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_0.2Ga_0.8As 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].

 

Fig. 1 Energy-level diagram and double Λ-type excitation scheme of the double quantum-well structure separated by a small tunneling barrier between a narrow well and a wide well. |3〉 and |4〉 are delocalized bounding and antibounding states of conduction band, which are coupled to a continuum by a thin tunneling barrier adjacent to the wide well. |1〉 and |2〉 are localized hole states of valence band. Ω_p (Ω_s) is the half Rabi frequency of the probe (signal) field, Δ_l (l = 1 – 4) are detunings.

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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_p, and amplitude E_p) propagates in z-direction and interacts with the double quantum-well and induces the transition |1〉 ↔ |3〉 and |1〉 ↔ |4〉 with the respective half Rabi frequencies (μ₃₁ · e_p)E_p/h̄ ≡ Ω_p and (μ₄₁ · e_p)E_p/h̄ = f₁Ω_p, where f₁ = (μ₄₁ · e_p)/(μ₃₁ · e_p), μ₃₁ and μ₄₁ are corresponding interband electric dipole moments. At the same time a weak signal optical field (with angular frequency ω_s, wavenumber k_s = ω_s/c, polarization vector e_s, and amplitude E_s) propagates also in z-direction and interacts with the double quantum-well and induces the transition |2〉 ↔ |3〉 and |2〉 ↔ |4〉 with the respective half Rabi frequencies (μ₃₂ · e_s)E_s/h̄ ≡ Ω_s and (μ₄₂ · e_s)E_s/h̄ = f₂Ω_s, where f₂ = (μ₄₂ · e_s)/(μ₃₂ · e_s), μ₃₂ and μ₄₂ are corresponding interband electric dipole moments. Detunings are defined by ω_p + Δ₁ = Δ₃ + (E₃ – E₁)/h̄, ω_p + Δ₄ + Δ₁ = (E₄ – E₁)/h̄, ω_s + Δ₂ = Δ₃ + (E₃ – E₂)/h̄, and ω_s + Δ₄ + Δ₂ = (E₄ – E₂)/h̄.

Defining the energy splitting Δ = (E₄ – E₃)/h̄, we can express the detunings as Δ₃ = Δ/2 + δ, Δ₄ = δ – Δ/2 with δ = ω_p – (E₃ + E₄ – 2E₁)/(2h̄), here E_j is the eigen energy of the state |j〉 (j = 1 – 4).

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

The evolution of the probe field and the signal field are governed by the Maxwell equation ∇²E – (1/c²)∂²E/∂t²=(1/ɛ₀c²)∂²P/∂t², where P = P_p + P_s with P_p ≡ N{μ₃₁σ₁₃ exp[i(k_pz – ω_pt)] + μ₄₁σ₁₄ exp[i(k_pz –ω_pt)]} and P_s ≡ N {μ₃₂σ₂₃ exp[i(k_sz – ω_st)] + μ₄₂σ₂₄ exp[i(k_sz – ω_st)] + c.c.}. Here 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

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 σ₁₁ and σ₂₂ are constants satisfying σ₁₁ + σ₂₂ ≈ 1. Taking σ₃₁, σ₄₁ and Ω_p to be proportional to exp[i(K_pz – ωt)] and σ₃₂, σ₄₂ and Ω_s to be proportional to exp[i(K_sz – 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^p,s and β^p,s 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.

Shown in Fig. 2(a) and Fig. 2(b) are the real part (Re(K_p,s)) and the imaginary part (Im(K_p,s)) of the linear dispersion relation as functions of 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_0.2Ga_0.8As buffer layer working at temperature of 10 K, with f₁ ≈ f₂ = 0.9 and B₁ ≈ B₂ = 1.5 × 10¹⁵cm⁻¹s⁻¹. The carrier density is chosen as N = 1.13×10¹⁸ cm⁻³ for the sheet carrier density of a quantum well N_s = 10¹² cm⁻². Thus the linear density of the semiconductor quantum wells is about N_L = 1.13 × 10⁶ cm⁻¹. Population decay rates and dephasing rates of the subbands |3〉 and |4〉 can be estimated according to Refs. [30, 31], i.e. Γ₃ = 1.28 ns⁻¹, Γ₄ = 1.28 ns⁻¹, ${\Gamma }_{31}^{\text{dph}}=0.125{\text{ps}}^{-1}$, ${\Gamma }_{41}^{\text{dph}}=0.125{\text{ps}}^{-1}$, ${\Gamma }_{32}^{\text{dph}}=0.125{\text{ps}}^{-1}$, ${\Gamma }_{42}^{\text{dph}}=0.125{\text{ps}}^{-1}$, ${\Gamma }_{42}^{\text{dph}}=0.25{\text{ps}}^{-1}$, Δ₁ = 1.0 × 10¹² s⁻¹, Δ₂ = 0 and Δ = 1.2 × 10¹² 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].

 

Fig. 2 Linear dispersion relations as functions of ω. (a): Linear dispersion Re(K_p(ω)) (Re(K_s(ω))) for the probe (signal) field. (b): Linear absorption Im(K_p(ω)) (Im(K_s(ω))) for the probe (signal) field. (c): Curves of the group velocities of both the probe and signal fields as functions of ω for Δ = 1.2 × 10¹² s⁻¹. The solid (dashed) line is for the probe (signal) field.

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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 Δ₁ = Δ₂, 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 × 10¹² s⁻¹. Other parameters are the same those given above. If taking Δ₁ = Δ₂ 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_s (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 E_p and E_s can be made. Keeping to the second-order approximation, one obtains expressions of the optical susceptibility χ_p (for the probe field) and χ_s (for the signal field)

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_s as functions of the detuning δ (defined above Eq. (1)). We have chosen Δ₁ = Δ₂ = 1.5 × 10¹⁰ 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_p and E_s. 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 δ. In this way we can obtain an enhanced cross-Kerr coefficient with a very small imaginary part.

 

Fig. 3 (a): Real (i.e. $\text{Re}\left({\chi }_{p,s}^{\left(3,S\right)}\right)$, the solid line) and imaginary (i.e. $\text{Im}\left({\chi }_{p,s}^{\left(3,S\right)}\right)$, the dashed line) parts of the self-Kerr susceptibilities of the probe field E_p and the signal field E_s as functions of the detuning δ. Parameters in all panels are given in text. (b): Real (i.e. $\text{Re}\left({\chi }_{p,s}^{\left(3,C\right)}\right)$, the solid line) and imaginary (i.e. $\text{Im}\left({\chi }_{p,s}^{\left(3,C\right)}\right)$, the dashed line) parts of the cross-Kerr susceptibilities of E_p and E_s.

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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 Δ₁ = Δ₂ = 1.5 × 10¹⁰ s⁻¹ and δ = 5 × 10¹⁰ 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. ${|\psi 〉}_j=\left(1/\sqrt{2}\right){|{\sigma }^{-}〉}_j+\left(1/\sqrt{2}\right){|{\sigma }^{+}〉}_j$ (j = p, s). Here ${|{\sigma }^{\pm }〉}_j=\int d\omega {\xi }_j\left(\omega \right)a_{\pm }^{†}\left(\omega \right)|0〉$ with ξ_j(ω) being a Gaussian frequency distribution of incident wave packet centered at frequency ω_j. The photon field operators undergo a transformation while propagating through the medium of length 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_j, one gets ${|{\sigma }^{\pm }〉}_j\to \text{exp}\left(-i{\varphi }_{\pm }^j\right){|{\sigma }^{\pm }〉}_j$, 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

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) = −xlog₂(x) – (1 – x)log₂(1 – x) is Shannon’s entropy function, C is the concurrence given by C(ρ̂) = max{0, λ₁ – λ₂ – λ₃ – λ₄}. 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 ρ̂, and σ̂_y denotes the 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 = 1.9×10¹² s⁻¹. We see that a nearly 100% degree of entanglement can be obtained at L = 6.0 μm. 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].

 

Fig. 4 The degree of entanglement versus the medium length L. The system parameters are taken the same as those given in section 3.2 except for Δ = 1.9 × 10¹² s⁻¹.

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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^lz (l = 0 to 2) and t_l = ɛ^lt (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

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

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 Δ₁ = Δ₂ = 0.1 × 10¹¹ s⁻¹, Δ = 1.9 × 10¹² s⁻¹, δ = 0.5 × 10¹¹ 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_ss = (−1.72+i0.03)× 10⁻²¹ cm⁻¹s², W_ps = W_sp = (−1.56 + i0.05) × 10⁻²¹ cm⁻¹s². 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_D = 3.0 μm with τ₀ = 2 × 10⁻¹² s, L₀ = 5.0 μm. The dimensionless coefficients read $g_0^{p,s}=0.25$ and g_jl ≈ −1 (j,l = 1,2). The propagating velocity of the coupled soliton pair

With the above parameters and using the initial condition u₁ = u₂ = 1.0sech(σ + 4)exp(i0.5σ), we numerically integrated Eqs. (10a) and (10b). The evolution of the probe-field half Rabi frequency |Ω_p/U₀| versus the dimensionless time t/τ₀ and the distance z/(2L_D) 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 z = 4L_D. 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 t/τ₀ and z/(2L_D) 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).

 

Fig. 5 (a): Time evolution of the dimensionless probe-field Rabi frequency |Ω_p/U₀| as a function of the dimensionless time t/τ₀ and the distance z/(2L_D), obtained by numerically integrating Eqs. (10a) and (10b). (b): Numerical solution obtained by a direct numerical integration of Eqs. (1) and (2) with the same parameters given in the panel (a).

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The generation power of the coupled optical soliton pair can be obtained by using Poynting’s vector. With the above parameters and taking |μ₃₁| ≈ |μ₃₂| = 2.688 × 10⁻²⁸ cm C and S₀ = πR² = π × 10⁻⁷ cm² (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.