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Phase shift of electron spin echo signal caused by polarization of nuclear spins: revealing the quantum nature of spin environment

Damian Kwiatkowski, Piotr Szańkowski, Łukasz Cywiński

Published Sep 13, 2019• 0 citations
OpenAlex

Abstract

We consider the spin echo decay of a nitrogen-vacancy center spin qubit, based on $m\! =\! 0$ and $m\! =\! 1$ levels of the $S\!= \! 1$ ground state spin manifold of the center, in presence of finite dynamic nuclear polarization of the nuclear environment. We show that the signal acquires a nontrivially time-dependent phase shift in presence of such polarization. In Gaussian approximation to calculation of dephasing this phase shift is absent when the NV center is prepared in a superposition of $m\! = \! 1$ and $m\! =\! -1$ states, i.e. when the qubit couples to the spin environment in a way analogous to that of spin-$1/2$. For nuclear environment devoid of spins strongly coupled to the qubit, the phase shift can be quite well described in Gaussian approximation, which makes it clear that it is related to a linear response function of the environment affected by an evolving qubit. Consequently, its observation signifies the presence of back-action of the qubit on the environment, and it thus excludes the possibility of physically well-motivated treatment of environment as a source of external classical noise affecting the qubit. Furthermore, when exact coherence dynamics is well described by weak coupling approximation, for qubit based on $m\! =\! 0$ and $m\! =\! 1$ levels observation of dependence of modulus of echo signal on nuclear polarization signifies non-Gaussian character of environmental influence. The same conclusion follows from observation of nonzero phase shift of spin echo signal of a qubit based on $m\! = \! \pm 1$ levels.

Related papers

Influence of nuclear spin polarization on the spin-echo signal of an NV-center qubit

Damian Kwiatkowski, Piotr Szańkowski, Łukasz Cywiński

We consider the spin-echo dynamics of a nitrogen-vacancy (NV)-center qubit based on the $S=1$ ground-state spin manifold, caused by a dynamically polarized nuclear environment. We show that the echo signal acquires then a nontrivially time-dependent phase shift. This effect should be observable for polarization $\ensuremath{\approx}0.5$ of nuclei within $\ensuremath{\sim}1$ nm from the qubit, and for the NV center initialized in a superposition of $m=0$ and either $m=1$ or $m=\ensuremath{-}1$ states. This phase shift is much smaller when the NV center is prepared in a superposition of $m=1$ and $m=\ensuremath{-}1$ states, i.e., when the qubit couples to the spin environment in a way analogous to that of spin-$1/2$. For a nuclear environment devoid of spins strongly coupled to the qubit, the phase shift is well described within Gaussian approximation, which provides an explanation for the dependence of the shift magnitude on the choice of states on which the qubit is based, and makes it clear that its presence is related to the linear response of the environment perturbed by an evolving qubit. Consequently, its observation signifies the presence of an environment-mediated self-interaction of the qubit and, hence, it invalidates the notion that the nuclear environment acts as a source of external noise driving the qubit. We also show how a careful comparison of the echo signal from qubits based on $m=0,1$ and $m=\ifmmode\pm\else\textpm\fi{}1$ manifolds can distinguish between effectively Gaussian and non-Gaussian environments.

arXiv:1909.06438•Physical review. B./Physical review. B•Apr 13, 2020•20 citations

Decoherence of two entangled spin qubits coupled to an interacting sparse nuclear spin bath: Application to nitrogen vacancy centers

