Experimental State Tomography of Itinerant Single Microwave Photons

A wide range of experiments studying microwave photons localized in superconducting cavities have made important contributions to our understanding of the quantum properties of radiation. Propagating microwave photons, however, have so far been studied much less intensely. Here we present measurements in which we reconstruct the quantum state of itinerant single photon Fock states and their superposition with the vacuum by analyzing moments of the measured amplitude distribution up to fourth order. Using linear amplifiers and quadrature amplitude detectors, we have developed efficient methods to separate the detected single photon signal from the noise added by the amplifier. From our measurement data we have also reconstructed the corresponding Wigner function.

Figure 1

Simplified schematic of the experimental setup. An optical analog of the photon source realized in our experiments is shown as a single sided cavity with one highly reflective and one partially transmitting mirror. A single photon is generated in the cavity by preparing the qubit in its excited state when detuned from the cavity and then tuning it into resonance with the cavity for half a vacuum Rabi period $\pi /2g$. The photon is then emitted at a rate $\kappa$ into the output mode $a_{\mathrm{out}}$ resulting in an exponentially decaying envelope of the single photon pulse while the mode $a_{\mathrm{in}}$ remains in the vacuum state. The signal is then amplified with effective gain $G$ and noise in mode $h$ is added [8, 19]. The amplified signal is down-converted in a microwave quadrature mixer using a local oscillator (LO). The two quadrature amplitudes $X$ and $P$ are recorded using an analog-to-digital converter and stored in real time in a two-dimensional histogram using field programmable gate array electronics. An optical frequency analog of our measurement scheme is discussed in detail in [8].

Figure 2

(a) Measured quadrature histogram $D^{[|0⟩⟨0|]}(S)$ for $a$ in the vacuum where $S=\sqrt{G}(X+iP)$. The inset shows a horizontal cut through the histogram (thick red curve). The distribution is well described by a normal distribution (thin blue curve) with width $\sigma =5.7$ (indicated by black arrows) corresponding to a system noise temperature of $T_{\mathrm{noise}}\approx 21\mathrm{K}$. (b) Quadrature histogram $D^{[|1⟩⟨1|]}(S)$ for preparation of single photon Fock states. (c) Difference of $D^{[|1⟩⟨1|]}(S)$ and $D^{[|0⟩⟨0|]}(S)$. Note the two different color scales, both given in units of $D^{[|1⟩⟨1|]}(0)$, indicating the small difference between the two histograms.

Figure 3

Absolute value of the normally ordered moments $|⟨(a^{†}{)}^n{a}^m⟩|$ up to fourth order for (a) a single photon Fock state, (b) a superposition state $(|0⟩-|1⟩)/\sqrt{2}$, and two coherent states with amplitude (c) $\alpha =1$ and (d) $\alpha =0.5$.

Figure 4

Wigner function $W$ $(\alpha =X+iP)$ for (a) a single photon Fock state and (b) a superposition state both reconstructed from the measured moments shown in Fig. 3.