Ramsey, Decoherence, and Echo

14.1 Separated oscillatory fields and Hahn spin echo

14.1.1 Converting free-evolution phase into population with two pulses

A single population measurement cannot read the equatorial phase \(\phi\) of a superposition \((|0\rangle + e^{i\phi}|1\rangle)/\sqrt{2}\) directly because \(P_1 = 1/2\) for every \(\phi\). Norman Ramsey’s method of separated oscillatory fields turns phase accumulation into a population signal using two resonant \(\pi/2\) pulses separated by a free-precession interval \(\tau\). The first \(\pi/2\) pulse rotates \(|0\rangle\) onto the Bloch equator; during the dark interval \(\tau\), a detuning \(\Delta = \omega - \omega_0\) rotates the equatorial state around the \(Z\) axis by angle \(\phi = \Delta\tau\); the second \(\pi/2\) pulse maps the resulting equatorial projection into the readout population \(P_1(\tau)\).

14.1.2 Inhomogeneous frequency spread versus irreversible environment coupling

As the free-evolution time \(\tau\) increases, the amplitude of Ramsey fringes decays with characteristic transverse coherence time \(T_2^*\). That decay combines two physically distinct mechanisms: reversible inhomogeneous broadening—where static or slowly drifting shifts \(\delta\omega\) vary from shot to shot or across an ensemble of atoms—and irreversible homogeneous dephasing (\(T_2\)) caused by high-frequency fluctuations and entangling interactions with an uncontrolled environment, alongside longitudinal energy relaxation (\(T_1\)).

14.1.3 Refocusing static detuning with a midpoint inversion pulse

Erwin Hahn’s spin-echo sequence separates static frequency offsets from genuinely stochastic dephasing by inserting a resonant \(\pi\) inversion pulse at the midpoint \(t = \tau/2\) of the dark interval. A static detuning \(\Delta\) accumulates phase \(+\Delta\tau/2\) during the first half \([0,\tau/2]\); the \(\pi\) pulse flips the equatorial phase angle \(\phi \mapsto -\phi\) by exchanging \(|0\rangle\) and \(|1\rangle\); during the second half \([\tau/2,\tau]\), the same static detuning adds \(+\Delta\tau/2\), bringing the net detuning phase back to zero at \(t = \tau\). Any fringe decay that remains at the echo time \(\tau\) measures dynamic noise that varied between the first and second halves of the trajectory.

14.2 Transverse decay envelopes and relaxation bounds

14.2.1 Exponential visibility decay, longitudinal relaxation, and pure dephasing

In the rotating frame of the drive, free evolution for duration \(\tau\) followed by a second \(\pi/2\) pulse with relative phase \(\phi_{\mathrm{ref}}\) produces the Ramsey population signal

where \(W(\tau) \in [0,1]\) is the coherence envelope equal to the magnitude of the normalized off-diagonal density-matrix element \(2|\rho_{01}(\tau)|\). For Markovian white-noise fluctuations, \(W(\tau) = \exp(-\tau/T_2)\) decays exponentially, and complete positivity of the quantum channel bounds the transverse dephasing rate \(\Gamma_2 = 1/T_2\) in terms of the longitudinal energy relaxation rate \(\Gamma_1 = 1/T_1\) and the pure dephasing rate \(\Gamma_\phi = 1/T_\phi\):

14.2.2 Gaussian envelopes from static ensemble detuning distributions

When each experimental shot or sub-ensemble member experiences a quasi-static frequency shift \(\delta\omega\) drawn from a zero-mean Gaussian distribution \(p(\delta\omega) = \exp[-\delta\omega^2/(2\sigma_\omega^2)]/\sqrt{2\pi\sigma_\omega^2}\), averaging the phase factor \(e^{i(\Delta + \delta\omega)\tau}\) over \(p(\delta\omega)\) yields a Gaussian envelope

with \(T_2^* = \sqrt{2}/\sigma_\omega\). Combined Markovian and quasi-static noise gives a Voigt-profile envelope \(W(\tau) = \exp(-\tau/T_2 - (\tau/T_2^*)^2)\).

14.2.3 Phase reversal under an ideal echo pulse

With a midpoint \(\pi\) pulse about the \(X\) axis at \(\tau/2\), a static frequency offset \(\Delta + \delta\omega\) evolves the initial superposition \((|0\rangle + |1\rangle)/\sqrt{2}\) through

Up to a global phase, every static \(\delta\omega\) refocuses at \(t = \tau\), extending the decay time from the inhomogeneous \(T_2^*\) to the homogeneous Hahn-echo coherence time \(T_{2\mathrm{E}} \le 2T_1\).

