Pure Dephasing Model

A pure dephasing model is an open-system model in which the environment destroys relative phases between preferred alternatives without changing their populations. When the coupling operator commutes with the system Hamiltonian, the problem reduces to conditional bath evolutions. Their overlaps can be evaluated exactly for important examples, including Gaussian phase noise and a linearly coupled oscillator bath.

The model is valuable because it separates three ideas that are often blended together:

• loss of coherence in a chosen basis;
• absence of energy relaxation in that basis;
• non-Markovian memory in the coherence factor.

The Markovian generator version is Pure Dephasing Master Equation. The finite-time channel version is Dephasing Channel. This page is the exactly solvable model behind both when the microscopic assumptions are simple enough.

Conditional Bath Hamiltonians

Let the system Hamiltonian and coupling operator share an eigenbasis:

$$H_S\lvert m\rangle = E_m\lvert m\rangle, \qquad A\lvert m\rangle = a_m\lvert m\rangle.$$

The pure dephasing Hamiltonian is

$$H = H_S + H_B + A\otimes B,$$

with

$$[A,H_S]=0.$$

Because the interaction is diagonal in the energy basis, it cannot by itself move the system from $\lvert m\rangle$ to $\lvert n\rangle$. It can, however, make different basis states imprint different phases and different bath states.

Equivalently, the full Hamiltonian can be written as a conditional bath Hamiltonian:

$$H = \sum_m \lvert m\rangle\langle m\rvert \otimes \left( E_m I_B+H_B+a_m B \right).$$

Each system alternative makes the bath evolve under a different Hamiltonian.

Exact Reduced Dynamics

Assume an initially factorized state,

$$\rho(0) = \rho_S(0)\otimes\rho_B.$$

For a system basis state $\lvert m\rangle$, define the conditional bath unitary

$$U_m(t) = \exp \left[ -\frac{i}{\hbar} \left( H_B+a_m B \right)t \right].$$

Then the reduced density-matrix elements evolve as

$$\rho_{mn}(t) = e^{-i(E_m-E_n)t/\hbar} \chi_{mn}(t) \rho_{mn}(0),$$

where the coherence factor is

$$\chi_{mn}(t) = \operatorname{Tr}_B \left[ U_m(t)\rho_B U_n^\dagger(t) \right].$$

Diagonal elements have $m=n$, so $\chi_{mm}(t)=1$ and

$$\rho_{mm}(t)=\rho_{mm}(0).$$

The magnitude of $\chi_{mn}$ is at most one, but it need not decrease monotonically: conditional bath evolutions can separate and later rephase. For an initially pure bath it is an overlap of two bath state vectors. For a mixed bath it also includes averaging over initially uncertain phases; distinguishability of the two bath density operators alone does not determine it. Its phase gives an additional system phase, which need not correspond to a constant energy shift.

Qubit Form

For a qubit in the dephasing basis, a common Hamiltonian is

$$H = \frac{\epsilon}{2}\sigma_z + H_B + \frac{\sigma_z}{2}B.$$

The two conditional bath Hamiltonians are

$$H_\pm = H_B \pm \frac12 B.$$

The populations in the $\sigma_z$ basis are constant, while the coherence obeys

$$\rho_{+,-}(t) = e^{-i\epsilon t/\hbar} \chi(t)\rho_{+,-}(0),$$

with

$$\chi(t) = \operatorname{Tr}_B \left[ e^{-iH_+t/\hbar} \rho_B e^{iH_-t/\hbar} \right].$$

If $|\chi(t)|=e^{-\Gamma_\phi t}$ over the time window of interest, the model reduces to a Markovian exponential dephasing law. If $|\chi(t)|$ is Gaussian, algebraic, oscillatory, or partially reviving, a one-rate Markovian description is not faithful.

Classical Gaussian Phase Noise

A classical version replaces the bath operator by a stochastic energy fluctuation, taking $A$ dimensionless:

$$H(t) = H_S + A\,\xi(t).$$

Here $\xi$ has units of energy. A frequency-noise convention instead writes $\hbar A\xi_\omega$, with $\xi_\omega=\xi/\hbar$. For one noise realization the evolution is unitary. After averaging over realizations,

$$\rho_{mn}(t) = e^{-i(E_m-E_n)t/\hbar} \left\langle e^{-i\phi_{mn}(t)} \right\rangle \rho_{mn}(0),$$

where

$$\phi_{mn}(t) = \frac{a_m-a_n}{\hbar} \int_0^t ds\,\xi(s).$$

If $\xi(t)$ is zero-mean Gaussian noise with correlation

$$C(\tau) = \langle \xi(\tau)\xi(0)\rangle,$$

then

$$\left\langle e^{-i\phi_{mn}(t)} \right\rangle = \exp[-\Gamma_{mn}(t)],$$

with

$$\Gamma_{mn}(t) = \frac{(a_m-a_n)^2}{2\hbar^2} \int_0^t ds \int_0^t ds'\, C(s-s').$$

This formula is often the simplest bridge between noise measurements and dephasing envelopes.

