Quantum Nonlinear Spectroscopy

• Quantum Nonlinear Spectroscopy (QNS) is a suite of experimental and theoretical techniques that probe multi-time correlation functions in complex quantum systems.
• It leverages controlled pulse sequences, phase cycling, and Fourier analysis to distinguish coherent from incoherent dynamics and reveal many-body effects.
• QNS is scalable for large quantum systems and is applied on platforms such as trapped ions and superconducting qubits to advance quantum simulation and error correction.

Quantum Nonlinear Spectroscopy (QNS) is a broad class of experimental and theoretical techniques for probing multi-time, higher-order correlation functions in complex quantum systems. By generalizing concepts from multidimensional optical spectroscopy and quantum sensing, QNS enables direct access to dynamics, coherence, and many-body effects—including discrimination between coherent and incoherent transport, anharmonicities, system-bath interactions, decoherence-free subspaces, and signatures of quantum chaos or symmetry. The QNS framework leverages precisely controlled quantum platforms (e.g., trapped ions, superconducting qubits, entangled photons) to engineer pulse sequences, quantum measurements, and analysis protocols that scale efficiently with system size, allowing multi-point correlation functions to be measured with moderate overhead even in large many-body settings. This capability is essential for studying quantum information processing, quantum thermodynamics, fundamental quantum correlations, and the emergent behavior of synthetic quantum matter.

1. Core Principles and Measurement Protocols

QNS extends the principles of multidimensional spectroscopy—originally developed in chemical physics—to precisely controllable quantum systems. At the protocol level, impulsive interactions (typically realized as short laser or microwave pulses with tunable phase, amplitude, and site-selectivity) are applied in sequence to the system, separated by controlled time delays. Each pulse is engineered to create either an excitation or de-excitation, with its effect modeled by displacement or spin-rotation operators, for example:

$$D_{(j)}(\alpha e^{i\phi}) \approx \mathbb{I} + \alpha e^{i\phi} a_j^\dagger - \alpha e^{-i\phi} a_j$$

where $a_j^\dagger$ is a local creation operator (e.g., for a phonon or spin excitation) at site $j$, $\alpha \ll 1$ is the pulse strength, and $\phi$ is a controllable phase. The resulting multi-pulse evolution of the density matrix is described as:

$$\rho^{(m)}_{i_1,\ldots,i_m}(t_1,\ldots,t_m) = \mathcal{G}(t_m) \mathcal{V}_{i_m}(\phi_m) \cdots \mathcal{G}(t_1) \mathcal{V}_{i_1}(\phi_1)[\rho(0)]$$

with $\mathcal{G}(t) = \exp(\mathcal{L} t)$ a Green's function including system Hamiltonian and dissipative (Lindbladian) terms, and $\mathcal{V}_j(\phi)$ the pulse superoperator. Readout is performed via spatially resolved fluorescence or direct measurement of system observables (e.g., population, current, or local spin projection). Phase cycling—scanning the phases of the applied pulses—isolates desired “quantum pathways” in the multidimensional response (Gessner et al., 2013, Schlawin et al., 2014).

This formalism generalizes Ramsey-type experiments (traditionally two-pulse) into fully multidimensional, phase-coherent protocols documented via Feynman ladder diagrams, which efficiently enumerate and separate different correlation channels. The general principle is that QNS reconstructs time-ordered multi-point correlation functions by mapping the temporal evolution to experimentally controllable delays and phases.

2. Multidimensional Spectra and Signal Analysis

A defining feature of QNS is the extraction of high-resolution multidimensional spectra—Fourier transforms of the measured signals with respect to pulse delay times—revealing the full structure of system correlations. For instance, the single quantum coherence (SQC) signal is expressed as:

$$S^{(\mathrm{SQC})}_{i_1, i_2; j}(t_1, t_2) = \mathrm{Tr}\left\{ A_j\, \mathcal{G}(t_2)\left[\mathcal{G}(t_1)\left[a^\dagger_{i_1}\, \rho(0)\right] a_{i_2} \right] \right\}$$

The two-dimensional Fourier transform:

$$S^{(\mathrm{SQC})}_{i_1, i_2; j}(\Omega_1, \Omega_2) = \int_0^\infty dt_1 \int_0^\infty dt_2\, e^{i(\Omega_1 t_1 + \Omega_2 t_2)} S^{(\mathrm{SQC})}_{i_1, i_2; j}(t_1, t_2)$$

provides axes $\Omega_1$ (single-exciton energies) and $\Omega_2$ (tunneling frequencies, energy differences), enabling the disambiguation of spatial energy transfer and local dynamics. Multidimensional spectra reveal quantum transport, resolve individual tunneling contributions, and expose the system's eigenmode structure (Gessner et al., 2013).

Higher-order signals, such as double quantum coherence (DQC) and photon echo (PE), access double-excited manifolds or population relaxation, respectively. Characteristic “quantum pathway” phase signatures (e.g., $\phi_1 + \phi_2 - \phi_3 - \phi_4$ for DQC) select particular correlations and excitation processes. Ladder and loop Feynman diagrams codify these pathways, while phase cycling and Fourier analysis separate signal contributions (Schlawin et al., 2014, Dorfman et al., 2016).

