Directional quantum scattering transducer in cooperative Rydberg metasurfaces

TL;DR

This work introduces a directional quantum transduction scheme based on driven four-wave mixing in cooperative two-dimensional Rydberg atom arrays, analyzed via a quantum S-matrix formalism that explicitly treats photonic fields. By enforcing a resonance on the signal transition and a criticality condition on the idler, the scheme channels single THz photons into highly directional optical idler modes, with infinite lattices achieving up to $1/2$ efficiency for a single mode and higher total efficiency when considering all idler channels. Finite arrays exhibit beam-like scattering lobes whose width narrows as $1/\,\sqrt{N}$, indicating practical scalability toward directed single-photon transduction. The framework connects quantum scattering with classical Maxwellian steady-state polarizabilities, revealing instabilities linked to dipolar surface modes and enabling mode-selective transduction useful for THz astronomy, quantum networks, and quantum sensing.

Abstract

We present a single-photon transduction scheme using 4-wave-mixing and quantum scattering in planar, cooperative Rydberg arrays that is both efficient and highly directional and may allow for terahertz-to-optical transduction. In the 4-wave-mixing scheme, two lasers drive the system, coherently trapping the system in a dark ground-state and coupling a signal transition, that may be in the terahertz, to an idler transition that may be in the optical. The photon-mediated dipole-dipole interactions between emitters generate collective super-/subradiant dipolar modes, both on the signal and the idler transition. As the array is cooperative with respect to the signal transition, an incident signal photon can efficiently couple into the array and is admixed into dipolar idler modes by the drive. Under specific criticality conditions, this admixture is into a superradiant idler mode which primarily decays into a specific, highly directional optical photon that propagates within the array plane. Outside of the array, this photon may then be coupled into existing quantum devices for further processing. Using a scattering-operator formalism we derive resonance and criticality conditions that govern this two-step process and obtain analytic transduction efficiencies. For infinite lattices, we predict transduction efficiencies into specific spatial directions of up to 50%, while the overall, undirected transduction efficiency can be higher. An analysis for finite arrays of $N^2$ emitters, shows that the output is collimated into lobes that narrow as $1/\sqrt{N}$. Our scheme combines the broadband acceptance of free-space 4-wave mixing with the efficiency, directionality and tunability of cooperative metasurfaces, offering a route towards quantum-coherent THz detection and processing for astronomical spectroscopy, quantum-networked sparse-aperture imaging and other quantum-sensing applications.

Figures (12)

• Figure 1: Schematic view of the experimental setup and level scheme of an individual atom site. (a) Rydberg atoms are arranged in a square lattice with spacing $d$ in the $x$-$y$ plane at $z=0$. The array is illuminated (ellipses) by two driving lasers, laser 1 (red) and laser 2 (blue), with Rabi polarizations $\mathbf{\Omega_1}$, $\mathbf{\Omega_2}$ and wavevectors $\mathbf{k}_{L_1}$, $\mathbf{k}_{L_2}$, respectively. (b) The level structure of each atom site features a double $\Lambda$-system which shares the ground states $g_1$ and $g_2$. In the lower $\Lambda$-system the lasers 1 and 2 couple the ground states to a manifold of excited states $(e_2)_n$. In the upper $\Lambda$-system the ground states couple to another manifold of excited states $(e_1)_n$. The $g_1 \leftrightarrow e_1$ and $g_2 \leftrightarrow e_1$ transitions are referred to as the signal and idler transition, which respectively couple to quantized electromagnetic fields $a$ and $b$ with vacuum coupling strength $g_a$ and $g_b$. Due to the strong drive on the lower $\Lambda$-system, the ground state manifold is coherently trapped in a dark state superposition $d$ of the two ground states, which now couples to the excited state manifold $(e_1)_n$ through the signal field $a$ and the idler field $b$ with effective coupling strengths $Ag_a$ and $Bg_b$, as described in the main text.
