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Nonlinear Response of a Quantum System to Impulsive Perturbations: A Non-Perturbative Real-Time Approach

Alberto Guandalini, Caterina Cocchi, Stefano Pittalis, Alice Ruini, Carlo Andrea Rozzi

Published Apr 15, 2020• 1 citation
OpenAlex

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Using the positive P-representation of the density matrix, we develop an algorithm for calculating the quantum many-body nonlinear response functions of a system of bosons driven impulsively by external fields. The formalism maps the quantum time evolution of N boson degrees of freedom into a stochastic dynamics of 4N classical degrees of freedom. The first- and the third-order response functions are calculated by propagating the parameters of the P-representation using a set of coupled Langevin equations with multiplicative noise. These parameters serve as classical variables. Two classical ways for computing the response functions are presented. In the nonequilibrium method, an observable is calculated for weak impulsive pulses, and the response functions are obtained by taking its derivatives with respect to the pulse amplitudes. In the alternative, equilibrium simulation, the response functions are expressed in terms of time-correlation functions involving the P-representation parameters and stability matrices representing the perturbation of the trajectories. The stability matrices can be propagated simultaneously with the Langevin equations for the parameters. The formalism is generalized for a many-body boson system coupled to a harmonic bath.
Nonlinear response functions, formulated as multipoint correlation functions or Volterra kernels, encode the dynamical and spectroscopic properties of physical systems and underpin a wide range of nonlinear transport and optical phenomena. However, their evaluation rapidly becomes prohibitive at high orders because of combinatorial (often factorial) scaling or severe numerical errors. Here, we establish a systematic and efficient framework to compute nonlinear response functions directly from real-time dynamics, without explicitly constructing multipoint correlators or relying on numerically unstable finite-difference methods for order-resolved extraction. Our approach is based on the Gateaux derivative with respect to the external field in function space, which yields a closed hierarchy of tangent equations of motion (TEOM). Propagating the TEOM alongside the original dynamics isolates each perturbative order with high accuracy, providing a term-by-term decomposition of physical contributions. The computational cost scales exponentially with response order in the fully general setting and reduces to polynomial complexity when all perturbation directions are identical; both regimes avoid the factorial scaling of explicit multipoint-correlator evaluations. We demonstrate the power of TEOM by computing frequency-resolved fifth-order response functions for a solid-state electron model and by obtaining nonlinear response functions up to the 49th order with controlled accuracy in a classical Duffing oscillator. We further show that our time-evolution formulation allows optical conductivities to be evaluated directly while remaining numerically stable even near zero frequency. TEOM can be incorporated seamlessly into existing real-time evolution methods, yielding a general framework for computing nonlinear response functions in quantum and classical dynamical systems.
We present a formal analysis of nonlinear response functions in terms of correlation functions in real- and imaginary-time domains. In particular, we show that causal nonlinear response functions, expressed in terms of nested commutators in real time, can be obtained from the analytic continuation of time-ordered response functions, which are more easily amenable to diagrammatic calculation. This generalizes the well-known result of linear response theory. We then use gauge invariance arguments to derive exact relations between second-order response functions in density and current channels. These identities, which are non-perturbative in the strength of inter-particle interactions, allow us to establish exact connections between nonlinear optics calculations done in different electromagnetic gauges.