudinal modes with this behavior are referred to herein as "transversely non-degenerate." Each focus can be used as an optical dipole trap, i.e., an optical tweezer, for trapping quantum emitters (e.g., neutral atoms, atomic ions, molecules, etc.). It can be considered that the plurality of foci thus form an array of optical tweezers for trapping several quantum emitters.

[0004] Multiple foci may be separated, for example, by several microns in the lateral direction. These embodiments advantageously avoid the aberration sensitivity and resonator stability limits that exist when many foci are generated using a recirculating resonator (also known as an "imaging resonator"). Each focus may be considered to be generated by its own optical resonator, i.e., a cavity, having its own unique path. However, most, if not all, of the optical components are bulk. These optical resonator paths can form an array that extends laterally to the optical axis in one or two dimensions. Each optical resonator path of the array can operate far from any mode degeneracy that would tend to make it susceptible to mode mixing, aberration-induced instability, or both, and be independently stable.

[0005] In an embodiment, the optical cavity array includes a plurality of mirrors that form the optical cavity, a first lens system positioned within the optical cavity, and a second lens system positioned within the optical cavity. The first lens system has a first output facing a first mirror of the plurality of mirrors and a first input facing a second mirror of the plurality of mirrors. The second lens system has a second input facing the first input and a second output facing the second mirror. The first and second lens systems are configured to support longitudinal modes in which the optical cavity is non-degenerate in the lateral direction. These longitudinal modes form spatially separated waists that exist along a focal plane axially positioned between the first input and the second input. When the longitudinal modes are excited, the waists can be used as an array of optical dipole traps (i.e., an optical tweezer array) or an array of optical lattices.

Brief Description of the Drawings

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DETAILED DESCRIPTION OF THE INVENTION

[0017] (Detailed Description) In cavity quantum electrodynamics (cQED), it is desirable to engineer the single-particle cooperativity η to be as large as possible at high frequencies.

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[0018] Another way to express the cooperativity is

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[0019] The prior art optical cavities used for cQED achieve a high cooperativity η by employing a relatively large waist w 0 and a very high finesse

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[0020] One aspect of this embodiment is the realization that increasing the single-particle cooperativity η does not impose any significant limitation on the cavity length L. The idea that cQED requires a small mode volume V resulted in an inappropriate way of speaking by describing the Purcell factor F P = 3λ 3 Q / (4π 2 V). The Purcell factor F P quantifies the amount by which the natural emission rate of a quantum emitter is enhanced when it is located inside a resonant cavity having a quality factor Q. The expression makes it seem advantageous for increasing the small mode volume V, but ignores the fact that Q drops as the cavity length L decreases. What is relevant for cQED is the finesse P which is given by, where f

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[0021] This embodiment achieves a waist w smaller than that achieved using the prior art optical cavity described above. 0 A lens inside the cavity is used to achieve this.

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[0022] The increase in the photon collection probability P C will substantially improve the state detection of cQED setups and other systems using optical tweezer arrays. Thus, this embodiment may be used to generate a photon-matter interface that efficiently converts quantum information between photonic qubits and matter-based qubits (e.g., trapped ions, neutral atoms, defects in diamonds, quantum dots, etc.). Such an optical coupling system can increase the number of qubits in a quantum computer, thereby improving qubit scaling. For example, an optical coupling setup can be used to efficiently transfer optical information between spatially heterogeneous ion traps, thereby enabling quantum computing beyond the melting size limit of a single ion crystal.

[0023] Another use of this embodiment is sensing using color centers. Here, the optical cavity improves the light collection, thereby enabling faster and more accurate readout of the color center state. Such a scanning cavity microscope is not of high finesse, but has a small waist w0 It will rely on 0 and greatly relax the material constraints. As another application, the optical cavity is used for accelerating the state detection digit difference related to atomic array quantum simulators and computers, thereby enabling optical medium non-local gates and real-time feedback-based error correction.

[0024] FIG. 1 is a side view of an optical cavity array 100 forming a plurality of longitudinal modes 140 that are non-degenerate in the lateral direction in an embodiment. FIG. 2 is a side view of the optical cavity array 100 of FIG. 1 showing in more detail how each longitudinal mode 140 forms an individual one of a plurality of foci 130 in the focal plane 108. FIGS. 1 and 2 are most clearly viewed together using the following description.

[0025] The optical cavity array 100 includes a first mirror 102 and a second mirror 116 that face each other and form a Fabry-Perot cavity. The mirrors 102 and 116 are opposing retroreflectors that define an optical axis 118 whose position is parallel to z (see the right coordinate system 120). For clarity, the directions along x and y are also referred to as the "lateral direction", while the direction along z is referred to as the "longitudinal direction" or the "axial direction". In the embodiment of FIG. 1, the first mirror 102 is a planar mirror that exists perpendicular to the optical axis 118. However, the first mirror 102 may have different geometries as described in more detail below. Further details regarding the geometry of the second mirror 116 are described below with respect to FIGS. 3-7.

