Tunneling-assisted optical information storage with lattice polariton solitons in cavity-QED arrays

# Tunneling-assisted optical information storage with lattice polariton solitons in cavity-QED arrays

## Abstract

Considering two-level media in the array of weakly coupled nano-cavities, we reveal a variety of dynamical regimes, such as diffusion, self-trapping, soliton, and breathers for the wave-packets in the presence of photon tunneling processes between the next-nearest cavities. We focus our attention on the low branch (LB) bright polariton soliton formation, due to the two-body polariton-polariton scattering processes. When detuning frequency is manipulated adiabatically, the low-branch lattice polariton localized states, i.e., that are solitons and breathers evolving between photon-like and matter-like states, are shown to act as carriers for spatially distributed storage and retrieval of optical information.

###### pacs:
42.50.Ex, 71.36.+c, 42.50.Pq, 05.45.Yv

## I Introduction

Nowadays, the elaboration and investigation of hybrid quantum devices and artificial nanostructures represent a huge area of experimental and theoretical research (1); (2). In particular, quantum memory devices are proposed for mapping the quantum state of light onto the matter by using a slow light phenomenon, through the coupling between matter excitation and quantized field (2); (3). In this sense, polaritons, linear superpositions of quantized field and collective excitations in matter, provide a very elegant way for optical information storage, where the group velocity of the wave-packet could be low enough due to a large value of polariton mass (4); (5).

Within the framework of modern scalable quantum technologies (6); (7); (8); (9), the arrays of cavities containing two-level systems (atoms, quantum dots, or Cooper pair boxes, referred as qubits) strongly interacting with a cavity field at each site, are theoretically supposed to provide a promising platform for quantum computing and quantum information processing (10); (11); (12). Moreover, a strong Kerr-type nonlinearity caused by two-body polariton-polariton interaction leads to the formation of bright polariton solitons (13); (14); (15).

In the experiments, great efforts have been aimed at the achievement of deterministic trapping of single qubits with a strong coupling to the quantized electromagnetic fields in nanocavities (16); (17); (18); (19). Especially, we stress here the recent challenging results established in Ref.  (19) with “ultracold” single rubidium atoms trapped in the vicinity of tapered fiber (about 100 nm far from) and their effective coupling with photonic crystal cavity. The obtained single photon Rabi frequency was in the range of few gigahertz for the cavity volume less than ( is light wavelength). Such results pave the way to the design of new scalable devices for quantum memory purposes being compatible with photonic circuits (20); these devices exploring two-level systems at their heart (5); (21).

Here, we apply full power of current quantum technological achievements obtained in the atomic optics area to provide theoretically an alternative approach to optical information storage and retrieval by using half-matter, half-photon property of polaritons and by investigating collective dynamics of coupled atom-light states in a qubit-cavity quantum electrodynamical (QED) array. Low branch (LB) polariton solitons, as well as different dynamical regimes for diffusion, self-trapping, and breather states occur through the interaction between atoms and quantized optical cavity field (22); (23). Considering the next-nearest tunneling effect for photonic fields while the distance between adjacent cavities is within the order of optical wavelength, lattice polariton soliton solutions are revealed to exist at the border of two different kinds of breather states. Due to the robustness in preserving the shape of wave-packets, by manipulating the detuning frequency adiabatically, optical information storage and retrieval are proposed to carry out through the transformation between photon-like and matter-like lattice polariton solitons.

This paper is arranged as follows. In Sec. II, we explain in details our model to realize atom-light interaction in a cavity array occurring at nanoscales. The extended tight-binding model will be established in this case, and In Sec. III, we introduce coupled atom-light excitation basis that is polariton basis for lattice system. Apart from results obtained by us previously (15), LB polaritons occurring at each site of the cavity array are a subject of our study at the rest of the paper. In Sec. IV, we use time dependent variational approach to obtain polariton wave-packet behavior. Basic equations for the wave-packet parameters and their general properties in the QED cavity array are established. In Sec. V, we establish results relaying to investigation of variety of 1D lattice polariton wave-packet dynamical regimes in the presence of next neighbor interaction in the lattice. In Sec. VI, we propose the physical algorithm of storage of optical information by using lattice polariton localized states that is soliton and breather states. In the conclusion, we summarize the results obtained.

