## Reconfigurable quantum metamaterials |

Optics Express, Vol. 19, Issue 12, pp. 11018-11033 (2011)

http://dx.doi.org/10.1364/OE.19.011018

Acrobat PDF (2865 KB)

### Abstract

By coupling controllable quantum systems into larger structures we introduce the concept of a quantum metamaterial. Conventional meta-materials represent one of the most important frontiers in optical design, with applications in diverse fields ranging from medicine to aerospace. Up until now however, metamaterials have themselves been classical structures and interact only with the classical properties of light. Here we describe a class of dynamic metamaterials, based on the quantum properties of coupled atom-cavity arrays, which are intrinsically lossless, reconfigurable, and operate fundamentally at the quantum level. We show how this new class of metamaterial could be used to create a reconfigurable quantum superlens possessing a negative index gradient for single photon imaging. With the inherent features of quantum superposition and entanglement of metamaterial properties, this new class of dynamic quantum metamaterial, opens a new vista for quantum science and technology.

© 2011 OSA

## 1. Introduction

*artificial magnetism*[1

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32. A. L. Rakhmanov, A. M. Zagoskin, S. Savelev, and F. Nori, “Quantum metamaterials: electromagnetic waves in a Josephson qubit line,” Phys. Rev. B. **77**, 144507 (2008). [CrossRef]

*quantum superlens*based on the CAM. We envisage a configuration depicted schematically in Fig. 1 where the JCH system is manipulated to produce a perfect image using single photons. The aim here is not build a better superlens, but to use the superlens as an casestudy of how CAMs can be used to exhibit metamaterial properties such as negative refraction. However as with photonic crystal (PhC) implementations, CAMs have the advantage of being ideally lossless and do not require the operating wavelength to be larger than the constitutent element spacing. Further, because the transition energy of each atom in the system can be individually controlled via a Stark shifting control voltage, CAMs have the distinct advantage of being highly tunable and reconfigurable: features not possible using conventional designs.

*diffraction limit*. Near-field scanning optical microscopy overcomes this problem by scanning a probe in close proximity to the object, but this is often undesirable for applications such as optical lithography and sensing. It has been proposed that a lens built from NIMs can produce

*perfect*far field imaging, exhibiting all-angle negative refraction (AANR) and evanescent wave enhancement (EWE) [33

33. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

*superlens*[33

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## 2. Jaynes-Cummings-Hubbard Hamiltonian

17. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. **2**, 856–861 (2006). [CrossRef]

20. M. I. Makin, C. Cole, C. D. Tahan, L. C. L. Hollenberg, and A. D. Greentree, “Quantum phase transitions in photonic cavities with two-level systems,” Phys. Rev. A **80**, 043842 (2009). [CrossRef]

**80**, 063838 (2009). [CrossRef]

43. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics **2**, 741–747 (2008). [CrossRef]

*a*) at site

_{r}*r*, the total JCH Hamiltonian reads (

*h̄*= 1),. where Σ

_{〈}

_{r,s}_{〉}is the sum over all nearest-neighbor cavities,

*κ*> 0 is the hopping frequency (requiring that the ground eigenstate to be symmetric and the first excited eigenstate to be anti-symmetric also means that

*κ*> 0),

*ε*is the atomic transition energy,

*ω*is the cavity resonance frequency,

*β*is the single-photon Rabi frequency, and the rotating wave approximation is assumed. The onsite terms can be diagonalized in a basis of mixed photonic and atomic excitations called dressed states or polaritons, |±,

*n*〉

*= sin Θ*

_{r}*|*

_{n}*g*,

*n*〉

*+ cos Θ*

_{r}*|*

_{n}*e,n*− 1〉

*, with energy*

_{r}*K*≡ 4

*κ*cos(

*k*) cos(

_{x}d*k*) and for the unrotated lattice (inset of Fig. 2(b))

_{y}d*K*≡ 2

*κ*[cos(

*k*) + cos(

_{x}d*k*)], where

_{y}d## 3. All-angle negative refraction

*k*-space. Given a dispersion relation

*E*(

*k*), an isoenergy contour defines the curves over which the energy is constant. In this representation the gradient of the energy surface is the vector field of group velocities, which points normal to the isoenergy contour as illustrated by the colored arrows in Fig. 2(a). Using Eq. (2), the group velocity

_{x}, k_{y}*k*is conserved at the interface, the group velocities associated with these contours determine the refraction angle. In the illustration, an incident photon with wavevector

_{y}*k⃗*

_{1}= (

*k*

_{1,}

*,*

_{x}*k*

_{1,}

*) and velocity*

_{y}*v⃗*

_{1}will couple to an allowed mode of the lattice, and propagate with

*k⃗*

_{2}= (

*k*

_{2,}

*,*

_{x}*k*

_{2,}

*) and*

_{y}*v⃗*

_{2}. The refraction angle is where

*k*

_{2}

*is given by Eq. (13). Since the isoenergy contours of the lattice are*

_{,x}*convex*, we have negative refraction (

*θ*< 0), and since the lattice contour is larger than the air contour, this occurs for all incident angles.

