Time-dependent quantum correlations in phase space

General quasiprobabilities are introduced to visualize time-dependent quantum correlations of light in phase space. They are based on the generalization of the Glauber-Sudarshan $P$ function to a time-dependent $P$ functional [W. Vogel, Phys. Rev. Lett. 100, 013605 (2008)], which fully describes temporal correlations of radiation fields on the basis of continuous phase-space distributions. This approach is nontrivial, as the $P$ functional itself is highly singular for many quantum states and nonlinear processes. In general, it yields neither a well-behaved nor an experimentally accessible description of quantum stochastic processes. Our regularized version of this multitime-dependent quasiprobability is a smooth function and applies to stronger divergences compared to the single-time and multimode scenario. The technique is used to characterize an optical parametric process with frequency mismatch and a strongly nonlinear evolution of the quantized center-of-mass motion of a trapped ion. A measurement scheme, together with a sampling approach, is provided which yields direct experimental access to the regularized $P$ functional from measured data.

Figure 1

Plot of the triangular function $\mathrm{tri}\left(\mathrm{Re}\right[\beta ]/w)$ multiplied by an exponential rising function $exp\left[\right|\beta {|}^m]$. We see that the resulting function is bounded, and thus, its Fourier transform is a continuous function.

Figure 2

Plot of the regularized and scaled phase-space distribution $P_{\mathrm{\Omega }}[\alpha ;\tau ]$ for several system times $\tau$. We chose $\mathrm{Im}\left[\alpha \right]=0, w=2.3$ for the triangular filter and $r=\delta /\kappa =10/\pi \approx 3.18$. The negativities for $\tau >0$ clearly show the evolution of the nonclassicality.

Figure 3

Plot of the two-time regularized functional $P_{\mathrm{\Omega }}({\alpha }_1,{\alpha }_2;{\tau }_1,{\tau }_2)$ for $w=2.9$. Here, we used ${\tau }_1=0.1,{\tau }_2=0.45$, and the cross section $\mathrm{Im}\left[{\alpha }_2\right]=\mathrm{Im}\left[{\alpha }_1\right]=0$. The negativities of the quasiprobability reveal nonclassical normally and time-ordered correlation properties of the considered system.

Figure 4

Plot of $\mathrm{\Delta }\mathrm{\Phi }$ as defined in Eq. (45) for fixed $|{\beta }_1|=|{\beta }_2|=1.3$ and ${\phi }_1={\phi }_2\equiv \phi$, where ${\beta }_j=\left|{\beta }_j\right|e^{i{\phi }_j}, j=1,2$. We vary the common phase $\phi$ and the dimensionless time $\tau =\varepsilon t$. As $\mathrm{\Delta }\mathrm{\Phi }$ clearly exceeds the value of 0 (pink areas), we confirm that the MTCF is differently bounded compared to the two-mode single-time case.

Figure 5

Experimental scheme to directly measure two-time quantum correlations in terms of two-time quasiprobabilities. The scheme consists of two balanced homodyne detection (BHD) setups whose difference signals are additionally correlated. The creation operators $\widehat{a}\left(t\right)$ and $\widehat{a}(t+\mathrm{\Delta }t)$ label the different travel times of the field. The resulting correlated difference statistics of the detector events can be directly related to $P_{\mathrm{\Omega }}[{\alpha }_1,{\alpha }_2;{\tau }_1,{\tau }_2]$, where the phases for the local oscillator (LO) of each BHD setup are controlled continuously through the phase shifters $\mathrm{\Delta }\phi$ and $\mathrm{\Delta }{\phi }^{\prime }$. The temporal matching of the LO to the signal fields is ensured through a proper path-length control.