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We consider pure dephasing of Bell states of electron spin qubits interacting with a sparse bath of nuclear spins. Using the newly developed two-qubit generalization of cluster-correlation expansion method, we calculate the spin-echo decay of $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ and $|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$ states for various interqubit distances. Comparing the results with calculations in which dephasing of each qubit is treated independently, we identify signatures of influence of common part of the bath on the two qubits. At large interqubit distances, this common part consists of many nuclei weakly coupled to both qubits, so that decoherence caused by it can be modeled by considering multiple uncorrelated sources of noise (clusters of nuclei), each of them weakly affecting the qubits. Consequently, the resulting genuinely two-qubit contribution to decoherence can be described as being caused by classical Gaussian noise. On the other hand, for small interqubit distances the common part of the environment contains clusters of spins that are strongly coupled to both qubits, and their contribution to two-qubit dephasing has visibly non-Gaussian character. We show that one can easily obtain information about non-Gaussianity of environmental noise affecting the qubits from the comparison of dephasing of $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ and $|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$ Bell states. Numerical results are obtained for two nitrogen vacancy centers interacting with a bath of $^{13}\mathrm{C}$ nuclei of natural concentration, for which we obtain that Gaussian description of a correlated part of environmental noise starts to hold for centers separated by about 3 nm.

arXiv:1806.06845•Physical review. B./Physical review. B•Oct 5, 2018•32 citations

Spin bath mediated long-lived coherent oscillations of NV centers in diamond

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Decoherence is the biggest bottleneck in all quantum technologies. For nitrogen-vacancy (NV) centers in diamond, the loss of coherence is caused by the electron and nuclear spin bath of the diamond lattice. Here, we demonstrate that the spin bath - that typically causes decoherence - entangles the spin states of the NV electron and the host $^{14}$N nucleus. The many-body interaction between the $^{14}$N nucleus - electron - bath spins at an energy level anti-crossing occurring for an applied magnetic field orientation perpendicular to the NV axis is responsible for this effect. This is observed experimentally in NV ensembles via electron spin-echo measurements, where the echo envelope is modulated at the frequency of a $^{14}$N nuclear spin transition. Using numerical simulations, we show that the spin bath coupling to the NV centers is essential for observing this modulation. Due to the zero first-order Zeeman effect at the anti-crossing, the observed oscillations have long spin-echo coherence times, 2--3 times those at the parallel magnetic field orientation. The oscillation frequency is highly stable and robust against environmental fluctuations. These findings provide new opportunities for fundamental studies of many-body physics and quantum sensing.

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Quantum control of nuclear-spin qubits in a rapidly rotating diamond

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Nuclear spins in certain solids couple weakly to their environment, making them attractive candidates for quantum information processing and inertial sensing. When coupled to the spin of an optically active electron, nuclear spins can be rapidly polarized, controlled, and read via lasers and radio-frequency (rf) fields. Possessing typical coherence times of several milliseconds at room temperature, nuclear spins hosted by a nitrogen-vacancy center in diamond are thus intriguing systems to observe how classical physical rotation at quantum timescales affects a quantum system. In this work, we demonstrate optical nuclear-spin polarization and rapid quantum control of nuclear spins in a diamond physically rotating at $1\phantom{\rule{0.16em}{0ex}}\mathrm{kHz}$, faster than the nuclear-spin coherence time. Free from the need to maintain strict field alignment, we are able to measure and control nuclear spins in the presence of a large, time-varying magnetic field that makes an angle of more than ${100}^{\ensuremath{\circ}}$ to the nitrogen-lattice vacancy axis. The field induces spin mixing between the electron and nuclear states of the qubits, decoupling them from oscillating rf fields. We are able to demonstrate that coherent spin state control is possible at any point of the rotation. We combine continuous dynamical decoupling with feed-forward control to eliminate decoherence induced by imperfect mechanical rotation. Our work liberates a previously inaccessible degree of freedom of the NV nuclear spin, unlocking new approaches to quantum control and rotation sensing.

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Environment spectroscopy with an NV center in diamond

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Nitrogen-vacancy (NV) centers in diamond have emerged in the last decade as a prominent platform for quantum technologies. As for any qubit system, a good understanding of their local environment is crucial to build quantum devices protected from detrimental noise. Here, we describe in detail a method to spectroscopically characterize the spin bath around an NV center, even when the NV coherence time is short, and identify the coherent coupling with the nearest nuclear spins. In the regime of weak qubit-bath coupling, the acquired knowledge of the bath reliably predicts the qubit dynamics under different controls.

