Quantum Metasurfaces

TL;DR

This paper introduces quantum metasurfaces that utilize entangled atomically thin reflectors to control quantum light properties, enabling advanced quantum operations and non-classical state generation for quantum information processing.

Contribution

It presents a novel approach to quantum metasurfaces using entangled superposition states of atomically thin reflectors, enabling quantum control and state generation.

Findings

Enables massively parallel quantum operations between atoms and photons.

Allows generation of non-classical states like GHZ and cluster states.

Analyzes effects of imperfections and decoherence.

Abstract

Metasurfaces mold the flow of classical light waves by engineering sub-wavelength patterns from dielectric or metallic thin films. We describe and analyze a method in which quantum operator-valued reflectivity can be used to control both spatio-temporal and quantum properties of transmitted and reflected light. Such a quantum metasurface is realized by preparing and manipulating entangled superposition states of atomically thin reflectors. Specifically, we show that such a system allows for massively parallel quantum operations between atoms and photons and for the generation of highly non-classical states of light, including photonic GHZ and cluster states suitable for quantum information processing. We analyze the influence of imperfections and decoherence, as well as specific implementations based on atom arrays excited into Rydberg states. Finally, the extension to quantum metamaterials and emulations of quantum gravitational background for light are discussed.

Figures

b): from a ground state $\ket{g}$ to a Rydberg state $\ket{r}$ via an excited state $\ket{e}$. When a weak (quantum) probe field and a strong (classical) control field are resonant with respective $\ket{g}\rightarrow\ket{e}$ and $\ket{e}\rightarrow\ket{r}$ transitions, the array is transparent due to electromagnetically induced transparency (EIT), realizing $\ket{U}$. In order to control its reflectivity, a proximal ancillary, individually controlled atom can be used: if such an atom is prepared in its ground state $\ket{g^{\prime}}$, it does not affect the array. However if this atom is prepa

b): from a ground state $\ket{g}$ to a Rydberg state $\ket{r}$ via an excited state $\ket{e}$. When a weak (quantum) probe field and a strong (classical) control field are resonant with respective $\ket{g}\rightarrow\ket{e}$ and $\ket{e}\rightarrow\ket{r}$ transitions, the array is transparent due to electromagnetically induced transparency (EIT), realizing $\ket{U}$. In order to control its reflectivity, a proximal ancillary, individually controlled atom can be used: if such an atom is prepared in its ground state $\ket{g^{\prime}}$, it does not affect the array. However if this atom is prepa

b,d). The graph representation (Fig 2b) presents a tree cluster states where specific single qubits are each connected to six others. These types of entangled states are important for overcoming photon loss errors Varnava *et al.* ([2006](#bib.bib23)), as part of quantum error correction schemes and can serve as a basic building block to a more general entangled state which is related to holographic high energy theories Pastawski *et al.* ([2015](#bib.bib24)). The protocol starts by initializing the quantum metasurface in the $\frac{1}{\sqrt{2}}(\ket{U}+\ket{C})$ state. After scattering a sing

b,d). The graph representation (Fig 2b) presents a tree cluster states where specific single qubits are each connected to six others. These types of entangled states are important for overcoming photon loss errors Varnava *et al.* ([2006](#bib.bib23)), as part of quantum error correction schemes and can serve as a basic building block to a more general entangled state which is related to holographic high energy theories Pastawski *et al.* ([2015](#bib.bib24)). The protocol starts by initializing the quantum metasurface in the $\frac{1}{\sqrt{2}}(\ket{U}+\ket{C})$ state. After scattering a sing

b. For less than $2\%$ of missing atoms, which is a typical error in the experimental systems Bernien *et al.* ([2017](#bib.bib28)); Cooper *et al.* ([2018](#bib.bib31)), the light state fidelity is still higher than $0.965$.