[0026] The optical cavity array 100 also includes a first lens system 122 located axially between the mirrors 102 and 118. The first lens system 122 has a first input facing in the +z direction (i.e., towards the second mirror 116) and a first output facing in the -z direction (i.e., towards the first mirror 102). In the example of FIG. 1, the first lens 106 defines the first input and the second lens 104 defines the first output. The optical cavity array 100 also includes a second lens system 124 located axially between the first lens system 122 and the second mirror 116. The second lens system 124 has a second input facing the first input (i.e., the first lens 106) and a second output facing the second mirror 116. In the embodiment of FIG. 1, the third lens 110 defines a second input and the fourth lens 112 defines a second output.

[0027] The first and second inputs (i.e., lenses 106 and 110) are axially separated and form a focal plane 108 therebetween. The first lens 106 has a first focal length f 1 while the second lens 104 has a first focal length f 1 The second focal length, f 2 The surface of the first lens 106 closest to the focal plane 108 is a first working distance WD therefrom. 1 Generally, the first working distance WD 1 is the first focal length f 1 However, among other parameters, the focal length f 1 and f 2 and a first lens spacing D between the lenses 104 and 106. 1 The first lens system 122 may be at a first rear distance B 1 In the embodiment of FIG. 1, the first mirror 102 is disposed behind the second lens 104 at a first rear distance B 1 However, the first mirror 102 does not need to be located exactly at that position. In general, the first rear distance B 1 is the second focal length f2 is not equal to. However, f 2 >>f 1 and D 1 =f 1 +f 2 Regarding the first back distance B 1 may have a value close to that of the second focal length f 2 .

[0028] Although not labeled in FIG. 1, the second lens system 124 is similar to the first lens system 122, although not exactly equal. The third lens 110 has a third focal length f 3 , and the fourth lens 112 has a fourth focal length f 3 in the embodiment of FIG. 2 that is greater than the third focal length f 4 . The surface of the third lens 110 closest to the focal plane 108 is located at a second working distance WD 2 from there. The lenses 110 and 112 are axially separated by a second lens interval D 2 . The second lens system 124 has a second back distance B 2 between the fourth lens 112 and the second mirror 116.

[0029] In FIG. 1, the lenses 106 and 104 are separated by approximately the sum of their focal lengths, i.e., D 1 ~f 1 +f 2 . However, the lenses 106 and 104 may be separated by different values of D 1 . Similarly, the lenses 110 and 112 are separated by approximately the sum of their focal lengths, i.e., D 2 ~f 3 +f 4 . However, the lenses 110 and 112 may be separated by different values of D 2 . Also, in FIG. 1, the first lens 106 has a higher NA than the second lens 104, and the third lens 110 has a higher NA than the fourth lens 112. As can be seen, the effective aperture of the second lens 104 is, in particular, when the first focal length f 1 is the second focal length f2 When it exceeds that of the first lens 106, it can be larger than that of the first lens 106. Similarly, the effective aperture of the fourth lens 112 may exceed that of the third lens 110.

[0030] Each of the lens systems 122 and 124 is shown in FIG. 1 together with two plano-convex lenses, but one or both of the lens systems 122 and 124 may have more than two lens elements, other types of lens elements, or both. As is known to those skilled in the art, such multi-lens systems may be used to correct aberrations (e.g., chromatic aberration, spherical aberration, coma aberration, astigmatism, etc.). Further, the lens systems 122 and 124 are not limited to thin lenses and may alternatively or additionally include thick lenses, compound lenses, objective lenses, GRIN lenses, aspherical lenses, and the like. Thus, one or both of the lens systems 122 and 124 may be configured differently from those shown in FIG. 1 without departing from the scope of this specification.

[0031] One or both of the lens systems 122 and 124 may have a finite conjugate ratio and thus a magnification. For example, when the second lens system 124 has a finite conjugate ratio, it may be configured to image the focal plane 108 onto the imaging plane 114. The second mirror 116 may be located at or near the imaging plane 114. However, the second mirror 116 does not necessarily have to be located exactly at the imaging plane 114. In fact, for example, it may be advantageous to intentionally position the second mirror 116 away from the imaging plane 114 in order to improve cavity stability, correct aberrations, or achieve a certain design specification.

[0032] Figure 1 shows an optical cavity array 100 that supports three longitudinal modes 140 that are non-degenerate in the lateral direction. Each of the longitudinal modes 140 corresponds to a standing wave that resonates with a Fabry-Perot cavity. Specifically, the first longitudinal mode 140(1) spatially overlaps with the optical axis 118 at all axial positions between the mirrors 102 and 116, the second longitudinal mode 140(2) is laterally positioned above the optical axis 118 (i.e., in the +x direction) between the first mirror 102 and the focal plane 108, and the third longitudinal mode 140(3) is laterally positioned below the optical axis 118 (i.e., in the -x direction) between the first mirror 102 and the focal plane 108. Between the focal plane 108 and the second mirror 116, the second longitudinal mode 140(2) and the third longitudinal mode 140(3) are reversed, with the second longitudinal mode 140(2) existing below the optical axis 118 and the third longitudinal mode 140(3) existing above the optical axis 118.

