Permutational symmetries for coincidence rates in multimode multiphotonic interferometry

We obtain coincidence rates for passive optical interferometry by exploiting the permutational symmetries of partially distinguishable input photons, and our approach elucidates qualitative features of multi-photon coincidence landscapes. We treat the interferometer input as a product state of any number of photons in each input mode with photons distinguished by their arrival time. Detectors at the output of the interferometer count photons from each output mode over a long integration time. We generalize and prove the claim of Tillmann et al. [Phys. Rev. X 5 041015 (2015)] that coincidence rates can be elegantly expressed in terms of immanants. Immanants are functions of matrices that exhibit permutational symmetries and the immanants appearing in our coincidence-rate expressions share permutational symmetries with the input state. Our results are obtained by employing representation theory of the symmetric group to analyze systems of arbitrary number of photons in arbitrarily sized interferometers.

[1]  B. Sanders,et al.  Coincidence landscapes for three-channel linear optical networks , 2014, 1402.2391.

[2]  Barry C. Sanders,et al.  Generalized Multiphoton Quantum Interference , 2014, 1403.3433.

[3]  Nicolò Spagnolo,et al.  Experimental validation of photonic boson sampling , 2014, Nature Photonics.

[4]  Jin-quan Chen.  Group Representation Theory For Physicists , 1989 .

[5]  Ish Dhand,et al.  Realization of arbitrary discrete unitary transformations using spatial and internal modes of light , 2015, 1508.06259.

[6]  Peter Bürgisser,et al.  The Computational Complexity of Immanants , 2000, SIAM J. Comput..

[7]  Nicolò Spagnolo,et al.  Experimental scattershot boson sampling , 2015, Science Advances.

[8]  L. Mandel.  Photon interference and correlation effects produced by independent quantum sources , 1983 .

[9]  장윤희,et al.  Y. , 2003, Industrial and Labor Relations Terms.

[10]  Andrew G. White,et al.  Photonic Boson Sampling in a Tunable Circuit , 2012, Science.

[11]  Representations of the Weyl group and Wigner functions for SU(3) , 1998, math-ph/9811012.

[12]  Jean-Luc Brylinski,et al.  Complexity and Completeness of Immanants , 2003, ArXiv.

[13]  V. Shchesnovich,et al.  Partial indistinguishability theory for multiphoton experiments in multiport devices , 2014, 1410.1506.

[14]  Yvonne Y Gao,et al.  SU(3) quantum interferometry with single-photon input pulses. , 2012, Physical review letters.

[15]  J Fan,et al.  Invited review article: Single-photon sources and detectors. , 2011, The Review of scientific instruments.

[16]  W Steven Kolthammer,et al.  Distinguishability and Many-Particle Interference. , 2016, Physical review letters.

[17]  Valery Shchesnovich,et al.  Sufficient condition for the mode mismatch of single photons for scalability of the boson-sampling computer , 2013, 1311.6796.

[18]  R. Loudon,et al.  Quantum theory of the lossless beam splitter , 1987 .

[19]  V. Shchesnovich.  Partial distinguishability and photon counting probabilities in linear multiport devices , 2017, 1712.03191.

[20]  E. Condon.  The Theory of Groups and Quantum Mechanics , 1932 .

[21]  E. Knill,et al.  A scheme for efficient quantum computation with linear optics , 2001, Nature.

[22]  Reck,et al.  Experimental realization of any discrete unitary operator. , 1994, Physical review letters.

[23]  Valery S. Shchesnovich,et al.  Tight bound on trace distance between a realistic device with partially indistinguishable bosons and the ideal Boson-Sampling computer , 2015, ArXiv.

[24]  Heidi Fearn,et al.  Theory of two-photon interference , 1989 .

[25]  Scott Aaronson,et al.  The Computational Complexity of Linear Optics , 2013, Theory Comput..

[26]  A. Bunse-Gerstner,et al.  Numerical Methods for Simultaneous Diagonalization , 1993, SIAM J. Matrix Anal. Appl..

[27]  B. J. Metcalf,et al.  Boson Sampling on a Photonic Chip , 2012, Science.

[28]  Philip Walther,et al.  Experimental boson sampling , 2012, Nature Photonics.

[29]  D. Rowe,et al.  Dual pairing of symmetry and dynamical groups in physics , 2012, 1207.0148.

[30]  Vincenzo Tamma,et al.  Multiboson Correlation Interferometry with Arbitrary Single-Photon Pure States. , 2014, Physical review letters.

[31]  S. Scheel,et al.  Permanents in linear optical networks , 2004, quant-ph/0406127.

[32]  Hong,et al.  Measurement of subpicosecond time intervals between two photons by interference. , 1987, Physical review letters.