2 One can consider the introduction of nonlocal potentials as a regularization of the classical theory (1) by smearing out the point interaction. In the limit of sharp potentials, V$_{i}$ ( z ) = V$_{i}$ δ ( z ), the regularized model reduces to Eq. (1) for XPM coefficient γ$_{i}$ = V$_{i}$ / 2. iments for quantum non-demolition (QND) measurement of photon numbers and CPG. This paper is organized as follows. In Sec. II we will present a phenomenological quantum model for XPM be- tween two pulses with different group velocities and dis- cuss the two cases of unitary and non-unitary dynamics in this model. In Sec. III we present proposals for QND measurement of photon numbers and CPG and show that they can have a high fidelity in this case. The analysis of these proposals for the non-unitary case is presented in Sec. IV. Several appendices contain details of our deriva- tion. Multi-photon pulse dynamics should account for SPM, but we omit this effect here in order to focus on XPM. When XPM is placed in the context of interferometry, which gives phase an operational meaning, pairs of non- linear media in both paths can offset SPM [36]. Alter- natively SPM is not present for single-photon pulses be- cause a single photon cannot induce SPM on itself. The multi-mode field operators E$_{i}$ ( z ) satisfy the com- mutation relations with ω$_{i}$ the central frequency of pulse i and A the trans- verse area of the pulses. The field is assumed to be in a single mode with the i th longitudinal mode function given by ψ$_{i}$ ( z ). Photons in this mode are annihilated by the operator We consider two quantized light pulses that travel through an XPM medium with different group velocities v$_{1}$ > v$_{2}$ . We assume that the light pulses are transver- sally confined so that we can restrict the model to one spatial dimension. In classical fiber optics this situation is commonly described by the set of equations [32–34] For later use we have included a possible shift u of the wavefunction. If the shift is zero we will sometimes sup- press this notation and write a$_{i}$ ≡ a$_{i}$ [0]. If Eq. (2) is taken to be the dynamical equation of the quantum fields E$_{i}$ one has to be careful with its interpre- tation. It can be shown that Eq. (2) can be expressed as ∂$_{t}$ E$_{i}$ = − i[ E$_{i}$ , H ] for some Hamiltonian H if and only if v$_{1}$V$_{1}$ ( z ) = v$_{2}$V$_{2}$ ( z ). In other words, a unitary evolution of the quantum fields will only occur if the two interac- tion potentials are proportional to each other, with the proportionality factor given by the ratio of the group ve- locities. In absence of absorption or other decohering processes one therefore would assume that in the quan- tum model this relation must be fulfilled. However, in the corresponding classical models for XPM pulse prop- agation this is generally not the case [32–34]. with imaginary unit i = √ − 1. Here E$_{i}$ is the field am- plitude of pulse i = 1 , 2, I$_{i}$ = |E$_{i}$| 2 is proportional to the intensity, and γ$_{i}$ is the self phase modulation (SPM) coef- ficient. The XPM coefficient is given by 2 γ$_{i}$ . This model is local and instantaneous; i.e., the light field E$_{1}$ ( z, t ) is only affected by E$_{2}$ ( z, t ) and vice versa. In the context of SPM, Joneckis and Shapiro have shown that in a local and instantaneous quantum model, where E$_{i}$ represents a field operator, infinite zero-point fluctuations do occur [35]. They suggested taking a fi- nite response time of the medium into account to avoid singularities. We instead propose a model in which the medium’s response is still instantaneous but spatially nonlocal, i.e., E$_{1}$ ( z, t ) can be affected by E$_{2}$ ( z $^{′}$, t ). The most general macroscopic model that describes such an instantaneous interaction between two light pulses of dif- ferent group velocities is then given by In our subsequent analysis we will therefore first as- sume that the evolution is unitary and analyze our pro- posals within this framework. In Sec. IV we then will develop a consistent phenomenological quantum model for non-unitary evolution using techniques of open quan- tum systems and reassess the proposals for this case [37]. II. PULSE PROPAGATION WITH MISMATCHED GROUP VELOCITIES [ E$_{i}$ ( z, t ) , E † $_{j}$( z $^{′}$, t )] = $_{ℏ}$ω$_{i}$ 2 ϵ$_{0}$A δ$_{ij}$δ ( z − z $^{′}$) ≡ ηδ$_{ij}$δ ( z − z $^{′}$) , (3) a$_{i}$ [ u ] = 1 √ η ∫ ∞ −∞ dz ψ ∗ $_{i}$( z − u ) ˆ E$_{i}$ ( z, 0) , (4) ( 1 v$_{i}$ ∂$_{t}$ + ∂$_{z}$$^{)}$ E$_{i}$ ( z, t ) = − i γ$_{i}$ ( I$_{i}$ ( z, t ) + 2 I$_{3}$$_{−}$$_{i}$ ( z, t )) E$_{i}$ ( z, t ) , (1) ( 1 v$_{i}$ ∂$_{t}$ + ∂$_{z}$$^{)}$ E$_{i}$ ( z, t ) = − i$^{∫}$ ∞ −∞ dz $^{′}$I$_{3}$$_{−}$$_{i}$ ( z $^{′}$, t ) × V$_{i}$ ( z − z $^{′}$) E$_{i}$ ( z, t ) . (2) To devise a scheme for CPG and QND measurements of the photon number we will assume that the dynamics is unitary; in this case the interaction potentials can gen- erally be written as V$_{i}$ ( z ) = V ( z ) /v$_{i}$ , with V ( z ) a Hamil- tonian interaction potential. However, we will make use Here and henceforth E$_{i}$ denotes the field operator and I$_{i}$ ( z, t ) = E † $_{i}$( z $^{′}$, t ) E$_{i}$ ( z $^{′}$, t ) the intensity operator. The quantities V$_{i}$ ( z − z $^{′}$) are spatially nonlocal interaction po- tentials between the two pulses generated by the atomic medium. To avoid causality violation its support must be much smaller than the wavelength of light. We will assume that V$_{i}$ ( z − z $^{′}$) is essentially zero if the distance between z and z ′ is much larger than the Bohr radius. III. XPM FOR UNITARY EVOLUTION