[0033] In FIGS. 1 and 2, each longitudinal mode 140 is represented by a shaded region that shows how its spot size (i.e., the lateral dimension along x) varies with the axial position z. Each longitudinal mode 140 is assumed to be in a TEM 00 transverse mode. Thus, the lateral intensity profile of each longitudinal mode 140 is a Gaussian distribution, and the spot size may be the 1 / e 2 intensity radius or diameter of the Gaussian intensity profile.

[0034] Figure 2 shows how each longitudinal mode 140 forms a focus 130 on or near the focal plane 108. Specifically, the first longitudinal mode 140(1) forms a first focus 130(1) that coincides with the optical axis 118, the second longitudinal mode 140(2) forms a second focus 130(2) that is laterally positioned above the optical axis 118, and the third longitudinal mode 140(3) forms a third focus 130(3) that is laterally positioned below the optical axis 118. Thus, the foci 130(1), 130(2), and 130(3) are laterally separated. For clarity, in FIG. 2, each of the foci 130 is enclosed by a small circle.

[0035] For clarity, in FIGS. 1 and 2, the respective lateral centers (i.e., the points of maximum intensity) of the longitudinal modes 140(1), 140(2), and 140(3) are identified using dashed lines. These dashed lines are also referred to as the central axes of the longitudinal modes 140. As can be seen in FIG. 2, these central axes do not intersect in the focal plane 108. Rather, the point at which they intersect is located at a distance Δz from the focal plane 108. This feature is thought to be related to the ability of the optical cavity array 100 to form longitudinal modes that are non-degenerate laterally and result from lens systems 122 and 124 that are configured differently. Thus, if the lens systems 122 and 124 were configured exactly the same, Δz would be zero and the optical cavity array 100 would no longer be able to support longitudinally non-degenerate modes (i.e., all of the longitudinal modes 140 would "collapse" into one degenerate mode similar to a prior art Fabry-Perot cavity).

[0036] There are many ways in which the lens systems 122 and 124 can be configured differently. For example, the lenses 104, 106, 110, and 112 may be selected such that D 1 ≠D 2 , WD 1 ≠WD 2 , B 1 ≠B 2 , or a combination thereof. In another example, the first mirror 102 may be axially positioned such that it is separated from the second lens 104 by a rear distance B 1 . Similarly, the second mirror 116 may also be axially positioned such that it is separated from the rear distance B 2 . In another example, the lenses 104 and 106 may be positioned such that D 1 ≠f 1 +f 2 . Similarly, the lenses 110 and 112 may be positioned such that D 2 ≠f 3 +f 4 .

[0037] For clarity, only three foci 130(1), 130(2), and 130(3) are shown in FIG. 2. However, the optical cavity array 100 may alternatively form only two foci 130 or more than three foci 130. The foci 130 are shown in FIGS. 1 and 2 as extending along the x, but the foci 130 may alternatively or additionally extend along the y. The number of foci 130 formed by the optical cavity array 100 can be on the order of hundreds, if not more.

[0038] For each longitudinal mode 140 having its waist (i.e., minimum spot size) at its respective focus 130 within the focal plane 108, each of the lenses 106 and 110 may have a high NA (0.5 or greater). As can be seen in FIG. 1, the spot size of each longitudinal mode 140 is also small at the mirrors 102 and 116. However, due to the relationship between the NAs of the lenses 104, 106, 110, and 112, the spot size of the longitudinal mode 140 may be larger at the mirrors 102 and 116 than at the focal plane 108. In general, one or both of the lenses 106 and 110 may have an NA less than 0.5 without departing from the scope of this specification.

[0039] As can be seen in FIG. 1, the lateral spacing of the longitudinal modes 140 is greater at the mirrors 102 and 116 than at the focal plane 108. The lateral spacing is measured between the central axes of adjacent longitudinal modes 140 (i.e., between the dashed lines). This effect is thought to result from the magnification of the lens systems 122 and 124. Specifically, the first lens system 122 has a first magnification M 1 >1 for light propagating through it in the -z direction. Similarly, the second lens system 122 has a second magnification M 2 >1 for light propagating through it in the +z direction. The longitudinal modes 140 have a first lateral spacing d 1 at the first mirror 102 and a second lateral spacing d 2 at the second mirror 116. In FIG. 1, d 1>d 2 and this can occur when M 1 >M 2 However, the lens systems 122 and 124 may alternatively be configured such that d 2 <d 1 is the case.

