Collisional Quantum Decoherence Time Calculator

Verified worked examples

Values below use the same equations and constants as the calculator. Results are rounded to three significant figures.

1. 50 nm nanoparticle in room-temperature nitrogen

Inputs: M = 6.30 × 10⁸ u, r_t = 50 nm, Δx = 100 nm, N₂ at 293.15 K and 101,325 Pa. This mass corresponds to a nominal spherical density of about 2,000 kg/m³. With r_g = 182 pm, μ = 4.65 × 10⁻²⁶ kg and σ = 7.91 × 10⁻¹⁵ m². Then n = 2.50 × 10²⁵ m⁻³, ⟨v_rel⟩ = 471 m/s, R = 9.32 × 10¹³ s⁻¹. Since Δx/λ_rel = 5.19 × 10³, L ≈ 1 and τ ≈ 1.07 × 10⁻¹⁴ s.

2. C₆₀-scale molecule in high vacuum

Inputs: M = 720 u, r_t = 0.355 nm, Δx = 100 nm, residual N₂ at 300 K and 10⁻⁸ Pa. The calculation gives σ = 9.06 × 10⁻¹⁹ m², n = 2.41 × 10¹² m⁻³, ⟨v_rel⟩ = 485 m/s and R = Γ = 1.06 × 10⁻³ s⁻¹. Therefore τ ≈ 942 s (15.7 min) from this channel alone.

3. 1 µm dust grain in room air approximated as nitrogen

Inputs: M = 10⁻¹⁴ kg, r_t = 1 µm, Δx = 1 µm, N₂ at 293.15 K and 101,325 Pa. Here σ = 3.14 × 10⁻¹² m² and R = Γ = 3.70 × 10¹⁶ s⁻¹, giving τ ≈ 2.70 × 10⁻¹⁷ s. “Air” remains an N₂-only simplification.

Same target and separation, different environments

The target is the 720 u, 0.355 nm molecule above with Δx = 100 nm. Changing only the bath conditions (including gas species for interstellar space) isolates the environmental contribution in this model.

What collisional decoherence time means

Spatial decoherence is the loss of observable interference between wave packets centered at different positions after the target becomes entangled with scattered environmental particles. The reported τ is the time for the modeled off-diagonal coherence at separation Δx to fall by a factor of e, assuming an exponential, memoryless decay.

How to use the calculator

1. Describe the target with its mass and an effective collision radius.
2. Enter the center-to-center superposition separation Δx.
3. Choose a gas, temperature and absolute pressure, or start from a compatible scenario.
4. Read τ first, then check the localization regime, collision rate and assumptions before interpreting it.

The collisional-decoherence approximation

The general collisional master equation has Γ(Δx) = R[1 − η(Δx)], where η is determined by the momentum-transfer distribution and differential scattering cross-section. This calculator uses η = exp[−2π(Δx/λ_rel)²] as a transparent Gaussian, isotropic momentum-transfer approximation. It recovers quadratic suppression at unresolved separations and saturation at one decohering event per collision, but it is not a fitted scattering law.

Variable definitions

M and m are target and gas-particle masses; μ is their reduced mass; r_t and r_g are effective hard-sphere radii; P and T are gas pressure and absolute temperature; n is number density; R is collision rate; λ_rel is the relative-coordinate thermal de Broglie wavelength; L is the modeled localization probability per collision; Γ is the spatial-decoherence rate.

How pressure and temperature affect the estimate

At fixed T, n and therefore Γ scale linearly with pressure. Temperature is not unconditionally harmful or helpful. At fixed pressure, n ∝ T⁻¹ and ⟨v_rel⟩ ∝ T^1/2, so R ∝ T^−1/2. Meanwhile λ_rel ∝ T^−1/2, so the localization factor grows with T until it saturates. Thus Γ grows roughly as T^1/2 in the unresolved limit but falls roughly as T^−1/2 after saturation. At fixed number density, the trend is different.

How to interpret the result

If Δx ≪ λ_rel, a collision poorly resolves the branches and L is small. If Δx ≫ λ_rel, L approaches one and τ approaches the mean time between modeled collisions. Compare this estimate with the experiment's timescale and with independently calculated channels; do not label it a system's total coherence time.

Assumptions and when not to use this model

Do not use it for liquids, dense or non-ideal gases, strongly forward-peaked or resonant scattering, chemically reactive collisions, a cross-section with important energy dependence, or where a measured differential cross-section is available. It excludes photon scattering, absorption and emission; phonons; electromagnetic and magnetic noise; material defects and two-level fluctuators; control and readout noise; surface interactions; vibration; gravity-related proposals; and every other platform-specific channel. It therefore does not predict superconducting-qubit T₁/T₂, trapped-ion coherence, or a complete experimental visibility curve.

Frequently asked questions

Methodology, sources and review

The general position-space collisional-decoherence rate and its short- and long-wavelength limits follow the modern scattering treatment summarized by Schlosshauer and the primary derivation by Hornberger and Sipe. The calculator then adds the stated hard-sphere and Gaussian/isotropic approximations so that every input maps to an explicit quantity.

• K. Hornberger and J. E. Sipe, “Collisional decoherence reexamined,” Physical Review A 68, 012105 (2003).
• M. Schlosshauer, “Quantum decoherence,” Physics Reports 831 (2019), especially the scattering master equation and wavelength limits.
• K. Hornberger et al., “Collisional Decoherence Observed in Matter Wave Interferometry,” Physical Review Letters 90, 160401 (2003), an experimental fullerene/background-gas benchmark.
• NIST SI defining constants: exact h = 6.62607015 × 10⁻³⁴ J s and k_B = 1.380649 × 10⁻²³ J K⁻¹.
• IBM Quantum noise-model documentation for the distinct roles of T₁ and T₂.
• NASA summary of New Horizons local interstellar hydrogen measurements; the preset rounds 0.1 cm⁻³ to 10⁵ m⁻³.

Author: Starlight Tools Editorial Team. Last methodology review: 17 July 2026. Reviewed against the cited sources; no independent technical reviewer is claimed. Preset gas radii are approximate kinetic hard-sphere inputs, not reference-quality scattering data. To report a scientific or calculation error, email [email protected].