Pure Dephasing Master Equation

The pure dephasing master equation describes loss of coherence in a preferred basis without population transfer in that basis. It is the continuous-time Markovian generator behind the phase-damping channel, unread monitoring of an observable, elastic environmental scattering, and many simple models of frequency noise. For the exactly solvable non-Markovian envelope behind this limit, see Pure Dephasing Model.

For a qubit with Hamiltonian

$$H = \frac{\hbar\omega_0}{2}\sigma_z,$$

one common convention is

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \frac{\Gamma_\phi}{2} \left( \sigma_z\rho\sigma_z-\rho \right).$$

Equivalently,

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + 2\Gamma_\phi\mathcal D[P_e]\rho,$$

where $P_e=\lvert e\rangle\langle e\rvert$ is diagonal in the energy basis. The important invariant statement is physical: populations in the dephasing basis are unchanged, while coherences decay exponentially. Different authors move factors of $2$ between the rate and the Lindblad operator.

Physical Meaning

Pure dephasing is not damping. A relaxation channel moves population between states, while a pure dephasing channel only destroys phase relations between alternatives that the environment or apparatus can distinguish.

For the qubit equation above,

$$\rho = \begin{pmatrix} \rho_{00} & \rho_{01}\\ \rho_{10} & \rho_{11} \end{pmatrix}$$

evolves as

$$\begin{aligned} \dot\rho_{00}&=0,\\ \dot\rho_{11}&=0,\\ \dot\rho_{01}&=-(i\omega_0+\Gamma_\phi)\rho_{01},\\ \dot\rho_{10}&=(i\omega_0-\Gamma_\phi)\rho_{10}. \end{aligned}$$

Thus

$$\rho_{01}(t) = e^{-i\omega_0 t} e^{-\Gamma_\phi t} \rho_{01}(0),$$

while $\rho_{00}$ and $\rho_{11}$ remain fixed. The corresponding finite-time channel is the Markovian case of the Dephasing Channel with coherence factor $\lambda(t)=e^{-\Gamma_\phi t}$ after separating the Hamiltonian phase.

For a notebook contract that checks this solution against ODE integration and a vectorized Liouvillian, see Solving Lindblad Equations.

Bloch-Vector Form

Write

$$\rho = \frac12 \left( I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z \right).$$

The pure dephasing master equation gives

$$\begin{aligned} \dot r_x&=-\Gamma_\phi r_x-\omega_0 r_y,\\ \dot r_y&=\omega_0 r_x-\Gamma_\phi r_y,\\ \dot r_z&=0. \end{aligned}$$

In a frame rotating with the Hamiltonian, the transverse Bloch components decay as

$$r_x(t),r_y(t)\propto e^{-\Gamma_\phi t},$$

and the longitudinal component $r_z$ is constant. The Bloch ball contracts toward the $z$ axis rather than toward a thermal state.

Relation to T-Times

In spectroscopy and qubit experiments, $T_1$ is the population-relaxation time, $T_2$ is the transverse coherence-decay time, and $T_\phi$ is the pure-dephasing time. In the standard weak-coupling Markovian qubit model,

$$\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi}.$$

The factor $1/(2T_1)$ appears because population relaxation also reduces coherence. Pure dephasing is the additional contribution that remains when energy relaxation is absent or has been separated.

With the convention used in the opening equation,

$$T_\phi^{-1}=\Gamma_\phi.$$

Other communities may put factors of $2$ into the Lindblad operator or into the symbol called $\gamma_\phi$. Always translate the equation, not just the name of the rate.

For how this bookkeeping fits with Ramsey $T_2^*$, echo times, filter functions, and spatial decoherence estimates, see Decoherence Timescales.

In a driven two-level optical or microwave transition, this same pure-dephasing contribution enters the transverse rate of the Optical Bloch Equations.

General Diagonal Lindblad Operator

The qubit model is a special case of dephasing by a Hermitian Lindblad operator. Suppose

$$L=L^\dagger, \qquad L\lvert m\rangle = \ell_m\lvert m\rangle.$$

For

$$\dot\rho = \gamma\mathcal D[L]\rho,$$

the matrix elements in the $L$ eigenbasis obey

$$\left(\dot\rho\right)_{mn} = - \frac{\gamma}{2} (\ell_m-\ell_n)^2 \rho_{mn}.$$

Diagonal entries have $m=n$ and are unchanged. Coherences between states with different $\ell_m$ decay. Coherences inside a degenerate eigenspace are protected from this dissipator.

