Quantization of nonlinear non-Hamiltonian systems

TL;DR

This work solves the long-standing problem of quantizing nonlinear, non-Hamiltonian 2D dynamics by formulating an inverse problem for Lindbladians. The authors introduce cascade quantization, an exact open-system method that maps any polynomial vector field $h(\alpha,\alpha^*)$ into a physically valid quantum generator ${\cal L}$, combining a Hamiltonian part with carefully chosen dissipators to ensure complete positivity and trace preservation. They provide an efficient quantization table for cubic ($m=3$) systems, develop general formulae for arbitrary degree $m$, and demonstrate the approach on a suite of nonlinear phenomena including bifurcations (saddle-node, transcritical, pitchfork, Hopf, SNIC), stochastic Fitzhugh–Nagumo dynamics, and Liénard limit cycles. The results show that cascade quantization yields exact quantum counterparts to classical nonlinear dynamics, revealing nonclassical features such as Wigner negativity and coherence-resonance-like effects, and it offers advantages over variational approaches by avoiding auxiliary constructions and preserving a true Lindbladian structure. Overall, the method provides a scalable, principled framework for translating classical polynomial dynamics into quantum dynamics with potential applications to quantum sensing, quantum information, and dissipative quantum dynamics.

Abstract

Several important dynamical systems are in $\mathbb{R}^2$, defined by the pair of differential equations $(x',y')=(f(x,y),g(x,y))$. A question of fundamental importance is how such systems might behave quantum mechanically. In developing quantum theory, Dirac and others realized that classical Hamiltonian systems can be mapped to their quantum counterparts via canonical quantization. The resulting quantum dynamics is always physical, characterized by completely-positive and trace-preserving evolutions in the Schrödinger picture. However, whether non-Hamiltonian systems can be quantized systematically while respecting the same physical requirements has remained a long-standing problem. Here we resolve this question when $f(x,y)$ and $g(x,y)$ are arbitrary polynomials. By leveraging open-systems theory, we prove constructively that every polynomial system admits a physical generator of time evolution in the form of a Lindbladian. We call our method cascade quantization, and demonstrate its power by analyzing several paradigmatic examples of nonlinear dynamics such as bifurcations, noise-activated spiking, and Liénard systems. In effect, our method can quantize any classical system whose $f(x,y)$ and $g(x,y)$ are analytic with arbitrary precision. More importantly, cascade quantization is exact. This means restrictive system properties usually assumed in the literature to facilitate quantization, such as weak nonlinearity, rotational symmetry, or semiclassical dynamics, can all be dispensed with by cascade quantization. We also highlight the advantages of cascade quantization over existing proposals, by weighing it against examples from the variational paradigm using Lagrangians, as well as non-variational approaches.

Figures (9)

