Optical Bloch Equations

The optical Bloch equations are the Bloch-vector equations for a driven two-level system with relaxation and dephasing. They are the workhorse model for driven atoms, resonance fluorescence, saturation spectroscopy, driven qubits, and many AMO control experiments.

They combine three ingredients:

coherent drive          Rabi frequency and detuning
population relaxation  spontaneous emission or T1 decay
transverse decay        relaxation plus pure dephasing

The equations are “optical” by history, but the same structure appears for microwave-driven superconducting qubits, spin resonance, color centers, trapped ions, and any controlled two-level transition whose environment is well approximated by a Markovian master equation. The closed-system finite-pulse limit and the distinction between Fourier, homogeneous, and drive-induced widths are organized in Resonant Driving.

Two-Level Atom owns the multilevel-to-two-state reduction and the closed rotating-frame Hamiltonian. This page begins after that reduction and adds relaxation and dephasing.

Rabi Oscillations develops the complementary experimental trace, pulse-area, readout, and chevron diagnostics, while this page owns the full Markovian equations and their steady state.

The AMO-facing Optical Bloch Equations uses the chapter convention $\Delta=\omega_0-\omega_L$ and owns the observable dictionary: atomic polarization, fluorescence and count models, saturation intensity, power broadening, branching, and spatial averaging. This page retains the general Lindblad derivation and unconditional open-system interpretation.

The Optical Bloch Equation Notebook implements the AMO sign convention and verifies transients, steady states, saturation, fluorescence, power broadening, RK4 convergence, and Bloch-ball physicality. Its declared detuning is the negative of the local convention used below.

Starting Master Equation

Let $\lvert g\rangle$ and $\lvert e\rangle$ be ground and excited states, with

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

and

$$\sigma_z = \lvert e\rangle\langle e\rvert - \lvert g\rangle\langle g\rvert.$$

In a frame rotating at the drive frequency $\omega_L$, use the convention

$$\Delta=\omega_L-\omega_0,$$

where $\omega_0$ is the transition frequency. A common rotating-frame Hamiltonian is

$$H_{\mathrm{rot}} = - \hbar\Delta\,\sigma_+\sigma_- + \frac{\hbar\Omega}{2} (\sigma_+ + \sigma_-),$$

where $\Omega$ is the Rabi frequency in this convention.

The corresponding Markovian master equation is

$$\dot\rho = - \frac{i}{\hbar} [H_{\mathrm{rot}},\rho] + \Gamma\,\mathcal D[\sigma_-]\rho + \frac{\gamma_\phi}{2} \mathcal D[\sigma_z]\rho,$$

with

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

Here $\Gamma$ is the population decay rate from $\lvert e\rangle$ to $\lvert g\rangle$, and $\gamma_\phi$ is the additional pure-dephasing rate. The transverse decay rate is

$$\Gamma_2 = \frac{\Gamma}{2} + \gamma_\phi.$$

This convention matches the driven two-level starting point in Quantum Optical Master Equation. Other books may define $\Delta$ with the opposite sign or absorb factors of $2$ into $\Omega$.

Density-Matrix Equations

Write the density matrix in the $\{\lvert e\rangle,\lvert g\rangle\}$ basis. The excited-state population satisfies

$$\dot\rho_{ee} = - \Gamma\rho_{ee} - i\frac{\Omega}{2} (\rho_{ge}-\rho_{eg}).$$

The ground-state population is fixed by normalization:

$$\rho_{gg}=1-\rho_{ee}.$$

The optical coherence obeys

$$\dot\rho_{eg} = - (\Gamma_2-i\Delta)\rho_{eg} - i\frac{\Omega}{2} (\rho_{gg}-\rho_{ee}).$$

The conjugate equation gives $\dot\rho_{ge}$. These equations show the basic competition:

• the drive converts population imbalance into coherence;
• the coherence drives population transfer;
• $\Gamma$ removes excited population;
• $\Gamma_2$ damps the optical coherence;
• detuning rotates coherence in the transverse plane.

When $\Gamma=\gamma_\phi=0$, these equations reduce to coherent Rabi dynamics. The closed-system first encounter is Rabi Oscillations.

Bloch-Vector Form

Define

$$u=\rho_{eg}+\rho_{ge}, \qquad v=i(\rho_{eg}-\rho_{ge}),$$

and

$$w=\rho_{ee}-\rho_{gg}.$$

Then $\rho_{ee}=(1+w)/2$, and the optical Bloch equations become

$$\begin{aligned} \dot u &= - \Gamma_2 u + \Delta v, \\ \dot v &= - \Gamma_2 v - \Delta u - \Omega w, \\ \dot w &= \Omega v - \Gamma(w+1). \end{aligned}$$

The ground state has $w=-1$. The excited state has $w=+1$. The drive rotates the Bloch vector, while the dissipative terms contract the transverse components and pull $w$ toward $-1$.

This is an affine flow, not just a rotation. The origin of the Bloch ball is not generally fixed because spontaneous emission is nonunital.