Equations

$$|\Psi ⟩=\frac{1}{\sqrt{2}}(|U⟩\otimes |\alpha ,0⟩+|C⟩\otimes |t\alpha ,r\alpha ⟩),$$

$$r=\frac{i(\gamma +\Gamma )\left({\delta }_r+V\right)}{-i\left({\delta }_r+V\right)(\gamma +\Gamma -2i(\delta -\Delta ))+2|{\Omega }_p{|}^2}$$

$$r\to -1+i\frac{|{\Omega }_p{|}^2}{(\gamma +\Gamma )/2}\frac{1}{V}.$$

$$\hat{\mathcal{U}}=\sum _{\Phi }|\Phi ⟩⟨\Phi |\underset{{\text{k}}_{\perp }}{⨂}{\mathcal{V}}_{{\text{k}}_{\perp },\Phi },$$

Full text

Recently, significant effort has been directed toward structuring classical light waves using spatial light modulators and metasurfaces Bomzon et al. (2002); Yu et al. (2011); Kildishev et al. (2013); Chen et al. (2016); Rubinsztein-Dunlop et al. (2016). Such structured light has many potential applications, ranging from micro-machining to manipulating particles Baumgartl et al. (2008); Yao and Padgett (2011); Mathis et al. (2012); Schley et al. (2014). Further, it can affect interactions in nonlinear optical systems Bekenstein et al. (2015). While previous efforts focus on structuring the spatial or temporal properties of the light Weiner (2000), we explore whether the quantum properties of light can be efficiently controlled in a similar manner. Apart from fundamental significance, such a technique could have important practical applications in fields ranging from quantum information processing and communication to metrology.

In this article, we introduce and analyze a method in which quantum operator-valued reflectivity can be used to control both spatio-temporal and quantum properties of transmitted and reflected light. Such quantum metasurfaces are realized by preparing and manipulating entangled states of atomically thin reflectors made of atom arrays and scattering light from it. This approach enables a cavity-free multi-mode optical scheme for highly parallel quantum operations on multiple photonic qubits. In particular, we analyze a protocol for preparing highly entangled photonic states relevant for quantum information and describe a method for quantum control over specific spatial modes of the light.

Key idea & proposed realization

The key idea of this work can be understood by considering a two dimensional array of $N$ atoms placed with sub-wavelength spacing in the $xy$ plane (Fig. 1). Such an array can act as an atomically thin mirror for the incident light tuned close to the $\ket{g}\rightarrow\ket{e}$ atomic transition. Near-perfect reflectivity originates from cooperative resonances arising from the dipole-dipole interaction between atoms resulting in collective scattering by the array Bettles et al. (2016); Shahmoon et al. (2017). Note that the scattering pattern of the weak incoming light can be effectively engineered via controlled variation of the atomic positions Shahmoon et al. (2017), in direct analogy to classical metasurfaces.

To realize a quantum metasurface, we consider an array prepared in superposition of states with different reflectivity. In such a case, the conditional reflection directly results in atom-photon entanglement and allows for generating and controlling quantum states of light. To illustrate the main idea, we introduce collective atomic states which are either reflective, since they are coupled ($\ket{C}$) to the light tuned to the collective $\ket{g}\rightarrow\ket{e}$ atomic transition, or transparent, since they are uncoupled ($\ket{U}$) to the latter. We consider an atomic array prepared in the quantum state $|\psi_{QMS}\rangle=\frac{1}{\sqrt{2}}(\ket{U}+\ket{C})$, a superposition of a reflective or transparent state. If a right propagating coherent light wave is launched perpendicular to the array: $\ket{\psi_{i}}=\ket{\alpha,0}$, where the first mode is right propagating and the second mode is left propagating, then the scattering results in the final state:

[TABLE]

where $r$ and $t$ are the reflection and transmission coefficients identified with the linear response of the atom array. In particular, when the light is tuned to the cooperative resonance of an infinite array, the perfect conditional reflectivity yields the entangled state: $\ket{\Psi}=\frac{1}{\sqrt{2}}(\ket{U}\otimes\ket{\alpha,0}+\ket{C}\otimes\ket{0,-\alpha})$ (see Fig. 1). Further control can be obtained by measuring the state of the array. Specifically, a projective measurement in the basis of entangled states $\frac{1}{\sqrt{2}}(\ket{U}\pm\ket{C}$ projects the conditionally scattered light into odd and even cat states for the light $\ket{\psi_{f}}=\frac{1}{\sqrt{2}}(\ket{\alpha,0}\pm\ket{0,-\alpha})$.

