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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 20475–20490
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Modal and polarization qubits in Ti:LiNbO3 photonic circuits for a universal quantum logic gate

Mohammed F. Saleh, Giovanni Di Giuseppe, Bahaa E. A. Saleh, and Malvin Carl Teich  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 20475-20490 (2010)
http://dx.doi.org/10.1364/OE.18.020475


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Abstract

Lithium niobate photonic circuits have the salutary property of permitting the generation, transmission, and processing of photons to be accommodated on a single chip. Compact photonic circuits such as these, with multiple components integrated on a single chip, are crucial for efficiently implementing quantum information processing schemes. We present a set of basic transformations that are useful for manipulating modal qubits in Ti:LiNbO3 photonic quantum circuits. These include the mode analyzer, a device that separates the even and odd components of a state into two separate spatial paths; the mode rotator, which rotates the state by an angle in mode space; and modal Pauli spin operators that effect related operations. We also describe the design of a deterministic, two-qubit, single-photon, CNOT gate, a key element in certain sets of universal quantum logic gates. It is implemented as a Ti:LiNbO3 photonic quantum circuit in which the polarization and mode number of a single photon serve as the control and target qubits, respectively. It is shown that the effects of dispersion in the CNOT circuit can be mitigated by augmenting it with an additional path. The performance of all of these components are confirmed by numerical simulations. The implementation of these transformations relies on selective and controllable power coupling among single- and two-mode waveguides, as well as the polarization sensitivity of the Pockels coefficients in LiNbO3.

© 2010 Optical Society of America

1. Introduction

We recently investigated the possibility of using spontaneous parametric down-conversion (SPDC) in two-mode waveguides to generate guided-wave photon pairs entangled in mode number, using a cw pump source. If one photon is generated in the fundamental (even) mode, the other will be in the first-order (odd) mode, and vice versa [1

1. M. F. Saleh, B. E. A. Saleh, and M. C. Teich, “Modal, spectral, and polarization entanglement in guided-wave parametric down-conversion,” Phys. Rev. A 79, 053 842 (2009). [CrossRef]

]. We also considered a number of detailed photonic-circuit designs that make use of Ti:LiNbO3 diffused channel, two-mode waveguides for generating and separating photons with various combinations of modal, spectral, and polarization entanglement [2

2. M. F. Saleh, G. Di Giuseppe, B. E. A. Saleh, and M. C. Teich, “Photonic circuits for generating modal, spectral, and polarization entanglement,” IEEE Photon. J. 2, 736–752 (2010). [CrossRef]

]. Selective mode coupling between combinations of adjacent single-mode and two-mode waveguides is a key feature of these circuits.

Although potassium titanyl phosphate (KTiOPO4, KTP) single- and multi-mode waveguide structures have also been used for producing spontaneous parametric down-conversion [3–7

3. M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals,” Opt. Express 15, 7479–7488 (2007). [CrossRef] [PubMed]

], it appears that only the generation process, which makes use of a pulsed pump source, has been incorporated on-chip. Substantial advances have also recently been made in the development of single-mode silica-on-silicon waveguide quantum circuits [8

8. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008). [CrossRef] [PubMed]

, 9

9. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photon. 3, 346–350 (2009). [CrossRef]

], with an eye toward quantum information processing applications [10–15

10. C. H. Bennett and P. W. Shor, “Quantum information theory,” IEEE Trans. Inform. Theory 44, 2724–2742 (1998). [CrossRef]

]. For these materials, however, the photon-generation process necessarily lies off-chip.

Lithium niobate photonic circuits have the distinct advantage that they permit the generation, transmission, and processing of photons all to be achieved on a single chip [2

2. M. F. Saleh, G. Di Giuseppe, B. E. A. Saleh, and M. C. Teich, “Photonic circuits for generating modal, spectral, and polarization entanglement,” IEEE Photon. J. 2, 736–752 (2010). [CrossRef]

]. Moreover, lithium niobate offers a number of ancillary advantages: 1) its properties are well-understood since it is the basis of integrated-optics technology [16

16. H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw—Hill, New York, 1989).

]; 2) circuit elements, such as two-mode waveguides and polarization-sensitive mode-separation structures, have low loss [2

2. M. F. Saleh, G. Di Giuseppe, B. E. A. Saleh, and M. C. Teich, “Photonic circuits for generating modal, spectral, and polarization entanglement,” IEEE Photon. J. 2, 736–752 (2010). [CrossRef]

]; 3) it exhibits an electro-optic effect that can modify the refractive index at rates up to tens of GHz and is polarization-sensitive [17

17. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, Hoboken, NJ, 2007).

