Coherently controlled quantum features in a coupled interferometric scheme
11 Coherently controlled quantum features in a coupled interferometric scheme
Byoung S. Ham
Center for Photon Information Processing, School of Electrical Engineering and Computer Science, Gwangju Institute of Science and Technology 123 Chumdangwagi-ro, Buk-gu, Gwangju 61005, South Korea [email protected] (submitted on Feb. 18, 2021)
Abstract:
Over the last several decades, entangled photon pairs generated by spontaneous parametric down conversion processes in both ππ ( ) and ππ ( ) nonlinear optical materials have been intensively studied for various quantum features such as Bell inequality violation and anticorrelation. In an interferometric scheme, anticorrelation results from photon bunching based on randomness when entangled photon pairs coincidently impinge on a beam splitter. Compared with post-measurement-based probabilistic confirmation, a coherence version has been recently proposed using the wave nature of photons. Here, the origin of quantum features in a coupled interferometric scheme is investigated using pure coherence optics. In addition, a deterministic method of entangled photon-pair generation is proposed for on-demand coherence control of quantum processing. Introduction
Quantum entanglement [1] is the heart of quantum technologies such as quantum computing [2], quantum communications [3,4], and quantum sensing [5,6]. Although intensive research has been performed in both interferometric and noninterferometric schemes for quantum features such as a the Hong-Ou-Mandel (HOM) dip [7-9], photonic de Broglie wavelength (PBW) [10-13], Bell inequality violation [14-16], and Franson-type nonlocal correlation [17-20], the fundamental physics of entangled photon-pair generation itself has still been vailed in terms of probabilistic measurements via coincidence detection of coupled photon pairs. Thus, nondeterministic measurement-based quantum technologies have prevailed, resulting in an extreme inefficiency compared with their classical counterparts. The classical technologies are of course deterministic and macroscopic. Recently, a novel method of deterministic quantum correlation has been proposed and demonstrated to unveil secretes of quantum entanglement for both HOM dip and PBW using the wave nature of photons [21-23]. As a result, the fundamental physics of quantum features has been found in the phase property of a coupled system, where the coupled system does not have to be confined by the Heisenbergβs uncertainty principle. Based on this wave nature of photons, collective control of coherent photons becomes a great benefit for macroscopic quantum technologies compatible with the classical counterparts. Here, the fundamental physics of quantum correlation is investigated using the wave nature of photons to find the origin of quantum features demonstrated in an interferometric scheme [24]. For typical ππ ( ) β generated entangled photon pairs, some misunderstanding regarding quantum correlation are pointed out not to criticize but to support the novelty of the wave nature of photons. Without violating quantum mechanics, a proper choice of photon property should depend on photon resources according to the wave-particle duality [25]. Finally, a coherence version of quantum feature generation