Quantum metamaterials in the microwave and optical ranges

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.


I. Introduction 2
II. Superconducting quantum metamaterials 5 III. Optical quantum metamaterials 12 IV. A review of the theoretical tools for quantum metamaterials in optics 17

I. 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 John Pendry 10 extended the results by Victor 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 single-electron 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.
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.
The full treatment of such a system should start from quantum description of both the electromagnetic field and the "atoms". In case when their role is played by qubits (that is, two-level quantum systems), a general enough Hamiltonian of a quantum metamaterial is given by 21,26Ĥ where the first term describes unperturbed photon modes, the second the qubit degrees of freedom, and the third their interaction. The direct control over (at least some) qubits is realized through the qubit Hamiltonian parameters (the bias ǫ or, sometimes, the tunneling matrix element ∆).
Forming photon wave packets with characteristic size Λ ≫ 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 "Λ-smooth" density matrix. It will describe the state of both the electromagnetic field and the quantum metamaterial, characterized by a nondiagonal, nonlocal, state-and position-dependent "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 quasi-classical treatment of the atom-light interaction 32 , but is more conveniently done using the lumped-elements description (see, e.g., 30 to the node voltages V j (t) and completely describe the classical electromagnetic field degrees of freedom (the current-voltage distribution) in the system. The lumped-elements 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).
Qubits are introduced in this scheme through their own Hamiltonians and coupling terms (e.g., flux qubits can be coupled inductively -through the magnetic flux penetrating their loop, -or galvanically -through sharing a current-carrying conductor with the circuit), which are usually cast in the formĤ The "quantum Routhian",R plays the role of the Hamiltonian for qubits (in Heisenberg representation), and its expectation value R = R produces the equations of motion for the classical field variables Φ, where we have included the dissipative function Q to take into account resistive losses in the circuit (see Refs. 30 , 2.3.3, and 33 ): one can take the continuum limit and, switching to the Schrödinger representation, obtain coupled systems of equations for the field variable Φ(r, t) and the qubit two-component "macroscopic wave function"Ψ(r, t). At any convenient stage the electromagnetic modes , leading back to a certain approximation of an approach based directly on Eq.(1).

II. SUPERCONDUCTING QUANTUM METAMATERIALS
The above scheme is most often applied -though not limited -to the case of superconducting metamaterials based on various types of superconducting qubits (see, e.g., Ref.30, Ch.2). The simplest case is given by the experimental setup of Ref. 34 (Fig.1a). There a single artificial atom (a flux qubit) is placed in a transmission line, and transmission and reflection coefficients for the microwave signal are measured. Here the equations (4) for the field in the continuum limit will yield free telegraph equations for the voltage and current everywhere except the point x = 0, where the qubit is situated: whereL,C are the inductance and capacitance per unit length of the transmission line.
Of course, for such a simple structure these equations can be written down directly. The influence of the flux qubit, which is coupled to the transmission line through the effective mutual inductance M (taking into account both magnetic and kinetic inductance) is through the matching conditions at x = 0, I(+0, t) = I(−0, t).
The qubit current operatorÎ q = I p σ z is governed by the qubit Hamiltonian (2) with the coupling term J q = I(0, t)M/c 2 . An explicit solution for the reflection/transmission amplitudes was found to be in a very good agreement with the experimental data in Ref. 34 (Fig.1b).
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 calcu-lations in the one-excitation approximation (with electromagnetic modes treated quantum mechanically) show that in such a structure arise long-living quasi-bound states of photons and the qubit, manifested as ultra-narrow resonances in the transmission coefficient.
Going from these "proto-metamaterials" to QMMs containing many artificial atoms, we return to the classical treatment of the electromagnetic field. Though 2D and 3D ver- Dimensionless equations of motion for the field vector-potential in the lowest order in field-qubits interaction yield the wave equation where where A(x), B(x) are periodic functions andhω is the energy splitting between the ground (|g ) and excited (|e ) state of the qubit. Quantum beats between these states will produce a "breathing" photonic band structure (Fig.3a); external control of qubit states allows to trap a portion of radiation in a pocket of such a structure and move it across the QMM at a desired speed 18 . In the absence of direct control over individual qubit states (apart from their initialization in the ground state) it is still possible to create a photonic crystal structure 35 by sending into the QMM specially shaped "priming" electromagnetic pulses from the opposite directions (Fig. 3b). Exiting the QMM, they leave behind a spatially periodic pattern of the probability of finding a qubit in the ground (excited) state.
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 electomagnetic field ( Fig. 3c. Remarkably, not only the process has a precipitous character, but its onset starts the sooner the greater the amplitude of the triggering pulse: In equilibrium a fully quantum treatment of a superconducting QMM becomes possible, which allows the investigation of phase transitions in the photon system 26  In case when P 0 is independent on the imaginary time τ , the photon system may undergo a second order classical phase transition: above a critical temperature T ⋆ the momentum P = 0, while below it the system can choose between two values ±P 0 . The transition details depend on the level of disorder in the qubit chain, which is modeled here by the random distribution of tunneling matrix elements ∆. In the case of a low disorder, ∆ k ≈ ∆ 0 , the transition occurs at the critical temperature where m is the transmission line inductance per unit cell, η parametrizes the qubit-field interaction, and N is the total number of qubits in the QMM. This phase transition occurs In the case of strong disorder with ∆ k distributed from zero to some ∆ 0 , the transition temperature and the transition occurs only if ∆ 0 > mη 2 N. As the authors of Ref. 26 note, the first case is similar to the metal-ferromagnet phase transition, while the second one is reminiscent of the normal metal-superconductor or Peierls metal-insulator transition. In either case a coherent state of the photon field emerges, with nonzero value of the order parameter P 0 , which in the case of low disorder is proportional to the number of qubits in the system, N. It turns out that there is the possibility of a quantum transition as well, into a state representing a superposition of semiclassical states ±P 0 . Then the order parameter becomes a periodic function of τ . In the case of low disorder and strong field-qubit coupling it occurs below the temperature (Fig.4) while in the opposite case it is given by The

