Distance-Dependent Real-Space Dephasing

• Distance-dependent real-space dephasing is the spatial variation of quantum coherence decay due to environmental noise, measurement selectivity, and intrinsic material properties.
• It is modeled using spatially resolved Lindblad operators and interferometric techniques to quantify layer-specific decoherence and transport transitions.
• Understanding this phenomenon aids in engineering quantum device performance, controlling decoherence in topological insulators, and optimizing transport in complex lattices.

Distance-dependent real-space dephasing refers to the spatial variation of quantum coherence decay due to underlying mechanisms whose rates, spatial profiles, or physical manifestations depend on real-space position or separation within a quantum system. This phenomenon emerges in diverse condensed matter, quantum optical, and quantum information systems, and is governed by the microscopic details of interactions, disorder, geometry, measurement protocols, or environmental couplings. A rigorous understanding of distance-dependent dephasing is crucial for interpreting transport, coherence, and decoherence phenomena in electronic, photonic, and atomic-scale devices.

1. Physical Origins and Definitions

Dephasing generally describes the decay of quantum mechanical coherences between states due to environment-induced noise, many-body interactions, or measurement back-action. When the spatial structure of the system, its environment, or the interaction processes induce position- or distance-dependent rates of decoherence, one obtains distance-dependent real-space dephasing. The physical origin can be:

• Spatially structured environment: The environmental correlation length, disorder, or fluctuations (e.g., fluctuating metal fields, background atoms) may be local or nonlocal, making the dephasing rate between two points in the system explicitly depend on their separation.
• Measurement-induced dephasing: Quantum measurement performed with spatial selectivity (e.g., local probes, weak continuous measurements) introduces decoherence that acts only in certain spatial regions or above certain distances.
• Intrinsic system properties: Layered or inhomogeneous materials (e.g., topological insulators, quantum wells), position-dependent density of states, or symmetry constraints give rise to spatially varying dephasing channels.
• Long-range interactions: Electromagnetic or dipolar couplings, which decay polynomially with distance, produce dephasing rates strongly dependent on interparticle separation.

Notably, the form of the Lindblad superoperator or dissipator in the quantum master equation is key to modeling this, with rates or operators carrying explicit spatial labels or dependence.

2. Paradigmatic Systems and Mechanisms

(A) Topological Insulators and Layered Materials

Spatially resolved studies of three-dimensional topological insulators (TIs) such as BiSbTeSe$_{1.25}$Se$_{1.75}$, using surface- versus edge-contacted devices, establish that the dominant dephasing mechanism changes as a function of depth relative to the surface (Banerjee et al., 2018):

• Surface-contacted (SC) devices probe the outermost quintuple layer, where variable range hopping (VRH) governs dephasing, with a gate voltage-independent $L_\phi \propto T^{-1/3}$ and marked WAL (weak anti-localization) suppression at charge neutrality.
• Edge-contacted (EC) devices probe deeper (subsurface) layers, revealing conventional Nyquist electron-electron interaction dephasing with $L_\phi \propto T^{-1/2}$ and strong gate voltage dependence.

The transition between VRH-induced, localized-state dominated decoherence at the surface, and conventional electron-electron dephasing in the TI bulk, illustrates layer-by-layer variation—a strong form of distance-dependent real-space dephasing.

(B) Electronic Transport in Networks and Lattices

Generalized Markovian models for dephasing in quadratic lattices, such as chains with long-range hopping or arbitrary network connectivity, permit explicit construction of spatially or distance-dependent dephasing operators (Sarkar et al., 5 Oct 2025). The dephasing superoperator,

$$\mathcal{L}_\sigma[\rho] = \sum_{mm'} \sigma_{mm'} \left( n_m \rho n_{m'} - \frac{1}{2}\{n_{m'} n_m, \rho\} \right)$$

where $n_m$ is a local occupation operator, accommodates both strictly local ($\sigma_{mm'} \propto \delta_{mm'}$) and fully nonlocal dephasing rates.

This framework allows the paper of how dephasing reverts anomalous (superdiffusive) current scaling due to long-range hopping to classical diffusive scaling, as well as phase transitions in transport exponents, all governed by the interplay of distance-dependent dephasing and Hamiltonian range (Sarkar et al., 5 Oct 2025, Sarkar et al., 2023).

