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the main impediment towards the use of single photons in schemes for deterministic quantum logic and teleportation @xcite as very robust and versatile carriers of quantum information @xcite is the weakness of optical nonlinearities in conventional media . a major trend aimed at the enhancement of optical nonlinearities exploits one - dimensional ( 1d- ) periodic distributed bragg reflectors and 2d- or 3d - periodic photonic crystals ( pcs ) , where light can be slowed down or trapped via multiple reflections in the vicinity of photonic bandgaps ( pbgs ) @xcite . giantly enhanced nonlinearity has been predicted when dopants with transition frequencies within the pbg are implanted in the structure , so that light near such frequencies resonantly interacts with the dopants and is concurrently affected by the pbg dispersion @xcite . the resulting soliton - like transmission of very weak pulses within the pbg while filtering out spurious noise is highly advantageous for classical communication @xcite . however , this mechanism is incompatible with the goals of quantum logic and communications , particularly with photon - photon entanglement , because of the quantum noise associated with resonant field - atom interactions . another pathway to enhance the nonlinearities is based on electromagnetically induced transparency ( eit ) in atomic media , which comes about when classical driving fields induce coherence between atomic levels and transform the weak fields into atom - dressed dark - state polaritons @xcite . the ultrahigh sensitivity of the eit polaritonic dispersion to a small field - induced stark shift of its atomic level can result in an appreciable nonlinear phase shift , impressed by one ultraweak field upon another @xcite . notwithstanding this promising sensitivity , eit - polariton entanglement by a large conditional phase shift of one photon in the presence of another ( also known as cross - phase modulation ) faces serious challenges in spatially uniform media . one drawback of these schemes has been the mismatch between the group velocities of the probe pulse moving as a slow eit polariton and the nearly - free propagating signal pulse , which severely limits their effective interaction length and the maximal conditional phase - shift @xcite . to enable long interaction times and thus large conditional phase shifts in a medium of finite length ( up to a few centimeters ) , the group velocities of _ both _ interacting pulses should be small @xcite . this , however , imposes a limitation on the photonic component of the signal polariton whose magnitude determines the phase shift . copropagating pulses pose yet another difficulty : since the phase - shift of each pulse is proportional to the local intensity of the other pulse , different parts of the interacting pulses acquire different phase shifts , which results in their spectral broadening .    here we put forward a mechanism that may produce strong photon - photon interactions along with suppressed quantum noise and give rise to their entanglement with high fidelity , by combining the advantages of their dispersion in pbg structures and of the strongly enhanced nonlinear optical coupling achievable via eit in an appropriately doped medium . the main idea is that a single - photon signal pulse is adiabatically converted into a standing - wave polaritonic excitation inside the periodic structure . this trapped polariton , having an appreciable photonic component , can impress a large , spatially - uniform phase shift upon a slowly propagating probe polariton . this task can further be facilitated by employing 2d- or 3d - periodic structures with defects where the two pulses interact via tightly confined modes @xcite . at frequency outside the pbg to the trapped field @xmath0 at frequency inside the pbg . atoms b provide eit for the probe field @xmath1 and its cross - coupling with the signal field @xmath0 . ( b ) 2d - periodic photonic crystal ( pc ) having the density of modes @xmath2 , is doped with atomic species a and b. ( c )  interaction and the resulting phase - shift of the probe and signal pulses.,width=529 ]    the proposed scheme is based upon a periodic structure containing uniformly distributed dopants atoms a and b ( see fig . [ as - pr ] ) . atoms a , having double-@xmath3 level configuration and interacting with classical driving fields on the transitions @xmath4 with the rabi frequencies @xmath5 , respectively , facilitate the trapping of the signal pulse inside the periodic structure , by converting its frequency from outside to inside of the pbg . on the other hand , atoms b , having n level configuration and interacting with the @xmath6 driving field on the transition @xmath7 , serve to simultaneously slowdown the probe pulse and cross - couple it with the signal . the following procedure is foreseen to this end . initially , all atoms a are in the ground state @xmath8 , the driving fields @xmath9 and @xmath10 , where @xmath11 is the temporal width of the input signal pulse @xmath12 . the carrier frequency of @xmath12 