OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 23 — Nov. 7, 2011
  • pp: 23364–23376
« Show journal navigation

Giant Kerr Nonlinearity, Controlled Entangled Photons and Polarization Phase Gates in Coupled Quantum-Well Structures

Chengjie Zhu and Guoxiang Huang  »View Author Affiliations


Optics Express, Vol. 19, Issue 23, pp. 23364-23376 (2011)
http://dx.doi.org/10.1364/OE.19.023364


View Full Text Article

Acrobat PDF (2132 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

1. R. W. Boyd, Nonlinear Optics (2cd edition) (Academic, San Diego, 2003).

] but also for many applications in quantum information processing, including quantum nondemolition measurement, quantum state teleportation, quantum logic gates, and so on [2

2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge, England, 2000).

]. 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

3. M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

]. 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

3. M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

] and ultra-slow optical solitons has been explored intensively [4

4. Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef] [PubMed]

, 5

5. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E72, 016617 (2005). [CrossRef]

]. Additionally, EIT schemes based on atomic ensembles for producing entangled photons and polarization qubit quantum phase gates (QPGs) have also been suggested [6

6. M. D. Lukin and A. Imamoǧlu, “Nonlinear Optics and Quantum Entanglement of Ultraslow Single Photons,” Phys. Rev. Lett. 84, 1419 (2000). [CrossRef] [PubMed]

, 7

7. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003). [CrossRef] [PubMed]

].

In a recent study [22

22. W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16, 17161 (2008). [CrossRef] [PubMed]

], 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

23. H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936 (1996). [CrossRef] [PubMed]

], 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

24. S. E. Harris and L. V. Hau, “Nonlinear Optics at Low Light Levels,” Phys. Rev. Lett. 82, 4611 (1999). [CrossRef]

] and later by by Lukin and Imamoglu [6

6. M. D. Lukin and A. Imamoǧlu, “Nonlinear Optics and Quantum Entanglement of Ultraslow Single Photons,” Phys. Rev. Lett. 84, 1419 (2000). [CrossRef] [PubMed]

], 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

25. Z.-B. Wang, K.-P. Marzlin, and B. Sanders, in Quantum Communications and Quantum Imaging, edited by R. E. Meyers, Y. Shih, and K. S. Deacon, Proc. of SPIE6305, 6305H1–8 (2006).

]).

In this work, we adopt the same SQW structure as in [22

22. W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16, 17161 (2008). [CrossRef] [PubMed]

] 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.

2. The MODEL

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. Ωps) is the half Rabi frequency of the probe (signal) field, Δl (l = 1 – 4) are detunings.

Different from Ref. [22

22. W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16, 17161 (2008). [CrossRef] [PubMed]

], we consider a double Λ-type excitation scheme, in which a weak probe optical field (with angular frequency ωp, wavenumber kp = ωp/c, polarization vector ep, and amplitude Ep) 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 · ep)Ep/ ≡ Ωp and (μ41 · ep)Ep/ = f1Ωp, where f1 = (μ41 · ep)/(μ31 · ep), μ31 and μ41 are corresponding interband electric dipole moments. At the same time a weak signal optical field (with angular frequency ωs, wavenumber ks = ωs/c, polarization vector es, and amplitude Es) 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 · es)Es/ ≡ Ωs and (μ42 · es)Es/ = f2Ωs, where f2 = (μ42 · es)/(μ32 · es), μ32 and μ42 are corresponding interband electric dipole moments. Detunings are defined by ωp + Δ1 = Δ3 + (E3E1)/, ωp + Δ4 + Δ1 = (E4E1)/, ωs + Δ2 = Δ3 + (E3E2)/, and ωs + Δ4 + Δ2 = (E4E2)/.

Defining the energy splitting Δ = (E4E3)/, we can express the detunings as Δ3 = Δ/2 + δ, Δ4 = δ – Δ/2 with δ = ωp – (E3 + E4 – 2E1)/(2), here Ej is the eigen energy of the state |j〉 (j = 1 – 4).

