Revealing photon's behaviors in a birefringent interferometer - arXiv

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Zhi-Yuan Zhou,1,2,3,†,* Shi-Kai Liu,1,2,† Shi-Long Liu,1,2 Yin-Hai Li,1,2,3 Yan Li,1,2 Chen Yang,1,2 Zhao-Huai. Xu,1,2 Guang-Can Guo,1,2 and Bao-Sen Shi1 ...
Tunable quantum beating in a birefringent interferometer Zhi-Yuan Zhou, 1,2,†,* Shi-Kai Liu, 1,2,† Shi-Long Liu, 1,2 Yin-Hai Li, 1,2 Yan Li, 1,2 Chen Yang, 1,2 Zhao-Huai Xu,1,2 Guang-Can Guo,1,2 and Bao-Sen Shi1,2,* 1

CAS Key Laboratory of Quantum Information, USTC, Hefei, Anhui 230026, China

2

Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China *Corresponding authors: [email protected]; [email protected]

These two authors contributed equally to this work.

The interferometer is one of the most important devices for study of the basic properties of light and for high-precision optical metrology. We observe quantum beating versus temperature in a birefringent crystal in a modified self-stable Mach-Zehnder interferometer for both single-photon and two-photon inputs. The beating intensity can be tuned by rotating the crystal and the two-photon interference fringes beat two times faster than the single-photon interference fringes. This beating effect is used to determine the thermal dispersion coefficients of the two principal refractive axes with a single measurement, while the two-photon input case shows super-resolution and high sensitivity, which cannot be achieved using other implementations. A quantum optical model is constructed to describe the phenomenon and agrees well with experimental observations. Our findings will be important for accurate measurement of the optical properties of birefringent materials and for high-precision optical sensing with quantum enhancement.

The optical interferometer is a basic tool of modern optical science that can be used to study the physical properties of light and to perform high-precision optical measurements [1]. In quantum optics, the interferometer is used to study the wave-particle duality of photons [2-4] and the nonclassical effects of quantum sources [5-7]. In precision optical metrology, most physical quantities, including position, displacement, distance, angle, optical dispersion and optical path length can be obtained via measurement of phase changes in an interferometer [8]. Quantum optical metrology uses the superpositioning of N-photon number states, which are referred to as N00N-states, for phase sensing with an N times improvement in sensitivity when compared with classical approaches involving N photons. This super-sensitive phase measurement method has many important applications, including imaging, microscopy, gravity-wave detection, material properties measurements, and chemical and biological sensing [9-12]. Photonic beating is an effect that occurs when two light fields with different frequencies interfere; the interference intensity shows beating behavior with a rapidly oscillating carrier frequency and a slowly oscillating envelope frequency [13-15]. This effect has many potential applications, such as synchronization of two lasers with different operating

frequencies, spectroscopy and heterodyne detection [16, 17]. While various configurations have been studied for Mach-Zehnder interferometers (MZIs) [18], complete quantum description of a birefringent MZI has not been performed or experimentally realized to date. Here, by inserting a birefringent crystal into a modified self-stable MZI, quantum beating versus crystal temperature is observed for both single-photon and two-photon inputs. In addition, the beating intensity can be tuned by rotating the crystal, and the two-photon interference fringes beat two times faster than the single-photon interference fringes. This beating effect can be used to determine the thermal dispersion coefficients of the two principal refractive axes with a single measurement, while the two-photon input cases also show super-resolution and high sensitivity; these advantages cannot be replicated when using traditional methods. A quantum optical model is constructed to describe this phenomenon fully and demonstrates complete agreement with the experimental observations. Theoretical model. The quantum optical description of our measurements will be given first. A simplified diagram is shown in Fig. 1, in which two birefringent potassium titanyl phosphate (KTP) crystals have been inserted into a modified MZI. One of these crystals, designated KTP2, is used to

compensate for the optical path length differences in the interferometer. The other KTP crystal is mounted on a rotation stage to rotate its position with respect to the horizontal polarization direction. The photon operators at the output ports (P4, P5) are connected to those at the input ports (P0, P1) through the transformations provided by the beam splitters and the birefringent crystals. In this derivation, we assume that the photon bandwidths are narrow enough that the coherence length of the photon is much greater than both the imbalance of the interferometer and the optical path length difference between the optical axes of the birefringent crystal. We will derive the final expressions for the operators

1 ˆ Aˆ 4  (A2  iAˆ3 )=F1 ( Aˆ0 , Aˆ1 ) e H  G1 ( Aˆ0 , Aˆ1 ) eV 2 (3) ˆA  1 (iAˆ   Aˆ  )=F ( Aˆ , Aˆ ) e  G ( Aˆ , Aˆ ) e 5 2 3 2 0 1 H 2 0 1 V 2 where

