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Dispersion cancellation with an energy-time entangled photon pair in Hong-Ou-Mandel (HOM) interference is one phenomenon that reveals the nonclassical nature of the entangled photon pair. This phenomenon has been observed in materials with very weak dispersions. If the higher-order dispersion coefficient is non-negligible, then the experiment must be modified to realize dispersion cancellation. All-order dispersion cancellation using balanced dispersion was suggested by Steinberg. However, the same phenomenon is expected to occur even if a photon pair is not entangled. This behaviour can be explained by path indistinguishability with identical dispersion. To achieve an all-order dispersion experiment that cannot be explained classically, we modified the experiment and performed another all-order dispersion cancellation experiment that cannot be explained by identical dispersion. This is the first demonstration of nonclassical all-order dispersion cancellation.
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Fig. 1 Nonclassicality of dispersion cancellation.(a) Nonlocal dispersion cancellation of Franson. Signal and idler photons pass through dispersive materials with opposite dispersion coefficients. The temporal correlation between the signal and idler photons is preserved. (b) Local dispersion cancellation of Steinberg. A dispersive material is placed in the path of a signal photon. The shape of the HOM dip is preserved, and only the position of the dip is shifted. (c) All-order dispersion cancellation with balanced dispersion. Identical dispersive materials are placed in the paths of the signal photon and the idler photon. Even if the material has an arbitrary dispersion property, the HOM dip will be preserved. (d) Balanced dispersion with a modified HOM interferometer. The Mach-Zender interferometer is placed in one path of the interferometer in (c).
Fig. 2 Experimental setups. (a) Signal and idler photons are generated via SPDC. The two photons are energy-time entangled. (b) HOM interference with balanced dispersion. Both signal and idler photons pass through respective Rb vapour cells and are then detected by a Single-photon Avalanche Detector (SPAD, Perkin Elmer SPCM-AQRH-1X). At the same temperature, the dispersion properties of the two cells are identical. (c) Balanced dispersion in a modified HOM interferometer. Mach-Zender interferometer is placed in one arm of the interferometer.
Fig. 3 Two-photon interference in a modified HOM interferometer with balanced dispersion. When photons are detected at D1 and D2, the two photons could have originated from four indistinguishable paths. The temporally advanced two-photon detection amplitude can be transmitted or reflected at the beam splitter. The temporally delayed amplitude also can be transmitted or reflected at the beam splitter. . Since the signal photon path length for the amplitude generated earlier is relatively long, the amplitudes can interfere at the detectors. The coincidence count when Lo = (LLong + LShort)/2 is determined by this interference. The effective photon wave function corresponding to each path is marked. (b) Biphoton wave packet after balanced dispersion. Only the width of the wave packet is increased via even-order dispersion. The effective two-photon wave functions corresponding to the paths in (a) are drawn in the temporal domain. Wave packets connected by the arrows are pairs that interfere when Lo = (LLong + LShort)/2. If the phase difference between them is π, they interfere destructively, resulting in a dip. If the phase difference is 0, they interfere constructively, generating a peak. T+ = t1 + t2 and L+ denotes sum of signal path length and idler path length. L+ = Ls + Li.
Fig. 4 Distortion of a wave packet by dispersion. If a Gaussian pulse passes through a Rb cell, its shape becomes distorted because of dispersion. At 60°C, the effect of dispersion is negligible, whereas at 140°C, the Gaussian wave packet is clearly distorted. After dispersion, the wave packet is highly asymmetric, and the width of the wave packet is increased by more than 10 times.
Fig. 5 Distortion and restoration of HOM interference pattern. (a) HOM interference patterns for different RB cell temperatur-es. The temperature of the Rb cell in the idler path is varied (142-150°C), whereas the temperature of the Rb cell in the signal-photon path is fixed at 140°C. (b) Visibility of the HOM dip. The visibility depends on the temperature of the Rb cell in the idler path. (c) Two-photon interference pattern in the modified HOM interferometer. As the phase difference changes in the Mach-Zender interferometer, the interference pattern near position 0 changes from a dip to a peak. (d) Restoration of the two-photon interference pattern in the modified HOM interferometer with balanced dispersion. The phase difference between the short and long paths in MZI is 0, and a peak is generated by interference.
Fig. 6 HOM interference and dispersion cancellation. (a) Dispersion cancellation experiment of Steinberg . The signal photon of the entangled photon pair travels in free space, whereas the idler photon passes through a dispersive material (glass sample L). The two photons meet at the beam splitter(BS), and HOM interference occurs. The shape of the interference pattern is not distorted, and only the position of the dip is shifted. μ, −μ are detunings from the half-centre frequencies of photons detected at detector 1(D1) and detec-tor 2(D2), respectively. (b) The dispersion experiment of Stei-nberg performed with two independent single-photon wave packets. The wave packet is distorted after it passes through the dispersive material. Different from the original experim-ent of Steinberg, dispersive material has strong dispersion property, i.e., high order dispersion coefficients are non-negligible, in this figure. Whether dispersion is weak or strong, dispersion cancellation does not occur because the two wave packets cannot overlap completely at the beam splitter. (c) Two independent and separable wave packets pass through a dispersive material. The dispersion of the two wave packets is identical. Therefore, the two wave packets are indistinguishable at the BS, and the HOM dip is restored without entanglement.