The equations of motion for density-matrix σ in interaction picture read
iσ˙11=iΓ13σ33+iΓ14σ44Ωp*σ31+Ωpσ13f1*Ωp*σ41+f1Ωpσ14,
(1a)
iσ˙22=iΓ23σ33+iΓ24σ44Ωs*σ32+Ωsσ23f2*Ωs*σ42+f2Ωsσ24,
(1b)
iσ˙33=iΓ3σ33+iΓ34σ44Ωpσ13+Ωp*σ31Ωsσ23+Ωs*σ32+U(σ34+σ43),
(1c)
iσ˙44=iΓ4σ44f1Ωpσ14+f1*Ωp*σ41f2Ωsσ24+f2*Ωs*σ42+U(σ34+σ43),
(1d)
iσ˙21=d21σ21+Ωpσ23+f1Ωpσ24Ωs*σ31f2*Ωs*σ41,
(1e)
iσ˙31=d31σ31+Ωp(σ33σ11)+f1Ωpσ34Ωsσ21+Uσ41,
(1f)
iσ˙41=d41σ41+Ωpσ43+f1Ωp(σ44σ11)f2Ωsσ21+Uσ31,
(1g)
iσ˙32=d32σ32+Ωs(σ33σ22)+f2Ωsσ34Ωpσ12+Uσ42,
(1h)
iσ˙42=d42σ42+Ωsσ43+f2Ωs(σ44σ22)f1Ωpσ12+Uσ32,
(1i)
iσ˙43=d43σ43+Ωp*σ41+Ωs*σ42f1Ωpσ13f2Ωsσ23+U(σ33+σ44),
(1j)
where dot above σjl denotes time derivative, djl ≡ Δj – Δl + jl (j, l = 1 – 4), γjl(Γj+Γl)/2+Γjldph, with Γjl being the rates at which population decays from the state |l〉 to the state |j〉, and Γjldph being the dephasing rates of coherence σjl (jl), 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 with κ=Γ3Γ4, 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 ∇2E – (1/c2)2E/∂t2=(1/ɛ0c2)2P/∂t2, where P = Pp + Ps with PpN{μ31σ13 exp[i(kpzωpt)] + μ41σ14 exp[i(kpzωpt)]} and PsN {μ32σ23 exp[i(kszωst)] + μ42σ24 exp[i(kszωst)] + 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
i(z+1ct)Ωp+B1(σ31+f1*σ41)=0,
(2a)
i(z+1ct)Ωs+B2(σ32+f2*σ42)=0.
(2b)
with B1 = p|μ31|2/(2h̄ɛ0c) and B2 = s|μ32|2/(2h̄ɛ0c). 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 σ11 and σ22 are constants satisfying σ11 + σ22 ≈ 1. Taking σ31, σ41 and Ωp to be proportional to exp[i(Kpzωt)] and σ32, σ42 and Ωs to be proportional to exp[i(Kszωt)], by using the Maxwell-Bloch Eqs. (2) and (1) we obtain
Kp(ω)=ωcB1(ω+d41)+|f1|2(ω+d31)+(f1+f1*)U(ω+d31)(ω+d41)U2,
(3a)
Ks(ω)=ωcB2(ω+d42)+|f2|2(ω+d32)+(f2+f2*)U(ω+d32)(ω+d42)U2,
(3b)

For pulsed fields, the linear dispersion relations Kp(ω) (“p” denotes the probe field) and Ks(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

29. The frequency and wavevector of the probe (signal) field in the quantum well are given by ωp + ω and kp + Kp(ω) (ωs + ω and ks + Ks(ω)), respectively. Thus ω = 0 corresponds to the center frequency of both the probe and signal fields.

], i.e. Kp,s(ω)=K0p,s+K1p,sω+(K2p,s/2)ω2+, with Kjp,s(jKp,s/ωj)|ω=0. In general, K0,p,s=ϕp,s+iβp,s with ϕp,s and βp,s describing respectively the phase shift per unit length and the absorption, 1/K1p,s(Vg(p,s)) represents (complex) group velocity, and K2p,s represents (complex) group-velocity dispersion.

Fig. 2 Linear dispersion relations as functions of ω. (a): Linear dispersion Re(Kp(ω)) (Re(Ks(ω))) for the probe (signal) field. (b): Linear absorption Im(Kp(ω)) (Im(Ks(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.

