# spin-echo

### Diffusion-assisted selective dynamical recoupling: A new approach to measure background gradients in magnetic resonance | J. Chem. Phys. (2014)

**Diffusion-assisted selective dynamical recoupling: A new approach to measure background gradients in magnetic resonance**

Gonzalo A. Álvarez, Noam Shemesh and Lucio Frydman

J. Chem. Phys. **140**, 084205 (2014); http://dx.doi.org/10.1063/1.4865335

Dynamical decoupling, a generalization of the original NMR spin-echo sequence, is becoming increasingly relevant as a tool for reducing decoherence in quantum systems. Such sequences apply non-equidistant refocusing pulses for optimizing the coupling between systems, and environmental fluctuations characterized by a given noise spectrum. One such sequence, dubbed Selective Dynamical Recoupling SDR [P. E. S. Smith, G. Bensky, G. A. Álvarez, G. Kurizki, and L. Frydman, Proc. Natl. Acad. Sci. **109**, 5958 (2012)], allows one to coherently reintroduce diffusion decoherence effects driven by fluctuations arising from restricted molecular diffusion [G. A. Álvarez, N. Shemesh, and L. Frydman, Phys. Rev. Lett. **111**, 080404 (2013)]. The fully-refocused, constant-time, and constant-number-of-pulses nature of SDR also allows one to filter out “intrinsic” T1 and T2 weightings, as well as pulse errors acting as additional sources of decoherence. This article explores such features when the fluctuations are now driven by unrestricted molecular diffusion. In particular, we show that diffusion-driven SDR can be exploited to investigate the decoherence arising from the frequency fluctuations imposed by internal gradients. As a result, SDR presents a unique way of probing and characterizing these internal magnetic fields, given an a priori known free diffusion coefficient. This has important implications in studies of structured systems, including porous media and live tissues, where the internal gradients may serve as fingerprints for the systems composition or structure. The principles of this method, along with full analytical solutions for the unrestricted diffusion-driven modulation of the SDR signal, are presented. The potential of this approach is demonstrated with the generation of a novel source of MRI contrast, based on the background gradients active in an ex vivo mouse brain. Additional features and limitations of this new method are discussed.

© 2014 AIP Publishing LLC

### Measuring small compartment dimensions by probing diffusion dynamics via Non-uniform Oscillating-Gradient Spin-Echo NOGSE NMR | J. Magn. Reson. (2013)

**Measuring small compartment dimensions by probing diffusion dynamics via Non-uniform Oscillating-Gradient Spin-Echo NOGSE NMR**

Noam Shemesh, Gonzalo A. Álvarez, Lucio Frydman.

Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel.

J. Magn. Reson. **237**, 49–62 (2013).

Highlights:

•NOGSE, a novel diffusion MR approach for measuring pore sizes, is presented and analyzed.

•NOGSE is based on a constant time and a constant number of oscillating gradients.

•Experiments on microstructural phantoms, spinal cords and brains, validate NOGSE.

Abstract:

Noninvasive measurements of microstructure in materials, cells, and in biological tissues, constitute a unique capability of gradient-assisted NMR. Diffusion–diffraction MR approaches pioneered by Callaghan demonstrated this ability; Oscillating-Gradient Spin-Echo OGSE methodologies tackle the demanding gradient amplitudes required for observing diffraction patterns by utilizing constant-frequency oscillating gradient pairs that probe the diffusion spectrum, Dω. Here we present a new class of diffusion MR experiments, termed Non-uniform Oscillating-Gradient Spin-Echo NOGSE, which dynamically probe multiple frequencies of the diffusion spectral density at once, thus affording direct microstructural information on the compartment’s dimension. The NOGSE methodology applies N constant-amplitude gradient oscillations; N − 1 of these oscillations are spaced by a characteristic time x, followed by a single gradient oscillation characterized by a time y, such that the diffusion dynamics is probed while keeping N − 1x + y ≡ TNOGSE constant. These constant-time, fixed-gradient-amplitude, multi-frequency attributes render NOGSE particularly useful for probing small compartment dimensions with relatively weak gradients – alleviating difficulties associated with probing Dω frequency-by-frequency or with varying relaxation weightings, as in other diffusion-monitoring experiments. Analytical descriptions of the NOGSE signal are given, and the sequence’s ability to extract small compartment sizes with a sensitivity towards length to the sixth power, is demonstrated using a microstructural phantom. Excellent agreement between theory and experiments was evidenced even upon applying weak gradient amplitudes. An MR imaging version of NOGSE was also implemented in ex vivo pig spinal cords and mouse brains, affording maps based on compartment sizes. The effects of size distributions on NOGSE are also briefly analyzed.