(C) Environmental Noise and Interferometry

In matter-wave interferometry and quantum spatial qubits, electromagnetic dephasing due to interactions with environmental charges, dipoles, or particles exhibits explicit inverse power-law scaling with the impact parameter, $b$, i.e., the spatial separation of superposed paths and environmental particle trajectories. Analytic dephasing rates for Coulomb, charge–dipole, dipole–dipole, etc., scale as $1/b^n$ with $n=4,5,6,7$ depending on the interaction channel, reflecting the fundamental real-space character of the decoherence process (Schut et al., 2023).

3. Theoretical Description and Key Equations

The mathematical formulation of distance-dependent dephasing is system-dependent but most fundamentally realized through Lindblad dissipators whose rates depend on spatial configuration:

• General Lindblad form (e.g. for lattice fermions):

$$\mathcal{L}_\sigma[\rho] = \sum_{m,m'} \sigma_{mm'} \left( n_m \rho n_{m'} - \frac{1}{2}\{n_{m'} n_m, \rho\} \right)$$

where $\sigma_{mm'}$ encodes the spatial profile.

• Distance-dependent rate functions: In Wannier (real-space) basis approaches for solid-state electron dynamics, dephasing can be encoded as

$$\mathcal{L}_{\mathrm{rs}}[\rho]_{ij} = -\gamma_{rs}(|\tau_i-\tau_j|) (\rho_{ij} - \rho^0_{ij}),$$

where $\gamma_{rs}(d)$ is a function of inter-site distance and vanishes for $d$ below a threshold $R_{rs,\mathrm{cut}}$ (Molinero et al., 24 Oct 2025, Brown et al., 2022).

• Hydrodynamic and noise kernel approaches: In interacting integrable systems with spatially correlated noise, a finite spatial correlation length $\ell$ of the environment leads to decay of density matrix off-diagonals at a rate set by the separation, with points within $\ell$ co-dephased and those at larger distances dephased independently (Bastianello et al., 2020).
• Master equation for motional states under dephasing

$$\dot{\rho}(x,x') = \cdots - \frac{1}{2} \gamma(x, x') \rho(x, x'),$$

with $\gamma(x, x')$ sharply peaked outside a chosen region, can trap quantum particles through decoherence-induced Zeno effect (Mukherjee et al., 2021).

4. Experimental Manifestations and Key Observables

• Phase coherence length ($L_\phi$): Extracted from magneto-conductance/weak anti-localization (WAL) measurements, $L_\phi$ can vary spatially, with extracted exponents and prefactors distinguishing different real-space dephasing mechanisms as a function of probe geometry and gate voltage (Banerjee et al., 2018).
• Variance decay in time-domain measurements: In strongly disordered quantum wires, site- and distance-resolved variances of the local density reveal a power-law dephasing, $F(t) \sim t^{-\zeta(W)}$, with the exponent explicitly determined by real-space localization length and multifractality (Nandy et al., 2020).
• Distance scaling of dephasing in interferometers: Experimental design in quantum gravity, C-NOT gates, and macroscopic superpositions must account for dephasing rates that drop rapidly with increased minimum impact parameter to environmental charges: e.g., for the Coulomb case, $\Gamma_n \propto 1/b^4$ (Schut et al., 2023).
• Spatial profiles of correlation functions: Spin wave dephasing in dense atomic media is fundamentally controlled by near-field dipole–dipole interactions, making the rate of decoherence contextually dependent on atomic separations (Grava et al., 2021).
• Abrupt localization and spatial jumps: In non-Hermitian disordered lattices, locally implemented dephasing results in abrupt spatial jumps of probability density, highlighting the nontrivial spatial structure induced by the combination of non-Hermiticity and incoherent evolution (Kokkinakis et al., 29 Dec 2024).

5. Physical Interpretation, Implications, and Applications

Spatial Gradients and Layer-Specific Engineering

The spatial inhomogeneity of dephasing, especially in layered or inhomogeneous materials (e.g., TIs), suggests the possibility of real-space engineering of quantum coherence. Understanding the spatial profile of dephasing mechanisms within TIs and related systems is essential for quantum device design—surface states may be subject to stronger and qualitatively distinct decoherence than deeper subsurface channels, affecting device robustness (Banerjee et al., 2018).

