Contour-Time-Ordered Correlations (CTOCs)

• CTOCs are generalized quantum correlation functions defined along a time contour with forward (+) and backward (-) segments, unifying conventional, advanced/retarded, and out-of-time-order correlators.
• They are derived using path-integral and Keldysh contour techniques, which enable rigorous treatment of nonequilibrium dynamics, thermalization, and quantum chaos.
• CTOCs facilitate experimental and theoretical breakthroughs in quantum nonlinear spectroscopy and information scrambling by revealing enhanced correlations under symmetry constraints.

Contour-Time-Ordered Correlations (CTOCs) are generalized quantum correlation functions whose operator insertions are arranged along a time contour with both forward and backward branches. CTOCs play a foundational role in the theory of quantum dynamical responses, thermalization, quantum chaos, and nonlinear spectroscopy. They subsume conventional time-ordered, advanced/retarded, and out-of-time-order correlators (OTOCs) within a unified contour path-integral framework, enabling an exponential proliferation of accessible correlations in quantum systems that are subject to driving, dissipation, entanglement, or unconventional symmetry constraints.

1. Formal Definition and Structure

CTOCs are defined by correlators in which operator time arguments are placed along a contour with both forward ($+$) and backward ($-$) segments. The generalized n-point Wightman correlator is written as:

$$W_n^{(\sigma)} \equiv \mathrm{Tr}[B_{\sigma(n)} \cdots B_{\sigma(1)} \rho_B]$$

where $\sigma$ specifies the operator permutation.

Rewriting in terms of superoperators, CTOCs are labeled by index strings $\eta_j \in \{+, -\}$, yielding:

$$C^{(\eta_n)} = \mathrm{Tr}\left[ \mathbb{B}_n^{\eta_n} \cdots \mathbb{B}_1^{\eta_1} \rho_B \right]$$

with superoperators $\mathbb{B}^{+} X = \tfrac{1}{2} \{B, X\}$ (anti-commutator) and $\mathbb{B}^{-} X = -i [B, X]$ (commutator).

The time-contour approach encompasses standard time ordering (all $\eta_j = -$), fully anti-commutator-ordered (all $\eta_j = +$), and mixed commutator/anti-commutator orderings, each corresponding to physically distinct dynamical processes.

2. Path-Integral and Keldysh Contour Construction

CTOCs are naturally derived from nonequilibrium Green's function techniques employing closed-time-path (Keldysh/Kostantinov-Perel') contours. On the Keldysh contour, operator insertions are ordered by a contour-ordering operator $\mathcal{T}_C$, reflecting the physical chronology of real-time evolution, and allow encoding both equilibrium and nonequilibrium cases. Multifold (k-OTO) contours generalize the construction, facilitating representation of all possible $n!$ time-orderings (see (Haehl et al., 2017, Secchi et al., 2017)).

The extended contour mesh may be discretized (cf. (Secchi et al., 2017), Eqs. (16–18)) to rigorously construct CTOCs in the path-integral formalism:

• Discrete real-time coordinates $t_j^{\pm}$ for forward/backward branches,
• Additional imaginary-time (Matsubara) branch for thermal/statistical initial states,

Enabling exact treatment of boundary conditions, initial state specifications, and generating functionals for higher-order correlators.

3. CTOCs in Quantum Versus Classical Spectroscopy

Classical nonlinear spectroscopy probes systems using commutator-only CTOCs, corresponding to the response to "classical forces" and yielding:

$$C^{(-\cdots-)} = (-i)^n \left\langle \left[ \left[ \cdots [B_{n+1}, B_n], \cdots, B_2 \right], B_1 \right] \right\rangle$$

Quantum nonlinear spectroscopy (QNS) induces responses involving both commutators and anti-commutators via "quantum forces" from a quantum sensor, opening up exponentially more correlations. For example, at second order:

• Propagation: $C^{(+-)} = -i \langle [B_2, B_1] \rangle$
• Noise correlation: $$C^{(++)} = \langle \frac{1}{2} \{ B_2, B_1\} \rangle$$

QNS thereby accesses the full CTOC family, making visible quantum correlations inaccessible to classical techniques (Sun et al., 28 Aug 2025).

4. Symmetry Constraints and Selection Rules

Non-spatial symmetries such as particle-hole (C), time-reversal (T), and chiral (S) constrain the CTOC landscape:

• C-symmetry: If $\mathcal{C} B_i^T \mathcal{C}^{-1} = \alpha_i B_i$ ($\alpha_i = \pm 1$), then $C^{(\eta_n)} = 0$ whenever $\prod_{j} \eta_j + \prod_j \alpha_j = 0$ ((Sun et al., 28 Aug 2025), Eq. (2)).
• T-symmetry: Relates time-reversed CTOCs via $$W_n^{(\sigma)} = W_n^{(\tilde{\sigma}(\{t_i \rightarrow -t_i\}))} \prod_j \beta_j$$, for $\mathcal{T} B_i^* \mathcal{T}^{-1} = \beta_i B_i$.
• S-symmetry: Similar constraints with $\gamma_i$ factors, Eq. $$W_n^{(\sigma)} = W_n^{(\sigma(\{t_i \rightarrow -t_i\}))} \prod_i \gamma_i$$.

