Nonclassicality in weak measurements

We examine weak measurements of arbitrary observables where the object is prepared in a mixed state and on which measurements with imperfect detectors are made. The weak value of an observable can be expressed as a conditional expectation value over an infinite class of different generalized Kirkwood quasiprobability distributions. “Strange” weak values for which the real part exceeds the eigenvalue spectrum of the observable can only be found if the Terletsky-Margenau-Hill distribution is negative or, equivalently, if the real part of the weak value of the density operator is negative. We find that a classical model of a weak measurement exists whenever the Terletsky-Margenau-Hill representation of the observable equals the classical representation of the observable and the Terletsky-Margenau-Hill distribution is non-negative. Strange weak values alone are not sufficient to obtain a contradiction with classical models. We propose feasible weak measurements of photon number of the radiation field. Negative weak values of energy contradict all classical stochastic models, whereas negative weak values of photon number contradict all classical stochastic models where the energy is bounded from below by the zero-point energy. We examine coherent states in particular and find negative weak values with probabilities of 16% for kinetic energy (or squared field quadrature), 8% for harmonic oscillator energy, and 50% for photon number. These experiments are robust against detector inefficiency and thermal noise.

Figure 1

Scheme for a weak measurement via cross-Kerr coupling in a nonlinear crystal.

Figure 2

The probability for observing a negative weak value of ${\widehat{p}}^2$ as a function of the detector efficiency $\eta$ and the average thermal occupation number $n_{\mathit{th}}$. It is assumed that ${\alpha }_i=0$, whereas ${\alpha }_r$ may take any value. This effect therefore is present even for macroscopic excitations ${\alpha }_r$.

Figure 3

The probability for observing a negative weak value of energy as a function of the coherent state amplitudes ${\alpha }_r$ and ${\alpha }_i$. We assume perfect detectors ${\sigma }_{\eta }=0$ and no thermal noise $n_{\mathit{th}}=0$. The probability has a maximum at ${\alpha }_r=1,{\alpha }_i=0$, at which the probability is $(1∕2)\mathrm{erfc}1\approx 0.079$.

Figure 4

The probability for observing a negative weak value of energy as a function of the coherent state amplitudes ${\alpha }_r$ and ${\alpha }_i$. We assume detectors with quantum efficiency $\eta =0.7$ and thermal occupation number $n_{\mathit{th}}=0.3$. There is a minimum ${\alpha }_r$ required in order to see nonclassical behavior.

Figure 5

The probability for observing a negative weak value of energy as a function of the detector efficiency $\eta$ and the average thermal occupation number $n_{\mathit{th}}$. The coherent excitation is assumed to be ${\alpha }_r=1,{\alpha }_i=0$.

Figure 6

The probability for observing a negative weak value of the photon number $\widehat{n}$ as a function of the coherent-state amplitudes ${\alpha }_r$ and ${\alpha }_i$. We assume perfect detectors ${\sigma }_{\eta }=0$ and no thermal noise $n_{th}=0$. The probability increases with decreasing $\mid \alpha \mid$, but has a singularity at $\mid \alpha \mid =0$.

Figure 7

The probability for observing a negative weak value of the photon number $\widehat{n}$ as a function of the coherent-state amplitudes ${\alpha }_r$ and ${\alpha }_i$. We assume detectors with quantum efficiency $\eta =0.7$ and thermal occupation number $n_{th}=0.3$. We see that there is a minimum ${\alpha }_r$ required in order to see nonclassical behavior.

Figure 8

The probability for observing a negative weak value of the photon number $\widehat{n}$ as a function of detector efficiency $\eta$ and average thermal occupation number $n_{th}$. The coherent excitation is assumed to be ${\alpha }_r=0.1,{\alpha }_i=0$. A minimum detector efficiency (and a maximum thermal occupation number) is required to observe a negative weak value.