## Ii Model of atom-light interaction beyond the tight-binding approximation

We consider a one-dimensional (1D) array of small (nanoscale) cavities, each containing the ensemble of a small but macroscopic number of non-interacting two-level atoms, see Fig. 1. The proposed structure in Fig. 1 can be realized by loading a small number of ultracold atoms via a dipole trap, slightly above the so-called collisional blockade regime which is practically valid for the beam waist (19); (24). The total Hamiltonian for our atom-light coupled system in Fig. 1 can be represented as (9); (25); (15)

 ^H=^HAT+^HPH+^HI, (1)

where is a quantum field theory Hamiltonian for non-interacting atoms, is responsible for the photonic field distribution, and the term characterizes the atom-light interaction in each cavity, respectively. In the second quantized form, the Hamiltonian in Eq. (1) can be written as

 ^HAT=∑i,j=1,2i≠j∫^Φ†j(−ℏ2Δ2Mat+V(j)ext)^Φjd→r, (2a) ^HPH=∫^Φ†ph(−ℏ2Δ2Mph+Vph)^Φphd→r, (2b) ^HI=ℏκ∫(^Φ†ph^Φ†1^Φ2+^Φ†2^Φ1^Φph)d→r, (2c)

with the mass of free atoms, , and the effective mass of trapped photons, . In Eq. (2), quantum field operators () and ( ) annihilate and create atoms (photons) at position ; and are trapping potentials for atoms and photons, respectively. Since each cavity contains a small number of atoms, one can safely neglect the terms responsible for atom-atom scattering processes in Eq. (2a(26).

In general, one can expand atomic () and photonic () field operators as follows

 ^Φj(→r)=∑n^aj,nφj,n(→r),j=1,2, (3a) ^Φph(→r)=∑n^ψnξn(→r), (3b)

with the real (Wannier) functions: , , responsible for the spatial distribution of atoms and photons at –site. They fulfill the normalization condition . Annihilation operators and in Eq. (3a) characterize the dynamical properties of atoms at internal lower () and upper () levels, respectively; meanwhile the annihilation operator in Eq. (3b) describing the temporal behavior of the photonic mode located at the -th lattice cavity.

Substituting Eq. (3) for Eq. (2), one can obtain

 ^HAT=^H1+^H2+^H12, (4a) ^Hj=ℏM∑n[ω(j)n^a†j,n^aj,n−βj,n(^a†j,n^aj,n+1+^a†j,n^aj,n−1)], ^Hj=j=1,2, (4b) ^HPH=ℏM∑n[ωn,ph^ψ†n^ψn−α(1)n(^ψ†n^ψn+1+^ψ†n^ψn−1) HPH=−α(2)n(^ψ†n^ψn+2+^ψ†n^ψn−2)], (4c) ^HI=ℏM∑ng√Nn[^ψ†n^a†1,n^a2,n+^a†2,n^a1,n^ψn], (4d)

where corresponds to the total number of atoms at the -th lattice cell. The frequencies , , hopping constants , , and nonlinearity strength are in the form

 ω(j)n=1ℏ∫(ℏ22Mat(→∇φj,n)2+φj,nV(j)extφj,n)d→r, (5a) ωn,ph=1ℏ∫(ℏ22Mat(→∇ξn)2+ξnVphξn)d→r, (5b) βj,n=−1ℏ∫(ℏ22Mat→∇φj,n⋅→∇φj,n+1 βj,n=+φj,nV(j)extφj,n+1)d→r, (5c) α(1)n=−1ℏ∫[ℏ22Mph→∇ξn⋅→∇ξn+1+ξnVphξn+1]d→r, (5d) α(2)n=−1ℏ∫[ℏ22Mph→∇ξn⋅→∇ξn+2+ξnVphξn+2]d→r, (5e) g=κ∫ξnϕ1,nϕ2,nd→r. (5f)

Thereafter, for simplicity we assume that all cavities are identical to each other and contain the same average number of atoms. In this case, it is convenient to suppose that functions are identical for all , that is . We are also working under the strong atom-field coupling condition, that is

 g≫Γat,Γph, (6)

where and are spontaneous emission and cavity decay rates, respectively. The parameter in Eq. (5d) describes overlapping of an optical field for the nearest-neighbor cavities; is responsible for the overlapping of photonic wave functions beyond the tight-binding approximation. Since the characteristic spatial scale of atomic localization is essentially smaller than cavity size , it seems reasonable to use the tight-binding approximation especially for atomic system in the cavity array. Coupling coefficients in Eq. (5) are the nearest-neighbor hopping constants for atoms in a 1D lattice structure.