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 Kp(ω) and Ks(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. Re(Vgp)/c, illustrated by solid line) and for the signal field (i.e. Re(Vgs)/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

32. Because Im(Vgp,s) is much less than Re(Vgp,s), we disregard Im(Vgp,s) and take (Vgp,s)Re(Vgp,s) here and in the following.

]
VgpVgs=1.2×103c.
(4)
From the figure, we see that, different from Ref. [22

22. W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16, 17161 (2008). [CrossRef] [PubMed]

], the group-velocity matching in our system can be realized in a fairly large parameter region, which comes from the symmetric configuration between the probe and signal fields in our system. We stress that the slowdown and matching of the group velocities are very essential for the two-qubit polarization QPGs, highly entangled photons, and coupled slow-light soliton pairs discussed in Secs. 4 and 5 below.

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 Pp (for the probe field) and Ps (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 Ep and Es can be made. Keeping to the second-order approximation, one obtains expressions of the optical susceptibility χp (for the probe field) and χs (for the signal field)
χp=N|μ31|2ɛ0h¯Ωp(σ31+f1*σ41)=χp(1)+χp(3,S)|Ep|2+χp(3,C)|Es|2,
(5a)
χs=N|μ32|2ɛ0h¯Ωs(σ32+f2*σ42)=χs(1)+χs(3,S)|Ep|2+χs(3,C)|Es|2,
(5b)
where χα(1)=2cKα(ω=0)/ωα, χα(3,S), and χα(3,C) are the linear, self-Kerr, and cross-Kerr susceptibilities for the probe field (for α = p) and the signal field (for α = s) respectively which read
χp(3,S)=N|μ31|4ɛ0h¯3Z1(U+f1*d31)+Z1*(f1d41+|f1|2U)d31d41U2,
(6a)
χs(3,S)=N|μ32|4ɛ0h¯3Z2(U+f2*d32)+Z2*(f2d42+|f2|2U)d32d42U2,
(6b)
χp(3,C)=κZ2(U+f1*d31)+Z2*(f1d41+|f1|2U)Z3[d41+f1*f2d31+U(f1*+f2)]d31d41U2,
(6c)
χs(3,C)=κZ1(U+f2*d32)+Z1*(f2d42+|f2|2U)Z3*[d42+f2*f1d32+U(f2*+f1)]d32d42U2,
(6d)
with κ = N|μ31|2|μ32|2/(ɛ03), Z1=[D1*(f1d31+U)D1(f1d41*+|f1|2U*)]/(2d43|D1|2), Z2=[D2*(f2d32+U)D1(f2d42*+|f2|2U*)]/(2d43|D2|2), Z3={D1[d42*+f1f2*d32*+U*(f1+f2*)]D2*[d41+f1f2d31+U(f1+f2)]}/(2d21D1D2*), D1 = U2d31d41 and D2 = U2d32d42.

Shown in the panel (a) of Fig. 3 are the real part (i.e. Re(χp,s(3,S)), the solid line) and the imaginary part (i.e. Im(χp,s(3,S)), the dashed line) of the self-Kerr susceptibilities experienced by the probe field Ep and the signal field Es 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 Re(χp,s(3,S)) 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 Im(χp,s(3,S)) is largely suppressed due to the Fano interference of the system. Panel (b) in the figure shows the real part (i.e. Re(χp,s(3,C)), the solid line) and the imaginary part (i.e. Im(χp,s(3,C)), the dashed line) of the cross-Kerr susceptibilities of Ep and Es. One sees that in the TIT transparency window the cross-phase modulation (CPM) determined by Re(χp,s(3,C)) can be enlarged much more rapidly than the nonlinear absorption (given by Im(χp,s(3,C))) 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. Re(χp,s(3,S)), the solid line) and imaginary (i.e. Im(χp,s(3,S)), the dashed line) parts of the self-Kerr susceptibilities of the probe field Ep and the signal field Es as functions of the detuning δ. Parameters in all panels are given in text. (b): Real (i.e. Re(χp,s(3,C)), the solid line) and imaginary (i.e. Im(χp,s(3,C)), the dashed line) parts of the cross-Kerr susceptibilities of Ep and Es.