[0040] One advantage of having a greater lateral separation at the mirrors 102 and 116 compared to the focal plane 108 is the ease of coupling light into the optical cavity array 100. The longitudinal mode 140 may be excited, for example, by transmitting light through one of the mirrors 102 and 116. The light may be a single monochromatic laser beam with a spot size large enough to cover all of the longitudinal modes. Alternatively, the light may be several smaller monochromatic laser beams that are laterally displaced from each other. Each of these several laser beams may be individually controlled (e.g., intensity, propagation direction, etc.) to couple into individual ones of the longitudinal modes 140. Such individual control may be easier to implement when the several laser beams are displaced from each other by a greater lateral distance.

[0041] Another advantage of having a greater lateral spacing in mirrors 102 and 116 is to process the light leaking out of the optical cavity array 100. This leaked light can result from fluorescence emitted into the longitudinal mode 140 by a quantum emitter that is either in, located at, or trapped in the waist of the longitudinal mode 140. The leaked light exits the optical cavity array 100 via transmission through one or both of mirrors 102 and 116. Alternatively, or in addition, light can be coupled out of the optical cavity array 100 using an intra-cavity beam sampler. To minimize aberration, this beam sampler may be placed within a low NA region of the optical cavity array 100 (e.g., between the fourth lens 112 and the second mirror 116, or between the second lens 104 and the first mirror 102). In either case, it may be necessary to process the leaked light differently depending on the resulting longitudinal mode 140. This different processing can be facilitated by spatially separating the leaked light, which is easier to do when the lateral spacing is greater.

[0042] FIG. 1 shows an optical cavity array 100 with mirrors 102 and 116 that form a Fabry-Perot cavity, although the optical cavity array 100 may alternatively be positioned and oriented to have three or more mirrors to form a ring cavity. In this case, each of the three or more mirrors may be a rotating mirror, as opposed to a retroreflector. The ring cavity also supports multiple longitudinal modes. However, in this case, each longitudinal mode corresponds to a traveling wave, as opposed to a standing wave.

[0043] FIG. 3 is a side view of a polygonal mirror 300, which is an embodiment of the second mirror 116 of FIG. 1. The polygonal mirror 300 has a first surface 304 that forms a first oblique angle with the optical axis 118, a second surface 302 that is perpendicular to the optical axis 118, and a third surface 306 that forms a second oblique angle with the optical axis 118. The surfaces 302, 304, and 306 are positioned to retroreflect light incident thereon back onto itself. More specifically, the first oblique angle is selected such that the third longitudinal mode 140(3) retroreflects from the first surface 304. Similarly, the second oblique angle is selected such that the second longitudinal mode 140(2) retroreflects from the third surface 306. The polygonal mirror 300 may be shaped with additional surfaces for when there are more than three longitudinal modes 140. The second surface 302 is part of the polygonal mirror 300 that cooperates with the first mirror 102 to define the optical axis 118.

[0044] FIG. 4 is a side view of a polygonal mirror 400, which is similar to the polygonal mirror 300 of FIG. 3 except that the second surface 302 is not axially recessed. Specifically, the second surface 302 is located further in the -z direction within the polygonal mirror 400 compared to the polygonal mirror 300. Due to the absence of this recess, the polygonal mirror 400 may be easier to fabricate than the polygonal mirror 300.

[0045] FIG. 5 is a side view of a cat's eye retroreflector array 500, which is another embodiment of the second mirror 116 of FIG. 1. The cat's eye retroreflector array 500 includes a microlens array 502 that extends in one or both of two lateral dimensions (i.e., x and y). Each longitudinal mode 140 uniquely interacts with one of the microlenses of the array 502. The microlens array 502 focuses the longitudinal modes 140 such that their central axes are all parallel to the optical axis 118. As a result, the longitudinal modes 140 can be reflected using a planar mirror 504 that is located behind the microlens array 502 (i.e., in the +z direction) and is oriented perpendicular to the optical axis 118.

[0046] FIG. 6 is a side view of a convex micro-mirror array 600, which is another embodiment of the second mirror 116 of FIG. 1. The micro-mirror array 600 is a one- or two-dimensional array of convex mirrors that can be fabricated, for example, by depositing a high-reflectivity coating on the convex surface of a microlens array (e.g., the microlens array 502 of FIG. 5). As can be seen in FIG. 6, the micro-mirror array 600 is configured to retroreflect the longitudinal mode 140 regardless of their different angles of incidence.

[0047] FIG. 7 is a side view of a retroreflector array 700, which is another embodiment of the second mirror 116 of FIG. 1. The retroreflector array 700 includes a microlens array 702 that is similar to the microlens array 502 of FIG. 5, except that it is axially positioned beyond the imaging plane 114 in the +z direction. The microlens array 702 collimates the longitudinal mode 140 and deflects the longitudinal mode 140 such that their central axes are parallel to the optical axis 118. In this case, a planar mirror 704 oriented perpendicular to the optical axis 118 can be used to retroreflect the longitudinal mode 140.