In particular, we examine the properties of parameters for the cavity-QED array containing two-level rubidium atoms. We take the D-line of rubidium atoms as an example, which gives the resonance frequency . The strength of the interaction of a single atom with a quantum optical field is taken as at each cavity with the effective volume of atom-field interaction . We assume , that is compatible with current experimental results (19); is the atomic dipole matrix element. To achieve a strong atom-field coupling regime – see Eq. (6) – one can propose a macroscopically large number of atoms at each cavity, say . This number implies a collective atom-field coupling parameter . The lifetime for rubidium atoms is , which corresponds to the spontaneous emission rate of about . The minimal value of each cavity field decay rate can be taken up to several hundred of megahertz that corresponds to cavity quality factor .

To get a variational estimate for the tunneling coefficients mentioned above, we assume that Wannier wave functions for the atomic and photonic parts localized at the jth cavity may be approached by

 φj,n(→r)=Cje−(x−xn)2/−(x−xn)22σ2x,j2σ2x,je−(y2+z2)/−(y2+z2)2σ2j2σ2j, (7a) Missing dimension or its units for \kern (7b)

where () and are relevant normalization constants, respectively. Taking into account the realistic values of atomic and photonic wave functions, we assume

 σx,j≪σj,ξx≪ξ. (8)

If the atoms are trapped in the vicinity of thin (tapered) optical fiber (that is not shown in Fig. 1), the trapping potential can be represented as a sum of that is Van der Waals potential occuring due to the closeness of atoms to the fiber surface, and that is a potential created by the optical field, cf. (18); (27). For current experiments the depth of total potential is of order of millikelvins (19); (28). Although the general (radial) dependence of on the distance from the surface is not so simple, however, it is possible to consider a harmonic traping potential at the bottom of by choosing the appropriate external laser field parameters. Roughly speaking, we consider atomic trapping potential represented as  (29):

 Vext≃Mat2[ω2x(x−xn)2+ω2⊥(y2+z2)], (9)

where and are relevant trapping axial and radial frequencies, respectively. We suppose that the minimum of a 2D periodic potential is located at a centers of the -th cavity. By substituting Eq. (7) and Eq. (9) into Eq. (5), we obtain

 β= −ℏ4Matσ2xe−d24σ2x(1−d22σ2x), (10)

for the atomic tunneling rate . The atomic tunneling rate is positive if a cavity effective size is . The latest one () is typically a few hundred nanometers in real experiments with ultracold atoms (29).

The calculation of photon tunneling rates () between the cavities can be performed in the same way. Thus, we have

 α(ζ)=−ℏ4Mphξ2xe−ζ2d24ξ2x(1−ζ2d22ξ2x), (11)

where enumerates the number of cavities. Taking into account a typical effective photon mass, for rubidium D-lines average wavelength , and assuming the width of photonic wave-packet to be for , it is possible to estimate photonic tunneling parameters as and respectively.

## Iii Polaritons in the nano-size cavity array

One of the main features of our approach is a strong nonlinearity due to small cavity volumes occupied by the optical field, that is , where is a light wavelength, is a refractive index (19); (28). In Schwinger representation, atom-field interaction in the lattice can be described by atomic excitation operators , and by operator of the population imbalance which are defined as

 ^S+,n=^a†2,n^a1,n, (12) ^S−,n=^a†1,n^a2,n, ^Sz,n=12(^a†2,n^a2,n−^a†1,n^a1,n).

The operators determined in Eq. (12) obey SU(2) algebra commutation relations

 [^S+,n,^S−,n]=2^Sz,n,[^Sz,n,^S±,n]=±^S±,n. (13)

Alternatively, it is possible to map operators in Eq. (12) onto the atomic excitation operators , applying the so-called Holstein–Primakoff transformation, i.e.,

 ^S+,n=^ϕ†n√N−^ϕ†n^ϕn, (14a) ^S−,n=√N−^ϕ†n^ϕn^ϕn, (14b) ^Sz,n=^ϕ†n^ϕn−N/N22. (14c)

It is worth noticing that the atomic excitation operators , obey the usual bosonic commutation relations . Strictly speaking, it is possible to treat the operators and describing particles at lower and upper levels, respectively, as ,  (15).

When number at each cavity is macroscopical but not so large, one can keep all the terms in the expansion of . In this limit, we get for an effective Hamiltonian ,

 ^HL=ℏ∑n[~ω12^ϕ†n^ϕn+ωn,ph^ψ†n^ψn+g(^ψ†n^ϕn+H.C.)], (15a) ^HTUN=−ℏ∑n[β(^ϕ†n^ϕn+1+H.C.) HC+α(1)(^ψ†n^ψn+1+H.C.)+α(2)(^ψ†n^ψn+2+H.C.)], (15b) ^HNL=−ℏ∑n[g2N(^ψ†n^ϕ†n^ϕ2n+H.C.)], (15c)

where we have introduced new parameters . Now, let us introduce the lattice polariton operators as follows

 ^Ξ1,n=Xn^ψn+Cn^ϕn,^Ξ2,n=Xn^ϕn−Cn^ψn, \ref{Xi_1_Xi_2}a,b

where and are Hopfield coefficients defined as

 Xn=1√2(1+2πδn√4g2+(2πδn)2)1/2, (17a) Cn=1√2(1−2πδn√4g2+(2πδn)2)1/2. (17b)

In Eq. (17), , is atom-light field detuning frequency at each cavity. Note that we consider parameters and to be the same for all cavities (sites ), assuming that and . This approach implies an equal atom-light detuning for all cavities too.

