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methods for preparation and manipulation of nonclassical states of light have become an important research area in quantum optics @xcite , especially in relation to possible optical implementations of quantum computers and systems for quantum communication and quantum cryptography @xcite . among various schemes for optical - qubit generation , the so - called _ quantum scissors _ device of pegg _ et al . _ @xcite produces a superposition of vacuum and single - photon states , @xmath0 , by optical - state truncation of an input single - mode coherent light . quantum scissors device was studied in numerous papers ( see , e.g. , @xcite ) , and tested experimentally by babichev _ _ @xcite and resch _ et al . this simple scheme and its generalizations for truncation of an input optical state to a superposition of @xmath1 fock states ( the so - called qudits ) @xcite are based on linear optical elements , and thus referred to as the _ linear quantum scissors _ devices . optical - state ` truncation ' can also be achieved in systems comprising nonlinear elements ( e.g. , kerr media ) @xcite , and thus will be referred to as the _ nonlinear quantum scissors _ devices . the above - mentioned schemes are restricted to the single - mode optical truncation . here , by generalizing our former scheme @xcite , we present a realization of nonlinear quantum scissors for optical - state truncation of two cavity modes by means of a pumped nonlinear coupler . two - mode nonlinear couplers have become , shortly after their introduction by jensen @xcite and maier @xcite , one of the important topics of photonics due to their wide potential applications and relative simplicity ( see , e.g. , reviews @xcite ) . among various types of the nonlinear optical couplers , those based on kerr effect have attracted especial interest both in classical @xcite and quantum @xcite regimes . the kerr nonlinear couplers can exhibit variations of self - trapping , self - modulation and self - switching effects . in quantum regime , they can also generate sub - poissonian and squeezed light @xcite . possibilities of entanglement generation were also studied in nonlinear couplers operating by means of the kerr effect @xcite or degenerate parametric down - conversion @xcite .    here , we analyze kerr nonlinear couplers , which can be modelled by systems composed of two quantum nonlinear oscillators linearly coupled to each other and placed inside a double - ring cavity . we discuss two schemes based on the coupler with an external excitation of a single mode and the coupler with two modes pumped . we show that the states generated in the excited nonlinear couplers under suitable conditions can be limited to a superposition of only vacuum and single - photon states , @xmath2 . we compare the possibilities of generation of maximally entangled states by the couplers excited in single and two modes . we also discuss effects of dissipation on the fidelity of truncation and suggest a method to achieve strong kerr interactions at low intensities in our system . we consider a system , referred to as the pumped kerr nonlinear coupler , which consists of two nonlinear oscillators , with kerr nonlinearities @xmath3 and @xmath4 , linearly coupled to each other and additionally linearly coupled to an external classical field . in this section , we assume that the field is coupled to one of the oscillators only . thus , the system can be described by the following hamiltonian @xcite ( @xmath5 ) : @xmath6 where @xmath7 @xmath8 and @xmath9 ( @xmath10 ) is the bosonic annihilation operator corresponding to the mode @xmath11 ( @xmath12 ) of frequency @xmath13 ( @xmath14 ) . hamiltonian ( [ n01 ] ) for @xmath15 describes the standard nonlinear coupler @xcite composed of two kerr nonlinear oscillators linearly coupled to each other , where the parameter @xmath16 is the strength of this coupling . however , our model differs from the standard by inclusion of the extra term @xmath17 , which describes linear coupling between the driving single - mode classical field ( say , with frequency @xmath18 ) and the cavity mode @xmath11 . the parameter @xmath19 describes the strength of this coupling and is proportional to the classical field amplitude . a possible physical realization of the model is presented in figure 1 , where an external pump , described by @xmath20 , is off in the present analysis . kerr media , marked by @xmath21 and @xmath22 , are linearly coupled as described by grey central region corresponding to @xmath23 . the evolution of our system in the interaction picture can be described by the schrdinger equation @xmath24 where @xmath25 thus , the complex probability amplitudes @xmath26 satisfy the set of equations of motion : @xmath27c_{mn } \notag \\   & & + \epsilon c_{m-1,n+1 } \sqrt{m(n+1 ) } + \epsilon ^{\ast } c_{m+1,n-1 } \sqrt{(m+1)n } \notag \\ & &   + \alpha c_{m-1,n } \sqrt{m } + \alpha ^{\ast } c_{m+1,n } \sqrt{m+1}. \label{n07}\end{aligned}\ ] ] a superficial analysis of ( [ n07 ] ) could lead to a conclusion that the evolution of the system pumped by classical external field can not be restricted to two lowest photon - numer states , but will also include states with a greater number of photons . however , by generalizing the method of single - mode nonlinear quantum scissors proposed in @xcite ( for a review see @xcite ) , we have observed in @xcite that evolution can be restricted to the only four states : @xmath28 , @xmath29 , @xmath30 and @xmath31 as a result of degeneracy of hamiltonians @xmath21 and @xmath22 . by assuming that the couplings @xmath32 , and @xmath33 are much smaller than the kerr nonlinearities @xmath3 and @xmath4 , we can interpret the evolution between the four states as resonant transitions , while the negligible evolution to other states as out of resonance , analogously to the single - mode case @xcite . this phenomenon can be shown explicitly as follows : under the assumption of @xmath34 and short evolution times , equation ( [ n07 ] ) for @xmath35 can be approximated by @xmath36 c_{mn } \label{n08}\ ] ] which has the simple solution @xmath37 t\}c_{mn}(0 ) . \label{n09}\ ] ] by setting the initial condition @xmath38 for @xmath35 , one gets @xmath39 . by contrast , for @xmath40 , the terms proportional to @xmath3 and @xmath4 are vanishing due to degeneracy of the kerr hamiltonian and so the remaining terms proportional to @xmath16 and @xmath19 are significant . thus , the ideally ` truncated ' two - mode state generated in the system has the following simple form @xmath41 where the evolution of @xmath42 , precisely given by ( [ n07 ] ) , can approximately be described by the following equations @xmath43 hereafter , in equations for the probability amplitudes under the discussed assumptions , sign ` = ' should be understood as ` @xmath44 ' . although approximate equations ( [ n11 ] ) are independent of @xmath3 , our derivation clearly shows that the kerr nonlinearity plays a crucial role in the truncation process . by assuming that both oscillators are initially in vacuum states , @xmath45 , and parameters @xmath19 and @xmath16 are real , we find the following solutions of ( [ n11 ] ) for the time - dependent probability amplitudes : @xmath46 , \notag \\ c_{01}&=&\frac{\alpha}{\gamma}\left(\cos\tau _ { 1}-\cos\tau _ { 2}\right ) , \notag \\ c_{10}&=&-\frac{i(\gamma+\epsilon)\omega_2}{4 \alpha \gamma}(\sin\tau _ { 1}+\sin\tau _ { 2 } ) , \notag \\ c_{11}&=&-\frac{i}{2\gamma}\left(\omega_2\sin\tau _ { 1}-\omega_1\sin\tau _ { 2}\right ) , \label{n12}\end{aligned}\ ] ] where @xmath47}$ ] , @xmath48 , and @xmath49 for @xmath50 . note that the solution for @xmath51 can be written in a more symmetric form since the properties hold : @xmath52 . in a special case of equal couplings @xmath19 and @xmath16 , ( [ n12 ] ) simplifies to our former solution @xcite : @xmath53 where @xmath54 . to estimate the quality of the optical - state truncation ( up to single - photon states ) of the generated light , we apply fidelity as a measure of discrepancy between the ideally truncated two - qubit state @xmath55 , given by ( [ n10 ] ) , and the actually generated output state @xmath56 calculated numerically from @xmath57 for a large ( practically infinite - dimensional ) two - mode hilbert space specifically , in our numerical analysis , we have chosen the dimension equal to @xmath58 for each subspace associated with single mode of the field . the fidelity , also referred to as the uhlmann s transition probability for mixed states , is defined by ( see , e.g. , @xcite ) :    = 4.4 cm = 4.4 cm    = 4.4 cm = 4.4 cm    = 4.4 cm = 4.4 cm    @xmath59\right\}^2 . \label{n15}\end{aligned}\ ] ]    by assuming that one of the states is pure ( say @xmath60 ) , then ( [ n15 ] ) simplifies to @xmath61 . instead fidelity , the bures distance @xmath62 is often applied as a measure of discrepancy between the states . note that the bures distance satisfies the usual metric properties including symmetry @xmath63 , contrary to the quantum kullback - leibler ` distance ' ( quantum relative entropy ) often used in quantum - information context @xcite . general expression ( [ n15 ] ) will be applied in our description of the effect of damping on the optical - state truncation in section 4 . in this section , we are focused on pure - state truncation , for which equation ( [ n15 ] ) simplifies to the familiar expression @xmath64 , where the ideally truncated state @xmath65 is given by ( [ n10 ] ) and the actually generated state @xmath66 is calculated numerically from ( [ n14 ] ) . the fidelity for perfect truncation is equal to one . figure 2 clearly indicates that the fidelity of truncation using the pumped coupler is close to one for short times and the coupling strengths much smaller than the nonlinearity parameters ( @xmath67 ) . the numerical results shown in figure 2 confirm the validity of our analytical approach at