1 i Fˆ1   (T   ) Aˆ0  (T   ) Aˆ1 2 2  Gˆ1  ( Aˆ0  iAˆ1 ) 2 i 1 Fˆ2  (T   ) Aˆ0  (T   ) Aˆ1 2 2 i Gˆ 2  ( Aˆ0  iAˆ1 ) 2

ˆ , Aˆ ) at ports P2 and step-by-step; the photon operators ( A 2 3

ˆ , Aˆ ) at P3 can be expressed in terms of the operators ( A 0 1 ports P0 and P1 using the following expression [19]:

ˆ  1 (A ˆ  iAˆ ) A 2 0 1 2 ˆ  1 (iA ˆ  Aˆ ) A 3 0 1 2

(1)

(KTP2) with a rotation angle  relative to the horizontal polarization, the transformation performed by the crystal can be expressed as follows:

ˆ  A ˆ [(cos 2  ei y  sin 2  eiz ) e A 2 2 H

Here, 

2

3

+  =1 ; 2

(2)

and

eV

represent the unit

vectors in the two orthogonal polarization directions; and

i  2 ni ( ,T) L/  (i  y, z)

1 = [1+cos 2  cos( y -c )+sin 2  cos( z -c )] 2

(4)

R5  0,1 Aˆ5 Aˆ5 0,1  0,1 Fˆ2 Fˆ2 0,1  0,1 Gˆ 2Gˆ 2 0,1 (5) 1 = [1  cos 2  cos( y  c )  sin 2  cos( z  c )] 2

R4,5 between ports 4 and 5 can be calculated as follows: R4,5  1,1 Aˆ4 Aˆ5 Aˆ5 Aˆ4 1,1

3

eH

at ports 4 and 5 can be calculated to be

For the two-photon input case, the corresponding count rate

i

3

R4 , R5

R4  0,1 Aˆ4 Aˆ4 0,1  0,1 Fˆ1 Fˆ1 0,1  0,1 Gˆ1Gˆ1 0,1

When the photon passes through the birefringent crystal

3

(4)

For the single-photon input case, the single-photon count rates

 cos  sin  (e y  eiz ) eV ] ˆ [ e   e ] =A 2 H V ik L Aˆ   Aˆ e y c  Aˆ eic  TAˆ

ˆ , Fˆ ,G ˆ are expressed as follows: Fˆ1 ,G 1 2 2

are the optical phase

changes along the y and z axes of the birefringent crystal, where L is the crystal length,  is the wavelength of the photon and T is the crystal temperature. When the photons interact at the BS for the second time, the output photon

ˆ , Aˆ ) can be expressed as follows: operators ( A 4 5

1 = {1+ cos 4  cos(2 y  2c )+ (6) 4 1 sin 4  [cos(2 z  2c )+ cos( y   z  2c )]} 2 It is clear from Eqs. (4), (5), and (6) that for both the single-photon and two-photon inputs, the interference fringes are dependent on both phase changes that occur in the two optical axes of the birefringent crystal, and the oscillation period of the two-input case is twice as fast as the single-photon input case. The deviations described above are the main results for input of the narrow-band photon pair. The results for input of a broad-band photon pair will be discussed in the following text and the supplementary material.

Experimental results. We now experimentally demonstrate the predictions that were described in the theoretical models. We first study the case in which the coherence length of the photon pair is greater than the optical path difference between the two arms of the structure. The experimental setup is shown in Fig. 2. The photon pair is generated using a type-II periodically-poled KTP crystal (PPKTP), which has a length of 2 cm. The type-II PPKTP crystal that we used has a poling period of 46.2 m, and the degenerate phase matching temperature for the 775 nm pump beam is 30.03°C. The orthogonally-polarized photon pair is separated by the polarizing beam splitter (PBS) and the photons are then coupled into single-mode fibers (SMFs). The pump beam is removed using long pass filters (LPFs) before the photons are coupled into the SMFs. The polarization of the photon pairs in the SMFs is controlled using two pairs of half-wave plates (HWPs) and quarter-wave plates (QWPs). The photons that are released from the two SMFs are first polarization-purified using two PBSs, and are then injected into a self-stable MZI, which contains two KTP crystals; one KTP crystal is used for the measurements, while the other compensates for the optical path length differences between the two arms of the MZI. The self-stable MZI is based on a tilted Sagnac loop, where the clockwise and counterclockwise beams travel for a distance of 10 mm. The two KTP crystals have dimensions of 5 mm5 mm8 mm, and both end faces are anti-reflection coated for 1550 nm. Both crystals are x-cut such that the beams propagate along the x-axes of the crystal. KTP1 is used for the measurements, while KTP2 is used for compensation, and the temperature of KTP2 is kept at a constant 22.3°C. The temperature of KTP1 can be tuned from 17.81°C to 45.67°C. The temperatures of the two crystals are controlled using two home-made temperature controllers with temperature stability of ±2 mK. The delay between the two-photon pairs is controlled using a one-dimensional translation stage. The output photons at ports 4 and 5 are connected to two free-running InGaAs single-photon detectors (SPDs; ID220; 20% quantum efficiency; 5 µs dead time). The output signals from the two SPDs are sent to a coincidence measurement device (Timeharp 260; 0.4 ns coincidence window).

