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 proposed the theoretical concept of a time qubit, a two-level control space H T= span, that selects between opposite Hamiltonian evolutions plus or minus= 0 plus or minus, 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. First, we show theoretically and verify experimentally that a controlled π pump--seed phase inversion during pair creation from the quantum vacuum flips the stimulated displacement generator to - on top of the spontaneous two-mode squeezing background 0, reversing the stimulated evolution time [stim,-(t)= stim,+(-t)] and enabling universal single-qubit SU(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 r= V=α 2/(1+α 2) for the purity of the arrow of time and enabling direct first-order measurement of the Loschmidt time-reversal echo.