•Jan 1, 2020•0 citations

Quantum and classical correlations in electron-nuclear spin echo

V. E. Zobov

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 β I in addition to the electron polarization β S 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 β S ≈ β I and small times. At large times, such correlations decay and multispin correlations ensuring nonzero mutual information and zero quantum discord become dominant.

Journal of Experimental and Theoretical Physics•Nov 1, 2014•2 citations
We consider the spin echo decay of a nitrogen-vacancy center spin qubit, based on m ⁣= ⁣0m\! =\! 0m=0 and m ⁣= ⁣1m\! =\! 1m=1 levels of the S ⁣= ⁣1S\!= \! 1S=1 ground state spin manifold of the center, in presence of finite dynamic nuclear polarization of the nuclear environment. We show that the signal acquires a nontrivially time-dependent phase shift in presence of such polarization. In Gaussian approximation to calculation of dephasing this phase shift is absent when the NV center is prepared in a superposition of m ⁣= ⁣1m\! = \! 1m=1 and m ⁣= ⁣−1m\! =\! -1m=−1 states, i.e. when the qubit couples to the spin environment in a way analogous to that of spin-1/21/21/2. For nuclear environment devoid of spins strongly coupled to the qubit, the phase shift can be quite well described in Gaussian approximation, which makes it clear that it is related to a linear response function of the environment affected by an evolving qubit. Consequently, its observation signifies the presence of back-action of the qubit on the environment, and it thus excludes the possibility of physically well-motivated treatment of environment as a source of external classical noise affecting the qubit. Furthermore, when exact coherence dynamics is well described by weak coupling approximation, for qubit based on m ⁣= ⁣0m\! =\! 0m=0 and m ⁣= ⁣1m\! =\! 1m=1 levels observation of dependence of modulus of echo signal on nuclear polarization signifies non-Gaussian character of environmental influence. The same conclusion follows from observation of nonzero phase shift of spin echo signal of a qubit based on m ⁣= ⁣±1m\! = \! \pm 1m=±1 levels.
We consider the spin-echo dynamics of a nitrogen-vacancy (NV)-center qubit based on the S=1S=1S=1 ground-state spin manifold, caused by a dynamically polarized nuclear environment. We show that the echo signal acquires then a nontrivially time-dependent phase shift. This effect should be observable for polarization $\ensuremath{\approx}0.5$ of nuclei within $\ensuremath{\sim}1$ nm from the qubit, and for the NV center initialized in a superposition of m=0m=0m=0 and either m=1m=1m=1 or $m=\ensuremath{-}1$ states. This phase shift is much smaller when the NV center is prepared in a superposition of m=1m=1m=1 and $m=\ensuremath{-}1$ states, i.e., when the qubit couples to the spin environment in a way analogous to that of spin-1/21/21/2. For a nuclear environment devoid of spins strongly coupled to the qubit, the phase shift is well described within Gaussian approximation, which provides an explanation for the dependence of the shift magnitude on the choice of states on which the qubit is based, and makes it clear that its presence is related to the linear response of the environment perturbed by an evolving qubit. Consequently, its observation signifies the presence of an environment-mediated self-interaction of the qubit and, hence, it invalidates the notion that the nuclear environment acts as a source of external noise driving the qubit. We also show how a careful comparison of the echo signal from qubits based on m=0,1m=0,1m=0,1 and $m=\ifmmode\pm\else\textpm\fi{}1$ manifolds can distinguish between effectively Gaussian and non-Gaussian environments.
We consider pure dephasing of Bell states of electron spin qubits interacting with a sparse bath of nuclear spins. Using the newly developed two-qubit generalization of cluster-correlation expansion method, we calculate the spin-echo decay of $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ and $|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$ states for various interqubit distances. Comparing the results with calculations in which dephasing of each qubit is treated independently, we identify signatures of influence of common part of the bath on the two qubits. At large interqubit distances, this common part consists of many nuclei weakly coupled to both qubits, so that decoherence caused by it can be modeled by considering multiple uncorrelated sources of noise (clusters of nuclei), each of them weakly affecting the qubits. Consequently, the resulting genuinely two-qubit contribution to decoherence can be described as being