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 χp,s(1)=2.44×106+i3.17×104mV4, χp,s(3,S)=3.06×1011+i1.05×109m2V2 and χp,s(3,C)=5.13×1010i3.22×1011m2V2. 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

7. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003). [CrossRef] [PubMed]

] 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 ϕ0j=kjL. Here kjωj/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 ϕ1s, while the probe field acquires only a vacuum phase shift ϕ0p. When only the probe and the signal fields have “right” polarizations, all of them will acquire nontrivial phase shifts ϕ2p and ϕ2s, 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. |ψj=(1/2)|σj+(1/2)|σ+j (j = p, s). Here |σ±j=dωξj(ω)a±(ω)|0 with ξj(ω) being a Gaussian frequency distribution of incident wave packet centered at frequency ωj. The photon field operators undergo a transformation while propagating through the medium of length L, i.e. a±(ω)a±(ω)exp{iω/c0Ldzn±(ω,z)}. Assuming n±(ω,z) (the real part of the refractive index) varies slowly over the bandwidth of the wave packet centered at ωj, one gets |σ±jexp(iϕ±j)|σ±j, with ϕ±j=ωjn±(ωj,z)L/c. Thus, the truth table for a polarization two-qubit QPG using the present configuration is given by
|σp|σ+sexp[i(ϕ0p+ϕ0s)]|σp|σ+s,
(7a)
|σp|σsexp[i(ϕ0p+ϕ1s)]|σp|σs,
(7b)
|σ+p|σ+sexp[i(ϕ1p+ϕ0s)]|σ+p|σ+s,
(7c)
|σ+p|σsexp[i(ϕ2p+ϕ2s)]|σ+p|σs,
(7d)
where ϕ0j=kjL, ϕ1j=kjL(1+2πχj(1))+ϕ(j,s) and ϕ2j=ϕ1j+ϕ(j,c), with
ϕ(j,s)=kjLπ3/2h¯2|Ωj|24|μ|2Re(χj(3,S)),
(8a)
ϕ(j,c)=kjLπ3/2h¯2|Ωj|24|μ|2Re(χj(3,c))erf(ξjj)ξjj,
(8b)
contributed respectively by the SPM and the CPM, where ξjj=2L(1vgj/vgj)/(τjvgj), with τj being the width of the pulse. If group velocity matching is satisfied, i.e. ξjj → 0, erf(ξjj)/ξjj reaches its maximum value 2/π.

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

33. V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000). [CrossRef]

] EF(C)=h(1+1C2/2), 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, λ1λ2λ3λ4}. Here λi’s are square roots of eigenvalues of the matrix ρ^ρ^˜=ρ^σ^ypσ^ysρ^*σ^ypσ^ys in decreasing order. The density matrix ρ̂ can be directly obtained by using Eq. (7), the quantity ρ^˜(ρ^*) means the transpose (complex conjugation) of ρ̂, and σ̂y denotes the y-component of the Pauli matrix.

Eq. (7) supports a universal QPG if the conditional phase shift (ϕ0p+ϕ0s)+(ϕ2p+ϕ2s)(ϕ0p+ϕ1s)(ϕ1p+ϕ0s)=ϕ(p,c)+ϕ(s,c) 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

22. W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16, 17161 (2008). [CrossRef] [PubMed]

], 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

22. W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16, 17161 (2008). [CrossRef] [PubMed]

].

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.

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

34. K.-P. Marzlin, Z.-B. Wang, S. A. Moiseev, and B. C. Sanders, “Uniform cross-phase modulation for nonclassical radiation pulses,” J. Opt. Soc. Am. B 27, A36–A45 (2010). [CrossRef]

].