These theorems reduce the number of independent CTOCs, link correlators of different rank (i.e., connect CTOCs and OTOCs), and enforce observable selection rules in quantum nonlinear spectroscopy experiments.

5. Spectral Representation and Computational Techniques

CTOCs admit compact spectral representations on multifold contours (Chaudhuri et al., 2018):

• Decomposition of the Wightman array onto tensor products of column vectors (thermal basis),
• Generalized spectral functions $\rho[12]$, $\rho[123]$, $\rho[4321]$ encode system-specific dynamics,
• Universal kinematic factors encode causality and Kubo-Martin-Schwinger (KMS) periodicity constraints,

For example, CTOCs in frequency space for two-point functions:

$${}_{k,2}(2\mathrm{-Pt}) = \rho[12] \sum_{r=1}^{k} (e^{(r+1)} - e^{(r)}) \otimes e^{(r)}$$

with $e^{(r)}$ thermal column vectors associated with contour legs—reducing enormous combinatorial tables to a handful of nonzero entries via symmetry and causality.

Diagrammatic rules, perturbative expansions, and kinetic theory treatments follow naturally from this compact representation, facilitating analytic and numerical evaluation of CTOCs and their role in transport, hydrodynamics, and information scrambling.

6. Experimental Realization and Measurement Protocols

A suite of experimental protocols have been developed to measure CTOCs:

• Quantum nonlinear spectroscopy schemes use entangled photon pairs, ultrafast interferometry, and symmetry phase cycling to induce and selectively probe CTOCs and OTOCs (Asban et al., 2021, Sun et al., 28 Aug 2025).
• Ultracold atom experiments can realize extended Kadanoff–Baym contours (via quench-reversal protocols (Tsuji et al., 2016)).
• Quantum optics implementations with Bell-state entanglement enable robust OTOC/CTOC measurement in systems with chiral or particle-hole symmetry (Sundar, 2020), leveraging invariance properties to avoid explicit time-reversal.

Experimental observables include nonlinear response signals, coincidence detection rates, and spectroscopically resolved time/frequency correlation maps, each revealing pathway-dependent quantum dynamics.

7. Operational Equivalences and Unified Frameworks

The pseudo-density matrix (PDM) formalism unifies CTOCs, OTOCs, process matrices, and consistent histories into a single operational framework (Zhang et al., 2020):

$$R = \frac{1}{2^n} \sum_{i_1\ldots i_n} \langle \sigma_{i_1} \otimes \cdots \otimes \sigma_{i_n} \rangle \sigma_{i_1} \otimes \cdots \otimes \sigma_{i_n}$$

PDMs encode space-time quantum correlations in the structure of measured probabilities, establishing that apart from amplitude-weighted correlations in path integrals, CTOCs, OTOCs, and related formalisms yield equivalent experimental observables.

This equivalence underpins theoretical, algorithmic, and platform-independent approaches for measuring, analyzing, and interpreting temporal quantum correlations in non-relativistic quantum mechanics.

8. Applications and Impact

CTOCs are central for:

• Quantum information scrambling, chaos diagnostics, and black hole analog studies,
• Dynamical mean-field theory (DMFT) approaches to many-body fermion systems (Tsuji et al., 2016),
• Real-time entanglement and Rényi entropy calculations (via k-OTO contours, (Haehl et al., 2017)),
• Nonequilibrium dynamics and phase boundaries in strongly correlated systems (Bose-Hubbard models, (Fitzpatrick et al., 2018)).

The symmetry-based classification, enhanced correlation accessibility in QNS, and unification across path-integral and operational frameworks position CTOCs as a cornerstone for both theoretical developments and experimental explorations in complex quantum dynamics.

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Table: CTOC Types and Physical Correspondence

CTOC Index String	Mathematical Form	Physical Process
$(-,-,\ldots,-)$	Nested commutators	Propagation/dissipation
$(+,+,\ldots,+)$	Nested anti-commutators	Fluctuation/noise correlation
Mixed	Mixed commutator/anticomm	Quantum dissipative dynamics

Experimental setups select different CTOC types by engineering the nature of the probe (classical/quantum, entangled/unentangled) and control of symmetry operations.

• "Dynamical equations for time-ordered Green's functions: from the Keldysh time-loop contour to equilibrium at finite and zero temperature" (Ness et al., 2012)
• "Exact out-of-time-ordered correlation functions for an interacting lattice fermion model" (Tsuji et al., 2016)
• "Classification of out-of-time-order correlators" (Haehl et al., 2017)
• "Discrete-time construction of nonequilibrium path integrals on the Kostantinov-Perel' time contour" (Secchi et al., 2017)
• "Spectral Representation of Thermal OTO Correlators" (Chaudhuri et al., 2018)
• "Quantum correlations in time" (Zhang et al., 2020)
• "Proposal to measure out-of-time-ordered correlations using Bell states" (Sundar, 2020)
• "Interferometric-Spectroscopy With Quantum-Light; Revealing Out-of-Time-Ordering Correlators" (Asban et al., 2021)
• "Non-spatial symmetries in quantum nonlinear spectroscopy" (Sun et al., 28 Aug 2025)