Keywords:

Restricted diffusion; Oscillating gradients; OGSE; Microstructure; Magnetic resonance imaging; CNS; Gradient echoes; Selective dynamical recoupling

Graphical abstract:

### Coherent dynamical recoupling of diffusion-driven decoherence in magnetic resonance | Phys. Rev. Lett. 111, 080404 (2013)

**Coherent dynamical recoupling of diffusion-driven decoherence in magnetic resonance**

Gonzalo A. Álvarez, Noam Shemesh, and Lucio Frydman

Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100, Israel

Received 13 May 2013; published 20 August 2013

During recent years, dynamical decoupling (DD) has gained relevance as a tool for manipulating and interrogating quantum systems. This is particularly relevant for spins involved in nuclear magnetic resonance (NMR), where DD sequences can be used to prolong quantum coherences, or to selectively couple or decouple the effects imposed by random environmental fluctuations. In this Letter, we show that these concepts can be exploited to selectively recouple diffusion processes in restricted spaces. The ensuing method provides a novel tool to measure restriction lengths in confined systems such as capillaries, pores or cells. The principles of this method for selectively recoupling diffusion-driven decoherence, its standing within the context of diffusion NMR, extensions to the characterization of other kinds of quantum fluctuations, and corroborating experiments, are presented.

© 2013 American Physical Society

arXiv: [1305.2794] Coherent dynamical recoupling of diffusion-driven decoherence in magnetic resonance.

### Random – but not quite: exploiting quantum decoherence as a tool for characterizing unknown systems | Seminar

**SEMINAR at IV Quantum Information Workshop – Paraty 2013**

Wednesday, August 14th 2013

See comments of the talk at the Paraty 2013’s Blog: Decoupling system and environment.

**SEMINAR at CBPF, Rio de Janeiro – Brazil, August 20th, 2013**

**SEMINAR at FaMAF, Córdoba – Argentina, August 27th, 2013**

**Abstract**

The ability to understand and manipulate the dynamics of quantum systems that interact with external degrees of freedom is a major challenge for fundamental quantum physics and its diverse applications, e.g., quantum information processing (QIP) or precision measurements. Progress in dynamical decoupling has lead to new ways to “protect” quantum bits from external degrees of freedom. Sometimes, however, a little bit of “recoupling” –i.e., exposure to the unknowns of the surrounding medium– can help. In this seminar, I will present a series of experimental methods implemented in NMR where by varying the “protection” given to the quantum states of ½-spins (qubits) can lead to new tools for characterizing unknown systems. In particular, I will show how Dynamical Decoupling noise spectroscopy can probe the spectrum of the environmental noise in order to find optimal methods for protecting the qubits [1]. In a new twist, I will present a method termed Selective Dynamical Recoupling (SDR), where suitable designed pulse sequence applied to the spins can selectively address specific information from the probed systems. SDR can be used to measure coupling strengths to the environment via oscillatory modulations that can serve for example to probe chemical identities derived from chemical shifts [2]. Alternatively, SDR can be designed to selectively measure the correlation time of the environmental noise where its value can be useful to probe diffusion processes in restricted spaces to extract the sizes of pores or cells in a non-invasive manner [3]. Applications of this new and simple approach can be found in materials sciences and biology and in particular it can be useful for investigating the nature of tissue compartmentalization *in vivo*, in manners which eventually could be useful in human and clinical settings.