Figure 1. Experimental scheme. A 775 nm Ti:sapphire laser is used to pump

a

type-II

PPKTP

crystal,

which

then

generates

orthogonally-polarized photon pairs. The photon pairs are separated using a polarizing beam splitter (PBS) and coupled into single-mode fibers (SMFs) via fiber collimators (FCs). Two groups of quarter-wave plates (QWPs) and half-wave plates (HWPs) are used to control the polarizations of the photons inside the SMFs. Two long pass filters (LPFs) are used to remove the pump beam. The SMF outputs are injected into a self-stable MZI that contains two KTP crystals. The photons that are output from the two ports of the MZI are coupled to the SMFs and detected using two single-photon detectors (SPDs). The outputs from the SPDs are sent to a coincidence counting device.

We first characterize the photonic beating effects for the heralded single-photon and two-photon N00N states. The bandwidth of the photon pairs is 1.3 nm without spectral filtering. The Hong-Ou-Mandel (HOM) interference characteristics for the photons without filtering and with a 0.5 nm filter are shown in Fig. 1, and we see that nearly perfect HOM interference characteristics are observed in both cases, with visibilities of 97.42±0.16% and 98.71±0.29%, respectively. The HOM dip shapes are determined by the spectrum of the photon pair [20]. In the unfiltered case, the pump power is 8.4 mW, the single count rates are approximately 62 kHz and 77 kHz for the signal and idler photons, respectively, and the dark count rate is approximately 3 kHz. For the filtered case, the pump power is approximately 16 mW, and the single count rates for the signal and idler photons are 90 kHz and 56 kHz, respectively. For the detailed performance characteristics of the photon sources, please refer to [21, 22].

(0.980±0.067)10−5/K [(0.928±0.104)10−5/K] and (1.592±0.013)10−5/K [(1.594±0.037)10−5/K], respectively. Another important feature of the beating curve is that the temperature oscillation period of the beating curve for the two-photon case is two times faster than that of the heralded single-photon cases, which indicates higher measurement resolution when a high-photon-number entanglement state is used in measurements.

Figure 2. HOM interference characteristics for the unfiltered and filtered cases. In the unfiltered case, the theoretical curve used to fit the data is a triangle function; in the filtered case, the theoretical curve used to fit the data is Gaussian.

When the photon pairs reach the BS simultaneously, two-photon N00N states are generated after the two output ports of the BS because of the HOM interference. The single-photon and two-photon beating curves versus temperature for the different rotation angles of the KTP1 crystal are shown in Fig. 3. The group of figures on the left (bottom to top) shows the beating curves of the two-photon input for rotation angles of   0,  / 6,  / 4,  / 3,  / 2 . The group of figures on

Figure 3. Photonic beating versus temperature for the two-photon input

the right shows the corresponding beating curves for the heralded single photon. The rotation angles of   0,  / 2 represent cases in which the input photon

case and the heralded single-photon input case. The group of figures on

polarization coincides with the y and z optical axes of the birefringent crystal. The two-photon and single-photon visibilities for the two cases are (98.01±0.18)% and (94.75±0.27)%, and (93.09±0.23)% and (90.87±0.34)%, respectively. In these two cases, the two (single)-photon

figures

cases yield thermal dispersions ( dn y

/ dT , dnz / dT ) of

(1.027±0.019)10−5/K [(1.041±0.044)10−5/K] and −5 −5 (1.680±0.019)10 /K [(1.651±0.035)10 /K] for the y and z axes, respectively. The other curves show the beating behavior of the optical properties along the two axes, and we can determine the optical properties along both axes from any single measurement of this type of beating curve. For example, when    / 3 , the thermal dispersions that were obtained for two (single) photons for axes y and z were

the left (from bottom to top) represents the two-photon cases at rotation angles of on

  0,  / 6,  / 4,  / 3,  / 2 the

right

represents

the

. The group of

corresponding

heralded

single-photon input case. The measurement time is 10 s. Different offsets in each of the interference fringes come from the different initial phases between the two arms of the interferometer.

