Degenerated Spontaneous Parametric Down Conversion in Semiconductor Waveguide with -43m Crystal Symmetry

Waveguides with -43m Crystal Symmetry.

Eleonora De Luca and Marcin Swillo ^∗

School of Engineering Sciences (SCI), KTH Royal Institute of Technology, Roslagstullsbacken 21, SE-10691 Stockholm, Sweden

In this paper, a configuration consisting of a pump propagating in an orthogonal direction to the surface of a nanowaveguide is studied. It is possible to show that the generation by spontaneous parametric down-conversion of a signal and an idler photons inside the nanowaveguide is strongly dependent on the waveguide thickness because the high refractive index of the waveguide core creates also an optical cavity for the pump field. Furthermore, it was evaluated the best condition for photon-pair generation in the case of a nanowaveguide in gallium indium phosphide, which has a −43m symmetry.

For the same waveguide geometry, but orientation parallel to the [110] plane, it is possible to use spontaneous parametric down-conversion to generate counter-propagating photon-pair with the same polarization. When the efficiency of the generated photon-pairs in TM

and TE

modes is the same, then it is possible to obtain polarization-entangled photons, by using this configuration.

Furthermore, by adding a mirror at one of the ends of the waveguide, the system can be used to produce a squeezed-state.

I. INTRODUCTION

Photons are indeed an excellent resource for several quantum applications, including quantum computing [1], quantum cryptography [2, 3] and communication [4–7], and quantum metrology [8]. This is due to the possibility of easily transmit and manipulate them in the open air or through optical fibers, as well as the advances in non- linear optics that have enabled the generation of single and entangled photons.

Photon pairs generated by optical spontaneous paramet- ric down-conversion (SPDC) are good examples of en- tangled states, i.e. pairs of photons that share a single and not factorized wave function, generated in a nonlin- ear medium. SPDC has been realized in bulk nonlinear crystals [9] and in microstructured fibers [4, 10], atomic ensembles [5], and microwaveguide and photonic crystals [6, 11–15].

In this process, a photon from an intense excitation field (known as pump photon) interacting with a nonlinear optical medium generates two photons, known as idler and signal [16]. As an alternative perspective, the seed fields of the process are provided by vacuum fluctuations, and only the signal and idler pairs that satisfy momen- tum and energy conservation are efficiently transformed in existing photons. The combination of the conservation laws and the vacuum fluctuations are the center of the entanglement between idler and signal.

SPDC is a versatile process since it makes possible to generate photons that can be polarization, space, time, etc. entangled. When a low mean photon number is con- sidered, a system known as a ”heralded” source, where one of the two generated photons is used for heralding the presence of the other photon can be also used [11, 12, 17].

∗

marcin@kth.se

In this work, we want to suggest a possible solution for generating polarization-entangled photons by using a rel- atively simple setup, based on the possibility of using a nanowaveguide made in a semiconductor with −43m symmetry [18], for the generation. In particular, the data were calculate by using Ga 0.51 In 0.49 P as illustra- tive material, since it allows the generation of photons in the near-infrared region (NIR, 0.75 − 1.4 µm), a pre- ferred region for using single-mode optical fibers, whose attenuation is below 2 dB/km in the NIR region [3].

Ga 0.51 In 0.49 P shows excellent properties for nonlinear ap- plications, such as large nonlinear coefficient [19] and large refractive index [20]. Furthermore, III - V alloys are of great interest due to the possibility of integration on chips [11, 12, 14, 15, 17], making those materials of great interest for future applications with integrated sys- tems on a single device.

In this work, the pump propagating in an orthogonal di- rection to the surface of a nanowaveguide is studied. We consider a waveguide with −43m crystal symmetry ro- tated 45 ^o around the waveguide axes, which results in 45 ^o to the two crystallographic axes of the semiconduc- tor. This configuration makes possible the generation of a pair of counter-propagating photons with the same polarization and, so with degenerated wavelengths. Fur- thermore, the crystal geometry of the material allows the generation of entangled states at specific conditions. This is possible by using one pump beam only, in contrast to what presented by [21]. Additionally, the study of the conditions to generate a squeezed state in this configura- tion is presented.

II. THEORY.

To generate pairs of counter-propagating photons in

the waveguide, we will use the scheme with the pump

propagation direction perpendicular to the waveguide.

This technique is similar to the one presented by A.