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 σij=n=0ɛnσij(n) (i, j = 1 to 3) and Ωp,s=n=1ɛnΩp,s(n), 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 zl = ɛlz (l = 0 to 2) and tl = ɛlt (l = 0 to 1). Substituting above expansion into Eqs. (1), we obtain a series of equations for σijn and Ωp,sn (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
i(s+g0p)u1+igδu1σg1p22u1σ2(g11|u1|2+g12|u2|2)u1=0,
(9a)
i(s+g0s)u2+igδu2σg1s22u2σ2(g22|u2|2+g21|u1|2)u2=0,
(9b)
where s = z/LD, σ=(tz/Vg)/τ0(Vg2VgpVgs/(Vgp+Vgs)), u1 = (Ωp/Up) exp (−iK̃pz), u2 = (Ωs/Us) exp (−iK̃sz) (p,s ≡ Re(Kp,s)), g0p,s=αp,sLD with αp,s = Im(Kp,s), g sgn(δ)LD/Lδ, g1p=K2p/|K2s|, g1s=sgn(K2s), g11 = Wpp/|Wss|, g22 = −1 g12 = Wps/|Wss|, g21 = Wsp/|Wss|, with (Wpp,Wps)=[ωp/(2c)](χp(3,S),χp(3,C)) and (Wsp,Wss)=[ωs/(2c)](χs(3,S),χs(3,C)). In these expressions we have defined the characteristic dispersion length LD=τ02/|K2s|, and the characteristic group velocity mismatch length Lδ=τ0/|δ|(δ(1/Vgp1/Vgs)/2), with τ0 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 gδ = 0. In addition, we also have K2p=K2s. As a result, the above coupled NLS equations can be reduced to
iu1s+122u1σ2(g11|u1|2+g12|u2|2)u1=ig0pu1,
(10a)
iu2s+122u2σ2(g22|u2|2+g21|u1|2)u2=ig0su2.
(10b)
The terms on the right contribute by small absorption, which can be taken as a perturbation.

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 K0p,s=(2.74+i0.21)×104cm1, K1p,s=(6.25+i0.48)×109cm1s, K2p,s=(1.33+i0.09)×1020cm1s2, Wpp = Wss = (−1.72+i0.03)× 10−21 cm−1s2, Wps = Wsp = (−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 LD = 3.0 μm with τ0 = 2 × 10−12 s, L0 = 5.0 μm. The dimensionless coefficients read g0p,s=0.25 and gjl ≈ −1 (j,l = 1,2). The propagating velocity of the coupled soliton pair
Vsol=7.1×103c
(12)
is after returning to the dimensional variables. We see that the coupled soliton pair propagates with matched, ultraslow group velocity in comparison with c.

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 |Ωp/U0| versus the dimensionless time t/τ0 and the distance z/(2LD) 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 = 4LD. 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/(2LD) 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 |Ωp/U0| as a function of the dimensionless time t/τ0 and the distance z/(2LD), 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).

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
P¯1maxP¯2max=38.76mW.
(13)
We see that to generate the ultraslow coupled optical soliton pair in this semiconductor quantum-well system very low input power is needed. This is drastically different from the optical soliton generation schemes in passive (e.g. fiber-based) media, where much higher input power is required in order to bring out the nonlinear effect required for soliton formation.

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.

References and links

1.

R. W. Boyd, Nonlinear Optics (2cd edition) (Academic, San Diego, 2003).

2.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge, England, 2000).

3.

M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

4.

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef] [PubMed]

5.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E72, 016617 (2005). [CrossRef]

6.

M. D. Lukin and A. Imamoǧlu, “Nonlinear Optics and Quantum Entanglement of Ultraslow Single Photons,” Phys. Rev. Lett. 84, 1419 (2000). [CrossRef] [PubMed]

7.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003). [CrossRef] [PubMed]

8.

D. E. Nikonov, A. Imamoglu, and M. O. Scully, “Fano interference of collective excitations in semiconductor quantum wells and lasing without inversion,” Phys. Rev. B59, 12212 (1999). [CrossRef]

9.

M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nat. Mater.5, 175–178 (2006). [CrossRef]

10.

M. Phillips and H. Wang, “Spin Coherence and Electromagnetically Induced Transparency via Exciton Correlations,” Phys. Rev. Lett. 89, 186401 (2002). [CrossRef] [PubMed]

11.

P. Palinginis, S. Crankshaw, F. Sedgwick, E.-T. Kim, M. Moewe, C. J. Chang-Hasnain, H. Wang, and S.-L. Chuang, “Ultraslow light (< 200 m/s) propagation in a semiconductor nanostructure,” Appl. Phys. Lett. 87, 171102 (2005). [CrossRef]

12.

