Absorption to fluctuating bunching states in nonunitary boson dynamics

We show that noisy nonunitary dynamics of bosons drives arbitrary initial states into a novel fluctuating bunching state, where all bosons occupy one time-dependent mode. We propose a concept of the noisy spectral gap, a generalization of the spectral gap in noiseless systems, and demonstrate that the exponentially fast absorption to the fluctuating bunching state takes place asymptotically. The fluctuating bunching state is unique to noisy nonunitary dynamics, with no counterpart in any unitary dynamics and nonunitary dynamics described by a time-independent generator. We also argue that the times of relaxation to the fluctuating bunching state obey a universal power law as functions of the noise parameter in generic noisy nonunitary dynamics.

Figure 1

Schematic picture of the fluctuating bunching state. Various initial states, $|{\psi }_{t=0}^a\rangle ,|{\psi }_{t=0}^b\rangle ,\cdots$, are absorbed into the fluctuating bunching state, where all bosons described by yellow circles occupy one dominant eigenmode of $V_t$. We can define the time-dependent gap ${\mathrm{\Delta }}_t$ from the eigenvalues of $ln\left(V_t\right)/t$.

Figure 2

(a) Schematic picture of the nonunitary dynamics of photons in an optical network. Photons evolve via the unitary matrix $U$ (blue rectangles) and then experience nonunitary dynamics by $G_t$ (orange rectangles) due to the photon loss effect and postselection. (b) The convergence of $\overline{{\mathrm{\Delta }}_t}$ into the noisy spectral gap $\mathrm{\Delta }\simeq 6\times {10}^{-4}$, where the number of samples is ${10}^5$. (c) The difference between the first and second Lyapunov exponents. Symbols with different colors correspond to different realizations of noises $\left\{z_{\eta ,t}\right\}$. For large $t$, $e_{t,1}-e_{t,2}$ for each trajectory converges to the sample-independent value that is the same as $\mathrm{\Delta }$. In (b) and (c), $\beta =0.3$ and $X=20$.

Figure 3

Decays of $f_t=-ln\left(\overline{|{\lambda }_{t,1}/{\lambda }_{t,2}{|}^2}\right)/2$ (red symbols), $\overline{ln({\mathrm{\Lambda }}_{t,2}/{\mathrm{\Lambda }}_{t,1})}$ (blue symbols), and $-\mathrm{\Delta }t$ (the green line), respectively, which indicate that the fluctuating bunching state emerges after long-time dynamics. The average is taken over ${10}^5$ samples with $\beta =0.3$ and $X=20$. The orange broken line is $f_t=-{\beta }^{2}t/6X-ln\left(\sqrt{2}\right)$ obtained from Eq. (15), which is calculated through the perturbation analysis on $\overline{ln({\mathrm{\Lambda }}_{t,2}/{\mathrm{\Lambda }}_{t,1})}$. The black line $f_t=tln(\sqrt{\nu }/\mu )$ represents the analytical upper bound on $-ln\left(\overline{|{\lambda }_{t,1}/{\lambda }_{t,2}{|}^2}\right)/2$, derived from Eq. (12).

Figure 4

Expectation values of ${\sum }_x{x}^2{\widehat{b}}_x^{†}{\widehat{b}}_x/n$ under nonunitary dynamics of three bosons with different initial states. Green, blue, and red symbols, respectively, correspond to $(x_1^{\mathrm{in}},x_2^{\mathrm{in}},x_3^{\mathrm{in}})=(-6,1,8)$, $(-1,0,1)$, and $(-6,-5,6)$ with a fixed sample $\left\{z_{\eta ,t}\right\}$. The three trajectories are absorbed into one time-dependent trajectory independent of initial states in the long-time regime, while they depend on initial states in the short-time regime. The black broken line represents $t={\tau }_{\mathrm{\Delta }}=|ln\left(c\right)|/\mathrm{\Delta }$ with $c={10}^{-2}$ at which $\overline{ln\left(\right|{\lambda }_{t,2}/{\lambda }_{t,1}\left|\right)}=ln\left(c\right)$ is satisfied. The parameters are $({\theta }_1,{\theta }_2,{\theta }_3)=(0.37\pi ,0.19\pi ,0.25\pi )$ and $\beta =0.3$, with $X=20$.