_{R}*k*condition, Fig. 2(b) shows that the unrotated lattice does not exhibit AANR. When AANR does occur, the propagating modes in free space can be brought into a focus to form an image on the other side of the lattice even with a planar lattice slab. Such a device therefore satisfies the first criterion of a superlens. In contrast, conventional lenses that rely on positive refraction must have curvatures to converge light.

_{y}*T*≡ 1 −

*R*. Comparing Eq. (4) and Eq. (5), there is a trade-off between refraction and reflection, i.e, large negative refraction is accompanied by large reflection. This trade-off is illustrated in a comparison plot (Fig. 4) of the refraction angle and reflection coefficient for different incident angles and varying detuning.

*ψ*(

*t*)〉 =

*e*|

^{iℋt}*ψ*(0)〉. We consider the case when the source is initialized in an equal superposition of atomic and photonic modes. It is instructive to use a directional pulse by specifying an initial state with a normalized Gaussian momentum distribution around

*k⃗*

_{1}= (

*π*/4,±

*π*/4) so that it is incident on the lens at ±45°, as shown in Fig. 5(a). The superposition of the two

*k⃗*modes manifests in a coherent interference pattern in the

*y*-direction.

*κ*, which by Eq. (4), predicts a refraction angle of

*θ*= −25°. Superimposing different time instances, the incident, reflected and refracted pulses in Fig. 5(a) follow the predicted refraction angle and the trajectory predicted by Eq. (3), to converge at a location on the image plane. The reflection and transmission coefficients are also found to be in good agreement with Eq. (20). The incident and reflected polariton (±

_{R}*π*/4,

*π*/4) coherently interfere near the interface to give an interference pattern along the

*x*-direction. Note that there is considerable reflection, so that the population density has been multiplied by a factor

*M*in Fig. 5(a) and Fig. 5(b) for clearer representation.

*ε*, after fabrication. Such manipulations can be achieved dynamically by, for example, a controlled external electric field via Stark shift. This control allows one to tailor the dispersion relation, and hence the light guiding properties and focal point of the lens. The effect of changing

*ε*is demonstrated in Fig. 5(b), where decreasing Δ

_{2}by 0.73

*κ*shifts the focus 56 sites to the right.

*k⃗*in Fig. 5(c). Taking a snapshot of the propagation at time,

*t*= 300/

*κ*, it shows that as the point source propagates into the planar lens, all components are negatively refracted, so that an image of the point source is successfully formed on the image plane. As with Fig. 5(a) and Fig. 5(b) there is considerable reflection at the lens’ interfaces.

*k*. This can be minimized if we provide an adiabatic spatial change of the atomic transition energy within the lens, in effect producing a gradient-index (GRIN) structure. In Fig. 5(d) the detuning distribution follows the form,

_{x}*x*′ is the number of sites from the interface,

*w*is the width of the GRIN region and

*W*is the total width of the lens. By fine tuning the GRIN region, the level of reflection can be made arbitrarily small, although the physical trade-off is a larger lens. The removal of reflection losses is an important development, demonstrating fine control of propagation possible in our system.

## 4. Evanescent wave enhancement

*k*>

_{y}*ω*exponentially decay away from the source along the

*x*-axis. Existing superlens proposals overcome this diffraction limit by amplifying the evanescent wave (EW) components.

*W*, derived from taking the summation of the multiple scattering events at the left and right interfaces is, where

*T*(

_{ij}*R*) is the transmission (reflection) amplitude at the interface between region

_{ij}*i*and

*j*. At the resonance condition

*π*.

*bound modes*[36

36. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B **68**, 045115 (2003). [CrossRef]

_{1}and Δ

_{3}respectively, are sufficiently different from the detuning in region 2, Δ

_{2}(see Fig. 6(a)), so that we can setup an eigenstate where evanescent tails exists in region 1 and 3. This setup is analogous to that of a square well. Region 4 is the lens, and region 5 our image plane.

_{4}, where the resonance condition is met.

_{2}= Δ

_{4}= 0, the population exchange occurs between the source and the lens is that of coupled homogenous resonators, i.e.,

*P*

_{4}(

*t*) = sin

^{2}(Ω

*t*), where Ω is the characteristic mutual coupling. Following the increase in

*P*

_{4}, the EW incident on the lens is transmitted amplified by 6 orders of magnitude after a time

*t*= 10

^{4}

*κ*

^{−1}.

