## Abstract

We study linear and nonlinear propagations of probe and signal pulses in a multiple quantum-well structure with a four-level, double Λ-type configuration. We show that slow, mutually matched group velocities and giant Kerr nonlinearity of the probe and the signal pulses may be achieved with nearly vanishing optical absorption. Based on these properties we demonstrate that two-qubit quantum polarization phase gates can be constructed and highly entangled photon pairs may be produced. In addition, we show that coupled slow-light soliton pairs with very low generation power can be realized in the system.

© 2011 OSA

## 1. Introduction

Kerr nonlinearity is important not only for most nonlinear optical processes [1] but also for many applications in quantum information processing, including quantum nondemolition measurement, quantum state teleportation, quantum logic gates, and so on [2]. However, the Kerr nonlinearity are usually produced in passive optical media such as glass-based optical fibers, in which far-off resonance excitation schemes are used to avoid optical absorption. As a result, the Kerr nonlinearity in passive optical media is very small. To acquire a significant nonlinear response a very high light-intensity is needed. In addition, light propagation in passive optical media generally travels close to the light speed in vacuum.

Kerr nonlinearity can be greatly enhanced if a system works near resonance. In the early days of nonlinear optics, there were substantial efforts for utilizing resonant atomic systems to obtain efficient nonlinear optical response. Unfortunately, resonance enhancement of the Kerr nonlinearity usually accompanies with serious optical absorption. For this reason, it was generally recognized that it is unrealistic to take advantages of such resonantly enhanced Kerr nonlinearity. In recent years such paradigm has been challenged due to the finding of electromagnetically induced transparency (EIT) in resonant atomic systems [3]. By use of the quantum interference effect induced by a control laser field, the absorption of a probe laser field can be largely eliminated, together with significant reduction of group velocity and giant enhancement of Kerr nonlinearity. Based on these characters, the possibility of highly efficient four-wave mixing [3] and ultra-slow optical solitons has been explored intensively [4, 5]. Additionally, EIT schemes based on atomic ensembles for producing entangled photons and polarization qubit quantum phase gates (QPGs) have also been suggested [6, 7].

Although atoms are ideal systems for realizing high-quality quantum coherence, solid state systems are more promising for practical applications, benefiting directly from the modern developments of micro- and nano-technologies. In recent years there have been a lot of efforts on the quantum interference effect in semiconductor heterostructures, including quantum wells and quantum dots. Many related phenomena (e.g. lasing without inversion, coherent population oscillation, EIT, etc) in semiconductor heterostructures have been investigated both theoretically and experimentally [8, 9, 10, 11, 12]. In particular, much effort has been paid to the study on the Fano interference in semiconductor quantum wells (SQWs) such as tunneling induced transparency (TIT) [13, 14, 15]. It has also been shown that enhanced Kerr nonlinearity, all optical switching, four-wave mixing, and slow optical solitons are possible in such systems [16, 17, 18, 19, 20, 21]. In many aspects, SQWs possess properties similar to atoms, but have advantages of large electro-dipole moments because of small effective electron mass. Moreover, their transition energies, electro-dipole moments, and symmetries can be engineered by materials selection and device design.

In a recent study [22], a scheme based on *N*-type four-level configuration for realizing large cross-phase modulation (CPM), controllable entanglement, and polarization phase gate was suggested based on the interaction between a probe field and a signal field in a coupled double SQW structure. However, such *N*-type scheme, first proposed by Schmidt and Imamoglu [23], is problematic for obtaining a large CPM for two pulsed fields, required for quantum information processing. The reason is that in this scheme the probe field propagates with a very low group velocity but the signal field propagates with a velocity near the light speed in vacuum, and hence the entanglement and polarization phase gate based on the CPM between the probe and the signal fields can not be realized efficiently. This point was indicated first by Harris and Hau [24] and later by by Lukin and Imamoglu [6], and many improvements have been made and new schemes have been suggested by many authors later on (For detail, see the excellent review by Wang, Marzlin, and Sanders [25]).

In this work, we adopt the same SQW structure as in [22] but with a different, double Λ-type excitation scheme. With this scheme, the shortcomings of the *N*-type scheme can be overcame. Our work includes: (i) By a calculation based on density-matrix formulation, we show that the absorption of both the probe and signal fields can be largely suppressed due to the TIT effect in the quantum well, and the group velocities of the both fields in the double Λ-type scheme may be reduced significantly, and can be mutually matched very well in a fairly large region of system parameters; (ii) We calculate the Kerr coefficients of the both fields, and show that giant cross-Kerr nonlinearity can be achieved in the system; (iii) Based on the slow, matched group velocities and the greatly enhanced cross-Kerr nonlinearity, which allow long-time interaction and effective entanglement between the probe and signal photons, we show that two-qubit QPGs can be constructed and highly entangled photon pairs may be produced easily. (iv) We study the nonlinear propagation of the probe and signal pulses, and demonstrate that a stable propagation of coupled slow-light soliton pairs with very low generation power are possible in the system. The results presented in this work are helpful for the application of optical and quantum information processing and transmission based on solid systems.

