Time-Resolved Faraday Ellipticity

• Time-resolved pump-probe Faraday ellipticity is a magneto-optical technique that uses a pump pulse to induce non-equilibrium states and a probe to measure resulting ellipticity.
• It captures key phenomena such as carrier scattering, spin precession, and nonlinear optical symmetry breaking across materials like graphene, EuTe, and quantum dots.
• Experimental setups range from Voigt to transmission geometries, emphasizing spectral selectivity, temporal dynamics, and ensemble averaging effects in the observed signals.

Time-resolved pump-probe Faraday ellipticity is a transient magneto-optical observable in which a pump pulse prepares a non-equilibrium state and a delayed linearly polarized probe acquires ellipticity during transmission through the sample. In the literature represented here, the underlying nonequilibrium state may be a carrier distribution in graphene, a spin-polarized or defect-mediated state in EuTe, a resident-electron spin ensemble in quantum dots or colloidal nanostructures, or a transiently chiral electronic state in a transparent dielectric. Faraday ellipticity is treated as the ellipticity component of a complex polarization response complementary to Faraday rotation, and its time dependence encodes carrier scattering, spin precession, spectral selectivity, ensemble dephasing, and nonlinear optical symmetry breaking (Heyman et al., 2014, Glazov et al., 2010, Golovatenko et al., 7 Feb 2025, Wismer et al., 2016).

1. Observable, notation, and constitutive description

In standard magneto-optics, the real part of the complex magneto-optical response is associated with rotation, while the imaginary part is associated with ellipticity. In thin-film or sheet geometries this response is commonly expressed through the longitudinal and off-diagonal conductivity components, and for a magnetized conducting sheet the transmitted polarization state is controlled by the complex ratio of orthogonal field components,

$$\frac{E_{\perp}}{E_{\parallel}} \sim -\frac{\sigma_{xy}}{\sigma_{xx}},$$

so that the same conductivity tensor that determines Faraday rotation also determines Faraday ellipticity (Heyman et al., 2014).

A complementary formulation uses circular probe components. For an initially linearly polarized probe,

$$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$

and a pump-induced difference between the effective susceptibilities of the two helicities,

$\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$

produces circular birefringence when the phases differ and circular dichroism when the imaginary parts differ. The latter is the direct route to ellipticity (Wismer et al., 2016).

In trion-based semiconductor formulations, the transient probe response is written as a complex signal

$\mathcal E+i\mathcal F,$

with $\mathcal E$ identified as Faraday ellipticity and $\mathcal F$ as Faraday rotation. This notation makes explicit that the two observables are different projections of the same pump-induced optical anisotropy rather than independent phenomena (Golovatenko et al., 7 Feb 2025).

2. Experimental configurations and readout geometries

A recurring implementation uses a circularly polarized pump and a linearly polarized probe in Voigt geometry, with the magnetic field orthogonal to light propagation. This configuration appears in singly charged $(\mathrm{In},\mathrm{Ga})$As/GaAs quantum dots, in n-GaAs time-resolved Faraday measurements, and in singly charged colloidal CdSe nanoplatelets and nanocrystals. By contrast, the graphene THz experiment places the magnetic field normal to the film and parallel to the beam direction, and the EuTe work uses transmission geometry with $\mathbf{k}\parallel(111)\perp\mathbf{H}$. A distinct all-optical configuration replaces the external magnetic field by a strong circularly polarized near-IR pump and probes the induced chirality with a weak linearly polarized ultraviolet pulse (Glazov et al., 2010, Trowbridge et al., 2015, Golovatenko et al., 7 Feb 2025, Heyman et al., 2014, Pavlov et al., 2018, Wismer et al., 2016).

System	Pump-probe arrangement	Principal readout
p-type CVD graphene	800 nm optical pump and polarization-sensitive THz transmission in a magnetic field normal to the film	$\Delta T_{\parallel}$, $\Delta T_{\perp}$, $$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$0, $$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$1 (Heyman et al., 2014)
EuTe thin films	Degenerate transmission geometry at $$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$2, up to 6 T in Voigt geometry	Pump-induced rotation angle $$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$3 (Pavlov et al., 2018)
$$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$4As/GaAs quantum dots	Two synchronized Ti:Sapphire lasers, circular pump, linearly polarized probe, Voigt geometry	Faraday rotation and ellipticity versus delay and detuning (Glazov et al., 2010)
Transparent dielectric	Circular $$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$5, $$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$6 near-IR pump and linear $$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$7, $$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$8 UV probe	Pump-induced rotation $$\mathbf{f}^{\mathrm{UV}}=\mathbf{e}_+ f_+^{\mathrm{UV}}+\mathbf{e}_- f_-^{\mathrm{UV}}, \qquad \mathbf{e}_\pm=\frac{\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}}{\sqrt{2}},$$9 and ellipticity $\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$0 (Wismer et al., 2016)

The readout can be framed directly in terms of the generated orthogonal probe polarization. In the ultraviolet optical Faraday-effect model, for an initially $\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$1-polarized probe the leading propagation relation is

$\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$2

so the experimentally accessible polarization change is sourced by the pump-induced nonlinear polarization orthogonal to the incident probe polarization (Wismer et al., 2016).

