1. Introduction

Efficient control over photonic degrees of freedom, including frequency, polarization, and spatial mode, has widespread applications in information and communication. Put simply: the more degrees of freedom one can manipulate, the more information one can encode in a single channel of light. This idea is utilized regularly in both classical and quantum communication, where light has been multiplexed in arrival time [1–3], frequency [1–4], polarization [3], quadrature [3], and most recently space [3,5–12] to substantially increase information transfer over a fiber [13–15] and free-space link [16]. Spatial information may be conveniently encoded within families of propagation eigenmodes; the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) families are appealing for their orthogonality and infinite-dimensionality, supporting the exploration of higher-dimensional Hilbert spaces for quantum computing [17,18], formation of orbital angular momentum qudits [7,18–23], improved quantum key distribution [7,19,23–27], lower-crosstalk quantum communication [25,28,29], and distribution of quantum information to multiple users in a quantum network [21].

High-dimensional optical information encoding requires the ability to manipulate the various photonic degrees of freedom through “mode conversion” [12,30–39]. Frequency and polarization mode conversion can be achieved quite flexibly at near-unit efficiency using electro-optic modulators [40] and waveplates. However, efficient spatial mode conversion is more challenging. In general, spatial mode conversion requires a spatially-dependent phase and amplitude modification of a photon’s electric field. While phase can be modified losslessly by a phase-imprinting device, amplitude modification occurs only through propagation or discarding amplitude via a physical barrier, limiting the efficiency with which spatial mode conversion can occur. For instance, devices such as spatial light modulators, digital micromirror devices, vortex plates, and liquid crystal q-plates [41–44] are excellent devices for generating modes with orbital angular momentum (OAM) by imprinting incident light with a spiral phase. While the resulting mode has the correct phase winding to be purely LG, its amplitude distribution does not. Rather, the resulting mode can be expressed as an expansion of the LG radial modes for a given OAM, illustrating that a phase imprint alone is insufficient for highly efficient spatial mode conversion to a single LG mode [25,45]. Thus, mode-converting devices have been designed to modify light in environments such as waveguides, cavities, and photonic crystals that limit the occupiable spatial modes to enhance conversion to a single target mode. Among these devices are a HG$\leftrightarrow$LG mode converter using an astigmatic microcavity [46], an arbitrary HG mode-order converter utilizing the impedance mismatches between coupled Fabry-Pérot resonators [47], design-by-specification converters based on computational methods [48], and an assortment of silicon photonic converters that harness refractive index variation to smoothly modify a propagating spatial mode [49–61].

In this paper, we explore a new method in which spatial and frequency mode conversion occur simultaneously in a single system with high efficiency. In effect, we create a rapidly sculptable, rotating optic inside of an optical cavity that converts photons between cavity modes. In practice, we modulate [62], in both space and time, the optical susceptibility of a stationary atomic sample at the waist of a twisted optical cavity using a strong auxiliary beam, inducing a coupling between cavity modes. This auxiliary beam Stark shifts the energy levels of the atomic sample to create a spatiotemporally-varying optical susceptibility across the atomic sample akin to a rotating optic. Photons that are incident on the atomic sample accrue a position-dependent phase that couples the incident mode to other modes of the cavity, which enables repeated light-atom interactions and preferentially enhances the emission of light into supported, resonant spatial modes. We measure the efficiency of this conversion process for increasing atom number and modulation beam intensity. We find a parameter regime in which the internal conversion efficiency saturates near unity.

2. Experimental overview

We demonstrate conversion between LG modes of orbital angular momenta $l=3 \rightarrow l=0$ (i.e., LG$_{30}\rightarrow$LG$_{00}$). Our optical cavity is a four-mirror twisted cavity, meaning one mirror lies outside of the plane formed by the remaining three [63]. As the eigenmodes of this cavity are non-degenerate LG modes, cavity photons require a change in both their spatial and frequency degrees of freedom to undergo mode conversion. This change can be accomplished by passage through the an atomic sample whose optical susceptibility varies in time and space. Provided the variation occurs at the frequency difference between $l=0$ and $l=3$ and imprints a phase on $l=0 (3)$ such that the resulting spatial mode has non-zero overlap with $l=3 (0)$, a coupling will be engineered between the $l=0$ and $l=3$ cavity modes.

