Quantum metamaterials in the microwave and optical ranges
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Abstract
Quantum metamaterials generalize the concept of metamaterials (artificial optical media) to the case when their optical properties are determined by the interplay of quantum effects in the constituent ‘artificial atoms’ with the electromagnetic field modes in the system. The theoretical investigation of these structures demonstrated that a number of new effects (such as quantum birefringence, strongly nonclassical states of light, etc.) are to be expected, prompting the efforts on their fabrication and experimental investigation. Here we provide a summary of the principal features of quantum metamaterials and review the current state of research in this quickly developing field, which bridges quantum optics, quantum condensed matter theory and quantum information processing.
Keywords
Photonic Crystal Transmission Line Polarization Field Quantum Coherence Qubit State1 Introduction
The turn of the century saw two remarkable developments in physics. First, several types of scalable solid state quantum bits were developed, which demonstrated controlled quantum coherence in artificial mesoscopic structures [1, 2, 3, 4] and eventually led to the development of structures, which contain hundreds of qubits and show signatures of global quantum coherence (see [5, 6] and references therein). In parallel, it was realized that the interaction of superconducting qubits with quantized electromagnetic field modes reproduces, in the microwave range, a plethora of effects known from quantum optics (in particular, cavity QED) with qubits playing the role of atoms (‘circuit QED’, [7, 8, 9]). Second, since Pendry [10] extended the results by Veselago [11], there was an explosion of research of classical metamaterials resulting in, e.g., cloaking devices in microwave and optical range [12, 13, 14]. The logical outcome of this parallel development was to ask, what would be the optical properties of a ‘quantum metamaterial’  an artificial optical medium, where the quantum coherence of its unit elements plays an essential role?
As could be expected, this question was arrived at from the opposite directions, and the term ‘quantum metamaterial’ was coined independently and in somewhat different contexts. In Refs. [15, 16, 17] it was applied to the plasmonic properties of a stack of 2D layers, each of them thin enough for the motion of electrons in the normal direction to be completely quantized. Therefore ‘the wavelike nature of matter’ had to be taken into account at a singleelectron level, but the question of quantum coherence in the system as a whole did not arise. In Refs. [18, 19] the starting point was the explicit requirement that the system of artificial atoms (qubits) maintained quantum coherence on the time scale of the electromagnetic pulse propagation across it, in the expectation that the coherent quantum dynamics of qubits interacting with the electromagnetic field governed the ‘optical’ properties of the metamaterial.
Currently the term ‘quantum metamaterial’ is being used in both senses (see, e.g., [20, 21, 22, 23, 24, 25]). We will follow the more restrictive usage and call quantum metamaterials (in the narrow sense) such artificial optical (in the broad sense) media that [23] (i) are comprised of quantum coherent unit elements with desired (engineered) parameters; (ii) quantum states of (at least some of) these elements can be directly controlled; and (iii) can maintain global coherence for the duration of time, exceeding the traversal time of the relevant electromagnetic signal. The totality of (i)(iii) (in short: controlled macroscopic quantum coherence) that makes a quantum metamaterial a qualitatively different system, with a number of unusual properties and applications.
A conventional metamaterial can be described by effective macroscopic parameters, such as its refractive index. (The requirement that the size of a unit cell of the system be much less  in practice at least twice less  than the wavelength of the relevant electromagnetic signal, is implied in its definition as an optical medium, and is inherited by quantum metamaterials.) From the microscopic point of view, these parameters are functions of the appropriately averaged quantum states of individual building blocks. In a quantum metamaterial, these states can be directly controlled and maintain phase coherence on the relevant spatial and temporal scale.
Forming photon wave packets with characteristic size \(\Lambda\gg a\) (where a is the unit cell size) and averaging over the quantum states of qubits on the scale of Λ, one should eventually arrive at the effective equation of motion for the ‘Λsmooth’ density matrix. It will describe the state of both the electromagnetic field and the quantum metamaterial, characterized by a nondiagonal, nonlocal, state and positiondependent ‘refractive index’ matrix.
Following through with this program involves significant technical difficulties, and the task is not brought to conclusion yet. Nevertheless certain key effects in quantum metamaterials can be investigated at a more elementary level using an approximate wave function (e.g., [26, 27, 28, 29]), or treating the electromagnetic field classically [18, 19, 30, 31]. The latter is like the standard quasiclassical treatment of the atomlight interaction [32], but is more conveniently done using the lumpedelements description (see, e.g., [30], Section 2.3). The state of a system of M nodes connected by capacitors, inductors and, if necessary, Josephson junctions, is described by a Lagrangian \({\mathcal {L}}(\{\Phi,\dot{\Phi}\})\). Here the ‘node fluxes’ \(\Phi_{j}(t) = c\int^{t} dt' V_{j}(t')\) are related to the node voltages \(V_{j}(t)\) and completely describe the classical electromagnetic field degrees of freedom (the currentvoltage distribution) in the system. The lumpedelements description is appropriate, since we are interested in signals with a wavelength much larger than the dimension of a unit element of the circuit (the condition of its serving as a metamaterial).
