How to administer an antidote to Schrödinger's cat

Let us consider two photons arriving simultaneously at the input ports A and B of a beam splitter (BS), as illustrated in the interference box of figure 1. The joint probability of detecting two photons at times t₀ and t₀ + τ at detectors placed at outputs C and D of the BS, respectively, can be written as [12, 18]

$$\begin{equation}{P}_{\text{joint}\;}\left({t}_{0},\tau \right)={\left\vert {\hat{E}}_{D}^{+}\left({t}_{0}+\tau \right){\hat{E}}_{C}^{+}\left({t}_{0}\right){\hat{a}}_{A}^{{\dagger}}{\hat{a}}_{B}^{{\dagger}}\left\vert {\text{0}}_{A}{0}_{B}\right\rangle \right\vert }^{2},\end{equation} \tag{ 1 }$$

where ${\hat{a}}_{A}^{{\dagger}}$ and ${\hat{a}}_{B}^{{\dagger}}$ correspond to the creation operators of a photon in the ports A and B (${\hat{a}}_{A}$ and ${\hat{a}}_{B}$ would correspond to the annihilation operators), ${\hat{E}}_{C}^{+}+{\hat{E}}_{C}^{-}$ and ${\hat{E}}_{D}^{+}+{\hat{E}}_{D}^{-}$ are the electric field operators at the output ports of the beam splitter, and $\left\vert {0}_{A}{0}_{B}\right\rangle$ corresponds to the vacuum state on the input ports A and B. In this context, τ can be positive or negative, and τ < 0 corresponds to the detector in port D clicking before that in port C. We emphasise that τ is a detection time difference giving rise to a time-resolved Hong–Ou–Mandel (HOM) signal [12, 18], and must not be confused with the photon arrival time difference $\left({\Delta}t\right)$ used in many other HOM experiments [19]. Here, we are only interested in the case for which the two photon wavepackets arrive simultaneously (Δt = 0). If Δt ≠ 0, parts of the photon envelope would not overlap, giving rise to random correlations.

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The electric field operators can be written as the sum of spatio-temporal functions in distinct modes k, ${\zeta }_{k}\left(t\right)={{\epsilon}}_{k}\left(t\right)\mathrm{exp}\left(-i{\phi }_{k}\left(t\right)\right)$:

$$\begin{equation}{\hat{E}}^{+}(t)=\sum\limits _{k}\;{\zeta }_{k}(t){\hat{a}}_{k}\text{,}\;{\hat{E}}^{-}(t)=\sum\limits _{k}\;{\zeta }_{k}^{\ast }(t){\hat{a}}_{k}^{{\dagger}},\end{equation} \tag{ 2 }$$

where ${{\epsilon}}_{k}\left(t\right)$ corresponds to the photon amplitude in mode k and ${\phi }_{k}\left(t\right)$ to its phase.

Since the absolute photon detection times are irrelevant, we calculate the joint detection probability as a function of τ only:

$$\begin{equation}{P}_{\text{joint}}\left(\tau \right)={\int }_{-\infty }^{\infty }\mathrm{d}{t}_{0}{P}_{\text{joint}}\left({t}_{0},\tau \right).\end{equation} \tag{ 3 }$$

Without loss of generality, we consider two linearly polarised photons with a relative polarisation angle θ. In this case, equation (1) can be written as [18]

$$\begin{equation}{P}_{\text{joint}}\left({t}_{0},\tau \right)={P}_{\text{joint}}^{(HV)}\left({t}_{0},\tau \right)-{\mathrm{c}\mathrm{o}\mathrm{s}}^{2}\,\theta \;F\left({t}_{0},\tau \right),\end{equation} \tag{ 4 }$$

where

$$\begin{align}\hfill {P}_{\text{joint}}^{(HV)}\left({t}_{0},\tau \right)& =\frac{1}{4}\left({\left\vert {{\epsilon}}_{A}\left({t}_{0}\right){{\epsilon}}_{B}\left({t}_{0}+\tau \right)\right\vert }^{2}\right.\hfill \\ \hfill & \left.\quad +{\left\vert {{\epsilon}}_{A}\left({t}_{0}+\tau \right){{\epsilon}}_{B}\left({t}_{0}\right)\right\vert }^{2}\right)\hfill \end{align} \tag{ 5 }$$

