Amplitude Damping Master Equation

The amplitude damping master equation is the standard Markovian equation for irreversible relaxation from an excited state to a lower state. It is the generator behind spontaneous emission, zero-temperature qubit $T_1$ decay, cavity photon loss, and many quantum-jump models.

For a two-level system with excited state $\lvert e\rangle$ and ground state $\lvert g\rangle$,

$$\sigma_-=\lvert g\rangle\langle e\rvert, \qquad \sigma_+=\lvert e\rangle\langle g\rvert.$$

The zero-temperature amplitude damping master equation is

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \Gamma_1\mathcal D[\sigma_-]\rho,$$

where

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac12\{L^\dagger L,\rho\}.$$

Here $\Gamma_1$ is the population-relaxation rate. The finite-time channel generated by this equation is the Amplitude-Damping Channel with decay probability $p(t)=1-e^{-\Gamma_1t}$.

For a notebook contract that validates this master equation by comparing ODE integration, matrix exponentials, and finite-time channel checks, see Solving Lindblad Equations.

Physical Meaning

Amplitude damping is energy relaxation into degrees of freedom that are not retained in the system description. For an atom, those degrees of freedom may be emitted radiation modes. For a superconducting qubit, they may be electromagnetic, dielectric, quasiparticle, or engineered reservoir modes. For a cavity oscillator, the dissipative operator is usually $a$, corresponding to loss of one quantum.

The word “amplitude” comes from the fact that a probability amplitude involving the excited state decays at half the population-relaxation rate. In density-matrix language, this is the relation between $T_1$ relaxation and transverse coherence decay.

Two-Level Matrix Equations

Take

$$H = \hbar\omega_0 \lvert e\rangle\langle e\rvert$$

or equivalently the same Hamiltonian up to an irrelevant multiple of the identity. In the interaction picture with respect to $H$, the dissipative equation gives

$$\begin{aligned} \dot\rho_{ee}&=-\Gamma_1\rho_{ee},\\ \dot\rho_{gg}&=\Gamma_1\rho_{ee},\\ \dot\rho_{eg}&=-\frac{\Gamma_1}{2}\rho_{eg},\\ \dot\rho_{ge}&=-\frac{\Gamma_1}{2}\rho_{ge}. \end{aligned}$$

Thus

$$\rho_{ee}(t) = e^{-\Gamma_1t}\rho_{ee}(0),$$

and

$$\rho_{eg}(t) = e^{-\Gamma_1t/2}\rho_{eg}(0)$$

apart from Hamiltonian phase rotation. The ground-state population increases by the lost excited-state population:

$$\rho_{gg}(t) = 1-\rho_{ee}(t)$$

for a normalized two-level state.

Schrödinger-Picture Coherence

If

$$H = \hbar\omega_0 \lvert e\rangle\langle e\rvert,$$

then the Schrödinger-picture coherence obeys

$$\dot\rho_{eg} = - \left( i\omega_0+\frac{\Gamma_1}{2} \right) \rho_{eg}.$$

Hence

$$\rho_{eg}(t) = e^{-i\omega_0t} e^{-\Gamma_1t/2} \rho_{eg}(0)$$

when no additional dephasing is present. If a pure-dephasing term with rate $\Gamma_\phi$ is also present, the transverse decay rate becomes

$$\frac{1}{T_2} = \frac{\Gamma_1}{2} + \Gamma_\phi.$$

See Pure Dephasing Master Equation for the separate generator that changes coherence without changing populations.

For the broader $T_1$, $T_2$, and protocol-dependent coherence-time bookkeeping, see Decoherence Timescales.

Adding a near-resonant coherent drive to this same two-level relaxation model gives the Optical Bloch Equations.

Bloch-Vector Form

Let the ground state $\lvert g\rangle$ be the north pole, so

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

with $r_z=+1$ for $\lvert g\rangle$ and $r_z=-1$ for $\lvert e\rangle$. In the interaction picture,

$$\begin{aligned} \dot r_x&=-\frac{\Gamma_1}{2}r_x,\\ \dot r_y&=-\frac{\Gamma_1}{2}r_y,\\ \dot r_z&=\Gamma_1(1-r_z). \end{aligned}$$

The fixed point is the ground state. The transverse components contract, while the longitudinal component relaxes toward $+1$.

This affine flow is the continuous-time version of the nonunital Bloch-ball map on the Amplitude-Damping Channel. It is not a Pauli channel and not depolarizing noise.

