Entanglement in a pair of photons can be manifested in degrees of freedom that are discrete, such as polarization [1], or in degrees of freedom that are continuous, such as spectral or spatial [2], [3], [4], [5]. However, a continuous degree of freedom may be binarized by selection of a pair of values, as in path entanglement [6], [7], or, as in orbital-angular-momentum entanglement [8], by expansion in some basis while retaining only two basis functions [9]. Alternatively, in place of truncation, the space of continuous functions of position may be mapped onto a binary set of even and odd functions, as in spatial-parity entanglement [10], [11], [12]. We recently investigated the possibility of using spontaneous parametric down-conversion (SPDC) in two-mode planar and circular waveguides to generate guided-wave photon pairs entangled in mode number [13]. If the photons are confined in this manner, the spatial variables are naturally binarized and can be used to represent a modal qubit [14].

In this paper, we consider the design of photonic circuits, based on $\hbox{Ti:LiNbO}_{3}$ diffused channel waveguides, that generate photon pairs endowed with various combinations of modal, spectral, and polarization entanglement. The on-chip generation of such photon pairs can serve as a basic resource for one-way quantum computation [15], as well as for linear optical quantum computing [16]. Down-converted photon pairs are generated via SPDC in a two-mode waveguide (TMW) using a CW pump source. The waveguide configuration confines the photons to a single direction of propagation. Any of the available degrees of freedom—mode number, frequency, or polarization—can be used to distinguish the down-converted photons, while the others can serve as carriers of entanglement. We study a class of photonic circuits that comprises: 1) a nonlinear periodically poled TMW structure; 2) a set of single-mode waveguide (SMW)- and TMW-based couplers arranged in such a way that they suitably separate the three photons comprising the SPDC process; and, for certain applications, 3) a holographic Bragg grating.

A good deal of effort has been devoted to examining the properties of single-mode silica-on-silicon waveguide quantum circuits [17], [18], with an eye toward applications in quantum-information processing [19], [20], [21], [22], [23], [24]. For these materials, the photon-generation process necessarily lies off-chip, however. Single- and multi-mode potassium titanyl phosphate ($\hbox{KTiOPO}_{4}$, KTP) waveguide structures have also been extensively studied for producing pulsed SPDC [25], [26], [27], [28], [29], but it appears that only the generation process has been incorporated on-chip. Earlier, periodically poled lithium niobate (PPLN) waveguide structures were suggested for producing SPDC [30], and the conditions required to generate counterpropagating entangled photons from an unguided pump field were established [31]. Furthermore, the generation of non-collinear and non-degenerate polarization-entangled photons via concurrent Type-I parametric down-conversion was demonstrated in a PPLN crystal [32].

The use of lithium niobate photonic circuits has a number of merits: 1) The properties of the material are well-understood since it has low loss and has long been the basis of integrated-optics technology [33], [34, ch. 8]; and 2) the material can easily be periodically poled [35], [36] for the purpose of phase matching parametric interactions. In addition, as we show in this paper, 3) circuit elements such as mode-separation components can readily be designed for TMWs; and 4) the generation, separation, and processing of entangled photons can all be accommodated on a single chip. Moreover, consistency between simulation and experimental measurement has been demonstrated in a whole host of configurations [37], [38], [39], [40], [41]. To enhance tolerance to fabrication errors, photonic circuits can be equipped with electrooptic adjustments. For example, an electrooptically switched coupler with stepped phase-mismatch reversal serves to maximize coupling between fabricated waveguides [42], [43].

The paper is organized as follows. In Section 2, the theory of modal entanglement in TMWs is reviewed. Section 3 is dedicated to examining the operation of a photonic circuit that produces a pair of frequency-degenerate photons at its output. As explained in Section 4, changing the parameters of the elements comprising the circuit considered in Section 3 allows it to produce nondegenerate down-converted photons that are entangled in frequency or simultaneously entangled in frequency and polarization. The second photonic circuit, which is examined in Section 5, distinguishes the photons on the basis of their frequencies and provides means for generating modal entanglement and, with some modifications, modal and polarization double entanglement. The third photonic circuit, which is considered in Section 6, is designed to manage the deleterious effects of dispersion. The conclusion is provided in Section 7.

The theory underpinning the generation of modal entanglement in waveguides was recently described [13]. Entanglement arises from the multiple possibilities for satisfying energy and momentum conservation, as required by the parametric interaction process. In this section, we proceed to apply the theory provided in [13] to diffused channel $\hbox{Ti:LiNbO}_{3}$ waveguides.

Consider a SPDC process in a periodically poled TMW structure, in which a CW pump wave $p$ propagating in the $y$-direction is down-converted into a signal wave $s$ and an idler wave $i$ [34, ch. 21]. The fundamental mode of this TMW $(m_{q} = 0)$ is even while the next-higher mode $(m_{q} = 1)$ is odd, where $q = p, s, i$. We seek to generate an entangled state such that if the signal is in the even mode, then the idler must be in the odd mode, and vice versa.

The biphoton state is written as [13] TeX Source $$\vert\Psi\rangle \sim \int d\omega_{s}\left[\Phi_{0, 1, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert\omega_{s}, 0, \sigma_{s}\rangle\vert\omega_{i}, 1, \sigma_{i}\rangle + \Phi_{1, 0, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert\omega_{s}, 1, \sigma_{s}\rangle\vert\omega_{i}, 0, \sigma_{i}\rangle\right] \eqno{\hbox{(1)}}$$ where the squared-magnitude of $\Phi_{m_{s}, m_{i}, \sigma_{s}, \sigma_{i}}$ represents the SPDC spectrum associated with the $(m_{s}, m_{i}, \sigma_{s}, \sigma_{i})$ component. The signal and idler mode numbers are $m_{s}$ and $m_{i}$, their polarization indexes are $\sigma_{s}$ and $\sigma_{i}$, and their angular frequencies are $\omega_{s}$ and $\omega_{i}$, respectively, with $\omega_{i} = \omega_{p} - \omega_{s}$. The angular frequency of the pump is denoted $\omega_{p}$. For ease of display, the output spectra for the two possibilities represented by the quantities $\vert\Phi_{0, 1, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert^{2}$ and $\vert \Phi_{1,0, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert^{2}$ are normalized to the maximum of their peak values. Moreover, these quantities are plotted only for signal frequencies above the degenerate frequency $(1/2)\omega_{p}$ by virtue of the fact that $\Phi_{0, 1, \sigma_{i}, \sigma_{s}}(-\omega_{s}) = \Phi_{1,0, \sigma_{s}, \sigma_{i}}(\omega_{s})$.

