Spin Qubits

Spin qubits encode quantum information in spin states of electrons, holes, nuclei, donors, or few-electron quantum dots. They are attractive because spin can be well isolated from electric noise, but real devices are controlled and read out electrically, embedded in solids, and coupled to nuclear spins, phonons, charge fluctuators, reservoirs, and sensors.

The examples below connect relaxation, charge and nuclear-spin noise, pulse sequences, and spin-to-charge readout. For confinement and charge states, see Quantum Dots and Coulomb Blockade. Silicon Spin Qubits develops encodings and processor architectures.

Minimal Qubit Model

A driven spin qubit is often modeled as

$$\frac{H(t)}{\hbar} = \frac{\omega_q+\delta\omega_z(t)}{2}\sigma_z + \frac{\Omega_x(t)}{2}\sigma_x + \frac{\Omega_y(t)}{2}\sigma_y + \frac{\delta\omega_x(t)}{2}\sigma_x + \frac{\delta\omega_y(t)}{2}\sigma_y .$$

The terms $\Omega_x$ and $\Omega_y$ are controlled drives. The noise terms have different physical consequences:

• longitudinal noise $\delta\omega_z(t)$ mainly causes dephasing;
• transverse noise $\delta\omega_x(t)$ and $\delta\omega_y(t)$ near $\omega_q$ causes relaxation;
• slow parameter drift produces inhomogeneous broadening and calibration error;
• strong noise or leakage invalidates the two-level model.

The same measured decay time can hide multiple mechanisms. A spin-qubit paper should say how the decay was measured, not only quote a number.

T1, T2, and Tphi

The standard weak-coupling Markovian qubit model is

$$\dot\rho = - \frac{i}{\hbar}[H_q,\rho] + \Gamma_\downarrow\mathcal D[\sigma_-]\rho + \Gamma_\uparrow\mathcal D[\sigma_+]\rho + \frac{\Gamma_\phi}{2}\mathcal D[\sigma_z]\rho .$$

The population-relaxation time is

$$\frac{1}{T_1} = \Gamma_\downarrow+\Gamma_\uparrow .$$

The transverse coherence time is

$$\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi},$$

where $T_\phi=1/\Gamma_\phi$ in this convention. This rate relation applies to the displayed Markovian two-level model. Slow drift, non-Gaussian nuclear noise, leakage, and pulse errors can make fitted coherence envelopes nonexponential.

See Dephasing Versus Dissipation for the distinction between population and phase relaxation.

Noise Spectra View

For a qubit splitting $\omega_q$, transverse noise at the transition frequency drives relaxation. Schematically,

$$\frac{1}{T_1} \propto S_\perp(\omega_q),$$

For quantum noise, excitation and relaxation sample opposite ordered frequencies, so $1/T_1$ contains both contributions. The one-frequency shorthand assumes a classical/symmetrized spectrum or negligible thermal excitation. Noise Spectra fixes the ordering and normalization.

Longitudinal noise near zero frequency dephases the qubit:

$$\frac{1}{T_\phi} \propto S_z(0)$$

in the simplest Markovian limit. In many spin-qubit devices, however, important dephasing noise is slow and non-Markovian. Then Ramsey, echo, and dynamical decoupling experiments probe different filtered parts of the noise spectrum rather than one universal $T_2$.

See Noise Spectra and Common Noise Spectra.

Charge Noise

Spin qubits are controlled by gates, exchange couplings, tunnel barriers, and electric fields. This makes them indirectly sensitive to charge noise even when the encoded degree of freedom is spin.

Suppose the qubit frequency depends on a gate-controlled parameter $\lambda$:

$$\omega_q = \omega_q(\lambda).$$

Small fluctuations give

$$\delta\omega_q \approx \frac{\partial\omega_q}{\partial\lambda} \delta\lambda .$$

This produces dephasing. A sweet spot is a bias point where

$$\frac{\partial\omega_q}{\partial\lambda} = 0,$$

so the leading sensitivity is quadratic:

$$\delta\omega_q \approx \frac{1}{2} \frac{\partial^2\omega_q}{\partial\lambda^2} (\delta\lambda)^2 .$$

The quadratic expression includes a mean frequency shift. If $\delta\lambda$ has zero mean and variance $v$, the centered fluctuating part is $\omega_q^{\prime\prime}[(\delta\lambda)^2-v]/2$. Even Gaussian parameter noise therefore produces non-Gaussian frequency noise at a sweet spot.