Finite-Time Channel

Solving the master equation gives a completely positive trace-preserving map. With

$$p(t)=1-e^{-\Gamma_1t},$$

the Kraus operators may be chosen as

$$K_0(t) = \lvert g\rangle\langle g\rvert + \sqrt{1-p(t)}\, \lvert e\rangle\langle e\rvert,$$

and

$$K_1(t) = \sqrt{p(t)}\, \lvert g\rangle\langle e\rvert.$$

This finite-time channel is useful for quantum information noise models, while the master equation is useful when composing with Hamiltonian dynamics, drives, other Lindblad terms, and time-dependent control.

Jump and No-Jump Interpretation

If the emitted quantum is monitored by an ideal photon counter, the jump operator is

$$J=\sqrt{\Gamma_1}\,\sigma_-.$$

During a short interval $dt$, the probability of observing a jump is

$$p_{\mathrm{jump}} = dt\, \operatorname{Tr}(J^\dagger J\rho) = \Gamma_1dt\,\rho_{ee}.$$

After a detected jump, the state updates to

$$\rho \longmapsto \frac{ \sigma_-\rho\sigma_+ }{ \operatorname{Tr}(\sigma_+\sigma_-\rho) }.$$

If no jump is detected, the unnormalized conditional state evolves under the effective Hamiltonian

$$H_{\mathrm{eff}} = H - \frac{i\hbar\Gamma_1}{2} \sigma_+\sigma_-.$$

The no-jump evolution is not the same as closed Hamiltonian evolution; absence of a click is itself information. Averaging the jump and no-jump conditioned updates recovers the unconditional amplitude damping master equation.

Finite Temperature

At nonzero temperature, the bath may also excite the system. The two-level finite-temperature equation is

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \Gamma_\downarrow\mathcal D[\sigma_-]\rho + \Gamma_\uparrow\mathcal D[\sigma_+]\rho.$$

The excited-state population obeys

$$\dot\rho_{ee} = - \Gamma_\downarrow\rho_{ee} + \Gamma_\uparrow\rho_{gg}.$$

The population relaxation time is

$$T_1 = \frac{1} {\Gamma_\downarrow+\Gamma_\uparrow}.$$

The steady excited-state population is

$$\rho_{ee}^{\mathrm{ss}} = \frac{\Gamma_\uparrow} {\Gamma_\uparrow+\Gamma_\downarrow}.$$

For a single thermal bath at inverse temperature $\beta$ and transition frequency $\omega_0>0$, Detailed Balance gives

$$\frac{\Gamma_\uparrow}{\Gamma_\downarrow} = e^{-\beta\hbar\omega_0}.$$

The zero-temperature amplitude damping equation is the limit $\Gamma_\uparrow\to0$.

Harmonic-Oscillator Damping

For a cavity or oscillator mode at zero temperature, amplitude damping is

$$\frac{d\rho}{dt} = - \frac{i}{\hbar} [\hbar\omega a^\dagger a,\rho] + \kappa\mathcal D[a]\rho.$$

The mean occupation obeys

$$\frac{d}{dt} \langle a^\dagger a\rangle = - \kappa \langle a^\dagger a\rangle,$$

while the field amplitude obeys

$$\frac{d}{dt}\langle a\rangle = - \left( i\omega+\frac{\kappa}{2} \right) \langle a\rangle.$$

Thus energy decays at rate $\kappa$, while field amplitude decays at rate $\kappa/2$. Confusing these two rates is a common linewidth-convention error.

At finite temperature, add the thermal excitation term $\kappa\bar n\mathcal D[a^\dagger]\rho$ as described in Thermal Master Equations and Quantum Optical Master Equation.

Microscopic Origin

In a weak-coupling derivation, amplitude damping comes from a transverse system–bath coupling with matrix elements between different energy eigenstates. A schematic interaction is

$$H_I = A\otimes B,$$

where $A$ has components such as

$$A(\omega_0) \propto \sigma_-, \qquad A(-\omega_0) \propto \sigma_+.$$

After Born, Markov, and secular approximations, the bath spectrum at the transition frequency gives the downward rate, and the bath spectrum at the negative transition frequency gives the upward rate.

At zero temperature for an ordinary passive bath, the upward rate vanishes. In structured reservoirs, strong coupling, non-Markovian environments, or driven frames, this simple separation can fail or need modification.

Validity Checks

Before using an amplitude damping master equation, check:

• Which state is excited and which state is lower?
• Is the bath effectively at zero temperature, or are upward transitions needed?
• Is the transition frequency inside a smooth reservoir spectrum?
• Are the Born, Markov, and secular approximations justified?
• Is the system Hamiltonian written in the basis used by $\sigma_-$?
• Are additional pure-dephasing terms separated from $T_1$ decay?
• Is leakage outside the modeled Hilbert space important?
• Are reported linewidths energy-decay rates, amplitude-decay rates, or full widths?