P. C. Ku, C. J. Chang-Hasnain, and S.-L. Chuang, “Slow light in semiconductor heterostructures,” J. Phys. D: Appl. Phys. 40, R93–R107 (2007), and references therein. [CrossRef]

13.

J. Faist, F. Capasso, A. L. Hutchinson, L. Pfeiffer, and K. W. West, “Suppression of optical absorption by electric-field-induced quantum interference in coupled potential wells,” Phys. Rev. Lett. 71, 3573 (1993). [CrossRef] [PubMed]

14.

J. Faist, C. Sirtori, F. Capasso, S.-N. Chu, L. N. Pfeiffer, and K. W. West, “Tunable Fano interference in intersubband absorption,” Opt. Lett. 21, 985–987 (1996). [CrossRef] [PubMed]

15.

H. Schmidt, K. L. Campman, A. C. Gossard, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett. 70, 3455 (1997). [CrossRef]

16.

H. Schmidt and A. Imamoglu, “Nonlinear optical devices based on a transparency in semiconductor intersubband transitions” Opt. Commun. 131, 333–338 (1996). [CrossRef]

17.

H. Sun, S. Gong, Y. Niu, R. Li, S. Jin, and Z. Xu, “Enhancing Kerr nonlinearity in an asymmetric double quantum well via Fano interference,” Phys. Rev. B 74, 155314 (2006). [CrossRef]

18.

J. H. Wu, J. Y. Gao, J. H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, and F. Bassani, “Ultrafast All Optical Switching via Tunable Fano Interference,” Phys. Rev. Lett. 95, 057401 (2005). [CrossRef] [PubMed]

19.

Y. Xue, X.-M. Su, G. W. Wang, Y. Chen, and J.-Y. Gao, “Photon switch in quantum well by quantum interference in interband transitions,” Opt. Commun. 249, 231–237 (2005). [CrossRef]

20.

X. Hao, J. Li, J. Liu, P. Song, and X. Yang, “Efficient four-wave mixing of a coupled double quantum-well nanostructure,” Phys. Lett. A 372, 2509–2513 (2008).

21.

C. Zhu and G. Huang, “Slow-light solitons in coupled asymmetric quantum wells via interband transitions,” Phys. Rev. B 80, 235408 (2009). [CrossRef]

22.

W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16, 17161 (2008). [CrossRef] [PubMed]

23.

H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936 (1996). [CrossRef] [PubMed]

24.

S. E. Harris and L. V. Hau, “Nonlinear Optics at Low Light Levels,” Phys. Rev. Lett. 82, 4611 (1999). [CrossRef]

25.

Z.-B. Wang, K.-P. Marzlin, and B. Sanders, in Quantum Communications and Quantum Imaging, edited by R. E. Meyers, Y. Shih, and K. S. Deacon, Proc. of SPIE6305, 6305H1–8 (2006).

26.

H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, and D. A.B. Miller, “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Phys. Rev. Lett. 68, 2216 (1992). [CrossRef] [PubMed]

27.

I. Waldmüller, J. Förstner, S.-C. Lee, A. Knorr, M. Woerner, K. Reimann, R. A. Kaindl, T. Elsaesser, R. Hey, and K. H. Ploog, “Optical dephasing of coherent intersubband transitions in a quasi-two-dimensional electron gas,” Phys. Rev. B 69, 205307 (2004). [CrossRef]

28.

We adopt phenomenological few-level model to study the optical response of SQWs. On the relation between such approach and microscopic theory, see N. H. Kwong, I. Rumyantsev, R. Binder, and A. L. Smirl, “Relation between phenomenological few-level models and microscopic theories of the nonlinear optical response of semiconductor quantum wells,” Phys. Rev. B 72, 235312 (2005). [CrossRef]

29.

The frequency and wavevector of the probe (signal) field in the quantum well are given by ωp + ω and kp + Kp(ω) (ωs + ω and ks + Ks(ω)), respectively. Thus ω = 0 corresponds to the center frequency of both the probe and signal fields.

30.

Y. Xue, X.-M. Su, G. W. Wang, Y. Chen, and J.-Y. Gao, “Photon switch in a quantum well by quantum interference in interband transitions,” Opt. Commun. 249, 231–237 (2005). [CrossRef]

31.