III. 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 ultra-long 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 Cu-CoPc2 (a chain of copper-phtalocyanine (CuPc) and cobalt phtalocyanine (CoPc)) possesses a relatively strong ferromagnetic interaction.
In the specific meaning used in this review, a proposal was made in ( 21,28 ) to study the full quantum processes that occurs between the quantized electromagnetic field and two-level atoms. The system studied there is a 2D network of coupled atom-optical cavities, called a cavity array metamaterial (CAM). The authors propose to realize the model by using a two-dimensional photonic crystal membrane. The quantum oscillators could be quantum dots or substitution centers. Under reasonable assumptions, this system can be described way, an effective permittivity that is non-local in time and space could be derived, in exactly the same way as for natural material. This is the line followed in a series of papers by G.
Weick where a collection of metallic nanoparticles is shown to exhibit collective plasmonic modes [43][44][45] . However, a genuine quantum metamaterial requires more than that, namely the active coherent control of the quantum state of the "atoms" inserted in the photonic structure, so as to induce a control over the collective properties of the medium.
Such a system could be implemented by considering, as above, a photonic crystal in which quantum oscillators are inserted, under the guise of quantum dots for instance. The quantum dots can be described semi-classically by a dielectric function ε QD that reads as 46 : The or emission). When the quantum dots are in the emission regime, a transmission peak appears in the transmission spectrum of the probe (Fig. 8). This somewhat simplified