Spectral Form

For stationary classical Gaussian noise, write

$$C(\tau) = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi}\, e^{-i\omega\tau} S_{\xi\xi}(\omega).$$

Then

$$\Gamma_{mn}(t) = \frac{(a_m-a_n)^2}{2\hbar^2} \int_{-\infty}^{\infty} \frac{d\omega}{2\pi}\, S_{\xi\xi}(\omega) \left| \int_0^t ds\,e^{-i\omega s} \right|^2.$$

Since

$$\left| \int_0^t ds\,e^{-i\omega s} \right|^2 = \frac{4\sin^2(\omega t/2)}{\omega^2},$$

the coherence envelope samples noise through a filter centered near zero frequency. Control pulses replace the constant integrand by a sign-changing modulation function; that is the entry point to Dynamical Decoupling.

For white noise,

$$C(\tau)=D\,\delta(\tau),$$

one finds exponential decay:

$$\Gamma_{mn}(t) = \frac{(a_m-a_n)^2D}{2\hbar^2}t.$$

For quasi-static Gaussian noise with $\xi$ constant during each run but distributed with variance $\sigma^2$,

$$\Gamma_{mn}(t) = \frac{(a_m-a_n)^2\sigma^2}{2\hbar^2}t^2.$$

The first is a Markovian-looking irreversible envelope. The second is inhomogeneous broadening and can often be refocused by echo sequences.

Oscillator-Bath Pure Dephasing

The pure-dephasing limit of the spin-boson model sets the tunneling amplitude to zero. A common qubit-bath Hamiltonian is

$$H = \frac{\epsilon}{2}\sigma_z + \sum_k \hbar\omega_k b_k^\dagger b_k + \frac{\sigma_z}{2} \sum_k g_k \left( b_k+b_k^\dagger \right).$$

Here the real $g_k$ have units of energy. The bath is initially thermal for $H_B$, uncorrelated with the qubit. Conservation of $\sigma_z$ and the Gaussian oscillator displacement identities make the overlap exactly calculable. Define

$$J(\omega) = \pi \sum_k g_k^2 \delta(\omega-\omega_k), \qquad \omega>0.$$

For a thermal Gaussian oscillator bath, the qubit coherence has the form

$$\rho_{+,-}(t) = \rho_{+,-}(0) e^{-i\epsilon t/\hbar} e^{-\Phi(t)}.$$

The dephasing exponent is

$$\Phi(t) = \frac{1}{\pi\hbar^2} \int_0^\infty d\omega\, \frac{J(\omega)}{\omega^2} \left[ 1-\cos(\omega t) \right] \coth \left( \frac{\beta\hbar\omega}{2} \right).$$

For this symmetric $\sigma_z/2$ coupling and the stated unshifted thermal preparation, the bath-induced phase is zero: both spin alternatives have the same displacement energy. A common displacement energy multiplies the identity and cannot shift their relative phase. Different preparations or couplings with unequal squared eigenvalues can produce a phase; those are different assumptions, not an arbitrary phase to append here.

Ohmic Example

Take an Ohmic zero-temperature spectrum in the convention

$$J(\omega) = 2\pi\alpha\hbar^2\,\omega e^{-\omega/\omega_c}.$$

Then

$$\Phi(t) = 2\alpha \int_0^\infty d\omega\, e^{-\omega/\omega_c} \frac{1-\cos(\omega t)}{\omega}.$$

Using

$$\int_0^\infty d\omega\, e^{-a\omega} \frac{1-\cos(b\omega)}{\omega} = \frac12 \ln \left( 1+\frac{b^2}{a^2} \right),$$

with $a=1/\omega_c$ and $b=t$, one obtains

$$\Phi(t) = \alpha \ln \left( 1+\omega_c^2t^2 \right).$$

Thus the zero-temperature Ohmic pure-dephasing envelope is algebraic:

$$e^{-\Phi(t)} = \left( 1+\omega_c^2t^2 \right)^{-\alpha}.$$

This is not a constant-rate exponential. At finite temperature, for $t\gg\omega_c^{-1},\hbar/(k_BT)$, the same Ohmic convention gives leading exponent $\Phi(t)\sim2\pi\alpha k_BTt/\hbar$. A nonexponential envelope does not by itself prove failure of CP divisibility; see What Non-Markovian Means.

Markovian Limit

A pure-dephasing model reduces to a Lindblad pure-dephasing master equation when the coherence factor is approximately exponential over the times being modeled:

$$\chi_{mn}(t) \approx e^{-i\delta\omega_{mn}t} e^{-\gamma_{mn}t}.$$

For a qubit, a nonnegative constant decay rate gives a dephasing semigroup. For a multilevel system, the collection of rates must also satisfy complete-positivity constraints; choosing each $\gamma_{mn}$ independently is not enough. The resulting matrix elements are

$$\rho_{mn}(t) \approx e^{-i(E_m-E_n)t/\hbar} e^{-i\delta\omega_{mn}t} e^{-\gamma_{mn}t} \rho_{mn}(0).$$

For a Hermitian Lindblad operator $L$ with eigenvalues $\ell_m$, the Markovian generator

$$\dot\rho = \gamma\mathcal D[L]\rho$$

gives

$$\gamma_{mn} = \frac{\gamma}{2} (\ell_m-\ell_n)^2.$$

This is a useful effective model, but it should not be assumed when the measured coherence has Gaussian decay, power-law decay, revivals, or strong dependence on pulse sequence.

