Single-photon time qubit: Quantum-vacuum control and complementarity of the arrow of time

Treating the arrow of time as a controllable quantum variable rather than a fixed external parameter is a central challenge in quantum physics. Recently, Bloch~\cite{Bloch2025} proposed the theoretical concept of a \emph{time qubit}, a two-level control space ${\mathcal{H}}_T=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{\mathrm{\mid }+⟩,\mathrm{\mid }-⟩\}$ that selects between opposite Hamiltonian evolutions ${\widehat{H}}_{\pm }={\widehat{H}}_0\pm \widehat{V}$, but this paradigm has remained experimentally unrealized. Here we formulate the quantum-optical theory of a single-photon time qubit—uncovering quantum-vacuum control and complementarity of the arrow of time—and experimentally probe and validate its predictions using phase-resolved single-photon interferometry from two coherently seeded entangled nonlinear biphoton sources~\cite{Lee2018,Yoon2021}. First, we show theoretically and verify experimentally that a controlled $\pi$ pump–seed phase inversion during pair creation from the quantum vacuum flips the stimulated displacement generator $\widehat{V}\to -\widehat{V}$ on top of the spontaneous two-mode squeezing background ${\widehat{H}}_0$, reversing the stimulated evolution time [${\widehat{U}}_{\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{m},-}(t)={\widehat{U}}_{\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{m},+}(-t)$] and enabling universal single-qubit $\mathrm{S}\mathrm{U}(2)$ control and postselection-free first-order tomography on the temporal Bloch sphere. Second, we prove that spontaneous idler vacuum fluctuations record complete which-branch (temporal-direction) information whereas the shared coherent seed erases it without cascading or aligning the idler beams, establishing an exact wave–particle complementarity law $\mathrm{\mid }\mathit{r}\mathrm{\mid }=\mathcal{V}=\mathrm{\mid }\alpha {\mathrm{\mid }}^2\mathrm{/}(1+\mathrm{\mid }\alpha {\mathrm{\mid }}^2)$ for the purity of the arrow of time and enabling direct first-order measurement of the Loschmidt time-reversal echo.