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 $\Gamma_1\ge0$ is the population-relaxation rate. The explicit solutions below assume an undriven Hamiltonian diagonal in $g,e$; adding a noncommuting drive changes the finite-time channel. The finite-time channel generated by this equation is the Amplitude-Damping Channel with decay probability $p(t)=1-e^{-\Gamma_1t}$.

Helpful background. The amplitude-damping channel describes one interval, while Lindblad operators explain the continuous generator. Solving Lindblad equations compares numerical evolution with this exact solution.

Energy relaxation and the excited-state amplitude

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.

In a joint system–environment or unnormalized no-emission description, the excited-state amplitude decays at half the population-relaxation rate. The unconditional system state is generally mixed and has no single state vector whose amplitude describes the whole evolution. 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 times, 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

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

in the interaction picture 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

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

For Schrödinger-picture evolution, multiply both Kraus operators on the left by $e^{-iHt/\hbar}$. For an initial $|+\rangle=(|g\rangle+|e\rangle)/\sqrt2$ and $t=\ln2/\Gamma_1$, the interaction-picture state is

$$\rho(t)=\begin{pmatrix}3/4&1/(2\sqrt2)\\1/(2\sqrt2)&1/4\end{pmatrix}_{g,e}.$$

Its purity is $7/8$: damping has created mixing at this intermediate time, even though the eventual ground state is pure. The energy has fallen from $\hbar\omega_0/2$ to $\hbar\omega_0/4$.

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

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

For a positive-probability 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

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

For the same $|+\rangle$ example at $t=\ln2/\Gamma_1$, the probability of no emission is $3/4$. Conditional on that result, the normalized state is $(|g\rangle+|e\rangle/\sqrt2)/\sqrt{3/2}$, with excited population $1/3$. The remaining probability $1/4$ gives an emission and the ground state. Averaging returns the unconditional excited population $1/4$, not $1/3$.

These no-detection formulas assume unit detection efficiency and no other unmonitored channels. With missed emissions, absence of a click produces a mixture that includes the missed-jump branch.

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

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

For a positive total rate and a diagonal undriven Hamiltonian, 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

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

For finite occupation moments and a well-defined oscillator generator, 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, replace the loss coefficient by $\kappa(\bar n+1)$ and add $\kappa\bar n\mathcal D[a^\dagger]\rho$. Adding gain alone while keeping loss at $\kappa$ would give the wrong thermal state and can even cause unbounded heating. The paired thermal rates are 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.

Damping within a subspace versus leakage

The qubit model places the decay product $|g\rangle$ inside the modeled Hilbert space. A transition to a third level outside a computational subspace is leakage. Either retain that level in a larger trace-preserving model or explicitly describe the trace-decreasing projected state and any conditioning on survival. Renormalizing the projected state changes the prediction to a conditional one.

The microscopic rate also assumes an appropriate reservoir spectrum. A narrow unretained mode, band edge, or appreciable bath return can produce nonexponential relaxation; a constant $\Gamma_1$ should not be inferred from the word “damping” alone.

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

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• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press (2010).
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012). Author manuscript.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).
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