The article is arranged as follows. In the next section, we present the model under study. In Sec. 3, we investigate the linear property of the system and analyze the enhancement of Kerr nonlinearity of the probe and signal fields. In Sec. 4, we describe a method to construct two qubit polarization QPGs and produce highly entangled photons. In Sec. 5, we study the formation and propagation of coupled slow-light soliton pairs in the system. In the last section we summarize the main results obtained in this work.

## 2. The MODEL

We consider a multiple quantum-well structure grown by molecular beam epitaxia. Each double quantum well consists of a narrow well and a wide well, separated by a Al_{0.2}Ga_{0.8}As barriers. (Fig. 1). In this structure, the first electron level in conduction band of the wide well can be energetically aligned with the first electron level of the narrow well through applying a static electric field with a given polarity, whereas the first hole levels in the valence bands of both the wide and narrow wells are never aligned for this polarity of the static electric field (for detail, see Ref. [26]). Because of the small tunneling barrier, electrons delocalize and the corresponding levels split into a bounding and an antibounding states (labeled |3〉 and |4〉). The holes remain localized, corresponding states in the narrow and wide wells are |1〉 and |2〉, respectively. The level splitting between |3〉 and |4〉 can be controlled by adjusting the height and width of the tunneling barrier with applied bias voltage. In order to have a Fano-type quantum interference, an additional thin barrier next to the wide well is introduced, which couples the states |3〉 and |4〉 to a continuum. As in Refs. [17, 18, 19, 20, 21], we assume (i) the system works at low-temperature (e.g. 10 K) [26] and the carrier density in the wells is low enough, and (ii) the difference between effective masses in different subbands is small. The latter condition is, strictly speaking, valid only for the situation when the subbands are parallel and Coulomb and Fröhlich matrix elements possess a two-dimensional character. In this situation, the quantized electron motion perpendicular to layers in growth direction of the wells is decoupled from the free motion in the well plane, and intrasubband electron-electron and electron-phonon interactions do not affect intersubband coherence (for detail, see Ref. [27]). Making such assumptions is for simplicity in theoretical analysis and we believe that these effects are of significance only in final engineering optimization without changing basic physical property under study [28].

Different from Ref. [22], we consider a double Λ-type excitation scheme, in which a weak probe optical field (with angular frequency *ω _{p}*, wavenumber

*k*=

_{p}*ω*/

_{p}*c*, polarization vector

**e**

*, and amplitude*

_{p}*E*) propagates in

_{p}*z*-direction and interacts with the double quantum-well and induces the transition |1〉 ↔ |3〉 and |1〉 ↔ |4〉 with the respective half Rabi frequencies (

*μ*

_{31}·

**e**

*)*

_{p}*E*/

_{p}*h̄*≡ Ω

*and (*

_{p}*μ*

_{41}·

**e**

*)*

_{p}*E*/

_{p}*h̄*=

*f*

_{1}Ω

*, where*

_{p}*f*

_{1}= (

*μ*

_{41}·

**e**

*)/(*

_{p}*μ*

_{31}·

**e**

*),*

_{p}*μ*

_{31}and

*μ*

_{41}are corresponding interband electric dipole moments. At the same time a weak signal optical field (with angular frequency

*ω*, wavenumber

_{s}*k*=

_{s}*ω*/

_{s}*c*, polarization vector

**e**

*, and amplitude*

_{s}*E*) propagates also in

_{s}*z*-direction and interacts with the double quantum-well and induces the transition |2〉 ↔ |3〉 and |2〉 ↔ |4〉 with the respective half Rabi frequencies (

*μ*

_{32}·

**e**

*)*

_{s}*E*/

_{s}*h̄*≡ Ω

*and (*

_{s}*μ*

_{42}·

**e**

*)*

_{s}*E*/

_{s}*h̄*=

*f*

_{2}Ω

*, where*

_{s}*f*

_{2}= (

*μ*

_{42}·

**e**

*)/(*

_{s}*μ*

_{32}·

**e**

*),*

_{s}*μ*

_{32}and

*μ*

_{42}are corresponding interband electric dipole moments. Detunings are defined by

*ω*+ Δ

_{p}_{1}= Δ

_{3}+ (

*E*

_{3}–

*E*

_{1})/

*h̄*,

*ω*+ Δ

_{p}_{4}+ Δ

_{1}= (

*E*

_{4}–

*E*

_{1})/

*h̄*,

*ω*+ Δ

_{s}_{2}= Δ

_{3}+ (

*E*

_{3}–

*E*

_{2})/

*h̄*, and

*ω*+ Δ

_{s}_{4}+ Δ

_{2}= (

*E*

_{4}–

*E*

_{2})/

*h̄*.

Defining the energy splitting Δ = (*E*_{4} – *E*_{3})/*h̄*, we can express the detunings as Δ_{3} = Δ/2 + *δ*, Δ_{4} = *δ* – Δ/2 with *δ* = *ω _{p}* – (

*E*

_{3}+

*E*

_{4}– 2

*E*

_{1})/(2

*h̄*), here

*E*is the eigen energy of the state |

_{j}*j*〉 (

*j*= 1 – 4).