Figure 1(a) illustrates our mode conversion scheme. A $^{87}$Rb atomic sample resides at the waist of our twisted optical cavity, which hosts modes at $780$ nm (near the $5\textrm {S}_{1/2}\leftrightarrow 5\textrm {P}_{3/2}$ transition of $^{87}$Rb) and at $1529$ nm (near the $5\textrm {P}_{3/2}\leftrightarrow 4\textrm {D}_{5/2}$ transition of $^{87}$Rb). The $5\textrm {P}_{3/2}\leftrightarrow 4\textrm {D}_{5/2}$ transition of the atomic sample is energetically modulated by a time-varying, spatially-dependent optical Stark shift generated by an auxiliary “modulation” beam at $1529$ nm whose intensity distribution is illustrated in Fig. 1(b). This pattern is achieved by overlapping $1529$ nm $l=0$ and $l=3$ modes, forming an intensity profile with three “holes” that rotates at the frequency difference ($\approx 65$ MHz) between the modes. Illuminating the atomic sample with this profile changes the resonance condition of individual atoms with intracavity $780$ nm photons, creating a spatiotemporally-varying optical susceptibility across the sample that adopts the modulation beam profile (Fig. 1(c)). As the modulation beam profile is comprised of both the $l=0$ and $l=3$ modes, a coupling is engineered between the $l=0$ and $l=3$ modes at $780$ nm. Note that the atomic sample is stationary whereas the modulation profile rotates, enabling far faster temporal modulation of incident probe light than that which can be achieved by a real, rotating optic. We utilize the atomic level scheme illustrated in Fig. 1(d), which may be understood as a near-resonant four-wave mixing process.

 

Fig. 1. Modulated atomic samples as sculptable optics. This work demonstrates the conversion of photons between two Laguerre-Gaussian modes of orbital angular momenta $l=0$ and $l=3$. These modes are the non-degenerate eigenmodes of the twisted cavity depicted in (a), which hosts $780$ nm probe modes (red) and slightly larger, copropagating $1529$ nm modulation modes (orange). We inject $l=3$ probe photons which are converted to $l=0$ via coupling to an atomic sample of $^{87}$Rb atoms at the waist of the cavity. (b) The optical susceptibility of this sample is modulated in space and time by the $1529$ nm modulation beam, effectively sculpting a rotating, mode-coupling optic from the atomic cloud with a spatiotemporally-varying refractive index. This coupling arises from the spatial profile used modulate the atomic sample, which is comprised of both $l=0$ and $l=3$ modes as illustrated in (c). When an $l=0$ mode of frequency $\omega _m$ is spatially overlapped with an $l=3$ mode of frequency $\omega _m+\delta$, the resulting interference profile is a three-holed pattern that rotates at the frequency difference between the two modes. When the atomic sample is illumated with this rotating profile, the optical susceptibility of the sample is modulated according to the profile of the modulation beam, effectively sculpting the stationary sample into a rotating, three-fold symmetric optic. The relevant atomic levels for this mode conversion process are illustrated in (d). We inject $780$ nm, $l=3$ probe photons in the dispersive regime, $130$ MHz detuned from the $5S_{1/2}\rightarrow 5P_{3/2}$ atomic resonance. These photons are coupled by the far-detuned $1529$ nm modulation beam to the $l=0$ cavity mode at $780$ nm. If mode conversion is successful, $l=0$ photons will emerge from the cavity at a frequency $65$ MHz lower than that of the injected $l=3$ photons as a result of the frequency difference between non-degenerate cavity eigenmodes. The optical susceptibility of the atomic cloud is modulated via the time-varying, spatially-dependent optical Stark shift of the $5P_{3/2}$ energy, which periodically shifts the $5S_{1/2}\rightarrow 5P_{3/2}$ atomic resonance further from the cavity resonances.

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We begin our experimental sequence by transporting a sample of laser-cooled $^{87}$Rb into the waist of our twisted optical cavity from a magneto-optical trap. The modulation beam and weak probe beam co-propagate through the cavity and illuminate the atomic sample for a probe time of $10$ ms. Probe photons are injected into the $l=3$ cavity eigenmode. These photons pass through the modulated atomic sample and the resulting photons are collected on the cavity output during the probe time. See Supplement 1 for additional details about the experimental setup.