and

$$\begin{align}\hfill F\left({t}_{0},\tau \right)& =\frac{{{\epsilon}}_{A}\left({t}_{0}\right){{\epsilon}}_{B}\left({t}_{0}+\tau \right){{\epsilon}}_{A}\left({t}_{0}+\tau \right){{\epsilon}}_{B}\left({t}_{0}\right)}{2}\hfill \\ \hfill & \quad \times \mathrm{cos}\left({\phi }_{A}\left({t}_{0}\right)-{\phi }_{A}\left({t}_{0}+\tau \right)\right.\hfill \\ \hfill & \left.\quad +{\phi }_{B}\left({t}_{0}+\tau \right)-{\phi }_{B}\left({t}_{0}\right)\right).\hfill \end{align} \tag{ 6 }$$

When both input photons have orthogonal polarisations $\left(\theta =\pi /2\right)$, ${P}_{\text{joint}}\left(\tau \right)$ can be written as the convolution of their spatio-temporal squared amplitudes, ${P}_{\text{joint}}\left(\tau \right)=\frac{1}{2}\left({\left\vert {\varepsilon }_{A}\right\vert }^{2}\ast {\left\vert {\varepsilon }_{B}\right\vert }^{2}\right)(\tau )$, rendering the choice of ϕ_A and ϕ_B irrelevant. This is in sharp contrast to the case for which both input photons have parallel polarisations $\left(\theta =0\right)$, where ϕ_A and ϕ_B become relevant. The expected behaviour of ${P}_{\text{joint}}\left(\tau \right)$ for the particular case of two photons of perpendicular polarisation with the temporal shape ${\epsilon}\left(t\right)={\mathrm{sin}}^{2}\left(2\pi t/\delta t\right)$, is shown as a dashed curve in figure 2(a) for δt = 400 ns. The three peaks that are shown in figure 2(a) correspond to the detection of coincidences in detectors C and D delayed by varying time differences $\tau \in \left[-400,400\right]\;\text{ns}$.

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We subdivide the overall duration of the photons into two distinct intervals (bins) of equal length labelled I₁ (early) and I₂ (late). Additionally, let us assume that, for the photon arriving through A, ${\phi }_{A}\left(t\right)=0$ for all times, and that, for the photon arriving through B, ${\phi }_{B}\left(t\right)=0$ for t ∈ I₁, and ${\phi }_{B}\left(t\right)=\phi$ for t ∈ I₂. In the Schrödinger picture [18], the state entering the BS is given by

$$\begin{equation}\left\vert {{\Psi}}_{\text{in}}\right\rangle =\frac{1}{2}\left({\hat{a}}_{A1}^{{\dagger}}+{\hat{a}}_{A2}^{{\dagger}}\right)\left({\hat{a}}_{B1}^{{\dagger}}+{\mathrm{e}}^{\mathrm{i}\phi }{\hat{a}}_{B2}^{{\dagger}}\right)\left\vert 0\right\rangle .\end{equation} \tag{ 7 }$$

Here, ${\hat{a}}_{Aj}^{{\dagger}}$ and ${\hat{a}}_{Bj}^{{\dagger}}$ correspond to the photon creation operators in ports A and B in the time intervals I_j , with j ∈ {1, 2}, and $\left\vert 0\right\rangle$ is the vacuum state in the basis of all the temporal and beam splitter input paths: $\left\vert 0\right\rangle =\left\vert {0}_{A1}{0}_{A2}{0}_{B1}{0}_{B2}\right\rangle$. For the cases ϕ = 0 and ϕ = π, the theoretical predictions of ${P}_{\text{joint}}\left(\tau \right)$ are shown as dashed curves in figures 2(b) and (c). For the case of figure 2(b), both input photons are identical and feature photon bunching in the output detectors. For this reason, the expected behaviour would not include any coincidences between detectors C and D. In contrast, in figure 2(c) the photon entering through port B acquired a π-phase change from I₁ to I₂. This is the only difference between the two photons. Therefore, if both photons are detected during either I₁ or I₂, the photons are indistinguishable and no correlations are found with τ ≃ 0. However, if the photons are detected in different intervals, the change in phase drives them to different outputs, resulting in a coincidence probability that is twice as large as the orthogonal polarisation case for τ = ±201 ns.