Next, we discuss the effects of the imbalance of the MZI on the interference fringes for both the two-photon input and the heralded single-photon input when the rotation angle of KTP1 is   0 . Three cases are studied: (I) when the optical lengths are equal, the compensating crystal KTP2 also aligned along the y axes, and the optical path length is 0 mm; (II) when the imbalance of the MZI is within the single-photon coherence length, which is realized by rotating KTP2 by 90°, and the optical path length difference is 0.66 mm; (III) when the imbalance of the MZI is greater than the coherence length of the single photon, which is realized by

removing the compensating crystal, and the optical path length difference is 13.87 mm. In these three cases, the two-photon interference fringes for both the two-photon and single-photon cases are shown in Fig. 4. The visibilities for these three cases for the two-photon and single-photon inputs are 97.98±0.19%, 94.26±0.46%, and 98.18±0.14%, and 93.09±0.23%, 54.20±0.97%, and 0, respectively. We can see obvious differences between the two-photon interference fringes and the single-photon interference fringes. The single-photon interference visibility decreases with increasing optical path difference in the MZI, while the two-photon interference visibility is immune to small optical path differences. These phenomena can be well explained in terms of quantum optics [18] because the two-photon coherence length is the same as the coherence length of the pump laser beam, which is at the ~km level. Therefore, small optical path differences in the MZI will have no effect on the two-photon interference fringes. In contrast, when the single-photon coherence length is at the 2 mm level, the interference fringes are very sensitive to optical path length differences in the MZI.

at port 4 and the coincidence count between port 4 and 5 are given as follows:

dk   1  cos2  cos( y LT )    1 dT  R4   2 2  dkz D 2 L2 2 )cos(  LT )   sin  exp( 4 dT  

(8)

dk 1 R45  {1  cos 4  cos(2 y LT )  2 dT dk sin 4  cos(2 z LT  2 )  (9) dT dk dk 1 4 D 2 L2 2 sin  cos[( z  y ) LT   ]exp( )} 2 dT dT 4 where

D

  [k z (0 , T0 )  k y (0 , T0 )]L

k z k y 1 1 , and      vgz vgy

T  T  T0

,

. These

parameters can be obtained from the Sellmeier equations for the y and z axes [23, 24]. The detailed derivations of Eqs. (8) and (9) can be found in the supplementary materials and [25]. For a photon spectrum with the form of a Sinc2 function, analytical expressions are given in the supplementary materials in Eqs. (s29)–(s34). The experimental results for the unfiltered case for both the two-photon and single-photon input cases when    / 4 are shown in Fig. 5. The interference visibility decreases as the bandwidth increases. For photons with large bandwidths, the last terms in Eqs. (8) and (9) can be ignored. The fitted curves based on Eqs. (8) and (9) show perfect agreement with the experimental results.

Figure 4. Interference fringes of the two-photon (left) and single-photon (right) cases for

 0

with increasing optical path

length difference. The optical path length differences in order from top to bottom are 0 mm, 0.66 mm and 13. 87 mm.

Finally, we study the effects of the photon bandwidth on the interference fringes. For photons with an angular spectral bandwidth of   2 ln 2 and a spectral distribution of

f 2 (i ) 

1 exp((i  i 0 )2 /  2 ) , the single count 

Figure 5. Two-photon and single-photon interferences without filtering when

   / 4 . The coincidence time is 10 s. The parameters used

for the fitting are L=8 mm, dny / dT , dnz / dT are 1.0310−5/K and

1.62010−5/K, respectively;  

 ln 2

 50GHz ; D=0.947

ps/mm.

Delayed-Choice

Experiment.

Phys.

Rev.

Lett.

107,

230406(2011). [4] Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, B.-S. Shi, Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality. Science Bulletin 62, 1185-1192(2017). [5] X. Ma, J. Kofler, A. Zeilinger, Delayed-choice gedanken experiments and their realizations. Rev. Mod. Phys. 88, 015005(2016).