, Sec. 20.1D]; and 4) periodic poling of the second-order nonlinear optical coefficient is straightforward so that phase-matched parametric interactions [18

18. A. C. Busacca, C. L. Sones, R. W. Eason, and S. Mailis, “First-order quasi-phase-matched blue light generation in surface-poled Ti:indiffused lithium niobate waveguides,” Appl. Phys. Lett. 84, 4430–4432 (2004). [CrossRef]

, 19

19. Y. L. Lee, C. Jung, Y.-C. Noh, M. Park, C. Byeon, D.-K. Ko, and J. Lee, “Channel-selective wavelength conversion and tuning in periodically poled Ti:LiNbO3 waveguides,” Opt. Express 12, 2649–2655 (2004). [CrossRef] [PubMed]

], such as SPDC and the generation of entangled-photon pairs [20

20. S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26–28 (2001). [CrossRef]

, 21

21. H. Guillet de Chatellus, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, and G. Di Giuseppe, “Non-collinear and non-degenerate polarization-entangled photon generation via concurrent type-I parametric downconversion in PPLN,” Opt. Express 14, 10 060–10 072 (2006).

], can be readily achieved. Moreover, consistency between simulation and experimental measurement has been demonstrated in a whole host of configurations [22–26

22. R. C. Alferness and R. V. Schmidt, “Tunable optical waveguide directional coupler filter,” Appl. Phys. Lett. 33, 161–163 (1978). [CrossRef]

]. To enhance tolerance to fabrication errors, photonic circuits can be equipped with electro-optic adjustments. For example, an electro-optically switched coupler with stepped phase-mismatch reversal serves to maximize coupling between fabricated waveguides [27

27. R. V. Schmidt and H. Kogelnik, “Electro-optically switched coupler with stepped Δβ reversal using Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 28, 503–506 (1976). [CrossRef]

, 28

28. H. Kogelnik and R. V. Schmidt, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396–401 (1976). [CrossRef]

].

Compact photonic circuits with multiple components integrated on a single chip, such as the ones considered here, are likely to be highly important for the efficient implementation of devices in the domain of quantum information science. The Controlled-NOT (CNOT) gate is one such device. It plays an important role in quantum information processing, in no small part because it is a key element in certain sets of universal quantum logic gates (such as CNOT plus rotation) that enable all operations possible on a quantum computer to be executed [11

11. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000).

,15

15. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45–53 (2010). [CrossRef] [PubMed]

,29

29. D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51, 1015–1022 (1995). [CrossRef] [PubMed]

,30

30. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). [CrossRef] [PubMed]

]. Two qubits are involved in its operation: a control and a target. The CNOT gate functions by flipping the target qubit if and only if the control qubit is in a particular state of the computational basis. Two separate photons, or, alternatively, two different degrees-of-freedom of the same photon, may be used for these two qubits. A deterministic, two-qubit, single-photon, CNOT gate was demonstrated using bulk optics in 2004 [31

31. M. Fiorentino and F. N. C. Wong, “Deterministic controlled-NOT gate for single-photon two-qubit quantum logic,” Phys. Rev. Lett. 93, 070 502 (2004). [CrossRef]

]. More recently, a probabilistic, two-photon, version of the CNOT gate was implemented as a silica-on-silicon photonic quantum circuit; an external bulk-optics source of polarization qubits was required, however [8

8. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008). [CrossRef] [PubMed]

]. It is worthy of mention that qubit decoherence is likely to be minimal in photonic quantum circuits; however, decoherence resulting from loss in long waveguides can be mitigated by the use of either a qubit amplifier [32

32. N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070 501 (2010). [CrossRef]

] or teleportation and error-correcting techniques [33

33. S. Glancy, H. M. Vasconcelos, and T. C. Ralph, “Transmission of optical coherent-state qubits,” Phys. Rev. A 70, 022 317 (2004). [CrossRef]

].

2. Diffused channel Ti:LiNbO waveguides

All of the simulations presented in this paper refer to structures that make use of Ti:LiNbO3 diffused channel waveguides, as illustrated in Fig. 1. These waveguides are fabricated by diffusing a thin film of titanium (Ti), with thickness δ ≈ 100 nm and width w, into a z-cut, y-propagating LiNbO3 crystal. The diffusion length D is taken to be the same in the two transverse directions: D = 3 µm. The TE mode polarized in the x-direction sees the ordinary refractive index no, whereas the TM mode polarized in the z-direction (along the optic axis) sees the extraordinary refractive index ne.

Fig. 1. Cross-sectional view of the fabrication of a diffused channel Ti:LiNbO3 waveguide (not to scale). A thin film of titanium of thickness δ ≈ 100 nm and width w is diffused into a z-cut, y-propagating LiNbO3 crystal. The diffusion length D = 3µm.