The equations of motion for density-matrix *σ* in interaction picture read

*σ*denotes time derivative,

_{jl}*d*≡ Δ

_{jl}*– Δ*

_{j}*+*

_{l}*iγ*(

_{jl}*j*,

*l*= 1 – 4), ${\gamma}_{jl}\equiv \left({\Gamma}_{j}+{\Gamma}_{l}\right)/2+{\Gamma}_{jl}^{\text{dph}}$, with Γ

*being the rates at which population decays from the state |*

_{jl}*l*〉 to the state |

*j*〉, and ${\Gamma}_{jl}^{\text{dph}}$ being the dephasing rates of coherence

*σ*(

_{jl}*j*≠

*l*), which may originate not only from electron-electron scattering and electron-phonon scattering, but also from inhomogeneous broadening due to scattering on interface roughness. The Fano-type quantum interference is reflected by the terms proportional to

*U*≡

*iκ*with $\kappa =\sqrt{{\Gamma}_{3}{\Gamma}_{4}}$, which represents the cross coupling of states |3〉 and |4〉 contributed by the process in which a phonon is emitted from the subband |3〉 and recaptured by the subband |4〉.

The evolution of the probe field and the signal field are governed by the Maxwell equation ∇^{2}**E** – (1/*c*^{2})*∂*^{2}**E**/*∂t*^{2}=(1/*ɛ*_{0}*c*^{2})*∂*^{2}**P**/*∂t*^{2}, where **P** = **P*** _{p}* +

**P**

*with*

_{s}**P**

*≡*

_{p}*N*{

*μ*

_{31}

*σ*

_{13}exp[

*i*(

*k*–

_{p}z*ω*)] +

_{p}t*μ*

_{41}

*σ*

_{14}exp[

*i*(

*k*–

_{p}z*ω*)]} and

_{p}t**P**

*≡*

_{s}*N*{

*μ*

_{32}

*σ*

_{23}exp[

*i*(

*k*–

_{s}z*ω*)] +

_{s}t*μ*

_{42}

*σ*

_{24}exp[

*i*(

*k*–

_{s}z*ω*)] + c.c.}. Here

_{s}t*N*is carrier density in the conduction band. For simplicity, we assume both the probe and signal fields are homogeneous in the transverse (i.e.

*x*and

*y*) directions. Then under the slowly-varying envelope approximation, the Maxwell equation reduces into

*B*

_{1}=

*Nω*|

_{p}*μ*

_{31}|

^{2}/(2

*h̄ɛ*

_{0}

*c*) and

*B*

_{2}=

*Nω*|

_{s}*μ*

_{32}|

^{2}/(2

*h̄ɛ*

_{0}

*c*). Here

*c*is the light speed in vacuum.

## 3. Linear and nonlinear optical susceptibilities

#### 3.1. Suppression of absorption and slowdown and matching of group velocities

We first examine the linear property of the probe and signal fields. We assume population initially occupies in the states |1〉 and |2〉, and both the probe- and signal-field intensities are very small. In this situation the population in the states |1〉 and |2〉 will not be depleted during time evolution, i.e. both *σ*_{11} and *σ*_{22} are constants satisfying *σ*_{11} + *σ*_{22} ≈ 1. Taking *σ*_{31}, *σ*_{41} and Ω* _{p}* to be proportional to exp[

*i*(

*K*–

_{p}z*ωt*)] and

*σ*

_{32},

*σ*

_{42}and Ω

*to be proportional to exp[*

_{s}*i*(

*K*–

_{s}z*ωt*)], by using the Maxwell-Bloch Eqs. (2) and (1) we obtain

For pulsed fields, the linear dispersion relations *K _{p}*(

*ω*) (“

*p*” denotes the probe field) and

*K*(

_{s}*ω*) (“

*s*” denotes the signal field) can be Taylor expanded around

*ω*= 0, which corresponds to the center frequency of both the probe and the signal field [29], i.e. ${K}_{p,s}\left(\omega \right)={K}_{0}^{p,s}+{K}_{1}^{p,s}\omega +\left({K}_{2}^{p,s}/2\right){\omega}^{2}+\cdots $, with ${K}_{j}^{p,s}\equiv {\left({\partial}^{j}{K}_{p,s}/\partial {\omega}^{j}\right)|}_{\omega =0}$. In general, ${K}_{0,}^{p,s}={\varphi}^{p,s}+i{\beta}^{p,s}$ with

*ϕ*and

^{p,s}*β*describing respectively the phase shift per unit length and the absorption, $1/{K}_{1}^{p,s}\left(\equiv {V}_{g}^{\left(p,s\right)}\right)$ represents (complex) group velocity, and ${K}_{2}^{p,s}$ represents (complex) group-velocity dispersion.