Figures 2(b) and (c) exhibit a form that corresponds to a different type of interference between the two photons depending on the value of ϕ. Bosonic (ϕ = 0) and fermionic interference (ϕ = π) is found with photons either bunching in the same outputs or avoiding each other and giving rise to coincidences, respectively. To see this, consider the following argument, using the Schrödinger picture: the operators after the beam splitter in the output channels C and D in I_j are linked to those before it by the standard unitary relation ${\hat{a}}_{Aj,Bj}^{{\dagger}}=\left({\hat{a}}_{Cj}^{{\dagger}}\pm {\hat{a}}_{Dj}^{{\dagger}}\right)/\sqrt{2}$. Therefore, after the beam splitter, equation (7) results in:

$$\begin{align}\hfill \left\vert {{\Psi}}_{\text{out}\;}\right\rangle & =\frac{1}{2\sqrt{2}}\left(\left({\hat{a}}_{C1}^{{\dagger}}{\hat{a}}_{C1}^{{\dagger}}+{\mathrm{e}}^{\mathrm{i}\phi }{\hat{a}}_{C2}^{{\dagger}}{\hat{a}}_{C2}^{{\dagger}}\right.\left.\left.-{\hat{a}}_{D1}^{{\dagger}}{\hat{a}}_{D1}^{{\dagger}}-{\mathrm{e}}^{\mathrm{i}\phi }{\hat{a}}_{D2}^{{\dagger}}{\hat{a}}_{D2}^{{\dagger}}\right)\right.\right.\hfill \\ \hfill & \left.\quad +\left({\mathrm{e}}^{\mathrm{i}\phi }+1\right)\left({\hat{a}}_{C1}^{{\dagger}}{\hat{a}}_{C2}^{{\dagger}}-{\hat{a}}_{D1}^{{\dagger}}{\hat{a}}_{D2}^{{\dagger}}\right)\right.\hfill \\ \hfill & \left.\quad +\left({\mathrm{e}}^{\mathrm{i}\phi }-1\right.\left.{\hat{a}}_{C2}^{{\dagger}}{\hat{a}}_{D1}^{{\dagger}}-{\hat{a}}_{C1}^{{\dagger}}{\hat{a}}_{D2}^{{\dagger}}\right)\right)\left\vert 0\right\rangle .\hfill \end{align} \tag{ 8 }$$

Here, $\left\vert 0\right\rangle$ is represented in the basis of all the output ports and temporal modes: $\left\vert 0\right\rangle =\left\vert {0}_{C1}{0}_{C2}{0}_{D1}{0}_{D2}\right\rangle$. The first four terms account for both photons arriving at the same detector in the same time bin. These terms lead to the standard boson bunching reported in the canonical HOM effect [19]. The last four terms correspond to cross-correlations between detectors and/or time bins. Unless the single photon detectors are number resolving, it is not possible to measure outcomes where both detections occur at the same detector in the same time bin. By eliminating the terms of the form ${\hat{a}}_{Xi}^{{\dagger}}{\hat{a}}_{Xi}^{{\dagger}}$ and rearranging the terms, the observable sub-state from equation (8) becomes

$$\begin{align}\hfill \left\vert {\tilde{{\Psi}}}_{\text{out}\;}\right\rangle & =\frac{1}{2\sqrt{2}}\left({\hat{a}}_{C1}^{{\dagger}}\left(\left({\text{e}}^{\text{i}\phi }+1\right){\hat{a}}_{C2}^{{\dagger}}-\left({\text{e}}^{\text{i}\phi }-1\right){\hat{a}}_{D2}^{{\dagger}}\right)\right.\hfill \\ \hfill & \left.\quad +{\hat{a}}_{D1}^{{\dagger}}\left(\left({\text{e}}^{\text{i}\phi }-1\right){\hat{a}}_{C2}^{{\dagger}}-\left({\text{e}}^{\text{i}\phi }+1\right){\hat{a}}_{D2}^{{\dagger}}\right)\right)\left\vert 0\right\rangle .\hfill \end{align} \tag{ 9 }$$