We have provided a full theoretical and experimental description of quantum photonic beating in a birefringent interferometer. The results show that on rotation of the crystal in a self-stable MZI, tunable beating behavior versus crystal temperature is observed. This beating behavior is used to determine the optical properties along both crystal axes with a single measurement. The two-photon interference fringes oscillate twice as fast as those of a single photon, which indicates super-resolution measurement capabilities in the multi-photon entangled state. The present system is not limited to determination of the thermal dispersion of a birefringent crystal and can also be used to determine the wavelength dispersion and electro-optical coefficient of the birefringent crystal. This study will thus be of great importance for characterization of the main optical properties of birefringent crystals; some of the obvious advantages of our method cannot be realized using traditional methods.

[6]C. K. Hong, Z. Y. Ou, L. Mandel, Measurement of Subpicosecond Time Intervals between Two Photons by Interference. Phys. Rev. Lett. 59, 2044-2046 (1987). [7] A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, Asking Photons Where They Have Been, Phys. Rev. Lett. 111, 240402(2013). [8]V.

Giovannetti, S. Lloyd, L. Maccone, Quantum-enhanced

measurements: beating the standard quantum limit, Science 306, 1330(2004). [9]T. Roger, S. Restuccia, A. Lyons, D. Giovannini, J. Romero, J. Jeffers, M. Padgett, and D. Faccio, Coherent absorption of N00N states, Phys. Rev. Lett. 117, 023601 (2016). [10] T.

Ono,

R.

entanglement-enhanced

Okamoto,

and

microscope,

S.

Takeuchi,

Nat.

Commun.

An 4,

2426(2013). [11] J. A. Jones, S. D. Karlen, J. Fitzsimons, A. Ardavan, S. C. Benjamin, G. A. D. Briggs, J. J. L. Morton, Magnetic field sensing beyond the standard quantum limit using 10-spin

Acknowledgments

NOON states, Science, 324, 1166(2009).

This work is supported by the National Natural Science

[12] F. Kaiser, P. Vergyris, D. Aktas, C. Babin, L. Labonté, S.

Foundation of China (NSFC) (61435011, 61525504, 61605194);

Tanzilli. Quantum enhancement of accuracy and precision in

the National Key Research and Development Program of China

optical interferometry. arXiv:1701.01621 [quant-ph].

(2016YFA0302600); the China Postdoctoral Science Foundation

[13]Z. Y. Ou, L. Mandel, Observation of Spatial Quantum

(2016M590570); and the Fundamental Research Funds for the

Beating with Separated Photodetectors. Phys. Rev. Lett. 61,

Central Universities. We thank David MacDonald, MSc, from

54-57(1988).

Liwen Bianji, Edanz Editing China (www.liwenbianji.cn/ac), for

[14] T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn,

editing the English text of a draft of this manuscript.

Quantum Beat of Two Single Photons. Phys. Rev. Lett. 93, 070503(2004). [15] J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H.

References [1]

Virgo

Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R.

Collaboration,“Observation of gravitational waves from a binary

LIGO

Scientific

H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L.

black hole merger,” Phys. Rev. Lett., vol. 116, 061102(2016).

O’Brien and M. G. Thompson, On-chip quantum interference

[2] V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A.

between

Aspect, J. Roch, Experimental Realization of Wheeler’s

104-108(2014).

Delayed-Choice

[16] T. Matsuoka, K. Sakai, and K. Takagi, Hyper-resolution

Gedanken

Collaboration

Experiment,

and

Science

315,

silicon

photon-pair

sources,

Nat.

Photon.

8,

966-968(2007).

Brillouin–Rayleigh spectroscopy with an optical beating

[3] R. Ionicioiu and D. R. Terno, Proposal for a Quantum

technique. Rev. of Sci. Instruments 64, 2136 (1993).

[17] S. T. Cundiff and J. Ye, Femtosecond optical frequency

[22] Z.-Y. Zhou, S.-L. Liu, S.-K. Liu, Y.-H. Li, D.-S. Ding, G.-C.

combs, Rev. Mod. Phys. 75, 325(2003).

Guo, and B.-S. Shi, Superresolving Phase Measurement with

[18]J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A.

Short-Wavelength NOON States by Quantum Frequency

Campos, M. C. Teich and B. E. A. Salch, Two-photon

Up-Conversion. Phys. Rev. Appl. 7, 064025 (2017).

interference in a Mach-Zehnder interferometer, Phys. Rev. Lett.

[23]F. Kong, F. N. C. Wong, Extended phase matching of

65, 1348(1990).

second-harmonic generation in periodically

[19] K. Edamatsu, R. Shimizu, and T. Itoh, Measurement of the

poled KTiOPO4 with zero group-velocity mismatch. Appl. Phys.

Photonic de Broglie Wavelength of Entangled Photon Pairs

Lett. 84, 1644 (2004).

Generated by Spontaneous Parametric Down-Conversion. Phys.

