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.

The coherent drive rotates the state, spontaneous emission lowers its excitation, and pure dephasing shortens transverse coherence. The same model applies to microwave or spin transitions when a two-level reduction and a Markovian reservoir are justified.

Helpful background. Two-level atom explains the reduction to two levels, and Rabi oscillations supplies the closed-system limit. Resonant driving distinguishes finite-pulse and steady-state linewidths.

The AMO optical Bloch equations relate the solution to polarization, fluorescence, intensity, branching, and spatial averaging. That article and the optical Bloch notebook use $\Delta_{\mathrm{AMO}}=\omega_0-\omega_L$, the negative of the detuning below. Translate the Hamiltonian before comparing components. AMO Rabi oscillations explains experimental pulse and readout traces.

Driven two-level 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

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

Choose the drive phase so that $\Omega$ is real. Frequencies and rates below use inverse-time units, with detuning and Rabi frequency as angular frequencies. The rotating-wave and broadband-reservoir approximations must hold over the driven transitions; slow noise or leakage outside the two-level subspace requires a different model.

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 rate 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

Assume $\Gamma>0$ and $\gamma_\phi\ge0$, so $\Gamma_2>0$. The singular closed-system limit has no generic attracting steady state and should be solved separately. 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

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

For detected fluorescence with efficiency $\eta$, the ideal signal rate is $\eta\Gamma\rho_{ee}^{\mathrm{ss}}$, with dark counts and unresolved backgrounds added separately. A photon-rate model is not yet a complete probability law for the correlated counting record.

As a numerical check, choose inverse-time units with $\Gamma=1$, $\gamma_\phi=1/4$, $\Omega=3/2$, and $\Delta=1/2$. Then $\Gamma_2=3/4$ and

$$(u_{\mathrm{ss}},v_{\mathrm{ss}},w_{\mathrm{ss}}) =(0.30,0.45,-0.325),\qquad p_e^{\mathrm{ss}}=0.3375.$$

Substitution into all three differential equations gives zero. The Bloch length squared is $0.398125<1$, as required for a mixed physical state. This sign-sensitive check also gives $\rho_{eg}=(u-iv)/2=0.15-0.225i$.

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 half width at half maximum in angular detuning is $\Gamma_2$; the full width is $2\Gamma_2$. At arbitrary drive strength the steady-state detuning scan has full width $2\sqrt{\Gamma_2^2+\Omega^2\Gamma_2/\Gamma}$ in this model.

Strong drive

At resonance, the two coupled $(v,w)$ transient eigenvalues are

$$\lambda_\pm=-\frac{\Gamma+\Gamma_2}{2} \pm\sqrt{\frac{(\Gamma-\Gamma_2)^2}{4}-\Omega^2}.$$

Damped oscillations occur when $|\Omega|>|\Gamma-\Gamma_2|/2$; below this threshold these modes are overdamped. With no pure dephasing, the threshold is $|\Omega|>\Gamma/4$ and the oscillation frequency is $\sqrt{\Omega^2-\Gamma^2/16}$. Its envelope decays at $3\Gamma/4$. Within this constant-coefficient model the oscillation frequency is fixed; damping does not by itself make it irregular.

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:

• with ideal photon detection and no additional unmonitored channels, no detected photon gives non-Hermitian conditional evolution;
• a detected photon applies a jump proportional to $\sigma_-$;
• unmonitored dephasing or missed photons add conditional mixing; averaging records recovers the optical Bloch master equation.

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

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 half width $\Gamma_2$ and full width $2\Gamma_2$ in angular frequency.

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; reprint of the 1975 Wiley edition).
• H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker–Planck Equations, Springer (1999).
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications, Wiley (1992).
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004). Publisher.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010). DOI.