De Rossi and V. Berger [21]; however, by changing the waveguide geometry and the crystal orientation, the gen- eration of particular quantum states, which were not con- sidered in [21], can be obtained.

For crystal symmetry −43m the second-order nonlinear susceptibility tensor (in contracted notation) has a form [18]:

d eff =





0 0 0 d 0 0 0 0 0 0 d 0 0 0 0 0 0 d



 . (1)

Applying the non-depleted pump approximation and as- suming counter-propagating signal and idler leads to two sets of equations:

dE _x,s

dz = iκE z,p E _y,i ^∗ e ^i∆kz , dE _y,i

dz = −iκE _z,p E _x,s ^∗ e ^i∆kz , (2) dE x,s

dz = −iκE z,p E ^∗ _y,i e ^i∆kz , dE _y,i

dz = iκE _z,p E _x,s ^∗ e ^i∆kz , (3) where κ = d q _µ

0

ε

ω

ω

ω

n

n

n

and ∆k = k i − k s . Since the crystallographic axes of the crystal are parallel to xyz- coordinates the excited counter-propagating modes need to have orthogonal polarization. In this configuration, it is impossible to have photons with degenerate wave- lengths: as from the graph presented in Fig. 1, it is pos- sible to see that for thicker waveguides, the wavelengths of emission of the photon pairs are closer as well as for waveguides which thickness is very small. In order to get polarization entangled photon-pair generation, the signal and the idler emitted at the end of the waveguide need to be indistinguishable. As suggested in [21], this can be solved by using two pump beams.

An alternative solution with a single pump beam can be obtained when the crystallographic axes of the nonlin- ear crystal are rotated 45 ^o around the z-axis (Fig. 2(a)).

Then, the dispersion problem can be avoided by the gen- eration of down-converted photons with the same polar- ization and the same wavelength. In Fig. 2(b) the mo- mentum diagram for considered photons in the recipro- cal space is presented: the profile of d _eff is a sinc func- tion due to the waveguide cross-section. However, the pump distribution, as well as the guided modes of the signal and the idler, contributes to the final conversion efficiency. Even if the waveguide geometry prevents effi- cient conversion (if d = _n ^λ

p

corresponding to k _p = ^2π _d ), we will demonstrate that the profiles of the guided modes and the pump distribution inside the waveguide will con- structively contribute to the photon-pair generation. It is also indicated the case presented in orange in Fig. 2(a), where the signal photon propagates along z and the idler

FIG. 1. Calculated dispersion inside a waveguide made of GaInP in a air cladding for different thicknesses of the waveg- uide for λ

= 700 nm.

FIG. 2. Schemes (a) for spontaneous parametric down- conversion process in a nonlinear waveguide with −43m crystal symmetry (crystal axes 45

to xyz) for counter- propagating signal and idler. Two orthogonal solutions for generating photon-pairs are indicated by orange and green.

The pump polarization is in brown. The direction of prop- agation of the different waves is shown by dotted lines. (b) Momentum conservation diagram presented in the reciprocal space, when the idler is propagating along −z and the signal is propagating along +z.

photon propagates along −z and where the momentum of the signal and the idler photons have to cancel each other, to satisfy the phase-matching condition for SPDC process. For the pump photons, the momentum is ab- sorbed by the waveguide geometry due to the nonlinear process. In the new xyz-coordinates system (Fig. 2(a)), d eff can be written as:

d eff =





0 0 0 0 d 0 0 0 0 −d 0 0 d −d 0 0 0 0



 . (4)

The Eq. 2 and Eq. 3 can be written as:

dE _x,s

dz = iκE z,p E _x,s ^∗ e ^i∆kz , dE _x,i

dz = −iκE _z,p E _x,i ^∗ e ^i∆kz , (5) dE y,s

dz = −iκE z,p E _y,i ^∗ e ^i∆kz , dE _y,i

dz = iκE _z,p E _y,s ^∗ e ^i∆kz , (6) Now, one can see that the signal and the idler have the same polarization, which leads to degenerated wave- length of emitted photons. As one can see from Eq. 5 and Eq. 6, an efficient light conversion requires a good overlap among the interacting fields. Therefore, we have to consider profiles for both the guided modes (of the sig- nal and the idler) as well as the pump field distribution inside the waveguide. The overlap η p,i,s among all the three interacting fields can be written as:

η p,i,s = Z

core

E ^∗ _p E i E s dx, (7) where E _s and E _i are the normalized field profile for the guided fundamental mode, and E _p is the electric field of the pump with normalized amplitude in the free space.