• Figure 2: Transduction efficiency at normal incidence for varying dark state mixing ratios $|A|$. For a vanishing grating ($\mathbf{k}_{L,ba}= \mathbf{0}$), the transduction efficiency $\left| S_b\ket{(\mathbf{k_\parallel},k_\perp) \mu a}\right|^2$ is shown for resonant photons at normal incidence as a function of the squared dark state mixing ratio $|A|^2$. Curves for $\omega_{\mathbf{k}}= 0.2 \times (2\pi/d)$ (solid) and $\omega_{\mathbf{k}}= 0.495 \times (2\pi/d)$ (dashed) are shown for $\omega_{g_1}- \omega_{g_2} =0$ (black), $\frac{5}{2} \times (2 \pi/d)$ (red) and $5 \times(2 \pi/d)$ (orange). The efficiency is peaked around a value of $|A|$ which approaches $|A|\to 1$ for increasing transduction energies. At normal incidence an efficiency of up to $\frac{1}{2}$ can be attained.
• Figure 3: Transduction efficiency at normal incidence for varying incoming photon energies. (a) The transduction efficiency of a resonant ($\rho=0+ i 0^+$) $a$ photon into $b$ modes at the optimal value of $|A|$ is shown for different transduction energies $\Delta \omega= \omega_{g_1}- \omega_{g_2}$ as a function of the energy of the incoming photon $\omega_{\mathbf{k}}$. The efficiency (solid) is shown for an $a$ photon at normal incidence ($\mathbf{k_\parallel}=\mathbf{0}, k_\perp= \omega_{\mathbf{k}}$) and a vanishing grating ($\mathbf{k}_{L,ba}=\mathbf{0}$) for transduction energies $\Delta \omega$ of $0$ (black), $\frac{1}{2}\frac{2\pi}{d}$ (purple), $\frac{2 \pi}{d}$ (red), $\frac{3}{2} \frac{2 \pi}{d}$ (orange) and $2 \frac{2\pi}{d}$ (yellow). At normal incidence, for $\omega_\mathbf{k}< 2 \pi/d$ the signal transition is fully cooperative and the efficiency can reach values up to $\frac{1}{2}$. For lesser degrees of cooperativity the maximum efficiency is lowered. The efficiency shows sharp spikes towards $0$ at the instabilities discussed in \ref{['sect:methods']} when $\omega_{\mathbf{k}'}= \omega_{\mathbf{k}} +\Delta \omega =\left| \mathbf{k}'(\mathbf{p}, s, b)_\parallel \right|= \left|\mathbf{k_\parallel} + \frac{2 \pi}{d} (m,n)+ \mathbf{k}_{L,ba} \right|$ for some $m,n\in \mathbb{Z}$, i.e. the perpendicular wavevector component of the outgoing photon is vanishingly small. For example, for $\Delta \omega= \frac{3}{2}\frac{2 \pi}{d}$, the shortest lattice vectors that can fulfill this condition are $[\pm 2,0] (2\pi/d)$ and $[0,\pm 2] (2\pi/d)$, which leads to the spiked feature at $\omega_\mathbf{k}= \frac{1}{2} \frac{2 \pi}{d}$. The next longer lattice vectors are $[\pm 2,\pm 1] (2\pi/d)$ and $[\pm 1,\pm 2] (2\pi/d)$, which lead to an instability at $\omega_\mathbf{k}= \left(\sqrt{5} - \frac{3}{2}\right) \frac{2 \pi}{d}\approx 0.74 \frac{2 \pi}{d}$. Additionally, the transduction energy is shown for when the real part of $\rho \mathbb{I}- \rho P(\mathbf{k_\parallel}, a)$ is set to zero (black, dashed). In that case, the efficiency does not depend on the transduction energy $\Delta\omega$ and the spiked features are no longer present. For any $\omega_\mathbf{k}\leq \frac{2 \pi}{d}$ the maximum efficiency is attained, while for $\omega_\mathbf{k}> \frac{2 \pi}{d}$, the efficiency depends on $\omega_\mathbf{k}$ and has discontinuities whenever new modes turn critical (i.e. they turn from evanescent to light-like) on the $a$ transition (as opposed to modes turning critical on the $b$ transition which leads to the spiked features). (b) Reciprocal lattice vectors for $\omega_{\mathbf{k}'}= 2 (2 \pi/d) \pm 0^{+}$ ($-$ dotted, $+$ dashed) in units of $2 \pi/d$ . Bound, evanescent modes are shown as black dots, while free space modes are shown in yellow. Modes inside the circle (light cone) have a real perpendicular momentum component $\mathbf{k}'_\perp= \pm \sqrt{(\omega_{\mathbf{k}'})^2 - |\mathbf{p}|^2}\in \mathbb{R}$ and are free space modes, while modes outside have an imaginary perpendicular component, making them evanescent. If a mode lies on the light cone, it has a vanishing perpendicular component and is therefore critical. Going from $\omega_{\mathbf{k}'}= 2 \times (2 \pi/d) - 0^{+}$ (dotted circle) to $\omega_{\mathbf{k}'}= 2 \times (2 \pi/d) + 0^{+}$ (dashed circle), the modes with $[\pm 2,0] (2\pi/d)$ and $[0,\pm 2] (2\pi/d)$ (red dots) turn into free space modes and may therefore be present in $\hat{S}_b\ket{(\mathbf{k_\parallel},k_\perp) \mu a}$. For $\omega_{\mathbf{k}'}= 2 (2 \pi/d)$ these modes are critical and lie along the light cone, leading to the instabilities and the spiked features in (a). Similarly, on the signal transition for $\omega_\mathbf{k}< 2 \pi/d$, only $\mathbf{p}=\mathbf{0}$ is radiant, while at $\omega_\mathbf{k}= 2 \pi/d$, $\omega_\mathbf{k}= \sqrt{2} 2 \pi/d$ and $\omega_\mathbf{k}= 4 \pi/d$, modes such as $\mathbf{p}= [1,0] \frac{2\pi}{d}$, $\mathbf{p}= [1,1] \frac{2\pi}{d}$ and $\mathbf{p}= [2,0] \frac{2\pi}{d}$, turn radiative. For $\mathbf{k_\parallel} \neq \mathbf{0}$ the same picture applies with the center of the circle displaced from the origin by $\mathbf{k_\parallel}$.
• Figure 4: Transduction efficiency near critical modes. For resonant, $y$-polarized $a$ photons at normal incidence ($\mathbf{k_\parallel}=\mathbf{0}$) the transduction efficiency into any $b$ mode (\ref{['conv_efficiency_atob']}, solid, black) along with the transduction efficiency into specific $b$ modes (\ref{['specific_efficiency']}, orange, red, purple) is shown. These efficiencies are shown as function of the incoming photon energy $\omega_{\mathbf{k}}$ for different transduction energies $\Delta \omega= \omega_{g_1}- \omega_{g_2}= \frac{3}{2} (2 \pi/d)$ (a), $\frac{5}{2} ( 2 \pi/d)$ (b), $1.8 ( 2 \pi/d)$ (c) and $2.8 (2 \pi/d)$ (d), such that the energy and momenta of the outgoing transduced modes are given by $\omega_{\mathbf{k}'}= \omega_\mathbf{k}+ \Delta \omega$ and $\mathbf{k}'(\mathbf{p}, s,b), \mathbf{p} \in (2 \pi/d) \mathbb{Z}_2$. Thus at $\omega_{\mathbf{k}}= \frac{1}{2} ( 2\pi/d)$ (a,b) and $0.2( 2 \pi/d)$ (c,d), modes with $|\mathbf{p}|=2 (2 \pi/d)$ (a,c) and $|\mathbf{p}|=3 (2 \pi/d)$ (b,d) turn critical. As a result, approaching the instability from below, the overall $a\to b$ efficiency (solid, black) drops to zero and rapidly increases again to values near $\frac{1}{2}$ beyond the instability. Below the instability, all of the transduction is into the lower modes inside the light cone (orange, dotted) which have $|\mathbf{p}|<2 \times (2\pi/d)$(a,c) and $|\mathbf{p}|<3\times (2\pi/d)$(b,d). Slightly, beyond the instability, transduction is predominantly into the new modes (solid, red). At normal incidence, if the incoming photon is $y$-polarized, then the polarization of the outgoing photon must have a polarization component in the $y$-direction. Hence, no weight is placed into modes such as the $\mathbf{p}= (0,\pm 3) (2\pi/d)$ mode (dotted, purple) in (d), as free-space photons are transversely-polarized.