3. Results

We search for $l=3\rightarrow l=0$ mode conversion for several different combinations of modulation beam intensity and atom number by collecting only $l=0$ light from the cavity using a single mode fiber as a filter (Fig. 2(a)). For each of these combinations, we scan the frequency of the probe beam about a point in the dispersive regime, where the $l=0$ and $l=3$ cavity resonances are detuned from the atomic $5P_{3/2}$ state as illustrated in Fig. 1(d). This scan generates the $l=0$ spectra in Fig. 2(b). We observe an increase in the $l=3\rightarrow l=0$ internal conversion efficiency, $\mathcal {E}_{3\rightarrow 0}$, for increasing $\Omega$ and $N\eta$, in effect the modulation beam intensity and resonant optical density, respectively. Note that the internal conversion efficiency does not account for losses induced by the cavity mirror surfaces and impedance matching, which is instead captured by the external conversion efficiency. Here, we quote the internal efficiency to better capture the inherent potential of this conversion method rather than experimental imperfections which may be corrected. See Supplement 1 for additional details about $\mathcal {E}_{3\rightarrow 0}$ and the external conversion efficiency. $N\eta$ is the collective cooperativity [64] where $N$ is the atom number and $\eta$ is the single atom cooperativity. This quantity can be generally interpreted as the number of times a photon is lensed by the atomic sample before it leaks out of the cavity. See Supplement 1 for additional details about $N\eta$ and $\Omega$, respectively. As $\Omega$ increases, we observe the $l=0$ cavity transmissions collapse leftward toward the location of the bare $l=3$ transmission at $\delta _p=0$. This behavior is a result of the $5P_{3/2}$ state energetically shifting away from the $l=0$ and $l=3$ cavity resonances at higher modulation beam intensities, reducing the dispersive shift of the resonances.

 

Fig. 2. Conversion in the cavity spectra. We inject $l=3$ probe photons into the twisted cavity and read out only on $l=0$ using a single mode fiber as illustrated in (a). For various combinations of $\Omega$ and $N\eta$, corresponding with the modulation beam intensity and atom number, respectively, we observe the $l=0$ transmission spectrum by scanning the frequency of the probe laser frequency, $\delta _p$, about the bare $l=3$ transmission frequency ($\delta _p=0$ MHz). In (b), we plot the $l=0$ transmission spectrum (normalized as an internal conversion efficiency) at $\Omega /(2\pi )=(0.7, 1.3, 2.1, 3.5)$ GHz (light to dark) for each value of $N\eta =(140, 250, 560)$. The $l=3\rightarrow l=0$ conversion efficiency, $\mathcal {E}_{3\rightarrow 0}$, increases for increasing $\Omega$ and $N\eta$, reaching near unity for the highest values of $\Omega$ and $N\eta$. Intuitively, $\mathcal {E}_{3\rightarrow 0}$ should increase for increasing modulation beam intensity and atom number, akin to increasing the refractive index variation and density of our effective, intracavity optic from zero. The $l=0$ transmission curves shift toward lower frequencies with increasing $\Omega$ due to the increasing Stark shift of the $5P_{3/2}$ resonance with higher modulation beam intensity, which lessens the dispersive shift of the $l=0$ transmission curves away from the bare $l=3$ transmission frequency.

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To verify photons are indeed converted into the $l=0$ mode of the cavity, we perform a spatial and frequency analysis of the cavity output. In principle, the modulated atomic sample induces a coupling between the $l=3$ mode and many other spatial modes. However, with the exception of the $l=6$ mode, these modes are Purcell suppressed because they are non-resonant. Despite a potential $3\leftrightarrow 6$ coupling, we do not observe $l=6$ light on the cavity output, likely because the $l=6$ mode is further detuned from the $5\textrm {S}_{1/2}\leftrightarrow 5\textrm {P}_{3/2}$ atomic resonance compared to $l=3$ and $l=0$ modes (see Supplement 1). Thus, in general, the non-degenerate mode structure of the cavity improves the isolation of a target mode by frequency discrimination.