This particular state requires that: 1) the pump mode be odd so that the spatial overlap integral does not vanish; and that 2) the phase matching conditions of the two possibilities at the preselected frequencies $\overline{\omega}_{s}$ and $\overline{\omega}_{i}$ be satisfied, i.e., TeX Source $$\eqalignno{\Delta\beta_{0, 1, \sigma_{s}, \sigma_{i}}(\overline{\omega}_{s}) - 2\pi k/\Lambda^{(k)} =&\, 0\cr \Delta\beta_{1, 0, \sigma_{s}, \sigma_{i}}(\overline{\omega}_{s}) - 2\pi k/\Lambda^{(k)} =&\, 0,& \hbox{(2)}}$$ where $\Delta\beta_{m_{s}, m_{i}, \sigma_{s}, \sigma_{i}} = \beta_{m_{p}, \sigma_{p}}(\omega_{p}) - \beta_{m_{s}, \sigma_{s}}(\overline{\omega}_{s}) - \beta_{m_{i}, \sigma_{i}}(\overline{\omega}_{i});$ $\overline{\omega}_{i} = \omega_{p} - \overline{\omega}_{s}; \quad \beta_{m_{q}, \sigma_{q}}$ is the propagation constant of wave $q$; and $\Lambda^{(k)}$ is the $k$th-order uniform poling period. The efficiency of the interaction decreases by a factor $1/k^{2}$ for the $k$th-order poling period with respect to the first-order one [44]. The choice of the TMW width $w_{1}$ determines the propagation constants of the interacting modes. The values of the waveguide width $w_{1}$ and poling period $\Lambda^{(k)}$ selected should ensure that these conditions are met or approximately met.

The indistinguishability between the down-converted photons is typically ascertained via a Hong–Ou–Mandel (HOM) interferometer [45], [46]. The rate $R(\tau)$ of photon coincidences at a pair of detectors placed at the two output ports of the interferometer is given by [13] TeX Source $$R(\tau) = \int\limits_{-\infty}^{\infty}d\omega_{s}\left\{\left\vert\Phi_{0, 1, \sigma_{s}, \sigma_{i}}(\omega_{s})\right\vert^{2} - \Phi_{0, 1, \sigma_{s}, \sigma_{i}}(\omega_{s})\Phi_{0, 1, \sigma_{s}, \sigma_{i}}^{\ast}(-\omega_{s})\exp\left[j(\omega_{p} - 2\omega_{s})\tau\right]\right\}\eqno{\hbox{(3)}}$$ where $\tau$ is the temporal delay between the down-converted photons, and the asterisk denotes complex conjugation.

All of the simulations presented in this paper refer to structures that make use of $\hbox{Ti:LiNbO}_{3}$ diffused channel waveguides, as illustrated in Fig. 1. These waveguides are fabricated by diffusing a thin film of titanium (Ti), with thickness $\delta = 100\ \hbox{nm}$ and width $w$, into a $z$-cut, $y$-propagating $\hbox{LiNbO}_{3}$ crystal. The diffusion length $D$ is taken to be the same in the two transverse directions: $D = 3\ \mu \hbox{m}$. The TE mode polarized in the $x$-direction sees the ordinary refractive index $n_{o}$, whereas the TM mode polarized in the $z$-direction sees the extraordinary refractive index $n_{e}$.

Fig. 1. Cross-sectional view of the fabrication of a diffused channel $\hbox{Ti:LiNbO}_{3}$ waveguide (not to scale). A thin film of titanium of thickness $\delta = 100\ \hbox{nm}$ and width $w$ is diffused into a $z$-cut, $y$-propagating $\hbox{LiNbO}_{3}$ crystal. The diffusion length $D = 3\ \mu\hbox{m}$.

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The ordinary and extraordinary refractive indexes may be calculated by making use of the Sellmeier equations [34, ch. 5], [47], [48]. The refractive-index increase introduced by Ti indiffusion is characterized by $\Delta n = 2\delta\rho\ {\rm erf}(w/2D)/ \sqrt{\pi}\,D$, where $\rho = 0.47$ and 0.625 for $n_{o}$ and $n_{e}$, respectively [49]. To accommodate wavelength dispersion, $\Delta n$ can be modified by incorporating the weak factor $\xi = 0.052 + 0.065/\lambda^{2}$, where the wavelength $\lambda$ is specified in $\mu \hbox{m}$ [50]. We calculate the effective refractive index $n_{\rm eff}$ of a confined mode in two ways: 1) by using the effective-index method described in [51]; and 2) by making use of the commercial photonic and network design software package RSoft. The propagation constant of a guided mode is related to $n_{\rm eff}$ via $\beta = 2\pi n_{\rm eff}/\lambda$.

In the following sections, we consider several photonic circuits for the generation, separation, and control of down-converted photons. In Section 3, we consider a circuit that generates degenerate photons in separated SMWs. In this case, the two conditions specified in (2) collapse to a single condition since $\overline{\omega}_{s} = \overline{\omega}_{i}$ and $\sigma_{s} = \sigma_{i}$. In Section 4, with a change of parameters, this same circuit is used to produce nondegenerate photons entangled in frequency or simultaneously entangled in frequency and polarization. Section 5 relates to photonic circuitry that produces nondegenerate photons distinguished by their frequencies, which leads to modal as well as modal and polarization double entanglement.

This section is dedicated to examining the operation of a photonic circuit that produces a pair of down-converted photons and then uses their differing spatial properties (mode numbers) to ultimately guide them into two separate SMWs. The output provides a pair of degenerate photons that have even parity and can be fed into an HOM interferometer [45], [46] integrated on the same chip (see Section 6).