• Figure 5: Transduction efficiency for $30^{\circ}$ degree incidence. (a) Transduction efficiency of a resonant ($\rho=i0^+$) $a$ photon into $b$ modes at the optimal value of $|A|$ as function of the incoming photon energy $\omega_{\mathbf{k}}$ for different wavevectors and polarizations. The transduction energy is given by $\Delta\omega=\omega_{g_1}-\omega_{g_2}= 3 (\frac{2 \pi}{ d})$. The incoming photon has a wavevector either along one of the lattice directions $(\mathbf{k_\parallel}, k_\perp)= (1/2,0,\sqrt{3/4} )\omega_\mathbf{k}$ (black, purple) or diagonal to it $(\mathbf{k_\parallel}, k_\perp)= (1/2\sqrt{2},1/2\sqrt{2},\sqrt{3/4} )\omega_\mathbf{k}$ (red, orange) with polarizations either parallel to the plane of incidence ("p", purple, orange) or parallel to the array ("s", black, red). The efficiency shows sharp spikes towards $0$ near the instabilities discussed in the main text. For polarizations parallel to the array, the efficiency is bounded by $1/2$, while for polarizations parallel to the plane of incidence higher efficiencies may be achieved. (b) Transduction efficiencies for the same parameters as in (a) with the real part of $(\rho\mathbb{I}- \rho P(\mathbf{k_\parallel},a))$ set to $0$. The efficiencies give an upper bound to the efficiencies shown in (a). In both (a) and (b), with a decreasing degree of cooperativity, the efficiency decreases rapidly and the critical energies for that are the same in (a) and (b). Note that the critical energy up to which incoming photons are still cooperative depends on $\mathbf{k_\parallel}$ and as a result the sharp drop in efficiency occurs at different energies for the two different wavevectors. When the real part is set to zero, the spikes stemming from $b$-modes turning critical are only observed for $p$-polarized modes, the transduction efficiency of $s$-polarized modes on the other hand, does not depend on the transduction energy as also seen in \ref{['transduction_bestA']}. (c) Transduction efficiency into different modes near the critical point at $\omega_{\mathbf{k}}= 0.3094 (2\pi/d)$ with $(\mathbf{k_\parallel}, k_\perp)= (1/2,0,\sqrt{3}/2 )\omega_\mathbf{k}$ and $\Delta \omega= 3 \times (2 \pi/d)$ (this corresponds to the black and purple curves shown in (a)), where the mode $\mathbf{k}'(\mathbf{p},s,b)=\left(\mathbf{k_\parallel}+ \mathbf{p},s \sqrt{(\omega_\mathbf{k} + \Delta \omega)^2-|\mathbf{k_\parallel}+ \mathbf{p}|^2 }\right)$ with $\mathbf{p}=[3, \pm 1] (2\pi/d)$ turns critical. Solid lines denote transduction into any $b$-mode, while dashed lines denote transduction into lower modes inside the light cone and dotted lines denote transduction into the new modes which turn critical at the critical energy. The efficiencies are given both for "$s$"- and "$p$"-polarized photons. Below the critical point, transduction is entirely into the lower modes, while slightly above the critical point the new modes are strongly scattered into. While for "$p$"-polarized modes the overall efficiency may be higher than $\frac{1}{2}$, transduction efficiency into the critical modes is lower than $\frac{1}{2}$ for both polarizations.