Spontaneous Beliaev-Landau scattering out of equilibrium

We investigate Beliaev-Landau scattering in a gas of interacting photons in a coherently driven array of nonlinear dissipative resonators, as described by the one-dimensional driven-dissipative Bose-Hubbard model. Due to the absence of detailed balance in such an out-of-equilibrium setup, steady-state properties can be much more sensitive to the underlying microscopic dynamics. Because the popular truncated Wigner approximation dramatically fails in capturing this physics, we present an alternative approach, based on a systematic expansion beyond the Bogoliubov approximation, which includes the third-order correlation functions in the dynamics. As experimentally accessible signatures of Beliaev-Landau processes, we report a small but non-negligible correction to the Bogoliubov prediction for the steady-state momentum distribution, in the form of a characteristic series of peaks and dips, as well as non-Gaussian features in the statistics of the cavity output field.

Figure 1

(a) Spectrum of excitations (14) for $(J,U{n}_0)=(30\gamma ,10\gamma )$ and for different values of the renormalized detuning $\mathrm{\Delta }$ [Eq. (6)]. If $\mathrm{\Delta }\to 0^{(-)}$, i.e., a drive exactly on the blueshifted resonance, the gap vanishes. Beliaev decay is sketched as an excitation at momentum $k$ that decays to $q$ and $k-q$ with conservation of energy. (b) Contours of energy conservation from (20) for the same parameters as in (a). The extremal momenta are found from the contours and are indicated for $\mathrm{\Delta }=-10\gamma$ (solid black line), the case we have considered for the rest of the analysis. (c) Shift of the extremal momenta as a function of $\mathrm{\Delta }$ for the positive contour. At $\mathrm{\Delta }<{\mathrm{\Delta }}_0<0$ the spectral gap becomes too large and no resonant scattering channels exist.

Figure 2

Momentum distribution of photons (in units of the number of photons per mode) as obtained (a) from the truncated Wigner approximation and (b) by truncating the hierarchy of correlation functions at the third order, as discussed in the text. The parameters are $(J,\mathrm{\Delta },U{n}_0)=(30\gamma ,-10\gamma ,10\gamma )$, $L=128$, and three different interaction strengths $U=0.02\gamma$ (blue solid lines), $U=0.1\gamma$ (red dashed lines), and $U=0.5\gamma$ (green dash-dotted lines) such that the average number of photons per cavity is 500, 100, and 20, respectively. The latter case is not shown for the TWA computation, because it is outside its regime of validity anyway. The Bogoliubov result (17) is also indicated (black dotted line). For clarity, we show the difference of the HOC from the Bogoliubov distribution $\delta n=n-n_{\text{Bog}}$ in the inset of (b). The redistribution of particles is in both cases the strongest around the extremal values $q_{\text{min}}$, $k_{\text{min}}$, $q_{\text{max}}$, and $k_{\text{max}}$ (vertical dotted lines) of the contour from Fig. 1. The inaccurate TWA result tremendously overestimates the corrections stemming from Beliaev-Landau decay, with a negative value for certain $k$ modes, while the HOC result predicts only a small deviation from the Bogoliubov distribution. This is a direct consequence of the unphysical decay of the quantum vacuum in the Wigner representation.

Figure 3

Absolute value of the third-order correlation matrix $M_{k,q}$ for (a) $U=0.02\gamma$, (b) $U=0.1\gamma$, and (c) $U=0.5\gamma$ for the same parameters as in Fig. 2. The curve representing energy and momentum conservation shown as a black solid line in Fig. 1 corresponds here to a line of enhanced scattering.

Figure 4

(a) Sketch of the standard setup for an angle-resolved measurement. While the cavity array is pumped in the $k=0$ mode, corresponding to a perpendicularly incident laser beam, quantum fluctuations are expected to leak out at a nonzero angle $sin{\theta }_k=ck/{\omega }_L\mathrm{\Delta }x$. In the setup we discuss in the text, all photons are expected to leak out within an angle of about ${23}^{\circ }$. Measuring the intensity at an angle ${\theta }_k$ with width $\delta \theta$ then allows for the measurement of $n_k$, for which the theoretical prediction is given in Fig. 2. (b) Possible setup to detect the non-Gaussianities in the cavity output field through the third-order correlator (23). Simultaneous clicks between detector 1 and the difference signal of detectors 2 and 3 allows for the measurement of the quantity from expression (32). Here PS stands for phase shifter and BS for a (50:50) beam splitter.

Figure 5

Momentum distribution (blue solid line) obtained from a TWA simulation with $(J,\mathrm{\Delta },U{n}_0)=(10\gamma ,-10\gamma ,10\gamma )$ and an interaction constant $U=0.1\gamma$. The result lies very close to the Bogoliubov prediction (black dotted line) when the spectrum does not allow for resonant Beliaev-Landau channels.

Figure 6

Comparison of different truncation schemes for the quantity $\delta n_k=n_k-n_k^{\text{Bog}}$, with $n_k^{\text{Bog}}$ the momentum distribution in the Bogoliubov approximation (19). We show the values of $U$ that have been studied in the main text. The chain consists of only ten cavities such that higher-order truncations can be computed and compared. Here FC stands for factorized cutoff, the method that has been outlined and employed in the main text; HC stands for hard cutoff and the integer indicates up to which order $N_c$ normal-ordered correlation functions have been included in the hierarchy.

Figure 7

Difference $\mathrm{\Delta }n=1/L{\sum }_k|n_k-n_k^{\text{Bog}}|/n_k^{\text{Bog}}$ as a function of the coupling constant $U$ for different truncation schemes for a chain comprising ten cavities.