Quantifying fermionic decoherence in many-body systems

TL;DR

This paper introduces practical measures called distilled purities, based on reduced density matrices, to quantify electronic decoherence and coherences in many-electron systems, facilitating analysis of quantum dynamics.

Contribution

It presents a new, accessible method for quantifying electronic coherences using reduced density matrices, extending previous purity measures to many-body systems.

Findings

Derived exact expressions for one- and two-body distilled purities.

Applied the method to analyze the quantum dynamics of oligo-acetylene.

Discussed advantages and limitations of various coherence measures.

Abstract

Practical measures of electronic decoherence, called distilled purities, that are applicable to many-body systems are introduced. While usual measures of electronic decoherence such as the purity employ the full $N$-particle density matrix which is generally unavailable, the distilled purities are based on the $r$-body reduced density matrices ($r$-RDMs) which are more accessible quantities. The $r$-body distilled purities are derivative quantities of the previously introduced $r$-body reduced purities [I. Franco and H. Appel, J. Chem. Phys. 139, 094109 (2013)] that measure the non-idempotency of the $r$-RDMs. Specifically, the distilled purities exploit the structure of the reduced purities to extract coherences between Slater determinants with integer occupations defined by a given single-particle basis that compose an electronic state. In this way, the distilled purities offer a practical platform to quantify coherences in a given basis that can be used to analyze the quantum dynamics of many-electron systems. Exact expressions for the one-body and two-body distilled purities are presented and the utility of the approach is exemplified via an analysis of the dynamics of oligo-acetylene as described by the Su-Schrieffer-Heeger Hamiltonian. Last, the advantages and limitations of the purity, reduced purity and distilled purity as measures of electronic coherence are discussed.

Tables

Table 1. Table 1: Basic features of the different ways to quantify electronic decoherence in many-body systems.

Type	Definition	Remarks
Purity	$P=\text{Tr}\left[{\hat{\rho }}_e^2\right]$	• Measures the non-idempotency of ${\hat{\rho }}_e$ • Well-defined and easy-to-interpret measure of coherence • Basis independent • Numerically removed for many-body systems
Reduced Purity	$P_r=\text{Tr}{[}^{\left(r\right)}{\hat{\Gamma }}^2]$	• Measures the non-idempotency of ${}_{}{}^{\left(r\right)}\hat{\Gamma }$ • Difficult to interpret as both decoherence and correlation among electrons lead to non-idempotency of ${}_{}{}^{\left(r\right)}\hat{\Gamma }$ • Basis independent • Easy to compute
Distilled Purity	${\tilde{P}}_1=P_1-\sum _{ϵ}{\left({}_{}{}^{\left(1\right)}\Gamma _{ϵ}^{ϵ}\right)}^2$ $\begin{array}{c}{\tilde{P}}_2=P_2-2\sum _{{ϵ}_1,{ϵ}_2}{\left({}_{}{}^{\left(2\right)}\Gamma _{{ϵ}_1{ϵ}_2}^{{ϵ}_1{ϵ}_2}\right)}^2\end{array}$	• Summarizes $S$-coherences (off-diagonal elements among Slater determinants defined by a given single particle basis) • Useful and easy to interpret, but not necessarily informative of state purity • Basis dependent • Easy to compute

Equations

$$P(t)=\text{Tr}[{\hat{\rho }}_e^2(t)]=\sum _i{\lambda }_i^2,$$

$$P_r(t)=\text{Tr}{[}^{(r)}{\hat{\Gamma }}^2]=\sum _i{}^{(r)}{\lambda }_i^2,$$

$$\begin{array}{rl}{\hat{\rho }}_e(t)& ={\text{Tr}}_N[|\Omega (t)⟩⟨\Omega (t)|]=\sum _i{\lambda }_i|{\mathcal{E}}_i⟩⟨{\mathcal{E}}_i|\\ & =\sum _{j,k}⟨{\chi }_k(t)|{\chi }_j(t)⟩|E_j⟩⟨E_k|,\end{array}$$

$$\begin{array}{rl}P(t)& =\text{Tr}[{\hat{\rho }}_e^2(t)]=\sum _i{\lambda }_i^2\\ & =\sum _{j,k}|⟨{\chi }_k(t)|{\chi }_j(t)⟩{|}^2\le 1.\end{array}$$

$${}^{(r)}{\Gamma }_{{ϵ}_1\dots {ϵ}_r}^{{ϵ}_1^{\prime }\dots {ϵ}_r^{\prime }}(t)$$

$${\hat{\rho }}_e$$

$$s_{nm}=N-\sum _{ϵ}{f}_n(ϵ)f_m(ϵ)$$

$$|{\Phi }_n⟩=c_{{\alpha }_2}^{†}{c}_{{\beta }_2}{c}_{{\alpha }_1}^{†}{c}_{{\beta }_1}|{\Phi }_m⟩.$$

$$\begin{array}{rl}{}^{(1)}{\Gamma }_{{ϵ}_1}^{{ϵ}_2}& =\sum _m{a}_{mm}{\delta }_{{ϵ}_1{ϵ}_2}{f}_m({ϵ}_1)+{\sum _{\begin{array}{c}n,m\\ n\ne m\end{array}}}^{\prime }{a}_{nm}{f}_m({\beta }_1)(1-f_m({\alpha }_1))\\ & \times (1-f_m({\alpha }_2)){\delta }_{{\alpha }_1{\beta }_2}{\delta }_{{ϵ}_1{\beta }_1}{\delta }_{{ϵ}_2{\alpha }_2}.\end{array}$$

$$\begin{array}{rl}{P}_1& =\sum _{{ϵ}_1,{ϵ}_2}{}^{(1)}{\Gamma }_{{ϵ}_1}^{{ϵ}_2}{}^{(1)}{\Gamma }_{{ϵ}_2}^{{ϵ}_1}=\sum _{ϵ}{\left(\sum _m{a}_{mm}{f}_m(ϵ)\right)}^2\\ & +{\sum _{\begin{array}{c}n,m\\ n\ne m\end{array}}}^{\prime }{\sum _{\begin{array}{c}p,q\\ p\ne q\end{array}}}^{\prime }{a}_{nm}{a}_{pq}{A}_1{\delta }_{{\alpha }_1{\beta }_2}{\delta }_{{\gamma }_1{\delta }_2}{\delta }_{{\alpha }_2{\delta }_1}{\delta }_{{\beta }_1{\gamma }_2},\end{array}$$

$$\begin{split}{}^{(2)}\Gamma_{\epsilon_{1},\epsilon_{2}}^{\epsilon_{4},\epsilon_{3}}&=\frac{1}{2}\Big{[}\sum_{m}a_{mm}f_{m}(\epsilon_{1})f_{m}(\epsilon_{2})(\delta_{\epsilon_{1}\epsilon_{4}}\delta_{\epsilon_{2}\epsilon_{3}}-\delta_{\epsilon_{1}\epsilon_{3}}\delta_{\epsilon_{2}\epsilon_{4}})\\ &+\displaystyle{\sum_{\begin{subarray}{c}n,m\\ n\neq m\end{subarray}}}^{\prime}a_{nm}f_{m}(\beta_{1})(1-f_{m}(\alpha_{1}))(1-f_{m}(\alpha_{2}))\\ &\times f_{m}(\epsilon_{1})f_{m}(\epsilon_{2})\left[\delta_{\alpha_{1}\beta_{2}}(\delta_{\epsilon_{1}\beta_{1}}(\delta_{\epsilon_{2}\epsilon_{3}}\delta_{\alpha_{2}\epsilon_{4}}-\delta_{\epsilon_{2}\epsilon_{4}}\delta_{\alpha_{2}\epsilon_{3}})\right.\\ &\left.\hskip 85.35826pt-\delta_{\epsilon_{2}\beta_{1}}(\delta_{\epsilon_{1}\epsilon_{3}}\delta_{\alpha_{2}\epsilon_{4}}-\delta_{\epsilon_{1}\epsilon_{4}}\delta_{\alpha_{2}\epsilon_{3}}))\right.\\ &\left.+(\delta_{\epsilon_{1}\beta_{2}}\delta_{\epsilon_{2}\beta_{1}}-\delta_{\epsilon_{1}\beta_{1}}\delta_{\epsilon_{2}\beta_{2}})(\delta_{\alpha_{1}\epsilon_{3}}\delta_{\alpha_{2}\epsilon_{4}}-\delta_{\alpha_{2}\epsilon_{3}}\delta_{\alpha_{1}\epsilon_{4}})\right]\Big{]}.\end{split}$$