^{p,s}Shown in Fig. 2(a) and Fig. 2(b) are the real part (Re(*K _{p,s}*)) and the imaginary part (Im(

*K*)) of the linear dispersion relation as functions of

_{p,s}*ω*, respectively. The solid (dashed) line in the figure is for the probe (signal) field. Parameters chosen are from a typical double quantum-well structure, which has a 51-monolayer (145 Å) wide well and a 35-monolayer (100 Å) narrow well, separated by a 9-monolayer (26 Å) Al

_{0.2}Ga

_{0.8}As buffer layer working at temperature of 10 K, with

*f*

_{1}≈

*f*

_{2}= 0.9 and

*B*

_{1}≈

*B*

_{2}= 1.5 × 10

^{15}cm

^{−1}s

^{−1}. The carrier density is chosen as

*N*= 1.13×10

^{18}cm

^{−3}for the sheet carrier density of a quantum well

*N*= 10

_{s}^{12}cm

^{−2}. Thus the linear density of the semiconductor quantum wells is about

*N*= 1.13 × 10

_{L}^{6}cm

^{−1}. Population decay rates and dephasing rates of the subbands |3〉 and |4〉 can be estimated according to Refs. [30, 31], i.e. Γ

_{3}= 1.28 ns

^{−1}, Γ

_{4}= 1.28 ns

^{−1}, ${\Gamma}_{31}^{\text{dph}}=0.125{\text{ps}}^{-1}$, ${\Gamma}_{41}^{\text{dph}}=0.125{\text{ps}}^{-1}$, ${\Gamma}_{32}^{\text{dph}}=0.125{\text{ps}}^{-1}$, ${\Gamma}_{42}^{\text{dph}}=0.125{\text{ps}}^{-1}$, ${\Gamma}_{42}^{\text{dph}}=0.25{\text{ps}}^{-1}$, Δ

_{1}= 1.0 × 10

^{12}s

^{−1}, Δ

_{2}= 0 and Δ = 1.2 × 10

^{12}s

^{−1}. We see that a Autler-Townes-like splitting (or called TIT transparency window) in both linear absorption curves (Fig. 2(b)) appears, i.e. in the region near

*ω*= 0, the absorption of both the probe and signal fields is largely suppressed. The reason of such absorption suppression is due to the Fano interference, which can be explained as follows. Because the wide well in the system (Fig. 1) is strongly coupled to the continuum through the thin barrier, the electron decay from the wide well to the continuum inevitably results in two dependent, in-distinguished decay pathways, i.e. from both |3〉 and |4〉 to the continuum. As a result, a quantum destructive interference on the absorption of both the probe and signal fields happens. Such TIT effect is mathematically characterized by the cross-coupling terms related to the parameter

*U*=

*iκ*appearing in the density-matrix Eq. (1). Note that companied with the appearance of the absorption suppression, the TIT effect also results in a significant change of the linear dispersion (Fig. 2(a)), a property the same as EIT-based atomic systems [3].

From Fig. 2(a) and Fig. 2(b) we see that the absorption and dispersion curves for both the probe and signal fields are very similar. In fact, these curves can be easily adjusted by choosing the system parameters. For example, if we take Δ_{1} = Δ_{2}, the curves for both the probe and signal fields will almost coincide with each other. Such similarity of the two linear dispersion relations *K _{p}*(

*ω*) and

*K*(

_{s}*ω*) is due to the symmetric configuration between the probe and signal fields we have chosen in the system.

Although generally the group velocities of both the probe and signal fields are complex, their imaginary parts are much less than their real parts due to the TIT effect and hence can be disregarded. Fig. 2(c) shows the real part of the group velocity for the probe field (i.e.
$\text{Re}\left({V}_{g}^{p}\right)/c$, illustrated by solid line) and for the signal field (i.e.
$\text{Re}\left({V}_{g}^{s}\right)/c$, illustrated by dashed line) as functions of *ω* for Δ = 1.2 × 10^{12} s^{−1}. Other parameters are the same those given above. If taking Δ_{1} = Δ_{2} the group velocity curves for the probe and signal fields nearly coincide. In this case, we can obtain the greatly reduced, mutually matched group velocities for both the probe and signal fields at *ω* = 0 (where absorption is minimum) [32]

#### 3.2. Giant Kerr nonlinearity

We now consider the linear and nonlinear optical susceptibilities of the system. Assuming that the probe and signal fields changing slowly enough so that the carriers can follow the variation of the fields adiabatically, we obtain the solution of the Bloch Eq. (1) under an adiabatic approximation. Then the analytical expressions of polarization intensity **P*** _{p}* (for the probe field) and

**P**

*(for the signal field) can be obtained analytically. In order to obtain the self- and cross-Kerr nonlinear coefficients, we assume both the probe and the signal fields are weak, so that a Taylor expansion with respect to*

_{s}*E*and

_{p}*E*can be made. Keeping to the second-order approximation, one obtains expressions of the optical susceptibility

_{s}*χ*(for the probe field) and

_{p}*χ*(for the signal field)