Relation to T1 and T2

With the conventions above,

$$T_1=\frac{1}{\Gamma}, \qquad T_2=\frac{1}{\Gamma_2}.$$

Thus

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

This is the same bookkeeping relation used in Decoherence Timescales. The optical Bloch equations are the driven version of that Markovian qubit model. Their magnetic-resonance interpretation, including CW saturation, pulsed signals, and the distinction between $T_2$ and $T_2^*$, is organized in Magnetic Resonance Overview.

Steady State

Set the time derivatives to zero. From

$$0=-\Gamma_2u+\Delta v,$$

one has

$$u_{\mathrm{ss}} = \frac{\Delta}{\Gamma_2}v_{\mathrm{ss}}.$$

The population equation gives

$$w_{\mathrm{ss}} = \frac{\Omega}{\Gamma}v_{\mathrm{ss}}-1.$$

Solving the three equations gives

$$\rho_{ee}^{\mathrm{ss}} = \frac{ \Omega^2\Gamma_2 }{ 2\left[ \Gamma(\Gamma_2^2+\Delta^2) + \Omega^2\Gamma_2 \right] }.$$

Equivalently,

$$w_{\mathrm{ss}} = - \frac{ \Gamma(\Gamma_2^2+\Delta^2) }{ \Gamma(\Gamma_2^2+\Delta^2) + \Omega^2\Gamma_2 }.$$

The emitted photon rate in an unconditioned fluorescence model is

$$R_{\mathrm{sc}} = \Gamma\rho_{ee}^{\mathrm{ss}}.$$

This is the basic saturation curve behind many spectroscopy and laser-cooling estimates.

No Extra Pure Dephasing

If $\gamma_\phi=0$, then $\Gamma_2=\Gamma/2$ and

$$\rho_{ee}^{\mathrm{ss}} = \frac{\Omega^2} {\Gamma^2+4\Delta^2+2\Omega^2}.$$

On resonance, $\Delta=0$:

$$\rho_{ee}^{\mathrm{ss}} = \frac{\Omega^2} {\Gamma^2+2\Omega^2}.$$

For weak drive, $\Omega\ll\Gamma$,

$$\rho_{ee}^{\mathrm{ss}} \approx \frac{\Omega^2}{\Gamma^2}.$$

For strong resonant drive, $\Omega\gg\Gamma$,

$$\rho_{ee}^{\mathrm{ss}} \to \frac12.$$

The two-level atom saturates at equal populations. It does not reach $\rho_{ee}=1$ in the steady state under continuous resonant driving with spontaneous emission.

Saturation Parameter

It is often useful to define a saturation parameter

$$s = \frac{ \Omega^2\Gamma_2 }{ \Gamma(\Gamma_2^2+\Delta^2) }.$$

Then

$$\rho_{ee}^{\mathrm{ss}} = \frac{s}{2(1+s)}.$$

When $\gamma_\phi=0$, this becomes

$$s = \frac{ 2\Omega^2 }{ \Gamma^2+4\Delta^2 }.$$

The precise definition of $s$ varies with the convention for $\Omega$ and $\Delta$. The invariant content is the steady-state population formula derived from the master equation.

Transient Regimes

The transient solution depends on the relative sizes of $\Omega$, $\Delta$, $\Gamma$, and $\gamma_\phi$.

Weak drive

When the drive is weak, the atom remains mostly in the ground state. The coherence follows the drive approximately linearly, and the steady-state excitation has a Lorentzian dependence on detuning:

$$\rho_{ee}^{\mathrm{ss}} \approx \frac{ \Omega^2\Gamma_2 }{ 2\Gamma(\Gamma_2^2+\Delta^2) }.$$

The linewidth is controlled by $\Gamma_2$ in this convention.

Strong drive

When $\Omega$ is large enough, the system undergoes damped Rabi oscillations before reaching steady state. The oscillations are not perfectly periodic because spontaneous emission and dephasing continually remove phase information.

Large detuning

When $|\Delta|$ is large compared with $\Omega$ and $\Gamma_2$, excitation is suppressed. The drive mainly produces an AC Stark shift and small virtual admixture rather than strong population transfer. A two-level optical Bloch model may still be useful, but off-resonant couplings to other levels may become important in real atoms and molecules.

Conditional Versus Unconditional Pictures

The optical Bloch equations are unconditional equations. They describe the density operator after emitted photons, scattered photons, or detector records have been ignored.

If photons are monitored, individual runs are better described by quantum trajectories:

• no detected photon gives non-Hermitian conditional evolution;
• a detected photon applies a jump proportional to $\sigma_-$;
• averaging many records recovers the optical Bloch master equation.

For the trajectory interpretation, see Quantum-Jump Trajectories. For input–output fields, see Input–Output Theory.

Common Mistakes

Mixing detuning signs

Some authors define $\Delta=\omega_0-\omega_L$ rather than $\omega_L-\omega_0$. This flips signs in the coherence equations. Always translate from the Hamiltonian.

Comparing Rabi frequencies without checking factors

Some conventions write the drive Hamiltonian as $\hbar\Omega\sigma_x$ rather than $\hbar\Omega\sigma_x/2$. The same symbol $\Omega$ can differ by a factor of $2$.

