We study the fluctuations of the autocorrelation and autoresponse functions and, in particular, their variances and co-variance. In a first general part of the Article, we show the equivalence of the variance of the response function with the second-order susceptibility of a composite operator, and we derive an equilibrium fluctuation-dissipation theorem beyond-linear order relating it to the other variances. In a second part of the paper we apply the formalism to the study to non-disordered ferromagnets, in equilibrium or in the coarsening kinetics following a critical or sub-critical quench. We show numerically that the variances and the non-linear susceptibility obey scaling with respect to the coherence length $ξ$ in equilibrium, and with respect to the growing length $L(t)$ after a quench, similarly to what is known for the autocorrelation and the autoresponse functions.We study the fluctuations of the autocorrelation and autoresponse functions and, in particular, their variances and co-variance. In a first general part of the Article, we show the equivalence of the variance of the response function with the second-order susceptibility of a composite operator, and we derive an equilibrium fluctuation-dissipation theorem beyond-linear order relating it to the other variances. In a second part of the paper we apply the formalism to the study to non-disordered ferromagnets, in equilibrium or in the coarsening kinetics following a critical or sub-critical quench. We show numerically that the variances and the non-linear susceptibility obey scaling with respect to the coherence length $ξ$ in equilibrium, and with respect to the growing length $L(t)$ after a quench, similarly to what is known for the autocorrelation and the autoresponse functions.

arXiv:1003.1065Journal of Statistical Mechanics Theory and ExperimentApr 1, 20107 citations

Commuting Heisenberg operators as the quantum response problem: Time-normal averages in the truncated Wigner representation

B. Berg, Lev I. Plimak, Anatoli Polkovnikov +3 more

The applicability of the so-called truncated Wigner approximation $(\ensuremath{-}W)$ is extended to multitime averages of Heisenberg field operators. This task splits naturally in two. First, what class of multitime averages the $\ensuremath{-}W$ approximates and, second, how to proceed if the average in question does not belong to this class. To answer the first question, we develop a (in principle, exact) path-integral approach in phase space based on the symmetric (Weyl) ordering of creation and annihilation operators. These techniques calculate a new class of averages which we call time-symmetric. The $\ensuremath{-}W$ equations emerge as an approximation within these path-integral techniques. We then show that the answer to the second question is associated with response properties of the system. In fact, for two-time averages, Kubo's renowned formula relating the linear-response function to two-time commutators suffices. The $\ensuremath{-}W$ is directly generalized to the response properties of the system allowing one to calculate approximate time normally ordered two-time correlation functions with surprising ease. The techniques we develop are demonstrated for the Bose-Hubbard model.The applicability of the so-called truncated Wigner approximation $(\ensuremath{-}W)$ is extended to multitime averages of Heisenberg field operators. This task splits naturally in two. First, what class of multitime averages the $\ensuremath{-}W$ approximates and, second, how to proceed if the average in question does not belong to this class. To answer the first question, we develop a (in principle, exact) path-integral approach in phase space based on the symmetric (Weyl) ordering of creation and annihilation operators. These techniques calculate a new class of averages which we call time-symmetric. The $\ensuremath{-}W$ equations emerge as an approximation within these path-integral techniques. We then show that the answer to the second question is associated with response properties of the system. In fact, for two-time averages, Kubo's renowned formula relating the linear-response function to two-time commutators suffices. The $\ensuremath{-}W$ is directly generalized to the response properties of the system allowing one to calculate approximate time normally ordered two-time correlation functions with surprising ease. The techniques we develop are demonstrated for the Bose-Hubbard model.

arXiv:0905.3127Physical Review ASep 29, 200943 citations

An efficient quantum algorithm for generation of ab initio n-th order susceptibilities for non-linear spectroscopies

