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

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 quasiprobability 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.

Figure 1

Schematic of a situation where two conjugate field quadatures, $\widehat{X}$ and $\widehat{P}$, of a radiation-field mode $a$ and an arbitrary spin component of a localized qubit $\sigma$ are measured.

Figure 2

Field quadrature detection schemes for optical and microwave fields. (a) Schematic of balanced optical homodyne detection. The signal field $a$ is combined with a coherent local oscillator (LO) field with controlled phase $\phi$ at a beam splitter and the quadrature amplitude ${\widehat{X}}_{\phi }$ is detected with photon counters (n) in the two output arms. (b) Double homodyne detection scheme. The signal field $a$ is split into two parts at a beam splitter while an additional vacuum mode $h$ is introduced. Placing a homodyne detector as described for (a) at each of the two beam-splitter outputs allows for measuring two conjugate quadratures (i.e., the complex amplitude $\widehat{S}$). (c) Measurement of the complex amplitude at microwave frequencies. The signal mode is amplified with a phase-insensitive linear amplifier introducing an additional noise mode $h_{\mathrm{amp}}$. At a microwave frequency mixer the amplified output is split into two parts, while adding the mode $h_{\mathrm{mix}}$, and multiplied by a coherent local oscillator field. The down-converted electrical field is sampled with analog-to-digital converters (ADCs).

Figure 3

(a) Absolute value of the experimentally reconstructed density matrices (gray scale) in comparison with the ideal ones (wire frames) for the four indicated quantum states. (b) Measured density matrices transformed into their corresponding Wigner functions $W(x,p)$. Fidelities between ideal states $|\psi \rangle$ and measured density matrices are evaluated as $F=\langle \psi \left|\rho \right|\psi \rangle$.

Figure 4

Diagonal matrix elements $p_n=\langle n|{\tilde{\rho }}_h|n\rangle$ of the detector state (filled circles) obtained with the iterative maximum-likelihood method from experimental data. The photon number distribution is well described by a thermal distribution (solid line) with mean photon number $N_0\approx 4.4$. Inset: Reconstructed density matrix with fidelity $F=95\%$ of a coherent state with $\alpha \approx 1.7$.

Figure 5

Two-channel detector with radiation incident from input mode $a$. Each of the beam-splitter outputs has an individual amplification stage adding noise in modes $h_1$ and $h_2$. In both channels the complex amplitude is measured [11, 43].