Applying a steady electric field to this structure in the z-direction (along the optic axis) changes the ordinary and extraordinary refractive indices of this uniaxial (trigonal 3m) material by 12no3r13Vd and 12ne3r33Vd , respectively [17

17. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, Hoboken, NJ, 2007).

, Example 20.2-1], where V is the applied voltage; d is the separation between the electrodes; and r 13 and r 33 are the tensor elements of the Pockels coefficient, which have values 10.9 and 32.6 pm/V, respectively [35

35. K. K. Wong, ed., Properties of Lithium Niobate (Institution of Electrical Engineers, Stevenage, U.K., 2002).

].

3. Modal qubits

4. Mode coupling between adjacent waveguides

The coupling between two lossless, single-mode waveguides is described by a unitary matrix T that takes the form [17

17. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, Hoboken, NJ, 2007).

, Sec. 8.5B]

T=[AjBjB*A*],
(1)

where A = exp(jΔβ L/2) [cos γLjβ/2γ) sin γL] and B = (κ/γ) exp(jΔβ L/2) sin γL. Here, Δβ is the phase mismatch per unit length between the two coupled modes; L is the coupling interaction length; κ is the coupling coefficient, which depends on the widths of the waveguides and their separation as well as on the mode profiles; γ2=κ2+14β2 ; and the symbol * represents complex conjugation.

This unitary matrix T can equivalently be written in polar notation as [53

53. L. L. Buhl and R. C. Alferness, “Ti:LiNbO3 waveguide electro-optic beam combiner,” Opt. Lett. 12, 778–780 (1987). [CrossRef] [PubMed]

]

T=[cos(θ2)exp(jϕA)jsin(θ2)exp(jϕB)jsin(θ2)exp(jϕB)cos(θ2)exp(jϕA)],
(2)

where θ = 2sin−1 [(κ/γ)sin γL]; ϕA = ϕB + tan−1 [(−Δβ/2γ) tan γL]; and ϕB = ΔβL/2. Using this representation, the coupling between the two waveguides can be regarded as a cascade of three processes: 1) phase retardation, 2) rotation, and 3) phase retardation. This becomes apparent if Eq. (2) is rewritten as

T=exp(jϕB)T3T2T1,
(3)

with

T1=[100ejΓ1];T2=[cos(θ2)jsin(θ2)jsin(θ2)cos(θ2)];T3=[ejΓ2001],
(4)

where Γ1 = ϕAϕB; Γ2 = −ϕAϕB; and T 1, T 2, and T 3 represent, in consecutive order, phase retardation, rotation, and phase retardation. The phase shift ϕB is a constant of no consequence.

For perfect phase matching between the coupled modes, i.e., for Δβ = 0 and an interaction coupling length L = /2κ, where q is an odd positive integer, the coupling matrix T reduces to

T=exp(jqπ2)[0110],
(5)

indicating that the modes are flipped. Applying this operation twice serves to double flip the vector, thereby reproducing the input, but with a phase shift twice that of /2. On the other hand, for γL = , with p an integer, the matrix becomes

T=(1)p[exp(jϕA)00exp(jϕA)].
(6)

Finally, for weak coupling (κ ≈ 0 or κ ≪ Δβ), we have ϕA ≈ 0, whereupon T reduces to the identity matrix.

Our interest is in three scenarios: 1) coupling between a pair of single-mode waveguides (SMWs); 2) coupling between a pair of two-mode waveguides (TMWs); and 3) coupling between a SMW and a TMW. The matrix described in Eq. (2) is not adequate for describing the coupling in the latter two cases; in general, a 4 × 4 matrix is clearly required for describing the coupling between two TMWs. However, for the particular cases of interest here, the coupling between the two waveguides is such that only a single mode in each waveguide participates; this is because the phase-matching conditions between the interacting modes are either satisfied — or not satisfied. As an example for identical waveguides, similar modes couple whereas dissimilar modes fail to couple as a result of the large phase mismatch. The net result is that, for the cases at hand, the general matrix described in Eq. (2) reduces to submatrices of size 2 × 2, each characterizing the coupling between a pair of modes.

