Demonstration of Weak Measurement Based on Atomic Spontaneous Emission

We demonstrate a new type of weak measurement based on the dynamics of spontaneous emission. The pointer in our scheme is given by the Lorentzian distribution characterizing atomic exponential decay via emission of a single photon. We thus introduce weak measurement, so far demonstrated nearly exclusively with laser beams and Gaussian statistics, into the quantum regime of single emitters and single quanta, enabling the exploitation of a wide class of sources that are abundant in nature. We describe a complete analogy between our scheme and weak measurement with conventional Gaussian pointers. Instead of a shift in the mean of a Gaussian distribution, an imaginary weak value is exhibited in our scheme by a significantly slower-than-natural exponential distribution of emitted photons at the postselected polarization, leading to a large shift in their mean arrival time. The dynamics of spontaneous emission offer a broader view of the measurement process than is usually considered within the weak measurement formalism. Our scheme opens the path for the use of atoms and atomlike systems as sensitive probes in weak measurements, one example being optical magnetometry.

Figure 1

(a) Atomic $\mathsf{V}$ system for a weak measurement using atomic line shapes. The atom is excited from the ground state to a superposition of the upper states by a short pulse at vertical polarization $V={\sigma }^{+}+{\sigma }^{-}$, and spontaneously decays back to the ground state. In an external magnetic field, the energies of the upper levels are Zeeman shifted by $\pm \hslash \Delta$. The two line shapes with width $\Gamma$ are shown, overlapping when $\Delta \ll \Gamma$. (b) For $\Delta \gg \Gamma$, interfering the two circular polarizations with a PBS leads to time-domain quantum beats. (c) A weak measurement corresponds to the case in which $\Delta \ll \Gamma$ and a linear polarization that is rotated by $ϵ$ from $H$ is postselected, with $\Delta /\Gamma \ll ϵ\ll 1$. The result is an exponentially decaying signal with a longer-than-natural decay time.

Figure 2

(a) Experimental setup. Ultracold ${}^{87}\mathrm{Rb}$ atoms are released from a MOT and a small magnetic field $\mathbf{B}$ is applied. The atoms are pumped to the $|F=2,m=0⟩$ state using a $\pi$-polarized pump beam (for which this state is a dark state due to selection rules). The atoms are then excited by a 4.2 ns pulse linearly polarized perpendicular to the magnetic field, inducing ${\sigma }^{\pm }$ transitions only. Spontaneous emission in the direction of the magnetic field is collected into a multimode fiber. Time is measured from the triggering of the excitation pulse. HWP, half-wave plate; POL, Glan-Thompson polarizer; SPCM, single-photon counting module. (b) ${}^{87}\mathrm{Rb}$ levels used in the experiment. The $\mathsf{V}$-system transitions are shown in straight solid arrows. Gray wavy arrows are $\pi$-decay paths eliminated by our observation direction. Black wavy arrows are undesired $\sigma$ decays, which add incoherent background to our measurements, at a level of $\sim 12\%$ of the total signal [38]. The $\pi$-polarized optical pumping is shown as dashed lines.

Figure 3

Experimental results. (a) Temporal distribution of arrived photons for various degrees of postselection $ϵ$. The case of no postselection (natural decay with lifetime $1/\Gamma$) is shown in dashed black lines, for comparison. The solid black lines are fits of the data to Eq. (2) with no free parameters except a scale factor. The shift of the mean to later times with smaller values of $ϵ$ is evident. Data were low-pass filtered using a Gaussian of 4.2 ns full width. Inset: Logarithmic presentation of the decaying part of the signal. (b) Mean arrival time computed from the experimental data for various values of $ϵ$. The solid curve is the theoretical mean based on exact solution for $\Delta =600\mathrm{kHz}$. The dashed horizontal line indicates the mean that corresponds to the natural lifetime. Inset: The two corresponding line shapes with separation equal to $1/5$ FWHM.