Observing Interferences between Past and Future Quantum States in Resonance Fluorescence

The fluorescence of a resonantly driven superconducting qubit is measured in the time domain, providing a weak probe of the qubit dynamics. Prior preparation and final, single-shot measurement of the qubit allows us to average fluorescence records conditionally on past and future knowledge. The resulting interferences reveal purely quantum features characteristic of weak values. We demonstrate conditional averages that go beyond classical boundaries and probe directly the jump operator associated with relaxation. The experimental results are remarkably captured by a recent theory, which generalizes quantum mechanics to open quantum systems whose past and future are known.

Figure 1

Principle of the experiment. A resonant field (blue) drives a qubit via a weakly coupled line $a$. While the off-resonant cavity reflects most of the driving field back on line $a$, the fluorescence signal (green) mostly exits through the strongly coupled line $b$ with an amplitude, proportional to ${\sigma }_{-}$, oscillating at the Rabi frequency ${\nu }_R$. Because of nonzero transmission from line $a$, it is displaced by a resonant field independent of the qubit state (blue on line $b$). This signal is then measured at time $t$ with a heterodyne detection setup including a phase-preserving quantum limited amplifier (triangle). At final time $T$, the qubit state is measured with high fidelity using a pulse at the bare cavity frequency (purple), enabling conditional averaging of the fluorescence signal depending on the measured state. In the quadrature phase space rotating at ${\nu }_q$ (right panel), resonance fluorescence is revealed by the time oscillation (green) of the voltage $V_{\mathrm{Re}}$, shifted by a constant value (blue).

Figure 2

Resonance fluorescence in time domain. (a) Average time traces of the heterodyned outgoing field on line $b$ represented in the Fresnel plane ($V_{\mathrm{Re}},V_{\mathrm{Im}}$), for driving amplitudes corresponding to ${\nu }_R=0.6$, 1, and 1.4 MHz (blue arrow represents increasing drive amplitudes). The unit of voltage $V_0$ corresponds to an average emitted photon rate equal to ${\gamma }_{1b}$. Purple (respectively, orange) lines correspond to a qubit prepared in $|g⟩$ (respectively, $|e⟩$) at $t=0$. Each trace is the sum of a term proportional to the drive amplitude and an oscillating part in $V_{\mathrm{Re}}$ which corresponds to the resonance fluorescence. The finite bandwidth (1.6 MHz) of the detection setup deforms the time traces (finite rise time and diminished oscillation amplitude). (b) Dots: Average fluorescence signal $s_{-}$ as a function of time $t$ for a Rabi frequency ${\nu }_R=1\mathrm{MHz}$. Lines: corresponding predicted fluorescence signal filtered by detection setup. (c) Dashed lines: measured values of $⟨{\sigma }_z⟩$ for ${\nu }_R=1\mathrm{MHz}$. Plain lines: predicted $⟨{\sigma }_{-}⟩$ leading to plain lines in (b). (d),(e) Average value of the fluorescence signal $s_{-}$ as a function of both time and Rabi frequency, for a qubit either prepared in $|e⟩$ (d) or postselected in $|g⟩$ (e). Both measured and predicted averages of $s_{-}$ are shown in separate regions. Absolute values remain well bellow 0.5, as expected for the measurement of $\mathrm{Re}[⟨{\sigma }_{-}⟩]=⟨{\sigma }_x⟩/2$. Each data point was averaged on at least $3\times 10^5$ experiments leading to a maximal standard deviation of 0.05 on $\overline{s_{-}}$.

Figure 3

Interferences between past and future states. (a) Average value of the measured fluorescence signal $s_{-}$ as a function of both time and Rabi frequency, for a qubit prepared in $|e⟩$ and postselected in $|g⟩$. Plain lines surround regions with weak values beyond the range allowed by macrorealism. (b) Dots: cuts of (a) as a function of ${\nu }_R$ for times $t=0.99\mu \mathrm{s}$ (green) and $t=1.44\mu \mathrm{s}$ (red). The maximal standard deviation on each average of $s_{-}$ is 0.05. Plain lines: prediction for the same curves using Eq. (4). Dashed lines: cuts of Fig. 2 at the same times. The gray region delimits the range of possible unconditional average values, like the contours in (a). (c) Theoretical counterpart of (a) assuming that the average of $s_{-}$ is a measure of $\mathrm{Re}(⟨{\sigma }_{-}{⟩}_w)$ and using Eq. (4). (d) Theoretical counterpart of (a) assuming that the average of $s_{-}$ is a measure of $⟨\mathrm{Re}{\sigma }_{-}{⟩}_w$.