Setting T2 equal to T1

Population relaxation contributes $\Gamma/2$ to transverse decay. Extra pure dephasing adds more. In general $T_2$ and $T_1$ are not equal.

Expecting full inversion from a resonant continuous drive

The steady state saturates at $\rho_{ee}=1/2$ in the ideal two-level spontaneous-emission model. Pulsed coherent control can invert the system, but the continuous dissipative steady state does not.

Treating the equations as microscopic proof of Markovianity

The optical Bloch equations already assume a Markovian reduced description. Structured reservoirs, strong coupling, multilevel leakage, nonradiative channels, or slow noise can require a different model.

Confusing unconditional fluorescence with a single run

The smooth optical Bloch solution is an ensemble average. A monitored single emitter produces stochastic records.

Exercises

1. Bloch equations from matrix equations. Starting from the density-matrix equations, derive

$$\dot w=\Omega v-\Gamma(w+1).$$

Solution

Since $w=\rho_{ee}-\rho_{gg}=2\rho_{ee}-1$ for a normalized two-level state,

$$\dot w=2\dot\rho_{ee}.$$

Using

$$\dot\rho_{ee} = - \Gamma\rho_{ee} - i\frac{\Omega}{2} (\rho_{ge}-\rho_{eg}),$$

and

$$v=i(\rho_{eg}-\rho_{ge}),$$

one finds

$$\dot w = - 2\Gamma\rho_{ee} + \Omega v.$$

Because $\rho_{ee}=(1+w)/2$,

$$\dot w = \Omega v-\Gamma(w+1).$$

2. Resonant steady state. With $\Delta=0$ and $\gamma_\phi=0$, show that

$$\rho_{ee}^{\mathrm{ss}} = \frac{\Omega^2}{\Gamma^2+2\Omega^2}.$$

What are the weak- and strong-drive limits?

Solution

Set $\Gamma_2=\Gamma/2$ and $\Delta=0$ in the general formula:

$$\rho_{ee}^{\mathrm{ss}} = \frac{ \Omega^2(\Gamma/2) }{ 2\left[ \Gamma(\Gamma^2/4) + \Omega^2(\Gamma/2) \right] }.$$

Canceling $\Gamma$ gives

$$\rho_{ee}^{\mathrm{ss}} = \frac{\Omega^2}{\Gamma^2+2\Omega^2}.$$

For $\Omega\ll\Gamma$,

$$\rho_{ee}^{\mathrm{ss}} \approx \frac{\Omega^2}{\Gamma^2}.$$

For $\Omega\gg\Gamma$,

$$\rho_{ee}^{\mathrm{ss}} \to \frac12.$$

3. Line shape in the weak-drive limit. Show that for weak drive,

$$\rho_{ee}^{\mathrm{ss}} \approx \frac{ \Omega^2\Gamma_2 }{ 2\Gamma(\Gamma_2^2+\Delta^2) }.$$

What sets the detuning scale of the line?

Solution

In the exact steady-state expression,

$$\rho_{ee}^{\mathrm{ss}} = \frac{ \Omega^2\Gamma_2 }{ 2\left[ \Gamma(\Gamma_2^2+\Delta^2) + \Omega^2\Gamma_2 \right] },$$

weak drive means the saturation term $\Omega^2\Gamma_2$ is small compared with $\Gamma(\Gamma_2^2+\Delta^2)$. Dropping it in the denominator gives the stated expression. The detuning dependence is Lorentzian, with scale set by $\Gamma_2$ in this convention.

4. Pure dephasing and saturation. At fixed $\Omega$, $\Gamma$, and $\Delta=0$, how does increasing $\gamma_\phi$ affect the steady-state excited population?

Solution

At resonance,

$$\rho_{ee}^{\mathrm{ss}} = \frac{ \Omega^2\Gamma_2 }{ 2\left[ \Gamma\Gamma_2^2 + \Omega^2\Gamma_2 \right] } = \frac{\Omega^2}{2(\Gamma\Gamma_2+\Omega^2)}.$$

Increasing $\gamma_\phi$ increases $\Gamma_2$, so the denominator increases and $\rho_{ee}^{\mathrm{ss}}$ decreases. Pure dephasing broadens the coherence and reduces the resonant steady-state excitation for fixed drive strength.

5. Sign convention check. If another book defines $\Delta'=\omega_0-\omega_L$, how should you translate the Hamiltonian and the Bloch equations?

Solution

The detunings are related by

$$\Delta'=-\Delta.$$

Every occurrence of $\Delta$ in the Hamiltonian and Bloch equations should be replaced by $-\Delta'$. For example,

$$\dot u=-\Gamma_2u+\Delta v$$

becomes

$$\dot u=-\Gamma_2u-\Delta' v.$$

Physical predictions are unchanged if the translation is made consistently.

References

• L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover (1987).
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Wiley (1992).
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 3rd ed. (2004).
• H. J. Carmichael, Statistical Methods in Quantum Optics 1, Springer (1999).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).