In particular, we consider the implementation of degenerate Type-0 $(e, e, e)$ SPDC in a TMW of width $w_{1}$, wherein a pump photon with $m_{p} = 1$ is split into a pair of down-converted photons with different spatial parities. The associated two-photon state is TeX Source $$\vert\Psi\rangle \sim \int d\omega_{s}\left[\Phi_{0, 1, e, e}(\omega_{s})\vert\omega_{s}, e\rangle_{0}\vert\omega_{p} - \omega_{s}, e\rangle_{1} + \Phi_{1, 0, e, e}(\omega_{s}) \vert\omega_{s}, e\rangle_{1}\vert\omega_{p} - \omega_{s}, e\rangle_{0}\right]\eqno{\hbox{(4)}}$$ where the two terms merge for the degenerate case, i.e., when $\omega_{s} = (1/2) \omega_{p}$. Fig. 2(a) represents a photonic circuit that can be used to generate photon pairs of this type. The circuit comprises three sequential stages: a periodically poled region, an odd-mode coupler, and an even-mode coupler. We consider these three stages in turn.

Fig. 2. (a) Sketch of a photonic circuit that can be used to generate two photons in different single-mode waveguides (not to scale). The circuit comprises three sequential stages: a periodically poled region, an odd-mode coupler, and an even-mode coupler. The parameters associated with these stages are defined in panels (b), (c), and (d), respectively. (b) The periodically poled waveguide has width $w_{1}$, length $L_{1}$, and a $k$th-order poling period $\Lambda^{(k)}$. (c) The odd-mode coupler is implemented by bringing a single-mode waveguide (SMW) of width $w_{2}$ and length $L_{2}$ into proximity with a two-mode waveguide (TMW) of width $w_{1}$. The two waveguides are separated by a distance $b_{1}$. An $S$-bend waveguide of initial and final width $w_{2}$ and bending length $L_{b}$ is attached to the end of the SMW. The center-to-center separation between the output of the $S$-bend and the TMW is denoted $S$. All $S$-bends considered in this paper have dimensions $L_{b} = 10\ \hbox{mm}$ and $S = 127\ \mu\hbox{m}$ (the standard spatial separation [41]). (d) The even-mode coupler is implemented by bringing a second TMW of width $w_{1}$ and length $L_{3}$ into proximity with the original TMW. The two waveguides are separated by a distance $b_{2}$. The end of this second TMW is adiabatically tapered over a length $L_{t}$ so that it matches the width $w_{2}$ at the output of the odd-mode coupler. The length of the taper $L_{t} = 1.5\ \hbox{mm}$, as it is throughout this paper.

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The periodically poled stage shown in Fig. 2(b) is characterized by its width $w_{1}$, length $L_{1}$, and $k$th-order poling period $\Lambda^{(k)}$. For certain values of the width, there will always be a single value of the poling period that satisfies (2) so that there is flexibility in choosing the TMW width $w_{1}$. Once the width is assigned, the poling period is determined via (2). There are no restrictions on the length of the nonlinear stage $L_{1}$, save those associated with the process of fabrication. Increasing $L_{1}$ does, of course, increase the flux of photon pairs, but we will see in the next section that there is a limitation on how large $L_{1}$ can be made for generating entangled photons.

The odd-mode coupler is used to extract only the down-converted photon with an odd spatial distribution. Its principle of operation is based on selective coupling between adjacent waveguides of different widths. The even and odd modes of the TMW are characterized by different propagation constants. As depicted in Fig. 2(c), an auxiliary SMW with appropriate width $w_{2}$, length $L_{2}$, and separation distance $b_{1}$ from the TMW can be used to phase-match the odd-mode in the TMW to the even-mode in the SMW. An $S$-bend waveguide in which the initial and final widths are both $w_{2}$ is attached to the end of the SMW; this obviates the possibility of further unwanted coupling to the TMW. In short, the odd-mode coupler distinguishes between the two down-converted photons based on their spatial profiles and delivers the initially odd-mode photon as an even-mode photon at its output.

The even-mode coupler is used to separate the down-converted photon with an even spatial profile from the pump wave. The principle of operation is based on the limited extent of the short-wavelength evanescent pump field, which precludes it from coupling to other waveguides of the same width located at typical distances. The result is that the even-mode down-converted photon alone is coupled into a second identical TMW, of length $L_{3}$ and separation distance $b_{2}$ from the original TMW, as shown in Fig. 2(d). The end of the second TMW is then coupled to an SMW of width $w_{2}$ via an adiabatically tapered waveguide region of length $L_{t}$. The tapering offers two benefits: 1) The even mode of the TMW is transformed into an even mode that matches that of the partner photon at the output of the odd-mode coupler; and 2) unwanted odd modes inadvertently coupled in from external radiation are eliminated. The down-converted degenerate photons produced by this photonic circuit have the same mode number at the output and can therefore interfere quantum mechanically [46]. It is worthy of mention that heralded single-photon pure states in well-defined spatiotemporal modes are required for many quantum information technology applications, such as quantum cryptography [52] and linear optical quantum computing [16].

Fig. 3 provides an example in which the performance of this photonic circuit is simulated. Fig. 3(a) displays the dependencies of the fundamental and first-order mode propagation constants $\beta$ on the waveguide widths $w$. These curves are used to select the widths of the TMW and SMW; the horizontal dotted line represents the loci of the phase-matching conditions between even and odd modes in waveguides of different widths. Fig. 3(b) shows the normalized output spectra $\vert\Phi_{0, 1, e, e}\vert^{2}$ and $\vert\Phi_{1, 0, e, e}\vert^{2}$ as a function of the signal wavelength $\lambda_{s}$. As a consequence of the large phase mismatch between the interacting modes, the value for the first-order poling period turns out to be small, but values in this range have been realized using the method of surface poling in diffused channel waveguides [35]. Using a pump source of longer wavelength, such as 532 nm, would permit regular poling techniques to be used. A salutary feature is that the Type-0 interaction makes use of the strongest nonlinear component of the second-order tensor $d_{33}$ so that high efficiency is expected from this interaction [53]. Figs. 3(c) and (d) display the evolution with distance of the normalized amplitudes of the interacting modes in the odd- and even-mode couplers, respectively. For the odd-mode coupler, the amplitude of the even excited mode in the SMW exhibits a dip that is associated with the tapered nature of the $S$-bend. The values $m = 1$ and 0 in Figs. 3(c) and (d), respectively, refer to the mode numbers at the point of generation. Fig. 3(e) displays the dependencies of the normalized coincidence rate $R$ on the temporal delay $\tau$, without insertion (left curve), and with insertion (right curve), of two narrowband filters at the degenerate frequency before the two output detectors. The reasons underlying the reduced visibility of the dip, and its displacement from $\tau = 0$, are considered in [13, Sec. V].