14.3 Noise spectroscopy and reduced-state interpretation

14.3.1 Fitting envelope shapes and non-Markovian revivals

More generally, for stationary Gaussian phase noise with power spectral density \(S_\omega(\omega)\), the coherence envelope obeys \(W(\tau) = \exp[-\chi(\tau)]\), where the attenuation integral \(\chi(\tau) = \frac{1}{2\pi}\int_0^\infty S_\omega(\omega)\,[F(\omega\tau)/\omega^2]\,\mathrm{d}\omega\) weights the noise spectrum by the sequence filter function \(F(\omega\tau)\). Free induction decay (Ramsey) has \(F_{\mathrm{FID}}(\omega\tau) = 4\sin^2(\omega\tau/2)\), which peaks at \(\omega = 0\) and is maximally sensitive to DC drift, whereas Hahn echo has \(F_{\mathrm{echo}}(\omega\tau) = 8\sin^4(\omega\tau/4)\), which vanishes at \(\omega = 0\) and acts as a bandpass filter centered near \(\omega \sim 2\pi/\tau\). If the qubit couples to a discrete coherent bath mode (such as a nearby nuclear spin), \(W(\tau)\) can exhibit collapse and revival oscillations rather than monotonic decay.

14.3.2 Shrinking Bloch vectors as mixed reduced states

When the fringe visibility \(W(\tau)\) decays toward zero, the reduced qubit density matrix at the end of the free-evolution window approaches

Geometrically, the Bloch vector shrinks radially inward along the equatorial plane from length \(|\vec{r}| = 1\) (pure state on the surface) toward \(|\vec{r}| = 0\) (maximally mixed state at the center of the ball).

14.3.3 Operational scope of coherence times

Because \(T_2^*\), \(T_{2\mathrm{E}}\), and dynamical-decoupling coherence times \(T_2^{(N)}\) correspond to different filter functions \(F(\omega\tau)\) and different ensemble averaging windows, quoting a single number “\(T_2\)” without specifying the pulse sequence and averaging time is incomplete. An operational report always pairs the fitted decay constant with the pulse protocol (\(T_2^*\) for Ramsey, \(T_{2\mathrm{E}}\) for single-pulse Hahn echo) and the exponent \(p\) of the stretched-exponential envelope \(\exp[-(\tau/T_2)^p]\).

Explore the Apparatus

Adjust the detuning \(\Delta\), the free-evolution time \(\tau\), and the inhomogeneous and homogeneous dephasing rates to compare Ramsey free-induction fringes against the Hahn spin-echo envelope. Observe how switching the midpoint \(\pi\) pulse on cancels quasi-static frequency offsets and restores coherence up to the irreversible \(T_{2\mathrm{E}}\) ceiling.

Worked Check

Problem: A qubit has energy relaxation time \(T_1 = 40\,\mu\text{s}\), Markovian pure dephasing time \(T_\phi = 80\,\mu\text{s}\), and Gaussian shot-to-shot frequency jitter that yields \(T_2^* = 2.5\,\mu\text{s}\) in a Ramsey scan. What homogeneous coherence time \(T_{2\mathrm{E}}\) is measured when a Hahn echo \(\pi\) pulse removes the static jitter, and what is the echo visibility at \(\tau = 16\,\mu\text{s}\)?

Practice Problems

Key Takeaways

• Ramsey interferometry brackets a free-evolution interval \(\tau\) between two \(\pi/2\) pulses, converting relative phase accumulation \(\phi = \Delta\tau\) into population oscillations \(P_1(\tau) = \frac{1}{2}[1 + W(\tau)\cos(\Delta\tau + \phi_{\mathrm{ref}})]\).
• Free-induction fringe decay (\(T_2^*\)) combines reversible quasi-static frequency inhomogeneity with irreversible dephasing.
• Inserting a Hahn spin-echo \(\pi\) pulse at \(\tau/2\) reverses static phase accumulation and measures the homogeneous coherence time \(T_{2\mathrm{E}}\), which satisfies \(T_{2\mathrm{E}} \le 2T_1\).
• Loss of Ramsey fringe contrast \(W(\tau) \to 0\) reflects radial shrinkage of the Bloch vector from a pure equatorial state to a mixed reduced state.

Sources and Further Reading

• N. F. Ramsey, A Molecular Beam Resonance Method with Separated Oscillating Fields, Physical Review 78, 695 (1950).
• E. L. Hahn, Spin Echoes, Physical Review 80, 580 (1950).
• Ł. Cywiński, R. M. Lutchyn, C. P. Nave, and S. Das Sarma, How to enhance dephasing time in superconducting qubits, Physical Review B 77, 174509 (2008).