[24] K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, Tunable

Rev. Lett. 89, 213601(2002).

midinfrared source by difference frequency generation in bulk

[20]A.

M.

Branczyk,

Hong-Ou-Mandel

interference.

periodically poled KTiOPO4. Appl. Phys. Lett. 74, 914 (1999).

arXiv:1711.00080 [quant-ph]

[25]Z.-Y. Zhou, D.-S. Ding, B.-S. Shi, X.-B. Zou, and G.-C.

[21] Y. Li, Z.-Y. Zhou, D.-S. Ding, B.-S. Shi, CW-pumped

Guo, Characterizing dispersion and absorption parameters of

telecom band polarization entangled photon pair generation in a

metamaterial using entangled photons, Phys. Rev. A 85, 023841

Sagnac interferometer. Opt. Express 23, 28792 (2015).

(2012).

Supplementary of materials Derivation of single count and coincidence counts for narrow bandwidth photon input For a narrow bandwidth photon pair input, the transformation of the annihilation operators when the photon interact at the beam splitter for the first time can be expressed as [1]

ˆ  1 (A ˆ  iAˆ ) A 2 0 1 2 (s1) 1 ˆ ˆ ˆ A3  (iA 0  A1 ) 2 The transformation of birefringence crystal can be expressed as

ˆ  A ˆ [(cos 2  ei y  sin 2  eiz ) e  cos  sin  (ei y  eiz ) e ] A 2 2 H V (s2) ˆ [ e   e ] =A 2

where

H

V

 +  =1 , eH and eV represent the unit vector in the two orthogonal polarization directions; 2

2

i  2 ni ( ,T) L/  (i  y, z ) are the optical phase changes along the y and z axes of the birefringence crystal, where L is the crystal length,

 is the wavelength of the photon and T is the temperature of the crystal. The

transformation of compensate crystal can be expressed as

A3  eik ( ) Lc A3  eic A3  T ( ) A3

(s3)

When photons encounter at the BS again, the transformation of the operators can be expressed as

ˆ  1 (A ˆ   iAˆ  ) A 4 2 3 2 (s4) 1 ˆ ˆ ˆ A5  (iA2  A3 ) 2 By using Eqs.(s1-s3), Eq. (s4) can be reduced as

ˆ  F ( Aˆ , Aˆ ) e  G ( Aˆ , Aˆ ) e A 4 1 0 1 H 1 0 1 V ˆ ˆ ˆ ˆ ˆ A  F (A , A )e  G (A , A )e 5

2

0

1

H

2

0

1

(s5)

V

ˆ , Fˆ ,G ˆ are as following: Where Fˆ1 ,G 1 2 2 1 i Fˆ1   (T   ) Aˆ0  (T   ) Aˆ1 2 2

 Gˆ1  ( Aˆ0  iAˆ1 ) 2 i 1 Fˆ2  (T   ) Aˆ0  (T   ) Aˆ1 2 2 i Gˆ 2  ( Aˆ0  iAˆ1 ) 2

(s6)

Based on Eqs. (s5) and s(6), single photon counts at port 4 and 5 can be expressed as

R4  0,1 Aˆ4 Aˆ4 0,1  0,1 Fˆ1 Fˆ1 0,1  0,1 Gˆ1Gˆ1 0,1 (s7)

1 = [1+ cos 2  cos( y -c )+ sin 2  cos(  z -c )] 2 R5  0,1 Aˆ5 Aˆ5 0,1  0,1 Fˆ2 Fˆ2 0,1  0,1 Gˆ 2Gˆ 2 0,1 1 = [1  cos 2  cos( y  c )  sin 2  cos( z  c )] 2

(s8)

Coincidence counts at port 4 and 5,

R4,5  1,1 Aˆ4 Aˆ5 Aˆ5 Aˆ4 1,1  1,1 ( Fˆ1 Fˆ2 Fˆ2 Fˆ1  Gˆ1Gˆ 2Gˆ 2Gˆ1 +Gˆ1 Fˆ2 Fˆ2Gˆ1  Fˆ1Gˆ 2Gˆ 2 Fˆ1 ) 1,1 (s9) 1 = [1+ cos 4  cos(2 y  2c )+sin 4  cos(2 z  2c )+2sin 2  cos 2  cos(  y   z  2c )] 2 At this point, we obtains all the equations as show in the main text for narrow bandwidth photon input for both single photon and two-photon cases. Derivation of single count and coincidence counts for broad bandwidth photon input The two photon wave function can be expressed as [2]