A. Neogi, “Transient interband light modulation via intersubband coupling light in undoped semiconductor quantum wells,” Opt. Commun. 133, 479–486 (1997); A. Neogi, H. Yoshida, T. Mozume, and O. Wada, “Enhancement of interband optical nonlinearity by manipulation of intersubband transitions in an undoped semiconductor quantum well,” Opt. Commun. 159, 225–229 (1999); A. Neogi, O. Wada, Y. Takahashi, and H. Kawaguchi, “Ultrashort-pulse-controlled all-optical modulation by interband and intersubband transitions in doped quantum wells,” Opt. Lett. 23, 1212–1214 (1998). [CrossRef]

32.

Because Im(Vgp,s) is much less than Re(Vgp,s), we disregard Im(Vgp,s) and take (Vgp,s)Re(Vgp,s) here and in the following.

33.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000). [CrossRef]

34.

K.-P. Marzlin, Z.-B. Wang, S. A. Moiseev, and B. C. Sanders, “Uniform cross-phase modulation for nonclassical radiation pulses,” J. Opt. Soc. Am. B 27, A36–A45 (2010). [CrossRef]

35.

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, 1995).

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Nonlinear Optics

Citation
Chengjie Zhu and Guoxiang Huang, "Giant Kerr Nonlinearity, Controlled Entangled Photons and Polarization Phase Gates in Coupled Quantum-Well Structures," Opt. Express 19, 23364-23376 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23364