$$\begin{array}{rl}{P}_2& =\sum _{{ϵ}_1,{ϵ}_2,{ϵ}_3,{ϵ}_4}{}^{(2)}{\Gamma }_{{ϵ}_1,{ϵ}_2}^{{ϵ}_4,{ϵ}_3}{}^{(2)}{\Gamma }_{{ϵ}_4,{ϵ}_3}^{{ϵ}_1,{ϵ}_2}\\ & =\frac{1}{2}\sum _{n,p}{a}_{nn}{a}_{pp}\left[{\left(\sum _{ϵ}{f}_n(ϵ)f_p(ϵ)\right)}^2-\sum _{ϵ}{f}_n(ϵ)f_p(ϵ)\right]\\ & +{\sum _{\begin{array}{c}n,m\\ n\ne m\end{array}}}^{\prime }{\sum _{\begin{array}{c}p,q\\ p\ne q\end{array}}}^{\prime }{a}_{nm}{a}_{pq}{A}_2[{\delta }_{{\alpha }_1{\beta }_2}{\delta }_{{\gamma }_1{\delta }_2}{\delta }_{{\alpha }_2{\delta }_1}{\delta }_{{\beta }_1{\gamma }_2}(N-s_{mq})\\ & +A_3({\delta }_{{\beta }_1{\gamma }_1}{\delta }_{{\beta }_2{\gamma }_2}-{\delta }_{{\beta }_1{\gamma }_2}{\delta }_{{\beta }_2{\gamma }_1})({\delta }_{{\alpha }_1{\delta }_1}{\delta }_{{\alpha }_2{\delta }_2}-{\delta }_{{\alpha }_1{\delta }_2}{\delta }_{{\alpha }_2{\delta }_1})],\end{array}$$

$${\tilde{P}}_1$$

$${\tilde{P}}_2$$

$${\tilde{P}}_1$$

$$\begin{array}{rl}{\tilde{P}}_2& ={\sum _{n\ne m}}^{\prime }{\sum _{p\ne q}}^{\prime }{a}_{nm}{a}_{pq}{A}_2[{\delta }_{{\alpha }_1{\beta }_2}{\delta }_{{\gamma }_1{\delta }_2}{\delta }_{{\alpha }_2{\delta }_1}{\delta }_{{\beta }_1{\gamma }_2}(N-s_{mq})\\ & +A_3({\delta }_{{\beta }_1{\gamma }_1}{\delta }_{{\beta }_2{\gamma }_2}-{\delta }_{{\beta }_1{\gamma }_2}{\delta }_{{\beta }_2{\gamma }_1})({\delta }_{{\alpha }_1{\delta }_1}{\delta }_{{\alpha }_2{\delta }_2}-{\delta }_{{\alpha }_1{\delta }_2}{\delta }_{{\alpha }_2{\delta }_1})],\end{array}$$

$$\begin{array}{rl}{\tilde{P}}_1^{\text{max}}& \le P_1^{\text{max}}-\sum _{ϵ}{\left({}^{(1)}{\Gamma }_{ϵ}^{ϵ}\right)}^2\\ & \le N-\sum _{ϵ}{\left(\sum _m{a}_{mm}{f}_m(ϵ)\right)}^2\\ & \le \sum _{m\ne n}{a}_{mm}{a}_{nn}{s}_{nm}\end{array}$$

$$\begin{array}{rl}{\tilde{P}}_1^{\text{max}}& \le N\sum _{m\ne n}{a}_{mm}{a}_{nn}\\ & \le N(1-\sum _n{a}_{nn}^2).\end{array}$$

$${\left(\sum _n{a}_{nn}\right)}^2=1\le \left(\sum _n{a}_{nn}^2\right)\left(\sum _{m}1\right)=K\sum _n{a}_{nn}^2$$

$$\begin{array}{rl}{\tilde{P}}_1^{\text{max}}& \le N\left(1-\frac{1}{K}\right).\end{array}$$

$$\begin{array}{rl}{\tilde{P}}_2^{\text{max}}& \le \sum _{n>m}{a}_{mm}{a}_{nn}{s}_{nm}(2N-s_{nm}-1)\\ & \le \frac{N(N-1)}{2}\left(1-\frac{1}{K}\right),\end{array}$$

$$|\Omega (0)⟩=\frac{1}{\sqrt{2}}(|{\Phi }_0⟩+|{\Phi }_1⟩)\otimes |{\chi }_0⟩,$$

$$\begin{array}{rl}{}^{(1)}{\Gamma }_{{ϵ}_1}^{{ϵ}_2}& =\text{Tr}[c_{{ϵ}_1}^{†}{c}_{{ϵ}_2}{\hat{\rho }}_e]\\ & =\sum _n{a}_{nn}⟨{\Phi }_n|c_{{ϵ}_1}^{†}{c}_{{ϵ}_2}|{\Phi }_n⟩+\sum _{\begin{array}{c}n,m\\ n\ne m\end{array}}{a}_{nm}⟨{\Phi }_m|c_{{ϵ}_1}^{†}{c}_{{ϵ}_2}|{\Phi }_n⟩\\ & =\sum _n{a}_{nn}⟨{\Phi }_n|c_{{ϵ}_1}^{†}{c}_{{ϵ}_2}|{\Phi }_n⟩\\ & +{\sum _{\begin{array}{c}n,m\\ n\ne m\end{array}}}^{\prime }{a}_{nm}⟨{\Phi }_m|c_{{ϵ}_1}^{†}{c}_{{ϵ}_2}{c}_{{\alpha }_2}^{†}{c}_{{\beta }_2}{c}_{{\alpha }_1}^{†}{c}_{{\beta }_1}|{\Phi }_m⟩,\end{array}$$

$$\begin{array}{rl}& ⟨{\Phi }_m|c_{{ϵ}_1}^{†}{c}_{{ϵ}_2}{c}_{{\alpha }_2}^{†}{c}_{{\beta }_2}{c}_{{\alpha }_1}^{†}{c}_{{\beta }_1}|{\Phi }_m⟩\\ & =f_m({\beta }_1)(1-f_m({\alpha }_1))(1-f_m({\alpha }_2))\\ & \times ⟨{\Phi }_m|({\delta }_{{ϵ}_2{\alpha }_2}{\delta }_{{\alpha }_1{\beta }_2}{c}_{{ϵ}_1}^{†}{c}_{{\beta }_1}+{\delta }_{{\alpha }_1{\beta }_2}{c}_{{ϵ}_1}^{†}{c}_{{\alpha }_2}^{†}{c}_{{\beta }_1}{c}_{{ϵ}_2}\\ & +{\delta }_{{\alpha }_2{ϵ}_2}{c}_{{ϵ}_1}^{†}{c}_{{\alpha }_1}^{†}{c}_{{\beta }_1}{c}_{{\beta }_2}+{\delta }_{{\alpha }_1{ϵ}_2}{c}_{{ϵ}_1}^{†}{c}_{{\alpha }_2}^{†}{c}_{{\beta }_2}{c}_{{\beta }_1}\\ & +c_{{ϵ}_1}^{†}{c}_{{\alpha }_2}^{†}{c}_{{\alpha }_1}^{†}{c}_{{\beta }_1}{c}_{{\beta }_2}{c}_{{ϵ}_2})|{\Phi }_m⟩\\ & ={\delta }_{{\alpha }_1{\beta }_2}{\delta }_{{ϵ}_1{\beta }_1}{\delta }_{{ϵ}_2{\alpha }_2}{f}_m({\beta }_1)(1-f_m({\alpha }_1))(1-f_m({\alpha }_2)).\end{array}$$

