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, rt = 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 rg = 182 pm, μ = 4.65 × 10−26 kg and σ = 7.91 × 10−15 m². Then n = 2.50 × 1025 m−3, ⟨vrel⟩ = 471 m/s, R = 9.32 × 1013 s−1. Since Δx/λrel = 5.19 × 103, L ≈ 1 and τ ≈ 1.07 × 10−14 s.
2. C60-scale molecule in high vacuum
Inputs: M = 720 u, rt = 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, ⟨vrel⟩ = 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, rt = 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.
| Environment | Gas conditions | Number density | Collision/decoherence rate | Estimated τ |
|---|
| Room air approximation | N₂, 293.15 K, 101,325 Pa | 2.50 × 1025 m−3 | 1.09 × 1010 s−1 | 9.19 × 10−11 s |
| High vacuum | N₂, 293.15 K, 10−8 Pa | 2.47 × 1012 m−3 | 1.07 × 10−3 s−1 | 931 s |
| Local interstellar gas | Atomic H, 7,000 K, 9.66 × 10−15 Pa | 1.00 × 105 m−3 | 8.60 × 10−10 s−1 | 1.16 × 109 s (36.8 yr) |
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
- Describe the target with its mass and an effective collision radius.
- Enter the center-to-center superposition separation Δx.
- Choose a gas, temperature and absolute pressure, or start from a compatible scenario.
- Read τ first, then check the localization regime, collision rate and assumptions before interpreting it.
The collisional-decoherence approximation
μ = Mm/(M + m)
n = P/(kBT)
σ = π(rt + rg)²
⟨vrel⟩ = √[8kBT/(πμ)]
R = nσ⟨vrel⟩
λrel = h/√(2πμkBT)
L(Δx) = 1 − exp[−2π(Δx/λrel)²]
Γ = RL; τ = 1/Γ
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/λrel)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; rt and rg are effective hard-sphere radii; P and T are gas pressure and absolute temperature; n is number density; R is collision rate; λrel 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 ⟨vrel⟩ ∝ T1/2, so R ∝ T−1/2. Meanwhile λrel ∝ T−1/2, so the localization factor grows with T until it saturates. Thus Γ grows roughly as T1/2 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 ≪ λrel, a collision poorly resolves the branches and L is small. If Δx ≫ λrel, 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
This is an order-of-magnitude dilute ideal-gas estimate. It assumes a classical dilute gas in thermal equilibrium, binary elastic hard-sphere collisions, a user-specified geometric/effective cross-section, isotropic Gaussian momentum transfer, Markovian independent collisions, and a spatial superposition. The relative-speed formula further assumes target and gas translational velocities are Maxwellian at the same T; for a stationary target, use the gas mass in place of μ (the difference is negligible when M ≫ m). The treatment is most defensible when collisions are well separated and gas-gas correlations are negligible.
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
What is quantum decoherence time?
It is a characteristic timescale over which environmental interaction suppresses coherence between specified quantum alternatives. There is no single universal value: it depends on the state, environment, coupling and measurement definition.
How is decoherence time calculated?
For this model, calculate the gas collision rate R, multiply it by the separation-dependent localization factor L, then take τ = 1/(RL). More complete calculations integrate the differential scattering cross-section over the gas momentum distribution.
Is decoherence time the same as T2?
No. T2 is the phase-coherence time of a two-level system under its relevant noise channels. This page estimates just one spatial-decoherence mechanism.
What is the difference between T1 and T2?
T1 describes energy relaxation toward equilibrium. T2 describes phase-coherence loss and includes the effect of T1 relaxation plus pure dephasing. IBM Quantum likewise parameterizes thermal-relaxation noise using T1 and T2.
Why does pressure shorten coherence?
At fixed temperature in the dilute ideal-gas limit, n = P/(kBT). More pressure means proportionally more particles and collisions, so this model's Γ rises linearly and τ falls inversely.
Does decoherence collapse the wavefunction?
Decoherence explains why local interference becomes inaccessible when a system entangles with an environment. On its own, it does not select a unique observed outcome, so it does not by itself solve every interpretation's measurement problem.
Can decoherence be reversed?
Controlled or slowly varying phase errors can sometimes be refocused by echo and error-control methods. Reversing uncontrolled entanglement dispersed among many environmental degrees of freedom is generally impractical, though the combined closed-system evolution remains quantum mechanical.
How accurate is this calculator?
Use it for scaling and order-of-magnitude comparisons only. Its largest uncertainties usually come from the effective hard-sphere cross-section, the Gaussian isotropic localization kernel, ideal-gas assumptions and omitted channels—not floating-point rounding.
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 kB = 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].