The key element of this scheme involves preparation Guimond et al. (2019), manipulation, and measurement of quantum superposition states of the metasurface. These can be implemented by using atom excitation into the Rydberg states. As a specific example, we consider atom array excitation using a two-photon cascade configuration (see Fig. 1b): from a ground state $\ket{g}$ to a Rydberg state $\ket{r}$ via an excited state $\ket{e}$. When a weak (quantum) probe field and a strong (classical) control field are resonant with respective $\ket{g}\rightarrow\ket{e}$ and $\ket{e}\rightarrow\ket{r}$ transitions, the array is transparent due to electromagnetically induced transparency (EIT), realizing $\ket{U}$. In order to control its reflectivity, a proximal ancillary, individually controlled atom can be used: if such an atom is prepared in its ground state $\ket{g^{\prime}}$, it does not affect the array. However if this atom is prepared in a Rydberg state $\ket{r^{\prime}}$, due to the Rydberg blockade effect it can eliminate EIT for the array atoms and make it highly reflective by shifting the $\ket{r}$ level out of resonance required for the EIT condition (see Fig. 1b). In such a case the array is in its reflective state ($\ket{C}$) (see Fig. 1c). Using this approach, the superposition is realized by preparing the ancillary atom in the superposition $\frac{1}{\sqrt{2}}(\ket{g^{\prime}}+\ket{r^{\prime}})$, while the measurement correspond to ancila measurement in the appropriate basis. While these considerations are strictly valid in the case of infinite arrays, as discussed below, in practice control over atom arrays of modest size is sufficient to produce relatively large cat-like states with high fidelity (Fig. 1d). At the same time, by appropriate choice of levels $\ket{r},\ket{r^{\prime}}$, Rydberg blockade can be enabled over $>30\mu$m length scales Barredo et al. (2014); Browaeys et al. (2016).

Specifically, we consider the collective effect of the atoms, as their proximity allows dominant dipole-dipole interactions for the $\ket{g}\rightarrow\ket{e}$ transition. The reflection coefficient arising from the collective response of the array is given by:

[TABLE]

where $\Delta,\Gamma$ are cooperative corrections to the detuning ($\delta$) and decay rate ($\gamma$) associated with individual atoms (see SI for details). $\delta_{r}$ is the detuning from the Rydberg state $\ket{r}$, and $\Omega_{p}$ is the Rabi frequency of the control field tuned close to the $\ket{e}\rightarrow\ket{r}$ transition. The above expression accounts both for the collective effects arising from the dipole-dipole interactions within the atom array, as well as for the shift of the Rydberg level induced by the ancilla ($V$).

Let us consider a situation where the incident field is tuned to the cooperative ressonance, i.e. $\delta=\Delta$, the frequency of the pump field is chosen accordingly to match the two photon resonance ($\delta_{r}=0$). If the ancilla is in $\ket{g^{\prime}}$ resulting in $V=0$ for all atoms, then Eq. (2) fulfills the EIT condition and the array is in its transparent state $\ket{U}$, realizing $r=0$. If the ancilla is in $\ket{r^{\prime}}$ it induces a shift of the Rydberg level $V$ that can eliminate the EIT. Then $r\xrightarrow{V\rightarrow\infty}-\frac{(\gamma+\Gamma)/2}{(\gamma+\Gamma)/2-i(\delta-\Delta)}$, which on resonance results in the collective mirror state $\ket{C}$, realizing $r\rightarrow-1$. Specifically, when $V\gg\frac{|\Omega_{p}|^{2}}{(\gamma+\Gamma)/2}$ for all atoms, namely larger than the EIT window of the collective resonance, the reflection coefficient is:

[TABLE]

Thus, the correction from the two photon process is negligible and the collective state of the array is highly reflective ($\ket{C}$). Thus, the reflection coefficient in Eq. (2) is conditioned on the ancilla state. It is important to note that $V$ is inhomogeneous within the array, however, for the specific collective states we are interested in, it is sufficient to fulfill the above equality for the minimal $V$.

Before proceeding we note that the present method is a generalization of corresponding techniques realized with single atoms in cavities or waveguides Duan and Kimble (2004); Ritter et al. (2012). By allowing to combine intrinsic atom-photon entanglement mechanism with powerful techniques for shaping light propagation in free space, we now show how quantum states of multi-mode light fields can be controlled using quantum metasurfaces.