5. Mode analyzer and modal Pauli spin operator

A mode analyzer is a device that separates the even and odd components of an incoming state into two separate spatial paths. It is similar to the parity analyzer of one-photon parity space [40

40. A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052 114 (2007). [CrossRef]

]. For the problem at hand, its operating principle is based on the selective coupling between adjacent waveguides of different widths. The even and odd modes of a TMW of width w 1 are characterized by different propagation constants. An auxiliary SMW (with appropriate width w 2, length L 2, and separation distance b 1 from the TMW) can be used to extract only the odd component [2

2. M. F. Saleh, G. Di Giuseppe, B. E. A. Saleh, and M. C. Teich, “Photonic circuits for generating modal, spectral, and polarization entanglement,” IEEE Photon. J. 2, 736–752 (2010). [CrossRef]

]. The result is a mode analyzer that separates the components of the incoming state, delivering the the odd mode as an even distribution, as shown in Fig. 2(a). The end of the SMW is attached to an S-bend waveguide, with initial and final widths w 2, to obviate the possibility of further unwanted coupling to the TMW and to provide a well-separated output port for the extracted mode. If it is desired that the output be delivered as an odd distribution instead, another SMW to TMW coupling region (with the same parameters) may be arranged at the output end of the S-bend, as illustrated in Fig. 2(b). This allows the propagating even mode in the SMW to couple to the odd mode of the second TMW, thereby delivering an odd distribution at the output. The appropriate coupler configuration is determined by the application at hand. It is important to note that the mode analyzer is a bidirectional device: it can be regarded as a mode combiner when operated in the reverse direction, as we will soon see.

Fig. 2. (a) Sketch of a photonic circuit that serves as a mode analyzer (not to scale). It 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 Lb, 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 Lb = 10 mm and S = 127µm (the standard spatial separation [26]). The odd mode is separated and delivered as an even distribution. (b) Sketch of a mode analyzer (not to scale) that separates the odd mode and delivers it as an odd distribution. It is more complex than the design presented in (a) because it incorporates a second TMW, again of width w 1, that is brought into proximity with a SMW of width w 2 and length L 2 placed at the output of the S-bend. These two waveguides are again separated by a distance b 1. (c) Sketch of a photonic circuit (not to scale) that changes the sign of the odd mode while leaving the even mode intact, thereby implementing the modal Pauli spin operator σz. An electro-optic phase modulator is used to compensate for any unintended differences in the phase delays encountered by the even and odd modes as they transit the circuit.

An example illustrating the operation of a mode analyzer, such as that shown in Fig. 2(a), is provided in Fig. 3. The behavior of the normalized propagation constants β of the even (m = 0) and odd (m = 1) modes before Ti indiffusion, as a function of the waveguide width w, is presented in Fig. 3(a) for TM polarization at a wavelength of λ = 0.812µm. The horizontal dotted line crossing the two curves represents the phase-matching condition for an even and an odd mode in two waveguides of different widths. The simulation presented in Fig. 3(b) displays the evolution of the normalized amplitudes of the two interacting modes with distance.

Fig. 3. (a) Dependencies of the normalized propagation constants β of the fundamental (m = 0) and first-order (m = 1) modes on the widths w of the diffused channel Ti:LiNbO3 waveguides. The input wave has wavelength λ = 0.812µm and TM polarization. The solid curves were obtained using the effective-index method described in [38], whereas the plus signs were computed using the software package RSoft. The dotted vertical lines represent the desired widths w 1 and w 2. (b) Simulated performance of a mode analyzer that takes the form displayed in Fig. 2(a). The blue curve represents the evolution with distance of the normalized amplitude of the odd mode in a TMW of width w 1 = 5.6µm, whereas the green curve shows the evolution of the even mode in a SMW of dimensions w 2 = 3.4µm and L 2 = 6.2 mm. The separation between the TMW and the SMW is b 1 = 4µm and the S-bend has dimensions Lb = 10 mm and S = 127µm. The dip in the curve for the SMW is associated with the tapered nature of the S-bend. The results were obtained with the help of the software package RSoft.

6. Mode rotator and modal Pauli spin operator σx

The mode rotator is an operator that rotates the state by an angle θ in mode space, just as a polarization rotator rotates the polarization state. It is also analogous to the parity rotator of one-photon spatial-parity space [40

40. A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052 114 (2007). [CrossRef]

]. It achieves rotation by cascading a mode analyzer, a directional coupler, and a mode combiner; the three devices are regulated by separate EO phase modulators to which external voltages are applied. The mode analyzer splits the incoming one-photon state into its even and odd projections; the directional coupler mixes them; and the mode combiner recombines them into a single output.

Implementation of the mode rotator is simplified by making use of the factorization property of the unitary matrix T that characterizes mode coupling in two adjacent waveguides (see Sec. 4). As shown in Eqs. (3) and (4), the coupling between two lossless waveguides can be regarded as a cascade of three stages: phase retardation, rotation, and phase retardation. If the phase-retardation components were eliminated, only pure rotation, characterized by the SO(2) operator, would remain.