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. R. W. Boyd, Nonlinear Optics (2cd edition) (Academic, San Diego, 2003).
  2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge, England, 2000).
  3. M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005). [CrossRef]
  4. Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett.93, 143904 (2004). [CrossRef] [PubMed]
  5. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E72, 016617 (2005). [CrossRef]
  6. M. D. Lukin and A. Imamoǧlu, “Nonlinear Optics and Quantum Entanglement of Ultraslow Single Photons,” Phys. Rev. Lett.84, 1419 (2000). [CrossRef] [PubMed]
  7. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett.90, 197902 (2003). [CrossRef] [PubMed]
  8. D. E. Nikonov, A. Imamoglu, and M. O. Scully, “Fano interference of collective excitations in semiconductor quantum wells and lasing without inversion,” Phys. Rev. B59, 12212 (1999). [CrossRef]
  9. M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nat. Mater.5, 175–178 (2006). [CrossRef]
  10. M. Phillips and H. Wang, “Spin Coherence and Electromagnetically Induced Transparency via Exciton Correlations,” Phys. Rev. Lett.89, 186401 (2002). [CrossRef] [PubMed]
  11. P. Palinginis, S. Crankshaw, F. Sedgwick, E.-T. Kim, M. Moewe, C. J. Chang-Hasnain, H. Wang, and S.-L. Chuang, “Ultraslow light (< 200 m/s) propagation in a semiconductor nanostructure,” Appl. Phys. Lett.87, 171102 (2005). [CrossRef]
  12. P. C. Ku, C. J. Chang-Hasnain, and S.-L. Chuang, “Slow light in semiconductor heterostructures,” J. Phys. D: Appl. Phys.40, R93–R107 (2007), and references therein. [CrossRef]
  13. J. Faist, F. Capasso, A. L. Hutchinson, L. Pfeiffer, and K. W. West, “Suppression of optical absorption by electric-field-induced quantum interference in coupled potential wells,” Phys. Rev. Lett.71, 3573 (1993). [CrossRef] [PubMed]
  14. J. Faist, C. Sirtori, F. Capasso, S.-N. Chu, L. N. Pfeiffer, and K. W. West, “Tunable Fano interference in intersubband absorption,” Opt. Lett.21, 985–987 (1996). [CrossRef] [PubMed]
  15. H. Schmidt, K. L. Campman, A. C. Gossard, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett.70, 3455 (1997). [CrossRef]
  16. H. Schmidt and A. Imamoglu, “Nonlinear optical devices based on a transparency in semiconductor intersubband transitions” Opt. Commun.131, 333–338 (1996). [CrossRef]
  17. H. Sun, S. Gong, Y. Niu, R. Li, S. Jin, and Z. Xu, “Enhancing Kerr nonlinearity in an asymmetric double quantum well via Fano interference,” Phys. Rev. B74, 155314 (2006). [CrossRef]
  18. J. H. Wu, J. Y. Gao, J. H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, and F. Bassani, “Ultrafast All Optical Switching via Tunable Fano Interference,” Phys. Rev. Lett.95, 057401 (2005). [CrossRef] [PubMed]
  19. Y. Xue, X.-M. Su, G. W. Wang, Y. Chen, and J.-Y. Gao, “Photon switch in quantum well by quantum interference in interband transitions,” Opt. Commun.249, 231–237 (2005). [CrossRef]
  20. X. Hao, J. Li, J. Liu, P. Song, and X. Yang, “Efficient four-wave mixing of a coupled double quantum-well nanostructure,” Phys. Lett. A372, 2509–2513 (2008).
  21. C. Zhu and G. Huang, “Slow-light solitons in coupled asymmetric quantum wells via interband transitions,” Phys. Rev. B80, 235408 (2009). [CrossRef]
  22. W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express16, 17161 (2008). [CrossRef] [PubMed]
  23. H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett.21, 1936 (1996). [CrossRef] [PubMed]
  24. S. E. Harris and L. V. Hau, “Nonlinear Optics at Low Light Levels,” Phys. Rev. Lett.82, 4611 (1999). [CrossRef]
  25. Z.-B. Wang, K.-P. Marzlin, and B. Sanders, in Quantum Communications and Quantum Imaging, edited by R. E. Meyers, Y. Shih, and K. S. Deacon, Proc. of SPIE6305, 6305H1–8 (2006).
  26. H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, and D. A.B. Miller, “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Phys. Rev. Lett.68, 2216 (1992). [CrossRef] [PubMed]
  27. I. Waldmüller, J. Förstner, S.-C. Lee, A. Knorr, M. Woerner, K. Reimann, R. A. Kaindl, T. Elsaesser, R. Hey, and K. H. Ploog, “Optical dephasing of coherent intersubband transitions in a quasi-two-dimensional electron gas,” Phys. Rev. B69, 205307 (2004). [CrossRef]
  28. We adopt phenomenological few-level model to study the optical response of SQWs. On the relation between such approach and microscopic theory, see N. H. Kwong, I. Rumyantsev, R. Binder, and A. L. Smirl, “Relation between phenomenological few-level models and microscopic theories of the nonlinear optical response of semiconductor quantum wells,” Phys. Rev. B72, 235312 (2005). [CrossRef]
  29. The frequency and wavevector of the probe (signal) field in the quantum well are given by ωp + ω and kp + Kp(ω) (ωs + ω and ks + Ks(ω)), respectively. Thus ω = 0 corresponds to the center frequency of both the probe and signal fields.
  30. Y. Xue, X.-M. Su, G. W. Wang, Y. Chen, and J.-Y. Gao, “Photon switch in a quantum well by quantum interference in interband transitions,” Opt. Commun.249, 231–237 (2005). [CrossRef]
  31. A. Neogi, “Transient interband light modulation via intersubband coupling light in undoped semiconductor quantum wells,” Opt. Commun.133, 479–486 (1997); A. Neogi, H. Yoshida, T. Mozume, and O. Wada, “Enhancement of interband optical nonlinearity by manipulation of intersubband transitions in an undoped semiconductor quantum well,” Opt. Commun.159, 225–229 (1999); A. Neogi, O. Wada, Y. Takahashi, and H. Kawaguchi, “Ultrashort-pulse-controlled all-optical modulation by interband and intersubband transitions in doped quantum wells,” Opt. Lett.23, 1212–1214 (1998). [CrossRef]
  32. Because Im(Vgp,s) is much less than Re(Vgp,s), we disregard Im(Vgp,s) and take (Vgp,s)≈Re(Vgp,s) here and in the following.
  33. V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A61, 052306 (2000). [CrossRef]
  34. K.-P. Marzlin, Z.-B. Wang, S. A. Moiseev, and B. C. Sanders, “Uniform cross-phase modulation for nonclassical radiation pulses,” J. Opt. Soc. Am. B27, A36–A45 (2010). [CrossRef]
  35. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, 1995).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited