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Journal of Modern Optics

Quantum interference between an arbitrary-photon Fock state and a coherent state

Xue-xiang Xu, Fang Jia, Liyun Hu, Zhenglu Duan, Qin Guo, Ma Shan-Jun

Published Oct 20, 2012• 18 citations
Publisher

Abstract

Using the phase space method, we study the quantum interference through a Mach–Zehnder interferometer with an arbitrary-photon Fock state and a coherent state as two inputs. The statistical properties, in terms of the Wigner function, average photon number, photon number distribution and parity, are derived analytically for the field of the individual output port. These results indicate that the output state is actually a statistical mixture between a bunch of Fock state and coherent states. In addition, the phase sensitivity is also examined by using the detection scheme of intensity difference measurement.

DOI
10.1080/09500340.2012.733435

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Using the phase space method, we study the quantum interference through a Mach–Zehnder interferometer with an arbitrary-photon Fock state and a coherent state as two inputs. The statistical properties, in terms of the Wigner function, average photon number, photon number distribution and parity, are derived analytically for the field of the individual output port. These results indicate that the output state is actually a statistical mixture between a bunch of Fock state and coherent states. In addition, the phase sensitivity is also examined by using the detection scheme of intensity difference measurement.
We derive a general expression of the expectation value of the parity operator in a Mach–Zehnder interferometer with a Fock state in one input port and an arbitrary state in the other input. We obtain the quantum Crámer–Rao bound via the error propagation method from parity detection for the phase shift around zero. In addition, for an arbitrary state with the same mean photon number, our results show that the same phase resolution and the same phase uncertainty can be obtained for the phase shift around zero. That is to say, when a Mach–Zehnder interferometer is powered by a Fock state into one input port, all quantum states with same mean photon number at the other input may have the same performance in the phase estimation via parity detection. As some specific applications, we consider the arbitrary state being a Fock state, a coherent state, and a squeezed-vacuum state.
Abstract We present a theoretical study of the interferences between hybrid single-photon state, coherent state, and vacuum state in the designed medium of a six-ports Mach-Zehnder Interferometer (6p-MZI). The hybrid three different quantum states were simultaneously injected into the input ports of the interferometer. The derivation result demonstrated that the output state showed the statistical blend states among a substituted single-photon state, coherent state, and vacuum state. Interestingly, our calculation results attested that the maximum probability of detected photons at the individual interferometer output ports were significantly dependents on the amplitude of the inputted coherent state.
We present a unified theoretical analysis of three-photon quantum interference in a Six-Port Mach-Zehnder Interferometer (6p-MZI) constructed from two cascaded tritters, with two independent phase modulators placed between the tritter arms. We analytically derive the transfer matrix of the 6p-MZI and show how they organize into three symmetry classes, governed by the discrete Fourier transform (DFT) structure of the tritter and the conjugate relations. Furthermore, we analyze two input regimes: First, three indistinguishable single photons are injected into the tritter, and the output probability distributions P[111]P_{[111]}P[111]​, P[{300}]P_{[\{300\}]}P[{300}]​, and P[{210}]P_{[\{210\}]}P[{210}]​ are derived as functions of the two relative phases (φ1,φ2)(φ_1, φ_2)(φ1​,φ2​). At φ2=0φ_2 = 0φ2​=0, the single-phase limit is recovered, which confirms 100\% visibility of the even-distribution fringe. Second, a hybrid coherent-Fock input ∣α⟩1∣α⟩2∣1⟩3|α\rangle_1|α\rangle_2|1\rangle_3∣α⟩1​∣α⟩2​∣1⟩3​ is analyzed via the density matrix formalism. The average photon number at each output port exhibits amplitude-dependent phase shifts. Our results establish the 6p-MZI as a programmable platform for tripartite quantum state manipulation and coherent amplitude sensing.
This study presents a theoretical analysis of three-photon interference in a six-port Mach-Zehnder Interferometer (6p-MZI) system, which incorporates two identical ideal six-port optical beam splitters (tritters) and phase modulators. We investigate interference patterns produced by hybrid input photon configurations that combine Fock state and coherent states, exploring the effects of both relative phase differences and average number of coherent photon state at the input ports. Our analysis demonstrates that the phase positions of constructive interference peaks systematically shift in response to the average number of coherent photons (∣α∣2)\left(|\alpha|^{2}\right)(∣α∣2) at the input ports. When ∣α∣2|\alpha|^{2}∣α∣2 increases from 0 to 0.5, the constructive interference peaks at output ports 2 and 3 undergo symmetric shifts of up to π/3\pi / 3π/3 radians relative to the extreme interference pattern of output port 3 at 5π/35 \pi / 35π/3. The results highlight the high sensitivity of interference patterns to both relative phase differences and coherent state photon amplitudes. These findings suggest novel approaches for controlling three-photon interference patterns in 6p-MZI systems through precise manipulation of phase and low-amplitude coherent photon states. This work advances our understanding of multi-photon interference in the quantum regime, with potential applications in quantum information processing, metrology, and fundamental quantum optics studies.
In this work we present the theory for the quantum interference of states with an arbitrary number of photons in a Mach-Zehnder interferometer. We express the mathematical description of the interference in an algebraic way. We show the interference pattern of an average of a n photons input state corresponds to the classical interference pattern, which tells us the last comes from a quantum interference statistical average. Then, we propose to use this scheme to study the statistical transition from quantum to classical interference.
A quantum-statistical description of noise in interferometers is given in terms of the Wigner distribution function. The interferometer may contain an amplifier in one of its arms. The input field can be in a variety of states such as a Fock state, a coherent state or a squeezed coherent state. The Wigner function of the output field at the detector is shown to have a general Gaussian form with non-zero complex field amplitude and with other parameters related to the characteristics of the input field, amplifier and beam splitters, etc. An explicit form of the photon-number distribution is given. Higher-order correlations of the output field can be obtained from the Gaussian property of the Wigner function.