The phase-retardation components can indeed be compensated by making use of a pair of EO phase modulators to introduce phase shifts of Γ1 and Γ2, before and after the EO directional coupler, respectively. These simple U(1) transformations convert T 1 and T 3 in Eq. (4) into identity matrices, whereupon Eq. (3) becomes the SO(2) rotation operator. For a mode of wavelength λ, and an EO phase modulator of length L and distance d between the electrodes, the voltage required to introduce a phase shift of Γ is V = λdΓ/πrn 3 L, where the Pockels co-efficient r assumes the values r 13 and r 33, for n = no and n = ne, respectively [17

17. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, Hoboken, NJ, 2007).

, Sec. 20.1B].

Fig. 4. Sketch of a photonic circuit that serves as a mode rotator (not to scale). It is implemented by sandwiching a directional coupler between a mode analyzer and a mode combiner. The coupling length of the directional coupler is π/2κ. To obtain a specified angle of rotation θ, voltages V 1, V 2, and V 3 are applied to the EO directional coupler, the input EO phase modulator, and the output EO phase modulator, respectively.

An example showing the operating voltages V 1,V 2, and V 3 required to obtain a specified angle of rotation θ is provided in Fig. 5. The directional-coupler voltage V 1 has an initial value (for θ = 0) that corresponds to a phase mismatch ∣ΔβL∣ = √3π; decreasing V 1 results in increasing θ. When V 1 = 0, the angle of rotation is π; the device then acts as the Pauli spin operator σx, which is a mode flipper (analogous to the parity flipper [40

40. A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052 114 (2007). [CrossRef]

,41

41. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and analysis of entangled photonic qubits in spatial-parity space,” Phys. Rev. Lett. 99, 250 502 (2007). [CrossRef]

]). For V 1 = 0, there are an infinite number of solutions for the values of V 2 and V 3, provided, however, that V 2 = −V 3.

7. Controlled-NOT (CNOT) gate

Deterministic quantum computation that involves several degrees-of-freedom of a single photon for encoding multiple qubits is not scalable inasmuch as it requires resources that grow exponentially [31

31. M. Fiorentino and F. N. C. Wong, “Deterministic controlled-NOT gate for single-photon two-qubit quantum logic,” Phys. Rev. Lett. 93, 070 502 (2004). [CrossRef]

]. Nevertheless, few-qubit quantum processing can be implemented by exploiting multiple-qubit encoding on single photons [54

54. Y. Mitsumori, J. A. Vaccaro, S. M. Barnett, E. Andersson, A. Hasegawa, M. Takeoka, and M. Sasaki, “Experimental demonstration of quantum source coding,” Phys. Rev. Lett. 91, 217 902 (2003). [CrossRef]

]. We propose a novel deterministic, two-qubit, single-photon, CNOT gate, implemented as a Ti:LiNbO3 photonic quantum circuit, in which the polarization and mode number of a single photon serve as the control and target qubits, respectively.

Fig. 5. Operating voltages for the mode rotator vs. the angle of rotation θ. Voltages V 1 (solid blue curve), V 2 (dashed green curve), and V 3 (dashed-dotted red curve) are applied to the EO directional coupler, the input EO phase modulator, and the output EO phase modulator, respectively. The input has wavelength λ = 0.812 µm and TM polarization. The directional coupler comprises two identical SMWs separated by d = 5µm; each SMW has width 2.2µm and length 1.73 mm. The input and output EO phase modulators have electrode lengths of 5 mm and electrode separations of 5µm. The curves represent theoretical calculations while the symbols represent simulated data obtained using the RSoft program.
Fig. 6. Sketch of a Ti:LiNbO3 photonic quantum circuit that behaves as a novel deterministic, two-qubit, single-photon, CNOT gate (not to scale). The control qubit is polarization and the target qubit is mode number. The circuit bears some similarity to the mode rotator shown in Fig. 4; both are implemented by sandwiching an EO directional coupler between a mode analyzer and a mode combiner. However, for the CNOT gate, the EO directional coupler comprises a pair of TMWs, whereas the mode rotator uses a conventional EO directional coupler utilizing a pair of SMWs.

The operation of this gate is implemented via a polarization-sensitive, two-mode, electrooptic directional coupler, comprising a pair of identical TMWs integrated with an electro-optic phase modulator, and sandwiched between a mode analyzer and a mode combiner. It relies on the polarization sensitivity of the Pockels coefficients in LiNbO3. A sketch of the circuit is provided in Fig. 6. The mode analyzer spatially separates the even and odd components of the state for a TM-polarized photon, sending the even component to one of the TMWs and the odd component to the other. At a certain value of the EO phase-modulator voltage, as explained below, the even and odd modes can exchange power. The modified even and odd components are then brought together by the mode combiner.

Fig. 7. Dependencies of the normalized propagation constants β on the voltage applied to an EO TMW directional coupler comprising two waveguides [WG1 and WG2]. The propagation constants differ for the even and odd modes except at one particular voltage (vertical dashed line) where the even mode in one waveguide can be phase-matched to the odd mode in the other waveguide. The TMWs are identical, each of width 4µm, and they are separated by 4µm. The input has wavelength λ = 0.812µm and TM polarization. The symbols represent simulated data obtained using the RSoft program.