Photon state control via quantum metasurface

To illustrate the key idea of photon state control we first consider photonic qubits encoded in the propagation direction of single photons - either right ($\ket{0}$) or left ($\ket{1}$). The free space allows us to utilize multiple qubits by exploiting the different transverse modes. In particular, for light propagating in the $z$ direction the natural modes can be labeled by the transverse momentum ($\textbf{k}_{\perp}$) and are defined by the electric field modes $\textbf{E}(\textbf{r},z)=\int\textbf{E}_{\textbf{k}_{\perp}}e^{\pm ikz+i\textbf{k}_{\perp}\textbf{r}}d\textbf{k}_{\perp}$, where $\textbf{k}_{\perp}$ is within the diffraction limit.

Parallel quantum gate operations between the collective states of the atom array and photonic qubits can be employed in preparation protocols of highly entangled photonic states, which are sources for quantum communication and quantum computation Knill et al. (2001); Raussendorf et al. (2003); Kuzmich et al. (2003); Kimble (2008). In these protocols, photonic qubits entangle with the quantum metasurface, which is then measured in a state that projects the system onto a desired photonic entangled state.

One example for information processing is a parallel CNOT gate. Here the photons that encode the quantum information are defined by specific transverse momentum ($m$). We initially prepare the quantum metasurface in the state: $\ket{\Psi_{QMS}}=\frac{1}{\sqrt{2}}(\ket{U}+\ket{C})$. A parallel CNOT gate is realized between the quantum metasurface and $m$ photonic qubits, each with defined transverse momentum ($\textbf{k}_{\perp}$), by scattering from the array. This is followed by a projective measurement of the quantum metasurface in the state: $\frac{1}{\sqrt{2}}(\ket{U}+\ket{C})$ which projects the system to a photonic GHZ state. (see Fig. 2c). This parallel gate can serve in preparation protocols of matrix product states, which involve scattering of photons in a sequential process, that can be employed with the quantum metasurface.

As a second example, we consider preparing strong correlations of photonic tensor network states, and in particular for preparing a tree cluster state (Fig. 2b,d). The graph representation (Fig 2b) presents a tree cluster states where specific single qubits are each connected to six others. These types of entangled states are important for overcoming photon loss errors Varnava et al. (2006), as part of quantum error correction schemes and can serve as a basic building block to a more general entangled state which is related to holographic high energy theories Pastawski et al. (2015). The protocol starts by initializing the quantum metasurface in the $\frac{1}{\sqrt{2}}(\ket{U}+\ket{C})$ state. After scattering a single photon, a Hadamard gate on the metasurface is applied (e.g. by driving the ancilla qubit with a $\pi$-pulse) and six new photons are launched to scatter from the quantum metasurface. Following a Hadamard gate on the ancilla, the second photon is re-scattered from the quantum metasurface for creating the desired correlations. This is followed by another Hadamard on the ancilla and scattering of five new photons from the quantum metasurface, which become entangled with the second photonic qubit. This procedure, consists of re-scattering specific photons from the quantum metasurface which can be implemented by spatial light modulators that are placed on both sides of the array to control the reflection of specific transverse modes, that need to be re-scattered (see Supplementary Information).

Importantly, for any given protocol, the quantum information can be imprinted on any other property of the photonic qubits (flying qubits) after the scattering, and relocalized to the same optical path for further processing, by conventional linear optics (see Supplementary Information).

The above protocols exploit the collective coupled and uncoupled states of the quantum metasurface. Since the state of the quantum metasurface controls the reflectivity of the photons in different modes, it allows realization of controlled multiqubit gate of the form:

[TABLE]

where $\mathcal{V}_{\textbf{k}_{\perp}}$ is a unitary that depends on the scattering properties of the transverse mode $\textbf{k}_{\perp}$ from a quantum metasurface in state $\ket{\Phi}$. In the Supplementary Information we describe how to prepare a collective state which is reflective to specific transverse modes and transparent to others. This enables quantum control over specific photonic qubits, by preparing superpositions of mirror and non-mirror states for the different transverse modes. Analogous to a classical spatial light modulator (SLM) that gives a specific phase or amplitude to specific transverse modes, the quantum metasurface controls quantum gate operations for specific transverse modes, enabling manipulation of both spatial structure and quantum properties of the photonic state.

