Theory of phonon-induced spin relaxation in a structured phononic reservoir

The gist: The authors develop a theory describing electron spin relaxation in structured phononic reservoirs by combining Markovian and non-Markovian open quantum system theory with finite-element simulations, revealing that relaxation rates can be suppressed by many orders of magnitude due to gaps in mode dispersion and selection rules imposed by mode symmetry

System Description

The research focuses on electron spins confined in self-assembled semiconductor quantum dots (QDs) embedded within a phononic crystal (PnC) waveguide, specifically the W1m line-defect waveguide. This setup is technologically relevant as it allows for the coupling of spin degrees of freedom with mechanical waves, which can be controlled by periodic patterning in the phononic crystal. The PnC structure has a lattice constant of a = 760 nm, a snowflake radius of r = 0.44a, an arm width of c = 0.19a, and a slab thickness of p = 220 nm. The QD is modeled as a point-like object located at position r0 within the waveguide, with r0 = (0, 0, 0) corresponding to the channel center at the slab mid-plane.

Spin-Phonon Coupling and Mode Classification

The spin-phonon coupling is described by an effective spin-phonon Hamiltonian Hsph(r) which accounts for the leading spin-relaxation mechanism in self-assembled QDs. The modification to the g-factor, denoted as δgˆi j, contains two contributions referred to as off-diagonal and deformation-potential after the distinct ways in which they emerge from the multiband k·p Hamiltonian. The eigenmodes of the waveguide satisfy a Hermitian eigenvalue problem where Einstein’s summation convention is used throughout. Guided modes are classified by a two-letter symmetry label (SS, SA, AS, AA) denoting the symmetry (S) and antisymmetry (A) of its displacement field under z and y-mirror reflections.

Relaxation Rates and Symmetry Effects

The Markovian spin-flip rate is resolved by mode symmetry, magnetic-field orientation, and the position of the QD within the waveguide. The paper shows that the spin relaxation rate mostly exceeds the bulk value but is suppressed by many orders of magnitude in certain ranges of magnetic field.

- The SS branches carry only diagonal strains which do not couple to the spin in either the Faraday or the Voigt configuration, and their contribution to the spin-flip rate is exactly zero. 5

- The SA mode is symmetric under z → −z, which forces εˆxz and εˆyz to vanish at the mid-plane and leaves εˆxy as the dominant component. 8

- The AS mode is antisymmetric under z → −z, which places the εˆxz antinode at the mid-plane and forces εˆxy to vanish there. 8

- The AA mode carries εˆxz at the mid-plane, with a node along the centerline y = 0 and antinodes near the lateral edges of the core. 8

Non-Markovian Dynamics at Singularities

Near van Hove singularities at the Brillouin zone (BZ) center and edges, slowmode effects lead to singularities in the relaxation rate, where a power-law analysis separates the universal density of states divergence from the mode-specific strain coupling. The Born–Markov description breaks down near these singularities because the bath memory time and spectral density both diverge as the group velocity vanishes.

- In this regime, exponential relaxation is replaced by a universal power-law decay to a finite constant occupation in which the spin is dressed into the slow mode to form a “spin polaron”. 5

- The resolvent approach based on Weisskopf–Wigner theory is applied near these van Hove singularities, generalizing it to finite temperature. 5

Key Findings and Implications

The work demonstrates that the spin relaxation rate can be modulated over several orders of magnitude through tunable external parameters, paving the way for new spin control schemes including hybrid protocols involving mechanical degrees of freedom. The non-Markovian dynamics enable the modulation of spin decay rates in phononic structures overseveral orders of magnitude through tunable external parameters. Furthermore, the analysis shows that every branch exhibits a power-law behavior at the van Hove singularities, where the spectral density scales as f − f0d−1/2, producing either a divergent peak or a vanishing notch. Finally, tuning the Zeeman frequency outside the band protects at least 4/9 of the population against spin flip and nearly all of it at deep detuning.

How it works

The theory utilizes a combination of Markovian and non-Markovian open quantum system theory with finite-element simulations to develop a theory of electron spin relaxation in a structured phononic reservoir. The spin dynamics are governed by the Lindblad master equation [63] within the Born–Markov and secular approximations, where the rate is given by γ↓ = 2π/(nB ΔEZ/hbar + 1) (ΔEZ/hbar). The spectral density of the spin-phonon coupling is defined as ζ(ω) = 1/(ħ2 / 8πμ2BħL M η2Bz1/ω × Σ λ εˆλkλωxz(r0)2 + εˆλkyz(r0)2) (dk dω).