TERIALS IN OPTICS
Here we review the tools of quantum optics for the description of the quantum dynamics of a collection of emitters (e.g. quantum dots) in a complex electromagnetic environment that can be constituted by dielectric and/or metallic elements as depicted in Fig: 9 For a single emitter in free space, the interaction with light is given by the minimalcoupling Hamiltonian that reads 48 : where p is the electron momentum, A the vector-potential, V (r) the binding potential for the electron, B the magnetic field and E ⊥ the transverse part of the electric field, satisfying ∇.E ⊥ = 0 (c.f. 49 p. 254). The emitters are coupled to each other through the electromagnetic field that comprises both radiating and evanescent terms. where 2ε 0 is the Hamiltonian describing the dynamics of the atom variables, P is the polarization field. Even if the term d r P ⊥2 /2ε 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 ultra-strong 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 .
2µ 0 is the Hamiltonian describing the electromagnetic field dynamics.
• H int = − P( r).D( r) d r is the Hamiltonian describing the interaction between light and matter. Concerning the field operators, it is assumed that they are related to each other through Maxwell equations. This assumption implies that there exists some commutators between the field operators (p.18 in 55 ): [D( r, t), A( r ′ , t)] = ih ε 0 δ T ( r − r ′ ) where δ T ( r) is the transverse delta distribution (See also 52 p. 233). Finally, one gets the following set of equations between the field operators: One arrives at the well-known wave equations satisfied by the electric-field operator: Concerning the dynamics of the matter degrees of freedom, some usual approximations are done. We write the polarization field as a sum of a polarization field due to the atoms P a and a polarization field due to the electromagnetic environment P ind : P = P a + P ind .
We assume that the polarization field due to the electromagnetic environment P ind responds linearly and locally to the electric field. It reads as 56 where χ( r, t ′ ) is the susceptibility at position r and time t.
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 α emitters, the polarization field due to the emitters is then written as P a ( r) = Nα α=1 d α δ( r − r α ) where d α and r α are respectively the dipole-moment operator and the position of the emitter labeled by α, and δ( r) is the usual Dirac distribution. It is convenient to express all matter-operators with the help of the basis defined by the eigenstates of the matter hamiltonian H mat . The matter hamiltonian H mat is written as H mat = Nα α=1 H α mat where H α mat is the hamiltonian of the α th emitter. We note {|α, i } the eigenbasis constructed from the eigenstates of H α mat that satisfied H mat |α, i = E α i |i and the completeness condition i |α, i α, i| = I α d , where I α d is the identity matrix acting on the subspace of the α th emitter. We now assume that emitters are two-level systems. The ground state is labelled by i = − whereas the excited level is labelled by i = +. Applying twice the completeness condition on the hamiltonian H α mat , one finds H α mat = E − +E + 2 I α d +h ω α 2 σ α z wherehω α = E α + − E α − and σ α z = |α, + α, +| − |α, − α, −|. σ α z acts on the subspace defined by the eigenvectors of the α th emitter and measured its population difference between the excited and the ground state. The first term in H α mat is a constant that can be omitted by choosing correctly the reference of the energy. We then write the matter hamiltonian as (see 58 p. 23 and 53 p.128): The polarization field due to the emitters can also be written with the help of the eigenbasis of each emitter by writing the dipole-moment operator in this basis: where d α +− = α, +| d α |α, + is the projection of the dipole-moment operator in the basis {|α, + , |α, − }. The diagonal elements are null because we assume that the emitters have no permanent dipole. We introduce the raising σ α + = |α, + α, −| and lowering operators σ α − = |α, − α, +|. Finally, the polarization field due to the atoms reads: If we assume that the dipole moment projection d α +− = ( d α −+ ) ⋆ = d 0 α is real, the polarization field simplifies with the help of the Pauli matrice σ x : 58 p.24, 59 p.35). With the help of a Fourier transform, the hamiltonian for the free electromagnetic field can be written H f ield = j,sh ω 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 E = 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) The behavior of the quantum metamaterial can be computed by solving simultaneously the equation eq.(17) and the equations of motions for the collection of two-level "atoms" given by 48 , p.37 and 59 :σ Where all terms proportional to P 2 ( r) have been neglected. These equations are nonlinear coupled differential equations since σ α x are sources of the electric field eq:(17). The electromagnetic environment contributes also as a source term in eq: (17). This source term is important since it is responsible for all the unusual effects demonstrated theoretically or experimentally with classical metamaterials. Its effect on quantum metamaterials has been barely studied since solving this system of equations is challenging, even for a very simple geometry 60 . New theoretical tools should be developed to accurately describe the behavior of a quantum metamaterial in a complex electromagnetic environment. Nevertheless more approximations can be done to solve these equations. One can use the single electromagneticmode approximation 21,28,61,62 or perform a semi-classical approximation 18,19,30,31,63 . In the latter, it is assumed that there are no correlations 59 between the electromagnetic field and the matter degrees of freedom.

V. 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 20-qubits prototype 36 is much smaller than a recently fabricated 1000+-qubits superconducting quantum annealer D-Wave 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 well-known 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 quantum-classical transition, which makes this class of structures interesting also from the fundamental point of view.