is outside the pbg , close to the @xmath13 transition frequency , so that the usual eit for the input signal due to the @xmath3 configuration @xmath14 is realized . upon entering the medium , the signal pulse is slowed down and spatially compressed , by a factor of @xmath15 , to the length @xmath16 , where @xmath17 is its group velocity inside the medium . once the signal pulse has fully accommodated in the medium of length @xmath18 , which requires that @xmath19 , the driving field @xmath20 is adiabatically switched off . as a result , the signal is stopped , its photonic component being converted into the stationary atomic ( raman ) coherence @xmath21 @xcite . next , the driving field @xmath22 is adiabatically switched on to a value @xmath23 , converting the atomic coherence into the signal field @xmath0 , whose frequency is inside of the pbg and amplitude is larger than that of the input signal @xmath12 by a factor of @xmath24 @xcite . due to the bragg scattering of the forward and backward propagating components of the signal pulse with the wave vectors @xmath25 , it remains localized ( trapped ) within the medium @xcite . both components @xmath0 of the signal field dispersively interact with atoms b via transition @xmath26 with the detuning @xmath27 . this off - resonant interaction causes an ac stark shift of level @xmath28 , thereby strongly affecting the eit dispersion for the probe field @xmath1 , which interacts with atoms b on the transition @xmath29 @xcite . for a large enough product of the signal - field intensity @xmath30 and interaction time @xmath31 ( @xmath32 being the probe group velocity ) , both pulses accumulate a _ uniform _ conditional phase - shift which can reach @xmath33 ( see fig . [ as - pr](c ) ) . finally , reversing the sequence that resulted in trapping of the signal pulse , its frequency is converted back to the original frequency and the @xmath34 pulse leaves the medium . let us now consider the scheme more quantitatively . to describe the quantum properties of the medium , we use collective slowly varying atomic operators @xmath35 , averaged over a small but macroscopic volume containing many dopants of species @xmath36 around position @xmath37 @xcite : @xmath38 , where @xmath39 is the total number of the corresponding dopants . the quantum radiation is described by the traveling - wave ( multimode ) electric field operators @xmath40 and @xmath41 , where @xmath42 are the annihilation operators for the field mode with the wave - vector @xmath43 , @xmath44 being the carrier wave vector of the corresponding field . these single - mode operators possess the standard bosonic commutation relations @xmath45={\delta}_{ij } { \delta}_{qq^{\prime}}$ ] , which yield @xmath46 = l { \delta}_{ij } { \delta}(z - z^{\prime})$ ] .    using the standard technique @xcite , we perturbatively solve the heisenberg equations for the atomic coherences @xmath47 under eit conditions @xcite and substitute the solution into the propagation equations for the slowly varying field operators @xmath48 . these operators are related to the polariton operators @xmath49 and @xmath50 , which represent the coupled excitation of the corresponding field and atomic raman coherence @xmath51 . the mixing angles @xmath52 are defined via @xmath53 , where @xmath54 are the atom - field coupling constants , @xmath55 being the corresponding atomic dipole matrix element and @xmath56 the cross - sectional area of the quantum fields . note that the amplitude of the photonic component of each polariton is proportional to the @xmath57 of the corresponding mixing angle @xmath58 . under the bragg resonance condition @xmath59 , where @xmath60 is the structure period , the equations of motion for polaritons are obtained as    [ pols ] @xmath61    here @xmath62 and @xmath63 are the group velocities , @xmath64 and @xmath65 are the intensities ( excitation numbers ) of the signal and probe polaritons , respectively ; @xmath66 are the absorption rates , @xmath67 being the raman coherence decay rate of the corresponding atoms , @xmath68 are the @xmath69-correlated noise operators associated with the relaxation ; @xmath70\cos^2\theta_a \sin^2\theta_b   g_b^{\prime 2 } /{\delta}_b$ ] is the cross - coupling rate between the polaritons , @xmath71 being the atom - field coupling on the transition @xmath72 and @xmath73 the decay rate of @xmath74 ( for @xmath75 , the cross - absorption vanishes @xcite and then @xmath76 , being purely real , represents the cross - phase modulation rate ) . finally , @xmath77 is the bragg reflection rate , @xmath78 being the pbg bandwidth .    