is proposed for potential applications of deterministic and macroscopic quantum information processing. Figure 1 shows a particular scheme of HOM-type quantum correlation in a coupled interferometric scheme, where entangled photon pairs are generated from spontaneous parametric down conversion (SPDC) processes in a ππ ( ) nonlinear material [24]. Due to the spontaneous emission decay process, an initially given phase is randomly assigned to each photon pair, where each photon pair has also a random frequency detuning from the fixed half-frequency of the pump photon used for ππ ( ) . In related HOM-type experiments, a typical line shape observed by coincidence measurements shows a broad dip, whose decay is the inverse of the photon pairsβ bandwidth. Unlike the theory in ref. [20] based on the wave nature of photons, however, Ξ» β dependent ππ ( ) correlation has never been observed. In the present paper, both reasons for the missing ππ ( ) correlation in a single Mach-Zehnder interferometer (MZI) and the revival of ππ ( ) correlation in a coupled MZI are investigated to unveil the physical origin of quantum features. Results
For the analytic discussion as to why there is no ππ ( ) correlation in a HOM dip, we start with a typical SPDC-generated entangled-photon system as shown in Fig. 1(a). Assuming there is a specific phase relation between the paired photons, signal ( πΈπΈ ππ ) and idler ( πΈπΈ πΌπΌ ) , the basic equations for coincidence detection measurements can be derived using general matrix representations of pure coherence optics, where οΏ½πΈπΈ πΌπΌ πΈπΈ π½π½ οΏ½ =[ π΅π΅π΅π΅ ΞΆ ] οΏ½πΈπΈ πΌπΌ πΈπΈ ππ οΏ½ and οΏ½πΈπΈ π΄π΄ πΈπΈ π΅π΅ οΏ½ = [ π΅π΅π΅π΅ Ο ][ π΅π΅π΅π΅ ΞΆ ] οΏ½πΈπΈ πΌπΌ πΈπΈ ππ οΏ½ , [ π΅π΅π΅π΅
2] = [
π΅π΅π΅π΅
1] = οΏ½ ππππ οΏ½ , [ ΞΆ ] = οΏ½ππ ππππ
00 1 οΏ½ , and [ Ο ] = οΏ½ ππ ππππ οΏ½ [21]. Here, introduction of coherence optic is a choice matter without violation of quantum mechanics [25]. The j th input photon pair πΈπΈ ππ ππ and πΈπΈ πΌπΌ ππ are described with the wave nature property, where πΈπΈ ππ ππ = πΈπΈ ππ ππ ( ππ ππππ ππβ2ππππ
ππππ π‘π‘+ππ
ππππ ) and πΈπΈ πΌπΌ ππ = πΈπΈ ππ ππ ( ππ πΌπΌππ ππβ2ππππ
πΌπΌππ π‘π‘+ππ
πΌπΌππ ) . The photon pair generation rate and bandwidth in SPDC can be controlled by adjusting a pump power and a spectral filter. In general, the detectable photon rate by a single photon detector module is far less than MHz. Considering a detection module speed larger than GHz, consecutive photon pairs are treated independently throughout the coincidence measurement process. The coherent property of each generated photon pair is determined by Heisenbergβs uncertainty principle in terms of the energy-time relation: π₯π₯π₯π₯π₯π₯π₯π₯ β₯ . For a typical THz bandwidth π₯π₯π₯π₯ , the coherence time
π₯π₯π₯π₯ is in the order of ps. Compared with the corresponding coherence length ππ πΆπΆ ~100 ππππ , the original wavelength ππ of the pump is far shorter than ππ πΆπΆ . In other words, ππ ( ) correlation is much sensitive than ππ ( ) correlation. Fig. 1. Interferometric quantum feature generation. (a) A SPDC-based photon-pair bandwidth. (b) a SPDC-based coupled interferometric scheme. BS: beam splitter, D: detector, M: mirror. ππ = ΞπΏπΏ ; ππ = ΞπΏπΏ ; β : bandwidth; πΏπΏ ππ : random symmetric detuning of the j th entangled pair. According to the energy