2 Superconducting quantum metamaterials
For a structure, where the role of the ‘artificial atom’ between 1D transmission lines is played by a single qubit surrounded by an array of N coupled photonic cavities [29] the calculations in the oneexcitation approximation (with electromagnetic modes treated quantum mechanically) show that in such a structure arise longliving quasibound states of photons and the qubit, manifested as ultranarrow resonances in the transmission coefficient.
Solving the coupled equations for the classical field and qubits numerically (still in the approximation of factorized qubit state) allows to investigate lasing in a QMM [31]. If the qubits are initialized in the excited state (e.g. by sending a priming pulse through the QMM), an initial pulse triggers a coherent transition of energy from qubits to the electromagnetic field (Figure 3(c)). Remarkably, not only the process has a precipitous character, but its onset starts the sooner the greater the amplitude of the triggering pulse: \(\sqrt{\mathrm{field}\ \mathrm{amplitude}} \times\tau_{\mathrm{onset}} \approx\mathrm{const}\).
In equilibrium a fully quantum treatment of a superconducting QMM becomes possible, which allows the investigation of phase transitions in the photon system [26]. The chosen model of a QMM (a series of RF SQUIDS coupled to the transmission line and considered in twolevel approximation) lead to a generic Hamiltonian (1), with only parameters being model dependent. Using the instanton approach, the effective action of the photon subsystem was obtained as a function of the photon field momentum P (in imaginary time).
3 Optical quantum metamaterials
The domain of quantum metamaterials in the optical, or near IR, region of the spectrum is still in its infancy. As it has already been stated, some authors use the term quantum metamaterial to denote a structure in which quantum degrees of freedom are inserted [15]. In some other cases, it is the expression ‘quantum dots metamaterials’ that is used: this is to stress that, although quantum dots are inserted in a metamaterial, one is not interested in the quantum coherence of the dots, but rather on the gain that they provide, to counteract the losses due to the presence of metallic inclusions [38]. In other proposals, it is quantum wells that are inserted in a photonic structures. The quantum well are described electromagnetically by a permittivity allowing some control over the behavior of the structure. In Ref. [17], a layered metamaterial is investigated, in which the period comprises two GaAs quantum wells. This structure results in an effective permittivity tensor allowing to obtain a negative refraction. The effective properties strongly depend upon the 2D electron density in the quantum well. In Ref. [15] the same kind of structure is investigated in order to control plasmon propagation, allowing to obtain ultralong propagation distances.
An original proposal was made in [39] to extend the concept of metamaterial to quantum magnetism. The idea is to use molecular engineering or organic synthesis to fabricate magnetic quantum metamaterials. It is shown theoretically, by ab initio calculations, that CuCoPc2 (a chain of copperphtalocyanine (CuPc) and cobalt phtalocyanine (CoPc)) possesses a relatively strong ferromagnetic interaction.
4 A review of the theoretical tools for quantum metamaterials in optics

\(H_{\mathrm{mat}} = \frac{\mathbf{p}}{2m} + V(r) + \int d \vec{r} \frac {\mathbf{P}^{\bot2}}{2\varepsilon_{0}} \) is the Hamiltonian describing the dynamics of the atom variables, P is the polarization field. Even if the term \(\int d \vec{r} \mathbf{P}^{\bot2}/2\varepsilon _{0}\) is important to reproduce the correct dynamics of the emitter [48], it is usually neglected when studying the interaction between the emitter and light. It is usually argued that this term merely shifts the energy levels, an effect that can be accounted for by a correct renormalization of the emitter energy levels. Nevertheless in the ultrastrong coupling regime, this term has to be taken into account [54]. It leads to a decoupling of matter and light states because of a screening of the incident light by the polarization field P, resulting for example in a reduction of the Purcell factor, while increasing the coupling between the field and the emitter [54].

\(H_{\mathrm{field}} = \int \frac{ \mathbf{D}^{2}(\vec{r})}{2\varepsilon _{0}} + \frac{\mathbf{B}^{2}(\vec{r})}{2\mu_{0}}\) is the Hamiltonian describing the electromagnetic field dynamics.