3. Ultrafast temporal regimes and microscopic origins

In graphene, the Faraday response is tied to magneto-transport. The conductivity tensor satisfies

$\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$3

and pump-probe measurements separately resolve changes in carrier density and scattering time through

$\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$4

At $\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$5 following photoexcitation with $\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$6 pulses at $\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$7, the effective hole scattering time decreased from $\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$8 to $\chi_\pm^{\mathrm{eff}}=\chi^{(1)}+\Delta\chi_\pm,$9, while the carrier concentration increased from $\mathcal E+i\mathcal F,$0 to $\mathcal E+i\mathcal F,$1, producing a transient decrease in conductivity; the recovery time constant of the conductivity change was $\mathcal E+i\mathcal F,$2 (Heyman et al., 2014).

In EuTe, the temporal structure separates into two regimes. At low magnetic field, the transient is dominated by a narrow peak centered at zero delay and assigned to the inverse Faraday effect,

$\mathcal E+i\mathcal F,$3

Above about $\mathcal E+i\mathcal F,$4, a slower picosecond tail appears and is attributed to optical orientation; the reported spin-relaxation time is

$\mathcal E+i\mathcal F,$5

The crossover is controlled by a magnetic-field-induced redshift of the absorption edge by about $\mathcal E+i\mathcal F,$6 between $\mathcal E+i\mathcal F,$7 and $\mathcal E+i\mathcal F,$8, bringing the EuTe band gap near the fixed photon energy $\mathcal E+i\mathcal F,$9 (Pavlov et al., 2018).

In transparent solids, the nonlinear optical Faraday response is confined to pump-probe overlap. The delay dependence of the weak-field rotation follows

$\mathcal E$0

and the induced chirality vanishes for probe delays exceeding the duration of the near-infrared pump pulse. The weak-field scaling reported there is

$\mathcal E$1

indicating that ellipticity is especially sensitive to higher-order nonlinear mixing and multiphoton absorption channels (Wismer et al., 2016).

These examples show that time-resolved Faraday ellipticity can originate from distinct microscopic mechanisms: transport-driven conductivity changes, prompt coherent nonlinear polarization, resonantly enhanced spin orientation, or transient chirality. A plausible implication is that interpretation must be mechanism-specific even when the laboratory observable is formally the same.

4. Spectral selectivity, pump-probe detuning, and the distinction from Faraday rotation

Faraday ellipticity and Faraday rotation need not have the same temporal profile. In singly charged $\mathcal E$2As/GaAs quantum dots, nearly degenerate pump and probe conditions produce a regular decay of the ellipticity signal with pump-probe delay, while the Faraday rotation amplitude first grows with increasing $\mathcal E$3 and only later decays. For larger detuning, such as $\mathcal E$4, this growth disappears and both signals become conventionally decaying. The measured response is described by a spectrally weighted convolution,

$\mathcal E$5

which makes explicit that the probe does not measure spin directly but rather a convolution of the spin ensemble with the optical response function $\mathcal E$6 (Glazov et al., 2010).

The distinction comes from spectral parity. In the quantum-dot analysis, ellipticity is sensitive mainly to the even spectral part of the optical response and therefore reflects the average spin polarization, whereas Faraday rotation is sensitive to the odd spectral part and therefore reflects asymmetry in the spin distribution relative to the probe frequency. For degenerate pump and probe, the initial spin distribution is symmetric around the probe resonance, so Faraday rotation is initially small; as time evolves, the dependence of the electron $\mathcal E$7-factor on transition energy creates asymmetry and the Faraday rotation can increase before dephasing dominates (Glazov et al., 2010).

The same separation appears in colloidal CdSe ensembles. There the complex trion-based signal again takes the form $\mathcal E$8, and the paper emphasizes that $\mathcal E$9 and $\mathcal F$0 differ because the real and imaginary parts of the spectral sensitivity $\mathcal F$1 are different. As a consequence, Faraday rotation can be very small or even vanish at exact pump-probe resonance, while ellipticity remains prominent (Golovatenko et al., 7 Feb 2025).

This distinction corrects a common simplification in the pump-probe literature: Faraday ellipticity is not merely a redundant alternative to Faraday rotation. In the systems discussed here, it samples a different spectral weighting of the same nonequilibrium ensemble.