[0048] Depending on the quality of the optical system, it may be necessary to individually adjust the resonance frequencies of each longitudinal mode 140. Such adjustment can be used, for example, to ensure that the atoms emit fluorescence that resonates with the longitudinal mode 140 in which they are trapped. This can be achieved, for example, using a phase-only spatial light modulator placed within the low-NA region of the optical cavity array 100 (e.g., between the fourth lens 112 and the second mirror 116, or between the second lens 104 and the first mirror 102). Alternatively, once the necessary phase shift is determined, a custom antireflection coating phase mask can be disposed within the optical cavity array 100, which can advantageously suffer lower optical insertion losses than a spatial light modulator.

[0049] When the optical cavity array 100 is excited using light (as described above), optical dipole traps are formed at each of the foci 130. The optical dipole traps may be standing wave optical lattices or traveling wave optical tweezers. The resulting plurality of optical dipole traps may be used to trap cold or ultra-cold atoms, or another type of optically-trappable quantum emitter. Advantageously, these optical dipole traps are located far enough from nearby physical surfaces (e.g., lenses 106 and 110) to ensure that the trapped atoms will not be ejected in response to collisions with such surfaces. Once the atoms are trapped, they may then be driven, measured, coupled, probed, or otherwise manipulated as needed for the application at hand. For example, fluorescence can be collected from at least one atom trapped in one of the optical dipole traps. As described above, the fluorescence can then be coupled into one of the longitudinal modes 140, where it can be transmitted through one of the mirrors 102 and 116.

[0050] To further facilitate cold atom trapping, the optical cavity array 100 may be arranged such that the array of foci 130 is located inside an ultra-high vacuum environment. However, some conventional vacuum chambers are very large and the lenses 106 and 110 cannot be placed outside the vacuum chamber, which would then cause them to be too far apart from each other to produce a focus 130 (i.e., waist) that is sufficiently tight for the application at hand. In such a situation, the lenses 106 and 110 can be brought closer to each other by mounting one or both of them inside the vacuum chamber (see, e.g., FIG. 10). Additional components of the optical cavity array 100 may be mounted inside the vacuum chamber. Vacuum windows may be used on the vacuum system to allow light to pass through it.

[0051] FIG. 8 is a side view of an optical cavity array 800 that is similar to the optical cavity array 100 of FIG. 1, except that the first lens system 122 is excluded and the first mirror 102 is positioned at or near the focal plane 108. FIG. 9 is a side view of the optical cavity array 800 of FIG. 8, showing in more detail how the focal point 830 is positioned near the first mirror 102. FIGS. 8 and 9 are best viewed together using the following description.

[0052] The optical cavity array 800 is solid state and is thus advantageous for quantum emitters that do not need to be magnetically or optically trapped. In FIGS. 8 and 9, a sample of a nonlinear emitter 802 (e.g., a wafer or substrate) is attached to the front of the first mirror 102. Examples of such nonlinear emitters 802 include, but are not limited to, rare earth ions, quantum dots, solid state color centers (e.g., silicon vacancy centers in diamond, nitrogen vacancy centers in diamond, etc.), and molecules embedded within a host matrix.

[0053] The optical cavity array 800 includes a lens system 824 that projects the focal plane 108 onto the imaging plane 114. FIGS. 8 and 9 show a first longitudinal mode 840(1) with a first focal point 830(1), a second longitudinal mode 840(2) with a second focal point 830(2), and a third longitudinal mode 840(3) with a third focal point 830(3). The longitudinal modes 840(1), 840(2), and 840(3) are non-degenerate in the transverse direction and thus do not couple to each other. Accordingly, the focal points 830(1), 830(2), and 830(2) are spatially separated as shown in FIG. 9.

[0054] The lens system 824 includes a first lens 810 proximate to the focal plane 108 and a second lens 812 proximate to the imaging plane 114. Different from the second lens system 124 of FIG. 1, the lens system 824 of FIG. 8 is configured such that the central axes of the longitudinal modes 840(1), 840(2), and 840(3) are parallel to the optical axis 118. As a result, the first mirror 102 retroreflects the longitudinal mode 840 at the focal plane 108. FIGS. 8 and 9 show only three longitudinal modes 840, but the optical cavity array 800 may be configured to support a different number of laterally non-occurring longitudinal modes.

[0055] In the embodiments of FIGS. 8 and 9, the first lens 810 has a high NA (e.g., 0.5 or greater) that helps achieve a small waist at the focal plane 108. The second lens 812 has a lower NA than the first lens 810 but has a larger effective aperture. In addition, the second lens 812 has a focal length greater than that of the first lens 810. The lenses 810 and 812 are separated by a lens spacing D whose value can be equal to or close to the sum of their individual focal lengths. 3 However, the lens spacing D 3 may be a different value than this sum to ensure that the central axis is parallel to the optical axis 118 at the focal plane 108 and that all of the longitudinal modes 840 are stable. Thus, the lens system 824 may be configured differently from that shown in FIG. 8 without departing from the scope of this specification.

