Central Spin Model Research Articles

Equilibration in closed quantum systems: Application to spin qubits

We study an observable-based notion of equilibration and its application to realistic systems like spin qubits in quantum dots. On the basis of the so-called distinguishability, we analytically derive general equilibration bounds, which we relate to the standard deviation of the fluctuations of the corresponding observable. Subsequently, we apply these ideas to the central spin model describing the spin physics in quantum dots. We probe our bounds by analyzing the spin dynamics induced by the hyperfine interaction between the electron spin and the nuclear spins using exact diagonalization. Interestingly, even small numbers of nuclear spins as found in carbon or silicon based quantum dots are sufficient to significantly equilibrate the electron spin.

Symmetry criteria for quantum simulability of effective interactions

What can one do with a given tunable quantum device? We provide complete symmetry criteria deciding whether some effective target interaction(s) can be simulated by a set of given interactions. Symmetries lead to a better understanding of simulation and permit a reasoning beyond the limitations of the usual explicit Lie closure. Conserved quantities induced by symmetries pave the way to a resource theory for simulability. On a general level, one can now decide equality for any pair of compact Lie algebras just given by their generators without determining the algebras explicitly. Several physical examples are illustrated, including entanglement invariants, the relation to unitary gate membership problems, as well as the central-spin model.

Dynamic Nuclear Polarization as Kinetically Constrained Diffusion.

Dynamic nuclear polarization (DNP) is a promising strategy for generating a significantly increased nonthermal spin polarization in nuclear magnetic resonance (NMR) and its applications that range from medicine diagnostics to material science. Being a genuine nonequilibrium effect, DNP circumvents the need for strong magnetic fields. However, despite intense research, a detailed theoretical understanding of the precise mechanism behind DNP is currently lacking. We address this issue by focusing on a simple instance of DNP-so-called solid effect DNP-which is formulated in terms of a quantum central spin model where a single electron is coupled to an ensemble of interacting nuclei. We show analytically that the nonequilibrium buildup of polarization heavily relies on a mechanism which can be interpreted as kinetically constrained diffusion. Beyond revealing this insight, our approach furthermore permits numerical studies of ensembles containing thousands of spins that are typically intractable when formulated in terms of a quantum master equation. We believe that this represents an important step forward in the quest of harnessing nonequilibrium many-body quantum physics for technological applications.

Quantum and classical correlations in electron-nuclear spin echo

The quantum properties of dynamic correlations in a system of an electron spin surrounded by nuclear spins under the conditions of free induction decay and electron spin echo have been studied. Analytical results for the time evolution of mutual information, classical part of correlations, and quantum part characterized by quantum discord have been obtained within the central-spin model in the high-temperature approximation. The same formulas describe discord in both free induction decay and spin echo although the time and magnetic field dependences are different because of difference in the parameters entering into the formulas. Changes in discord in the presence of the nuclear polarization βI in addition to the electron polarization βS have been calculated. It has been shown that the method of reduction of the density matrix to a two-spin electron-nuclear system provides a qualitatively correct description of pair correlations playing the main role at βS ≈ βI and small times. At large times, such correlations decay and multispin correlations ensuring nonzero mutual information and zero quantum discord become dominant.

Cooperation of different exchange mechanisms in confined magnetic systems

The diluted Kondo lattice model is investigated at strong antiferromagnetic local exchange couplings J, where almost local Kondo clouds drastically restrict the motion of conduction electrons, giving rise to the possibility of quantum localization of conduction electrons for certain geometries of impurity spins. This localization may lead to the formation of local magnetic moments in the conduction-electron system, and the inverse indirect magnetic exchange (IIME), provided by virtual excitations of the Kondo singlets, couples those local moments to the remaining electrons. Exemplarily, we study the one-dimensional two-impurity Kondo model with impurity spins near the chain ends, which supports the formation of conduction-electron magnetic moments at the edges of the chain for sufficiently strong J. Employing degenerate perturbation theory as well as analyzing spin gaps numerically by means of the density-matrix renormalization group, it is shown that the low-energy physics of the model can be well captured within an effective antiferromagnetic RKKY-like two-spin model ("RKKY from IIME") or within an effective central-spin model, depending on edge-spin distance and system size.

Conservation laws protect dynamic spin correlations from decay: Limited role of integrability in the central spin model

Mazur's inequality renders statements about persistent correlations possible. We generalize it in a convenient form applicable to any set of linearly independent constants of motion. This approach is used to show rigorously that a fraction of the initial spin correlations persists indefinitely in the isotropic central spin model unless the average coupling vanishes. The central spin model describes a major mechanism of decoherence in a large class of potential realizations of quantum bits. Thus the derived results contribute significantly to the understanding of the preservation of coherence. We will show that persisting quantum correlations are not linked to the integrability of the model, but caused by a finite operator overlap with a finite set of constants of motion.