This formula is often the most compact way to identify pointer subspaces: the environment distinguishes eigenvalues of $L$, not necessarily individual basis vectors.

Microscopic Origin

A simple microscopic source of pure dephasing is longitudinal system–bath coupling:

$$H = H_S+H_B+H_I, \qquad H_I = A\otimes B,$$

with

$$[A,H_S]=0.$$

Because $A$ is diagonal in the energy basis, the bath does not induce transitions between energy eigenstates to leading order. It instead makes the energy splittings fluctuate or records which energy eigenstate is occupied.

For a qubit, one often writes

$$H_I = \frac{\hbar}{2}\sigma_z\otimes B.$$

Under weak-coupling, short-memory assumptions, the dephasing rate is controlled by the bath noise near zero frequency. Schematically,

$$\Gamma_\phi \propto S_{BB}(0),$$

where $S_{BB}(\omega)$ is a bath spectrum with convention-dependent normalization. The proportionality can include factors of $1/2$, $\hbar$, and coupling constants depending on how $B$ is defined.

The zero-frequency statement has a simple physical meaning: pure dephasing comes from slow or elastic fluctuations of the relative phase, not from bath modes that must absorb or provide the transition energy $\hbar\omega_0$.

Pulse sequences can reshape this low-frequency sensitivity; see Dynamical Decoupling for spin echo, CPMG, and filter-function language.

Classical Phase Noise

Pure dephasing can also arise from a classical stochastic Hamiltonian,

$$H(t) = \frac{\hbar}{2} \left[ \omega_0+\xi(t) \right]\sigma_z.$$

For a single noise realization, the state evolves unitarily. After averaging over noise realizations, the off-diagonal element becomes

$$\rho_{01}(t) = e^{-i\omega_0 t} \left\langle e^{-i\int_0^t \xi(s)\,ds} \right\rangle \rho_{01}(0).$$

If the accumulated phase is Gaussian with variance proportional to $t$, the average produces exponential decay:

$$\left\langle e^{-i\phi(t)} \right\rangle = e^{-\Gamma_\phi t}.$$

Classical phase noise is not the only origin of dephasing. A quantum environment can produce the same reduced master equation while retaining quantum correlations with the system.

Measurement Interpretation

An unread measurement of a diagonal observable causes dephasing. In the Markovian continuous-measurement limit, the unconditional state often obeys a dephasing equation of the form

$$\dot\rho = \Gamma_m\mathcal D[A]\rho,$$

where $A$ is the monitored observable. The measurement record contains information about $A$; if the record is ignored, the ensemble state loses coherence between distinct $A$ eigenvalues.

This does not mean that every dephasing process is literally a performed measurement. It means the reduced dynamics has the same structure as ignoring a record carried away by an apparatus or environment.

For outcome-resolved measurements, see Quantum Instruments. For the unconditional channel, see Selective and Nonselective Measurements.

Oscillator Number Dephasing

For a harmonic oscillator, phase diffusion in the energy basis is often modeled by

$$\dot\rho = \gamma_n\mathcal D[n]\rho, \qquad n=a^\dagger a.$$

In the number basis,

$$n\lvert m\rangle=m\lvert m\rangle,$$

so

$$\dot\rho_{mn} = - \frac{\gamma_n}{2} (m-n)^2\rho_{mn}.$$

Number states are unaffected by this dissipator, while superpositions of different number states lose relative phase. This model appears in cavity and circuit-QED contexts as number dephasing, dispersive measurement backaction, or phase diffusion, depending on the physical origin.

Do not confuse number dephasing with photon loss. Photon loss uses $L=a$ and changes number populations; number dephasing uses $L=n$ and leaves number populations fixed.

Validity Checks

The pure dephasing master equation is appropriate only after specifying the basis and assumptions.

