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arXiv:quant-ph/0212097 · Physical Review B

Parity effects in spin decoherence

Akakii Melikidze, V. V. Dobrovitski, Hans A. De Raedt, Mikhail I. Katsnelson, B. N. Harmon

Published Jul 30, 2004• 64 citations
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Abstract

We demonstrate that decoherence of many-spin systems can drastically differ from decoherence of single-spin systems. The difference originates at the most basic level, being determined by parity of the central system, i.e., by whether the system comprises even or odd number of spin-$1∕2$ entities. Therefore, it is very likely that similar distinction between the central spin systems of even and odd parity is important in many other situations. Our consideration clarifies the physical origin of the unusual two-step decoherence found previously in the two-spin systems.

DOI
10.1103/physrevb.70.014435

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We demonstrate that decoherence of many-spin systems can drastically differ from decoherence of single-spin systems. The difference originates at the most basic level, being determined by parity of the central system, i.e., by whether the system comprises even or odd number of spin-1∕21∕21∕2 entities. Therefore, it is very likely that similar distinction between the central spin systems of even and odd parity is important in many other situations. Our consideration clarifies the physical origin of the unusual two-step decoherence found previously in the two-spin systems.
We study decoherence of central spins by a spin bath, focusing on the difference between measurement of a single central spin and measurement of a large number of central spins (as found in typical spin-resonance experiments). For a dilute spin bath, the single spin demonstrates Gaussian free-induction decay, in contrast to exponential decay characteristic of spin ensembles. A strong difference between a single spin and a spin ensemble also exists for the Rabi oscillation decay: for a repeated Rabi oscillation experiment, suppression of decoherence happens for a single spin while acceleration takes place for a spin ensemble. The mathematical origin of such behavior is similar to quantum Zeno/anti-Zeno effects.
We revisit the concepts of spin relaxation and spin decoherence of two-level (spin-1∕21∕21∕2) systems. From two toy models, we clarify two issues related to the spin relaxation and decoherence: (1) For an ensemble of two-level particles each subjected to a different environmental field, there exists an ensemble relaxation time T1∗{T}_{1}^{*}T1∗​, which is fundamentally different from T1{T}_{1}T1​. When the off-diagonal coupling of each particle is in a single mode with the same frequency but a random coupling strength, we show that T1∗{T}_{1}^{*}T1∗​ is finite while the spin-relaxation time of a single-spin T1{T}_{1}T1​ and the usual ensemble decoherence time T2∗{T}_{2}^{*}T2∗​ are infinite. (2) For a two-level particle under only a random diagonal coupling, its relaxation time T1{T}_{1}T1​ shall be infinite, but its decoherence time T2{T}_{2}T2​ is finite.
In the ordered phase of a Z2\mathbb{Z}_2Z2​-symmetric collective spin system, two natural bases -- localised pointer states {∣P⟩,∣R⟩}\{|P\rangle,|R\rangle\}{∣P⟩,∣R⟩} and energy eigenstates {∣E0⟩,∣E1⟩}\{|E_0\rangle,|E_1\rangle\}{∣E0​⟩,∣E1​⟩} -- yield Lindblad dephasing rates that differ by a factor approaching 222 as N→∞N\to\inftyN→∞ and reaching 2.422.422.42 near the quantum-critical crossover. The discrepancy has a single algebraic origin: parity forces ⟨Ei∣J^z∣Ei⟩=0\langle E_i|\hat{J}_z|E_i\rangle=0⟨Ei​∣J^z​∣Ei​⟩=0 exactly, eliminating the cross-term that doubles the localised-state rate. Two distinct protection factors are identified: ηMF=(Nm∗)2/(2G01)≈2.42η_{\rm MF}=(Nm_*)^2/(2G_{01})\approx2.42ηMF​=(Nm∗​)2/(2G01​)≈2.42, where m∗m_*m∗​ is the order parameter and G01=12(⟨E0∣J^z2∣E0⟩+⟨E1∣J^z2∣E1⟩)G_{01}=\frac{1}{2}(\langle E_0|\hat{J}_z^2|E_0\rangle+\langle E_1|\hat{J}_z^2|E_1\rangle)G01​=21​(⟨E0​∣J^z2​∣E0​⟩+⟨E1​∣J^z2​∣E1​⟩) (advantage over the classical mean-field estimate), and ηexact=(G01+J012)/G01≈1.86η_{\rm exact}=(G_{01}+J_{01}^2)/G_{01}\approx1.86ηexact​=(G01​+J012​)/G01​≈1.86, where J01=⟨E0∣J^z∣E1⟩J_{01}=\langle E_0|\hat{J}_z|E_1\rangleJ01​=⟨E0​∣J^z​∣E1​⟩ (exact physical ratio of pointer-state to eigenstate decay rate). In the thermodynamic limit the secular approximation fails, the doublet degenerates, and both rates converge. The three-regime structure is demonstrated in the Lipkin-Meshkov-Glick model via exact diagonalisation, and the algebraic origin of the discrepancy is established via the Z2\mathbb{Z}_2Z2​ parity of the Lindblad jump operator.
Spin parity effects refer to those special situations where a dichotomy in the physical behavior of a system arises, solely depending on whether the relevant spin quantum number is integral or half-odd integral. As is the case with the Haldane conjecture in antiferromagnetic spin chains, their pursuit often derives deep insights and invokes new developments in quantum condensed matter physics. Here, we put forth a simple and general scheme for generating such effects in any spatial dimension through the use of anisotropic interactions, and a setup within reasonable reach of state-of-the-art cold-atom implementations. We demonstrate its utility through a detailed analysis of the magnetization behavior of a specific one-dimensional spin chain model, an anisotropic antiferromagnet in a transverse magnetic field, unraveling along the way the quantum origin of finite-size effects observed in the magnetization curve that had previously been noted but not clearly understood.
We study the decoherence of two coupled spins that interact with a chaotic spin-bath environment. It is shown that connectivity of spins in the bath is of crucial importance for the decoherence of the central system. The previously found phenomenon of two-step decoherence (Phys. Rev. Lett. {\bf 90}, 210401 (2003)) turns out to be typical for the bath with a slow enough dynamics or no dynamics. For a generic random system with chaotic dynamics a conventional exponential relaxation to the pointer states takes place. Our results confirm a conjecture of Paz and Zurek (Phys. Rev. Lett. {\bf 82}, 5181 (1999)) that for weak enough interactions the pointer states are eigenstates of the central system.
We study the decoherence of a central spin 1/21/21/2 due to a closed environment composed of spin-1/21/21/2 particles. It is known that a frustrated spin environment, such as a spin glass, is much more efficient for decoherence of the central spin than a similar-size environment without frustration. We construct a Hamiltonian where the degree of frustration is parametrized by a single parameter $\ensuremath{\kappa}$. By use of this model we find that the environment can be classified by two distinct regimes with respect to the strength of level repulsion. These regimes behave qualitatively different with respect to decoherence of the central spin and might explain the strong enhancement of decoherence observed for frustrated environments.