least for short evolution times . thus , we can refer to the system as a kind of nonlinear ( as operating by means of kerr nonlinearity ) quantum scissors device . solutions for probability amplitudes of the truncated states enable a simple calculation of quantum entanglement , which is one of the most fundamental resources of quantum information theory @xcite . it is well known that the entanglement of a bipartite pure state , described by a density matrix @xmath68 or @xmath69 or , equivalently , by the shannon entropy of the squared schmidt coefficients @xmath70 @xcite : @xmath71 for bipartite pure states this measure is often referred to as the _ entropy of entanglement_. in a special case of two qubits in a pure state , the entropy of entanglement @xmath72 ranges from zero for a separable state to 1 ebit for a maximally entangled state , and it is simply given in terms of the binary entropy @xmath73 . in fact , for a general two - qubit pure state , given by ( [ n10 ] ) with arbitrary amplitudes @xmath42 ( @xmath74 ) , the entropy of entanglement given by ( [ n16 ] ) can simply be calculated as @xmath75 where @xmath76 and @xmath77 is the binary entropy . if the probability amplitudes @xmath26 evolve according to ( [ n12 ] ) then the evolution of the entropy of entanglement is given by @xmath78 \nonumber \\ \times \cos({{\textstyle{\frac{1}{2 } } } } \omega_{j}t)\big \ }   \sin({{\textstyle{\frac{1}{2 } } } } \omega_{3-j}t)\big ) . \label{n19}\end{aligned}\ ] ] solution ( [ n19 ] ) is further simplified by assuming real @xmath79 then the amplitudes @xmath42 are given by ( [ n13 ] ) . thus , we obtain @xmath80 \sin ( \alpha t ) \nonumber \\   -\sqrt{5}\cos ( \alpha t)\sin ( \sqrt{5}\alpha t)\}\big ) . \label{n20}\end{aligned}\ ] ] figures 3(a ) and 4(a ) show the plots of the entropy of entanglement for the case discussed here , i.e. , for the single - mode pumped coupler with the coupling parameters @xmath32 and @xmath33 much smaller than the nonlinearities @xmath81 . we see in figure 3(a ) that the rapid oscillations in time ( with a period @xmath82 ) are modulated by oscillations of low frequency ( with period @xmath83 ) . as a consequence , their maxima are of various values but some of them approach 1 ebit corresponding to the formation of bell states . to show this explicitly , we represent the generated state in the basis @xmath84 spanned by the bell - like states :    = 4.4 cm = 4.4 cm    = 4.4 cm = 4.4 cm    = 4.4 cm = 4.4 cm    @xmath85    which differ from the standard bell states only by the phase factor @xmath86 . clearly , @xmath87 and @xmath88 for @xmath89 , so states ( [ n21 ] ) will shortly be referred to as the bell states . figures 5 and 6 show the probabilities for the generation of the bell states as a function of time for the single - mode driven couplers , when the initial state is the two - mode vacuum state . it is seen that states @xmath90 and @xmath91 , being superpositions of @xmath28 and @xmath31 , are generated with a high accuracy . in detail , the maxima of ( [ n20 ] ) occur approximately at times @xmath92 $ ] if @xmath79 . since the frequencies of oscillations in ( [ n20 ] ) are incommensurate , by waiting long enough ( assuming no dissipation ) , we can achieve 1 ebit with high precision . for example , in the first four periods of @xmath93 , one could observe generation of entangled states approaching bell state @xmath90 with @xmath94=0.994 $ ] and @xmath95=0.997 $ ] ebits , and approaching @xmath91 with @xmath96=0.992 $ ] and @xmath97=0.998 $ ] . note that only the first period is shown in figures 3(a ) and 5(a ) , for which the highest entanglement of 0.994 ebits occurs approximately at the time @xmath98 . thus , we can effectively treat our system as a source of maximally entangled states . on the other hand , the bell states @xmath99 and @xmath100 , being superpositions of @xmath29 and @xmath30 , are not generated if the initial state is vacuum and @xmath101 . this conclusion can be drawn by observing that the probabilities for @xmath99 and @xmath100 can reach only @xmath102 in figure 5(b ) . however , by relaxing the condition of equal couplings @xmath16 and @xmath19 , as shown in figure 6(b ) , bell states @xmath99 and @xmath100 can be generated from vacuum with high precision in the dissipation - free system . hitherto , we have assumed that both cavity modes were initially in vacuum states . now , we analyze a more general evolution when the cavity modes are initially not only in vacuum but also in single - photon fock states , i.e. , @xmath103 , where @xmath104 . thus , by assuming as usual that @xmath105 , the evolutions of the initial states are found to be @xmath106 in terms of the time - dependent amplitudes @xmath107 given by ( [ n13 ] ) and @xmath108 where , as usual , @xmath109 . please note that @xmath110 and @xmath111 differ in sign . we find that the generalized expression for the entropies of entanglement for the initial states @xmath112 with @xmath104 reads