\(H_{\mathrm{int}} = \int \mathbf{P}(\vec{r})\cdot\mathbf{D}(\vec{r}) \,d\vec {r} \) is the Hamiltonian describing the interaction between light and matter.
\(\mathbf{D}(\vec{r})\) is the displacement vector. Note that the Hamiltonian given by the equation Eq. (16) is exact. It is completely equivalent to the minimalcoupling Hamiltonian (Cohen [52], p.298). From Hamiltonian Eq. (16), with the help of the Heisenberg equation, one can find the dynamical equations satisfied by each operators (field and matter operators).
Concerning the polarization field due to emitters, we work in the usual dipole approximation and approximate the polarization field by keeping only the first term in the multipolar expansion even if this approximation can be crude for quantum dots [57]. If there are \(N_{\alpha}\) emitters, the polarization field due to the emitters is then written as \(\mathbf{P}_{a}(\vec{r}) = \sum_{\alpha=1}^{N_{\alpha}} \mathbf{d}^{\alpha} \delta(\vec{r}  \vec{r_{\alpha}})\) where \(\mathbf{d}^{\alpha}\) and \(\vec{r_{\alpha}}\) are respectively the dipolemoment operator and the position of the emitter labeled by α, and \(\delta(\vec{r})\) is the usual Dirac distribution. It is convenient to express all matteroperators with the help of the basis defined by the eigenstates of the matter Hamiltonian \(H_{\mathrm{mat}}\). The matter Hamiltonian \(H_{\mathrm{mat}}\) is written as \(H_{\mathrm{mat}} = \sum_{\alpha=1}^{N_{\alpha}} H_{\mathrm{mat}}^{\alpha}\) where \(H_{\mathrm{mat}}^{\alpha}\) is the Hamiltonian of the \(\alpha^{th}\) emitter. We note \(\{\alpha,i\rangle \}\) the eigenbasis constructed from the eigenstates of \(H_{\mathrm{mat}}^{\alpha}\) that satisfied \(H_{\mathrm{mat}}\alpha,i\rangle = E^{\alpha}_{i} i\rangle \) and the completeness condition \(\sum_{i} \alpha, i\rangle \langle \alpha,i = I_{d}^{\alpha}\), where \(I_{d}^{\alpha}\) is the identity matrix acting on the subspace of the \(\alpha^{th}\) emitter.
With the help of a Fourier transform, the Hamiltonian for the free electromagnetic field can be written \(H_{\mathrm{field}} = \sum_{j,s} \hbar\omega _{j} b_{j,s}^{+}b_{j,s}\) where \(b_{j}\), s (resp. \(b^{+}_{j,s}\)) is the annihilation (resp. creation) operator of the electromagnetic mode j with polarization s. Within this decomposition the electric field on its own reads \(\mathbf{E} = \sum_{j,s} e_{j,s}(b_{j,s}  b^{+}_{j,s})\). Finally, following these successive approximations one finds the Hamiltonian given by the equation Eq. (1)
5 Conclusions
The field of quantum metamaterials research arose at the intersection of quantum optics, microwave and Josephson physics, and quantum information processing. One of its rather paradoxical feature is that, while the theoretical progress in this area still significantly outweighs the experiment, the theoretical challenges seem more significant. Indeed, the existing experimental techniques, especially in case of superconducting structures, already allow creating massive arrays. The 20qubits prototype [36] is much smaller than a recently fabricated 1000+qubits superconducting quantum annealer DWave 2X. Given a simpler structure, and less strict demands to a quantum metamaterial than to a quantum computer, making and testing quantum metamaterials on this scale is a question of time and funding. On the other hand, the theoretical analysis of quantum metamaterials produces promising results, already using simple approximations. Nevertheless the understanding of the full scale of effects which can be expected in these systems requires a more detailed analysis of large scale quantum coherences and entanglement. Because of the wellknown impossibility to effectively simulate a large quantum system by classical means, a direct approach to this is currently limited to structures containing (optimistically) less than a hundred qubits. New theoretical tools need to be developed, generalizing the methods of quantum theory of solid state [5].
These challenges also present alluring opportunities. Developing and testing new theoretical methods applicable to large quantum coherent systems would be valuable for the whole field of quantum technologies, including quantum computing. Optical elements based on quantum metamaterials would provide new methods for image acquisition and processing. Last but not least, a quantum metamaterial would be a natural test bed for the investigation of quantumclassical transition, which makes this class of structures interesting also from the fundamental point of view.
Notes
Acknowledgements
AZ was supported in part by the EPSRC grant EP/M006581/1 and by the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUSTMISiS (No. K22014015).
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