[0056] FIGS. 8 and 9 also show how the longitudinal modes 840 have a lateral spacing d f that is larger than the lateral spacing d i near the imaging plane 114 than near the focal plane 108. Similar to what was described above with respect to FIGS. 1 and 2, these different lateral spacings are thought to result from the magnification of the lens system 824. Thus, the lens system 824 may be configured with magnification to achieve this effect.

[0057] FIG. 8 shows a lens system 824 with two plano-convex lenses, although the lens system 824 may alternatively have more than two lenses, different types of lenses, or both. Such a multi-lens system may be used to correct aberrations. Further, the lens system 824 is not limited to thin lenses and may alternatively or additionally include thick lenses, compound lenses, objective lenses, GRIN lenses, aspherical lenses, and the like.

[0058] FIG. 10 is a side view of an optical cavity array 1000 that is similar to the optical cavity array 100 of FIG. 1, except that the first lens system 122 and the first mirror 102 are replaced by a curved mirror 1010. FIG. 11 is a side view of the optical cavity array 1000 of FIG. 10 that shows in more detail how a focal point 1030 is positioned on a focal plane 108. FIGS. 10 and 11 are most clearly viewed together using the following description.

[0059] In FIG. 10, the curved mirror 1010 is a concave spherical mirror with a radius of curvature R. The curved mirror 1010 is axially positioned at a distance R from, or in the vicinity of, the focal plane 108. However, the curved mirror 1010 need not be positioned exactly at a distance R from the focal plane 108 in order to ensure the stability and formation of a longitudinally non-degenerate longitudinal mode.

[0060] Figures 10 and 11 show an optical cavity array 1000 that supports a first longitudinal mode 1040(1) that is laterally centered on the optical axis 118. In the vicinity of the focal plane 108, the first longitudinal mode 1040(1) has a first focus 1030(1) that is located where the focal plane 108 and the optical axis 118 intersect. The optical cavity array 1000 also supports a second longitudinal mode 1040(2) that has an upper leg 1140(1) and a bottom leg 1140(2). In the vicinity of the focal plane 108, the upper leg 1140(1) is lateral above the optical axis 118 (i.e., in the +x direction), which forms a second focus 1030(2) that is laterally separated from the first focus 1030(1). Also, in the vicinity of the focal plane 108, the bottom leg 1140(2) is lateral below the optical axis 118 (i.e., in the -x direction), which also forms a third focus 1030(3) that is laterally separated from the first focus 1030(1).

[0061] In FIG. 11, the central axis of the first longitudinal mode 1040(1) coincides with the optical axis 118 and is thus parallel to the optical axis 118. However, the central axes of the legs 1140(1) and 1140(2) are not parallel to the optical axis 118. As shown, the central axis is located at a point behind the curved mirror 1010 (i.e., in the -z direction with respect to the curved mirror 1010). The curved mirror 1010 reflects the upper leg 1140(1) downward (i.e., in the -x direction) over a short distance before reflecting the bottom leg 1140(2) back to form it. Thus, the curved mirror 1010 rather than completely retroreflecting the second longitudinal mode 1040(2), translates it along the x such that the upper leg 1140(1) and the bottom leg 1140(2) are laterally separated from each other. Thus, the upper leg 1140(1) and the bottom leg 1140(2) exhibit mirror symmetry about the optical axis 118.

[0062] Since both foci 1030(2) and 1030(3) are part of the second longitudinal mode 1040(2), the atoms trapped at these two foci will be bonded to each other. To avoid this bonding, the atoms should not be trapped at one of these two foci. In FIG. 11, the third focus 1030(3) is shown with an "x", indicating that this can be excluded for trapping. Alternatively, the first focus 1030(1) can be excluded. Thus, out of the three foci 1030(1), 1030(2), and 1030(3), at most two can be used for atom trapping without bonding. Extending this to N laterally non-degenerate longitudinal modes 1040, the optical cavity array 1000 forms 2N - 1 foci 1030, at most N of which can be used for atom trapping without bonding between the traps.

[0063] In FIG. 10, the central axes of the longitudinal modes 1040 do not all need to be parallel to the optical axis 118 in the vicinity of the second mirror 116. Thus, the second mirror 116 can be the polygonal mirror 300 of FIG. 3, the polygonal mirror 400 of FIG. 4, the cat's eye retroreflector array 500 of FIG. 5, the micromirror array 600 of FIG. 6, the retroreflector array 700 of FIG. 7, or another type of retroreflector known in the art. Further, since the curved mirror 1010 does not retroreflect any of the longitudinal modes 1040 that are laterally offset from the optical axis 118, the second mirror 116 retroreflects each of these off-axis longitudinal modes 1040 twice (in contrast to once for the optical cavity array 100 of FIG. 1).

