Wigner negativity in the steady-state output of a Kerr parametric oscillator

The output field from a continuously driven linear parametric oscillator may exhibit considerably more squeezing than the intracavity field. Inspired by this fact, we explore the nonclassical features of the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporal wave packet mode description. Utilizing a new numerical method, we have access to the density matrix of arbitrary wave packet modes. Remarkably, we find that even though the steady-state cavity field is always characterized by a positive Wigner function, the output may exhibit Wigner negativity, depending on the properties of the selected mode.

Figure 1

Filtered output field populations of the first 6 number states $|n\rangle$ excluding the vacuum. Even-number states in solid lines and odd-number states in dashed lines [(a), (b)]. Squeezing for both the output filtered field in solid lines and the cavity field in dashed lines [(c), (d)]. Determinant of the covariance matrix in solid lines. The dashed lines signal the minimum uncertainty condition [(e), (f)]. Everything for $\beta =0.2$ and $\beta =0.4$, and as a function of the boxcar field width $T$ in units where $\gamma =1$. Note the logarithmic scale on the horizontal axis.

Figure 2

PO output field Fock state populations for two different drive strengths $\beta$ and boxcar filter times $T$ (in units where $\gamma =1$). For the very large filter time $T=500$, only even Fock states are present, and squeezing is enhanced.

Figure 3

A drive photon of frequency $2\omega$ is down-converted into two photons each with frequency $\omega$ in the cavity with resonance frequency $\omega$ and single-photon loss rate $\gamma$. For a longer observation or filtering time $T$, more and more photon pairs are detected.

Figure 4

Klyshko coefficients $B_n$ corresponding to the number state $|n\rangle$ with $n=2$ (solid line), $n=4$ (dashed line), and $n=6$ (dot-dashed line) for drive strengths (a) $\beta =0.2$ and (b) $\beta =0.4$, for a boxcar filter of width $T$ (in units where $\gamma =1$). Note the logarithmic scale on the horizontal axis.

Figure 5

(a) Output two-photon population for $K=0.1$ as a function of $\beta$ for $T=0.1$, 0.5, and 1.0 in units where $\gamma =1$. The dotted line at ${\rho }_2=0.27$ represents the theoretical maximum two-photon population in the Poissonian regime. Note that the two-photon population is decreased when moving out of the Poissonian regime. Photon number distributions at the two-photon peaks for (b) $T=0.1$ fitted to a Poissonian distribution with $n_{\mathrm{cav}}=1.9$ and (c) $T=0.5$ fitted to a Poissonian distribution with $n_{\mathrm{cav}}=2.2$. (d) $T=1.0$ does not fit a Poissonian distribution.

Figure 6

(a) Output two-photon population for $K=0.5$ as a function of $\beta$ for $T=0.5$, 2.0, and 5.0 in units where $\gamma =1$. The dotted line at ${\rho }_2=0.27$ represents the theoretical maximum two-photon population in the Poissonian regime. The two-photon populations are close to the maximum both within and outside of the Poissonian regime (in stark contrast to the behavior in Fig. 5) with a maximum of ${\rho }_2=0.28$ obtained for the most non-Poissonian state corresponding to $T=5.0$. Photon number distributions at the two-photon peaks for (b) $T=0.5$ fitted to a Poissonian distribution with $n_{\mathrm{cav}}=2.0$. (c) $T=2.0$ does not quite fit a Poissonian distribution. (d) $T=5.0$ shows clearly non-Poissonian statistics with population oscillations.

Figure 7

When the nonlinearity $K$ is increased from zero, the Klyshko $B_2$ coefficient rapidly drops to its minimum for $K=0.7$, corresponding to the “most nonclassical” state as determined by the Klyshko criterion. As $K$ is further increased, the value of $B_2$ increases slightly before saturating. The minimum values were obtained by sweeping over a grid of $\beta$ and $T$ for each $K$.

Figure 8

Wigner functions and density matrices for $\beta =0.65$. (a) Wigner function and (b) density matrix for the steady-state cavity field. It is clearly Wigner-positive. (c) Wigner function and (d) density matrix for the output state with a $T=2.5$ boxcar filter. Negativity appears in the Wigner function, and in the density matrix, the vacuum population is reduced and the two-photon population is increased compared to the cavity field. (e) Wigner function and (f) density matrix for the output field with a $T=5$ boxcar filter, which gives the largest WLN (0.05). Here, the vacuum population is further reduced and the two-photon population is even more prominent. For clarity in the figure we truncate the density matrices at $n=3$, but the full states occupy a larger Fock space.

Figure 9

(a) Contour map of the WLN as a function of $\beta$ and $T$ for $K=0.5$. The dashed line indicates $T=5$. (b) shows the WLN and Klyshko $B_2$ coefficient as a function of $\beta$ for a fixed $T=5$. The WLN and $B_2$ are clearly related.

Figure 10

Numerical optimization of the filter function at discrete time points $t_i$. The initial filter function array was constructed by generating random numbers in the interval [0,1] and then normalizing. The filter array was then optimized for maximum WLN. A Gaussian curve can be well fitted to the optimized filter array, here giving $\sigma =2.2$ and $\mu =17.0$. In this example with system parameters $\beta =0.65$ and $K=0.5$, the filter array consists of 18 points and the obtained WLN is 0.1.

Figure 11

A comparison of the maximum WLN for the boxcar and Gaussian filters, as a function of $\beta$ with the filter widths fixed to the values that give the largest possible WLN: $T=5$ for the boxcar and $\sigma =2.3$ for the Gaussian filter. With the Gaussian filter the maximum WLN is doubled compared to the boxcar.