*η*is the difference in the eigenenergy of the source and lens. By fitting

*P*

_{4}(

*t*) to numerical results, we find that for Δ

_{4}= 0.305

*κ*,

*η*∼10

^{−3}

*κ*. This is in good agreement with the minimum energy difference between source and lens obtained by solving the lens Hamiltonian

*ℋ*directly.

*t*= 10

^{3}

*κ*

^{−1}the incident EW is amplified by a third of the exact resonance case. Thus, although exact resonance is an optimal condition for EWE it is not a necessary condition for enhancement. When the lens is tuned away from resonance, the degree of EWE can quickly diminish as seen in Fig 6(e).

_{0}. Our lens’ resolution,

*δ*= 2

*π*/

*k*

_{max}, is determined by the maximum

*k*that still satisfies the resonant condition. However we would also like to resolve all the

*k*-components leading up to

*k*

_{max}. This implies minimizing the bulk energy band spectrum so that the deviation from the resonant energy is always small. The drawback of this is a reduction in sharpness of focus. A better solution is to introduce surface mode resonance. This can be achieved by having a different

*ε*at the lens surfaces from that of the bulk. As shown in Fig. 7, the

*flatter*surface mode band (see appendix) provides the necessary minimal deviation from resonance to maximize

*k*

_{max}, leaving the bulk mode to provide the AANR and focal sharpness.

*k*

_{max}= 2

*π*/

*d*−

*ω*

_{0}, because beyond this the evanescent modes fold back into the light cone and the associated bound modes become

*leaky*states [36

36. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B **68**, 045115 (2003). [CrossRef]

*d*< Λ

_{0}/2, the resolution exceeds that of conventional lenses.

_{0}. Typically cavity resonant wavelength is twice the cavity size, so the subwavelength resolution condition becomes,

*ω*>

*ω*

_{0}. The non-linear interaction introduced by the cavity atom allows, beyond that which is available through just inter-cavity hopping, the cavity resonance frequency to be greater than the operating frequency.

*E*

^{±}=

*ω*

_{0}, the relative resolution of our lens can be approximated by, where we have assumed small detuning and that the minimum possible spacing between sites is half the resonant wavelength. Equation (10) gives the factor by which our lens beats the diffraction limit. Since

*δ*

_{0}

*/δ*

^{+}< 1, only the resolution from the negative energy branch,

*δ*

^{−}, can better the diffraction limit. Conventionally

*β*/

*ω*, restricted by the so-called fine structure constant limit, is of the order 0.01 (although larger values are possible for unconventional coupling mechanisms [44

44. M. Devoret, S. Girvin, and R. Schoelkopf, “Circuit-QED: how strong can the coupling between a Josephson junction atom and a transmission line resonator be?,” Ann. Phys. (Leipzig) **16**, 767–779 (2007). [CrossRef]

## 5. Experimental feasibility and outlook

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*Q*. WGM microcavities have experimentally produced

*Q ∼*10

^{9}in microspheres [56

56. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. **21**, 453–455 (1996). [CrossRef] [PubMed]

^{8}in microtoroids [57

57. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultrahigh-Q toroid microcavity on a chip,” Nature **421**, 925–928 (2003). [CrossRef] [PubMed]

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^{3}

*μ*m

^{3}and 180

*μ*m

^{3}respectively. PBG microcavities have achieved

*Q ∼*10

^{7}with cavity mode volume,

*V ∼*(

*λ*/2)

^{3}, where

*λ*is the operating wavelength [59

59. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics **1**, 449–458 (2007). [CrossRef]

*β*∼ 10

^{10}Hz [60

60. M. P. Hiscocks, C.-H. Su, B. C. Gibson, A. D. Greentree, L. C. L. Hollenberg, and F. Ladouceur, “Slot-waveguide cavities for optical quantum information applications,” Opt. Express **17**, 7295–7303 (2009). [CrossRef] [PubMed]

*Q*, we can approximate the inter-cavity tunneling frequency as

*κ*=

*ω*/

*Q*. For the superlensing properties presented in this work,

*β*= 100

*κ*, requiring in the visible light regime

*Q ∼*10

^{7}, which is at current experimental limits. In PBG arrays with over 100 microcavities however, only

*Q ∼*10

^{6}as yet been experimentally verified [43

43. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics **2**, 741–747 (2008). [CrossRef]