[1] G.A. Alvarez, and D. Suter. Phys. Rev. Lett. **107**, 230501 (2011).

[2] P.E.S. Smith, G. Bensky, G.A. Alvarez, G. Kurizki, and L. Frydman. Proc. Natl. Acad. Sci. U. S. A. **109**, 5958 (2012).

[3] G.A. Alvarez, N. Shemesh, and L. Frydman. Phys. Rev. Lett. **111**, 080404 (2013).

### Random – but not quite: exploiting quantum decoherence as a tool for characterizing unknown systems | Paraty 2013

**SEMINAR at IV Quantum Information Workshop – Paraty 2013**

Wednesday, August 14th 2013

See comments of the talk at the Paraty 2013’s Blog: Decoupling system and environment.

**Abstract**

The ability to understand and manipulate the dynamics of quantum systems that interact with external degrees of freedom is a major challenge for fundamental quantum physics and its diverse applications, e.g., quantum information processing (QIP) or precision measurements. Progress in dynamical decoupling has lead to new ways to “protect” quantum bits from external degrees of freedom. Sometimes, however, a little bit of “recoupling” –i.e., exposure to the unknowns of the surrounding medium– can help. In this seminar, I will present a series of experimental methods implemented in NMR where by varying the “protection” given to the quantum states of ½-spins (qubits) can lead to new tools for characterizing unknown systems. In particular, I will show how Dynamical Decoupling noise spectroscopy can probe the spectrum of the environmental noise in order to find optimal methods for protecting the qubits [1]. In a new twist, I will present a method termed Selective Dynamical Recoupling (SDR), where suitable designed pulse sequence applied to the spins can selectively address specific information from the probed systems. SDR can be used to measure coupling strengths to the environment via oscillatory modulations that can serve for example to probe chemical identities derived from chemical shifts [2]. Alternatively, SDR can be designed to selectively measure the correlation time of the environmental noise where its value can be useful to probe diffusion processes in restricted spaces to extract the sizes of pores or cells in a non-invasive manner [3]. Applications of this new and simple approach can be found in materials sciences and biology and in particular it can be useful for investigating the nature of tissue compartmentalization *in vivo*, in manners which eventually could be useful in human and clinical settings.

[1] G.A. Alvarez, and D. Suter. Phys. Rev. Lett. **107**, 230501 (2011).

[2] P.E.S. Smith, G. Bensky, G.A. Alvarez, G. Kurizki, and L. Frydman. Proc. Natl. Acad. Sci. U. S. A. **109**, 5958 (2012).

[3] G.A. Alvarez, N. Shemesh, and L. Frydman. Phys. Rev. Lett. (2013) – in press. arXiv:1305.2794.

### Robustness of dynamical decoupling sequences | Phys. Rev. A 87, 042309 (2013)

Robustness of dynamical decoupling sequences

Mustafa Ahmed Ali Ahmed [1,2], Gonzalo A. Álvarez [1,3], and Dieter Suter [1]

1Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany

2Department of Physics, International University of Africa, Khartoum, Sudan

3Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel

Active protection of quantum states is an essential prerequisite for the implementation of quantum computing. Dynamical decoupling (DD) is a promising approach that applies sequences of control pulses to the system in order to reduce the adverse effect of system-environment interactions. Since every hardware device has finite precision, the errors of the DD control pulses can themselves destroy the stored information rather than protect it. We experimentally compare the performance of different DD sequences in the presence of an environment that was chosen such that all relevant DD sequences can equally suppress its effect on the system. Under these conditions, the remaining decay of the qubits under DD allows us to compare very precisely the robustness of the different DD sequences with respect to imperfections of the control pulses.