What the Bath Overlap Means

For a pure initial bath state, different system alternatives create different bath state vectors. Their overlap multiplies the system coherence. For a mixed bath, use a purification to recover this picture, or work directly with $\operatorname{Tr}(U_m\rho_BU_n^\dagger)$. For example, a maximally mixed bath can stay maximally mixed under either conditional unitary while the system still loses coherence through phase averaging. No human observation is required.

Pure dephasing also depends on the Hamiltonian and basis. If a transverse system term is added, $\sigma_z$ no longer commutes with $H_S$; the same coupling can then cause energy relaxation. That is the full spin-boson problem.

Exercises

Conditional bath evolution

Starting from

$$H = \sum_m \lvert m\rangle\langle m\rvert \otimes \left( E_m I_B+H_B+a_m B \right),$$

derive the coherence factor $\chi_{mn}(t)$.

Solution

The full unitary is block diagonal:

$$U(t) = \sum_m e^{-iE_mt/\hbar} \lvert m\rangle\langle m\rvert \otimes U_m(t),$$

with

$$U_m(t) = \exp \left[ -\frac{i}{\hbar} \left( H_B+a_mB \right)t \right].$$

For an initially factorized state, the $mn$ reduced matrix element is

$$\rho_{mn}(t) = e^{-i(E_m-E_n)t/\hbar} \rho_{mn}(0) \operatorname{Tr}_B \left[ U_m(t)\rho_B U_n^\dagger(t) \right].$$

Thus

$$\chi_{mn}(t) = \operatorname{Tr}_B \left[ U_m(t)\rho_B U_n^\dagger(t) \right].$$

A mixed bath can hide the phase information

Take a bath qubit with $\rho_B=I/2$, $H_B=0$, and interaction $H_I=\hbar g\,\sigma_z\otimes\tau_z/2$. Find the system coherence factor. Do the two conditional bath density operators differ?

Solution

The conditional unitaries are $U_\pm=e^{\mp igt\tau_z/2}$, so

$$\chi(t)=\frac12\operatorname{Tr}(e^{-igt\tau_z})=\cos(gt).$$

Both conditional bath states remain $U_\pm(I/2)U_\pm^\dagger=I/2$. Nevertheless, the system coherence vanishes at $gt=\pi/2$ and later revives. Bath-state distinguishability without the purification or correlations is therefore not a sufficient explanation for mixed-bath dephasing.

White noise versus quasi-static noise

For classical Gaussian phase noise, compare $C(\tau)=D\delta(\tau)$ with a quasi-static random variable of variance $\sigma^2$.

Solution

The Gaussian dephasing exponent is

$$\Gamma_{mn}(t) = \frac{(a_m-a_n)^2}{2\hbar^2} \int_0^t ds \int_0^t ds'\, C(s-s').$$

For white noise,

$$\int_0^t ds \int_0^t ds'\, D\delta(s-s') = Dt,$$

so the envelope is exponential. For quasi-static noise, $C(s-s')=\sigma^2$ over one run, so

$$\int_0^t ds \int_0^t ds'\, \sigma^2 = \sigma^2t^2,$$

and the envelope is Gaussian in time. The quasi-static case is reversible in principle by echo techniques, while ideal white-noise decay is not.

Zero-temperature Ohmic integral

Show that for

$$J(\omega) = 2\pi\alpha\hbar^2\,\omega e^{-\omega/\omega_c},$$

the zero-temperature exponent is $\Phi(t)=\alpha\ln(1+\omega_c^2t^2)$ in the convention used on this page.

Solution

At zero temperature, $\coth(\beta\hbar\omega/2)\to1$. Substitute the Ohmic $J(\omega)$ into

$$\Phi(t) = \frac{1}{\pi\hbar^2} \int_0^\infty d\omega\, \frac{J(\omega)}{\omega^2} \left[ 1-\cos(\omega t) \right].$$

This gives

$$\Phi(t) = 2\alpha \int_0^\infty d\omega\, e^{-\omega/\omega_c} \frac{1-\cos(\omega t)}{\omega}.$$

Using

$$\int_0^\infty d\omega\, e^{-a\omega} \frac{1-\cos(b\omega)}{\omega} = \frac12 \ln \left( 1+\frac{b^2}{a^2} \right),$$

with $a=1/\omega_c$ and $b=t$ yields

$$\Phi(t) = \alpha \ln \left( 1+\omega_c^2t^2 \right).$$

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• L. 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).
• G. Ithier et al., “Decoherence in a superconducting quantum bit circuit,” Physical Review B 72, 134519 (2005).
• A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative two-state system,” Reviews of Modern Physics 59, 1–85 (1987).
• E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler, “1/f noise: Implications for solid-state quantum information,” Reviews of Modern Physics 86, 361–418 (2014).
• M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer (2007).
• U. Weiss, Quantum Dissipative Systems, 4th ed., World Scientific (2012).