Characterizing quantum microwave radiation and its entanglement with superconducting qubits using linear detectors

Abstract

Recent progress in the development of superconducting circuits has enabled the realization of interesting sources of nonclassical radiation at microwave frequencies. Here, we discuss field quadrature detection schemes for the experimental characterization of itinerant microwave photon fields and their entanglement correlations with stationary qubits. In particular, we present joint state tomography methods of a radiation field mode and a two-level system. Including the case of finite quantum detection efficiency, we relate measured photon field statistics to generalized quasi-probability distributions and statistical moments for one-channel and two-channel detection. We also present maximum-likelihood methods to reconstruct density matrices from measured field quadrature histograms. Our theoretical investigations are supported by the presentation of experimental data, for which microwave quantum fields beyond the single-photon and Gaussian level have been prepared and reconstructed.

Correlations, indistinguishability and entanglement in Hong–Ou–Mandel experiments at microwave frequencies

“…Using the dispersive interaction between qubit and resonator, we tune the emission frequencies of the two sources to an identical value of ν r = 7.2506 GHz. For our experiments, we sequentially create 20 single photons in each source at a rate 1/t r = 1/512 ns ∼ 1.95 MHz in a sequence repeated every 12.5 µs.To probe the photon statistics in the beam-splitter output modesâ andb we use two spatially separated heterodyne detection channels 21,22 (see dashed rectangle in Fig. 1b).…”

“…To probe the photon statistics in the beam-splitter output modesâ andb we use two spatially separated heterodyne detection channels 21,22 (see dashed rectangle in Fig. 1b).…”

“…In contrast to many other HOM experiments in which photons in the beam-splitter outputs are detected by single-photon counters, our measurement of the complex envelope is intrinsically photonnumber resolving for averaged measurements and allows us to measure coherences of the electromagnetic field. The complex envelopes S a/b (t ) are used to compute the statistics required to extract all relevant quantum correlations measured here 21,23 . To analyse the digital data, signal processing and statistical analysis are performed in real-time using field-programmable gate arraybased electronics 24 .…”

“…To analyse the digital data, signal processing and statistical analysis are performed in real-time using field-programmable gate arraybased electronics 24 . The noise added by the detection chain is fully characterized by measuring its statistical properties when the output modesâ andb of the beam splitter are left in the vacuum state with both sources idle 21,23 . To verify the single-photon character of each source individually, we have measured their second-order cross-correlation functions G (2) ab (τ ).…”

“…Each resonator output is coupled to the input modeâ (b ) (inset) of a microwave beam splitter (dark red) with two output modesâ (b). b, Schematic of the HOM experiment with two individual cavity quantum electrodynamics systems for photon generation, a beam splitter (BS), linear amplification and heterodyne detection using a pair of mixers (⊗) pumped by a common local oscillator (LO) for correlation function measurements and full quantum state analysis 21 .…”