3. Discriminating Dynamics: Coherence, Incoherence, and Many-Body Effects

QNS protocols distinguish between coherent (unitary) and incoherent (dissipative, dephasing) transport by analyzing the spectral features of the measured multidimensional signals. Under purely coherent evolution, spectra display sharp oscillatory peaks corresponding to well-defined energy splittings:

$$p_{ab}(t) = |\langle 1_b| e^{-iHt}|1_a\rangle|^2 = \sum_{ij} c_{ij} e^{-i\omega_{ij} t}$$

Here, Fourier analysis reveals the precise tunnel splitting $\omega_{ij}$. When local dephasing or decoherence is present (modeled by Lindbladian terms or local noise operators, e.g., $L_i = \sqrt{\gamma}\, a_i^{\dagger} a_i$), coherent oscillations are suppressed and broadened, and additional zero-frequency (static) components emerge in the spectrum, as observed in numerically simulated data (Gessner et al., 2013).

The fourth-order DQC signal uncovers spectral degeneracies and anharmonic mode shifts, allowing quantification of trap anharmonicities by the splitting and redistribution of spectral lines. In spin systems, differentiated forms of dephasing (local versus collective) are directly identified: collective dephasing preserves so-called decoherence-free subspaces, resulting in a sharply localized spectral response along a frequency axis (manifest as a narrow $\delta$-like line in the multidimensional spectra), while local dephasing causes spectral broadening along both axes. QNS thus provides both qualitative and quantitative diagnostics for coherence preservation, error protection, and the identification of many-body effects and collective quantum phenomena.

4. Scalability and Experimental Implementation

A key attribute of QNS protocols developed for controllable systems (notably trapped ion arrays and cold atoms) is their experimental scalability. The number of required pulses or measurement repetitions for reconstructing $m$-time correlation functions is independent of the total number of particles or system size, a central advantage over quantum state tomography, which scales exponentially. The combination of single-site addressability, fluorescence-based projective readout, and reliable phase cycling ensures that QNS can efficiently probe quantum correlations in large-scale systems (Gessner et al., 2013, Schlawin et al., 2014).

Implementations exploit platform-specific control—for instance, using localized carrier or sideband pulses in ion traps, or weak vibrational displacements via Raman transitions for motional-state spectroscopy. Electronic and vibrational degrees of freedom can be interrogated via similar protocols, enabling joint studies of spin-phonon coupling and other multi-mode interactions (Schlawin et al., 2014). The practical overhead is thus moderate, with complexity limited primarily by the stages of phase cycling and detector integration time, rather than exponential state complexity.

5. Extensions, Signal Classes, and Advanced Applications

QNS frameworks accommodate a range of advanced measurement protocols and analysis strategies. Signal classes include:

• Single Quantum Coherence (SQC): Probes single-exciton dynamics, eigenmode spectra, and energy transport.
• Double Quantum Coherence (DQC): Probes two-exciton sectors, anharmonic shifts, and interaction-induced level structure.
• Photon Echo (PE): Probes population relaxation, dephasing times, and bath-coupling effects.
• Steady-State Current Detection: By imposing boundary-driven gradients (e.g., temperature differences across ion chains), QNS protocols can detect steady-state heat or particle flows via spatially-resolved SQC measurements.
• Probing Chaotic Dynamics: In critical or disordered systems, redistribution of spectral weight and broadening of peaks indicate quantum phase transitions, many-body localization, or transition from integrable to non-integrable regimes (Schlawin et al., 2014).

Comparison with bulk optical spectroscopy underscores unique attributes: spatial (single-site) addressability replaces phase-matching, enabling high spatial and temporal resolution, while phase cycling substitutes for macroscopic field coherence selection. Theoretical treatments based on Green’s function evolution, superoperator diagrammatics, and Liouvillian dynamics provide strong formal groundwork, supporting generalizations to open quantum systems and scenarios with engineered reservoirs (Dorfman et al., 2016, Fetherolf et al., 2017).

6. Impact and Perspective

QNS represents an expansive toolkit for studying non-equilibrium quantum dynamics and many-body correlations. Its capabilities in directly probing multi-point, multi-time quantum correlations at high spatial and temporal resolution, combined with scalability for synthetic quantum matter platforms, are central for advancing quantum simulation, error correction, thermodynamic diagnostics, and materials science. Applications extend to probing decoherence-free subspaces, quantifying transport and scattering phenomena, detecting anharmonicities or many-body couplings, and identifying robust features of quantum information flow.

In summary, QNS enables direct access to complex temporal and spatial correlation structure in quantum systems, serves as a diagnostic and control tool for coherent and incoherent dynamics, and addresses outstanding questions in quantum science related to scalability, many-body dynamics, and quantum control in realistic experimental platforms (Gessner et al., 2013, Schlawin et al., 2014).