FR16 · Recover a fading optical coherence

Photon Echo Pulse Sequence

How can three light pulses make a dephased ensemble emit an echo? The rephasing pathway reverses the first interval of phase accumulation. Fixed frequency differences then cancel after an equal second interval, producing an echo whose strength still decays through irreversible dephasing.

Move an optical delay and watch three pulses meet in a sample. Recover the echo field with phase cycling, then transform the saved quadrature samples into a spectrum.

Refocusing time after pulse 3 $0.750\,\mathrm{ps}$

Recorded peak field $0.60$

Selected angular frequency spread $6.44\,\mathrm{ps}^{-1}$

Retained complex samples $201$

Let the ensemble rephase

Coherence pathway

First coherence delay

0.75

t_1/\mathrm{ps}

Static angular frequency spread

7

\Delta/\mathrm{ps}^{-1}

Homogeneous dephasing rate

0.22

\Gamma/\mathrm{ps}^{-1}

Drag the glowing delay carriage along its rail to move the echo.

A synthetic phase-cycled recording is loaded.

Pulse width, waiting & receiver

Pulse field RMS duration

0.035

\sigma_t/\mathrm{ps}

Population waiting time

0.3

T/\mathrm{ps}

Receiver angular frequency offset

3

\delta_0/\mathrm{ps}^{-1}

Raw quadrature noise RMS

0.035

\sigma_n

Noise seed

4616

s

The carrier follows the mean resonance. Receiver offset rotates the measured quadratures. Pulse width selects frequencies; overlap dynamics are omitted.

Phase cycling & saved samples

Sixteen phase steps reject constant and first-order leakage. Each retained point comes from the same raw complex samples exported below.

P_{\mathrm R}\propto e^{-i\phi_1+i\phi_2+i\phi_3},\qquad P_{\mathrm{NR}}\propto e^{i\phi_1-i\phi_2+i\phi_3}

Continue with the response-function course

Model assumptions

This is a weak third-order two-level saturation response with static Gaussian frequency disorder, exponential homogeneous dephasing and a three-picosecond population lifetime. Pulse widths enter through Gaussian spectral selection; pulse overlap and propagation are omitted. The carrier follows the ensemble mean; the receiver offset changes only the rotating frame. A calibrated complex heterodyne receiver adds independent Gaussian noise. The optical paths and delay travel are schematically magnified. The sample dipoles display a magnified rotating-frame coherence with the optical carrier removed. All recordings are synthetic. Vibrational excited-state absorption and its harmonic-limit cancellation are treated in the coupled-vibration Lab.

Synthetic heterodyne recording

The echo is in the saved electric field

Inspect time after pulse 3

0.750\\,\\mathrm{ps}

Phase-cycled field quadratures

Cyan: real · coral: imaginary · dots: extracted samples

Fourier transform of that recording

Positive transform sign · frequency offset in terahertz

Watch the ensemble phases

These arrows show model phases, separate from the noisy detector recording. In the rephasing pathway the static frequency differences cancel at the first delay; irreversible dephasing still reduces the field.

Start with the essentials

Focus question: How can three light pulses make a dephased ensemble emit an echo?
One-sentence intuition: The rephasing pathway reverses the sign of the first phase accumulation, so static frequency differences cancel after equal coherence times.

Core mathematical model

Opposite phase signs

P_{\mathrm R}(t_3;t_1,T)=A e^{-\Gamma(t_1+t_3)-T/T_1}e^{-\frac12\Delta_{\mathrm{sel}}^2(t_3-t_1)^2}e^{-i\delta_0(t_3-t_1)}

Times are in picoseconds. Gamma is a coherence decay rate; Delta is an angular frequency standard deviation. The static phase spread cancels at equal coherence times. The peak of the decaying envelope can occur slightly before that refocusing time.

Finite bandwidth selects the ensemble

A=\left(1+3\Delta^2\sigma_t^2\right)^{-1/2},\qquad\Delta_{\mathrm{sel}}=A\Delta

Three identical Gaussian field envelopes weight the static distribution. The pulse carrier tracks its mean. This spectral-selection approximation omits overlap dynamics and is used only for well-separated pulse centres.

Extract the signal from sixteen phase steps

\widehat P_{\mathrm R}(t_3)=\frac1{16}\sum_{m,n=0}^3 E_{mn}(t_3)e^{i(\phi_m-\phi_n)},\qquad\phi_m=\frac{m\pi}2

Each complex sample is phase-rotated before averaging. The rephasing phase survives, whereas a constant background and fields depending on only one pulse phase cancel. The nonrephasing pathway uses the conjugate weight.

Transform the retained observations

\widetilde P(f)=\Delta t\sum_j\widehat P(t_j)e^{+2\pi i f t_j}

The spectrum is a direct discrete Fourier sum of the saved complex observations, with no analytic replacement. Frequency is in cycles per picosecond, equal to terahertz. The finite window broadens features and the sample interval sets the Nyquist limit.

Common difficulties

An echo restores only reversible phase spreading

Typical misconceptionThe echo means the sample has recovered all the coherence it initially had.

Better mental modelFrequency differences that stay fixed during the sequence can cancel. Exponential homogeneous dephasing and population loss continue to reduce the recovered field.

Run the experiment

01
Move the refocusing time
Choose Recover an echo and drag the glowing delay carriage along its rail.
What to observe: The recorded field appears later when the first delay is longer. The ensemble arrows align when the two coherence times match.
02
Remove the phase reversal
Choose No phase reversal, then compare the recorded field with Recover an echo.
What to observe: In the nonrephasing pathway, static frequency differences accumulate with the same sign and the delayed echo disappears.
03
Separate two sources of broadening
Choose Mostly homogeneous. Increase the homogeneous dephasing rate, then restore Recover an echo and increase the static spread.
What to observe: Homogeneous dephasing reduces the field even at refocusing. Static spread mainly narrows the echo when the pulses remain broadband.
04
Keep the actual samples
Open Phase cycling & saved samples, save a comparison and export the raw CSV. Change the noise seed, then restore the saved run.
What to observe: Restoring brings back the same raw phase steps and Fourier spectrum. A new seed changes the noise realization.

Photon Echo Pulse Sequence

The echo is in the saved electric field

Recovering a phase memory

Core mathematical model

Opposite phase signs

Finite bandwidth selects the ensemble

Extract the signal from sixteen phase steps

Transform the retained observations

Common difficulties

An echo restores only reversible phase spreading

Run the experiment

Move the refocusing time

Remove the phase reversal

Separate two sources of broadening

Keep the actual samples

Related experiments and articles

Vibrations, echoes and frequency memory