Geometric control of collective spontaneous emission

Dipole spin-wave states of atomic ensembles with wave vector ${\bf k}(\omega)$ mismatched from the dispersion relation of light are difficult to access by far-field excitation but may support rich phenomena beyond the traditional phase-matched scenario in quantum optics. We propose and demonstrate an optical technique to efficiently access these states. In particular, subnanosecond laser pulses shaped by a home-developed wideband modulation method are applied to shift the spin wave in ${\bf k}$ space with state-dependent geometric phase patterning, in an error-resilient fashion and on timescales much faster than spontaneous emission. We verify this control through the redirection, switch off, and recall of collectively enhanced emission from a $^{87}$Rb gas with $\sim 75\%$ single-step efficiency. Our work represents a first step toward efficient control of electric dipole spin waves for studying many-body dissipative dynamics of excited gases, as well as for numerous quantum optical applications.