Non-Markovian Treatment

When the Born–Markov approximation fails near van Hove singularities, the resolvent approach is used, leading to a Volterra convolution equation for the amplitude C˜e(t). The self-energy E(s) is obtained as the Laplace transform of the memory kernel Ξ(ω) = nB(ω0) + 1/ζ(ω), and near a dispersion extremum, the spectral density scales as Ξ(ω) = A (ω − ω0)−1/2. The resulting dynamics show that the long-time spin amplitude is C˜e(t → ∞) ≃ 2w20(3w20 + Δ0)eiΩt + e-i3π/42π3/(2A t3 / 2)eiΔ0t.

Experimental Relevance

The sharpness of the non-Markovian window is a direct experimental consequence, as tuning the Zeeman frequency to within 20 Hz of a van Hove singularity requires fixing the magnetic field to ΔB ≈ 0.7 nT. This narrowness certifies the Markovian rates across the rest of the gap. The analysis shows that everywhere except within this extremely narrow range of ∼ 200 Hz around each van Hove singularity, the Markovian rates are exact to better than 10−5.

Conclusion

The authors have formulated a theoretical description of spin relaxation in structured phononic reservoirs based on a combination of Markovian and non-Markovian open-system theory and finite-element computations for strain profiles. The critical dependence of the spin-flip rates on the QD position, as well as on the orientation and magnitude of external fields makesthese findingsimportant for designing quantum devices. The results show that spins coupled to structured phononic reservoirs manifest relaxation dynamics much richer than those in plain bulk.

How it works

The system is modeled using the W1m line-defect waveguide in a free-standing GaAs slab, with a lattice constant of a = 760 nm and slab thickness p = 220 nm.

1. Improve spin control by exploiting parameter windows where the relaxation is suppressed by many orders of magnitude due to gaps in mode dispersion and selection rules imposed by mode symmetry. This allows for designing devices where spin relaxation in such environments is much more complex than in bulk, enabling suppression of relaxation rates through structural engineering rather than just material choice.

2. Develop hybrid quantum protocols by leveraging the non-Markovian dynamics to achieve spin control over several orders of magnitude through tunable external parameters. This enables hybrid protocols involving mechanical degrees of freedom for spin manipulation, a capability currently unattainable in simple bulk systems.

3. Design novel spin storage and transduction architectures by utilizing the fact that the relaxation rates are typically an order of magnitude higher than in bulk, while simultaneously identifying regions where relaxation is suppressed to achieve high spin lifetimes. This directly addresses the limitation that long spin lifetimes make QDs attractive for spin-based hybrid acousto-optical applications, they remain finite.

4. Implement a predictive model for device performance by using the derived power-law scaling and propose a non-Markovian description of the dynamics near van Hove singularities. This allows engineers to predict how relaxation rates will behave as magnetic fields are tuned across specific frequencies, guiding the design of structures optimized for coherence.

5. Create high-fidelity simulations for device characterization by using finite-element simulations to obtain elastic eigenmodes and classify them according to symmetry. This capability allows researchers to calculate the exact strain profiles that determine the spin-phonon coupling for specific QD positions and field orientations, providing a precise blueprint for fabrication.

By combining Markovian and non-Markovian open quantum system theory with finite-element simulations, we develop a theory of electron spin relaxation in a structured phononic reservoir. This problem is crucial for understanding spin dynamics in hybrid systems involving mechanical modes, as well as for the design of devices combining spin degrees of freedom with photonic and phononic architectures, where the phonon density of states is modulated in the relevant spectral range corresponding to moderate magnetic fields. Taking a QD in a phononic waveguide as a representative and technologically relevant example, we show that spin relaxation in such environments is much more complex than in bulk. While the relaxation rates are several times to an order of magnitude higher than in bulk, there are parameter windows where the relaxation is suppressed by many orders of magnitude due to gaps in mode dispersion and selection rules imposed by mode symmetry. At the border between these two sectors, van Hove singularities in phonon dispersion lead to singularities in relaxation rates, for which we develop power-law scaling and develop a non-Markovian description of the dynamics, revealing polaronic dressing of the spin into slow acoustic modes.