5. Ensemble averaging, spin precession, and repetition-rate effects

In anisotropic colloidal nanoplatelets and nanocrystals, time-resolved Faraday ellipticity depends not only on spin dynamics but also on orientation disorder and $\mathcal F$2-tensor anisotropy. For a magnetic field applied along the laboratory $\mathcal F$3-axis, the precession frequency for an individual tilted nanostructure is

$\mathcal F$4

and the spin component relevant for optical readout evolves as

$\mathcal F$5

The nonoscillating term arises whenever $\mathcal F$6, so randomly oriented ensembles generically show both an oscillating contribution and an offset. The paper distinguishes two damping mechanisms: anisotropy plus random orientation causes only partial damping of oscillation amplitude, whereas $\mathcal F$7-factor dispersion can fully quench the oscillations on nanosecond timescales. Its central conclusion is that, regardless of the anisotropy degree, the oscillation frequency of the Faraday rotation and ellipticity signals for a randomly oriented ensemble is determined by the transverse electron $\mathcal F$8-factor component $\mathcal F$9 (Golovatenko et al., 7 Feb 2025).

Long-lived spin systems introduce a second ensemble effect: coherent accumulation over the entire pump-pulse train. When the spin lifetime is comparable to or longer than the repetition period, the measured signal is the phasor sum of all previous pulses,

$(\mathrm{In},\mathrm{Ga})$0

which reduces to

$(\mathrm{In},\mathrm{Ga})$1

with

$(\mathrm{In},\mathrm{Ga})$2

$(\mathrm{In},\mathrm{Ga})$3

In n-GaAs, fitting this model gave $(\mathrm{In},\mathrm{Ga})$4, $(\mathrm{In},\mathrm{Ga})$5, optical-carrier lifetime $(\mathrm{In},\mathrm{Ga})$6, and $(\mathrm{In},\mathrm{Ga})$7, while an experimentally observed maximum phase angle of $(\mathrm{In},\mathrm{Ga})$8 implied a lifetime of $(\mathrm{In},\mathrm{Ga})$9 (Trowbridge et al., 2015).

Together, these results show that the measured ellipticity can be shaped by orientation averaging in a single pump-probe cycle and by coherent memory across many cycles. This suggests that time-domain fitting which ignores either effect can misattribute phase shifts or damping to intrinsic dephasing alone.

6. Inverse Faraday effect, light-induced Faraday effect, and conceptual boundaries

A persistent interpretive issue is whether a pump-probe Faraday-ellipticity signal necessarily implies a real magnetization. The works considered here answer this negatively in some regimes. In EuTe, the prompt component is explicitly attributed to the inverse Faraday effect, while the slower component is assigned to optical orientation; the two processes have different temporal profiles and different spectral conditions, so the same measured rotation-like transient can mix coherent nonlinear response and spin-relaxation dynamics (Pavlov et al., 2018).

For transparent solids, the predicted ultrafast optical Faraday effect is described as too large to be explained by any realistic magnetic field. It arises from transient chirality induced by a strong circularly polarized pump, not from magnetization, and depends on the non-instantaneity of the nonlinear medium response. The effect exists only while the pump field is present, reinforcing that the observable polarization change can be a purely dynamical optical property of the driven medium (Wismer et al., 2016).

This point is sharpened by the later theory of the light-induced Faraday effect from dynamical breakdown of Kleinman symmetry. There the balanced-detection pump-probe signal is proportional to the nonlinear polarization orthogonal to the probe,

$\mathbf{k}\parallel(111)\perp\mathbf{H}$0

and the quasi-static component generated by a circular pump is

$\mathbf{k}\parallel(111)\perp\mathbf{H}$1

The antisymmetric susceptibility $\mathbf{k}\parallel(111)\perp\mathbf{H}$2 vanishes under Kleinman symmetry and obeys the low-frequency scaling

$\mathbf{k}\parallel(111)\perp\mathbf{H}$3

so the response is inherently dynamical. The paper states that a static polarization rotation can originate entirely from this antisymmetric component of the third-order susceptibility, without generating a macroscopic magnetization; the corresponding effective magnetic-field analogy is explicitly described as fictitious and confined to the optical response of the medium. In its minimal model, $\mathbf{k}\parallel(111)\perp\mathbf{H}$4, and resonant phonon coupling can significantly enhance the pump-probe response (Sellati et al., 26 May 2026).

The conceptual boundary is therefore system- and mechanism-dependent. In some materials, time-resolved Faraday ellipticity directly tracks spin polarization and Larmor precession; in others, it reads out a transiently chiral nonlinear susceptibility. The experimental observable is the same class of polarization change, but the microscopic meaning need not be.