$${}^{(1)}{\Gamma }_{{ϵ}_1}^{{ϵ}_2}$$

$${}^{(1)}{\Gamma }_{{ϵ}_2}^{{ϵ}_1}$$

$$\begin{array}{rl}{P}_1& =\text{Tr}{[}^{(1)}{\hat{\Gamma }}^2]=\sum _{{ϵ}_1,{ϵ}_2}{}^{(1)}{\Gamma }_{{ϵ}_1}^{{ϵ}_2}{}^{(1)}{\Gamma }_{{ϵ}_2}^{{ϵ}_1}\\ & =\sum _{{ϵ}_1,{ϵ}_2}\left(\sum _m{a}_{mm}X+{\sum _{\begin{array}{c}n,m\\ n\ne m\end{array}}}^{\prime }{a}_{nm}Y\right)\left(\sum _q{a}_{qq}Z+{\sum _{\begin{array}{c}p,q\\ p\ne q\end{array}}}^{\prime }{a}_{pq}W\right)\\ & =\sum _{{ϵ}_1,{ϵ}_2}[\sum _{m,q}{a}_{mm}{a}_{qq}XZ+{\sum _{\begin{array}{c}n,m\\ n\ne m\end{array}}}^{\prime }\sum _q{a}_{nm}{a}_{qq}YZ\\ & +{\sum _m\sum _{\begin{array}{c}p,q\\ p\ne q\end{array}}}^{\prime }{a}_{mm}{a}_{pq}XW+{{\sum _{\begin{array}{c}n,m\\ n\ne m\end{array}}}^{\prime }\sum _{\begin{array}{c}p,q\\ p\ne q\end{array}}}^{\prime }{a}_{nm}{a}_{pq}YW],\end{array}$$

$$\begin{array}{rl}XZ& ={\delta }_{{ϵ}_1{ϵ}_2}{f}_m({ϵ}_1)f_q({ϵ}_2),\\ YZ& ={\delta }_{{ϵ}_1{ϵ}_2}{\delta }_{{\alpha }_1{\beta }_2}{\delta }_{{ϵ}_1{\beta }_1}{\delta }_{{ϵ}_2{\alpha }_2}{f}_q({ϵ}_2)f_m({\beta }_1)\\ & \times (1-f_m({\alpha }_1))(1-f_m({\alpha }_2))=0,\\ XW& ={\delta }_{{ϵ}_1{ϵ}_2}{\delta }_{{\gamma }_1{\delta }_2}{\delta }_{{ϵ}_2{\delta }_1}{\delta }_{{ϵ}_1{\gamma }_2}{f}_m({ϵ}_1)f_q({\delta }_1)\\ & \times (1-f_q({\gamma }_1))(1-f_q({\gamma }_2))=0,\\ YW& =A_1{\delta }_{{\alpha }_1{\beta }_2}{\delta }_{{ϵ}_1{\beta }_1}{\delta }_{{ϵ}_2{\alpha }_2}{\delta }_{{\gamma }_1{\delta }_2}{\delta }_{{ϵ}_2{\delta }_1}{\delta }_{{ϵ}_1{\gamma }_2}\end{array}$$

$$\begin{array}{rl}{P}_1& =\sum _{ϵ}{\left(\sum _m{a}_{mm}{f}_m(ϵ)\right)}^2\\ & +{\sum _{\begin{array}{c}n,m\\ n\ne m\end{array}}}^{\prime }{\sum _{\begin{array}{c}p,q\\ p\ne q\end{array}}}^{\prime }{a}_{nm}{a}_{pq}{A}_1{\delta }_{{\alpha }_1{\beta }_2}{\delta }_{{\gamma }_1{\delta }_2}{\delta }_{{\alpha }_2{\delta }_1}{\delta }_{{\beta }_1{\gamma }_2}.\end{array}$$

$$\begin{array}{rl}{P}_1& =\sum _{ϵ}{\left(\sum _m{a}_{mm}{f}_m(ϵ)\right)}^2\\ & +\sum _{\begin{array}{c}n,m\\ n\ne m\end{array}}\sum _{\begin{array}{c}p,q\\ p\ne q\end{array}}{a}_{nm}{a}_{pq}{A}_1{\delta }_{s_{nm},1}{\delta }_{s_{pq},1}{\delta }_{{\alpha }_2{\delta }_1}{\delta }_{{\beta }_1{\gamma }_2}.\end{array}$$

$$\sum _{ϵ}{\left({}^{(1)}{\Gamma }_{ϵ}^{ϵ}\right)}^2=\sum _{ϵ}{\left(\sum _m{a}_{mm}{f}_m(ϵ)\right)}^2.$$

Full text

A. Kar and I. Franco, J. Chem. Phys. 146, 214107 (2017)

I Introduction

Decoherence refers to the change of a state of a system from a pure state $\hat{\rho}_{e}=|\psi\rangle\langle\psi|$ to a mixed state $\hat{\rho}_{e}=\sum_{i}w_{i}|\psi_{i}\rangle\langle\psi_{i}|$ due to interactions with an environment Breuer and Petruccione (2002); Joos et al. (2003); Schlosshauer (2008); Scholes et al. (2017). In isolated electron-nuclear systems, electronic decoherence arises due to the entanglement of the electrons with the nuclear degrees of freedom caused by electron-nuclear couplings. As such, electronic decoherence is a basic feature of correlated electron-nuclear states Izmaylov and Franco (2017) and accompanies most dynamical processes in molecules. In addition to its interest at a fundamental level, determining mechanisms for electronic decoherence Kar et al. (2016) is central to interpreting coherence phenomena in matter Chenu and Scholes (2015); Neill et al. (2013); Pachón and Brumer (2011); Collini et al. (2010); Prokhorenko et al. (2006); Scholes et al. (2017), to the development of methods to follow correlated electron–nuclear dynamics Kapral (2015); Jaeger et al. (2012), and is the starting point to develop potential protocols to protect coherences for quantum control applications Nielsen and Chuang (2010); Shapiro and Brumer (2012).

In this paper, we focus on the problem of how to quantify electronic coherences Hwang and Rossky (2004); Wong and Rossky (2002); Smyth et al. (2012) in many-body systems by using purity and purity-related measures. The purity of an electronic state is defined as

[TABLE]

where $\hat{\rho}_{e}(t)$ is the $N$-particle electronic density matrix (obtained by tracing out the nuclear coordinates from the electron-nuclear density matrix $\hat{\rho}$, i.e. $\hat{\rho}_{e}(t)=\text{Tr}_{N}[\hat{\rho}(t)]$) and $\lambda_{i}$ its eigenvalues (the Schmidt coefficients squared). The purity is a well-defined measure of electronic decoherence. It is basis-independent, and easy to interpret ($P=1$ for pure systems where the density matrix is idempotent, $\hat{\rho}_{e}^{2}=\hat{\rho}_{e}$, and $P<1$ for mixed states Rajibul et al. (2015); Gneiting et al. (2016)). In spite of these advantages, it is often not possible to apply the purity to quantify coherence in many-electron systems because it requires knowing the $N$-particle electronic density matrix $\hat{\rho}_{e}(t)$ which is an experimentally and computationally challenging quantity to obtain except for few-level problems Tegmark (1996).

Thus, it is desirable to develop measures of coherence that are based on more accessible quantities such as the $r$-body electronic reduced density matrices ($r$-RDMs), ${}^{(r)}\hat{\Gamma}$. The $r$-RDMs are defined by tracing out $(N-r)$ electronic degrees of freedom out of $\hat{\rho}_{e}$. In particular, the one and two-body RDMs are useful targets as they can be propagated directly in state-of-the-art simulations of many-body systems Lackner et al. (2015), and they determine most observable quantities of physical interest. In analogy with the purity [Eq. (1)], one can define a reduced purity Franco and Appel (2013)

[TABLE]

that measures the non-idempotency of the $r$-RDM, where ${}^{(r)}\lambda_{i}$ are the eigenvalues of the $r$-RDM. As is the case of purity, the reduced purities are representation independent and decay with electronic coherence loss in the system Franco and Appel (2013). Nevertheless, contrary to the purity, as a measure of coherence these quantities can be difficult to interpret. This is because the non-idempotency of the $r$-RDMs induced by decoherence can also arise due to increased electronic correlation Ziesche (1995). In fact, due to the reduced information contained in the $r$-RDMs, at the $r$-RDM level it is very challenging to distinguish contributions of electronic correlation and decoherence to the non-idempotency of ${}^{(r)}\hat{\Gamma}$. This makes it difficult to distinguish, for instance, a pure state of an electronically correlated molecule from a mixed state of an electronically uncorrelated molecule.