 (2) (t )   ds di (0 p  s  i ) f (s , i )ei (s i )t aˆs? (s )aˆi† (i ) 0, 0 (s10) Where f (s , i ) is determined by the phase matching condition, and is expressed as

f (s , i )  sin c(

kL ) 2

(s11)

The phase mismatching k  k p (s  i )  ks (s )  ki (i ) 

2 . When heralded single photon is used, the spectral 

distribution of signal and idler can be expressed as

f s (s )   di (0 p  s  i ) f (s , i )  f (s , 0 p  s ), fi (i )   ds (0 p  s  i ) f (s , i )  f (i , 0 p  i )



Therefore (t ) i  d i f (i )e

ii t

(s12)

aˆi f (i ) 0

Aˆ 4 (t  t0 )   di [ F1 (i )eH  G1 (i )eV ]e ii (t t0 ) aˆi (i )

K  0 Aˆ 4 (t  t0 ) (t0 ) i  0

 dd [ F ()e i

i

1

i

H

 G1 (i)eV ]e ii (t t0 ) f (i )eiit0 aˆi (i)aˆi (i ) 0

  di [ F1 (i )eH  G1 (i )eV ] f (i )e iit For single photon input at port 1, the single count at port 4 can be expressed as

(s13)

R4   dti K   dti 2



 d [ F ( )e i

1

i

H

 G1 (i )eV ] f (i )e iit

2



2 2 1 d i T i    i    i  f 2 (i ) (s14)  4 1   d i 1  cos 2  cos( y  c )  sin 2  cos( z  c )  f 2 (i ) 2



When narrow band photon pairs are used, the frequency distribution of the photon pairs can be expressed as

f 2 (i )   (  i 0 ) , and the above expression can be expressed as

R4 



2 2 1 T 0    0    0  4



1  1  cos 2  cos( y  c )  sin 2  cos( z  c )  2

(s15)

Equation (s16) recovery the case of narrow bandwidth photon input case. For a single photon with Gaussian spectrum f (i )  2

1



exp((i  i 0 )2 /  2 ) , and bandwidth of photon is

  2 ln 2 . The phase changes are depended on frequency and temperature, therefore we can expanded them at the central frequency and the temperature of the compensate crystal.

 y  k y ( , T ) L  [k y (0 , T0 )  c  k y ( , T0 ) L  [k y (0 , T0 ) 

k y  k y

(  0 ) 

k y T

(T  T0 )]L

(  0 )]L  k k  z  k z ( , T ) L  [k z (0 , T )  z (  0 )  z (T  T0 )]L  T

(s16)

Therefore the single count rate at port 4 for photon with broad bandwidth can be expressed as

dk y  dk 1 D2 L2 2 R4  1  cos2  cos( LT )  sin 2  exp( ) cos(  z LT )  (s17) 2 dT 4 dT  Where   [k z (0 , T0 )  k y (0 , T0 )]L , D 

k z k y 1 1    , T  T  T0 .   vgz vgy

Similar to the derivation of equation (s18), the single count rate at port five can be expressed as

 Aˆ5 Aˆ5    dti  di [ F2 (i )eH  G2 (i )eV ] f (i )e iit 





2 2 1 di T i  - i    i  f 2 (i )  4

2

(s18)

dk y  dk 1 D2 L2 2 R5  1  cos 2  cos( LT )  sin 2  exp( ) cos(  z LT )  (s19) 2 dT 4 dT 

For two-photon input case,the coincidence count at port 4 and 5 can be expressed as

 dt dt   t , t  Aˆ Aˆ   dt dt   t , t  ( Fˆ s

i

s

s

i

 4

i

s

i

 5

Aˆ5 Aˆ 4   t s , ti 

 1

Fˆ2 Fˆ2 Fˆ1  Gˆ1Gˆ 2Gˆ 2Gˆ1 +Gˆ1 Fˆ2 Fˆ2Gˆ1  Fˆ1Gˆ 2Gˆ 2 Fˆ1 )   t s , ti 

(s20)

Where

 dt dt s

i

  ts , ti  Fˆ1 Fˆ2 Fˆ2 Fˆ1   t s , ti  

(s21) 1 2 2 d  d  f (  ,  )  (      ) T (  ) T (  )   (  )  (  ) s i s i 0p s i s i s i 4 

 dt dt s



i

  ts , ti  Gˆ1Gˆ 2Gˆ 2Gˆ1   ts , ti 

(s22) 1 2 2 d  d  f (  ,  )  (      )  (  )  (  ) s i s i 0p s i s i 4 

1 2 Gˆ1 Fˆ2 Fˆ2Gˆ1  T (i )  (s )   (i )  (s )  T (s )  (i )   (s )  (i ) (s23) 16 1 2 Fˆ1 Gˆ 2 Gˆ 2 Fˆ1   (s )T (i )   (s ) (i )   (i )T (s )   (s )  (i ) (s24) 16

 dt dt s



i

  ts , ti  Gˆ1 Fˆ2 Fˆ2Gˆ1  Fˆ1Gˆ 2Gˆ 2 Fˆ1   ts , ti 

1 2 2 ds di f 2 (s , i ) (0 p  s  i )(  (s )T (i )  T (s )  (i )   (i )  (s )   (s )  (i ) )  8