equations ( [ pols ] ) constitute the starting point of our analysis . their general solution , for arbitrary initial / boundary conditions of the traveling - wave quantized polaritons @xmath79 is not known . when absorption is negligible ( see below ) , for a given time- and space - dependence of the signal - polariton intensity @xmath80 , the solution for the probe is @xmath81,\ ] ] where @xmath82 is the retarded time . an analytic solution for the two counter - propagating components of the signal polariton can be obtained in the long probe limit , i.e. , when the spatial dependence of the probe - polariton intensity is negligible on the scale of @xmath83 : @xmath84 . this requires that @xmath85 , where @xmath86 is the duration of the probe pulse ( its spectral width being @xmath87 ) . then eq . ( [ cpol ] ) is soluble by the fourier transform @xmath88 , with the result    [ spolms ] @xmath89 , \\ { \hat{\psi}}_{-}(q , t ) & = & i { \hat{\psi}}_{+}(-q,0 ) \ : e^{i { \hat{\phi}}_s(t ) }   \frac{\beta}{\chi } \sin(\chi t),\end{aligned}\ ] ]    where @xmath90 and @xmath91 , with @xmath92 . thus all the spatial modes @xmath93 of the signal polariton acquire the same @xmath94-independent phase shift @xmath95 . it follows from eqs . ( [ spolms ] ) that a signal polariton composed of modes with @xmath96 will be strongly trapped inside the medium , its wavepacket periodically cycling between the forward and backward components while interacting with the probe polariton , yielding    [ polsol ] @xmath97    where @xmath98 , with @xmath99 , is the probe phase - shift . let us dwell upon the approximations involved in the derivation of eqs . ( [ polsol ] ) . during the conversion of the signal pulse into a standing - wave polaritonic excitation inside the periodic structure , the nonadiabatic corrections resulting in its dissipation are negligible provided the medium is optically thick , @xmath100 , where @xmath101 is the resonant absorption cross - section for the transition @xmath13 and @xmath102 is the density of atoms @xmath103 @xcite . after the signal pulse has been trapped in the pbg , due to nonzero values of @xmath94 , its gets spatially distorted ( spreads ) at a rate @xmath104 ( @xmath105 ) . we can estimate the bandwidth of the signal pulse from its spatial extent as @xmath106 , thus obtaining the upper limit for the distortion rate @xmath107 . on the other hand , the bandwidth of the probe is limited by the length of the medium via @xmath108^{-1/2 }   = |{\omega}_d^{(b)}|^2 k_p [ { \gamma}_a^{(b ) } \sqrt{3 \pi/2 \rho_b l}]^{-1}$ ] , where @xmath109 is the density of atoms @xmath110 @xcite . finally , the interaction time @xmath111 is limited by @xmath112 , and so the fidelity of the cross - phase modulation is given by @xmath113 $ ] . consider first the classical limit of eqs . ( [ polsol ] ) , where the operators @xmath79 and @xmath114 are replaced by c - numbers . then for two single - photon pulses @xmath115 , the conditional phase shift accumulated by the probe and signal fields during the interaction is given by @xmath116 note that the phase shift is proportional to the intensity of the photonic component of the signal polariton , as attested by the presence of the @xmath117 term in the nominator of eq . ( [ phsh_cls ] ) . for realistic experimental parameters , relevant to a doped periodic structure discussed below , one can obtain @xmath118 ( see fig . [ as - pr](c ) ) with the fidelity @xmath119 . we now turn to the fully quantum treatment of the system . to compare the classical and quantum pictures , we consider first the evolution of input wavepackets composed of the multimode coherent states @xmath120 . states @xmath121 and @xmath122 are the eigenstates of the input fields operators @xmath123 and @xmath124 with eigenvalues @xmath125 and @xmath126 , respectively . the expectation values for the fields are then obtained as    [ expval ] @xmath127 \ , , \\   { \langle { \hat{\cal e}}_{\pm } ( z , t ) \rangle } & = & \alpha_{+ } ( z ) \exp \left [ \frac{e^{i \phi}-1}{l } c \int_0^t |\alpha_p(\tau^{\prime})|^2 \ , \upd t^{\prime } \right ] \times   \left [ \begin{array}{c } \cos(\beta t ) \\ i \sin(\beta t ) \end{array }   \right ] \ , .\end{aligned}\ ] ]    these equations are notably different from those obtained for single - mode @xcite and multimode copropagating fields @xcite because all parts of the probe pulse interact with the whole signal pulse ( and vice versa ) , as is manifest in the space ( time ) integration . similarly to the cases discussed in @xcite , eqs . ( [ expval ] ) reproduce the classical result only in the limit @xmath128 , yielding @xmath129 and @xmath130 . a @xmath33 phase shift is then obtained for @xmath131 . this restriction on the classical correspondence of coherent states comes about since , for large enough cross - phase modulation rates @xmath76 , these states exhibit periodic collapses and revivals as @xmath132 changes from 0 to @xmath133 , which limits their usefulness for quantum information applications . consider finally the input state @xmath134 , consisting of two single - excitation polariton wavepackets @xmath135 , where the fourier amplitudes , normalized as @xmath136 , define the spatial envelopes @xmath137 of the probe and forward signal pulses that initially ( at @xmath138 ) are localized around @xmath139 and @xmath140 , respectively ( fig . [ as - pr](c ) ) . during the evolution , the state of the system evolves according to @xmath141 , where @xmath142 $ ] is the effective interaction hamiltonian whose first and second terms commute . the implicit time - dependence of the effective hamiltonian , due