η p,i,s has a complex value. However, to simplify the equa- tions we will use the absolute value:

η = q

η p,i,s · η _p,i,s ^∗ . (8) This approximation will only affect the phase of the gen- erated light. Since the refractive index of the nanowaveg- uide core is larger than the cladding, the waveguide can create an optical cavity for the pump wavelength. In this case, the enhancement of the pump field, due to the res- onance effect will contribute to the overlap η, where η ² is proportional to the efficiency of generation of the photon pairs.

The Eq. 5 and Eq. 6 describe an optical parametric os- cillator with the gain threshold condition satisfied for the pump:

E z,p = π

2κLη (9)

where L is the length of the waveguide. For the pump power below the threshold, the classical description can not be used. Then the quantum formalism has to be ap- plied. Since in both polarizations the generated photons have the same energy, their superposition represents a polarization-entangled state:

Ψ out = 1

√ 2 (|X s i F |X i i B − |Y s i F |Y i i B ), (10) where the subscripts s and i represent the signal and the idler photons, and X and Y represent the polarization of

FIG. 3. Overlap η as in Eq. 8 among the modes for a waveg- uide made of GaInP in a air cladding for different thicknesses of the waveguide for λ

= 700 nm.

FIG. 4. Nanowaveguide geometry including a mirror at the end of the waveguide.

the photons. The polarization entangled photons can be generated when η of the two cases, i.e. TE _0,F TE _0,B and TM _0,F TM _0,B , are equal, as shown in Fig. 3.

Since the TE 0,F TE 0,F overlap is generally dominating, it is a better option to consider when the generation in- terests only a photon-pair. Therefore, it is important to remind that the SPDC process with the signal and the idler generated at degenerated wavelength with the same polarization can be used to generate squeezed-state. By placing a mirror at one end of the waveguide as in Fig.4, the phase relation between forward and backward prop- agating modes can be defined by the mirror position, allowing the idler and signal to propagate in the same direction. The generation of vacuum squeezed states can be evaluated by considering the time dependent Hamil- tonian of the electromagnetic field [22]. The pump, the signal, and the idler can be written as:

E z,p = i r

~ω p

2V  (a p (t)e ^iky − a ^† _p (t)e ^−iky ), (11)

E _x,s = i r

~ω

2V  (a(t)e ^ikz − a ^† (t)e ^−ikz ), (12)

E x,i = i r

~ω

2V  (a(t)e i[k(2L−z)+φ

] −a ^† (t)e −i[k(2L−z)+φ

] ),

(13)

where φ m is the phase due to the reflection from the mirror, V is the quantization volume,  is the dielec- tric constant, L is the length of the waveguide. Since the pump is in a non-depleted pump approximation and strong pump field, then it can be used as a classical field (ε p ), which has the units of the square root of the mean photon number. In this case, the Heisenberg equations of motion need to be solved only for a and a ^† [22]. Fi- nally, by applying energy and momentum conservation and considering κ > 0 and a real number, it is possible to obtain:

a(t) = e ^iφ

a(0) cosh(ε p ηκt) + e ^−iφ

a ^† (0) sinh(ε p ηκt), (14) where φ 0 = −kL − φ m /2. This confirms the squeezing of the generated light.

III. CONCLUSIONS

In conclusion, we applied a configuration that includes the pump propagating in the orthogonal direction to the nanowaveguide. In this configuration of the pump, we used a crystal with a −43m symmetry, and we applied a

rotation of an angle of 45 ^o in the xy-plane. This allowed us to prove the generation of photons with degenerate wavelength and polarization, due to the new components of the second-order nonlinear susceptibility tensor. While it can be proven that the overlap η between the degener- ate case of photons that propagates in TE 0 mode inside the waveguide is on average larger, it is possible anyway to generate entangled photon pairs, when the overlap η of the pairs generated in TE ₀ mode and the ones gener- ated in TM ₀ mode are equal.

Moreover, we proved that when a mirror is added at the end of the nanowaveguide, it is possible to obtain a vac- uum squeezed-state. The best configuration, in this case, includes having both the idler and the signal in TE 0 mode when the ratio is close to 1 ^λ _n

p

, due to the largest over- lap in this condition. Such a source can have potential applications in integrated optical sensors.

ACKNOWLEDGMENTS

Swedish Research Council (VR) (2018-03457).

[1] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Quantum computers, Na- ture 464, 45 (2010).

[2] B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J.

Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, Provably secure and practical quantum key distribution over 307km of optical fibre, Nature Photonics 9, 163 (2015).

[3] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quan- tum cryptography, Rev. Mod. Phys. 74, 145 (2002).

[4] A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A.

Walmsley, Efficient Conditional Preparation of High- Fidelity Single Photon States for Fiber-Optic Quantum Networks, Physical Review Letters 93 (2004).

[5] H. J. Kimble, A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, and L.-M. Duan, Generation of nonclassical photon pairs for scalable quantum commu- nication with atomic ensembles., Nature 423, 731 (2003).

[6] M. Bock, A. Lenhard, C. Chunnilall, and C. Becher, Highly efficient heralded single-photon source for telecom wavelengths based on a ppln waveguide, Opt. Express 24, 23992 (2016).

[7] I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legre, and N. Gisin, Distribution of time-bin qubits over 50 km of optical fiber, Physical Review Letters 93, 180502 (2004), 0404124 [quant-ph].

[8] V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nature Photonics 5, 222 (2011).

[9] P. S. Kuo, T. Gerrits, V. Verma, S. W. Nam, O. Slattery, L. Ma, and X. Tang, Characterization of type-II sponta- neous parametric down-conversion in domain-engineered PPLN, in Advances in Photonics of Quantum Comput- ing, Memory, and Communication IX , Vol. 9762 (NIH

Public Access, 2016) p. 976211.

[10] J. A. Slater, J.-S. Corbeil, S. Virally, F. Bussi` eres, A. Kudlinski, G. Bouwmans, S. Lacroix, N. Godbout, and W. Tittel, Microstructured fiber source of photon pairs at widely separated wavelengths., Optics letters 35, 499 (2010).

[11] J. Belhassen, F. Baboux, Q. Yao, M. Amanti, I. Favero, A. Lemaˆıtre, W. S. Kolthammer, I. A. Walmsley, and S. Ducci, On-chip III-V monolithic integration of her- alded single photon sources and beamsplitters, Applied Physics Letters 112, 71105 (2018).

[12] R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, A semiconductor source of triggered entangled photon pairs, Nature 439, 178 (2006).

[13] H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J.

Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal, Physical Review Letters 111, 10.1103/PhysRevLett.111.023603 (2013).

[14] A. Orieux, A. Eckstein, A. Lemaˆıtre, P. Filloux, I. Favero, G. Leo, T. Coudreau, A. Keller, P. Milman, and S. Ducci, Direct bell states generation on a iii-v semiconductor chip at room temperature, Phys. Rev. Lett. 110, 160502 (2013).

[15] R. T. Horn, P. Kolenderski, D. Kang, P. Abolghasem, C. Scarcella, A. D. Frera, A. Tosi, L. G. Helt, S. V.

Zhukovsky, J. E. Sipe, G. Weihs, A. S. Helmy, and T. Jennewein, Inherent polarization entanglement gen- erated from a monolithic semiconductor chip, Scientific Reports 3, 2314 (2013).

[16] D. C. Burnhamand D. L. Weinberg, Observation of si-

multaneity in parametric production of optical photon

pairs, Phys. Rev. Lett. 25, 84 (1970).

[17] L. Caspani, C. Reimer, M. Kues, P. Roztocki, M. Clerici, B. Wetzel, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, L. Razzari, B. E. Little, S. T. Chu, D. J.

Moss, and R. Morandotti, Multifrequency sources of quantum correlated photon pairs on-chip: a path toward integrated Quantum Frequency Combs, Nanophotonics 5, 10.1515/nanoph-2016-0029 (2016).

[18] A. Yarivand P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed., The Oxford Series in Electrical and Computer Engineering (Oxford University Press, 2007) p. 363.

[19] I. Shoji, T. Kondo, and R. Ito, Second-order nonlin- ear susceptibilities of various dielectric and semiconduc- tor materials, Optical and Quantum Electronics 34, 797 (2002).

[20] M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, and J. A. Woollam, Optical constants of Ga

In

P lattice matched to gaas, Journal of Applied Physics 77, 3416 (1995).

[21] A. De Rossiand V. Berger, Counterpropagating Twin Photons by Parametric Fluorescence, Physical Review Letters 88, 043901 (2002).

[22] D. F. Wallsand R. Barakat, Quantum-Mechanical Am-

plification and Frequency Conversion with a Trilinear

Hamiltonian, Phys. Rev. A 1, 446 (1970).