Transport and Phase Transitions in Lattices

Markovian dephasing with programmable spatial profiles allows for the tuning of phase transitions in steady-state transport, including superdiffusive-to-diffusive crossovers and novel phase diagrams in networks with long-range hopping (Sarkar et al., 5 Oct 2025, Sarkar et al., 2023). This is particularly relevant for quantum network engineering, energy transport in biological complexes, and quantum reservoir computations.

Suppression and Control of Unphysical Dynamics

In real-time ultrafast optical simulations (e.g., solid-state HHG), macroscopic propagation and real-space dephasing mechanisms serve to suppress long-range, unphysical coherences more selectively than phenomenologically ultrafast, reciprocal-space dephasing. This distinction underlies the physical basis for robust harmonics, correct plateau structure, and improved numerical stability in real-space approaches (Molinero et al., 24 Oct 2025, Brown et al., 2022, Brown et al., 2023).

Quantum Information and Entanglement Protection

Spatially selective dephasing can be used to stabilize or generate long-range entanglement in open systems, when combined with lattice symmetries or strong sector constraints. For instance, local dephasing at a central lattice site, under reflection symmetry, leads to distance-independent steady-state correlations between symmetric sites—constituting robust entangled pairs spanning macroscopic distances (Saha et al., 10 Dec 2024).

Trapping and Binding via Dephasing Barriers

Spatially selective dephasing can induce effective trapping (single particle) or binding (two particles) in open quantum systems—even without conservative force potentials. The decoherence profile, not the energy landscape, becomes the controlling potential, leading to Zeno-like trapping and "decoherence-induced dimers" (Mukherjee et al., 2021).

6. Representative Table: System Types and Dephasing Mechanisms

System/Context	Dominant Mechanism / Scaling	Spatial Profile / Consequence
3D topological insulators (Banerjee et al., 2018)	VRH (surface), Nyquist e-e (subsurface)	Dephasing changes with layer; $L_\phi(T)$
Long-range hopping lattices (Sarkar et al., 5 Oct 2025, Sarkar et al., 2023)	Power-law/spatially tunable Lindblad dephasing	Controls superdiffusive–diffusive transitions
Quantum interferometers (Schut et al., 2023)	$1/b^n$ EM dephasing (Coulomb/dipole, $n=4,5,6,7$)	Sensitive to minimal separation from noise
Dense atomic media (Grava et al., 2021)	$1/r^3$ near-field dipole–dipole	Non-exponential decay, RG-resolved broadening
Solid-state HHG (Molinero et al., 24 Oct 2025, Brown et al., 2022, Brown et al., 2023)	Real-space decoherence kernel	Preferential suppression of long-range coherences
Symmetry-protected lattices (Saha et al., 10 Dec 2024)	Central-site dephasing + reflection symmetry	Distance-independent long-range entanglement
Trapping/binding by dephasing (Mukherjee et al., 2021)	Spatial/distance-gated coherent loss	Zeno trapping, dephasing-induced molecular states

7. Future Directions and Open Problems

• Microscopically accurate noise modeling: Determining accurate spatial and spectral properties of environmental noise sources in realistic quantum devices to inform distance-dependent dephasing models.
• Experimental characterization: Layer-specific probes, ultrafast microscopy, and experimental platforms with engineered decoherence profiles are essential for direct observation and control of distance-dependent dephasing.
• Quantum control and error correction: Leveraging spatially engineered dephasing profiles, possibly combined with symmetry-protected subspaces, to stabilize desired quantum correlations or to protect entanglement in open-system quantum devices.
• Relation to measurement-induced transitions: The interplay between distance-dependent dephasing, network structure, and many-body entanglement remains active, particularly concerning phase transitions in open quantum circuits and feedback networks.

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Distance-dependent real-space dephasing is a unifying concept connecting spatially structured decoherence phenomena across condensed matter, quantum optics, and information science. The theoretical and experimental insights into its mechanisms, modeling, and consequences are central to advancing both fundamental understanding and technological exploitation of coherence in extended quantum systems.