Fig. 3. Simulated performance of the photonic circuit presented in Fig. 2. A pump at wavelength $\lambda_{p} = 406\ \hbox{nm}$ and in an odd spatial mode $(m_{p} = 1)$ is used to produce degenerate Type-0 $(e, e, e)$ down-conversion with an effective nonlinear optical coefficient $d_{\rm eff} = d_{33}$ [53]. The circuit dimensions are $w_{1} = 4\ \mu\hbox{m}$, $w_{2} = 2.2\ \mu\hbox{m}$, $L_{1} = 20\ \hbox{mm}$, $L_{2} = 0.85\ \hbox{mm}$, $L_{3} = 21\ \hbox{mm}$, and $b_{1} = b_{2} = 5\ \mu\hbox{m}$. The poling period $\Lambda^{(1)} = 2.644\ \mu\hbox{m}$ is designed for operation at a temperature of 80 °C and can be temperature tuned. Values for $L_{b}$, $L_{t}$, and $S$ are specified in the caption of Fig. 2. (a) Dependencies of the propagation constants $\beta$ of the two modes for the down-converted photons on the waveguide widths $w$. The solid curves were obtained using the effective-index method described in [51], whereas the plus signs were computed using the software package RSoft. The dotted vertical lines represent the widths $w_{1}$ and $w_{2}$ that were chosen. (b) Dependence of the normalized output spectra on the wavelength of the signal $\lambda_{s}$. The solid blue and dotted green curves represent $\vert\Phi_{0, 1, e, e}\vert^{2}$ and $\vert\Phi_{1,0, e, e}\vert^{2}$, respectively, for wavelengths below the degenerate value of 812 nm. (c) Performance of the odd-mode coupler. The blue and green curves represent the evolution with distance of the normalized amplitudes of the odd mode in the TMW and the even mode in the SMW, respectively. (d) Performance of the even-mode coupler. The blue, green, and red curves represent the evolution with distance of the normalized amplitudes of the even modes in the original two-mode waveguide TMW(1), in the second two-mode waveguide TMW(2), and in the SMW, respectively. The even mode of TMW(2) is defined as extending to the end of the tapered region, whereas that for the SMW is defined as initiating at the beginning of the tapered region. (e) Dependencies of the normalized coincidence rate $R$ on the temporal delay $\tau$, without insertion (left curve), and with insertion (right curve), of two 10-nm narrowband filters centered about 812 nm, before the two output detectors.

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Changing the values of the parameters provided in Fig. 3 allows the circuit shown in Fig. 2 to produce two nondegenerate down-converted photons that are entangled in frequency, or simultaneously in frequency and polarization, via a Type-0 or Type-II interaction, respectively.

Using the mode numbers to distinguish the photons, (1) takes the form TeX Source $$\vert\Psi\rangle \sim \int d\omega_{s}\left[\Phi_{0, 1, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert\omega_{s}, \sigma_{s}\rangle_{0}\vert\omega_{i}, \sigma_{i}\rangle_{1} + \Phi_{1, 0, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert\omega_{s}, \sigma_{s}\rangle_{1}\vert\omega_{i}, \sigma_{i}\rangle_{0} \right].\eqno{\hbox{(5)}}$$ However, some restrictions apply to the generation and separation portions of the circuit, as discussed in the following paragraphs.

With respect to the generation region, values of the waveguide width $w$ and poling period $\Lambda^{(k)}$ govern how the conditions in (2) are satisfied. Simulations show that: 1) for certain values of the waveguide width, slightly different poling periods are required to satisfy these conditions; and 2) the absolute difference between the quantities $[\beta_{0, \sigma_{s}}(\overline{\omega}_{s}) + \beta_{1, \sigma_{i}}(\overline{\omega}_{i})]$ and $[\beta_{0, \sigma_{i}}(\overline{\omega}_{i}) + \beta_{1, \sigma_{s}}(\overline{\omega}_{s})]$, which appear in (2), is nearly independent of waveguide width. Consequently, the choice of $w_{1}$ can be left to other considerations, such as fabrication limitations and subsequent design requirements.

Our strategy is to satisfy the sum condition, rather than each condition individually, for a specified value of the waveguide width, so that TeX Source $$\Delta\beta_{\rm avg}(\overline{\omega}_{s}) - 2\pi k/\Lambda^{(k)} = 0,\eqno{\hbox{(6)}}$$ where $\Delta\beta_{\rm avg} = (1/2)[\Delta\beta_{0, 1, \sigma_{s}, \sigma_{i}}(\overline{\omega}_{s}) + \Delta\beta_{1, 0, \sigma_{s}, \sigma_{i}}(\overline{\omega}_{s})]$. Substituting the value of the poling period $\Lambda^{(k)}$ obtained in both conditions of (2) leads to small errors in satisfying the phase-matching conditions for the two interactions. These errors can be used to determine the coherence lengths for the two interactions, which then provide upper bounds to the length of the periodically poled region $L_{1}$. The choice of $L_{1}$ must also result in good spectral overlap between $\vert\Phi_{0, 1, \sigma_{s}, \sigma_{i}}(\omega_{s}) \vert^{2}$ and $\vert\Phi_{1, 0, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert^{2}$ around the preselected frequencies, since $L_{1}$ determines the widths of these functions. It turns out that this strategy leads to almost maximally entangled states at the preselected frequencies.

