Ramps

A Ramp separates the shape of a sweep from its endpoints. A RampShape is a pure profile on the unit interval — s: [0,1] → [0,1] with s(0) = 0 and s(1) = 1 — and Ramp maps that profile onto physical time and physical values:

Ramp(shape, t_start, t_stop, value_start, value_stop)

Evaluating ramp(t) normalizes the time, applies the profile, and rescales to the value range:

τ = (t - t_start) / (t_stop - t_start)         # normalized time
s = shape(τ)                                    # normalized progress (clamped)
value_start + (value_stop - value_start) * s    # physical value

Clamping happens inside the shape, so outside [t_start, t_stop] the ramp holds at its endpoint value rather than extrapolating — you can keep evolving past t_stop and the parameter stays put.

ramp = Ramp(SmoothstepRamp(), 0.0, 10.0, 0.0, 2.0)
ramp(0.0)   # 0.0
ramp(5.0)   # 1.0   (midpoint of a symmetric shape)
ramp(10.0)  # 2.0
ramp(50.0)  # 2.0   (held, not extrapolated)

Ramps run downward as readily as upward (value_start > value_stop), and nothing requires a Ramp at all — any f(t) is a valid driving, as is a plain number for a constant.

Shapes

Shape	`s(τ)`	endpoint slope
`LinearRamp()`	`τ`	nonzero at both ends
`PowerLawRamp(p)`	`τᵖ`	zero at start for `p > 1`
`ExponentialRamp(k)`	`(e^{kτ} − 1)/(e^k − 1)`	asymmetric; `k = 0` is linear
`SineRamp()`	`(1 − cos πτ)/2`	zero at both ends
`SineSquaredRamp()`	`sin²(πτ/2)`	identical to `SineRamp` (half-angle identity)
`SmoothstepRamp()`	`3τ² − 2τ³`	zero at both ends
`SmootherstepRamp()`	`6τ⁵ − 15τ⁴ + 10τ³`	zero slope and curvature at both ends

For adiabatic sweeps this choice is substantive. LinearRamp switches the sweep on discontinuously in Ḣ, and that kink is a broadband perturbation which drives diabatic transitions no matter how slowly you ramp. Shapes with s′(0) = s′(1) = 0 turn the drive on and off smoothly and suppress those endpoint excitations; SmootherstepRamp additionally kills the curvature. PowerLawRamp and ExponentialRamp are the asymmetric options — useful when you want to move slowly through a gap minimum at one end only.

Reference

ITensorTDMPO.RampShape — Type

Ramp shapes map a normalized time τ ∈ [0, 1] to a normalized progress s ∈ [0, 1], with s(0) == 0 and s(1) == 1. A Ramp combines a shape with a time window and a value range to produce a function of physical time, e.g. for use as a coefficient in a time-dependent Hamiltonian term.

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ITensorTDMPO.Ramp — Type

Ramp(shape::RampShape, t_start, t_stop, value_start, value_stop)

A scalar ramp that follows the normalized shape from value_start at t_start to value_stop at t_stop. Calling ramp(t) gives the ramp value at time t, clamped to value_start/value_stop outside [t_start, t_stop].

Examples

ramp = Ramp(SmoothstepRamp(), 0.0, 10.0, 0.0, 1.0)
ramp(0.0)  # 0.0
ramp(5.0)  # 0.5
ramp(10.0) # 1.0

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ITensorTDMPO.LinearRamp — Type

LinearRamp()

s(τ) = τ.

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ITensorTDMPO.SmoothstepRamp — Type

SmoothstepRamp()

Cubic smoothstep, s(τ) = 3τ² - 2τ³. Zero slope at both endpoints.

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ITensorTDMPO.SmootherstepRamp — Type

SmootherstepRamp()

Quintic smootherstep, s(τ) = 6τ⁵ - 15τ⁴ + 10τ³. Zero slope and curvature at both endpoints.

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ITensorTDMPO.SineRamp — Type

SineRamp()

Raised-cosine ramp, s(τ) = (1 - cos(πτ)) / 2. Zero slope at both endpoints.

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ITensorTDMPO.SineSquaredRamp — Type

SineSquaredRamp()

s(τ) = sin²(πτ/2). Zero slope at both endpoints.

By the half-angle identity this is exactly equal to SineRamp; both names are provided since either convention is common in the literature.

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ITensorTDMPO.PowerLawRamp — Type

PowerLawRamp(exponent)

s(τ) = τ^exponent. exponent = 1 is equivalent to LinearRamp; exponent > 1 starts slow and accelerates, exponent < 1 starts fast and decelerates.

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ITensorTDMPO.ExponentialRamp — Type

ExponentialRamp(rate)

Exponential ramp normalized to s(0) == 0 and s(1) == 1: s(τ) = (exp(rate * τ) - 1) / (exp(rate) - 1). rate > 0 starts slow and accelerates, rate < 0 starts fast and decelerates, and rate == 0 reduces to LinearRamp.

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