_{s}*α*=

*p*) and the signal field (for

*α*=

*s*) respectively which read

*κ*=

*N*|

*μ*

_{31}|

^{2}|

*μ*

_{32}|

^{2}/(

*ɛ*

_{0}

*h̄*

^{3}), ${Z}_{1}=\left[{D}_{1}^{*}\left({f}_{1}{d}_{31}+U\right)-{D}_{1}\left({f}_{1}{d}_{41}^{*}+{\left|{f}_{1}\right|}^{2}{U}^{*}\right)\right]/\left(2{d}_{43}{\left|{D}_{1}\right|}^{2}\right)$, ${Z}_{2}=\left[{D}_{2}^{*}\left({f}_{2}{d}_{32}+U\right)-{D}_{1}\left({f}_{2}{d}_{42}^{*}+{\left|{f}_{2}\right|}^{2}{U}^{*}\right)\right]/\left(2{d}_{43}{\left|{D}_{2}\right|}^{2}\right)$, ${Z}_{3}=\left\{{D}_{1}\left[{d}_{42}^{*}+{f}_{1}{f}_{2}^{*}{d}_{32}^{*}+{U}^{*}\left({f}_{1}+{f}_{2}^{*}\right)\right]-{D}_{2}^{*}\left[{d}_{41}+{f}_{1}{f}_{2}{d}_{31}+U\left({f}_{1}+{f}_{2}\right)\right]\right\}/\left(2{d}_{21}{D}_{1}{D}_{2}^{*}\right)$,

*D*

_{1}=

*U*

^{2}–

*d*

_{31}

*d*

_{41}and

*D*

_{2}=

*U*

^{2}−

*d*

_{32}

*d*

_{42}.

Shown in the panel (a) of Fig. 3 are the real part (i.e.
$\text{Re}\left({\chi}_{p,s}^{\left(3,S\right)}\right)$, the solid line) and the imaginary part (i.e.
$\text{Im}\left({\chi}_{p,s}^{\left(3,S\right)}\right)$, the dashed line) of the self-Kerr susceptibilities experienced by the probe field *E _{p}* and the signal field

*E*as functions of the detuning

_{s}*δ*(defined above Eq. (1)). We have chosen Δ

_{1}= Δ

_{2}= 1.5 × 10

^{10}s

^{−1}so that the curves for both the probe and signal fields coincide with each other. The other parameters in all panels are the same as those in Fig. 2. we see that the self-phase modulation (SPM) determined by $\text{Re}\left({\chi}_{p,s}^{\left(3,S\right)}\right)$ can become vanishing when the both fields coupled to the double quantum-well are resonant (i.e.

*δ*= 0). Simultaneously, the nonlinear absorption from the SPM determined by $\text{Im}\left({\chi}_{p,s}^{\left(3,S\right)}\right)$ is largely suppressed due to the Fano interference of the system. Panel (b) in the figure shows the real part (i.e. $\text{Re}\left({\chi}_{p,s}^{\left(3,C\right)}\right)$, the solid line) and the imaginary part (i.e. $\text{Im}\left({\chi}_{p,s}^{\left(3,C\right)}\right)$, the dashed line) of the cross-Kerr susceptibilities of

*E*and

_{p}*E*. One sees that in the TIT transparency window the cross-phase modulation (CPM) determined by $\text{Re}\left({\chi}_{p,s}^{\left(3,C\right)}\right)$ can be enlarged much more rapidly than the nonlinear absorption (given by $\text{Im}\left({\chi}_{p,s}^{\left(3,C\right)}\right)$) for non-zero

_{s}*δ*. In this way we can obtain an enhanced cross-Kerr coefficient with a very small imaginary part.

In order to show clearly the enhancement of the cross-Kerr effect of the system, we give the values of the linear, self-Kerr, and cross-Kerr susceptibilities of both the probe and signal fields which read
${\chi}_{p,s}^{\left(1\right)}=2.44\times {10}^{-6}+i3.17\times {10}^{-4}{\text{mV}}^{-4}$,
${\chi}_{p,s}^{\left(3,S\right)}=-3.06\times {10}^{-11}+i1.05\times {10}^{-9}{\text{m}}^{2}{\text{V}}^{-2}$ and
${\chi}_{p,s}^{\left(3,C\right)}=-5.13\times {10}^{-10}-i3.22\times {10}^{-11}{\text{m}}^{2}{\text{V}}^{-2}$. The system parameters are taken the same as those given in Fig. 2, except that by choosing Δ_{1} = Δ_{2} = 1.5 × 10^{10} s^{−1} and *δ* = 5 × 10^{10} s^{−1}. We see that the cross-Kerr susceptibilities have very small imaginary parts, and their real parts are two orders of magnitude larger than those of the self-Kerr susceptibilities.