With respect to the coupling region, the design must optimize coupling at both preselected frequencies, $\overline{\omega}_{s}$ and $\overline{\omega}_{i}$. The choice of $w_{1}$ and $w_{2}$ for the odd-mode coupler is based on satisfying the phase matching conditions between an odd mode in a TMW of width $w_{1}$ and an even mode in a SMW of width $w_{2}$, for both $\overline{\omega}_{s}$ and $\overline{\omega}_{i}$: TeX Source $$\eqalignno{\beta_{1, \sigma_{s}}(\overline{\omega}_{s}, w_{1}) =&\, \beta_{0, \sigma_{s}}(\overline{\omega}_{s}, w_{2})\cr \beta_{1, \sigma_{i}}(\overline{\omega}_{i}, w_{1}) =&\, \beta_{0, \sigma_{i}}(\overline{\omega}_{i}, w_{2}).& \hbox{(7)}}$$ Since the down-conversion generation process depends only weakly on waveguide width, we select the TMW width in the generation region to be equal to $w_{1}$; this obviates the need to taper the waveguide from the periodically poled region to the odd-mode coupler. The coupling length is frequency and polarization dependent, however, so that the SMW length $L_{2}$, in combination with the separation distance $b_{1}$, must be carefully chosen to maximize the coupling for both $\overline{\omega}_{s}$ and $\overline{\omega}_{i}$. The same considerations apply for the even-mode coupler so that the second TMW length $L_{3}$ and separation distance $b_{2}$ are selected similarly.

An example, provided in Fig. 4, illustrates the simulated performance of the photonic circuit shown in Fig. 2 to generate spectrally entangled photons. Consider nondegenerate Type-0 $(e, e, e)$ SPDC, in which a pump laser of wavelength 406 nm, in mode $m_{p} = 1$, generates two photons centered about the wavelengths 780 nm and 846.7 nm. The TMW width $w_{1}$ and the SMW width $w_{2}$ are determined via Fig. 4(a), which represents the dependencies of the propagation constants on waveguide width. The values $w_{1}$ and $w_{2}$ are chosen to satisfy (7) as closely as possible. Fig. 4(b) displays the output normalized spectra, $\vert\Phi_{0, 1, e, e}(\omega_{s})\vert^{2}$ and $\vert\Phi_{1,0, e, e}(\omega_{s})\vert^{2}$, for an SPDC poling length $L_{1}$ that was selected to attain maximum entanglement and high efficiency. The performance of the odd- and even-mode couplers are displayed in Figs. 4(c) and (d), respectively; each panel contains plots for the two frequencies $\overline{\omega}_{s}$ and $\overline{\omega}_{i}$. The values $m = 1$ and 0 in Figs. 4(c) and (d), respectively, refer to the mode numbers at the point of generation, and the curves are color-coded in the same way as the corresponding curves in Fig. 3. The values of $L_{2}\ (L_{3})$ and $b_{1}\ (b_{2})$ are selected to optimize coupling of the odd (even) modes for both $\overline{\omega}_{s}$ and $\overline{\omega}_{i}$. Fig. 4(e) displays the behavior of the normalized coincidence rate function $R(\tau)$ for the down-converted photons, without insertion (left curve), and with insertion (right curve), of two 10-nm narrowband filters, centered about 780 and 846.7 nm, before the two output detectors.

Fig. 4. Simulated performance of the photonic circuit portrayed in Fig. 2 when configured to generate spectrally entangled photons. An odd-mode $(m_{p} = 1)$ TM laser pump at $\lambda_{p} = 406\ \hbox{nm}$ is used to generate nondegenerate Type-0 $(e, e, e)$ SPDC with a nonlinear optical coefficient $d_{\rm eff} = d_{33}$. The center wavelengths of the two down-converted photons are 780 nm and 846.7 nm. The circuit dimensions are $w_{1} = 4.2\ \mu\hbox{m}$, $w_{2} = 2.4\ \mu\hbox{m}$, $L_{1} = 2\ \hbox{mm}$, $L_{2} = 0.18\ \hbox{mm}$, $L_{3} = 11\ \hbox{mm}$, and $b_{1} = b_{2} = 4\ \mu\hbox{m}$. The poling period $\Lambda^{(1)} = 2.588\ \mu\hbox{m}$ is designed for operation at a temperature of 80 °C and can be temperature tuned. The panels in this figure are similar to those displayed in Fig. 3, except that panels (a), (c), and (d) each contain separate plots for the two wavelengths. Furthermore, in panel (e), the two filters placed in front of the detectors are centered about 780 nm and 846.7 nm, instead of both being centered about 812 nm, as for degenerate down-conversion.

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An example based on nondegenerate Type-II $(o, e, o)$ SPDC, which is illustrated in Fig. 5, suggests that simultaneous spectral and polarization entanglement can be obtained by again making use of the photonic circuit shown in Fig. 2. By convention, the notation $(\cdot, \cdot, \cdot)$ indicates, in consecutive order, the polarization of the down-converted photon whose frequency lies above the degenerate frequency, the down-converted photon whose frequency lies below the degenerate frequency, and the pump photon. Type-II behavior is obtained by using a TE pump in place of the TM pump employed in Fig. 4 and by choosing the circuit-element parameters, including $\Lambda^{(k)}$, appropriately. The panels displayed in Fig. 5 are similar to those shown in Fig. 4. Again, the values $m = 1$ and 0 in Figs. 5(c) and (d), respectively, refer to the mode numbers at the point of generation, and the curves are color-coded in the same way as the corresponding curves in Figs. 3 and 4. In principle, spectral and polarization double entanglement [13] could also be attained with this arrangement, but the parameters that achieve it are not easily established.

Fig. 5. Simulated performance of the photonic circuit displayed in Fig. 2 when configured to generate photons simultaneously entangled in frequency and polarization. A laser pump at $\lambda_{p} = 406\ \hbox{nm}$, in mode $m_{p} = 1$, is used to produce nondegenerate Type-II $(o, e, o)$ SPDC, with a nonlinear optical coefficient $d_{\rm eff} = d_{31} \ll d_{33}$ [53]. A TE pump is used in place of the TM pump employed in Fig. 4, and $\Lambda^{(k)}$ is chosen appropriately. The center wavelengths of the two down-converted photons are 780 nm and 846.7 nm. The circuit dimensions are $w_{1} = 4.4\ \mu\hbox{m}$, $w_{2} = 2.4\ \mu\hbox{m}$, $L_{1} = 1\ \hbox{mm}$, $L_{2} = 1.95\ \hbox{mm}$, $L_{3} = 11\ \hbox{mm}$, and $b_{1} = b_{2} = 4\ \mu\hbox{m}$. The poling period $\Lambda^{(1)} = 1.869\ \mu\hbox{m}$ is designed for operation at a temperature of 80 °C and can be temperature tuned. All panels are similar to those displayed in Fig. 4 for nondegenerate Type-0 $(e, e, e)$ SPDC.