as @xmath113 \sin ( \alpha t)\hspace{1.5 cm } \nonumber \\ -(-1)^{k - l}\sqrt{5}\cos ( \alpha t)\sin ( \sqrt{5}\alpha t)\big ) , \label{n24}\end{aligned}\ ] ] which implies that @xmath114 it is worth noting that our system for the initial states @xmath30 or @xmath29 quasi - periodically evolves into the bell states @xmath99 and @xmath100 with high precision but does not evolve into @xmath90 or @xmath91 assuming @xmath79 . this is by contrast to the evolutions of the initial states @xmath28 ( see figure 5 ) or @xmath31 also for @xmath101 . for brevity , we will not present any graphs of the evolutions of the initial states @xmath30 , @xmath29 , and @xmath31 , which would correspond to figures 25 plotted for the initial vacua . the above solutions for the initial single - photon states are included for completeness of our mathematical approach to show that , in principle , all bell states can be generated in our system even for @xmath101 . but it should be stressed that the system with the initial fock states is much more experimentally challenging than that assuming initially the vacuum states only . despite of experimental difficulty , our system enables generation of the one - photon fock states from vacuum assuming no coupling @xmath16 between the modes , which corresponds to having two independent pumped cavities with nonlinear kerr media . a possibility of producing single - photon states in such systems was demonstrated in @xcite ( for a review see @xcite and references therein ) . this section is devoted to the most general scheme presented in figure 1 , namely that involving two external excitations . we assume here that both modes of the coupler are excited by external fields , whereas for the case discussed previously we assumed that only one of the modes was coupled to the external field . the hamiltonian describing such a system is of the form @xmath115 which is the same as that defined by ( [ n01])([n04 ] ) , except for the extra term given by @xmath116 corresponding to the coupling of the cavity mode @xmath12 with an external driving single - mode classical field ( say , with frequency @xmath117 ) , where the parameter @xmath118 describes the strength of this interaction being proportional to the classical field amplitude . the evolution of our system , described by hamiltonian ( [ n26 ] ) , can be given by the schrdinger equation from which we find the following set of equations for the amplitudes @xmath26 of the wavefunction ( [ n06 ] ) in the interaction picture : @xmath119c_{mn }   \notag \\ & & + \epsilon c_{m-1,n+1 } \sqrt{m(n+1 ) } + \epsilon ^{\ast } c_{m+1,n-1 } \sqrt{(m+1)n } \notag \\ & & + \alpha c_{m-1,n } \sqrt{m } + \alpha ^{\ast } c_{m+1,n } \sqrt{m+1}\notag \\   & & + \beta c_{m , n-1 } \sqrt{n } + \beta ^{\ast } c_{m , n+1 } \sqrt{n+1}. \label{n28}\end{aligned}\ ] ] analogously to the analysis in the former section , we assume short evolution times as well as the couplings @xmath32 , @xmath120 , and @xmath33 to be much smaller than the kerr nonlinearities @xmath3 and @xmath4 . then , equation ( [ n28 ] ) for @xmath121 can be approximated by ( [ n08 ] ) with the solution ( [ n09 ] ) , which vanishes for the initial condition @xmath38 . thus , under the above assumptions , the infinite set of equations ( [ n28 ] ) reduces to the following four equations : @xmath122 assuming that at the time @xmath123 , both oscillator modes are in the vacuum states , i.e. , @xmath124 and @xmath125 , we can obtain analytical solutions of ( [ n29 ] ) . to solve ( [ n29 ] ) we need to find zeros of a fourth - order polynomial and , hence , the solutions in their general form are rather complicated and unreadable . however , if we assume that all coupling constants are real and the couplings with two external fields are of the same strength ( @xmath126 ) then the solutions become much simpler and easier to interpret . thus , under these assumptions , the solutions are found to be : @xmath127e^{-i\epsilon t/2}\right\ } , \notag \\ c_{01 } & = & c_{10}=-i2\frac{\alpha } { \lambda } \sin ( \frac{\lambda t}{2})\,e^{-i\epsilon t/2 } , \notag \\ c_{11 } & = & c_{00}-1 , \label{n30}\end{aligned}\ ] ] where we have introduced the effective coupling constant @xmath128 . as in the case discussed in the previous section , the system s dynamics is closed within the finite set of @xmath129-photon states . figure 2 shows that for the assumed parameters and evolution times shorter than @xmath130s , the fidelity between the ideally truncated states and the actually generated states by means of the coupler pumped in two modes deviates from 1 by the values less than @xmath131 in figure 2(a ) or even @xmath132 in figure 2(b ) . this again confirms the validity of our analysis and justifies referring to this system as a kind of the quantum scissors device . for the parameters assumed in figure 2 , truncation with higher fidelity is usually observed for the coupler pumped in a single mode rather than in two modes . in the