Charge noise can also affect relaxation by modulating spin–orbit admixture, valley splitting, micromagnet gradients, or exchange. In singlet-triplet and exchange-only qubits, exchange noise is often the central dephasing mechanism because the exchange splitting is electrically controlled.

Hyperfine and Nuclear-Spin Noise

Electron spins interact with host nuclear spins through hyperfine coupling. In a simplified central-spin picture,

$$H_{\mathrm{hf}} = \sum_k A_k \mathbf S\cdot\mathbf I_k .$$

Take $\mathbf S$ and $\mathbf I_k$ dimensionless here, so $A_k$ has energy units. The nuclear bath produces an effective Overhauser field. Its longitudinal component shifts the electron spin splitting and causes dephasing; transverse components can contribute to relaxation when energy conservation and bath dynamics allow.

For a quasi-static Gaussian Overhauser shift $\delta\omega$ with variance $\sigma^2$, a Ramsey coherence has the form

$$\left\langle e^{-i\delta\omega t} \right\rangle = e^{-\sigma^2t^2/2}.$$

This is an inhomogeneous dephasing envelope, often associated with $T_2^*$. Echo and dynamical decoupling can refocus quasi-static components but cannot remove all bath dynamics.

Material matters. GaAs has many nuclear spins and strong hyperfine effects. Isotopically enriched silicon can greatly reduce nuclear-spin noise, but charge noise, valley physics, interface disorder, and residual nuclei still matter.

Example: Singlet-Triplet Qubit

A two-electron singlet-triplet qubit in a double quantum dot is often described in the $\{|S\rangle,|T_0\rangle\}$ subspace by an effective Hamiltonian

$$\frac{H_{\mathrm{ST}}}{\hbar} = \frac{J(\epsilon)}{2}\sigma_z + \frac{\Delta\omega_z}{2}\sigma_x .$$

Here $J(\epsilon)$ is the exchange splitting in angular-frequency units, controlled by electrostatic detuning $\epsilon$, and $\Delta\omega_z$ is the difference of Zeeman angular frequencies between dots. A magnetic-field difference must be multiplied by the gyromagnetic ratio before entering this Hamiltonian. The open-system reading is immediate:

• charge noise in $\epsilon$ creates exchange noise and dephasing;
• nuclear or micromagnet gradient noise changes $\Delta\omega_z$;
• relaxation and leakage can occur through coupling to other charge or triplet states;
• readout often uses spin-to-charge conversion followed by charge sensing.

This example illustrates why “spin qubit” does not mean “immune to electric noise.” Electrical control makes electric noise visible.

Dynamical Decoupling

Dynamical decoupling uses control pulses to average or filter dephasing noise. The simplest example is spin echo: a $\pi$ pulse reverses the phase accumulated from quasi-static detuning noise.

For longitudinal Gaussian noise, the coherence follows an integral of the noise spectrum weighted by the sequence filter. The definitions, prefactors, and Ramsey/echo comparison are developed in Dynamical Decoupling. Decoupling suppresses only noise in the appropriate frequency bands; it cannot generally remove relaxation or pulse imperfections.

Readout Backaction

Solid-state spin readout often converts spin information into charge information:

• spin-selective tunneling to a reservoir;
• Pauli spin blockade in double dots;
• coupling to a quantum point contact or single-electron transistor;
• dispersive sensing through a resonator;
• latched readout where a spin-dependent event becomes a longer-lived charge configuration.

The measurement is therefore an open-system process. A detector gains information by becoming correlated with the spin state, but the same coupling can cause relaxation, dephasing, heating, or tunneling.