61. C.-H. Su, M. P. Hiscocks, B. C. Gibson, A. D. Greentree, L. C. L. Hollenberg, and F. Ladouceur, “Coupling slot-waveguide cavities for large-scale quantum optical devices,” Opt. Express **19**, 6362–6373 (2011). [CrossRef]

32. A. L. Rakhmanov, A. M. Zagoskin, S. Savelev, and F. Nori, “Quantum metamaterials: electromagnetic waves in a Josephson qubit line,” Phys. Rev. B. **77**, 144507 (2008). [CrossRef]

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*g*〉) in the individual cavity Λ-system would be strongly coupled to the cavity modes while the other (|

*f*〉) is not (Fig. 8). The atom-photon coupling offers intriguing potential for entirely new quantum devices. For example, if some of the atoms are prepared in a Greenberger-Horne-Zeilinger–like (GHZ-like) state |

*gg...g*〉 +

*|ff...f*〉, then the system will exhibit a superposition of dispersion relations. This in turn implies a superlens with two focal points in quantum superposition.

## 6. Conclusion

## Appendix

**80**, 063838 (2009). [CrossRef]

*E*are no longer the polaritonic energies

_{r}*H*is the Hamiltonian that relates site

_{rs}*r*to site

*s*. Employing Bloch’s theorem for periodic structures, where

*d⃗*is the displacement to site

_{r}*r*, and

*k⃗*≡ (

*k*,

_{x}*k*) is the wavevector associated with the crystal momentum, Eq. (11) becomes an energy eigenequation whose eigenvalues are the energy band structure or the dispersion relation of the medium.

_{y}*E*

_{1}and wavevector

*k⃗*

_{1}= (

*k*

_{1,}

*,*

_{x}*k*

_{1,}

*) with transmitted field of*

_{y}*E*

_{2}and

*k⃗*

_{2}= (

*k*

_{2,}

*,*

_{x}*k*

_{2,}

*) at the interface, energy conservation (*

_{y}*E*

_{1}=

*E*

_{2}) and phase matching (

*k*

_{1,}

*=*

_{y}*k*

_{2,}

*=*

_{y}*k*) requires that

_{y}*k*

_{2,}

*satisfies the condition, where*

_{x}*K*≡ 4

*κ*cos(

*k*

_{2,}

*) cos(*

_{x}d*k*

_{1,}

*) for the rotated lattice.*

_{y}d*g,n*〉

*and*

_{r}*|e*,

*n*〉

*denote the ground and excited state respectively, with*

_{r}*n*photonic excitations at site

*r*. Given the symmetry, we consider a 5-site unit cell in X-configuration that is translational-invariant along the

*y*-direction. Using the standard eigenenergy equation

*ℋ*|

*ψ*〉 =

*E*|

*ψ*〉, one arrives at a discrete scattering equation for each region of the lattice and the interfaces. In particular in region

*j*with associated parameters (

*ω*,

_{j}*ε*,

_{j}*β*,

_{j}*κ*), we have where a given site at coordinate (

_{j}*p,q*) is surrounded by four nearest-neighboring sites at coordinates (

*p*± 1,

*q*± 1) and the conservation of energy requires

*E*=

_{j}*E*. At the interface centered at the origin (0,0), We make the typical assumption that region 1 consists of an incident and a reflected wave component, where

*k*=

_{j,p}*k*–

_{j,x}*k*and

_{y}*k*=

_{j,q}*k*+

_{j,x}*k*. In region 2, the transmitted wave has the form, where

_{y}*r*and

*t*are used here to denote reflection and transmission amplitudes respectively. Substituting these solutions in to the interface equation (Eq. (16)) and applying continuity condition

*t*= 1 +

*r*, we arrive at the reflection coefficient

*R*≡ |

*r*|

^{2}, Assuming uniform coupling

*κ*=

_{j}*κ*, we retrieve the required expression. Finally since a polaritonic pulse has a momentum distribution

*G*(

*k⃗*), we define an effective reflection coefficient,

*ε*and

_{s}*ε*are the surface and bulk atomic transition energies respectively. We solve for the eigenvalues to get the surface mode energy bands.

_{b}## Acknowledgments

## References and links

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**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(080.0080) Geometric optics : Geometric optics

(190.0190) Nonlinear optics : Nonlinear optics

(270.0270) Quantum optics : Quantum optics

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Metamaterials

**History**

Original Manuscript: February 3, 2011

Revised Manuscript: May 4, 2011

Manuscript Accepted: May 10, 2011

Published: May 23, 2011

**Citation**

James Q. Quach, Chun-Hsu Su, Andrew M. Martin, Andrew D. Greentree, and Lloyd C. L. Hollenberg, "Reconfigurable quantum metamaterials," Opt. Express **19**, 11018-11033 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11018

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### References

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