[0064] FIG. 10 also shows a way in which the present embodiment can be mounted inside the vacuum chamber 1004. In this example, the curved mirror 1010 and the third lens 110 are located inside the vacuum chamber 1004, and they are installed close to the focal plane 108 (e.g., within a few millimeters) to ensure high NA waist formation and fluorescence collection. The fourth lens 112 and the second mirror 116 are located outside the vacuum chamber 1004. The vacuum window 1008 provides optical access between those components inside the vacuum chamber 1004 and those outside the vacuum chamber 1004. The vacuum window 1008 may be located within the low NA region of the optical cavity array 1000. For example, in FIG. 10, the vacuum window 1008 is located between the lenses 110 and 112. Alternatively, the vacuum window 1008 may be located between the lens 112 and the second mirror 116, in which case the lens 112 would be present inside the vacuum chamber 1004.

[0065] Alternatively, the lens 112 and the second mirror 116 may also be located inside the vacuum chamber 1004, in which case the vacuum window 1008 is not located inside the optical cavity array 1000. In fact, the vacuum window 1008 may not be required in this case, such as when light for coupling into the optical cavity array 1000 is guided into the vacuum chamber 1004 via an optical fiber. However, the vacuum window 1008 can still be used, for example, to couple a free space light beam into the optical cavity array 1000, to collect light from the optical cavity array 1000, or both.

[0066] (Combination of features) The features described above as well as those claimed below can be combined in various ways without departing from the scope of this specification. The following examples illustrate non-limiting combinations that are considered possible with the features and embodiments described above. It should be apparent that other changes and modifications can be made to this embodiment without departing from the spirit and scope of the present invention.

[0067] (A1) The optical cavity array includes a plurality of mirrors positioned and oriented to form an optical cavity. The optical cavity array also includes a first lens system located within the optical cavity. The first lens system has a first output facing a first mirror of the plurality of mirrors and a first input facing a second mirror of the plurality of mirrors. The optical cavity array also includes a second lens system located within the optical cavity. The second lens system has a second input facing the first input of the first lens system and a second output facing the second mirror. The optical cavity supports a plurality of longitudinal modes, and the plurality of longitudinal modes are non-degenerate in the lateral direction and form corresponding multiple foci along a focal plane axially positioned between the first input of the first lens system and the second input of the second lens system.

[0068] (A2) In the optical cavity array shown in (A1), the plurality of mirrors includes three or more mirrors that form a ring cavity. Each of the plurality of longitudinal modes corresponds to a traveling wave propagating around the ring cavity.

[0069] (A3) In the optical cavity array shown in (A1), the first and second mirrors are first and second retroreflectors facing each other to form a Fabry - Perot cavity, respectively. Each of the plurality of longitudinal modes corresponds to a standing wave resonating with the Fabry - Perot cavity.

[0070] (A4) In the optical cavity array shown in (A3), the second lens system images the focal plane onto an image plane. The second retroreflector is axially positioned near the image plane.

[0071] (A5) In either of the optical cavity arrays shown in (A3) and (A4), the second retroreflector is a polygonal mirror, a cat's eye array, or a convex mirror array.

[0072] (A6) In any of the optical cavity arrays shown in (A3)-(A5), the first retroreflector is a planar mirror oriented perpendicular to the optical axis of the optical cavity.

[0073] (A7)(A3)-(A6) In any of the optical cavity arrays shown, the first lens system has a first focal length f 1 and includes a first lens having a first focal length f 2 and a second lens having a second focal length f 1 The second lens is axially located behind the first lens by f 2 only.

[0074] (A8)(A7) In the optical cavity array shown, the second focal length f 2 exceeds the first focal length f 1

[0075] (A9)(A7) and (A8) In any of the optical cavity arrays shown, the first lens has a numerical aperture greater than that of the second lens.

[0076] (A10)(A7)-(A9) In any of the optical cavity arrays shown, the second lens system includes a third lens having a third focal length f 3 and a fourth lens having a fourth focal length f 4 The fourth lens is axially located behind the third lens by f 3 +f 4 only.

[0077] (A11)(A10) In the optical cavity array shown, the fourth focal length f 4 exceeds the third focal length f 3

[0078] (A12)(A10) and (A11) In any of the optical cavity arrays shown, the third lens has a numerical aperture greater than that of the fourth lens.

[0079] (A13)(A1)-(A12) In any of the optical cavity arrays shown, each of the first and second lens systems has a finite conjugate ratio.

[0080] ​​In any of the optical cavity arrays shown in (A14)(A1)-(A13), the optical cavity array further includes a vacuum chamber. The focal plane is present within the vacuum chamber.

