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arXiv:1711.08919 · Physical review. B./Physical review. B

Quantum mechanical treatment of large spin baths

Robin Röhrig, Philipp Schering, Lars B. Gravert, Benedikt Fauseweh, Götz Silvester Uhrig

Published Apr 23, 2018• 9 citations
arXivHTMLPDF

Abstract

The electronic spin in quantum dots can be described by central spin models (CSMs) with a very large number ${N}_{\text{eff}}\ensuremath{\approx}{10}^{4}$ to ${10}^{6}$ of bath spins posing a tremendous challenge to theoretical simulations. Here, a fully quantum mechanical theory is developed for the limit ${N}_{\text{eff}}\ensuremath{\rightarrow}\ensuremath{\infty}$ by means of iterated equations of motion (iEoM). We find that the CSM can be mapped to a four-dimensional impurity coupled to a noninteracting bosonic bath in this limit. Remarkably, even for infinite bath the CSM does not become completely classical. The data obtained by the proposed iEoM approach are tested successfully against data from other, established approaches. Thus the iEoM mapping extends the set of theoretical tools that can be used to understand the spin dynamics in large CSMs.

DOI
10.1103/physrevb.97.165431

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The electronic spin in quantum dots can be described by central spin models (CSMs) with a very large number Neff\ensuremath≈104{N}_{\text{eff}}\ensuremath{\approx}{10}^{4}Neff​\ensuremath≈104 to 106{10}^{6}106 of bath spins posing a tremendous challenge to theoretical simulations. Here, a fully quantum mechanical theory is developed for the limit Neff\ensuremath→\ensuremath∞{N}_{\text{eff}}\ensuremath{\rightarrow}\ensuremath{\infty}Neff​\ensuremath→\ensuremath∞ by means of iterated equations of motion (iEoM). We find that the CSM can be mapped to a four-dimensional impurity coupled to a noninteracting bosonic bath in this limit. Remarkably, even for infinite bath the CSM does not become completely classical. The data obtained by the proposed iEoM approach are tested successfully against data from other, established approaches. Thus the iEoM mapping extends the set of theoretical tools that can be used to understand the spin dynamics in large CSMs.
We investigate the time dependence of correlation functions in the central spin model, which describes the electron or hole spin confined in a quantum dot, interacting with a bath of nuclear spins forming the Overhauser field. For large baths, a classical description of the model yields quantitatively correct results. We develop and apply various algorithms in order to capture the long-time limit of the central spin for bath sizes from 1000 to infinitely many bath spins. Representing the Overhauser field in terms of orthogonal polynomials, we show that a carefully reduced set of differential equations is sufficient to compute the spin correlations of the full problem up to very long times, for instance up to ${10}^{5}\ensuremath{\hbar}/{J}_{\mathrm{Q}}$ where JQ{J}_{\mathrm{Q}}JQ​ is the natural energy unit of the system. This technical progress renders an analysis of the model with experimentally relevant parameters possible. We benchmark the results of the algorithms with exact data for a small number of bath spins and we predict how the long-time correlations behave for different effective numbers of bath spins.
We investigate models of nonlinear qubit evolution based on mappings to an nnn-qubit central-spin model (CSM) in the large-nnn limit, where mean-field theory is exact. Extending a theorem of Erd\"os and Schlein [J. Stat. Phys. 134, 859 (2009)], we establish that the CSM is rigorously dual to a nonlinear qubit when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. The duality supports a type of nonlinear quantum computation in systems, such as a condensate, where a large number of ancillas couple symmetrically to a central qubit. It also enables a gate-model implementation of nonlinear quantum simulation with a rigorous error bound. Two variants of the model, with and without coupling between ancillas, map to effective models with different nonlinearity and symmetry. Without coupling the CSM simulates initial-condition nonlinearity, where the Hamiltonian is a linear combination of $\mathrm{tr}({\ensuremath{\rho}}_{0}{\ensuremath{\sigma}}^{x}){\ensuremath{\sigma}}^{x}, \mathrm{tr}({\ensuremath{\rho}}_{0}{\ensuremath{\sigma}}^{y}){\ensuremath{\sigma}}^{y}$, and $\mathrm{tr}({\ensuremath{\rho}}_{0}{\ensuremath{\sigma}}^{z}){\ensuremath{\sigma}}^{z}$, where ${\ensuremath{\sigma}}^{x}, {\ensuremath{\sigma}}^{y}$, and ${\ensuremath{\sigma}}^{z}$ are Pauli matrices and ${\ensuremath{\rho}}_{0}$ is the initial density matrix. With symmetric ancillas coupling it simulates linear combinations of $\mathrm{tr}(\ensuremath{\rho}{\ensuremath{\sigma}}^{x}){\ensuremath{\sigma}}^{x}, \mathrm{tr}(\ensuremath{\rho}{\ensuremath{\sigma}}^{y}){\ensuremath{\sigma}}^{y}$, and $\mathrm{tr}(\ensuremath{\rho}{\ensuremath{\sigma}}^{z}){\ensuremath{\sigma}}^{z}$, where $\ensuremath{\rho}$ is the current state. This case can simulate qubit torsion, which has been shown by Abrams and Lloyd [Phys. Rev. Lett. 81, 3992 (1998)] to enable an exponential speedup for state discrimination in an idealized setting. The duality discussed here might also be interesting from a quantum foundation perspective. There has long been interest in whether quantum mechanics might possess some type of small unobserved nonlinearity. If not, what is the principle prohibiting it? The duality implies that there is not a sharp distinction between universes evolving according to linear and nonlinear quantum mechanics: A one-qubit universe prepared in a pure state $|\ensuremath{\varphi}\ensuremath{\rangle}$ at the time of the big bang and symmetrically coupled to ancillas prepared in the same state would appear to evolve nonlinearly for any finite time t>0t>0t>0 as long as there are exponentially many ancillas $n\ensuremath{\gg}\mathrm{exp}[O(t)]$.
The decohering environment of a quantum bit is often described by the coupling to a large bath of spins. The quantum bit itself can be seen as a spin S=12S=\frac{1}{2}S=21​ which is commonly called the central spin. The resulting central spin model describes an important mechanism of decoherence. We provide mathematically rigorous bounds for a persisting magnetization of the central spin in this model with and without magnetic field. In particular, we show that there is a well-defined limit of infinite number of bath spins. Only if the fraction of very weakly coupled bath spins tends to 100% does no magnetization persist.
We solve the long-standing central spin problem for a general set of inhomogeneous bath couplings and a large class of initial bath states. We compute the time evolution of the coherence of a central spin coupled to a spin bath by resumming all orders of the time-convolutionless master equation, thus avoiding the need to assume weak coupling to the bath. The fully quantum, non-markovian solution is obtained in the large-bath limit and is valid up to a time scale set by the largest coupling constant. Our result captures the full decoherence of an electron spin qubit coupled to a nuclear spin bath in a GaAs quantum dot for experimentally relevant parameters. In addition, our solution is quite compact and can readily be used to make quantitative predictions for the decoherence process and to guide the design of nuclear state preparation protocols.
Central spin systems, in which a central spin is singled out and interacts nonlocally with several bath spins, are paradigmatic models for nitrogen-vacancy centers and quantum dots. They show complex emergent dynamics and stationary phenomena which, despite the collective nature of their interaction, are still largely not understood. Here, we derive exact results on the emergent behavior of open quantum central spin systems. The latter crucially depends on the scaling of the interaction strength with the bath size. For scalings with the inverse square root of the bath size (typical of one-to-many interactions), the system behaves, in the thermodynamic limit, as an open quantum Jaynes-Cummings model, whose bosonic mode encodes the quantum fluctuations of the bath spins. In this case, non-Gaussian correlations are dynamically generated and persist at stationarity. For scalings with the inverse bath size, the emergent dynamics is instead of mean-field type. Our Letter provides a fundamental understanding of the different dynamical regimes of central spin systems and a simple theory for efficiently exploring their nonequilibrium behavior. Our findings may become relevant for developing fully quantum descriptions of many-body solid-state devices and their applications.
We discuss the semiclassical and classical character of the dynamics of a single spin 1/21/21/2 coupled to a bath of noninteracting spins 1/21/21/2. On the semiclassical level, we extend our previous approach presented in D. Stanek, C. Raas, and G. S. Uhrig, Phys. Rev. B 88, 155305 (2013) by the explicit consideration of the conservation of the total spin. On the classical level, we compare the results of the classical equations of motions in absence and presence of an external field to the full quantum result obtained by density-matrix renormalization (DMRG). We show that for large bath sizes and not too low magnetic field the classical dynamics, averaged over Gaussian distributed initial spin vectors, agrees quantitatively with the quantum-mechanical one. This observation paves the way for an efficient approach for certain parameter regimes.