conservation law, the signal and idler photons in each pair are symmetrically detuned by Β± πΏπΏ ππ from the half-frequency ( π₯π₯ /2) of the pump laser as shown in Fig. 1(a). Due to spontaneous emission processes, however, the frequencies π₯π₯ ππ ππ and π₯π₯ πΌπΌ ππ of the j th photon pair are random within the bandwidth π₯π₯π₯π₯ . Likewise, the initially given phases ππ ππ ππ and ππ πΌπΌ ππ are not determined, either. As analyzed, however, the difference phase πΏπΏππ ππ between ππ ππ ππ and ππ πΌπΌ ππ is fixed at Ο /2 [20]. This fact will be derived differently below based on Fig. 1(a). Figure 1(b) originats in ref. 24 and is used to understand important quantum features. The first (second) MZI is controlled by βπΏπΏ ( βπΏπΏ ) , where ππ ππ = ππ ΞπΏπΏ οΏ½Ο ππ = ππ ΞπΏπΏ οΏ½ , and ππ ππ is the j th photonβs wavelength. Regardless of nondegeneracy in Ο ( ) , all pairs are symmetrically detuned, whose corresponding phase difference is Β± Ξ΄ ππ ππ = Β± ππ ππ , where ππ is the relative delay between paired photons for measurements. The coincidence measurements between output ports Ξ± and Ξ² on a beam splitter BS1 are for intensity correlation ππ ( ) ( Ο ) , where the j th output intensities are as follows (see Fig. S1 of the Supplementary Information): πΌπΌ πΌπΌππ ( ππ , π₯π₯ ) = πΌπΌ οΏ½ π π πππ π οΏ½ΞΆ ππβ² οΏ½οΏ½ , (1) πΌπΌ π½π½ππ ( ππ , π₯π₯ ) = πΌπΌ οΏ½ β π π πππ π οΏ½ΞΆ ππβ² οΏ½οΏ½ . (2) Here, the phase ΞΆ ππβ² is described as: ΞΆ ππβ² ( ππ , π₯π₯ ) = οΏ½ ππ β πΏπΏππ ππ οΏ½ ΞπΏπΏ β οΏ½ ππ β πΏπΏππ ππ οΏ½ ππ + πΏπΏππ ππ β οΏ½πΏπΏππ ππ ππ π π β πΏπΏππ ππ π₯π₯ π π οΏ½ . (3) For all πΏπΏ ππ β dependent photon pairs, πΌπΌ πΌπΌ = β πΌπΌ πΌπΌππππ and πΌπΌ π½π½ = β πΌπΌ π½π½ππππ . Equation (3) represents four different sources of the induced phase ΞΆ ππβ² . The first one οΏ½ ππ ΞπΏπΏ οΏ½ is a center frequency-related fundamental oscillation as a function of ΞπΏπΏ : ππ β dependent fast oscillation. The second one οΏ½πΏπΏππ ππ ΞπΏπΏ οΏ½ is the detuning-caused slow oscillation, resulting in π₯π₯π₯π₯ β1 ( ππ ) β dependent decoherence. The third one οΏ½πΏπΏππ ππ οΏ½ is for a fixed relative phase Ο /2 between the signal and idler photons in each pair. The last one οΏ½ πΏπΏππ ππ ππ π π οΏ½ is for ΞπΏπΏ β independent frequency beating between the paired photons, resulting in a fixed phase. Due to the wide spectrum in Fig. 1(b), this beating results in a π₯π₯π₯π₯ β1 ( ππ ) β dependent wide envelope. Thus, equation (3) becomes a function of ΞπΏπΏ (or Ο ) only. However, all Ξ΄π₯π₯ ππ β caused phase factors in equation (3) cancel each other out due to the Β± Ξ΄π₯π₯ ππ distribution of all photon pairs except for the fixed πΏπΏππ ππ at coincidence detection. Thus, the mean values of the output intensities are uniform, resulting in β©πΌπΌ πΌπΌ βͺ = β©πΌπΌ βͺ = πΌπΌ due to β©π π πππ π οΏ½ΞΆ ππβ² οΏ½βͺ = 0 , where the signal and idler photons are interchangeable. This is the physical origin why there is no ππ ( ) correlation in ππ ( ) ( ππ ) in the first MZI. As analyzed for the second MZI below, this fact also becomes the physical origin as to how ππ ( ) is retrieved in ππ ( ) ( ππ ) as observed in ref. 24. Fig. 2. Numerical simulations of intensity correlation in a typical HOM dip with πΏπΏππ ππ = Β± ππ /2 . (a) Photon distribution. (b) Ο vs. πΏπΏ ππ . (c) and (d) Sum ππ ( ) ( ππ ) for all πΏπΏ ππ for different coverage π₯π₯π₯π₯ . Dotted: πΏπΏππ ππ = 0 . According to the definition of intensity correlation ππ πΌπΌπ½π½ ( ) ( ππ ) = β©πΌπΌ πΌπΌ πΌπΌ π½π½ βͺβ©πΌπΌ πΌπΌ βͺβ©πΌπΌ π½π½ βͺ , the following equation results in: ππ πΌπΌπ½π½ ππ ( ) ( ππ , πΏπΏ ππ ) = πππππ π οΏ½ΞΆ ππβ² οΏ½ . (4) To satisfy anticorrelation of ππ πΌπΌπ½π½ ππ ( ) οΏ½ππ = 0, πΏπΏ ππ οΏ½ = 0 , in a SPDC-based HOM dip, πΏπΏππ ππ = Β± ππ /2 must be satisfied for each photon pair [20]. If Ο β , equation (4) gradually decays and shows a typical HOM dip curve as a function of delay time Ο ππ (= Ξπ₯π₯ β1 ) , where decay time Ο ππ in ππ ( ) ( ππ ) οΏ½ = β ππ ππ ( ) ( ππ , πΏπΏ ππ ) ππ οΏ½ is preset according to the inverse of the SPDC-generated photon bandwidth βπ₯π₯ as shown in Fig. 2. Figure 2 shows numerical calculations for equation (4). Figure 2(a) shows the Gaussian distribution of SPDC-generated photon pairs with the bandwidth of
π₯π₯π₯π₯ = 0.5x10 radians. According to Fig. 1, the j th photon pair has different detuning at Ξ΄π₯π₯ ππ , whose corresponding ππ ( ) ( ππ , πΏπΏπ₯π₯ ππ ) is shown in Fig. 2(b). In equation (4), the j th photon pair must have different ππ ( ) ( ππ ) only due to the detuning dependent ΞΆ ππβ² . By definition, ππ ( ) ( ππ ) is obtained via averaging all πΏπΏπ₯π₯ ππ β dependent coincidence measurements for a fixed Ο . As shown in Figs. 2(c) and (d), maximum ππ ( ) ( ππ ) is bound to ππ ( ) ( ππ ) = 0.5 , where ππ ( ) ( ππ ) = 0.5 is a classical lower bound [20]. This upper limit of ππ ( ) ( ππ ) = 0.5 strongly supports the nonclassical phenomenon of entangled photon pairs [24]. If all of the spectral photon pairs are not fully covered for measurements, there is a wiggle in ππ ( ) ( ππ ) as shown in Fig. 2(c). This wiggle is due to incomplete coherence washout in the summation process [24]. Disappearance of the ππ ( ) fringe in a HOM dip is not due to the measurement process or artifacts, but instead due to the inherent properties of the symmetrically detuned photon pairs in SPDC. If there is no relative phase between signal and idler photons, then there is no nonclassical feature in ππ ( ) ( ππ ) as indicated by the dotted curve in Fig. 2(d). If the relative phase πΏπΏππ ππ is random for all pairs, ππ ( ) ( ππ ) = 1/2 regardless of Ο , representing the property of individual particle ensemble [20]. In the second MZI in Fig. 1(b), the ΞπΏπΏ effect can be classified for bunched photons only on BS1 if ΞπΏπΏ ~0 . According to equation (3), all other terms become zero except for Ξ΄ππ ππ , which is Ο /2 for all j. Here, it should be noted that the bunched photons in each path of the MZI are composed of signal and idler photon pairs, whose detuning is exactly opposite across the center frequency π₯π₯ /2 . Thus, whenever a nonzero ΞπΏπΏ occurs, the detuning Ξ΄π₯π₯ ππ β caused phase terms in equation (3) are cancelled out automatically due to the +/ β relation in Ξ΄π₯π₯ ππ . As a result, only the original ππ β dependent fast oscillation survives in the output fields. This is the unspoken secretes in the SPDC-based ππ ( ) features observed in ref. 24 for ππ ( ) measurements. In the second MZI of Fig. 1(b), the following amplitude relations are obtained for the final outputs E π΄π΄ and E π΅π΅ : οΏ½πΈπΈ π΄π΄ πΈπΈ π΅π΅ οΏ½ ππ = οΏ½ β ππ ππππ ππ ππππ ππππ ππβ² οΏ½ ππ ππππ ππ οΏ½πποΏ½ ππ ππππ ππ οΏ½ βππ ππππ ππβ² οΏ½ β ππ ππππ ππ οΏ½οΏ½ οΏ½πΈπΈ ππ πΈπΈ πΌπΌ οΏ½ ππ . (5) From equation (5), the corresponding intensities are as follows (see Fig. S2 of the Supplementary Information): πΌπΌ π΄π΄ππ = πΌπΌ (1 β πππππ π ππ ππβ² sin ππ ππ ) , (6) πΌπΌ π΅π΅ππ = πΌπΌ (1 + πππππ π ππ ππβ² sin ππ ππ ) . (7) The anticorrelation condition ππ ππβ² = Β± ππ /2 in equation (3), however, results in independence of ππ ππ . If ππ ππβ² = 0 , πΌπΌ π΄π΄ππ = πΌπΌ οΏ½ β sin ππ ππ οΏ½ and πΌπΌ π΅π΅ππ = πΌπΌ οΏ½ ππ ππ οΏ½ are obtained. In this case, however, the photon bunching or anticorrelation in equations (3) and (4) is violated, resulting in the classical feature of ππ πΌπΌπ½π½ ππ ( ) ( ππ , πΏπΏ ππ ) = 1 from equation (4) (see Fig. S3 of the Supplementary Information). Although the normalized coincidence detection measurement becomes π π π΄π΄π΅π΅ππ = οΏ½ πππππ π ππ ππ οΏ½ , ππ π΄π΄π΅π΅ ππ ( ) ( ππ ππ , ππ ππ , πΏπΏ ππ ) = 1 shows a classical feature. In fact, the πππππ π ππ ππ modulation term in π π π΄π΄π΅π΅ is a typical classical feature of the intensity product from a single MZI. In other words, satisfying ππ πΌπΌπ½π½ ππ ( ) οΏ½ππ , πΏπΏ ππ οΏ½ = 0 for photon bunching violates π π π΄π΄π΅π΅ππ = οΏ½ πππππ π ππ ππ οΏ½ (see Fig. S3 of the Supplementary Information). Thus, the observations of πππππ π ππ ππ modulation in ref. 24 are not quantum but classical as shown in Fig. 3(d). To be quantum, both MZI paths must have bunched photons within coherence time as demonstrated [12,13]. To work with N β₯ for PBW, an inter-MZI superposition scheme can be a quantum solution as proposed [21,22] and demonstrated [23]. Figure 3 shows a coherence version of the entangled photon-pair generation comparable with Fig. 1. Because MZI works for either a single photon or coherence light equivalently [26], the results have no difference for the photon characteristics. The photons propagating along different paths of MZI 1 is strongly coupled by the relative phase of Ο /2 created from the first BS, regardless of the input photonβs wavelength [27]. The matrix representations for Fig. 3(a) are as follows without considering ΞπΏπΏ : πΌπΌ πΌπΌ = πΌπΌ (1 β πππππ π ππ ) , πΌπΌ π½π½ = πΌπΌ (1 + πππππ π ππ ) , πΌπΌ π΄π΄ = πΌπΌ [1 β π π πππ π πππ π πππ π ππ ] , and πΌπΌ π΅π΅ = πΌπΌ [1 + π π πππ π πππ π πππ π ππ ] (see Fig. S4 of the Supplementary Information). Using an acousto-optic modulator (AOM) driven by an rf pulse generator with an rf frequency of π₯π₯ ππππ , the role of Ξ΄π₯π₯ ππ β caused random phases in Fig. 1 can be satisfied by a 50% duty cycle of AOM between 0 and π₯π₯ ππππ , as shown in Fig. 3(b). In other words, the zeroth (original π₯π₯ ) and first-order ( π₯π₯ + π₯π₯ ππππ ππ ) diffracted light pulses are used, where T is the rf pulse duration. If ππππ ππ = ππ , the output direction is reversed. Thus, the average of each output intensity becomes uniform, β©πΌπΌ πΌπΌ βͺ = β©πΌπΌ π½π½ βͺ = β©πΌπΌ π΄π΄ βͺ = β©πΌπΌ π΅π΅ βͺ = πΌπΌ , satisfying randomness. Including the ΞπΏπΏ effect in ΞΆ , the revised output intensities are as follows: πΌπΌ πΌπΌ = πΌπΌ (1 β πππππ π ππ β² ) , (8) πΌπΌ π½π½ = πΌπΌ (1 + πππππ π ππ β² ) , (9) πΌπΌ π΄π΄ = πΌπΌ [1 β π π πππ π πππ π πππ π ππ β² ] , (10) πΌπΌ π΅π΅ = πΌπΌ [1 + π π πππ π πππ π πππ π ππ β² ] , (11) where ΞΆ β² = ππ + ππΞπΏπΏ ( ππππ ππ ) . Figure 3(c) shows numerical calculations