As an alternative, here we introduce distilled purities [defined by Eqs. (13) and (14)] as a practical measure of fermionic decoherence in many-body systems. The distilled purities are based on the $r$-RDMs and are derivative quantities of the reduced purities. They summarize the coherence content between single Slater determinants that form the electronic state. These quantities are easy to calculate and useful in interpreting the quantum dynamics of many body systems in a situation where well-defined measures of decoherence are not accessible. Nevertheless, as discussed below, they have the limitation of being manifestly basis-dependent and of not being simply related to well-defined measures of coherence such as the purity. As such, the distilled purities should be seen as useful quantities to interpret dynamics, and not as fundamental quantities. This is akin to atomic population analysis in molecular systems, which are of significant interpretative value but for which there is no apparent correct definition.

This paper is organized as follows: In Sec. II, we define different kinds of coherences that can be associated with a many-body electronic density matrix. As the mathematical formalism of the reduced purities is necessary to introduce the distilled purities, in Sec. III we briefly review the reduced purities and present exact expressions for the one- and two-body reduced purities. Then, in Sec. IV we define the distilled purities, derive exact expressions for the one-body and two-body distilled purities and isolate their limiting values. In Sec. V, the behavior of the reduced and distilled purities is exemplified for a model molecular system with electron-vibrational interactions. Last, in Sec. VI we summarize our findings and discuss the merits and limitations of the purity, reduced purity and distilled purity as measures of electronic decoherence.

II Definitions of coherence

We begin by defining different types of coherences that can be studied in the context of many body systems. For definitiveness, consider a pure state of an electron-nuclear system in Schmidt form $|\Omega\rangle=\sum_{i}\sqrt{\lambda_{i}}|\mathcal{E}_{i}\rangle|R_{i}\rangle$ where $\sqrt{\lambda_{i}}$ are the Schmidt coefficients, and $|\mathcal{E}_{i}\rangle$ and $|R_{i}\rangle$ are orthonormal states of the electronic and nuclear subsystems. In the eigenbasis of the electronic Hamiltonian for a fixed nuclear configuration {$|E_{j}\rangle$}, $|\Omega\rangle=\sum_{j}|E_{j}\rangle|\chi_{j}\rangle$ where $|\mathcal{E}_{i}\rangle=\sum_{j}c_{ij}|E_{j}\rangle$ and $|\chi_{j}\rangle=\sum_{i}\sqrt{\lambda_{i}}c_{ij}|R_{i}\rangle$. The $|\chi_{j}\rangle$ can be viewed as the nuclear wave packets associated with the $j$-th electronic state. In this context, the $N$-body electronic density matrix is given by

[TABLE]

where the partial trace is over the nuclear degrees of freedom Franco and Appel (2013). In the {$|E_{j}\rangle$} eigenbasis, the off-diagonal elements of the electron density matrix are determined by the overlap $|\langle\chi_{k}(t)|\chi_{j}(t)\rangle|$ between the nuclear wavepackets associated with different electronic states. In turn, in the Schmidt basis $\hat{\rho}_{e}(t)$ is a diagonal matrix with the squared Schmidt coefficients in the diagonal ($\sum_{i}\lambda_{i}=1$).

Coherence.

We define the degree of coherence of $\hat{\rho}_{e}(t)$ through the purity

[TABLE]

For pure systems, only one Schmidt coefficient is non-zero and the purity is one. For mixed states with reduced coherence properties, there is more than one non-zero Schmidt coefficients, and the purity is less than one. From the perspective of electron-nuclear dynamics, such decoherence occurs due to a decay of the overlaps $|\langle\chi_{k}(t)|\chi_{j}(t)\rangle|$ between the nuclear wavepackets associated with different electronic states.

$B$-Coherence.

The purity definition of coherence is well defined and basis-independent. Nevertheless, often by coherences it is simply meant the non-zero off-diagonal elements of the subsystem’s density matrix expressed in a given basis Kassal et al. (2013); Mukamel (1999), typically the position or the energy basis. For instance, the coherences of $\hat{\rho}_{e}$ in the {$|E_{j}\rangle$} eigenbasis [see Eq. (3)] are given by the nuclear wavepacket overlaps $|\langle\chi_{k}(t)|\chi_{j}(t)\rangle|$. We will refer to this type of basis-dependent coherences as $B$-coherences. While a decay in the $B$-coherences can lead to decay in the state purity, the absence of $B$-coherences does not necessarily signal an incoherent state. In this sense, the $B$-coherences are useful in interpreting the dynamics of the system, but they are not necessarily well-defined measures of coherence.

S-Coherence.

As a particular class of $B$-coherences, in many-fermion systems one is often interested in the $B$-coherences of the electronic density matrix expressed in a basis of single Slater determinants. We will refer to this type of coherence as $S$-coherences. These Slater determinants are defined by a given single-particle orbital basis and refer to the anti-symmetrized products of such orbitals with integer occupation numbers. A complete basis of single Slater determinants is constructed by considering all possible distribution of the electrons among the single-particle states.

These three definitions of coherences will be of importance when discussing the utility of the reduced and the distilled purities.

III Reduced purities

The distilled purities are derivative quantities of the reduced purities. To define them and understand their significance, it is necessary to review basic aspects of the reduced purities Franco and Appel (2013). Below we present exact expressions for the one and two-body reduced purities, which are the most important and readily applicable reduced purities. The final expression for these reduced purities will lead to expressions for the associated one- and two-body distilled purities. The expressions below generalize the developments in Franco and Appel (2013). As discussed, equations 19 and 21 in Ref. Franco and Appel, 2013 apply to many-body states with no distinct pairs of Slater determinants that differ by the same particle transition. The equations below overcome this limitation and apply to general electronic states.

The reduced purities are a hierarchy of measures of decoherence that are based on the well-known hierarchy of $r$-RDM, $\{^{(r)}\hat{\Gamma}\}$ ($r=1,2,\cdots$). The $r$-RDMs that define the reduced purities are obtained by tracing out $(N-r)$ electronic degrees of freedom out of $\hat{\rho}_{e}$. Specifically,

[TABLE]

where $c_{\epsilon_{i}}^{\dagger}$ (or $c_{\epsilon_{i}}$) creates (or annihilates) a fermion in the $i$-th spin-orbital, i.e. $|\epsilon_{i}\rangle=c_{\epsilon_{i}}^{\dagger}|0\rangle$ where $|0\rangle$ is the vacuum state Davidson (2012); Fetter and Walecka (1971). The creation and annihilation operators satisfy the usual fermionic anticommutation relations, $\{c_{\epsilon_{i}},c_{\epsilon_{j}}^{\dagger}\}=\delta_{\epsilon_{i}\epsilon_{j}}$ and $\{c_{\epsilon_{i}},c_{\epsilon_{j}}\}=\{c_{\epsilon_{i}}^{\dagger},c_{\epsilon_{j}}^{\dagger}\}=0$. The $r$-body reduced purity measures the non-idempotency of ${}^{(r)}\hat{\Gamma}$, as defined in Eq. (2). In doing so, it captures coherences in the system that manifest at the $r$-particle level.

At this point, it is convenient to express $\hat{\rho}_{e}$ in terms of a basis of Slater determinants

[TABLE]

where $|\Phi_{n}\rangle$ corresponds to a single Slater determinant with integer occupation numbers in a given single particle basis $\{|\epsilon_{i}\rangle\}$. $a_{nn}$ in Eq. (6) denote the population of Slater determinant $n$, while $a_{nm}$ refer to the $S$-coherences between the $n,m$ pair.

As in Franco and Appel (2013), we define the order $s_{nm}$ of the $S$-coherence $a_{nm}$ as the number of single particle transitions required to do a $|\Phi_{n}\rangle\to|\Phi_{m}\rangle$ transition. Specifically,

[TABLE]

where $f_{n}(\epsilon)=\langle\Phi_{n}|c^{\dagger}_{\epsilon}c_{\epsilon}|\Phi_{n}\rangle$ is the distribution function of the state $|\Phi_{n}\rangle$, and adopts values of 0 or 1 based on whether the spin-orbital $|\epsilon\rangle$ is unoccupied or occupied, such that $f_{n}^{2}(\epsilon)=f_{n}(\epsilon)$. The quantity $s_{nm}\in[0,N]$ and takes the value 1 for pairs of states that differ by single excitations, 2 for doubles, etc. Note that $P_{r}$ in Eq. (2) captures $S$-coherences of order $r$ or less. This is because $S$-coherences between states differing by more than $r$-body transitions do not appear in ${}^{(r)}\hat{\Gamma}$ [Eq. (5)] and are thus not reflected in the $r$-body purity.

