Measurement of the two-time intensity-correlation function of arbitrary states

For light-intensity-correlation measurements, different methods are used in the high-photon-number or high-intensity regime and in the single- and two-photon regime. Hence, there is an unfortunate measurement “gap” primarily for multiphoton, quantum states. These states, for example, multiphoton Fock states, will be increasingly important in the realization of quantum technologies and in exploring the boundaries between quantum and classical optics. We show that a naïve approach based on attenuation, state splitting, and two-detector correlation can give the correct two-time intensity correlation for any state. We analyze how added losses decrease the measurement systematic error. The price to be paid is that the losses increase the measurement statistical error or, alternatively, increase the acquisition time for a given tolerable level of statistical error. We have experimentally demonstrated the feasibility of the method for a coherent state and a quasithermal state. The method is easy to implement in any laboratory and will simplify characterization of medium and highly excited nonclassical states as they become experimentally available.

Figure 1

Proposed measurement setup to measure the fourth-order correlation function of the state $|\psi \rangle$. BS denotes the beam splitter.

Figure 2

Number states: calculated normalized coincidences ${\gamma }_n^{\left(4\right)}\left(0\right)$ for different $n$, as a function of attenuation [Eq. (20)]. Here we assume ${\eta }_1^2{\mu }_1^2={\eta }_2^2{\mu }_2^2$ and that they are included in ${\epsilon }^2$ for simplicity.

Figure 3

Number states: calculated normalized coincidences ${\gamma }_n^{\left(4\right)}\left(0\right)$ when using ordinary “click” detectors (red) and photon number resolving detectors (blue).

Figure 4

Coherent state: Measured normalized coincidences ${\gamma }^{\left(4\right)}\left(\tau \right)$, for several attenuations ${\epsilon }^2$ (different colored dots, $T=0.16$–1), as a function of delay time show no correlations. The inset shows measured ${\gamma }^{\left(4\right)}(\tau =0)$ for different attenuations ${\epsilon }^2$. The red line is the theoretically expected value. Bars represent the statistical error.

Figure 5

Setup to generate and detect ${\gamma }^{\left(4\right)}\left(\tau \right)$ for a quasithermal state. $v_{\mathrm{rot}}=105$ rad/s, the average diffuser film grain size is 5 $\mu \mathrm{m}$.

Figure 6

Thermal state: calculated normalized coincidences ${\gamma }^{\left(4\right)}\left(0\right)$ for thermal states with varying $\langle \widehat{n}\rangle$. Measured data from a quasithermal state (black squares, bars represent statistical errors), and the best fit to a rational (red, details in text). Here we assume ${\eta }_1^2{\mu }_1^2={\eta }_2^2{\mu }_2^2$ bunched into the expression ${\epsilon }^2$ for simplicity.

Figure 7

Quasithermal state: Measured ${\gamma }^{\left(4\right)}\left(\tau \right)$, for several attenuation strengths ($T$=transmission through filter). Inset: enlargement around $\tau =0$.