©2013 American Physical Society

via Phys. Rev. A 87, 042309 (2013): Robustness of dynamical decoupling sequences.

### Experimental protection of quantum gates against decoherence and control errors | Phys. Rev. A 86, 050301(R) 2012

Experimental protection of quantum gates against decoherence and control errors

Alexandre M. Souza, Gonzalo A. Álvarez, and Dieter Suter

Fakultät Physik, Technische Universität Dortmund, D-44221, Dortmund, Germany

One of the biggest challenges for implementing quantum devices is the requirement to perform accurate quantum gates. The destructive effects of interactions with the environment present some of the most difficult obstacles that must be overcome for precise quantum control. In this work we implement a proof of principle experiment of quantum gates protected against a fluctuating environment and control pulse errors using dynamical decoupling techniques. We show that decoherence can be reduced during the application of quantum gates. High-fidelity quantum gates can be achieved even if the gate time exceeds the free evolution decoherence time by one order of magnitude and for protected operations consisting of up to 330 individual control pulses.

©2012 American Physical Society

### Robust dynamical decoupling | Review Article | Phil. Trans. R. Soc. A 370, 4748 (2012)

#### Review article: Robust dynamical decoupling

Alexandre M. Souza, Gonzalo A. Álvarez and Dieter Suter

Fakultät Physik, Technische Universität Dortmund, 44221 Dortmund, Germany

Abstract

Quantum computers, which process information encoded in quantum mechanical systems, hold the potential to solve some of the hardest computational problems. A substantial obstacle for the further development of quantum computers is the fact that the lifetime of quantum information is usually too short to allow practical computation. A promising method for increasing the lifetime, known as dynamical decoupling (DD), consists of applying a periodic series of inversion pulses to the quantum bits. In the present review, we give an overview of this technique and compare different pulse sequences proposed earlier. We show that pulse imperfections, which are always present in experimental implementations, limit the performance of DD. The loss of coherence due to the accumulation of pulse errors can even exceed the perturbation from the environment. This effect can be largely eliminated by a judicious design of pulses and sequences. The corresponding sequences are largely immune to pulse imperfections and provide an increase of the coherence time of the system by several orders of magnitude.

#### via Robust dynamical decoupling: Phil. Trans. R. Soc. A 370, 4748 (2012).

### Featured in PRA Kaleidoscope for Vol. 85 Iss.5 (May 2012)

Image from “Iterative rotation scheme for robust dynamical decoupling.” [Gonzalo A. Álvarez, Alexandre M. Souza, and Dieter Suter, Phys. Rev. A 85, 052324 (2012)]

### Iterative rotation scheme for robust dynamical decoupling | Phys. Rev. A 85, 052324 (2012)

Gonzalo A. Álvarez, Alexandre M. Souza, and Dieter Suter

Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany

Received 1 March 2012; published 29 May 2012

The loss of quantum information due to interactions with external degrees of freedom, which is known as decoherence, remains one of the main obstacles for large-scale implementations of quantum computing. Accordingly, different measures are being explored for reducing its effect. One of them is dynamical decoupling DD which offers a practical solution because it only requires the application of control pulses to the system qubits. Starting from basic DD sequences, more sophisticated schemes were developed that eliminate higher-order terms of the system-environment interaction and are also more robust against experimental imperfections. A particularly successful scheme, called concatenated DD CDD, gives a recipe for generating higher-order sequences by inserting lower-order sequences into the delays of a generating sequence. Here, we show how this scheme can be improved further by converting some of the pulses to virtual and thus ideal pulses. The resulting scheme, called (XY4)^n, results in lower power deposition and is more robust against pulse imperfections than the original CDD scheme.

©2012 American Physical Society

URL: http://link.aps.org/doi/10.1103/PhysRevA.85.052324

DOI: 10.1103/PhysRevA.85.052324

via Phys. Rev. A 85, 052324 2012: Iterative rotation scheme for robust dynamical decoupling.