To obtain expressions for $P_{1}$ and $P_{2}$ we adopt the expansion in Slater determinants of $\hat{\rho}_{e}$ in Eq. (6) and focus on the case where distinct pair of states differ by at most two-particle transitions. Higher-order $S$-coherences do not contribute to $P_{1}$ and $P_{2}$, and can be ignored. Thus, given state $|\Phi_{m}\rangle$ in Eq. (6), all the other states $|\Phi_{n}\rangle$ $(n\neq m)$ in $\hat{\rho}_{e}$ that contribute to the reduced purity are supposed to be at most two-particle transitions away from $|\Phi_{m}\rangle$ such that $s_{nm}\leq 2$. That is,

[TABLE]

Here the choice $\alpha_{1}\neq\beta_{1},\alpha_{2}\neq\beta_{2}$ is made to guarantee that $|\Phi_{n}\rangle\neq|\Phi_{m}\rangle$. Also, $\alpha_{1}\neq\alpha_{2},\beta_{1}\neq\beta_{2}$ and $c^{\dagger}_{\beta_{1}}|\Phi_{m}\rangle=c_{\alpha_{1}}|\Phi_{m}\rangle=c_{\alpha_{2}}|\Phi_{m}\rangle=0$ is chosen to prevent $|\Phi_{n}\rangle$ from vanishing. Note that no particular requirement on the occupation of state $\beta_{2}$ is adopted to be able to capture states that differ by a both a single and two-particle transition within this framework. Further note that the labels $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ depend on the pair of states $|\Phi_{n}\rangle$ and $|\Phi_{m}\rangle$, and they should always be thought as having an implicit $n,m$ dependence. Such a dependence is not made explicit in the interest of simplicity in the notation.

The detailed calculation of the reduced purities is included in the Appendix. The one-body reduced density matrix (1-RDM) associated to $\hat{\rho}_{e}$ is given by

[TABLE]

Here and throughout the prime in the second sum indicates that only pairs of states that differ by at most two-particle transitions should be considered. The resulting expression for the one-body purity is given by

[TABLE]

where $A_{1}=f_{m}(\beta_{1})(1-f_{m}(\alpha_{1}))(1-f_{m}(\alpha_{2}))f_{q}(\delta_{1})(1-f_{q}(\gamma_{1}))(1-f_{q}(\gamma_{2}))$. Here we have used the labels $\hat{\rho}_{e}=\displaystyle{\sum_{p,q}a_{pq}|\Phi_{p}\rangle\langle\Phi_{q}|}$ with $|\Phi_{p}\rangle=c^{\dagger}_{\gamma_{2}}c_{\delta_{2}}c^{\dagger}_{\gamma_{1}}c_{\delta_{1}}|\Phi_{q}\rangle$ $(p\neq q)$ to obtain the transpose of 1-RDM, ${}^{(1)}\Gamma_{\epsilon_{2}}^{\epsilon_{1}}$, as required to evaluate $P_{1}$. The populations of the Slater determinants contribute to the first part of $P_{1}$, while the $S$-coherences are captured by the second part. $\delta_{\alpha_{1}\beta_{2}}\,(\delta_{\gamma_{1}\delta_{2}})$ in the second term shows that $P_{1}$ captures $S$-coherences of order 1 between the pair of states $|\Phi_{n}\rangle$ and $|\Phi_{m}\rangle$ $(|\Phi_{p}\rangle$ and $|\Phi_{q}\rangle)$. Further, distinct pairs of states $n,m$ and $p,q$ contribute to $P_{1}$ provided they differ by the same one-particle transition as indicated by $\delta_{\alpha_{2}\delta_{1}}\delta_{\beta_{1}\gamma_{2}}$. These contributions due to distinct pairs of states differing by the same one-particle transition were absent in the previously derived Eq. (19) in Ref. Franco and Appel (2013). When these states are not present in $\hat{\rho}_{e}$, Eq. (10) reduces to Eq. (19) in Ref. Franco and Appel (2013).

Similarly, the 2-RDM associated with the many-body density matrix in Eq. (6) is

[TABLE]

Adopting the same labels as in $P_{1}$, the final expression for the two-body purity is given by

[TABLE]

where $A_{2}=f_{m}(\beta_{1})(1-f_{m}(\alpha_{1}))(1-f_{m}(\alpha_{2}))f_{q}(\delta_{1})(1-f_{q}(\gamma_{1}))(1-f_{q}(\gamma_{2}))$ and $A_{3}=f_{m}(\beta_{2})f_{q}(\delta_{2})$. The diagonal terms of $\hat{\rho}_{e}$ in the single-Slater determinant basis [Eq. (6)] appear in the first square bracket of the expression whereas the $S$-coherences are present in the second square bracket. In addition to population-dependent terms and the $S$-coherences already captured by Eq. (21) in Ref. Franco and Appel (2013), Eq. (12) captures $S$-coherences of order 1 (or 2) between distinct pairs of states that differ by the same one-body (or two-body) transition from each other. This is imposed by the terms $\delta_{\alpha_{2}\delta_{1}}\delta_{\beta_{1}\gamma_{2}}$ and $(\delta_{\beta_{1}\gamma_{1}}\delta_{\beta_{2}\gamma_{2}}-\delta_{\beta_{1}\gamma_{2}}\delta_{\beta_{2}\gamma_{1}})(\delta_{\alpha_{1}\delta_{1}}\delta_{\alpha_{2}\delta_{2}}-\delta_{\alpha_{1}\delta_{2}}\delta_{\alpha_{2}\delta_{1}})$ for the one-body and two-body transition respectively.

IV Distilled Purities

IV.1 Definition and basic properties

The reduced purities, while numerically accessible, are typically difficult to interpret because the non-idempotency of ${}^{(r)}\hat{\Gamma}$ can arise due to decoherence or due to electronic correlation Franco and Appel (2013). In fact, the degree to which ${}^{(1)}\hat{\Gamma}$ deviates from idempotentency is an established measure of electronic correlation in isolated many-electron systems Ziesche (1995); Huang and Kais (2005). Further, even the less demanding goal of attempting to isolate $B$-coherences among general many-particle states at the $r$-RDM level is quite challenging because these $B$-coherences can get all mixed up in ${}^{(r)}\hat{\Gamma}$ when invoking a particular single-particle basis to construct the $r$-RDM.

One type of $B$-coherences among many-particle states that can, in fact, be isolated at the $r$-RDM level is that among Slater determinants defined by a given single-particle basis, i.e. the $S$-coherences. As can be seen in Eqs. (10) and (12), both the one-body and two-body reduced purity are composed of a term that depends on the populations of the single Slater determinants, while the second term is completely determined by the $S$-coherences. Note that while the $S$-coherences are not necessarily indicative of the degree of purity of the system, they can provide useful information to interpret the dynamics. The role of the distilled purities introduced below is precisely to extract the contributions of the $S$-coherences to the reduced purities.

The one-body and two-body distilled purities, $\tilde{P}_{1}$ and $\tilde{P}_{2}$, are defined as follows:

[TABLE]

In essence, the second term in this expression distills the contributions of the $S$-coherences to $P_{r}$ by removing the term dependent on the populations of the Slater determinants. Specifically, the one-body distilled purity is given by

[TABLE]

where $A_{1}=f_{m}(\beta_{1})(1-f_{m}(\alpha_{1}))(1-f_{m}(\alpha_{2}))f_{q}(\delta_{1})(1-f_{q}(\gamma_{1}))(1-f_{q}(\gamma_{2}))$, and the sums are over pairs of states that differ by at most two particle transitions. In obtaining Eq. (15), we have used the fact that ${}^{(1)}\Gamma_{\epsilon}^{\epsilon}=\sum_{n}a_{nn}f_{n}(\epsilon)$, as can be verified from Eq. (9). In turn, from Eqs. (11) and (12) it follows that the two-body purity is given by

[TABLE]

where $A_{2}=f_{m}(\beta_{1})(1-f_{m}(\alpha_{1}))(1-f_{m}(\alpha_{2}))f_{q}(\delta_{1})(1-f_{q}(\gamma_{1}))(1-f_{q}(\gamma_{2}))$ and $A_{3}=f_{m}(\beta_{2})f_{q}(\delta_{2})$. A detailed calculation of the one-body and two-body distilled purities is provided in the Appendix.