• Figure 1: Efficient cascade quantization for $\alpha'=h(\alpha,\alpha^*) \in \mathbb{P}_3$ (polynomials of degree three). The full solution for $\mathbb{P}_m$ (polynomials of degree $m$) with arbitrary $m$ is deferred to Sec. \ref{['GenSoln']}, with details provided in Appendix \ref{['ConsProof']}. Arbitrary complex coefficients are denoted by the letter $z$ (differentiated with subscripts when more than one such coefficient appears). We have also used the step function defined by $\theta(x)=0$ for $x\le0$, and $\theta(x)=1$ for $x>0$. For each $h(\alpha,\alpha^*)$, the corresponding Hamiltonian and Lindblad terms are indicated respectively in the green and blue boxes on the right. By linearity of the Lindbladian, the contributions for each monomial of $h(\alpha,\alpha^*) \in \mathbb{P}_m$ are simply summed together to obtain the full Lindbladian.
• Figure 2: Saddle-node bifurcation. (a)--(c) Phase portraits of \ref{['SadNodeNF']} as $\mu$ varies from a negative to positive value. The bifurcation occurs at $\mu=0$. In (a)--(c) horizontal dashed lines denote $y$ nullclines [all $(x,y)$ for which $y'=0$], while vertical dashed lines indicate $x$ nullclines [all $(x,y)$ for which $x'=0$]. The origin of phase space is placed at the middle of the $y$ nullcline. We have used the symbol $\triangleright\triangleright$ to denote the bottleneck/ghost region around the origin in (a). The closer $\mu$ is to zero, the greater the slowdown is in the bottleneck. In (d)--(h) we plot the steady-state Wigner function obtained from \ref{['SadNodeBifL']} and \ref{['SadNodeBifH']} for different values of $\mu$. These and other plots of steady-state Wigner functions in this paper were obtained using the QuTiP package in Python JNN12JNN13LGM+24. We have scaled the Wigner function so that its value is in $[-1,1]$. It is clear from inspecting (d) to (h) that a saddle-node bifurcation has occurred in the quantum system. However, the exact quantum bifurcation point cannot be inferred from \ref{['SadNodeNF']}, and a more nuanced study is required. Despite the simplicity of \ref{['SadNodeNF']}, the Wigner negativity seen in (g) and (h) means its quantum analog in \ref{['SadNodeBifL']} and \ref{['SadNodeBifH']} exhibits fundamentally different behavior, which cannot be understood even from stochastic generalizations of \ref{['SadNodeNF']}.
• Figure 3: Transcritical bifurcation. (a)--(c) Phase portraits of \ref{['TransCritNF']} as $\mu$ varies from a negative to positive value. The bifurcation occurs at $\mu=0$. In (a)--(c) horizontal dashed lines denote $y$ nullclines [all $(x,y)$ for which $y'=0$], while vertical dashed lines indicate $x$ nullclines [all $(x,y)$ for which $x'=0$]. We have marked the fixed point at the origin by an $\odot$ symbol. In (d)--(h) we plot the steady-state Wigner function defined by \ref{['TransCritBifL']} and \ref{['TransCritBifH']} corresponding to different values of $\mu$. We have scaled the Wigner function so that its value is in $[-1,1]$. While it is clear that a quantum transcritical bifurcation has occurred, the exact moment at which it happens is not derivable from \ref{['TransCritNF']}. Particularly interesting are the regions in (g) and (h) where the Wigner function becomes negative, highlighting again that even extremely simple dynamical systems can have nonclassical features when quantized.
• Figure 4: Pitchfork bifurcation. (a)--(c) Phase portraits of \ref{['PitchForkNF']} as $\mu$ varies from a negative to positive value. The bifurcation occurs at $\mu=0$. In (a)--(c) horizontal dashed lines denote $y$ nullclines [all $(x,y)$ for which $y'=0$], while vertical dashed lines indicate $x$ nullclines [all $(x,y)$ for which $x'=0$]. The origin of phase space is at the center of the $y$ nullcline. In (d)--(h) we plot the steady-state Wigner function corresponding to \ref{['PitchForkBifL']} and \ref{['PitchForkBifH']} for different values of $\mu$. We have scaled the Wigner function so that its value is in $[-1,1]$. A clear resemblance of the classical pitchfork bifurcation can be seen in the quantized system where a single peak in the steady-state Wigner function (before the bifurcation) splits into two lobes (after the bifurcation) that move apart as $\mu$ increases.
• Figure 5: Hopf bifurcation. (a)--(c) Phase portraits of \ref{['HopfNFx']} and \ref{['HopfNFy']} as $\mu$ varies from a negative to positive value. The bifurcation occurs at $\mu=0$. The origin of phase space is at the center, always occupied by a fixed point. Given the rotational symmetry of \ref{['HopfNFx']} and \ref{['HopfNFy']}, it is actually best to express them in polar coordinates. Doing so gives $r'=\mu\,r-r^3$ and $\phi'=1$ for the radial and phase variables, which are related to the Cartesian coordinates by $x=r \cos\phi$ and $y=r\sin\phi$. The radial equation is thus nothing more than a one-dimensional pitchfork bifurcation with $r$ restricted to be positive. It is then straightforward to see that only one stable fixed point exists at $r=0$ for $\mu\le0$, while there are two fixed points for $\mu>0$, one being unstable at the origin, and the other stable at $r=\sqrt{\mu}\,$. In (d)--(h) we plot the steady-state Wigner functions generated from \ref{['HopfBifL']} with different values of $\mu$. We have scaled the Wigner function so that its value is in $[-1,1]$. The limit cycle is created in passing from (e) to (f), when the Wigner-function peak starts to dip and a crater starts to form around the origin.