From quantum-mechanical to classical dynamics in the central-spin model

We discuss the semiclassical and classical character of the dynamics of a single spin 1/2 coupled to a bath of noninteracting spins 1/2. 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.

Control of open quantum systems: case study of the central spin model

We study the controllability of a central spin guided by a classical field and interacting with a spin bath and show that the central spin is fully controllable independently of the number of bath spins. Additionally we find that for unequal system-bath couplings even the bath becomes controllable by acting on the central spin alone. We then analyze numerically how the time to implement gates on the central spin scales with the number of bath spins and conjecture that for equal system-bath couplings it reaches a saturation value. We provide evidence that sometimes noise can be effectively suppressed through control.

Long time evolution of a spin interacting with a spin bath in arbitrary magnetic field

We introduce a completely different method to calculate the evolution of a spin interacting with a sufficient large spin bath, especially suitable for treating the central spin model in a quantum dot (QD). With only an approximation on the envelope of central spin, the symmetry can be exploited to reduce a huge Hilbert space which cannot be calculated with computers to many small ones which can be solved exactly. This method can be used to calculate spin-bath evolution for a spin bath containing many (say, 1000) spins, without a perturbative limit such as strong magnetic field condition, and works for long-time regime with sufficient accuracy. As the spin-bath evolution can be calculated for a wide range of time and magnetic field, an optimal dynamic of spin flip-flop can be found, and more sophisticated approaches to achieve extremely high polarization of nuclear spins in a QD could be developed.

Exact out-of-equilibrium central spin dynamics from integrability

We consider a Gaudin magnet (central spin model) with a time-dependent exchange couplings. We explicitly show that the Schrödinger equation is analytically solvable in terms of generalized hypergeometric functions for particular choices of the time dependence of the coupling constants. Our method establishes a new link between this system and the Wess–Zumino–Witten model, and sheds new light on the implications of integrability in out-of-equilibrium quantum physics. As an application, a driven four-spin system is studied in detail.

Spin noise in the anisotropic central spin model

Spin-noise measurements can serve as direct probe for the microscopic decoherence mechanism of an electronic spin in semiconductor quantum dots (QD).We have calculated the spin-noise spectrum in the anisotropic central spin model using a Chebyshev expansion technique which exactly accounts for the dynamics up to an arbitrary long but fixed time in a finite size system. In the isotropic case, describing QD charged with a single electron, the short-time dynamics is in good agreement with a quasi-static approximation for the thermodynamic limit. The spin-noise spectrum, however, shows strong deviations at low frequencies with a power-law behavior. In the Ising limit, applicable to QDs with heavy-hole spins, the spin-noise spectrum exhibits a threshold behavior above the Larmor frequency. In the generic anisotropic central spin model we have found a crossover from a Gaussian type of spin-noise spectrum to a more Ising-type spectrum with increasing anisotropy in a finite magnetic field. In order to make contact with experiments, we present ensemble averaged spin-noise spectra for QD ensembles charged with single electrons or holes. The Gaussian-type noise spectrum evolves to a more Lorentzian shape spectrum with increasing spread of characteristic time-scales and g-factors of the individual QDs.

Dynamics and decoherence in the central spin model in the low-field limit

We present a combination of analytic calculations and a powerful numerical method for large spin baths in the low-field limit. The hyperfine interaction between the central spin and the bath is fully captured by the density matrix renormalization group. The adoption of the density matrix renormalization group for the central spin model is presented and a proper method for calculating the real-time evolution at infinite temperature is identified. In addition, we study to which extent a semiclassical model, where a quantum spin-1/2 interacts with a bath of classical Gaussian fluctuations, can capture the physics of the central spin model. The model is treated by average Hamiltonian theory and by numerical simulation.

Spin decoherence due to a randomly fluctuating spin bath

We study the decoherence of a spin in a quantum dot due to its hyperfine coupling to a randomly fluctuating bath of nuclear spins. The system is modelled by the central spin model with the spin bath initially being at infinite temperature. We calculate the spectrum and time evolution of the coherence factor using a Monte Carlo sampling of the exact eigenstates obtained via the algebraic Bethe ansatz. The exactness of the obtained eigenstates allows us to study the non-perturbative regime of weak magnetic fields in a full quantum mechanical treatment. In particular, we find a large non-decaying fraction in the zero-field limit. The crossover from strong to weak fields is similar to the decoherence starting from a pure initial bath state treated previously. We compare our results to a simple semiclassical picture [Merkulov et al., Phys. Rev. B 65, 205309 (2002)] and find surprisingly good agreement. Finally, we discuss the effect of weakly coupled spins and show that they will eventually lead to complete decoherence.