The increase of $\mathcal {E}_{3\rightarrow 0}$ with $\Omega$ and $N\eta$ can be interpreted intuitively in the context of sculpting an effective optic from the atomic sample. For $\Omega =0$, there is no modulation of the atomic sample. Probe photons pass through an effective optic that imparts an almost completely flat phase, providing essentially no coupling between the $l=3$ and $l=0$ modes. For $N\eta =0$, no atoms are present; there is no effective optic. Thus, $\mathcal {E}_{3\rightarrow 0}$ regardless of $\Omega$. For $\Omega >0$ and $N\eta >0$, we begin to observe $l=3$ to $l=0$ conversion as the effective optic acquires density and a spatially-dependent optical susceptibility.

Figure 3 is a more in-depth investigation of $\mathcal {E}_{3\rightarrow 0}$ as a function of $\Omega$ and $N\eta$. $\mathcal {E}_{3\rightarrow 0}$ increases for increasing $\Omega$ and $N\eta$ and saturates near unity. In a double-ended cavity like ours, where light can leak out one of two cavity mirrors, $\mathcal {E}_{3\rightarrow 0}$=1 corresponds to a maximum external efficiency of 25% for lossless mirrors. For a general double-ended cavity comprised of two equally-transmissive cavity mirrors, incident light can be fully transmitted as the cavity is impedance matched. If a mode-converting element is placed within the cavity, this impedance matching condition is broken, limiting the amount of light, both converted and unconverted, that exits the cavity through the output mirror. In a single-ended cavity, the maximum external efficiency increases to 100% (see Supplement 1).

 

Fig. 3. Saturation in conversion efficiency. We further examine $l=3\rightarrow l=0$ conversion in two-dimensional sweeps over $\delta _p$ and $\Omega$ for additional values of $N\eta$. In (a), $\mathcal {E}_{3\rightarrow 0}$ increases for increasing $\Omega$ and $N\eta$. Plotting the numerical values of $\mathcal {E}_{3\rightarrow 0}$ in (b) at each slice of $\Omega$ for all $N\eta$ (left) and for each $N\eta$ at maximum $\Omega$ (right) elucidates both the attainment and saturation of conversion near $\mathcal {E}_{3\rightarrow 0}$=1. This internal efficiency corresponds to a maximum external efficiency of 25% due to the double-ended nature of our cavity. In general, light is fully transmitted through a double-ended cavity when the reflected light cancels with the light that leaks out of the cavity. This idea assumes the two cavity ends, or mirrors, have equal transmission coefficients and light drives the cavity from one side. Even though the two ends of our twisted cavity have equal transmission coefficients, the conversion of light from the injected mode to another mode acts as loss which breaks the cavity impedance matching that enables full transmission. See Supplement 1 for more details. Points are larger than their error bars of one standard deviation.

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4. Conclusion

We have demonstrated a highly efficient method to simultaneously manipulate photonic degrees of freedom by spatiotemporally modulating the optical susceptibility of an atomic sample. In our twisted optical cavity, we observe $l=3\rightarrow l=0$ conversion at an internal efficiency near unity. Extending this method to a low loss, single-ended cavity will provide conversion near 100% efficiency for both internal and external efficiencies. As the main import of our twisted cavity is to limit the occupiable cavity modes and enable repeated interactions of a cavity photon with the atomic sample, other cavity geometries such as non-twisted two- or three-mirror cavities containing an atomic sample would suffice as a platform for carrying out this mode conversion method. This method is additionally extendable to other atomic species, different propagation eigenmodes, polarization and $l=0\rightarrow l=3$ conversion (see Supplement 1), and the coherent conversion of single photons [65]. While the external conversion efficiency of this method can, in principle, be quite high, higher-order modes may be more difficult to convert to and from due to their large physical size compared to the atomic sample and minimal overlap with smaller spatial modes. Additionally, the conversion bandwidth is comparable to the linewidth of the cavity, which may be quite narrow relative to current commercial devices and not ideal for all applications. Nevertheless, mode conversion via optical susceptibility modulation might find applications in quantum state preparation, quantum information, and development as a tabletop device. One might use this method to grow topological few-body states of light by controllably adding orbital angular momentum to intracavity photons [66], convert within mode pairs for mode-division multiplexed transmission [25], or create a miniaturized device based on intracavity electro-optic elements whose refractive indices are modulated in space and time.