Fig. 7 (a) Biphoton wave packet. If ts−ti has the same value, the photons are detected with the same probability, regardl-ess of ts, ti. The amplitudes at different times exhibit only a phase difference. The amplitudes at different times exhibit two-photon mutual temporal coherence. (b)If the signal and idler photons travel different distances, the temporal distribution of the amplitude is shifted. The shift is determined by the length of the path difference.
Fig. 8 (a) Photo-detection amplitude distribution along the t1-t2 axis. The wave packets from the RR and TT paths are shifted in the opposite directions and are anti-symmetric around the t1-t2 = 0 point. The two wave packets have opposite phase, and the whole wave packet is the superposition of the two wave packets. (b) If the path length difference is decreased, the two wave packets overlap. The overlapping region causes interference terms, and the coincidence count rate is changed. This results in HOM interference. When the path length difference is 0, the two wave packets overlap completely and cancel each other out. Thus, the coincidence count rate is 0. (c) HOM interference after dispersion. Because the amplitudes of the TT and RR paths are asymmetric, they do not interfere completely. Thus, the interference pattern is distorted. (d) HOM interference after odd-order dispersion cancellation. Because two wave packets with even-order dispersion have symmetric distributions, complete destructive interference occurs when the path length difference is 0.
Fig. 9 (a) If a Gaussian wave packet passes through a medium with strong dispersion, the wave packet becomes distorted. Additionally, because of odd-order dispersion, the shape of the wave packet becomes asymmetrical. (b) Nonlocal dispersion cancellation of odd-order dispersion. When only one photon passes through a dispersive material, the biphoton wave packet is distorted. If another photon also passes through an identical dispersive material, then odd-order dispersion is cancelled out because of frequency anti-correlation. Thus, only the even-order dispersion effect remains, and the wave packets become symmetric.
Fig. 10 (a) Biphoton wave packet distribution after the BS when L0 = (LLong + LShort)/2 in modified HOM inteference. The arrow indicates that the amplitude generated earlier and the amplitude generated later meet and interfere. The amplitude generated earlier, which travelled along the TT path, interferes with the amplitude generated later that travelled along the RR path and vice versa. T+ = t1 + t2 and L+ denotes sum of signal path length and idler path length. L+ = Ls + Li. (b) Biphoton wave packet after the BS in the presence of a dispersive material. The amplitudes that travelled along the TT and RR paths do not overlap completely. Constructive or destructive interference cannot occur completely, and the interference pattern is distorted. (c) Biphoton wave packet after the BS with balanced dispersion. Because the wave packets are symmetric, the amplitudes that travelled along the TT and RR paths overlap completely. Thus, constructive or destructive interference can occur completely, and the interference pattern is restored.
(1) Δ ( t s − t i ) Δ ( ω s + ω i − ω p ) ≤ 1.
(4) P 12 = 1 2 ∫ − ∞ ∞ d μ | F ( μ ) | 2 ( 1 − cos ( 2 μ Δ L c ) ) = 1 2 ( 1 − ∫ − ∞ ∞ d μ | F ( μ ) | 2 cos ( 2 μ Δ L c ) ) .
(5) P 12 = 1 2 ∫ − ∞ ∞ d μ | F ( μ ) | 2 ( 1 − cos ( 2 μ ( Δ L − L c + α L ) ) ) = 1 2 ( 1 − ∫ − ∞ ∞ d μ | F ( μ ) | 2 cos ( 2 μ ( Δ L − L c + α L ) ) ) .
(7) P 12 = 1 2 ∫ − ∞ ∞ d μ | F ( μ ) | 2 ( 1 − cos ( 2 μ ( Δ L − L c + ( α + γ μ 2 ) L ) ) ) = 1 2 ( 1 − ∫ − ∞ ∞ d μ | F ( μ ) | 2 cos ( 2 μ ( Δ L − L c + ( α + γ μ 2 ) L ) ) ) .
(9) k s ( μ ) + k i ( − μ ) = 2 k 0 + ( α s − α i ) μ + ( β s + β i ) μ 2 + ( γ s − γ i ) μ 3 + ... .
(10) k s ( μ ) + k i ( − μ ) = 2 k 0 + 2 β s μ 2 + ... .
(11) | ψ 〉 μ , - μ = | μ 〉 s | - μ 〉 i | ψ ' 〉 μ , - μ = exp [ i ( k s ( μ ) + k i ( - μ ) L ) ] | μ 〉 s | - μ 〉 i = exp 2 i [ k 0 + 2 β s μ 2 ] | μ 〉 s | - μ 〉 i | ψ ' 〉 - μ , μ = exp 2 i [ k 0 + 2 β s μ 2 ] | - μ 〉 s | μ 〉 i .