## 4. Two-Qubit Polarization Phase Gates and highly entangled photons

We now consider the possibility of producing entangled photons and quantum phase gates in the coupled double quantum-well system based on the suppressed absorption and enhanced cross-Kerr nonlinearity obtained above. As in Ref. [7] for atomic systems, we choose two orthogonal polarization states |*σ*^{−}〉 (corresponding to the signal field) and |*σ*^{+}〉 (corresponding to the probe field) to encode binary information for each qubit. The scheme shown in Fig. 1 is completely implemented only if both the probe and signal fields have the “right” polarization states. When the both fields have “wrong” polarizations, there is no sufficient close excited states to which levels |1〉 and |2〉 can couple, and hence the probe and signal fields will only acquire the trivial vacuum phase shift
${\varphi}_{0}^{j}={k}_{j}L$. Here *k _{j}* ≡

*ω*/

_{j}*c*(

*j*=

*p*,

*s*), and

*L*denotes the length of the medium. When one of the two fields have “wrong” polarization state, say for a

*σ*

^{−}-polarized probe field, there is no sufficiently close excited state to which levels |1〉 can couple. Thus the signal field experiences a self-Kerr effect and acquires a nontrivial phase shift ${\varphi}_{1}^{s}$, while the probe field acquires only a vacuum phase shift ${\varphi}_{0}^{p}$. When only the probe and the signal fields have “right” polarizations, all of them will acquire nontrivial phase shifts ${\varphi}_{2}^{p}$ and ${\varphi}_{2}^{s}$, respectively.

Assume that the input probe and signal pulses can be treated as polarized single photon wave packets, expressed as a superposition of the circularly polarized states, i.e.
${|\psi \u3009}_{j}=\left(1/\sqrt{2}\right){|{\sigma}^{-}\u3009}_{j}+\left(1/\sqrt{2}\right){|{\sigma}^{+}\u3009}_{j}$ (*j* = *p, s*). Here
${|{\sigma}^{\pm}\u3009}_{j}=\int d\omega {\xi}_{j}\left(\omega \right){a}_{\pm}^{\u2020}\left(\omega \right)|0\u3009$ with *ξ _{j}*(

*ω*) being a Gaussian frequency distribution of incident wave packet centered at frequency

*ω*. The photon field operators undergo a transformation while propagating through the medium of length

_{j}*L*, i.e. ${a}_{\pm}\left(\omega \right)\to {a}_{\pm}\left(\omega \right)\text{exp}\left\{i\omega /c{\int}_{0}^{L}\mathit{dz}{n}_{\pm}\left(\omega ,z\right)\right\}$. Assuming

*n*

_{±}(

*ω,z*) (the real part of the refractive index) varies slowly over the bandwidth of the wave packet centered at

*ω*, one gets ${|{\sigma}^{\pm}\u3009}_{j}\to \text{exp}\left(-i{\varphi}_{\pm}^{j}\right){|{\sigma}^{\pm}\u3009}_{j}$, with ${\varphi}_{\pm}^{j}={\omega}_{j}{n}_{\pm}\left({\omega}_{j},z\right)L/c$. Thus, the truth table for a polarization two-qubit QPG using the present configuration is given by

_{j}*τ*

_{j}_{′}being the width of the pulse. If group velocity matching is satisfied, i.e.

*ξ*

_{jj′}→ 0, erf(

*ξ*

_{jj′})/

*ξ*

_{jj′}reaches its maximum value $2/\sqrt{\pi}$.

From Eq. (7), we can compute the degree of entanglement of the two-qubit state by using the entanglement of formation. For an arbitrary two-qubit system, it is given by[33]
${E}_{F}\left(C\right)=h\left(1+\sqrt{1-{C}^{2}}/2\right)$, where *h*(*x*) = −*x*log_{2}(*x*) – (1 – *x*)log_{2}(1 – *x*) is Shannon’s entropy function, *C* is the concurrence given by *C*(*ρ̂*) = max{0, *λ*_{1} – *λ*_{2} – *λ*_{3} – *λ*_{4}}. Here *λ _{i}*’s are square roots of eigenvalues of the matrix
$\widehat{\rho}\tilde{\widehat{\rho}}=\widehat{\rho}{\widehat{\sigma}}_{y}^{p}\otimes {\widehat{\sigma}}_{y}^{s}{\widehat{\rho}}^{*}{\widehat{\sigma}}_{y}^{p}\otimes {\widehat{\sigma}}_{y}^{s}$ in decreasing order. The density matrix

*ρ̂*can be directly obtained by using Eq. (7), the quantity $\tilde{\widehat{\rho}}\left({\widehat{\rho}}^{*}\right)$ means the transpose (complex conjugation) of

*ρ*̂, and

*σ̂*denotes the

_{y}*y*-component of the Pauli matrix.

Eq. (7) supports a universal QPG if the conditional phase shift $\left({\varphi}_{0}^{p}+{\varphi}_{0}^{s}\right)+\left({\varphi}_{2}^{p}+{\varphi}_{2}^{s}\right)-\left({\varphi}_{0}^{p}+{\varphi}_{1}^{s}\right)-\left({\varphi}_{1}^{p}+{\varphi}_{0}^{s}\right)={\varphi}^{\left(p,c\right)}+{\varphi}^{\left(s,c\right)}$ is non-zero. From this formula, we see that only the phase shifts due to the CPM effect contribute to the conditional phase shift.

