Regeneration amplitudes f21= | f21 | exp(i φ21)

N.D. Galanina

N.D. Galanina

Z. Ficek, Mohamed Ridza Wahiddin

Suppose, the field Eˆ2 is a coherent laser field (local oscillator) of a large amplitude α, whereas the signal (detected) field is weak. As a result, the terms proportional to 〈bˆ†bˆ〉 and 〈bˆ†〉, 〈bˆ〉 dominate over those without bˆ† and bˆ. Hence, the terms independent of the amplitude of the local oscillator can be discarded. Thus, we can ignore the term β〈aˆ†aˆ〉, and denoting 〈bˆ〉 = |α| exp(iφ), 〈bˆ†〉 = |α| exp(−iφ), the resultant light intensity at the detector is then I ≈ (1− β) |α|2 + 2 |α| √ β (1− β) 〈Eˆφ〉 , (6.102) where 〈 Eˆφ 〉 = 1 2i (〈aˆ〉e−iφ − 〈aˆ†〉eiφ) , (6.103) and φ is the phase of the laser. The first term on the right-hand side of Eq. (6.102) is equal to the intensity of the reflected coherent beam. The second term is an interference term between the coherent and the signals beams. This term contains the phase-dependent quadrature amplitude of the signal beam.

Andrey Zhukov, Andrey Zhukov, Andrey Lebedev +1 more

This archive contains the numerical data and scripts used to generate the figures for the paper “Energy-selective quantum search with Ising Hamiltonian phase oracles”. The main numerical dataset is used for the scaling analysis of the correlation-induced resonance displacement φ₀ in the zero-field random Ising ensemble. The simulations use dimensionless units with σ_J = 1 and the Hamiltonian H(s) = 2 sum_{i<j} J_ij s_i s_j, where the couplings are independent Gaussian random variables, J_ij ~ N(0, 1), and h_i = 0. For each random Ising instance, the archive stores the truncated characteristic-function coefficients g_m = 2^{-n} Tr exp(-i m T H), m = 0, ..., 15. These coefficients are used to extract the local displacement φ₀ from the zero of the imaginary part of the response function C(φ) = 2 sum_{m=0}^{15} g_m (-1)^m exp(i m φ) - 1. The quantity plotted in the paper is the disorder-averaged second moment <φ₀²> and the scaled combination n⁶<φ₀²>. The Monte Carlo average is over independent disorder realizations only. For each realization, the trace over spin configurations is evaluated deterministically. The data cover even system sizes n = 18, 20, 22, ..., 36. For the largest systems, the computation splits the spins into two equal halves, performs GPU block contractions, and uses the zero-field spin-flip symmetry H(s)=H(-s) to reduce the sum by a factor of two. Archive contents - scripts/01_generate_phi0_coefficients.py Generates the characteristic-function coefficient files g_m for the random Ising ensemble. - scripts/02_analyze_phi0_scaling.py Computes φ₀, bootstrap confidence intervals, the finite-n covariance-sum comparison, and the scaling plots. - scripts/03_make_gaussian_amplification.py Generates the Gaussian amplification figure used as Fig. 1 in the paper. - data/phi0_coefficients/ Directory for the NumPy .npz files containing the characteristic-function coefficients. The production files have names of the form Nq_<n>_K_<K>.npz. The .npz files are saved without pickle objects. They can be read with standard NumPy using np.load. Typical usage To regenerate the Gaussian amplification figure: python scripts/03_make_gaussian_amplification.py --output-dir output/fig1_gaussian To analyze the φ₀ scaling data: python scripts/02_analyze_phi0_scaling.py \ --data-dir data/phi0_coefficients \ --output-dir output/phi0_scaling To run a short test calculation of the characteristic-function coefficients: python scripts/01_generate_phi0_coefficients.py \ --smoke-test \ --output-dir output/smoke_test_coefficients To regenerate the full production dataset on a CUDA-capable GPU: python scripts/01_generate_phi0_coefficients.py \ --production \ --output-dir data/phi0_coefficients The production calculations were performed using CuPy on an NVIDIA GPU. The analysis and plotting scripts use NumPy, pandas, and Matplotlib.

Henryk Paul

Abstract A novel approach to the quantum mechanical description of phase properties of a one‐mode low‐intensity radiation field is presented. The procedure is based upon the correspondence between Glauber states and classical waves of fixed phase ϕ and amplitude a, and, in contrast to the conventional quantum mechanical phase concepts, allows to make contact between certain operators ℰm corresponding to the classical quantities exp {imφ} (m = ± 1, ± 2, …) and realistic measurements. In the observation, a large amplification of the microscopic field by means of a linear quantum amplifier plays a decisive role. Since the amplified field is a macroscopic one, the measurement of phase is not problematic, and hence, in particular, average values «exp {imϕ}» can be determined experimentally. The main result of our analysis is that the mean values «exp {imϕ}» are equal to the quantum mechanical expectation values for the operators ℰm with respect to the field before amplification. To obtain this result it became necessary to calculate Glauber's P function for the amplified field in case of an arbitrary state for the initial field.

Stanley Gudder