In this sub-state, the phase appears explicitly. By setting ϕ = 0, equation (9) reads

$$\begin{equation}\left\vert {\tilde{{\Psi}}}_{\text{out}\;}^{(\phi =0)}\right\rangle =\frac{1}{\sqrt{2}}\left({\hat{a}}_{C1}^{{\dagger}}{\hat{a}}_{C2}^{{\dagger}}-{\hat{a}}_{D1}^{{\dagger}}{\hat{a}}_{D2}^{{\dagger}}\right)\left\vert 0\right\rangle .\end{equation} \tag{ 10 }$$

This state again leads to the canonical HOM effect, yet across the time bins this time. Therefore, both photons arrive at in the same detector, but in separate time intervals. In contrast, setting ϕ = π, equation (9) now gives

$$\begin{equation}\left\vert {\tilde{{\Psi}}}_{\text{out}\;}^{(\phi =\pi )}\right\rangle =\frac{1}{\sqrt{2}}\left({\hat{a}}_{C1}^{{\dagger}}{\hat{a}}_{D2}^{{\dagger}}-{\hat{a}}_{C2}^{{\dagger}}{\hat{a}}_{D1}^{{\dagger}}\right)\left\vert 0\right\rangle ,\end{equation} \tag{ 11 }$$

which is strikingly different to equation (10), as both photons exhibit fermionic behaviour by arriving at different detectors.

Switching between the bosonic and fermionic behaviour by choice of ϕ allows one to put the system into a pre-defined quantum state conditioned on the random outcome of the first measurement. This is implemented in the form of a feedback mechanism, where a photon detection at either detector in I₁ reveals the required change of ϕ to steer the remaining photon in I₂ to a specific detector. For instance, if detector D clicks in I₁, the second detection would normally occur in detector D, due to the canonical HOM effect. However, as the second halves of the photons have not reached the input ports at the moment of detection, we can instantly change the phase between I₁ and I₂ for one of the incoming photons to change the port in which the second detection occurs. By choosing ϕ = π and projecting the output state in equation (11) onto the measured state $\left\vert {0}_{C1}{1}_{D1}\right\rangle ={\hat{a}}_{D1}^{{\dagger}}\left\vert {0}_{C1}{0}_{D1}\right\rangle$ (involving only the first time bin, since the second one has not yet occurred), the state reduces to

$$\begin{equation}\left\langle {0}_{C1}{0}_{D1}\left\vert {\hat{a}}_{D1}\right\vert {\tilde{{\Psi}}}_{\text{out}\;}^{(\phi =\pi )}\right\rangle =\frac{1}{\sqrt{2}}{\hat{a}}_{C2}^{{\dagger}}\left\vert {0}_{C2}{0}_{D2}\right\rangle .\end{equation} \tag{ 12 }$$

This state clearly shows that, when a measurement occurs in I₂, any click will be recorded in detector C. Analogously, if detector C clicks in I₁ and ϕ = 0 is chosen, both photons are identical in their two halves and any detection in I₂ will always occur at detector C. This reduces the quantum state to

$$\begin{equation}\left\langle {0}_{C1}{0}_{D1}\left\vert {\hat{a}}_{C1}\right\vert {\tilde{{\Psi}}}_{\text{out}\;}^{(\phi =0)}\right\rangle =\frac{1}{\sqrt{2}}{\hat{a}}_{C2}^{{\dagger}}\left\vert {0}_{C2}{0}_{D2}\right\rangle .\end{equation} \tag{ 13 }$$

Again, when a measurement occurs in I₂, any click will be recorded by detector C. Equations (12) and (13) imply that ϕ can be used as a parameter for feedback control, to steer the remaining photon to a desired output.

Returning to equations (3) and (4) with a value of ϕ conditional on the measurement of a photon in I₁, we observe the expected behaviour of ${P}_{\text{joint}}\left(\tau \right)$, as illustrated in figure 2(d). Figure 2(d) shows increased coincidences with τ < 0, for which a detection in D occurs before a detection in C. There are significantly less coincidences with τ > 0, as this would correspond to detector C firing first and detector D second, which cannot occur with an active feedback that always prompts the second detection in C. Figure 2 shows the subset of probabilities for cross-detector correlations only. For this reason, the value $\int {P}_{\text{joint}}\left(\tau \right)d\tau$ is bounded by 1/2. The theoretical curves for same-detector correlations are shown in the supplementary material (https://stacks.iop.org/JPB/55/054001/mmedia).

