Schemes for the observation of photon correlation functions in circuit QED with linear detectors

Correlations are important tools in the characterization of quantum fields, as they can be used to describe statistical properties of the fields, such as bunching and antibunching, as well as to perform field state tomography. Here we analyze experiments by Bozyigit et al. [Nat. Phys. (to appear), e-print arXiv:1002.3738] where correlation functions can be observed using the measurement records of linear detectors (i.e., quadrature measurements), instead of relying on intensity or number detectors. We also describe how large amplitude noise introduced by these detectors can be quantified and subtracted from the data. This enables, in particular, the observation of first- and second-order coherence functions of microwave photon fields generated using circuit quantum electrodynamics and propagating in superconducting transmission lines under the condition that noise is sufficiently low.

Figure 1

Experimental setups illustrating different degrees of optical coherence with intensity detectors. The red lines represent quantum fields, and the black lines represent measurement records.

Figure 2

Standard experiments for the observation of $G^{(1)}$ and $G^{(2)}$ using intensity detectors: (a) a Mach-Zender interferometer with a variable delay $\tau$ and a variable phase shift $\phi$, and (b) a Hanbury Brown and Twiss (HBT) interferometer with a variable delay $\tau$. The light-blue components are balanced beam splitters. The cavity is taken to be one-sided, with one mirror being perfectly reflective.

Figure 3

Optical analog of an IQ mixer as an eight-port homodyne detector. The field $\mathrm{r̂}$ is fed into a balanced beam splitter along with vacuum ${\mathrm{v̂}}_r$. The $\mathrm{X̂}$ quadrature of one of the beam-splitter outputs is measured while the $\mathrm{P̂}$ quadrature of the other output is measured. The classical outcomes are the real and imaginary parts of the complex envelope $S_r(t)$.

Figure 4

Hanbury Brown and Twiss interferometer with complex envelope measurement used to measure the first-order coherence function $G^{(1)}$. The output of the cavity field is separated by an on-chip beam splitter and the resulting fields measured by an IQ mixer.

Figure 5

The setup for the observation of coherence functions using a two-sided cavity.

Figure 6

Distortion of the $e^{-\kappa \tau }$ pulses in $G^{(1)}(\tau )$ due to Gaussian filters of different bandwidths. The solid blue line is the unfiltered function, and the others are filtered bandwidth decreasing progressively: $31/t_p$ for the dashed purple line, $14/t_p$ for the dot-dashed yellow line, and $10/t_p$ for the dotted green line.

Figure 7

The pulse train for unfiltered $G^{(1)}(\tau )$ (top) and $G_{\mathrm{fil}}^{(1)}(\tau )$ after Gaussian filtering (bottom) under the periodic preparation of $|1\rangle$ (solid line), $\frac{1}{\sqrt{2}}(|0\rangle +|1\rangle )$ (dashed line), and $|0\rangle$ (dotted line). Each plot is offset for clarity, with the true value for the baseline being 0 in all cases. The Gaussian filter used here has a bandwidth of $31/t_p$.

Figure 8

The heights of the center peak (solid line) and the side peaks (dashed line) of $G^{(1)}(\tau )$ as a function of the Rabi angle of the prepared state.

Figure 9

The pulse train for unfiltered $G^{(2)}(\tau )$ (top) under the periodic preparation of $|1\rangle$ (solid line), $\frac{1}{\sqrt{2}}(|0\rangle +|1\rangle )$ (dashed line), and $|0\rangle$ (dotted line). Each plot is offset for clarity, with the true value for the baseline being 0 in all cases.

Figure 10

The heights of the side peaks (solid line) of $G^{(2)}(\tau )$ as a function of the Rabi angle of the prepared state. For the states considered here, the center peak is always zero.