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This section focuses on the operation of a photonic circuit that generates modal entanglement by distinguishing two nondegenerate photons on the basis of their frequencies, rather than their mode numbers. Modally entangled photons in a TMW have the following property: A photon whose frequency lies above the degenerate frequency and appears in the fundamental (even) mode is always accompanied by another whose frequency lies below the degenerate frequency and appears in the first-order (odd) mode, and vice versa. Alternatively, the photons could be distinguished on the basis of their polarizations in the context of a Type-II configuration. When the photons are distinguished on the basis of their frequencies, the biphoton state (1) is written as TeX Source $$\vert\Psi \rangle \sim \int d\omega_{s}\left[\Phi_{0, 1, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert 0, \sigma_{s}\rangle_{\omega_{s}}\vert 1, \sigma_{i}\rangle_{\omega_{i}} + \Phi_{1, 0, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert 1, \sigma_{s}\rangle_{\omega_{s}}\vert 0, \sigma_{i}\rangle_{\omega_{i}}\right].\eqno{\hbox{(8)}}$$

Modal entanglement is generated by the photonic circuit displayed in Fig. 6. It comprises three stages. The first periodically poled stage is a nonlinear region, as in Fig. 2(b), that produces a pair of nondegenerate down-converted photons via SPDC. As discussed in Section 3, this stage is characterized by its width $w_{1}$, length $L_{1}$, and $k$th-order poling period $\Lambda^{(k)}$. The choice of these parameters is governed by the same considerations as those attendant to the discussion in Section 4.

Fig. 6. (a) Sketch of a photonic circuit that can be used to generate two photons entangled in mode number (not to scale). The circuit comprises three stages: a periodically poled region, a holographic Bragg grating, and two couplers (a low-frequency coupler located to the left and a high-frequency coupler located to the right). The periodically poled waveguide is identical to that portrayed in Fig. 2(b), with width $w_{1}$, length $L_{1}$, and poling period $\Lambda^{(k)}$. The holographic Bragg grating reflects the low-frequency photon and transmits the high-frequency photon. (b) Definitions of the parameters associated with the couplers. As in Fig. 2(c), $L_{b} = 10\ \hbox{mm}$ and $S = 127\ \mu\hbox{m}$.

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The second stage is a holographic Bragg grating that distinguishes the down-converted photons based on their frequencies, such as by reflecting the low-frequency photon while transmitting the high-frequency photon so that they travel in opposite directions. The third stage consists of two couplers that transfer the even and odd down-converted photons from the original waveguide in which they are generated to the output-port waveguides of the circuit, via a low-frequency coupler to the left and a high-frequency coupler to the right.

The holographic Bragg grating can be inscribed on the $\hbox{Ti:LiNbO}_{3}$ waveguide by superposing two coherent plane waves and making use of proton exchange [54] or a dopant such as Cu together with thermal fixing [39]. The sinusoidal light-intensity pattern generated by the superposition of the plane waves [34, ch. 2] is impressed on the waveguide as a sinusoidal modulation of the refractive index. The wavelength of light at which the reflected intensity is maximum (for normal incidence) is $\lambda = 2n_{\rm eff}\ \Lambda_{B}$, where $\Lambda_{B}$ is the grating period, and $n_{\rm eff}$ is the effective refractive index of the guided mode. The bandwidth of the reflected light decreases with increasing grating length. The fundamental and first-order modes can be reflected at a given wavelength by making use of either a single hologram of short length or two multiplexed holograms of longer length. Multiplexing is most effectively achieved by recording the holograms sequentially [39].

The low- and high-frequency couplers are designed to couple the idler-frequency $(\overline{\omega}_{i})$ and signal-frequency $(\overline{\omega}_{s})$ photons, respectively, into separate output TMWs that exclude the pump, as shown in Fig. 6. This is achieved by bringing two auxiliary TMWs into proximity with the principal TMW; all are of the same width $w_{1}$; therefore, the spatial profiles of both the even and odd modes can be maintained in the course of coupling. The pump fails to couple to these auxiliary TMWs because it has a short-range evanescent field by virtue of its high frequency. The lengths of the auxiliary TMWs and their separation distances are designed to optimize coupling of both the even and the odd modes. For the low-frequency (high-frequency) coupler, the auxiliary TMW is of length $L_{2}\ (L_{3})$, and the separation distance is $b_{1}\ (b_{2})$ from the principal waveguide, as illustrated in Fig. 6(b).

A simulation in which the photonic circuit in Fig. 6 is used to generate a pair of photons entangled in mode number is presented in Fig. 7. The nonlinear SPDC source is identical to that displayed in Fig. 4. The Bragg grating consists of two multiplexed holograms, with periods of 83.8 and 83.7 nm, to reflect the fundamental and first-order modes of the low-frequency down-converted photon, respectively. Figs. 7(a) and (b) displays the performance of the low- and high-frequency couplers, respectively. Each panel has two subplots, for the even $(m = 0)$ and odd modes $(m = 1)$. For both frequencies, the coupling length of the first-order mode is substantially shorter than that of the fundamental mode since the peaks of the field for the former lie closer to the transverse edges of the waveguide. Since the coupling length is fixed, multiple cycles of energy transfer of the odd modes (right panels) are required to assure full coupling of the even modes (left panels).

Fig. 7. Simulated performance of the photonic circuit described in Fig. 6 for generating photons entangled in mode number. The Type-0 $(e, e, e)$ SPDC source is identical to that used in Fig. 4; nondegenerate photon pairs with center wavelengths of 846.7 nm and 780 nm are generated using a poling period of $\Lambda^{(1)} = 2.588\ \mu\hbox{m}$. The circuit dimensions are $w_{1} = 4.2\ \mu\hbox{m}$, $L_{1} = 2\ \hbox{mm}$, $L_{2} = 4.1\ \hbox{mm}$, $L_{3} = 8.25\ \hbox{mm}$, and $b_{1} = b_{2} = 4\ \mu\hbox{m}$. The Bragg grating comprises two multiplexed holograms, with periods of 83.8 and 83.7 nm, so that both the even and odd modes associated with the low-frequency photon can be reflected. (a) and (b) Performance of the low-and high-frequency couplers, respectively. Each panel contains separate plots for the fundamental $(m = 0)$ and first-order $(m = 1)$ modes. For the left (right) subplots, the blue and green curves represent the evolution with distance of the normalized even (odd) modes in the original and auxiliary TMWs, respectively.