A simple spin-selective tunneling model has a collapse operator

$$L_{\mathrm{tun}} = \sqrt{\Gamma_{\mathrm{tun}}} |\mathrm{out}\rangle\langle \uparrow |.$$

If a tunneling event is detected, the state is updated conditionally. If the event is unobserved or the detector has finite bandwidth, the same process contributes to decoherence and readout error.

Readout fidelity is therefore not just detector signal-to-noise. It also depends on measurement time, relaxation during readout, thermal tunneling, sensor backaction, and the probability that spin-to-charge conversion itself fails.

Leakage and Valley Physics

Many spin-qubit devices have nearby states outside the ideal qubit subspace: valley states in silicon, orbital excited states, charge states, triplet states, donor states, or sensor states. Leakage matters because it is not captured by a two-level $T_1$ and $T_2$ model.

If $P_Q$ projects onto the qubit subspace, leakage can be tracked by

$$P_{\mathrm{leak}}(t) = 1-\operatorname{Tr}[P_Q\rho(t)].$$

Leakage may be coherent, caused by a pulse that drives unwanted levels, or dissipative, caused by relaxation into states outside the computational basis. Both forms need explicit modeling when gate and readout errors are being interpreted.

Exercises

Combining Coherence Decay Rates

In the standard Markovian qubit model, population relaxation has time $T_1$ and pure dephasing has time $T_\phi$. Derive

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

Solution

Population relaxation contributes a coherence decay factor

$$e^{-t/(2T_1)}.$$

Pure dephasing contributes

$$e^{-t/T_\phi}.$$

Multiplying the independent factors gives

$$e^{-t/T_2} = e^{-t/(2T_1)} e^{-t/T_\phi} = \exp \left[ - t \left( \frac{1}{2T_1} + \frac{1}{T_\phi} \right) \right].$$

Therefore

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

Sweet-Spot Expansion

Let $\omega_q(\lambda)$ be a qubit frequency controlled by a noisy parameter $\lambda=\lambda_0+\delta\lambda$. If $\partial_\lambda\omega_q(\lambda_0)=0$, find the leading frequency fluctuation.

Solution

Taylor expand around $\lambda_0$:

$$\omega_q(\lambda_0+\delta\lambda) = \omega_q(\lambda_0) + \left. \frac{\partial\omega_q}{\partial\lambda} \right|_{\lambda_0} \delta\lambda + \frac{1}{2} \left. \frac{\partial^2\omega_q}{\partial\lambda^2} \right|_{\lambda_0} (\delta\lambda)^2 + \cdots .$$

At a sweet spot, the linear term vanishes. The leading fluctuation is therefore

$$\delta\omega_q \approx \frac{1}{2} \left. \frac{\partial^2\omega_q}{\partial\lambda^2} \right|_{\lambda_0} (\delta\lambda)^2 .$$

Quasi-Static Gaussian Dephasing

Assume a Ramsey experiment has a random detuning $\delta\omega$ drawn from a Gaussian distribution with mean zero and variance $\sigma^2$. Show that

$$\left\langle e^{-i\delta\omega t}\right\rangle = e^{-\sigma^2t^2/2}.$$

Solution

The average is the characteristic function of a zero-mean Gaussian:

$$\left\langle e^{-i\delta\omega t}\right\rangle = \frac{1}{\sqrt{2\pi\sigma^2}} \int_{-\infty}^{\infty} d\delta\omega\, \exp \left[ - \frac{\delta\omega^2}{2\sigma^2} - i\delta\omega t \right].$$

Completing the square gives

$$\left\langle e^{-i\delta\omega t}\right\rangle = e^{-\sigma^2t^2/2}.$$

This Gaussian decay is not the same as an exponential Markovian $T_2$ envelope.

Spin-to-Charge Readout

Why can spin-selective tunneling improve measurement visibility while also increasing backaction?

Solution

Spin-selective tunneling converts a microscopic spin state into a charge configuration that a nearby sensor can distinguish more easily. This increases measurement visibility because charge sensors often have strong signals.

The same tunneling is a real physical transition out of the original spin state. It can cause loss from the qubit subspace, relaxation during readout, heating, and detector-induced dephasing. Thus the mechanism that makes the record visible also creates backaction.

References

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