Figure 5

Relaxation times as functions of the noise parameter, which exhibit the power law in Eq. (6). Green, blue, red, and purple symbols represent ${\tau }_{\mathrm{\Delta }}$, ${\tau }_{\mathrm{\Lambda }}$, ${\tau }_{\lambda }$, and ${\tau }_x$, respectively, with $c={10}^{-6}$ and $X=20$. Here, the average in $f_t$ is taken over ${10}^2$ samples for obtaining one $\tau$, and then we iterate it ${10}^2$ times and obtain the averaged $\tau$ over them. Expectation values are computed with $n=2$, where initial states are $(x_1^{\mathrm{in}},x_2^{\mathrm{in}})=(-5,5)$ for ${\langle x^2\rangle }_{t,a}$ and $(-1,0)$ for ${\langle x^2\rangle }_{t,b}$. The black symbols and orange broken line correspond to ${\tau }_{\mathrm{\Omega }}^{\lambda }$ and ${\tau }_{\mathrm{\Omega }}^{\mathrm{\Lambda }}$, in Eqs. (13) and (16), respectively, where the latter is exactly proportional to ${\beta }^{-2}$.

Figure 6

The noisy spectral gap as a function of $X$. The blue symbols are numerically obtained $\mathrm{\Delta }$ with $({\theta }_1,{\theta }_2,{\theta }_3)=(0.37\pi ,0.19\pi ,0.25\pi )$ and $\beta =0.3$. The green broken line is ${10}^{-2}/X$.

Figure 7

(a) The averaged inverse participation ratio (IPR) of the dominant mode, $\overline{{\sum }_x|{\varphi }_t^1{\left(x\right)|}^4/[{\sum }_x|{\varphi }_t^1\left(x\right){{|}^2]}^2}$, as a function of the system size $X$. The IPR is almost independent of $X$, which indicates that ${\varphi }_t^1$ is localized. The average is taken over 40 samples and 100 time steps after a time at which $\overline{|{\lambda }_{t,2}/{\lambda }_{t,1}|}<c={10}^{-6}$ is satisfied. (b) The probability distribution $P_t\left(\left\{x_q^{\mathrm{out}}\right\}\right)$ of a bunching state with $n=3$, $(x_1^{\mathrm{in}},x_2^{\mathrm{in}},x_3^{\mathrm{in}})=(-6,1,8)$, $\beta =0.3$, and $t=42418$. The value of $P_t\left(\left\{x_q^{\mathrm{out}}\right\}\right)$ is represented as the size and color of the symbols: the big (tiny) and dense green (shallow gray) symbols correspond to large (small) $P_t\left(\left\{x_q^{\mathrm{out}}\right\}\right)$.

Figure 8

Various relaxation times as functions of the noise parameter $\beta$, all of which exhibit the power law $\tau \propto {\beta }^{-2}$. Blue, red, purple, green, cyan, and gray symbols, respectively, represent relaxation times defined through $f_{\tau }\le ln\left(c\right)$ with $c={10}^{-6}$, where $f_t=\overline{ln({\mathrm{\Lambda }}_{t,2}/{\mathrm{\Lambda }}_{t,1})}$, $f_t=-ln\left(\overline{|{\lambda }_{t,1}/{\lambda }_{t,2}{|}^2}\right)/2$, $f_t=ln\left(\overline{|{\langle x^2\rangle }_{t,a}-{\langle x^2\rangle }_{t,b}|}\right)$ with $n=2$, $f_t=-\mathrm{\Delta }t$, $f_t=ln\left(\overline{|{\lambda }_{t,2}/{\lambda }_{t,1}{|}^2}\right)/2$, and $f_t=-ln\left(\overline{|{\mathrm{\Lambda }}_{t,1}/{\mathrm{\Lambda }}_{t,2}{|}^2}\right)/2$. The average included in $f_t$ is taken over ${10}^2$ samples for obtaining one $\tau$. Then, we reiterate obtaining $\tau$ for ${10}^2$ times and take the average over them. The subscripts $a$ and $b$ for ${\langle x^2\rangle }_t$ correspond to two initial states of bosons, $(x_1^{\mathrm{in}},x_2^{\mathrm{in}})=(-5,5)$ and $(-1,0)$. Regarding ${\tau }_{\mathrm{\Delta }}$ with $f_t=-\mathrm{\Delta }t$, $\mathrm{\Delta }$ is obtained through the ensemble average over ${10}^4$ samples. The orange broken line corresponds to the relaxation time ${\tau }_{\mathrm{\Omega }}^{\mathrm{\Lambda }}$ defined through the perturbative analysis on $\overline{ln({\mathrm{\Lambda }}_{t,2}/{\mathrm{\Lambda }}_{t,1})}$. Since this relaxation time is obtained as $-{\beta }^2{\tau }_{\mathrm{\Omega }}^{\mathrm{\Lambda }}/6X-ln\left(\sqrt{2}\right)=ln\left(c\right)$, the orange broken line is exactly proportional to ${\beta }^{-2}$. The black symbols represent ${\tau }_{\mathrm{\Omega }}^{\lambda }=ln\left(c\right)/ln\left(\right|\sqrt{\nu }/\mu \left|\right)$, which is above the red symbols for large $t$, due to the inequality obtained in Eq. (E4). The parameters are $({\theta }_1,{\theta }_2,{\theta }_3)=(0.37\pi ,0.19\pi ,0.25\pi )$ with the system size $X=20$.