The distilled purities provide a succinct way to summarize the $S$-coherences in the system in a particular basis. They are easy to obtain by simple matrix manipulations of the $r$-RDMs as indicated in Eqs. (13) and (14). The one-body distilled purity can capture $S$-coherences between pairs of Slater determinants that differ by at most a one-body transition. In fact, the terms $\delta_{\alpha_{1}\beta_{2}}$ and $\delta_{\gamma_{1}\delta_{2}}$ guarantee that this is the case. In turn, $\tilde{P}_{2}$ captures $S$-coherences of order 1 or 2. When distinct pairs of states (i.e. $\{p,q\}\neq\{n,m\}$) appear, they contribute to $\tilde{P}_{1}$ (or $\tilde{P}_{2}$) only if they differ by the same one-body (or one and two-body) transition. The distilled purities provide a manifestly basis dependent measure of coherence that succinctly captures the behavior of the off-diagonal elements of the few-body reduced density matrices expressed in a given single-particle basis.

IV.2 Limiting values

To aid the interpretation of the dynamics of $\tilde{P}_{1}$ and $\tilde{P}_{2}$ in Eqs. (13) and (14) it is useful to determine a few limiting values. The minimum value for the distilled purities is, of course, $\tilde{P}_{r}=0$ $(r=1,2)$. This occurs when all $S$-coherences of order $r$ or less are zero, i.e. $a_{nm}=0,\,\forall\,n\neq m$. For example, when the state can be described as a single-Slater determinant in the given basis, or when the $S$-coherences in the system are of order greater than $r$. A non-zero distilled purity signals $S$-coherences in the particular basis. The maximum reduced purity is achieved for a pure electronic state that can be described as a single Slater determinant in some basis. In this case, $P_{1}=N$ and $P_{2}=N(N+1)/2$ Franco and Appel (2013) (when contrasting with the result in Franco and Appel (2013) note that a superposition of single Slater determinant in which all the determinants differ by one particle transitions, $s_{nm}=1$, must also be a single Slater determinant Zhang and Kollar (2014); Ando (1963)). Thus, the maximum value for the distilled purity is given by

[TABLE]

where we have used Eq. (7) and the fact that $s_{nn}=0$. Note that $s_{nm}$ in the equation above can have any value $s_{nm}\in[0,N]$ because it arises from the contribution of the populations of the Slater determinants. A less restricting inequality can be obtained by taking into account that $s_{nm}^{\text{max}}=N$,

[TABLE]

By the Cauchy-Schwarz inequality [$(\sum_{i}u_{i}v_{i})^{2}\leq(\sum_{i}u_{i}^{2})(\sum_{i}v_{i}^{2})$],

[TABLE]

where $K$ is the total number of determinants that can be constructed in the given basis. Using this inequality,

[TABLE]

By a similar argument,

[TABLE]

where the first inequality is significantly more restrictive than the second one.

An increase in the distilled purities from their minimum value of 0 indicates the creation of $S$-coherences in the given basis. An increase (or decrease) in the value of the $S$-coherences will generally lead to an increase (or decrease) in the distilled purities. Exceptions can arise in the case where there are distinct pairs of states that differ by the same single or double particle transition in the superposition as the phase, and not just the magnitude, of these $S$-coherences influence the distilled purities, see Eqs. (34) and (45).

V Numerical Examples

We now exemplify the behavior of the distilled purities, and contrast it to that of the reduced purities, in the context of a Su-Schrieffer-Heeger (SSH) model for oligoacetylene Heeger et al. (1988); a tight-binding model of non-interacting electrons with electron-vibrational couplings. Specifically, we consider the dynamics of a SSH chain composed of four carbon atoms and four $\pi$ electrons. The four $\pi$ electrons are distributed in four molecular orbitals $|e_{i}\rangle$ of energy $e_{i}$, leading to 19 possible configurations (without counting spin degeneracies). A detailed discussion of the SSH model and the Ehrenfest mixed quantum-classical technique used to follow the vibronic dynamics has been presented before Franco et al. (2008); Franco and Brumer (2012); Franco et al. (2013). Here the electrons are the system of interest, the nuclei are the bath, and the electron-ion coupling is the source of electronic decoherence.

To test the utility of the distilled purity to inform about dynamical processes in the system, we consider the following exemplifying cases: (i) An initial separable vibronic state in which the electrons are in a superposition of energy eigenstates, i.e.

[TABLE]

where $|\chi_{0}\rangle$ is the ground vibrational state associated with the ground electronic state $|\Phi_{0}\rangle$, and $|\Phi_{1}\rangle$ is an excited electronic state. Both $|\Phi_{0}\rangle$ and $|\Phi_{1}\rangle$ are taken to be single Slater determinants. (ii) A chain in the ground vibronic state subject to a laser pulse that is resonant with a specific electronic transition. Case (i) is simple to interpret using reduced purities because the population of the Slater determinants remains approximately constant, and thus the dynamics of the reduced purities reflect the dynamics of the $S$-coherences Franco and Appel (2013). By contrast, in case (ii) the populations of the Slater determinants involved change in time making it challenging to separate the dynamics of the $S$-coherences from the dynamics of the populations using the reduced purities. Through (i), we illustrate how the distilled purities reflect the decay of the initial $S$-coherences. Through (ii) we test the ability of the distilled purities to monitor laser-induced $S$-coherences that are obscured by the population dynamics in the reduced-purities.

To explore the effect of changing the basis, the distilled and reduced purities are computed in the molecular orbital (energy) $|e_{i}\rangle=c^{\dagger}_{e_{i}}|0\rangle$ and the site $|n\rangle=c^{\dagger}_{n}|0\rangle$ basis, where $|0\rangle$ is the vacuum state. The molecular orbitals are the eigenstates of the single-particle SSH Hamiltonian in the optimal geometry of the chain. In turn, the sites refer to the spatially localized orbitals located at the positions of the carbon atoms in the chain. Naturally, these two basis are connected via a unitary transformation: $c^{\dagger}_{n}=\sum_{i}\langle e_{i}|n\rangle c^{\dagger}_{e_{i}}$. By applying this transformation to the 1-RDM and 2-RDM, the distilled and reduced purities can be computed in either the site or energy basis using Eqs. (2), (13) and (14). Roughly speaking, the $S$-coherences in site representation signal spatial coherences in the state, while the $S$-coherences in energy representation signal dynamics.

Figure 1 shows the dynamics of the distilled purities for the SSH chain prepared in an initial superposition of the form Eq. (22) where $|\Phi_{0}\rangle$ and $|\Phi_{1}\rangle$ differ by (a) one or (b) two-particle transitions in the molecular orbital basis as specified in the figure. During the vibronic dynamics of such states, there is evolution of the nuclear wavepacket in the excited state potential energy surface. Such evolution leads to a decay of the nuclear wavepacket overlap $|\langle\chi_{0}(t)|\chi_{1}(t)\rangle|$ associated with the ground $|\Phi_{0}\rangle$ ($|\chi_{0}\rangle$) and excited electronic state $|\Phi_{1}\rangle$ ($|\chi_{1}\rangle$). Such overlap determines the $S$-coherences between $|\Phi_{0}\rangle$ and $|\Phi_{1}\rangle$ and its decay leads to a decay of the purity of the electronic subsystem (cf. Eq. (4)), and thus to a decay in the reduced purity Franco and Appel (2013). As shown in Figure 1, the distilled purities capture the wavepacket evolution that leads to such decoherence. In both the energy and site basis, the distilled purities display a fast initial decay with recurrences every $\sim 30$ fs. These recurrences arise from the time dependence of the overlap of the nuclear wavefunctions in the ground and excited electronic states (see Eq. (4)), and signal the oscillatory motion of the nuclear wavepacket in the excited state potential. Between consecutive recurrences the amplitude of the distilled purity diminishes and eventually reaches an asymptotic value.