[0081] (A15) In the optical cavity array shown in (A14), one or both of the first and second mirrors are located inside the vacuum chamber.

[0082] (A16) In any of the optical cavity arrays shown in (A1)-(A15), the optical cavity array further includes a phase plate or a phase modulator located within the optical cavity.

[0083] (B1) The method includes coupling laser light into any one of the optical cavity arrays shown in (A1)-(A16) and exciting a plurality of longitudinal modes.

[0084] (B2) In the method shown in (B1), each of the plurality of longitudinal modes, when excited, forms an individual one of a plurality of optical dipole traps located at the focal plane. The method further includes trapping at least one atom in each of one or more of the plurality of optical dipole traps.

[0085] (B3) In the method shown in (B2), the method further includes collecting fluorescence emitted by at least one atom trapped in one of the plurality of optical dipole traps, and the fluorescence is transmitted through one of the first and second mirrors of the optical cavity array.

[0086] (B4) In any of the methods shown in (B1)-(B3), the method further includes changing the optical path length of only one of the plurality of longitudinal modes.

[0087] The modifications can be made in the above methods and systems without departing from the scope of the present specification. Therefore, it should be noted that the subject matter contained in the above description or shown in the accompanying drawings should be construed as illustrative and not in a limiting sense. The following claims are intended to cover all the general and specific features described in this specification, which may be said to be in between in terms of expression, as well as all the recitations of the scope of the present methods and systems.

Claims

1. An optical cavity array, comprising: a plurality of mirrors positioned and oriented to form an optical cavity; a first lens system positioned within the optical cavity, the first lens system having a first output facing a first mirror of the plurality of mirrors and a first input facing a second mirror of the plurality of mirrors; a second lens system positioned within the optical cavity, the second lens system having a second input facing the first input of the first lens system and a second output facing the second mirror; wherein the optical cavity supports a plurality of longitudinal modes, the plurality of longitudinal modes being non-degenerate in the transverse direction and forming corresponding pluralities of foci along a focal plane axially positioned between the first input of the first lens system and the second input of the second lens system.

2. The plurality of mirrors includes three or more mirrors that form a ring cavity, and each of the plurality of longitudinal modes corresponds to a traveling wave propagating around the ring cavity, according to the optical cavity array of claim 1.

3. The first and second mirrors are first and second retroreflectors facing each other to form a Fabry-Perot cavity, respectively, and each of the plurality of longitudinal modes corresponds to a standing wave resonating with the Fabry-Perot cavity, according to the optical cavity array of claim 1.

4. The second lens system images the focal plane onto an imaging plane, and the second retroreflector is axially positioned near the imaging plane, according to the optical cavity array of claim 3.

5. The second retroreflector comprises a polygonal mirror, a cat's eye array, or a convex mirror array, according to the optical cavity array of claim 3.

6. The first retroreflector comprises a planar mirror oriented perpendicular to the optical axis of the optical cavity, according to the optical cavity array of claim 3.

7. The first lens system The first focal length f １ having a first lens, and Second focal length f ２ a second lens having comprises The second lens is axially positioned behind the first lens by f １ + f ２ The optical cavity array according to claim 3

8. the second focal length f ２ is greater than the first focal length f １ The optical cavity array according to claim 7.

9. The first lens has a numerical aperture greater than that of the second lens, according to the optical cavity array of claim 7.

10. The second lens system The third focal length f ３ A third lens having, and The fourth focal length f ４ having a fourth lens and comprises The fourth lens is axially positioned behind the third lens by f ３ + f ４ The optical cavity array according to claim 7.

11. the fourth focal length f ４ is greater than the third focal length f ３ The optical cavity array according to claim 10.

12. The optical cavity array according to claim 10, wherein the third lens has a numerical aperture greater than that of the fourth lens.

13. The optical cavity array according to claim 1, wherein each of the first and second lens systems has a finite conjugate ratio.

14. The optical cavity array according to claim 1, further comprising a vacuum chamber, wherein the focal plane is present within the vacuum chamber.

15. The optical cavity array according to claim 14, wherein one or both of the first and second mirrors are located inside the vacuum chamber.

16. The optical cavity array according to claim 1, further comprising a phase plate or a phase modulator located within the optical cavity.

17. A method comprising coupling laser light into the optical cavity array according to claim 1 and exciting the plurality of longitudinal modes.

18. When each of the plurality of longitudinal modes is excited, it forms an individual one of a plurality of optical dipole traps located at the focal plane. The method according to claim 17, further comprising trapping at least one atom in each of one or more of the plurality of optical dipole traps.

19. The method according to claim 18, further comprising collecting fluorescence emitted by at least one atom trapped in one of the plurality of optical dipole traps, the fluorescence being transmitted through one of the first and second mirrors of the optical cavity array.

20. The method according to claim 17, further comprising changing the optical path length of only one of the plurality of longitudinal modes.

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