Shown in Fig. 4 is the calculating result of the degree of entanglement as a function of *L*. The system parameters are taken the same as those given in section 3.2 except for Δ = 1.9×10^{12} s^{−1}. We see that a nearly 100% degree of entanglement can be obtained at *L* = 6.0 *μ*m. Our result is different from that obtained in Ref. [22], where a much smaller degree of entanglement was realized. The reason is that in our system, the group velocities of the probe and signal fields are matched very well, resulting in a much larger CPM and hence the interaction time between the two laser fields is much longer that that in Ref. [22].

It should be pointed out that though the large CPM in the double quantum well is promising for designing a deterministic optical quantum phase gate, it still faces some challenges, including, e.g., how to achieve sufficiently high single-photon intensity, how to overcome the phase noise induced by non-instantaneous nonlinear response inherent in the system, and how to obtain a spatially homogeneous CPM necessary for effective entanglement between the probe and signal pulses, etc. The same as in atomic systems, these problems deserve to be investigated further [34].

## 5. Coupled slow-light soliton pairs

It is important to have a shape-preserving optical pulse propagation for light information processing and transmission. We hence consider the possibility to realize ultraslow optical solitons in the present quantum-well system based again on the suppressed absorption and the enhanced Kerr nonlinearity contributed by the TIT effect. From Sec. 3 we know that the system has also large dispersion near the transparency window, which will distort the probe and signal pulses during their propagation. It is natural to use the enhanced Kerr nonlinearity to balance the dispersion effect in the system. For this aim, we apply the standard method of multiple-scales to derive the nonlinear envelope equation of both the probe and signal fields by taking the asymptotic expansion
${\sigma}_{ij}={\sum}_{n=0}^{\infty}{\varepsilon}^{n}{\sigma}_{ij}^{\left(n\right)}$ (*i, j* = 1 to 3) and
${\Omega}_{p,s}={\sum}_{n=1}^{\infty}{\varepsilon}^{n}{\Omega}_{p,s}^{\left(n\right)}$, where *ɛ* is a small parameter characterizing the amplitude of the both fields. To obtain a divergence-free expansion, all quantities on the right hand side of the expansion are considered as functions of the multi-scale variables *z _{l}* =

*ɛ*(

^{l}z*l*= 0 to 2) and

*t*=

_{l}*ɛ*(

^{l}t*l*= 0 to 1). Substituting above expansion into Eqs. (1), we obtain a series of equations for ${\sigma}_{ij}^{n}$ and ${\Omega}_{p,s}^{n}$ (

*i, j*=1 to 4;

*n*=1 to 3), which can be solved order by order.

The nonlinear envelope equation governing the dynamics of the probe and signal fields obtained by the asymptotic expansion can be written the following dimensionless form

*s*=

*z/L*, $\sigma =\left(t-z/{V}_{g}\right)/{\tau}_{0}\left({V}_{g}\equiv 2{V}_{g}^{p}{V}_{g}^{s}/\left({V}_{g}^{p}+{V}_{g}^{s}\right)\right)$,

_{D}*u*

_{1}= (Ω

*/*

_{p}*U*) exp (−

_{p}*iK̃*),

_{p}z*u*

_{2}= (Ω

*/*

_{s}*U*) exp (−

_{s}*iK̃*) (

_{s}z*K̃*≡ Re(

_{p,s}*K*)), ${g}_{0}^{p,s}={\alpha}_{p,s}{L}_{D}$ with

_{p,s}*α*= Im(

_{p,s}*K*),

_{p,s}*g*sgn(

*δ*)

*L*/

_{D}*L*, ${g}_{1}^{p}={K}_{2}^{p}/\left|{K}_{2}^{s}\right|$, ${g}_{1}^{s}=\text{sgn}\left({K}_{2}^{s}\right)$,

_{δ}*g*

_{11}=

*W*/|

_{pp}*W*|,

_{ss}*g*

_{22}= −1

*g*

_{12}=

*W*/|

_{ps}*W*|,

_{ss}*g*

_{21}=

*W*/|

_{sp}*W*|, with $\left({W}_{pp},{W}_{ps}\right)=\left[{\omega}_{p}/\left(2c\right)\right]\left({\chi}_{p}^{\left(3,S\right)},{\chi}_{p}^{\left(3,C\right)}\right)$ and $\left({W}_{sp},{W}_{ss}\right)=\left[{\omega}_{s}/\left(2c\right)\right]\left({\chi}_{s}^{\left(3,S\right)},{\chi}_{s}^{\left(3,C\right)}\right)$. In these expressions we have defined the characteristic dispersion length ${L}_{D}={\tau}_{0}^{2}/\left|{K}_{2}^{s}\right|$, and the characteristic group velocity mismatch length ${L}_{\delta}={\tau}_{0}/\left|\delta \right|\left(\delta \equiv \left(1/{V}_{g}^{p}-1/{V}_{g}^{s}\right)/2\right)$, with

_{ss}*τ*

_{0}being the characteristic pulse length of the probe and signal fields.