The operators and in Eq. (16) characterize two types of bosonic quasiparticles, i.e., upper and lower branch polaritons occurring at each site of the lattice. At the low density limit, Eqs. (16-17) represent the exact solution that diagonalizes a linear part of the total Hamiltonian .

At equilibrium the lowest polariton branch is much more populated. Here, we use the mean-field approach to replace the corresponding polariton field operator by its average value , which simply characterizes the LB polariton wave function at the -th cavity. In particular, for further processing we introduce the -th normalized polariton amplitude , where is the total number of LB polaritons at the array. For this approach, by substituting Eq. (16) into Eq. (15) and keeping LB polariton terms only, we arrive at

 H =ℏM∑n[12ΩLB|Ψn|2−Ω1(Ψ∗nΨn+1+C.C.) (18) −Ω2(Ψ∗nΨn+2+C.C.)+12Ω3|Ψn|4],

where we have introduced characteristic frequencies

 ΩLB=12(~ω12+ωn,ph−√(2πδ)2+4g2), (19a) Ω1=βX2+α(1)C2, (19b) Ω2=α(2)C2, (19c) Ω3=2gCX3NpolN. (19d)

The nearest and next-nearest tunneling energies, and , are shown in Fig. 2, as a function of the characteristic cavity size for different detuning frequencies . For a large enough () cavity size both tunneling rates are positive, and condition is fulfilled, see Fig. 2. The overlapping of neighbor polariton wave-functions is a dominant term, and our lattice system is reduced to the typical tight-binding approach. On the other hand, the properties of polariton tunneling energies change dramatically for the small sized cavities, , where coefficient becomes much more important.

The main features of the polaritonic lattice are connected with the properties of atom-light detuning . In the limit of a negative and large atom-light field detuning chosen as , , LB polaritons behave as photons, i. e. . Thus, we can represent the parameters (19) as , , , . However, in another limit, we can take , (, ) and LB polaritons behave as excited atoms, i. e., . We readily find the coefficients (19) as , , , .

In practice, instead of using Eq. (18), it is useful to work with the dimensionless Hamoltonian in the form

 H =ℏM∑n[12ωLB|Ψn|2−ω1(Ψ∗nΨn+1+C.C.) (20) −ω2(Ψ∗nΨn+2+C.C.)+2√πω3|Ψn|4],

where , , , are normalized parameters characterizing polariton properties in the cavity QED chain. Eq. (20) is the starting model equation for the present work and is used to study the nonlinear phase diagrams, as well as the related optical information storage with lattice polariton solitons in the following sections.

## Iv Time-dependent variational approach

To analyze different regimes of polaritons in the cavity-QED arrays, we study the dynamical evolution of in-site Gaussian shape wave-packet,

 Ψn =Nexp[−(n−ξ(t))2γ(t)2+ip(t)(n−ξ(t)) (21) +iη(t)2(n−ξ(t))2(n−ξ(t))2γ(t)2],

where and are the time dependent dimensionless center and width of the wave-packet, respectively; is momentum and is curvature, is a normalization constant (a wave-packet amplitude). Lattice coordinate relates to the number of sites (cavities) as . The wave-packet dynamical evolution can be obtained from the corresponding Lagrangian density

 L= M∑n[i2(Ψ∗n∂Ψn∂t−Ψn∂Ψ∗n∂t)−ωLB|Ψn|2 (22) +ω1(Ψ∗nΨn+1+C.C.)+ω2(Ψ∗nΨn+2+C.C.) −2√πω3|Ψn|4].