latter case , the truncation fidelity depends on the relative phase between the pumping - field couplings @xmath19 and @xmath118 . note that state @xmath133 for @xmath134 , contrary to @xmath135 , can be directly calculated from solution ( [ n12 ] ) . for brevity , we have not presented here an analogous solution for @xmath135 , but we have used it for plotting the corresponding curves in figures 2 and 3(d ) . it is seen , by comparing figures 2(a ) and 2(b ) , that by decreasing @xmath16 in comparison to @xmath19 , the fidelity of truncation can be improved for @xmath136 . thus , pumping the coupler in two modes can lead sometimes to truncation better than that for the system driven in single mode as presented in figure 2(b ) by dot - dashed curve corresponding to @xmath135 and @xmath137 . the evolution of the pure - state entanglement , given by ( [ n16 ] ) , generated in the coupler pumped in two modes can be calculated from ( [ n17 ] ) with the probability amplitudes given by ( [ n30 ] ) . thus , we obtain @xmath138\big|\big).\label{n31}\ ] ] figures 3(b - d ) and 4(b ) show the evolution of the entropy of entanglement measured in ebits as a function of time for the system pumped in two modes in comparison to the results for the single - mode driven coupler shown in figures 3(a ) and 4(a ) . it is seen that the first maximum in figures 3(b - d ) is the highest , contrary to the case shown in figures 3(a ) for the single - mode pumping . nevertheless , the most important fact is that the value of the entropy of entanglement @xmath72 can approach unity to a high precision . so , as in the case of the single - mode pumping , the two - mode driven system effectively generates bell states . to find which bell states are generated , we can also transform the resulting wavefunction into the bell basis . thus , figure 7 depicts probabilities for the four bell states . as expected from the form of our analytical solutions for @xmath42 ( @xmath74 ) , the entanglement occurs for the states @xmath28 and @xmath31 , and leads to the generation of the states @xmath90 and @xmath91 ( with some unimportant global phase factor ) . clearly , the highest peaks of the entropy @xmath139 and those of the probabilities @xmath140 of the bell state generation occur at the same evolution times , as seen by comparing figures : 3(a ) with 5(a ) , 4(a ) with 6(a ) , and 4(b ) with 7 . analysis of the evolutions of the probabilities @xmath140 and the entropy of entanglement @xmath139 for relatively longer times reveals that the oscillations are modulated and some long - time oscillations occur in the system , which can be interpreted as a result of quantum beats . it is seen from ( [ n31 ] ) that two various frequencies appear in our solution and one of them is considerably greater than the other , e.g. , the effective coupling constant @xmath141 is @xmath142 times greater than the internal coupling constant of the coupler @xmath101 . we are not presenting here dissipation - free evolutions exhibiting modulated oscillations at times longer than those in figure 4 . it would be meaningless since the entanglement is lost at such evolution times due to dissipation , which inevitably occurs in real physical implementations of the coupler as will be discussed in the next section . although our analysis , including all figures , is focused on evolution of the initial vacuum states , we present shortly some results for other states too . we find that the evolution of the initial fock state @xmath143 with @xmath104 is of the form ( [ n22 ] ) but with the probability amplitudes @xmath107 given by ( [ n30 ] ) and @xmath144 equal to @xmath145 \big\ } , \notag \\ \widetilde{c}_{11}(\tau ) & = & \widetilde{c}_{00}(\tau ) -e^{i\epsilon t}. \label{n32}\end{aligned}\ ] ] with the help of these formulae , we can calculate the entropies of entanglement explicitly as @xmath146 \hspace{0.5 cm } \nonumber \\ + \big[16 \alpha^2 + \epsilon^2 \cos(\lambda t)(-1)^{k - l}i \epsilon \lambda \sin(\lambda t)\big]\big|\big ) \label{n33}\end{aligned}\ ] ] implying the same properties as those given by ( [ n25 ] ) for the single - mode excited system . we point out that both single- and two - mode pumped couplers for @xmath79 and the initial states @xmath30 or @xmath29 evolve into the bell states @xmath99 and @xmath100 but do not evolve into @xmath90 or @xmath91 . this is contrary to the evolutions of the initial vacuum states ( or @xmath31 ) as , for example , presented in figures 5 and 7 . in a more realistic description both fields @xmath11 and @xmath12 lose their photons from the cavities . according to the standard techniques in theoretical quantum optics , dissipation of our system can be modelled by its coupling to reservoirs ( heat baths ) as described by the interaction hamiltonian @xmath147 where @xmath148 are the reservoir operators ; @xmath149 are the boson annihilation operators of the reservoir oscillators coupled with mode @xmath11 or @xmath12 , respectively ; @xmath150 are the coupling constants of the interaction with the reservoirs ; @xmath151 is given either by ( [ n01 ] ) or ( [ n26 ] ) dependent on the system analyzed . we assume two kinds of functions @xmath152 and @xmath153 to describe standard damping and dephasing . the standard description of a damped system is obtained for ( [ n35 ] ) with @xmath154 and @xmath155 , or explicitly @xmath156 corresponding to energy transfer between the system and reservoirs . it should be stressed that the process described by ( [ n37 ] ) leads to combined effect of amplitude and phase damping . the evolution in the markov approximation of the reduced density operator @xmath157 of the two cavity modes after tracing out over the reservoirs can be described in the interaction picture by the following master equation ( see , e.g. , @xcite ) @xmath158 + \hat { \cal l}_{\rm loss}\hat \rho , \label{n38}\end{aligned}\ ] ] where the liouvillian @xmath159+[\hat a,\hat \rho\hat a^\dagger ] )   + \frac{\gamma_b}{2 }   ( [ \hat b\hat \rho,\hat b^\dagger ]   \nonumber \\   \quad + [ \hat b,\hat \rho\hat b^\dagger ] )   + \gamma_a \bar n_a [ [ \hat a,\hat \rho],\hat a^\dagger ]   + \gamma_b \bar n_b [ [ \hat b,\hat \rho],\hat b^\dagger ] \label{n39}\end{aligned}\ ] ] is the usual loss term corresponding to @xmath160 , given by ( [ n37 ] ) ; @xmath161 is the damping rate of the @xmath162th ( @xmath163 ) ( ring ) cavity , and @xmath164^{-1}$ ] is the mean number of thermal photons at the reservoir temperature @xmath165 . we will analyze both  noisyreservoirs ( at @xmath166 implying @xmath167 ) and ` quiet ' reservoirs ( at @xmath168 , so @xmath169 ) . the latter assumption implies that diffusion of fluctuations from the reservoirs into the system modes is negligible . but still this simplified master equation describes the loss of photons from the system modes to the reservoirs . one can raise some doubts @xcite against using liouvillian ( [ n39 ] ) in a description of lossy anharmonic oscillator models given by ( [ n02 ] ) . also the approximation of @xmath170 in ( [ n39 ] ) is problematic . nevertheless , master equation ( [ n38 ] ) with liouvillian ( [ n39 ] ) and hamiltonian @xmath171 set to @xmath21 was used in a number of works both for @xmath166 ( see , e.g. , @xcite ) but also for @xmath170 ( see , e.g. , @xcite ) . moreover , the same master equation , given by ( [ n38 ] ) for @xmath151 set to ( [ n02 ] ) , for coupled anharmonic oscillators was applied in , e.g. , @xcite . liouvillian ( [ n39 ] ) for @xmath170 was also used in ( @xcite p. 210 ) to describe a model essentially similar to ours comprising a nonlinear system , in which two quantized field modes in a cavity interact with a classical pump field . the standard liouvillian for @xmath170 was also used in @xcite to describe a system of kerr nonlinearity , given by @xmath21 , and a parametric amplifier driven by a pulsed classical field . nevertheless , it should be noted that a realistic master equation for kerr medium @xcite is more complicated . phase damping ( also referred to as dephasing ) can be described by ( [ n35 ] ) assuming @xmath172 and @xmath173 , which gives the following loss hamiltonian @xcite : @xmath174 this interaction can be interpreted as a scattering process , where the number of photons remains unchanged contrary to the interaction described by ( [ n37 ] ) . phase damping is essential in a fully quantum picture of dissipation of our system . as a simple generalization of the gardiner - zoller master equation for a single harmonic oscillator ( @xcite , equation ( 6.1.15 ) ) , we describe phase damping of our two - mode nonlinear system by master equation ( [ n38 ] ) for the liouvillian @xmath175 \notag \\ \quad   + \frac{\gamma_b}{2 } ( 2 \bar n_b+1 ) [ 2\hat b^\dag \hat b \hat \rho \hat b^\dag \hat b - ( \hat b^\dag \hat b)^2 \hat \rho-\hat \rho ( \hat b^\dag \hat b)^2 ] , \label{n41}\end{aligned}\ ] ] which is significantly different from ( [ n39 ] ) . to analyze dissipative evolution , governed by master equations ( [ n38 ] ) for liouvillians ( [ n39 ] ) and ( [ n41 ] ) , we apply standard numerical procedures for solving ordinary differential equations with constant coefficients as an exponential series . the entropy of entanglement @xmath72 , given by ( [ n16 ] ) , is valid for qudits of arbitrary dimension in a pure state , but it fails to determine entanglement of a system in a mixed state . thus , for a two - qubit mixed state @xmath176 , we have to apply more general measure , e.g. , the wootters measure of entanglement of formation given by @xcite : @xmath177 where @xmath178 is given by ( [ n18 ] ) with the argument being the concurrence @xmath179 defined as @xmath180 and @xmath181 are the square roots of the eigenvalues of @xmath182 , while @xmath183 is the pauli spin matrix of the @xmath162th qubit ( @xmath163 ) . it is well - known that the entanglement of formation goes into the entropy of entanglement for any two - qubit pure states . examples of evolution of the entanglement of formation and fidelity for the dissipative systems are shown in figures 811 both for quiet and noisy reservoirs . by analyzing the figures , the most important observation is that the generated two - qubit entangled states are very fragile to the leakage of photons from the cavities in the analyzed couplers . our system is more fragile to losses described by the standard master equation , given by ( [ n38 ] ) and ( [ n39 ] ) , rather than the losses due to only phase damping as described by the gardiner - zoller master equation , given by ( [ n38 ] ) and ( [ n41 ] ) . inclusion of reservoir noise , at least for the assumed low mean numbers @xmath184 of thermal photons , does not cause a dramatic deterioration of fidelity and entanglement in comparison to the losses caused by coupling the system to the zero - temperature reservoirs . note that we have chosen @xmath185 in the dephasing model shown in figure 11 and much smaller value of @xmath186 in the standard dissipation model in figure 9 . the reason is that for the latter dissipation model , the number of photons in the system can be increased by absorbing thermal photons from the reservoirs . if this absorption of thermal photons exceeded the loss of the system photons to the reservoirs then the generated fields could not be approximated as two - qubit states and the wootters function @xmath187 , given by ( [ n42 ] ) , would fail to be a good entanglement measure . for the parameters chosen in figure 9 , the generated states are well approximated by two - qubit density matrices and thus its entanglement of formation is well described by ( [ n42 ] ) . by contrast , the number of system photons is not affected by the reservoirs in the dephasing model ; thus ( [ n42 ] ) can be used for any number @xmath186 of thermal photons . finally , it should be noted that fragility of our system to dissipation seems to be a serious drawback from an experimental point of view . nevertheless , a method which enables a significant improvement of the entanglement robustness of the generated states , has recently been suggested for a similar system @xcite . one of the crucial conditions for the successful truncation and generation of the bell states in our scheme concerns the couplings @xmath32 , @xmath120 and @xmath33 to be much smaller than the kerr nonlinearities @xmath3 and @xmath4 . this implies that the kerr interaction should be strong at very low light intensities . thus , it is desirable to discuss an implementation , in which such stringent conditions can experimentally be satisfied . a possible realization can be based on the effect of the electromagnetically - induced transparency ( eit ) or atomic dark resonances as proposed by schmidt and imamolu @xcite ( see also @xcite ) and observed experimentally @xcite . the schmidt - imamolu eit scheme can be realized in a low density system of four - level atoms , which level structure is shown in figure 12 , exhibiting giant resonantly - enhanced kerr nonlinearity at very low intensities . the atoms are placed in cavity @xmath11 ( and analogously in cavity @xmath12 ) tuned to frequency @xmath13 of the mode @xmath11 resonant with the transition @xmath188 and detuned by @xmath189 of the transition @xmath190 . the eit effect is created by a classical pumping field of frequency @xmath18 resonant with the transition @xmath191 ( see @xcite ) , all the atomic levels can adiabatically be eliminated , which results in the following formula for the kerr nonlinearity @xcite : @xmath192 where @xmath193 is third - order nonlinear susceptibility , @xmath194 are the coupling coefficients , @xmath195 is the rabi frequency of the classical driving ( and coupling ) field , @xmath196 is the electric dipole matrix element between the states @xmath197 and @xmath198 , @xmath199 is the total number of atoms contained in the cavity of volume @xmath200 , and @xmath201 is the permittivity of free space . by replacing subscript @xmath11 by @xmath12 in ( [ n44 ] ) , an analogous expression for @xmath4 can be obtained . by putting the stringent limit on the required cavity parameters @xcite , imamolu _ et al . _ estimated @xmath202 rad / sec @xcite . note that some quantum information applications of these giant kerr nonlinearities have already been studied @xcite . more details about application of the schmidt - imamolu scheme of resonantly - enhanced kerr nonlinearity in the analysis of dissipation effects on entanglement generation are given by one of us in @xcite . it is worth stressing the differences between the present paper and our former works . ( i ) in @xcite , only the single - mode pumped systems were studied . here , we analyze also two - mode pumped systems , which is basically a different model . ( ii ) the model described in @xcite is crucially different from the one used here as was based on the two - mode _ nonlinear _ interaction term @xmath203 . in the present work , as well as in @xcite , we apply the two - mode _ linear _ interaction hamiltonian @xmath204 , which is the same ( by neglecting the pump terms @xmath205 and @xmath206 ) as that used in @xcite . the other novelties of the present work in comparison to @xcite can be summarized as follows : ( 1 ) for single - mode pumped system , we found a new generalized solution ( [ n12 ] ) , which , in a special case of equal couplings @xmath19 and @xmath16 , simplifies to our former solution obtained in @xcite . ( 2 ) analytical approximate solutions were found here for various initial fock states @xmath207 . in @xcite , solutions were given for initial vacuum states only . ( 3 ) here , we describe a possible realization of the model based on the effect of the electromagnetically - induced transparency . ( 4 ) in present numerical analysis based on the eit scheme , in comparison to @xcite , more realistic parameters were chosen for the coupling constants , kerr nonlinearities , and damping constants . ( 5 ) the deterioration of the fidelity of the generated bell states due to the standard dissipation and phase damping was analyzed here within two types of master equations both for quiet and noisy reservoirs . by contrast , analysis of losses in @xcite was limited to the standard master equation and for the quiet reservoir only . no effect of dissipation was studied in @xcite . ( 6 ) in the present manuscript , entanglement of formation was calculated analytically for dissipation - free systems and calculated numerically for dissipative systems . no analytical formulae were given in @xcite , while the entanglement of formation was not at all studied in @xcite . ( 7 ) the quality of truncation was described here , but it was not studied in @xcite . discrepancy between the really generated states and the exact truncated states was measured here by the fidelity .    in conclusion , we have described a realization of the generalized two - mode optical - state truncation of two coherent modes via a nonlinear process . our system is a two - mode generalization of the single - mode nonlinear quantum scissors device described in @xcite . we have described an implementation of the kerr nonlinear couplers , where resonantly enhanced nonlinearities can be achieved in the schmidt - imamolu eit scheme . we have compared kerr nonlinear couplers linearly excited in one or two modes by external classical fields . we have shown under assumption of the coupling strengths to be much smaller than the nonlinearity parameters @xmath3 and @xmath4 that the optical states generated by the couplers are the two - qubit truncated states spanned by vacuum and single - photon states . although our approximate solutions of the schrdinger equation are independent of @xmath208-constants , the kerr nonlinearity plays a crucial role in the physics as we have derived from the complete @xmath208-dependent hamiltonian . in fact , the kerr interaction is the mechanism in our model , which enables truncation of the generated state at some energy level . by contrast , the system without the kerr nonlinearities and pumped by an external field would gain more and more energy . to confirm our predictions , we have compared ` exact ' ( accurate up to double - precision ) direct numerical solutions of the schrdinger equation and compared with our approximate analytical solutions . the discrepancies between the exact and approximate solutions are relatively small as shown by fidelities in figures 24 . we have demonstrated that our system initially in vacuum state or single - photon fock states @xmath209 ( @xmath74 ) evolves into bell states . we have discussed the fragility of the entanglement of formation of the generated states due to the standard dissipation and dephasing in the two distinct master equation approaches . we thank professor ryszard tana , professor ryszard horodecki and professor robert alicki for discussions . this work was supported by the polish state committee for scientific research under grant no . 1 p03b 064 28 .
schemes for optical - state truncation of two cavity modes are analyzed . 
 the systems , referred to as the nonlinear quantum scissors devices , comprise two coupled nonlinear oscillators ( kerr nonlinear coupler ) with one or two of them pumped by external classical fields . 
 it is shown that the quantum evolution of the pumped couplers can be closed in a two - qubit hilbert space spanned by vacuum and single - photon states only . 
 thus , the pumped couplers can behave as a two - qubit system . 
 analysis of time evolution of the quantum entanglement shows that bell states can be generated . 
 a possible implementation of the couplers is suggested in a pumped double - ring cavity with resonantly enhanced kerr nonlinearities in an electromagnetically - induced transparency scheme . 
 the fragility of the generated states and their entanglement due to the standard dissipation and phase damping are discussed by numerically solving two types of master equations .