B-field dependence of the electron spin resonance line shape in a quantum dot

I investigate the line shape of the electron magnetic resonance broadened by an unpolarized bath of nuclear spins using an exact diagonalization method for the central spin model. While the shape is invariably Gaussian above the maximum Overhauser field of fully polarized nuclei, I find that the shape becomes non-trivial and magnetic field dependent in an intermediate range of external fields above a typical Overhauser field at high nuclear spin temperature and below the maximum Overhauser field. This effect can be observed in semiconductor quantum dots using electric transport or quantum optics techniques.

Loschmidt echo with a nonequilibrium initial state: Early-time scaling and enhanced decoherence

We study the Loschmidt echo (LE) in a central spin model in which a central spin is globally coupled to an environment (E) which is subjected to a small and sudden quench at $t=0$ so that its state at $t=0^+$, remains the same as the ground state of the initial environmental Hamiltonian before the quench; this leads to a non-equilibrium situation. This state now evolves with two Hamiltonians, the final Hamiltonian following the quench and its modified version which incorporates an additional term arising due to the coupling of the central spin to the environment. Using a generic short-time scaling of the decay rate, we establish that in the early time limit, the rate of decay of the LE (or the overlap between two states generated from the initial state evolving through two channels) close to the quantum critical point (QCP) of E is independent of the quenching. We do also study the temporal evolution of the LE and establish the presence of a crossover to a situation where the quenching becomes irrelevant. In the limit of large quench amplitude the non-equilibrium initial condition is found to result in a drastic increase in decoherence at large times, even far away from a QCP. These generic results are verified analytically as well as numerically, choosing E to be a transverse Ising chain where the transverse field is suddenly quenched.

Scaling of the decoherence factor of a qubit coupled to a spin chain driven across quantum critical points

We study the scaling of the decoherence factor of a qubit (spin-$1/2$) using the central spin model in which the central spin (qubit) is globally coupled to a transverse XY spin chain. The aim here is to study the nonequilibrium generation of decoherence when the spin chain is driven across (along) quantum critical points (lines) and derive the scaling of the decoherence factor in terms of the driving rate and some of the exponents associated with the quantum critical points. Our studies show that the scaling of the logarithm of the decoherence factor is identical to that of the defect density in the final state of the spin chain following a quench across isolated quantum critical points for both linear and nonlinear variations of a parameter, even if the defect density may not satisfy the standard Kibble-Zurek scaling. However, one finds an interesting deviation when the spin chain is driven along a critical line. Our analytical predictions are in complete agreement with numerical results. Our study, though limited to integrable two-level systems, points to the existence of a universality in the scaling of the decoherence factor which is not necessarily identical to the scaling of the defect density.

Entanglement and relaxation in exactly solvable models

A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρℐ(t) approaches an equilibrium state ρ eq ℐ , the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ eq ℐ , which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.

Master equation approach to the central spin decoherence problem: Uniform coupling model and role of projection operators

The generalized Master equation of the Nakajima-Zwanzig (NZ) type has been used extensively to investigate the coherence dynamics of the central spin model with the nuclear bath in a narrowed state characterized by a well defined value of the Overhauser field. We revisit the perturbative NZ approach and apply it to the exactly solvable case of a system with uniform hyperfine couplings. This is motivated by the fact that the effective Hamiltonian-based theory suggests that the dynamics of the realistic system at low magnetic fields and short times can be mapped onto the uniform coupling model. We show that the standard NZ approach fails to reproduce the exact solution of this model beyond very short times, while the effective Hamiltonian calculation agrees very well with the exact result on timescales during which most of the coherence is lost. Our key finding is that in order to extend the timescale of applicability of the NZ approach in this case, instead of using a single projection operator one has to use a set of correlated projection operators which properly reflect the symmetries of the problem and greatly improve the convergence of the theory. This suggests that the correlated projection operators are crucial for a proper description of narrowed state free induction decay at short times and low magnetic fields. Our results thus provide important insights toward the development of a more complete theory of central spin decoherence applicable in a broader regime of timescales and magnetic fields.

Hyperfine Interaction-Dominated Dynamics of Nuclear Spins in Self-Assembled InGaAs Quantum Dots

We measure the dynamics of nuclear spins in a single-electron charged self-assembled InGaAs quantum dot with negligible nuclear spin diffusion due to dipole-dipole interaction and identify two distinct mechanisms responsible for the decay of the Overhauser field. We attribute a temperature-independent decay lasting ∼100 sec at 5 T to intradot diffusion induced by hyperfine-mediated indirect nuclear spin interaction. By repeated polarization of the nuclear spins, this diffusion induced partial decay can be suppressed. We also observe a gate voltage and temperature-dependent decay stemming from cotunneling mediated nuclear spin flips that can be prolonged to ∼30 h by adjusting the gate voltage and lowering the temperature to ∼200 mK. Our measurements indicate possibilities for exploring quantum dynamics of the central spin model.

Gaudin models solver based on the correspondence between Bethe ansatz and ordinary differential equations

We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a new set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model and generalized Dicke model.