Implementation, imperfections, and decoherence

We now discuss a specific realization of a quantum metasurface by an array of trapped neutral atoms Barredo et al. (2016); Endres et al. (2016); Kim et al. (2016) whose many-body state can be manipulated by exploiting Rydberg interactions (see Supplementary Information) Bernien et al. (2017); Lienhard et al. (2018).

To realize the collective effect of the quantum metasurface atoms have to be trapped at distances that are smaller than the wavelength of the incident light. For trapping with optical tweezers the minimal spacing between atoms is above the diffraction limit. However, this requirement can be met by using, for example, 171Yb which has a telecom transition of $1389$nm and was recently trapped in optical tweezers using a laser of $470$nm Covey et al. (2018). This enables the following realization of the electronic levels: $\ket{g}\equiv{6s6p}$ ${}^{\,3}P_{0}$ and $\ket{e}\equiv{5d6s}$ ${}^{3}D_{1}$.

The photonic state fidelity is directly affected by the atom array quality. For the EIT implementation, we estimate the error in the reflection coefficient $r$, given by Eq. (2). The latter grows inversely with the shift of the Rydberg level induced by the ancilla, which is not uniform within the array (which is $\mu m$ scale). For the photonic cat state generation the resulting fidelity is $|\braket{\psi_{f_{0}}}{\psi_{f}}|^{2}\rightarrow 1-\frac{1}{8}|r+1|^{2}|\alpha|^{2}$, where $\ket{\psi_{f_{0}}}$ is the perfect photonic cat state. Imperfect projective measurement and the decoherence of the ancilla also affect the fidelity. Errors associated with imperfect preparation of atomic states or missing atoms (resulting from unsuccessful trapping) result in different scattering properties. $|\braket{\psi_{f_{0}}}{\psi_{f}}|^{2}$ for different percentages of missing atoms is presented in Fig. 3b. For less than $2\%$ of missing atoms, which is a typical error in the experimental systems Bernien et al. (2017); Cooper et al. (2018), the light state fidelity is still higher than $0.965$.

Outlook

The above considerations can be extended by combining techniques of classical metasurfaces for controlling the light degrees of freedom, such as angular momentum or polarization Yu et al. (2012). Our analysis can be further extended to interactions between photons beyond the weak field limit Lang et al. (2018a, b). This work can also serve as the basis for quantum metamaterials and quantum transformation optics. Similar to their classical counterpart Kildishev et al. (2013); Kildishev and Shalaev (2011) these can be realized with systems of different dimensionality. Finally, we note that other possible experimental realizations including excitons in atomically thin semiconductors, such as transition metal dichalcogenides Zhou et al. (2017); Scuri et al. (2017, 2018) can also be explored to realize quantum metasurfaces.

Quantum metamaterials and metasurfaces also offer an intriguing possibility for studying analogue systems for quantum gravity. Classically, the propagation of light in a dielectric maps mathematically to motion on a space-time background, thus engineering a specific permittivity distribution which creates an analogue system for curved space Landau and Lifshitz (1971); Genov et al. (2009); Narimanov and Kildishev (2009); Fernández-Núñez and Bulashenko (2016); Sheng et al. (2013). The present analysis shows how one can obtain quantum control of the background itself. Thus quantum metasurfaces described here can be viewed as being analogous to a gravitational background that is itself in a quantum state. While no full theory of quantum gravity is yet fully developed, quantum metamaterials offer a novel framework to study analogues to explore consequences of possible quantum gravitational effects, such as quantum fluctuations of the gravitational background Wheeler (1957); Barr (2005) or superpositions of background metrics Zych et al. (2017) in a table-top system.

Acknowledgments. We thank Dikla Oren for helpful discussions. This work was supported through the National Science Foundation (NSF), the Center for Ultracold Atoms, the Air Force Office of Scientific Research via the Multidisciplinary University Research Initiative, and the Vannevar Bush Faculty Fellowship. R.B., I.P. and H.P. are supported by the NSF through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian Astrophysical Observatory. I.P. also acknowledges funding by the Society in Science, The Branco Weiss Fellowship, administered by the ETH Zurich.