Note that in this case the dynamics of the distilled purities closely mimic that of the reduced purities. The reason for this is because in this particular case there are no appreciable changes in the populations of the two Slater determinants involved ($a_{mm}$ in Eq. (10)), and thus the dynamics of both quantities is determined by the $S$-coherences. Nevertheless, while the reduced purities are basis independent, the value of the distilled purities depend on the single-particle basis employed. In fact, in the energy basis the distilled purities asymptotically go to zero signaling the fact that the $S$-coherences between the Slater determinants constructed using the molecular orbitals basis decays to zero upon time evolution. However, the distilled purities in the site basis do not go to zero indicating that even for the asymptotic state some spatial coherences remain, as is expected for a quantum mechanical system.

Consider now how the distilled purities change with the coherence order. The initial superposition in Figure 1 (a) is of order one, while that of (b) is of order 2. The superposition in (a) is visible both in $\tilde{P}_{1}$ and $\tilde{P}_{2}$, and the fall of $\tilde{P}_{2}$ is $(N-1)$ times larger than that of $\tilde{P}_{1}$. By contrast, in (b) $\tilde{P}_{2}$ follows the decay of $S$-coherences while $\tilde{P}_{1}$ remains constant because it cannot distinguish a coherence of second order from a mixture of such states. At initial time, $\tilde{P}_{2}$ takes its maximum value that is consistent with the superposition in question and evolves with the vibronic evolution.

Figure 2 shows the dynamics of the polarization, distilled and the reduced purities during resonant photoexcitation with a 10 fs laser pulse. Such laser creates a superposition of single Slater determinants that is then subject to decoherence due to vibronic couplings. During photoexcitation the one and two body purity decay, as a result of the population of other possible Slater determinants and subsequent decoherence processes after photoexcitation. Such a decay is mirrored by the distilled purities in the site basis that signal the decay of spatial $S$-coherences that is onset by photoexcitation. Interpreting the dynamics of the reduced purities is quite challenging as it involves determining all the Slater determinants that participate in the dynamics, their populations and the $S$-coherences among them. By contrast, the distilled purities in the energy basis clearly show the $S$-coherences that are created by the laser pulse and their eventual decay due to decoherence, as signaled by the growth of the distilled purities and their decay in the energy basis. The distilled purities in the energy basis attains a maximum at 50 fs when the laser pulse is at its maximum, and follows the dynamics of the polarization as both quantities depend on the $S$-coherences in the energy basis. This example clearly shows how the distilled purities can aid the interpretation of the dynamics of many-body systems by signaling $S$-coherences that are created/destroyed during evolution.

VI Final Remarks

The basic features of the three measures of electronic decoherence discussed in this paper- purity, reduced purity and distilled purity are summarized in Table 1. The purity is a well-defined basis-independent measure of coherence that directly signals the extent to which the electronic subsystem is described as a mixed state. Whenever possible, this is our preferred quantity to interpret decoherence. However, to obtain it one needs the $N$-particle electronic density matrix which is generally inaccessible, making the purity often impractical to measure electronic decoherence in many body systems.

The reduced purities introduced in Ref. Franco and Appel (2013) measure the non-idempotency of the $r$-RDMs. These quantities are basis independent and accessible from simulations that propagate the 1-RDM and 2-RDM directly. For non-interacting electronic systems the decay of the reduced purity directly signals coherence loss. Nevertheless, in the general case where both electron-nuclear and electron-electron interactions play a role in the dynamics, the decay of the reduced purity can come from electronic correlation or from decoherence. Since these two effects are challenging to separate at the $r$-RDM level, the reduced purities are of limited applicability as a measure of electronic coherence or correlation.

As a practical alternative, here we have introduced the one- and two-body distilled purities in Eqs. (13) and (14) as a tool to interpret the dynamics of many-body systems in the presence of decoherence. The distilled purities are derivative quantities of the reduced purities that distill the contributions of the $S$-coherences to the reduced purities. That is, the distilled purities summarize the $S$-coherences among $N$-particle single Slater determinant states with integer occupations as defined by a given single particle basis. In this analysis, we have derived exact expression for the one-body and two-body distilled purities for general electronic states. For this, we generalized the expressions for the one- and two-body reduced purities in Ref. Franco and Appel (2013), by capturing possible contributions coming from two distinct pairs of states that differ by the same one- or two-particle transition.

The distilled purities are manifestly basis-dependent quantities that are useful in interpreting the dynamics of many-body systems. As an example, the distilled purities were shown to be able to signal $S$-coherences that are generated during resonant photoexcitation of a model molecule, which are obscure in the reduced purities. Further, since the $r$-body distilled purities can capture $S$-coherences of order $r$ or less, investigating the behavior of the distilled purities of different orders can aid the interpretation of the many-body dynamics. In spite of these advantages, the distilled purity is not simply related to the $N$-body purity of the system, and thus it is not indicative of the degree of coherence of the system. For example, a pure electronic state that can be described as a single Slater determinant in a given basis will have a distilled purity of zero in such basis. This limitation is shared with other basis-dependent measures of coherence. For instance, in the energy eigenbasis a ground state molecule in a pure state will have no off-diagonal elements in the density matrix and thus zero $B$-coherences, even when it is in a pure state. Albeit not necessarily indicative of whether there is actual decoherence in the system, these quantities are useful in analyzing the quantum dynamics of many-body systems in a situation where the purity is an inaccessible quantity.

Acknowledgements.

This material is based upon work supported by the National Science Foundation under CHE - 1553939.

Appendix A Derivation of the reduced and distilled purities

Below we derive the one- and two-body reduced and distilled purities (Eqs. (10) and (12) and Eqs. (15) and (16)) for the general electronic state in Eq. (6).

A.1 One-body reduced and distilled purity

A.1.1 1-RDM

The 1-RDM for a general electronic state of the form in Eq. (6) is given by

[TABLE]

where we have used Eqs. (5) and (8), and where the prime indicates that the sum goes over pairs of states that differ by at most two-particle transitions. Note that in the last summation only those states that differ by a single particle transition contribute, as pairs with coherences of higher order are not visible in the 1-RDM. As mentioned in the text, the labels $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ have an implicit dependence on $n$ and $m$. The second term can be developed further by first taking the creation and annihilation operators into normal ordering and then employing the restrictions on the $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ detailed under Eq. (8) in Sec. III:

[TABLE]

By inserting Eq. (24) into Eq. (23), it then follows that

[TABLE]

where $X=\delta_{\epsilon_{1}\epsilon_{2}}f_{m}(\epsilon_{1}),Y=f_{m}(\beta_{1})(1-f_{m}(\alpha_{1}))(1-f_{m}(\alpha_{2}))\delta_{\alpha_{1}\beta_{2}}\delta_{\epsilon_{1}\beta_{1}}\delta_{\epsilon_{2}\alpha_{2}}$ which is Eq. (9) in the main text. For obtaining the one body purity, it is also useful to express the transpose of the 1-RDM in Eq. (23) with a different set of labels $p$ and $q$ ($\hat{\rho}_{e}=\displaystyle{\sum_{p,q}a_{pq}|\Phi_{p}\rangle\langle\Phi_{q}|}$ and $|\Phi_{p}\rangle=c^{\dagger}_{\gamma_{2}}c_{\delta_{2}}c^{\dagger}_{\gamma_{1}}c_{\delta_{1}}|\Phi_{q}\rangle$) as follows:

[TABLE]

where $Z=\delta_{\epsilon_{2}\epsilon_{1}}f_{q}(\epsilon_{2}),W=f_{q}(\delta_{1})(1-f_{q}(\gamma_{1}))(1-f_{q}(\gamma_{2}))\delta_{\gamma_{1}\delta_{2}}\delta_{\epsilon_{2}\delta_{1}}\delta_{\epsilon_{1}\gamma_{2}}$. Note that the labels $\gamma_{1},\gamma_{2},\delta_{1},\delta_{2}$ depend on $p,q$ implicitly. The expressions in Eqs. (25) and (26) for 1-RDM are now employed to find $P_{1}$.

