Experimental and theoretical investigations of quantum state transfer   and decoherence processes in quasi-one-dimensional systems in   multiple-quantum NMR experiments

G. A. Bochkin; S. I. Doronin; S. G. Vasil'ev; A. V. Fedorova; E. B.; Fel'dman

arXiv:1701.03588·quant-ph·January 16, 2017

Experimental and theoretical investigations of quantum state transfer and decoherence processes in quasi-one-dimensional systems in multiple-quantum NMR experiments

G. A. Bochkin, S. I. Doronin, S. G. Vasil'ev, A. V. Fedorova, E. B., Fel'dman

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TL;DR

This paper combines experimental and theoretical approaches to study quantum state transfer and decoherence in quasi-one-dimensional spin systems using multiple quantum NMR, revealing insights into quantum information processing.

Contribution

It establishes a connection between MQ NMR dynamics and one-dimensional spin chain models, and develops a theory of decoherence that matches experimental results.

Findings

01

Quantum state transfer can be organized along a linear chain.

02

Theoretical predictions agree well with experimental data.

03

Decoherence modeled by dipolar relaxation in 1D systems.

Abstract

Multiple quantum (MQ) NMR methods \cite{Baum} are applied to the analysis of various problems of quantum information processing. It is shown that the two-spin/two-quantum Hamiltonian \cite{Baum} describing MQ NMR dynamics is related to the flip-flop Hamiltonian of a one-dimensional spin system in the approximation of the nearest neighbor interactions. As a result, it is possible to organize quantum state transfer along a linear chain. MQ NMR experiments are performed on quasi-one-dimensional chains of \BPChem{\^{19}F} nuclei in calcium fluorapatite \BPChem{Ca\_5(PO\_4)\_3F}. Relaxation of the MQ NMR coherences is considered as the simplest model of decoherence processes. A theory of the dipolar relaxation of the MQ NMR coherences in one-dimensional systems is developed. A good agreement of the theoretical predictions and the experimental data is obtained.

Equations50

H^{(0)} = H^{(+ 2)} + H^{(- 2)},

H^{(0)} = H^{(+ 2)} + H^{(- 2)},

H^{(+ 2)} = - \frac{1}{2} i, j \sum D_{ij} I_{i}^{+} I_{j}^{+}, H^{(- 2)} = - \frac{1}{2} i, j \sum D_{ij} I_{i}^{-} I_{j}^{-} .

H^{(+ 2)} = - \frac{1}{2} i, j \sum D_{ij} I_{i}^{+} I_{j}^{+}, H^{(- 2)} = - \frac{1}{2} i, j \sum D_{ij} I_{i}^{-} I_{j}^{-} .

D_{ij} = \frac{γ ^{2} ℏ ( 1 - 3 cos ^{2} θ )}{2 a ^{3} ∣ i - j ∣ ^{3}},

D_{ij} = \frac{γ ^{2} ℏ ( 1 - 3 cos ^{2} θ )}{2 a ^{3} ∣ i - j ∣ ^{3}},

H_{Φ}^{(0)} = e^{- 2 i Φ} H^{(+ 2)} + e^{2 i Φ} H^{(- 2)} .

H_{Φ}^{(0)} = e^{- 2 i Φ} H^{(+ 2)} + e^{2 i Φ} H^{(- 2)} .

G_{0} (τ) = \frac{1}{2} + \frac{1}{2} J_{0} (4 D τ),

G_{0} (τ) = \frac{1}{2} + \frac{1}{2} J_{0} (4 D τ),

G_{\pm 2} (τ) = \frac{1}{4} - \frac{1}{4} J_{0} (4 D τ),

H_{f f} = i \sum D_{i, i + 1} (I_{i}^{+} I_{i + 1}^{-} + I_{i}^{-} I_{i + 1}^{+}) .

H_{f f} = i \sum D_{i, i + 1} (I_{i}^{+} I_{i + 1}^{-} + I_{i}^{-} I_{i + 1}^{+}) .

U = exp (- iπ I_{2 x}) \cdot exp (- iπ I_{4 x}) \dots,

U = exp (- iπ I_{2 x}) \cdot exp (- iπ I_{4 x}) \dots,

U H^{(0)} U^{+} = i \sum D_{i, i + 1} (I_{i}^{+} I_{i + 1}^{-} + I_{i}^{-} I_{i + 1}^{+}) = H_{f f} .

U H^{(0)} U^{+} = i \sum D_{i, i + 1} (I_{i}^{+} I_{i + 1}^{-} + I_{i}^{-} I_{i + 1}^{+}) = H_{f f} .