Tyler Kharazi, Torin F. Stetina, Liwen Ko +2 more

We develop and analyze a fault-tolerant quantum algorithm for computing $n$ -th order response properties necessary for analysis of non-linear spectroscopies of molecular and condensed phase systems. We use a semi-classical description in which the electronic degrees of freedom are treated quantum mechanically and the light is treated as a classical field. The algorithm we present can be viewed as an implementation of standard perturbation theory techniques, focused on {\it ab initio} calculation of $n$ -th order response functions. We provide cost estimates in terms of the number of queries to the block encoding of the unperturbed Hamiltonian, as well as the block encodings of the perturbing dipole operators. Using the technique of eigenstate filtering, we provide an algorithm to extract excitation energies to resolution $γ$ , and the corresponding linear response amplitude to accuracy $ε$ using ${O}\left(N^{6}η^2{γ^{-1}ε^{-1}}\log(1/ε)\right)$ queries to the block encoding of the unperturbed Hamiltonian $H_0$ , in double factorized representation. Thus, our approach saturates the Heisenberg $O(γ^{-1})$ limit for energy estimation and allows for the approximation of relevant transition dipole moments. These quantities, combined with sum-over-states formulation of polarizabilities, can be used to compute the $n$ -th order susceptibilities and response functions for non-linear spectroscopies under limited assumptions using $\widetilde{O}\left({N^{5n+1}η^{n+1}}/{γ^nε}\right)$ queries to the block encoding of $H_0$ .We develop and analyze a fault-tolerant quantum algorithm for computing $n$-th order response properties necessary for analysis of non-linear spectroscopies of molecular and condensed phase systems. We use a semi-classical description in which the electronic degrees of freedom are treated quantum mechanically and the light is treated as a classical field. The algorithm we present can be viewed as an implementation of standard perturbation theory techniques, focused on {\it ab initio} calculation of $n$-th order response functions. We provide cost estimates in terms of the number of queries to the block encoding of the unperturbed Hamiltonian, as well as the block encodings of the perturbing dipole operators. Using the technique of eigenstate filtering, we provide an algorithm to extract excitation energies to resolution $γ$, and the corresponding linear response amplitude to accuracy $ε$ using ${O}\left(N^{6}η^2{γ^{-1}ε^{-1}}\log(1/ε)\right)$ queries to the block encoding of the unperturbed Hamiltonian $H_0$, in double factorized representation. Thus, our approach saturates the Heisenberg $O(γ^{-1})$ limit for energy estimation and allows for the approximation of relevant transition dipole moments. These quantities, combined with sum-over-states formulation of polarizabilities, can be used to compute the $n$-th order susceptibilities and response functions for non-linear spectroscopies under limited assumptions using $\widetilde{O}\left({N^{5n+1}η^{n+1}}/{γ^nε}\right)$ queries to the block encoding of $H_0$.

arXiv:2404.01454Apr 1, 20241 citation

A non-linear response theory

A.‐S. F. Obada, Saad S. M. Hassan, M. H. Mahran

Physica A Statistical Mechanics and its ApplicationsApr 1, 19872 citations

ξ

in equilibrium, and with respect to the growing length

L(t)

after a quench, similarly to what is known for the autocorrelation and the autoresponse functions.

We develop and analyze a fault-tolerant quantum algorithm for computing

n

-th order response properties necessary for analysis of non-linear spectroscopies of molecular and condensed phase systems. We use a semi-classical description in which the electronic degrees of freedom are treated quantum mechanically and the light is treated as a classical field. The algorithm we present can be viewed as an implementation of standard perturbation theory techniques, focused on {\it ab initio} calculation of

n

-th order response functions. We provide cost estimates in terms of the number of queries to the block encoding of the unperturbed Hamiltonian, as well as the block encodings of the perturbing dipole operators. Using the technique of eigenstate filtering, we provide an algorithm to extract excitation energies to resolution

γ

, and the corresponding linear response amplitude to accuracy

ε

using

{O}\left(N^{6}η^2{γ^{-1}ε^{-1}}\log(1/ε)\right)

queries to the block encoding of the unperturbed Hamiltonian

H_0

, in double factorized representation. Thus, our approach saturates the Heisenberg

O(γ^{-1})

limit for energy estimation and allows for the approximation of relevant transition dipole moments. These quantities, combined with sum-over-states formulation of polarizabilities, can be used to compute the

n

-th order susceptibilities and response functions for non-linear spectroscopies under limited assumptions using

\widetilde{O}\left({N^{5n+1}η^{n+1}}/{γ^nε}\right)

queries to the block encoding of

H_0