A.1.2 One-body reduced purity

The one-body reduced purity is given by

[TABLE]

where

[TABLE]

and $A_{1}=f_{m}(\beta_{1})(1-f_{m}(\alpha_{1}))(1-f_{m}(\alpha_{2}))f_{q}(\delta_{1})(1-f_{q}(\gamma_{1}))(1-f_{q}(\gamma_{2}))$. By removing the terms that vanish, and simplifying one obtains a final expression for the one-body purity [Eq. (10)]

[TABLE]

This equation can be simplified further by noticing that $\delta_{\alpha_{1}\beta_{2}}(\delta_{\gamma_{1}\delta_{2}})$ implies that the pair $n,m\,(p,q)$ are connected by a one-body transition. Thus

[TABLE]

A.1.3 One-body distilled purity

To calculate the one-body distilled purity in Eq. (13), it is necessary to obtain the square of the diagonal element of ${}^{(1)}\hat{\Gamma}$. From Eq. (25),

[TABLE]

This is the exactly same as the first term in Eq. (30). Thus the one-body distilled purity in Eq. (13) can be simplified to

[TABLE]

where $A_{1}=f_{m}(\beta_{1})(1-f_{m}(\alpha_{1}))(1-f_{m}(\alpha_{2}))f_{q}(\delta_{1})(1-f_{q}(\gamma_{1}))(1-f_{q}(\gamma_{2}))$. The equation above is equivalent to Eq. (15).

A.1.4 Example

As a simple example, consider $P_{1}$ for a 2-particle system with $\hat{\rho}_{e}=\sum_{n,m=1}^{4}a_{nm}|\Phi_{n}\rangle\langle\Phi_{m}|$ where $|\Phi_{1}\rangle=c^{\dagger}_{1}c^{\dagger}_{2}|0\rangle,|\Phi_{2}\rangle=c^{\dagger}_{3}c^{\dagger}_{4}|0\rangle,|\Phi_{3}\rangle=c^{\dagger}_{1}c^{\dagger}_{4}|0\rangle,|\Phi_{4}\rangle=c^{\dagger}_{2}c^{\dagger}_{3}|0\rangle$. In this case, Eq. (30) yields

[TABLE]

The associated distilled purity is then:

[TABLE]

Notice that the reduced purity is composed of a part that depends on the populations of the Slater determinants and another one on the $S$-coherences. The distilled purities extract the contributions due to the $S$-coherences. The $S$-coherences between each pair of states that differ by a one body transition contribute to $P_{1}$ and $\tilde{P}_{1}$. In addition, there are additional contributions in $P_{1}$ that arise when two distinct pair of states differ by the same one-particle transition. For example, the term $a_{14}a_{23}$ appears in $P_{1}$ because both pairs of states ($|\Phi_{1}\rangle$ and $|\Phi_{4}\rangle$, and $|\Phi_{2}\rangle$ and $|\Phi_{3}\rangle$) differ by the same one-body transition as $|\Phi_{2}\rangle=c^{\dagger}_{3}c_{1}|\Phi_{3}\rangle$ and $|\Phi_{4}\rangle=-c^{\dagger}_{3}c_{1}|\Phi_{1}\rangle$. The negative sign in the expression arises from the ordering of the states.

A.2 Two-body reduced and distilled purity

A.2.1 2-RDM

The 2-RDM for the state in Eq. (6) is given by

[TABLE]

where we have used Eqs. (5) and (8). Note that in the last summation only those states that differ by one or two particle transitions contribute, as pairs with coherences of higher order are not visible in the 2-RDM. This summation can be developed further by adopting normal ordering and imposing the restrictions on $\alpha_{1},\alpha_{2},\beta_{1}$, and $\beta_{2}$:

[TABLE]

Inserting this expression into Eq. (35) we obtain a final expression for the 2-RDM [Eq. (11)]:

[TABLE]

To calculate the purity it is also useful to obtain an expression for the transpose of Eq. (37) with alternative indexes. Specifically, we employ $\hat{\rho}_{e}=\displaystyle{\sum_{p,q}a_{pq}|\Phi_{p}\rangle\langle\Phi_{q}|}$ and $|\Phi_{p}\rangle=c^{\dagger}_{\gamma_{2}}c_{\delta_{2}}c^{\dagger}_{\gamma_{1}}c_{\delta_{1}}|\Phi_{q}\rangle$). In this case

[TABLE]

The expressions in Eqs. (37) and (38) are now employed to find $P_{2}$.

A.2.2 Two-body reduced purity

The two-body reduced purity is given by

[TABLE]

where

[TABLE]

with $A_{2}=f_{m}(\beta_{1})f_{q}(\delta_{1})(1-f_{m}(\alpha_{1}))(1-f_{m}(\alpha_{2}))(1-f_{q}(\gamma_{1}))(1-f_{q}(\gamma_{2}))$ and $A_{3}=f_{m}(\beta_{2})f_{q}(\delta_{2})$. The terms $BC$ and $AD$ vanish after simplification because of the constraints on $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},\gamma_{1},\gamma_{2},\delta_{1},\delta_{2}$. Thus, the final expression for the two-body reduced purity is [Eq. (12)]:

[TABLE]

A.2.3 Two-body distilled purity

To calculate the distilled purity using Eq. (14), it is necessary to first determine the square of the diagonal elements of ${}^{(2)}\hat{\Gamma}$, i.e. $\left({}^{(2)}\Gamma_{\epsilon_{1}\epsilon_{2}}^{\epsilon_{1}\epsilon_{2}}\right)^{2}$. From Eq. (37),

[TABLE]

Thus,

[TABLE]

Note that this term is identical to the term in the first square bracket of Eq. (40). Inserting Eqs. (A.2.3) and (40) into Eq. (14), we arrive at the final form of the two-body distilled purity in Eq. (16):

[TABLE]

where $A_{2}=f_{m}(\beta_{1})f_{q}(\delta_{1})(1-f_{m}(\alpha_{1}))(1-f_{m}(\alpha_{2}))(1-f_{q}(\gamma_{1}))(1-f_{q}(\gamma_{2}))$ and $A_{3}=f_{m}(\beta_{2})f_{q}(\delta_{2})$. Both $S$-coherences of order 1 and 2 are captured by $\tilde{P}_{2}$.

A.2.4 Example

As an example, consider the 3-particle system with $\hat{\rho}_{e}=\sum_{n,m=1}^{4}a_{nm}|\Phi_{n}\rangle\langle\Phi_{m}|$ where $|\Phi_{1}\rangle=c^{\dagger}_{1}c^{\dagger}_{2}c^{\dagger}_{3}|0\rangle,|\Phi_{2}\rangle=c^{\dagger}_{1}c^{\dagger}_{2}c^{\dagger}_{6}|0\rangle,|\Phi_{3}\rangle=c^{\dagger}_{3}c^{\dagger}_{4}c^{\dagger}_{5}|0\rangle,|\Phi_{4}\rangle=c^{\dagger}_{4}c^{\dagger}_{5}c^{\dagger}_{6}|0\rangle$. In this case, Eq. (40) yields

[TABLE]

The associated distilled purity is:

[TABLE]

The reduced purity has contributions from the populations of the Slater determinant states and the $S$-coherences between them, while the distilled purities captures just the $S$-coherences. The $S$-coherences between the states that differ by one-body transition and two-body transitions contribute to $P_{2}$ and $\tilde{P}_{2}$ . For example, the term $|a_{12}|^{2}$ appears due to a one-body between $|\Phi_{1}\rangle$ and $|\Phi_{2}\rangle$ while $|a_{13}|^{2}$ appears due to a two-body transition between $|\Phi_{1}\rangle$ and $|\Phi_{3}\rangle$. Moreover, two distinct pairs of states that differ by the same two-body transitions also contribute to the two-body reduced and distilled purities. The term $a_{13}a_{42}$ appears as both pairs of states ($|\Phi_{1}\rangle$ and $|\Phi_{3}\rangle$, and $|\Phi_{2}\rangle$ and $|\Phi_{4}\rangle$) differ by the same two-body transition as $|\Phi_{3}\rangle=c^{\dagger}_{5}c_{2}c^{\dagger}_{4}c_{1}|\Phi_{1}\rangle$ and $|\Phi_{4}\rangle=c^{\dagger}_{5}c_{2}c^{\dagger}_{4}c_{1}|\Phi_{2}\rangle$.