If the environment is monitored, also specify the detection scheme. The same unconditional master equation can correspond to different trajectory descriptions depending on how the environment is measured.

Common Mistakes

Treating amplitude damping as pure dephasing

Amplitude damping changes populations. Pure dephasing does not. Both can reduce coherence, but they represent different physical processes and different Lindblad operators.

Forgetting the half-rate coherence decay

Zero-temperature $T_1$ decay with rate $\Gamma_1$ gives transverse coherence decay $\Gamma_1/2$ before additional pure dephasing. The square root in the finite-time channel is the same fact.

Using zero-temperature damping at finite temperature

If $\Gamma_\uparrow$ is appreciable, the zero-temperature equation predicts the wrong steady state. Use upward and downward rates satisfying the appropriate thermal or nonequilibrium relation.

Confusing damping with leakage

Amplitude damping maps the excited state into a lower state inside the modeled Hilbert space. Leakage into a state outside the computational subspace requires a larger Hilbert space or a leakage channel.

Treating no-jump evolution as no effect

Conditioned no-jump evolution changes the state because the absence of an event updates the observer’s information. The unconditional equation is recovered only after averaging over all records.

Exercises

Population decay

For $\dot\rho=\Gamma_1\mathcal D[\sigma_-]\rho$, derive $\dot\rho_{ee}=-\Gamma_1\rho_{ee}$.

Solution

Use $\sigma_+\sigma_-=\lvert e\rangle\langle e\rvert$ and $\sigma_-\rho\sigma_+=\rho_{ee}\lvert g\rangle\langle g\rvert$. The positive term has no excited-state component:

$$\langle e|\sigma_-\rho\sigma_+|e\rangle=0.$$

The anticommutator gives

$$\left\langle e\left| \frac12\{\sigma_+\sigma_-,\rho\} \right|e\right\rangle = \rho_{ee}.$$

Therefore

$$\dot\rho_{ee} = -\Gamma_1\rho_{ee}.$$

Coherence half-rate

For the same equation, show that $\dot\rho_{eg}=-(\Gamma_1/2)\rho_{eg}$ in the interaction picture.

Solution

The positive term $\sigma_-\rho\sigma_+$ is proportional to $\lvert g\rangle\langle g\rvert$, so it has no $eg$ matrix element. The anticommutator gives

$$\left( \{\sigma_+\sigma_-,\rho\} \right)_{eg} = \rho_{eg}.$$

Thus

$$\dot\rho_{eg} = - \frac{\Gamma_1}{2} \rho_{eg}.$$

Finite-time probability

Solve $\dot\rho_{ee}=-\Gamma_1\rho_{ee}$ and identify the finite-time decay probability $p(t)$ in the amplitude-damping channel.

Solution

The population solution is

$$\rho_{ee}(t) = e^{-\Gamma_1t}\rho_{ee}(0).$$

The probability that an initially excited population has decayed by time $t$ is therefore

$$p(t) = 1-e^{-\Gamma_1t}.$$

Thermal steady state

For finite-temperature rates, solve for the steady excited-state population of

$$\dot\rho_{ee} = - \Gamma_\downarrow\rho_{ee} + \Gamma_\uparrow(1-\rho_{ee}).$$

Solution

Set the derivative to zero:

$$0 = - \Gamma_\downarrow\rho_{ee}^{\mathrm{ss}} + \Gamma_\uparrow \left( 1-\rho_{ee}^{\mathrm{ss}} \right).$$

Rearranging gives

$$(\Gamma_\downarrow+\Gamma_\uparrow) \rho_{ee}^{\mathrm{ss}} = \Gamma_\uparrow.$$

Thus

$$\rho_{ee}^{\mathrm{ss}} = \frac{\Gamma_\uparrow} {\Gamma_\downarrow+\Gamma_\uparrow}.$$

Oscillator linewidth

For $\dot\rho=\kappa\mathcal D[a]\rho$, use the adjoint dissipator to show that $\langle a\rangle$ decays at rate $\kappa/2$.

Solution

The adjoint dissipator is

$$\mathcal D^\dagger[a]O = a^\dagger Oa - \frac12\{a^\dagger a,O\}.$$

For $O=a$, use $[a^\dagger a,a]=-a$ to obtain

$$\mathcal D^\dagger[a]a = - \frac12 a.$$

Therefore

$$\frac{d}{dt}\langle a\rangle = \kappa \langle \mathcal D^\dagger[a]a\rangle = - \frac{\kappa}{2} \langle a\rangle$$

in the interaction picture.

References

• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 3rd ed. (2004).
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2010).
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012).