To show that the device portrayed in Fig. 6 operates as a CNOT gate, we first demonstrate that the target qubit is indeed flipped by a TM-polarized control qubit, so that ∣1〉 ≡ ∣TM〉. The polarization sensitivity of the Ti:LiNbO3 TMWs resides in the values of their refractive indices n, which depend on the polarizations of the incident waves and the voltage applied to its EO phase modulator; and on their Pockels coefficients r, which depend on the polarization [17

17. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, Hoboken, NJ, 2007).

, Example 20.2-1]. For a photon with TM polarization, the two-mode EO directional coupler offers two operating regions with markedly different properties. At low (or no) applied voltage, interaction and power transfer take place only between like-parity modes in the two waveguides because the propagation constants of the even and odd modes are different, so they are not phase-matched. However, at a particular higher value of the applied voltage, the behavior of the device changes in such a way that only the even mode in one waveguide, and the odd mode in the other, can interact and exchange power. This arises because the refractive indices of the two waveguides depend on the voltage applied to the device; they move in opposite directions as the voltage increases since the electric-field lines go downward in one waveguide and upward in the other. Figure 7 provides an example illustrating the dependencies of the propagation constants of the even and odd modes, in the two TMWs, as a function of the applied voltage.

A drawback of the photonic circuit illustrated in Fig. 6 is that it suffers from the effects of dispersion, which is deleterious to the operation of circuits used for many quantum information applications. Dispersion results from the dependence of the propagation constant β on frequency, mode number, and polarization. Polarization-mode dispersion generally outweighs the other contributions, especially in a birefringent material such as LiNbO3.

Fig. 8. Sketch of a Ti:LiNbO3 photonic quantum circuit that behaves as a novel dispersionmanaged, deterministic, two-qubit, single-photon, CNOT gate (not to scale). The control qubit is polarization and the target qubit is mode number. The design is more complex than that shown in Fig. 6 because it accommodates dispersion management via path-length adjustments of the upper, middle, and lower paths. An EO TMW directional coupler is sandwiched between polarization-sensitive mode analyzers and polarization-sensitive mode combiners. The lower and upper waveguides of the two-mode directional coupler are denoted WG1 and WG2, respectively. The paths taken by the components of the input state ∣Ψi〉 are shown, as is the output state ∣Ψo〉.

Fortunately, however, it is possible to construct a photonic circuit in which the phase shifts introduced by dispersion can be equalized. A Ti:LiNbO3 photonic quantum circuit that behaves as a novel dispersion-managed, deterministic, two-qubit, single-photon, CNOT gate is sketched in Fig. 8. It makes use of three paths (upper, middle, and lower), in which the path-lengths of the three arms are carefully adjusted to allow for dispersion management. The third path provides the additional degree-of-freedom that enables the optical path-lengths to be equalized.

The design relies on the use of polarization-dependent mode analyzers at the input to the circuit. The TM-mode analyzer couples the odd-TM component of the state to the upper path, while the TE-mode analyzer couples the odd-TE component to the lower path. The even-TM and even-TE components continue along the middle path. Polarization-dependent mode combiners are used at the output of the circuit.

If the control qubit is in a superposition state, the general quantum state at the input to the circuit, which resides in a 4D Hilbert space (2D for polarization and 2D for mode number), is expressed as

Ψi=α1e,TM+α2o,TM+α3e,TE+α4o,TE
=TM[α1e+α2o]+TE[α3e+α4o]
=e[α1TM+α3TE]+o[α2TM+α4TE],
(7)

Ψo=α1o,TM+α2e,TM+α3e,TE+α4o,TE
=TM[α1o+α2e]+TE[α3e+α4o],
(8)

where it is clear that the two terms in the input state, α 1e,TM〉 and α 2o,TM〉, are converted to α 1o,TM〉 and α 2e,TM〉, respectively, at the output, exemplifying the operation of this CNOT gate. Figure 8 displays the paths taken by the components of the input state provided in Eq. (7); the output state set forth in Eq. (8) is also indicated.