to the propagation of the probe - polariton pulse with the group velocity @xmath32 , is contained in the operator @xmath143 , with @xmath144 . after the interaction , at time @xmath145 , the output state of the system is @xmath146 , \label{sph}\ ] ] where @xmath147 . thus , while the signal pulse periodically cycles between the forward and backward modes , the combined state of the system acquires an overall conditional phase shift @xmath148 . when @xmath149 , transformation ( [ sph ] ) corresponds to the truth table of the _ universal _ controlled - phase ( cphase ) logic gate between the two photons representing qubits , which can be used to realize arbitrary unitary transformation @xcite . an @xmath150 2d - periodic lattice of dielectric rods or semiconductor stacks ( fig . [ as - pr](b ) ) @xcite , with controlled structural defects and mirror confinement in @xmath151 , appears to be the most suitable structure for realizing polaritonic entanglement of two single - photons , since both the signal and the probe pulses may be confined in the @xmath152 directions in the vicinity of a `` defect '' row forming a pc waveguide , thereby avoiding diffraction losses and focusing the fields to a radius of @xmath153 m @xcite . using the double-@xmath3 dopants ( atoms a ) , the signal pulse can be trapped in the pc , with the localization distance @xmath83 extending over many periods , and interact with the probe pulse via atoms b. expressing the atom - field coupling constants @xmath154 through the decay rate @xmath155 of the corresponding excited state as @xmath156 , and assuming that @xmath157 and @xmath158 ( @xmath159 ) , we have from eq . ( [ phsh_cls ] ) @xmath160 . among the possible dopants , iii - v or ii - vi @xmath161-doped semiconductor quantum dots , having large dipole moments and level structure conducive to eit @xcite , could be the best choice for our scheme , provided high densities can be achieved . in such single - electron doped qds , the spin degeneracy of the ground and the lowest and higher excited ( charged exciton or trion ) states can be lifted with a magnetic field @xcite , realizing the level scheme of fig . [ as - pr](a ) , where atoms a and b can spectroscopically be selected ( via optical pumping or spectral hole burning ) from the inhomogeneous ensemble of qds . thus states @xmath162 and @xmath163 are represented by the zeeman - split spin - up and spin - down states of the conduction - band electron in the qd . the excited states @xmath164 and @xmath165 of atoms a can be the first and the second ( or higher ) exciton states , while states @xmath166 and @xmath74 of atoms b are the zeeman sublevels of the lowest excitonic state . assuming the parameters @xmath167 cm , @xmath168@xmath169 , @xmath170rad / s @xcite , @xmath171@xmath172 , @xmath173rad / s , @xmath174rad / s , @xmath175rad / s , @xmath176rad / s , @xmath177s@xmath178 , and @xmath179s@xmath178 @xcite , we obtain @xmath118 with the fidelity @xmath180% , the main limiting factor being the decay of raman coherence @xmath181 . other contenders for observing the proposed effects include periodic structures fabricated from rare - earth doped crystals , such as pr : ysp , in which high - fidelity eit has experimentally been demonstrated @xcite , or cryogenically cooled diamond with high density of nitrogen - vacancy defect centers @xcite . to summarize , we have proposed a new class of multimode quantum - field interactions involving quantized eit - polaritons in pbg structures . we have shown that such interactions allow efficient cross - phase modulation between a propagating probe pulse and a trapped signal pulse , whose localization is achieved by an adiabatic four - wave mixing process that pulls its frequency into the pbg . this localization allows multiply repeated interaction of the signal with the entire probe pulse . as a result , the combined two - photon state of the system can acquire a conditional @xmath33 phase - shift , which corresponds to the universal cphase logic gate . the phase shift is spatially - uniform and the process may have high fidelity . the experimental realization of the predicted effects requires the fabrication of periodic structures with large densities of optically active dopants @xcite , which may also find useful applications in laser technology , optical communication or quantum computation . we note that a similar regime of giant cross - phase modulation with suppressed spectral broadening is also realizable in cold atomic vapors using optically - induced pbgs @xcite , on which we intend to report elsewhere . the proposed scheme may pave the way to quantum information applications such as deterministic all - optical quantum computation , dense coding and teleportation @xcite .
giantly enhanced cross - phase modulation with suppressed spectral broadening is predicted between optically - induced dark - state polaritons whose propagation is strongly affected by photonic bandgaps of spatially periodic media with multilevel dopants . 
 this mechanism is shown to be capable of fully entangling two single - photon pulses with high fidelity .