Because in our system the group velocities of the probe and signal pulses can be matched very well based on the discussions given in Sec. 3, thus we have *g _{δ}* = 0. In addition, we also have
${K}_{2}^{p}={K}_{2}^{s}$. As a result, the above coupled NLS equations can be reduced to

Equations (10a) and (10b) when disregarding the perturbation have various exact coupled soliton pair solutions [35]. The bright-bright coupled soliton pair reads

*u*

_{1}) travels at the same velocity with that of the soliton related to the signal field (i.e.

*u*

_{2}), except for different amplitudes, which can be adjusted to the same by choosing system parameters. Other coupled soliton-pair solutions (including bright-dark and dark-dark ones) can also be obtained under different conditions.

We now give a practical example to confirm that the coupled optical soliton pair given above can be realized in our semiconductor double quantum-well system. We take Δ_{1} = Δ_{2} = 0.1 × 10^{11} s^{−1}, Δ = 1.9 × 10^{12} s^{−1}, *δ* = 0.5 × 10^{11} s^{−1} and other parameters are the same as those in Fig. 2. With these parameters, we obtain
${K}_{0}^{p,s}=\left(-2.74+i0.21\right)\times {10}^{4}{\text{cm}}^{-1}$,
${K}_{1}^{p,s}=\left(6.25+i0.48\right)\times {10}^{-9}{\text{cm}}^{-1}\text{s}$,
${K}_{2}^{p,s}=\left(-1.33+i0.09\right)\times {10}^{-20}{\text{cm}}^{-1}{\text{s}}^{2}$, *W _{pp}* =

*W*= (−1.72+

_{ss}*i*0.03)× 10

^{−21}cm

^{−1}s

^{2},

*W*=

_{ps}*W*= (−1.56 +

_{sp}*i*0.05) × 10

^{−21}cm

^{−1}s

^{2}. Notice that the imaginary parts of these quantities are indeed much smaller than their relevant real parts due to the TIT effect. Then we have

*L*= 3.0

_{D}*μ*m with

*τ*

_{0}= 2 × 10

^{−12}s,

*L*

_{0}= 5.0

*μ*m. The dimensionless coefficients read ${g}_{0}^{p,s}=0.25$ and

*g*≈ −1 (

_{jl}*j,l*= 1,2). The propagating velocity of the coupled soliton pair

*c*.

With the above parameters and using the initial condition *u*_{1} = *u*_{2} = 1.0sech(*σ* + 4)exp(*i*0.5*σ*), we numerically integrated Eqs. (10a) and (10b). The evolution of the probe-field half Rabi frequency |Ω* _{p}*/

*U*

_{0}| versus the dimensionless time

*t*/

*τ*

_{0}and the distance

*z*/(2

*L*) is shown in Fig. 5(a) (The evolution of the signal field is the same, and thus not shown). We see that the soliton propagates stably up to

_{D}*z*= 4

*L*. There is however a small attenuation contributed from the small absorption. For comparison, Fig. 5(b) shows the evolution of the probe field half Rabi frequency as a function of the

_{D}*t*/

*τ*

_{0}and

*z*/(2

*L*) by a direct numerical integration based on the Eqs. (1) and (2). One sees that initially the soliton is fairly stable, but it radiates small-amplitude continuous waves when propagating to a large distance. This can be understood as the contribution by the effects of high-order dispersion and high-order nonlinearity that are not included in the approximations when obtaining the coupled NLS Eqs. (10a) and (10b).

_{D}The generation power of the coupled optical soliton pair can be obtained by using Poynting’s vector. With the above parameters and taking |*μ*_{31}| ≈ |*μ*_{32}| = 2.688 × 10^{−28} cm C and *S*_{0} = *πR*^{2} = *π* × 10^{−7} cm^{2} (*R* is the transverse radius of the probe and signal beams), we obtain the peak power

## 6. Conclusions

In this work, we have investigated the linear and nonlinear propagation of probe and signal pulses in a coupled double quantum-well structure with a four-level, double Λ-type configuration. In our scheme, slow, mutually matched group velocities are realized in a very large parameter region and giant cross Kerr nonlinearities of the probe and the signal pulses is obtained with nearly vanishing optical absorption. Such properties comes from the quantum interference effect of the system and the symmetric configuration between the probe and signal fields in our system. Based on these novel properties we have shown that two-qubit quantum polarization phase gates can be constructed and highly entangled photon pairs may be produced easily. For obtaining a shape-preserving optical pulse propagation, we have considered the possibility of realizing coupled optical soliton pairs. We have shown that a stable propagation of coupled optical soliton pairs with very slow propagating velocities and very low generation power can be achieved in the system. The results obtained in the presented work are helpful for the experimental realization of the giant Kerr nonlinearity, two-qubit polarization quantum phase gates, highly entangled photon pairs, and weak-light ultraslow soliton pairs in coupled quantum-well structures, and promising for practical applications in optical and quantum information processing and transmission based on solid-state devices.

## Acknowledgments

Authors thank Y. Li for useful discussions. This work was supported by NSF-China under Grant No. 10874043, and by the Ministry Reward for Excellent Doctors in Academics under Grant No. MXRZZ2010007.

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