Pure dephasing and coherence times

Take the energy basis as the eigenbasis of $Z=\operatorname{diag}(1,-1)$. Conjugation by Z leaves diagonal density entries unchanged and reverses off-diagonal signs. A dissipator proportional to the difference between that conjugation and the original state therefore damps only coherence. The coefficient convention matters: the same symbol is sometimes used for the dissipator prefactor and sometimes for the actual off-diagonal decay rate, producing a factor-of-two ambiguity unless the equation is written explicitly.

Here define the pure-dephasing rate as the actual exponential coherence-decay rate. A Hamiltonian diagonal in the same basis contributes only a deterministic phase, removed in an interaction frame. Since the dissipator does not change populations, it exchanges no mean energy with the system Hamiltonian in this ideal model. This statement does not mean that all microscopic environment degrees of freedom are unchanged, nor that a more general noise coupling cannot cause both phase and energy effects.

Add upward and downward amplitude damping with constant nonnegative rates. Their sum governs the return of excited population to its stationary value, and half their sum contributes to the coherence rate. Independent pure dephasing adds directly to that coherence rate. These statements follow from acting with each dissipator on a diagonal population mode and on an off-diagonal matrix unit. They do not require assigning the same decay constant to every matrix element.

Defining the longitudinal and transverse times by these exponentials yields the usual bound that T2 cannot exceed twice T1 within this model. A reported violation may signal inconsistent fitted definitions, nonexponential decay, correlated noise, experimental error, or a model outside these assumptions. It does not license a negative constant Lindblad dephasing rate. At infinite T1, pure dephasing still gives finite T2 while every initial energy population remains stationary.

An ensemble can show reduced transverse signal because its members precess at different but nearly static frequencies. Each member may remain pure, yet their averaged density matrix loses coherence. An ideal refocusing pulse can reverse this accumulated static phase spread, whereas a single deterministic phase correction cannot undo unknown fluctuating noise. The often-used free-induction time T2-star therefore need not equal the homogeneous transverse time extracted after appropriate refocusing.

Echo success is not a universal proof that the environment is classical or that all decoherence is reversible locally. Pulse errors, noise correlation times, and the measured ensemble determine the result. Pure-state Bloch geometry can display individual phases but cannot show an averaged vector shrinking without an explicit ensemble calculation. Diagnose a loss of coherence by measuring transverse observables or reconstructing the density matrix; computational-basis population measurements alone miss pure dephasing entirely.