It remains to demonstrate the manner in which dispersion management can be achieved in the CNOT gate displayed in Fig. 8. The phase shift φ acquired by each component at the output is given by

φe,TM=βe,TM1+βo,TM2+βLD(2q1+q2)π2
φo,TM=φe,TM
φe,TM=2βe,TE1+βLD
φo,TE=2βo,TE3q3π+2ϕA,
(9)

where the β’s are the mode propagation constants; β′ is the propagation constant of either the TM-even mode in WG1 or the TM-odd mode in WG2; β″ is the propagation constant of the TE-even mode in WG1; q 1, q 2, and q 3 are odd positive integers that depend on the lengths of the TM-mode analyzer, directional coupler, and TE-mode analyzer, respectively; LD is the length of the directional-coupler electrode; ℓ1 is the path-length for the even modes before and after the directional coupler, ℓ2 is the path-length for the odd-TM mode before and after the directional coupler; and 2ℓ1 + LD, 2ℓ2 + LD, and 2ℓ3 are the overall physical lengths of the middle, upper, and lower paths, respectively. The phase shift ϕA arises from the coupling that affects the odd-TE component as it travels through the TM-mode analyzer. Phase shifts that accrue for the even modes as they pass through the mode analyzers and mode combiners are neglected because of large phase mismatches and weak coupling coefficients. By adjusting the lengths ℓ1, ℓ2, and ℓ3, we can equalize the phase shifts encountered by each component of the state. Imperfections in the fabrication of the circuit may be compensated by making use of EO phase modulators.

Fig. 9. Simulation demonstrating the performance of the polarization-dependent mode analyzers and the EO TMW directional coupler associated with the dispersion-managed, deterministic, two-qubit, single-photon, CNOT gate set forth in Fig. 8. The input wavelength is λ = 0.812µm. The TM-mode-analyzer and mode-combiner parameters are w 1 = 5.6µm, w 2 = 3.4µm, b 1 = 4µm, and L 2 = 6.2 mm; the TE-mode-analyzer and mode-combiner parameters are w 2 =3µm, b 1 =4µm, and L 2 =3.7 mm (see Fig. 2 for symbol definitions). The S-bends have dimensions Lb = 10 mm and S = 127µm. The TMW directional-coupler has length L 1 = 2.2 mm, waveguide width w 1 = 5.6µm, electrode separation d = 4µm, and an EO phase-modulator voltage V = 36 V applied to WG2, with WG1 at ground potential. All panels display the spatial evolution of the normalized amplitudes of the interacting modes. (a) The curves display strong coupling between the odd and even modes for TM-polarization inside the TM-mode analyzer. The input odd mode in the TMW is shown in blue and the even mode transferred to the SMW is shown in green [the same color conventions are used in panels (b) and (c)]. The even mode is ultimately coupled to another TMW at the output of the TM-mode analyzer and once again becomes odd. (b) The curves show negligible coupling between the odd and even modes for TE-polarization inside the TM-mode analyzer. (c) The curves display good coupling between the odd and even modes for TE-polarization inside the TE-mode analyzer. At the TE-mode combiner, the even mode in the SMW once again becomes an odd mode in the TMW. Panels (d), (e), and (f) display the performance of the directional coupler for modal inputs that are TM-even, TM-odd, and TE-even, respectively. For a given polarization, the blue and green curves represent the amplitudes of the even [denoted Even(1)] and odd [denoted Odd(1)] modes in WG1, respectively, while the the red and black curves are the amplitudes of the even [denoted Even(2)] and odd [denoted Odd(2)] modes in WG2, respectively. All simulated data in this figure were obtained using the RSoft program.

The absence of a total power transfer from one waveguide to another in Figs. 9(d) and 9(e) can be ascribed to sub-optimal simulation parameters. The conversion efficiency can be expected to improve upon: 1) optimizing the length of the two-mode directional coupler; 2) minimizing bending losses by increasing the length of the S-bend; 3) mitigating the residual phase mismatch by more careful adjustment of the voltage; and 4) improving numerical accuracy. Moreover, the deleterious effects of dc drift and temperature on the operating voltage and stability of the two-mode directional coupler can be minimized by biasing it via electronic feedback [56

56. A. Djupsjobacka and B. Lagerstrom, “Stabilization of a Ti:LiNbO3 directional coupler,” Appl. Opt. 28, 2205–2206 (1989). [CrossRef] [PubMed]

]; a novel technique based on inverting the domain of one of its arms can also be used to reduce the required operating voltage [57

57. F. Lucchi, D. Janner, M. Belmonte, S. Balsamo, M. Villa, S. Giurgola, P. Vergani, and V. Pruneri, “Very low voltage single drive domain inverted LiNbO3 integrated electro-optic modulator,” Opt. Express 15, 10 739–10 743 (2007). [CrossRef]

]. Finally, it is worthy of note that decoherence associated with the use of a cascade of CNOT gates, such as might be encountered in carrying out certain quantum algorithms, may be mitigated by the use of either a qubit amplifier [32

32. N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070 501 (2010). [CrossRef]

] or teleportation and error-correcting techniques [33

33. S. Glancy, H. M. Vasconcelos, and T. C. Ralph, “Transmission of optical coherent-state qubits,” Phys. Rev. A 70, 022 317 (2004). [CrossRef]

].

8. Conclusion

Acknowledgments

This work was supported by the Bernard M. Gordon Center for Subsurface Sensing and Imaging Systems (CenSSIS), an NSF Engineering Research Center; by a U.S. Army Research Office (ARO) Multidisciplinary University Research Initiative (MURI) Grant; and by the Boston University Photonics Center.

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OCIS Codes
(130.3730) Integrated optics : Lithium niobate
(230.7380) Optical devices : Waveguides, channeled
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: July 19, 2010
Revised Manuscript: September 3, 2010
Manuscript Accepted: September 3, 2010
Published: September 10, 2010

Citation
Mohammed F. Saleh, Giovanni Di Giuseppe, Bahaa E. A. Saleh, and Malvin Carl Teich, "Modal and polarization qubits in Ti:LiNbO3 photonic circuits for a universal quantum logic gate," Opt. Express 18, 20475-20490 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20475


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References

  1. M. F. Saleh, B. E. A. Saleh, and M. C. Teich, “Modal, spectral, and polarization entanglement in guided-wave parametric down-conversion,” Phys. Rev. A 79, 053842 (2009). [CrossRef]
  2. M. F. Saleh, G. Di Giuseppe, B. E. A. Saleh, and M. C. Teich, “Photonic circuits for generating modal, spectral, and polarization entanglement,” IEEE Photon. J. 2, 736–752 (2010). [CrossRef]
  3. M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals,” Opt. Express 15, 7479–7488 (2007). [CrossRef] [PubMed]
  4. M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79, 043836 (2009). [CrossRef]
  5. P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett. 103, 233901 (2009). [CrossRef]
  6. T. Zhong, F. N. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber coupled periodically poled KTiOPO4 waveguide,” Opt. Express 17, 12019–12030 (2009).
  7. J. Chen, A. J. Pearlman, A. Ling, J. Fan, and A. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17, 6727–6740 (2009). [CrossRef]
  8. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008). [CrossRef] [PubMed]
  9. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009). [CrossRef]
  10. C. H. Bennett, and P. W. Shor, “Quantum information theory,” IEEE Trans. Inf. Theory 44, 2724–2742 (1998). [CrossRef]
  11. M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
  12. J. L. O’Brien, A. Furusawa, and J. Vuˇckovi’c, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). [CrossRef]
  13. A. Politi, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Integrated quantum photonics,” IEEE J. Sel. Top. Quantum Electron. 15, 1673–1684 (2009). [CrossRef]
  14. G. Cincotti, “Prospects on planar quantum computing,” J. Lightwave Technol. 27, 5755–5766 (2009). [CrossRef]
  15. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45–53 (2010). [CrossRef] [PubMed]
  16. H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw–Hill, 1989).
  17. B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  18. A. C. Busacca, C. L. Sones, R. W. Eason, and S. Mailis, “First-order quasi-phase-matched blue light generation in surface-poled Ti:indiffused lithium niobate waveguides,” Appl. Phys. Lett. 84, 4430–4432 (2004). [CrossRef]
  19. Y. L. Lee, C. Jung, Y.-C. Noh, M. Park, C. Byeon, D.-K. Ko, and J. Lee, “Channel-selective wavelength conversion and tuning in periodically poled Ti:LiNbO3 waveguides,” Opt. Express 12, 2649–2655 (2004). [CrossRef] [PubMed]
  20. S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26–28 (2001). [CrossRef]
  21. H. Guillet de Chatellus, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, and G. Di Giuseppe, “Non-collinear and non-degenerate polarization-entangled photon generation via concurrent type-I parametric downconversion in PPLN,” Opt. Express 14, 10060–10072 (2006).
  22. R. C. Alferness, and R. V. Schmidt, “Tunable optical waveguide directional coupler filter,” Appl. Phys. Lett. 33, 161–163 (1978). [CrossRef]
  23. R. C. Alferness, “Efficient waveguide electro-optic TETM mode converter/wavelength filter,” Appl. Phys. Lett. 36, 513–515 (1980). [CrossRef]
  24. J. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D Appl. Phys. 36, R1–R16 (2003). [CrossRef]
  25. D. Runde, S. Brunken, S. Breuer, and D. Kip, “Integrated-optical add/drop multiplexer for DWDM in lithium niobate,” Appl. Phys. B 88, 83–88 (2007). [CrossRef]
  26. D. Runde, S. Breuer, and D. Kip, “Mode-selective coupler for wavelength multiplexing using LiNbO3:Ti optical waveguides,” Cent. Eur. J. Phys. 6, 588–592 (2008). [CrossRef]
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