To demonstrate this phenomenon experimentally, we generate photons in an atom-cavity system using a standard V-STIRAP scheme [20] coupling the hyperfine levels of the D₂ line of ⁸⁷Rb. Specifically, the $\left\vert e\right\rangle =\left\vert {5}^{2}{S}_{1/2},F=1\right\rangle$ and $\left\vert g\right\rangle =\left\vert {5}^{2}{S}_{1/2},F=2\right\rangle$ ground states are coupled in a Λ-scheme to the excited state $\left\vert x\right\rangle =\left\vert {5}^{2}{P}_{3/2},{F}^{\prime }=3\right\rangle$ using a driving laser Ω(t), as illustrated in figure 3. The cavity has a decay rate of κ = 2π × 12 MHz and a maximum coupling strength of g₀ = 2π × 15 MHz. With the system initially prepared in $\left\vert e,0\right\rangle$, the laser adiabatically drives the system to $\left\vert g,1\right\rangle$, whereupon the photon is emitted from the cavity mode, leaving the system in $\left\vert g,0\right\rangle$, decoupled from further evolution. Atoms are loaded into the cavity with an atomic fountain, from a magneto-optic trap (MOT) located directly under the cavity.

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The spatio-temporal profile $\zeta \left(t\right)$ of the emitted photons can be directly controlled by shaping the driving pulse [21] using an acousto-optic modulator (AOM). For our purposes, it is most crucial that the phase of the driving laser, which we control via a separate electro-optic modulator (EOM), is directly mapped to the phase of the emitted photon. Figure 3(b) shows the driving pulse shape and figure 3(c) the resulting doubly-humped photon profile, evenly distributed across two 200 ns time-bins, for a total coherence time of δt = 400 ns.

The experimental sequence for the production of photons is as follows: photons are emitted at a repetition rate of 1 MHz, with approximately 400 ns of the cycle used for the production of a single photon. The remaining 600 ns of the cycle are used to optically repump the atom to $\left\vert e\right\rangle$ to repeat the process. The photon production and repumping timings can be changed at will. After being emitted, the photons impinge on a polarising beam splitter (PBS), and are randomly routed into two paths, one of which incorporates a fibre loop of 300 m of optical path length to induce a 1 μs delay. This ensures that a pair of subsequently emitted photons arrives simultaneously at the beam splitter, which is the only case of interest. A half wave plate (HWP) sets the relative polarisation of both photons.

Note from figure 3(c), that the first halves of the emitted photons span 60 m (∼200 ns), despite the optical path length between the cavity and the detectors being only 1.5 m or 301.5 m, depending on the path taken. This means that the first time interval of the two photon state is already being measured before the second half of the second photon leaving the cavity (travelling along the shorter path) has been fully generated. The length of the interfering photons is sufficient for a feedback loop to alter the phase ϕ of the second half of the photon under production, conditioned on the measurement outcome within the first time interval (figure 1).

The feedback is implemented using a home-built circuit controller and single photon counting modules (SPCMs) with a quantum efficiency of 60%–65% and a resolution of $< 300\enspace \mathrm{p}\mathrm{s}$ (Excelitas SPCM-AQRH-780-14-FC). The total feedback loop latency, from the EOM via the cavity to the SPCM, then back to the EOM via the controller circuitry is 97.0 ± 0.2 ns. Therefore, a conditioned phase change cannot be realised in time if a photon detection in I₂ occurs less than 97 ns after a detection in I₁. The resulting error rate is limited to 0.2% for a ${\mathrm{sin}}^{4}\left(t\right)$ photon intensity envelope. All photon counts are recorded with 81 ps accuracy using a qutools quTAU time-to-digital converter (TDC). Detections within the 400 ns photon window are further processed for a dark count correction for each SPCM. A detailed description on the error rates, time budgets, the implementation of the logical feedback circuit and background correction is provided in the supplemental material.