(s25)

(s , i )  Fˆ1 Fˆ2 Fˆ2 Fˆ1  Gˆ1Gˆ 2Gˆ 2Gˆ1 +Gˆ1 Fˆ2 Fˆ2Gˆ1  Fˆ1Gˆ 2Gˆ 2 Fˆ1 1  {1  cos 4  cos( ys  yi  ys 0  yi 0 )  sin 4  cos( zs  zi  ys 0  yi 0 )  2 (s26) z  z  ys  yi z  z  ys  yi  2 ys 0  2 yi 0 2 cos 2  sin 2  cos( s i ) cos( s i ) 2 2 z  yi z  ys z  z  ys  yi z  z  ys  yi  2 ys 0  2 yi 0 2 cos 2  sin 2  sin( i ) sin( s )[cos( s i )  cos( s i )]} 2 2 2 2 Where the symbols in Eq. (s26) are as follows:

y j  k y ( j , T ) L  [k y ( j 0 , T0 ) 

k y

( j   j 0 ) 

k y

(T  T0 )]L  T k y j 0  k y ( j , T0 ) L  [k y ( j 0 , T0 )  y ( j   j 0 )]L ( j = s, i ) (s27)  k k z j  k z ( j , T ) L  [k z ( j 0 , T )  z ( j   j 0 )  z (T  T0 )]L  T After some detail calculation, the coincidence count at port 4 and 5 can be expressed as

 dt dt   t , t  Aˆ Aˆ Aˆ Aˆ   t , t    d d f ( ,  ) (     )  ,   s

i

s

 4

i

 5

5

4

s

i

2

s

i

s

i

0p

s

i

s

i

dk dk 1  {1  cos 4 ( ) cos(2 y LT )  sin 4 ( ) cos(2 z LT  2 )  2 dT dT dk z dk y D 2 L2 2 2 2 2 cos ( ) sin ( ) cos[(  ) LT   ]exp( )} dT dT 4

(s28)

The above equations (s17), (s19) and (s28) show the influence of bandwidth of photon pairs to single counts and coincidence counts. Eq. (s26) is a general equation to describe the basic properties of a birefringence MZI, this formula can be used to study other optical properties of a birefringence crystal such as wavelength dispersion and electrical-optical effects. The above formula are calculated for photon pairs with Gaussian spectrum. Actually for photon with spectrum of Sinc2 function f (i )  2

 sinc 2 [ (i  i 0 )] , where   DLspdc / 2 is determined by the 2

parameters of the SPDC crystal. The expressions for single count rate and coincidence count rate is as following for

Lspdc  L :

 dk L L dk 1 R4  1  cos 2  cos( y LT )  sin 2  spdc cos(  z LT )   2  dT Lspdc dT 

(s29)

 dk L L dk 1 R5  1  cos 2  cos( y LT )  sin 2  spdc cos(  z LT )  (s30)  2  dT Lspdc dT  dk dk 1 R45  {1  cos 4 ( ) cos(2 y LT )  sin 4 ( ) cos(2 z LT  2 )  2 dT dT (s31) Lspdc  L dk z dk y 2 2 2 cos ( )sin ( ) cos[(  ) LT   ] } dT dT Lspdc While for Lspdc  L ,expressions (s29) to (s31) become

dk y  1 R4  1  cos2  cos( LT )  (s32) 2 dT  dk y  1 R5  1  cos2  cos( LT )  (s33) 2 dT  dk dk 1 R45  [1  cos 4 ( ) cos(2 y LT )  sin 4 ( ) cos(2 z LT  2 )] (s34) 2 dT dT

References [1] K. Edamatsu, R. Shimizu, and T. Itoh, Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion. Phys. Rev. Lett. 89, 213601 (2002). [2] Z.-Y. Zhou, D.-S. Ding, B.-S. Shi, X.-B. Zou, and G.-C. Guo, Characterizing dispersion and absorption parameters of metamaterial using entangled photons, Phys. Rev. A 85, 023841 (2012).