Check:

• What observable or basis is being dephased?
• Does the Lindblad operator commute with the Hamiltonian, or is the basis only approximate?
• Is energy exchange negligible on the time scale of interest?
• Is the bath or noise approximately Markovian?
• Are low-frequency noise correlations short enough to justify exponential decay?
• Are static inhomogeneous broadening and ensemble averaging being separated from irreversible decoherence?
• Are the reported rates $T_2^{-1}$, $T_\phi^{-1}$, linewidth half-widths, or angular-frequency rates?

Static disorder can cause reversible ensemble dephasing. A spin echo can refocus that part. A Markovian Lindblad pure-dephasing term represents irreversible loss in the reduced description, not merely ignorance of a fixed detuning.

Common Mistakes

Calling all coherence decay pure dephasing

Amplitude damping also damps coherences. Pure dephasing means the populations in the preferred basis are unchanged by the dephasing term.

Forgetting the basis

Pure dephasing is basis-dependent. A term diagonal in the energy basis may not be diagonal in a driven rotating frame or dressed-state basis.

Mixing up T-two and T-phi

$T_2$ includes both population relaxation and pure dephasing. $T_\phi$ is the extra dephasing contribution after subtracting the relaxation part in the model where that subtraction is justified.

Treating inhomogeneous broadening as irreversible dephasing

An ensemble with different static detunings can lose contrast, but echo sequences may reverse that contrast loss. A Lindblad dephasing term models irreversible or effectively irreversible phase randomization.

Reading the Lindblad operator as a unique mechanism

The same dephasing generator can be produced by unread measurement, classical phase noise, elastic scattering, or a quantum bath. The generator alone does not identify the microscopic cause.

Exercises

Off-diagonal decay

For

$$\dot\rho = \frac{\Gamma_\phi}{2} \left( \sigma_z\rho\sigma_z-\rho \right),$$

show that $\dot\rho_{01}=-\Gamma_\phi\rho_{01}$.

Solution

Since $\sigma_z=\operatorname{diag}(1,-1)$,

$$(\sigma_z\rho\sigma_z)_{01} = -\rho_{01}.$$

Therefore

$$\dot\rho_{01} = \frac{\Gamma_\phi}{2} \left( -\rho_{01}-\rho_{01} \right) = -\Gamma_\phi\rho_{01}.$$

Bloch-vector contraction

Use the same master equation to derive $\dot r_x=-\Gamma_\phi r_x$ and $\dot r_y=-\Gamma_\phi r_y$ in the interaction picture.

Solution

The density matrix is

$$\rho = \frac12 \begin{pmatrix} 1+r_z & r_x-ir_y\\ r_x+ir_y & 1-r_z \end{pmatrix}.$$

The off-diagonal relation gives

$$\frac{d}{dt}(r_x-ir_y) = -\Gamma_\phi(r_x-ir_y).$$

Equating real and imaginary parts gives

$$\dot r_x=-\Gamma_\phi r_x, \qquad \dot r_y=-\Gamma_\phi r_y.$$

General Hermitian dephasing

Let $L=L^\dagger$ and $L\lvert m\rangle=\ell_m\lvert m\rangle$. Derive the decay rate of $\rho_{mn}$ under $\dot\rho=\gamma\mathcal D[L]\rho$.

Solution

The positive term gives

$$(L\rho L)_{mn} = \ell_m\ell_n\rho_{mn}.$$

The anticommutator gives

$$\frac12(L^2\rho+\rho L^2)_{mn} = \frac12 (\ell_m^2+\ell_n^2) \rho_{mn}.$$

Thus

$$(\dot\rho)_{mn} = \gamma \left[ \ell_m\ell_n - \frac12(\ell_m^2+\ell_n^2) \right] \rho_{mn} = - \frac{\gamma}{2} (\ell_m-\ell_n)^2 \rho_{mn}.$$

T-time relation

An undriven qubit has population relaxation rate $T_1^{-1}$ and pure-dephasing rate $T_\phi^{-1}$. What is the standard weak-coupling expression for $T_2^{-1}$?

Solution

Population relaxation contributes half its rate to transverse coherence decay, and pure dephasing contributes directly:

$$\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi}.$$

This relation assumes the usual Markovian qubit model with relaxation and pure dephasing separated in the same basis.

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 3rd ed. (2004).
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer (2007).
• U. Weiss, Quantum Dissipative Systems, World Scientific, 4th ed. (2012).
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012).