We use permutation-group methods plus SU(3) group-theoretic methods to determine the action of a three-channel passive optical interferometer on controllably delayed single-photon pulse inputs to each channel. Permutation-group techniques allow us to relate directly expressions for rates and, in particular, investigate symmetries in the coincidence landscape. These techniques extend the traditional Hong-Ou-Mandel effect analysis for two-channel interferometry to valleys and plateaus in three-channel interferometry. Our group-theoretic approach is intuitively appealing because the calculus of Wigner D functions partially accounts for permutational symmetries and directly reveals the connections among D functions, partial distinguishability, and immanants.
This thesis reports advances in the theory of design, characterization and simulation of multi-photon multi-channel interferometers. I advance the design of interferometers through an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. This procedure effects an arbitrary
nsnptimesnsnp unitary matrix on the state of light in
ns spatial and
np internal modes. I devise an accurate and precise procedure for characterizing any multi-port linear optical interferometer using one- and two-photon interference. Accuracy is achieved by estimating and correcting systematic errors that arise due to spatiotemporal and polarization mode mismatch. Enhanced accuracy and precision are attained by fitting experimental coincidence data to a curve simulated using measured source spectra. The efficacy of our characterization procedure is verified by numerical simulations. I develop group-theoretic methods for the analysis and simulation of linear interferometers. I devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU
(n) basis states, which reduce the canonical subgroup chain, for arbitrary
n. The boson realizations are employed to construct
mathcalD-functions, which are the matrix elements of arbitrary irreducible representations, of SU
(n) in the canonical basis. I show that immanants of principal submatrices of a unitary matrix
T are a sum of the diagonal
mathcalD(Omega)-functions of group element
Omega over
t determined by the choice of submatrix and over the irrep
(lambda) determined by the immanant under consideration. The algorithm for
mathrmSU(n) mathcalD-function computation and the results connecting these functions with immanants open the possibility of group-theoretic analysis and simulation of linear optics.
This thesis reports advances in the theory of design, characterization and simulation of multi-photon multi-channel interferometers. I advance the design of interferometers through an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. This procedure effects an arbitrary
nsnp×nsnp unitary matrix on the state of light in
ns spatial and
np internal modes. I devise an accurate and precise procedure for characterizing any multi-port linear optical interferometer using one- and two-photon interference. Accuracy is achieved by estimating and correcting systematic errors that arise due to spatiotemporal and polarization mode mismatch. Enhanced accuracy and precision are attained by fitting experimental coincidence data to a curve simulated using measured source spectra. The efficacy of our characterization procedure is verified by numerical simulations. I develop group-theoretic methods for the analysis and simulation of linear interferometers. I devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU
(n) basis states, which reduce the canonical subgroup chain, for arbitrary
n. The boson realizations are employed to construct
D-functions, which are the matrix elements of arbitrary irreducible representations, of SU
(n) in the canonical basis. I show that immanants of principal submatrices of a unitary matrix
T are a sum of the diagonal
D(Ω)-functions of group element
Ω over
t determined by the choice of submatrix and over the irrep
(λ) determined by the immanant under consideration. The algorithm for
SU(n) D-function computation and the results connecting these functions with immanants open the possibility of group-theoretic analysis and simulation of linear optics.
Abstract We have studied the scattering of bi-photon input pulses from a six-port interferometer given by 3x3 unitary matrix, with different coincidences at the three output ports. The distinguishability of photons is introduced by taking time delays in the input. We observe different patterns of dips and peaks in coincidence probabilities by varying the time delays, manifesting destructive and constructive interference. The coincidence probability comes out to be octic function of the elements of scattering matrix U 3 and provides a means to learn matrix elements. This study provides insight into the interferometric properties of two independent photons in the same port. This is specifically useful in the case of chips where geometry of the chip forces photons in the same spatial input port.
Quantum information holds enormous potential to achieve communication and computational tasks that are impossible using classical systems. Many quantum information processing tasks can be implemented by exploiting the passive interference of photons. In this thesis, we reports advances towards the analysis, simulation, characterization, and design of passive interferometry experiments for quantum information processing applications. First, we develop a theory of passive optical interferometry experiments that relates coincidence rates at the output of an interferometer to the permutational symmetries of partially distinguishable photons. In our formalism, coincidence rates are elegantly expressed in terms of immanants, which are matrix functions that exhibit permutational symmetries, and the immanants appearing in our coincidence-rate expressions share permutational symmetries with the input state. Our formalism thus provides an intuitive qualitative understanding of how the permutational symmetries of distinguishable photons determine their interference. Our approach can be used to better understand the effect of partial distinguishability in modeling passive interferometry experiments aiming to perform quantum information tasks such as linear optical quantum computing and BosonSampling. By exploiting symmetries, we can also reduce some calculational tasks and provide additional insights into relations between the coincidence rates in various situations. Second, we tackle the problem of characterizing passive interferometers, essential for modelling any passive interferometry experiment. We develop a characterization procedure that uses only one- and two-photon coincidence data, and makes a number of advancements over existing procedures. We estimate the errors in the characterization via bootstrapping, a statistical technique with the advantage of not assuming any particular error model on the experimental data. We also introduce a scattershot approach to reduce the experimental data collection time quadratically in the number of modes of the interferometer. Third, we study a relativistic quantum information task called quantum summoning. We cast quantum summoning as an interactive protocol between a verifier and a possibly dishonest prover. For the prover, we present a protocol and quantum codes to summon in any valid configuration of causal diamonds, and design the encoding and decoding circuits for our code. Our protocol decreases the space complexity for encoding by a factor of two compared to the previous best result and reduces the gate complexity from scaling as the cube to the square of the number of causal diamonds. Within our interactive protocol framework, we construct an operational definition of quantum summoning, and use this definition to present a verification test. We prove that our verification test is sound and complete, thereby showing that quantum summoning can be verified by a resource-limited verifier in an adversarial setting. Our protocol and verification procedures are amenable to implementation by passive interferometry, thereby extending the reach of passive interferometry in the field of relativistic quantum information processing.