(13) ψ 12 ( t 1 , t 2 ) = ψ ( t 1 0 T , t 2 L T ) + ψ ( t 1 L R , t 2 0 R ) + ψ ( t 1 0 T , t 2 S T ) + ψ ( t 1 S R , t 2 0 R ) .
(14) ∫ − ∞ ∞ d ω 1 Δ ω 2 π e − ( ω − ω 0 ) 2 2 ( Δ ω ) 2 e i ω c ( n ( ω ) − 1 ) L c e l l e − i ω t .
(15) n ( ω ) = ∫ - ∞ ∞ d ω G ( ω ) [ 1 - ∑ j = 1 2 N | μ j | 2 2 ε h ∑ k = 1 4 g j k ω - ω j k + i ( γ j + γ c ) ] .
(19) | Ψ 〉 = ∫ d μ F ( μ ) a ^ s † ( ω s 0 + μ ) a ^ i † ( ω i 0 − μ ) | 0 〉 .
(23) | Ψ 〉 μ = e i ϕ ( μ ) ( | μ 〉 s | − μ 〉 i + e − i ( 2 μ ( Δ L − L c + α L ) ) | − μ 〉 s | μ 〉 i ) .
(24) P 12 ( μ , − μ ) = 1 2 ( 1 − cos ( 2 μ ( Δ L − L c + α L ) ) ) .
(25) P 12 = 1 2 ∫ − ∞ ∞ d μ | F ( μ ) | 2 ( 1 − cos ( 2 μ ( Δ L − L c + α L ) ) ) = 1 2 ( 1 − ∫ − ∞ ∞ d μ | F ( μ ) | 2 cos ( 2 μ ( Δ L − L c + α L ) ) ) .
(29) 〈 0 | E ^ D 2 ( + ) ( t 2 ) E ^ D 1 ( + ) ( t 1 ) | Ψ 〉 = 1 2 ( g d ( t 2 − L i c ) g ( t 1 − L s c ) − g ( t 2 − L s c ) g d ( t 1 − L i c ) ) .
(30) ∬ d t 1 d t 2 | 〈 0 | E ^ D 2 ( + ) ( t 2 ) E ^ D 1 ( + ) ( t 1 ) | Ψ 〉 | 2 = 1 2 ( 1 - Re ∬ d t 1 d t 2 g d * ( t 2 , i ) g ( t 2 , s ) g d * ( t 1 , s ) g ( t 1 , i ) ) .
(31) ∬ d t 1 d t 2 | 〈 0 | E ^ D 2 ( + ) ( t 2 ) E ^ D 1 ( + ) ( t 1 ) | Ψ 〉 | 2 = 1 2 ( 1 - Re ∬ d t 1 d t 2 g d * ( t 2 , i ) g d ( t 2 , s ) g d * ( t 1 , s ) g d ( t 1 , i ) ) .
(32) ∫ d t 1 g d * ( t 1 , s ) g d ( t 1 , i ) = ∫ d t 1 , s g d * ( t 1 , s ) g d ( t 1 , s + L s − L i c ) .
(34) ∫ d t 1 g d * ( t 1 , s ) g d ( t 1 , i ) = ∫ d t 1 , s g * ( t 1 , s ) g ( t 1 , s + L s − L i c ) .
(35) 〈 0 | E ^ s ( + ) ( t s ) E ^ i ( + ) ( t i ) | Ψ 〉 = e i ω p 2 ( t s + t i ) g ( t s − t i ) .
(36) 〈 0 | E ^ s ( + ) ( t 1 ) E ^ i ( + ) ( t 2 ) | Ψ 〉 = e i ω p 2 ( t 1 + t 2 − L s c − L i c ) g ( t 1 − L s c − t 2 + L i c ) = e i ω p 2 ( t 1 + t 2 − L s + L i c ) g ( t 1 − t 2 − Δ L c ) .
(39) | ∫ − ∞ ∞ d t φ ( t ) * φ ( − t ) | < 1.
(40) v ( τ ) = ∫ − ∞ ∞ d t φ * ( t + τ ) φ ( − t ) .
(41) Φ ( t 1 , t 2 ) = e i ω p ( t 1 + t 2 ) g ( t 1 − t 2 ) .
(43) V = | − 2 T Re ( ∫ 0 T d t 1 ∫ 0 T d t 2 Φ T T * ( t 1 , t 2 ) Φ R R ( t 1 , t 2 ) ) | .
(44) V = | 1 T Re ( ∫ 0 T d t 1 ∫ 0 T d t 2 g * ( t 1 − t 2 ) g ( − ( t 1 − t 2 ) ) ) | = | Re ( ∫ d t − g * ( t − ) g ( − t − ) ) | .
(45) V = | ∫ d t − g * ( t − ) g ( − t − ) | .
(46) D . A . = 1 − V .

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