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A modified version of the photonic circuit displayed in Fig. 6 can also be used to generate photons, via a Type-II interaction, that are doubly entangled in mode number and polarization. The biphoton state in that case is TeX Source $$\displaylines{\vert \Psi\rangle \sim \int d\omega_{s}\left[\Phi_{0, 1, o, e}(\omega_{s})\vert 0, o\rangle_{\omega_{s}}\vert 1, e \rangle_{\omega_{i}} + \Phi_{1, 0, o, e}(\omega_{s})\vert 1, o\rangle_{\omega_{s}}\vert 0, e\rangle_{\omega_{i}} \right.\hfill\cr\hfill\left. +\ \Phi_{0, 1, e, o}(\omega_{s})\vert 0, e\rangle_{\omega_{s}}\vert 1, o\rangle_{\omega_{i}} + \Phi_{1, 0, e, o}(\omega_{s})\vert 1, e\rangle_{\omega_{s}}\vert 0, o\rangle_{\omega_{i}}\right].\quad \hbox{(9)}}$$ This interaction can be viewed as a superposition of Type-II $(o, e, o)$ and Type-II $(e, o, o)$. To generate such doubly entangled photons, the periodically poled nonlinear region portrayed in Fig. 6 would have to be replaced by nonuniform (e.g., linearly chirped) poling, or aperiodic poling [55], to satisfy the phase matching conditions for the four possibilities. Also, the holographic Bragg grating would then have to consist of four multiplexed holograms of long length or two multiplexed holograms of short length. Since the Bragg-grating outputs in both directions could be a photon in the fundamental- or first-order mode, with transverse-electric or transverse-magnetic polarization, the low- and high-frequency couplers would both have to be designed to optimize coupling for all of the possibilities. It is worthy to mention that states entangled in more than one degree of freedom are the primary resource for one-way computation [15].

An example to illustrate the simulated performance of such a photonic circuit for generating photons that are doubly entangled in mode number and polarization is displayed in Fig. 8. As shown in Fig. 8(a), the use of linearly chirped poling leads to broadband spectra. The low-frequency photon would be filtered using the Bragg grating, while the high-frequency photon could be filtered by placing a narrowband filter at the output detector. The generation of narrowband spectra could, in principle, be achieved by making use of aperiodic poling; however, this would require poling periods so small that they might be beyond the capabilities of current fabrication techniques.

Fig. 8. Simulated performance of a modified version of the photonic circuit displayed in Fig. 6 for generating photons that are doubly entangled in mode number and polarization. An $o$-polarized pump at $\lambda_{p} = 406\ \hbox{nm}$, and in mode $m_{p} = 1$, is used to produce nondegenerate photons via Type-II $(o, e, o)$ and $(e, o, o)$ SPDC processes. The preselected center wavelengths of the down-converted photons are again 846.7 nm and 780 nm. The periodically poled region is linearly chirped over a distance $L_{1} = 20\ \hbox{mm}$ with an initial poling period of 1.84 $\mu\hbox{m}$ and a final poling period of 1.87 $\mu\hbox{m}$. The Bragg grating comprises four multiplexed holograms, with periods 83.7, 83.8, 80.8 and 80.9 nm, designed to reflect the low-frequency photons in the backward direction. Other circuit dimensions are $w_{1} = 4.4\ \mu\hbox{m}$, $L_{2} = 4.6\ \hbox{mm}$, $L_{3} = 3.8\ \hbox{mm}$, $b_{1} = 4\ \mu\hbox{m}$, and $b_{2} = 3.5\ \mu\hbox{m}$. (a) Normalized spectra at the output of the nonlinear region versus the signal wavelength $\lambda_{s}$. The solid blue and dashed green curves represent the output spectra associated with Type-II $(o, e, o)$ SPDC, whereas the dash-dotted black and dotted red curves represent the output spectra associated with Type-II $(e, o, o)$ SPDC. (b) and (c) Performance of the low- and high-frequency couplers, respectively. Each of these panels contains four subplots: The top (bottom) subplots represent TE (TM) polarization, while the left (right) subplots represent the fundamental (first-order) mode. The blue and green curves represent the evolution with distance of the normalized modes in the original and auxiliary TMWs, respectively.

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The presence of dispersion is deleterious to the operation of photonic circuits used for many quantum-information applications. Dispersion results from the dependence of the propagation constant $\beta$ on frequency, mode number, and polarization. Polarization-mode dispersion generally provides the strongest contribution, especially in a birefringent material such as $\hbox{LiNbO}_{3}$. However, the Type-0 $(e, e, e)$ interaction, by virtue of its three parallel electric field vectors, suffers only from frequency and modal dispersion, and these are often sufficiently weak that they can be neglected for short devices. This is not the case for Type-II circuits, however, where polarization-mode dispersion is indeed dominant.

In this section, we address the management of dispersion and consider possible ways of mitigating it in photonic circuits, whatever the type of interaction. The optimal approach for circumventing dispersion depends on the particular application.

An example that proves instructive is the photonic-circuit HOM interferometer, in which a simple adjustment of the path length can be effective. The photonic circuit shown in Fig. 2 provides for the generation, separation, and guidance of two SPDC photons into two distinct SMW output ports. An HOM interferometer can be fabricated on the same chip by feeding these outputs into a 3-dB coupler endowed with an electrooptic phase modulator in one of its input arms. This allows a variable phase delay to be introduced between the photons, thereby enabling an interferogram to be generated.