By plugging Eq. (21) intor Eq. (22), one can have an effective Lagrangian by averaging the Lagrangian density Eq. (22), as

 ¯L=[p˙ξ−˙ηγ28−ω3γ+ω1cos(p)e−σ+ω2cos(2p)e−4σ],

where we made the following denotation . Dots denote derivatives with respect to dimensionless time . It is remarked that Eq. (IV) is valid when parameter is not too small, that is  (22); (23). With the Lagrangian in Eq. (IV), one can obtain the following variational equations for the canonically conjugate polariton wave-packet parameters

 ˙p=0, (24a) ˙ξ=ω1sin(p)e−σ+2ω2sin(2p)e−4σ, (24b) ˙γ=γηm∗, (24c) ˙η=1m∗(4γ4−η2)+4ω3γ3, (24d)

where is a dimensionless polariton mass.

Phase diagrams for various dynamical regimes are determined by the property of polariton mass and by the sign of Hamiltonian that is a conserved quantity. At a polariton wave-packet exhibits diffusive and self-trapping regimes for which , and in the limit of infinite time scales (), respectively.

For an untitled trap of polaritons in the lattice, the momentum is conserved. By introducing the dimensionless polariton mass , one can have the effective Hamiltonian function in the dimensionless coordinates,

 H=−ω1cos(p)e−σ−ω2cos(2p)e−4σ+ω3γ, (25)

where . The transition between different regimes is governed by an equation that implies

 cos(pH01,2)≃−2ε10±√4ε210+0.5, (26)

where ; (we suppose that initially at and ), , and we denote as initial value of the Hamiltonian that is, obviously, conserved quantity in the absence of dissipation.

Both of the roots (26) are located within the domain if conditions and are fulfilled simultaneously. Otherwise Eqs. (26) impose only one root. It occurs for the tunneling rates . Practically, this situation corresponds to large enough cavity sizes for which both of the tunneling rates are positive and vanishes rapidly.

Physically important bound state for our problem occurs in the domain of negative polariton mass and can be associated with soliton formation for the polariton wave-packet. The polariton (bright) soliton wave-packet propagates with initial width , mass and velocity unchanged in time. The mass of the soliton wave-packet can be found from

 1m∗0=ω1cos(p0)e−σ0+4ω2cos(2p0)e−4σ0. (27)

Strictly speaking, Eq. (27) defines characteristic domain

 cos(p1,2)=0.5[−ε10±√2+ε210] (28)

of allowed wave-packet momentum where solitonic regime can be achieved. It can be obtained under the conditions  and .

Solitons exist within the region for which inequalities are hold for the positive tunneling rates () and for (), . Contrary, at solitons can be obtained at the outside of the named region.

In the limit of tight binding approximation () Eq. (26) implies only one solution that corresponds to the physical situation described in (22); (23) for atomic BEC lattice solitons. In this case polariton solitons exist only for wave-packet momentum that obeys to inequality .

## V Polariton wave-packet dynamics

It is much better to provide the analysis of polariton wave-packet dynamics in the dynamical phase diagram picture, that reflects particular features of polaritons in the lattice. In Fig. 3, we represented the corresponding dynamical phase diagram, the related polariton effective mass, and the Hamiltonian energy contours, as functions of the momentum parameter . For , the initial polariton mass is positive and one can expect self-trapping and diffusive regimes only. The most important results can be obtained for LB polariton dynamics in other domains of momentum .

In the region we have , for all values of detuning – see Fig. 3. Figure 4 demonstrates typical temporal dynamics of the wave-packet group velocity in this case. In the inset the dependence of detuning as a function of initial width of the wave-packet is presented. For we deal with the diffusive regime for which the group velocity tends to the constant value asymptotically. On the other hand the group velocity oscillates in time within the window . For , i.e. for the self-trapping regime rapidly vanishes and goes to zero. The soliton regime occurs for atom-field detuning and obviously is characterized by a constant value of the group velocity – dotted line in Fig. 4.

Essentially new results can be obtained when momentum obeys to the condition and represents a narrow area in Fig. 3. Analysis of the polariton wave-packet dynamics in the discussed domain is straightforward. Due to the energy conservation law it is possible to establish a simple inequality , where can be recognized as a shifted Hamiltonian in this case. The self-trapping regime for the wave-packet can be found out for – upper (red) curves in Fig. 5(a). However, since we can suppose that in this limit. The maximal value of the width of the polariton wave-packet for the self-trapping regime can be obtained as a result. The set of other regimes are achieved at , or simply, at . By using the Hamiltonian we can arrive at an equation for the curvature parameter that describes wave-packet behavior at its large width for . Since the system supports bright polariton soliton solutions and breather regimes as well. In particular, from the energy conservation law we can establish a relation . Hence, the lower diffusive regime with and an equation occurs for .

On the other hand, if , the width has to remain finite that corresponds to breather regimes. Transition between two regimes can be found out from an equation . It implies the critical number of polaritons that is characterized by the critical two-body polariton-polariton scattering parameter