caused by classical Gaussian noise. On the other hand, for small interqubit distances the common part of the environment contains clusters of spins that are strongly coupled to both qubits, and their contribution to two-qubit dephasing has visibly non-Gaussian character. We show that one can easily obtain information about non-Gaussianity of environmental noise affecting the qubits from the comparison of dephasing of $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ and $|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$ Bell states. Numerical results are obtained for two nitrogen vacancy centers interacting with a bath of 13C^{13}\mathrm{C}13C nuclei of natural concentration, for which we obtain that Gaussian description of a correlated part of environmental noise starts to hold for centers separated by about 3 nm.
Decoherence is the biggest bottleneck in all quantum technologies. For nitrogen-vacancy (NV) centers in diamond, the loss of coherence is caused by the electron and nuclear spin bath of the diamond lattice. Here, we demonstrate that the spin bath - that typically causes decoherence - entangles the spin states of the NV electron and the host 14^{14}14N nucleus. The many-body interaction between the 14^{14}14N nucleus - electron - bath spins at an energy level anti-crossing occurring for an applied magnetic field orientation perpendicular to the NV axis is responsible for this effect. This is observed experimentally in NV ensembles via electron spin-echo measurements, where the echo envelope is modulated at the frequency of a 14^{14}14N nuclear spin transition. Using numerical simulations, we show that the spin bath coupling to the NV centers is essential for observing this modulation. Due to the zero first-order Zeeman effect at the anti-crossing, the observed oscillations have long spin-echo coherence times, 2--3 times those at the parallel magnetic field orientation. The oscillation frequency is highly stable and robust against environmental fluctuations. These findings provide new opportunities for fundamental studies of many-body physics and quantum sensing.
Nuclear spins in certain solids couple weakly to their environment, making them attractive candidates for quantum information processing and inertial sensing. When coupled to the spin of an optically active electron, nuclear spins can be rapidly polarized, controlled, and read via lasers and radio-frequency (rf) fields. Possessing typical coherence times of several milliseconds at room temperature, nuclear spins hosted by a nitrogen-vacancy center in diamond are thus intriguing systems to observe how classical physical rotation at quantum timescales affects a quantum system. In this work, we demonstrate optical nuclear-spin polarization and rapid quantum control of nuclear spins in a diamond physically rotating at 1kHz1\phantom{\rule{0.16em}{0ex}}\mathrm{kHz}1kHz, faster than the nuclear-spin coherence time. Free from the need to maintain strict field alignment, we are able to measure and control nuclear spins in the presence of a large, time-varying magnetic field that makes an angle of more than ${100}^{\ensuremath{\circ}}$ to the nitrogen-lattice vacancy axis. The field induces spin mixing between the electron and nuclear states of the qubits, decoupling them from oscillating rf fields. We are able to demonstrate that coherent spin state control is possible at any point of the rotation. We combine continuous dynamical decoupling with feed-forward control to eliminate decoherence induced by imperfect mechanical rotation. Our work liberates a previously inaccessible degree of freedom of the NV nuclear spin, unlocking new approaches to quantum control and rotation sensing.
Nitrogen-vacancy (NV) centers in diamond have emerged in the last decade as a prominent platform for quantum technologies. As for any qubit system, a good understanding of their local environment is crucial to build quantum devices protected from detrimental noise. Here, we describe in detail a method to spectroscopically characterize the spin bath around an NV center, even when the NV coherence time is short, and identify the coherent coupling with the nearest nuclear spins. In the regime of weak qubit-bath coupling, the acquired knowledge of the bath reliably predicts the qubit dynamics under different controls.
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 β I in addition to the electron polarization β S 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 β S ≈ β I and small times. At large times, such correlations decay and multispin correlations ensuring nonzero mutual information and zero quantum discord become dominant.