The presence of dispersion alters the coincidence rate $R(\tau)$ associated with the HOM interferometer. The origin of this effect can be understood by casting the interferometer as a four-port system with each arm behaving as an ideal phase filter. Referring to Fig. 2, the matrix ${\bf T}$ that relates the output and input of the system then takes the form TeX Source $${\bf T} = \left[\matrix{H(\omega) & 0\cr 0 & H(\omega)} \right]\left[\matrix{1 & j\cr j & 1}\right]\left[\matrix{H(\omega)H_{o}(\omega) & 0\cr 0 & H(\omega)H_{e}(\omega)}\right]\eqno{\hbox{(10)}}$$ with TeX Source $$\eqalignno{H(\omega) =&\, \exp\left[-j\beta_{1, \sigma}(\omega)\,l\right]\cr H_{o}(\omega) =&\, \exp \left[-j\beta_{1, \sigma}(\omega)\,l_{o}\right]\cr H_{e}(\omega) =&\, \exp\left[-j\beta_{1, \sigma}(\omega) \,L_{t}/2 -j\beta_{0, \sigma}(\omega)(l_{e} + L_{t}/2)\right]\qquad&\hbox{(11)}}$$ where $l$ is the length of an arm of the 3-dB coupler; $l_{e}$ and $l_{o}$ are the distances traveled by the even and odd modes from the generation region to the 3-dB coupler, respectively; $L_{t}$ is the length of the tapered region; and the sandwiched matrix characterizes the lossless beam splitter. The mode in the adiabatically tapered region is assumed to propagate with propagation constant $(1/2)[\beta_{0, \sigma}(\omega) + \beta_{1, \sigma}(\omega)]$.

The presence of dispersion results in a modification of the coincidence rate $R(\tau)$. In particular, the second term of (3) is modified by the quantity TeX Source $$D(\omega_{s}) = T_{11}^{\ast}(\omega_{s})T_{22}^{\ast}(\omega_{i})T_{21}(\omega_{i})T_{12}(\omega_{s})\eqno{\hbox{(12)}}$$ where $\omega_{i} = \omega_{p} - \omega_{s}$, and the $T_{uv}$ are elements of the ${\bf T}$ matrix with $u$ and $v$ representing row and column indexes, respectively. Substituting (10) and (11) into (12) yields TeX Source $$D(\omega_{s}) = \exp\left\{j\left[\beta_{1, \sigma_{s}}(\omega_{s}) - \beta_{1, \sigma_{i}}(\omega_{i})\right][l_{o} - L_{t}/2] - j \left[\beta_{0, \sigma_{s}}(\omega_{s}) - \beta_{0, \sigma_{i}}(\omega_{i})\right][l_{e} + L_{t}/2]\right\}\eqno{\hbox{(13)}}$$ which, together with (2), gives rise to TeX Source $$D(\overline{\omega}_{s}) = \exp\left\{j\left[\beta_{1, \sigma_{s}}(\overline{\omega}_{s}) - \beta_{1, \sigma_{i}}(\overline{\omega}_{i})\right][l_{o} - l_{e} - L_{t}]. \right.\eqno{\hbox{(14)}}$$ It is apparent that choosing $l_{o} = l_{e} + L_{t}$ results in dispersion cancellation at the preselected frequencies. Moreover, it leads to even-order dispersion cancellation for other frequency components as well.

More generally, the deleterious effects of dispersion can be mitigated by mapping every possibility for each available degree of freedom into a different path, resulting in the conversion of modal, spectral, and polarization entanglement into path entanglement.

Consider, as an example, a Type-II $(e, o, o)$ interaction. In this case, frequency and polarization are effectively only a single degree of freedom since the photon with frequency above (below) the degenerate value is always $e(o)$. Thus, two degrees of freedom suffice for characterizing the system: frequency or polarization and mode number. A photonic circuit for converting these two degrees of freedom into photons entangled in path is sketched in Fig. 9. The photons are first separated on the basis of their frequencies and then on the basis of their mode numbers. Each output port thus carries only a particular degree of freedom, thereby skirting the distinguishability engendered by dispersion-induced time delay. The circuit displayed in Fig. 9 can be viewed as a superposition of those illustrated in Figs. 2 and 6.

Fig. 9. Photonic circuit that generates photons entangled in path (not to scale). After generation in the nonlinear region, they are separated first on the basis of their frequencies and then on the basis of their mode numbers.

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The biphoton state entangled in path can be written as TeX Source $$\vert \Psi\rangle \sim \int d\omega_{s}\left[\Phi_{0, 1, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert\omega_{s}, 0, \sigma_{s} \rangle_{1}\vert\omega_{i}, 1, \sigma_{i}\rangle_{3} + \Phi_{1, 0, \sigma_{s}, \sigma_{i}}(\omega_{s})\vert \omega_{s}, 1, \sigma_{s}\rangle_{2}\vert\omega_{i}, 0, \sigma_{i}\rangle_{4}\right]\eqno{\hbox{(15)}}$$ where the subscripts 1, 2, 3, and 4 refer to the output ports labeled in Fig. 9. The photon of frequency $\omega_{i}$ is assumed to be back-reflected by the holographic Bragg grating.

In the case of double entanglement, each output port in a circuit such as that shown in Fig. 9 would deliver photons with either TE- or TM-polarization. To avoid distinguishability as a result of polarization-mode dispersion, two methods could be used: 1) Each output port could be split into a pair of ports, resulting in eight doubly entangled paths; or 2) an electrooptic TE $\rightleftharpoons$ TM mode converter [38] could be used at the path center of each output port, thereby advancing the slow, and retarding the fast, photons in such a way that they would arrive simultaneously at the output detectors.

We have presented several photonic-circuit designs based on $\hbox{Ti:LiNbO}_{3}$ diffused channel waveguides for generating photons with various combinations of modal, spectral, and polarization entanglement. These circuits contain, in various forms, a nonlinear periodically poled TMW structure, SMW- and TMW-based couplers, and multiplexed holographic Bragg gratings. Depending on the choice of parameters, the first photonic circuit produces frequency-degenerate down-converted photons with even spatial parity, nondegenerate down-converted photons entangled in frequency, or nondegenerate down-converted photons simultaneously entangled in frequency and polarization. The second photonic circuit produces modal entanglement or photons that are doubly entangled in mode number and polarization. The third photonic circuit converts modal, spectral, and polarization entanglement into path entanglement to mitigate the effects of dispersion. Simulations have been carried out, with the help of the commercial photonic and network design software package RSoft, to gauge the performance of these circuits. Furthermore, since $\hbox{Ti:LiNbO}_{3}$ photonic circuits have the salutary property of permitting the generation, transmission, and processing of photons to be accommodated on a single chip, we expect that they will find use in the efficient implementation of various quantum information processing schemes [14].