Figure 9

Results of a model introduced in Appendix pp7. (a) Schematic picture of the dynamics for one step. Blue rectangles and orange squares correspond to unitary dynamics by $U_1,U_2$, and nonunitary dynamics by $G_t$, respectively. (b) Ratios of $|{\lambda }_{t,i}|$ and ${\mathrm{\Lambda }}_{t,i}$, which indicate that the fluctuating bunching state emerges after long-time dynamics. Red and blue symbols show $-ln\left(\overline{|{\lambda }_{t,1}/{\lambda }_{t,2}{|}^2}\right)/2$ and $\overline{ln({\mathrm{\Lambda }}_{t,2}/{\mathrm{\Lambda }}_{t,1})}$, respectively. The average is taken over ${10}^5$ samples. The orange broken line is $-{\beta }^{2}t/6X-ln\left(\sqrt{2}\right)$, which is derived through the perturbation analysis on $\overline{ln\left({\mathrm{\Omega }}_t^{\mathrm{\Lambda }}\right)}$. While this perturbation analysis is justified if ${\beta }^{2}t/X$ is small, as discussed in Appendix pp6, we can see that its prediction turns out to be valid even for the long-time regime. The black line represents $tln\left(\right|\sqrt{\nu }/\mu \left|\right)$ obtained through numerical diagonalization of $\mathcal{Q}$ and $\mathbb{Q}$. Here, $\mu =\sqrt{\nu }$, and thus $tln\left(\right|\sqrt{\nu }/\mu \left|\right)$ is always zero, making the bound in Eq. (E4) useless. The noise parameter is $\beta =0.3$. (c) Relaxation times as functions of the noise parameter $\beta$ with $c={10}^{-6}$, which exhibit the power law $\tau \propto {\beta }^{-2}$. Blue, red, and purple symbols, respectively, represent ${\tau }_{\mathrm{\Lambda }}$, ${\tau }_{\lambda }$, and ${\tau }_x$ with $n=2$. The average in $f_t$ is taken over ${10}^2$ samples for obtaining one $\tau$. We then reiterate obtaining $\tau$ for ${10}^2$ times and take the average over them. The orange broken line corresponds to relaxation times defined through the perturbative analysis on $\overline{ln({\mathrm{\Lambda }}_{t,2}/{\mathrm{\Lambda }}_{t,1})}$. Since this relaxation time is obtained by $-{\beta }^2{\tau }_{\mathrm{\Omega }}^{\mathrm{\Lambda }}/6X-ln\left(\sqrt{2}\right)=ln\left(c\right)$, the orange broken line is exactly proportional to ${\beta }^{-2}$. In (b) and (c), the parameters are $({\theta }_{11},{\theta }_{21},{\theta }_{31},{\theta }_{12},{\theta }_{22},{\theta }_{32})=(0.33\pi ,0.41\pi ,0.25\pi ,0.22\pi ,0.13\pi ,0.25\pi )$, where the system size is $X=20$.