for equations (8)-(11) (see also Figs. S4 and S5 of the Supplementary Information). For ΞΆ β² = ππ + ππ /2 , equations (8)-(11) are rewritten as πΌπΌ πΌπΌ = πΌπΌ (1 + π π πππ π ππ ) , πΌπΌ π½π½ = πΌπΌ (1 β π π πππ π ππ ) , πΌπΌ π΄π΄ = πΌπΌ [1 + π π πππ π ππ cos ( ππ )] , and πΌπΌ π΅π΅ = πΌπΌ [1 β π π πππ π ππ cos ( ππ )] . The normalized intensity product π π ππππ between πΌπΌ ππ and πΌπΌ ππ is the same as ππ πΌπΌπ½π½ ( ) ( ππ ) = (1 β πππππ π ππ ) for MZI 1 and ππ π΄π΄π΅π΅ ( ) ( ππ ) = (1 β π π πππ π πππ π πππ π ππ ) for MZI 2 due to the randomness by AOM. To satisfy the anticorrelation condition for ππ πΌπΌπ½π½ ( ) ( ππ ) , ππ = Β± ππ /2 is obtained as shown in the top panels of Fig. 3(c). For the same conditions of ππ = Β± ππ /2 , however, there is no way to satisfy the quantum feature between πΌπΌ π΄π΄ and πΌπΌ π΅π΅ , unless ΞπΏπΏ is changed. For π π π΄π΄π΅π΅ = 0 , ππ = Β± π π ππ must be satisfied as shown in the bottom panels, where n=0,1,2β¦ As analyzed in Fig. 2, this also proves the violation of quantum feature analysis in ref. 24. In a short summary, the correct condition for the quantum feature generation in Fig. 3(a) for the final outputs is to break the anticorrelation condition in ΞΆ . Neither way, the PBW cannot be possible in the directly coupled MZI scheme due to this reason, where Fig. 3(c) is just for the diffraction limit of the Rayleigh criterion in the intensity product: π π π΄π΄π΅π΅ = (1 + πππππ π ππ )/2 . As presented elsewhere, such PBW can be achieved by CBW via path superposition [28]. For this, an intermediate dummy MZI must be inserted between two MZIs in Fig. 3(a). Fig. 3. Schematic of deterministic entangled photon-pair generations. (a) A coupled MZI structure. (b) Basis randomness for ΞΆ (0; Ο ) . (c) Numerical calculations for π π ππππ = πΌπΌ ππ πΌπΌ ππ β at k βπΏπΏ = ππ2 . (Top row) For πΌπΌ πΌπΌ and πΌπΌ π½π½ . (bottom row) For πΌπΌ π΄π΄ and πΌπΌ π΅π΅ . πΌπΌ ππ and πΌπΌ ππ are interchangeable on behalf of AOM. Conclusion
In conclusion, the quantum features of anticorrelation and PBW were analyzed in a directly coupled MZI system using pure coherence optics, where SPDC-generated symmetrically distributed entangled photon pairs played an essential role in both ππ ( ) disappearance in the first MZI and ππ ( ) retrieval in the second MZI. Based on the ππ ( ) β generaged entangled photon-pair distribution, the relative Ο /2 phase difference between all paired photons was derived as an essential condition for anticorrelation. Moreover, anticorrelation condition in the first MZI violated quantum feature generation conditions in the second MZI. In other words, satisfying the anticorrelation in one MZI resulted in destruction of quantum features in the other MZI. By this reason, PBW could not be generated from the directly coupled MZI system. Instead, quantum superposition between MZIs is the solution of creation of PBW as presented in refs. 21-23 and 28. Finally, a deterministic coherence version of entangled light pair generation was proposed and analyzed using pure coherence optics applicable to both single photons and coherent light without violation of quantum mechanics. Acknowledgment BSH acknowledges that this work was supported by GIST via GRI 2021. Reference 1. J. R. Wheeler and W. H. Zurek,
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