ρ (0) = \frac{1}{Z} e^{β I_{l z}} \otimes E_{N \neq = l},

ρ (0) = \frac{1}{Z} e^{β I_{l z}} \otimes E_{N \neq = l},

ρ (t) = e^{- i H^{(0)} t} ρ (0) e^{i H^{(0)} t} .

ρ (t) = e^{- i H^{(0)} t} ρ (0) e^{i H^{(0)} t} .

⟨ I_{N z} ⟩

⟨ I_{N z} ⟩

Tr {U ρ (t) U^{+} U I_{N z} U^{+}} = Tr {e^{- i H_{f f} t} ρ (0) e^{i H_{f f} t} I_{N z}} .

\frac{⟨ I _{m z} ⟩ ( t )}{⟨ I _{l z} ⟩ ( t )} = \frac{4}{( N + 1 ) ^{2}} k \sum exp (- i ϵ_{k} t) sin (k l) \cdot sin (k m)^{2},

\frac{⟨ I _{m z} ⟩ ( t )}{⟨ I _{l z} ⟩ ( t )} = \frac{4}{( N + 1 ) ^{2}} k \sum exp (- i ϵ_{k} t) sin (k l) \cdot sin (k m)^{2},

ϵ_{k} = D cos (k) + ω_{0},

ϵ_{k} = D cos (k) + ω_{0},

k = \frac{π n}{N + 1}, n = 1, 2, \dots N .

k = \frac{π n}{N + 1}, n = 1, 2, \dots N .

σ (τ) = σ_{0} (τ) + σ_{2} (τ) + σ_{- 2} (τ),

σ (τ) = σ_{0} (τ) + σ_{2} (τ) + σ_{- 2} (τ),

σ_{0} (τ)

σ_{0} (τ)

σ_{2} (τ)

σ_{- 2} (τ)

H_{d z} = i < j \sum D_{ij} (3 I_{i z} I_{j z} - I_{i} I_{j}) = i < j \sum D_{ij} (2 I_{i z} I_{j z} - I_{i x} I_{j x} - I_{i y} I_{i y})

H_{d z} = i < j \sum D_{ij} (3 I_{i z} I_{j z} - I_{i} I_{j}) = i < j \sum D_{ij} (2 I_{i z} I_{j z} - I_{i x} I_{j x} - I_{i y} I_{i y})

H_{Z Z} = 2 i < j \sum D_{ij} I_{i z} I_{j z} = i \neq = j \sum D_{ij} I_{i z} I_{j z} .

H_{Z Z} = 2 i < j \sum D_{ij} I_{i z} I_{j z} = i \neq = j \sum D_{ij} I_{i z} I_{j z} .

F_{0} (τ, t) = \frac{Tr { e ^{- i H_{Z Z} t} σ _{0} ( τ ) e ^{i H_{Z Z} t} σ _{0} ( τ )}}{Tr ( I _{z}^{2} )} = \frac{Tr { e ^{- i H_{Z Z} t} σ _{0} ( τ ) e ^{i H_{Z Z} t} σ _{0} ( τ )}}{N \cdot 2 ^{N - 2}} .

F_{0} (τ, t) = \frac{Tr { e ^{- i H_{Z Z} t} σ _{0} ( τ ) e ^{i H_{Z Z} t} σ _{0} ( τ )}}{Tr ( I _{z}^{2} )} = \frac{Tr { e ^{- i H_{Z Z} t} σ _{0} ( τ ) e ^{i H_{Z Z} t} σ _{0} ( τ )}}{N \cdot 2 ^{N - 2}} .

F_{\pm 2} (τ, t) = \frac{Tr { e ^{- i H_{Z Z} t} σ _{2} ( τ ) e ^{i H_{Z Z} t} σ _{- 2} ( τ )}}{Tr ( I _{z}^{2} )} .

F_{\pm 2} (τ, t) = \frac{Tr { e ^{- i H_{Z Z} t} σ _{2} ( τ ) e ^{i H_{Z Z} t} σ _{- 2} ( τ )}}{Tr ( I _{z}^{2} )} .

F_{0}^{s t} = \frac{J _{0}^{2} ( 2 D τ )}{G _{0} ( τ )} = \frac{2 J _{0}^{2} ( 2 D τ )}{1 + J _{0} ( 4 D τ )} .

F_{0}^{s t} = \frac{J _{0}^{2} ( 2 D τ )}{G _{0} ( τ )} = \frac{2 J _{0}^{2} ( 2 D τ )}{1 + J _{0} ( 4 D τ )} .

F_{\pm 2} (τ, t) = \frac{1}{8 N} m, m^{'} \sum n \neq = (m, m^{'}) \prod cos [(D_{nm} + D_{n m^{'}}) t] \cdot [1 - (- 1)^{m - m^{'}}]^{2} J_{m - m^{'}}^{2} (2 D τ) .

F_{\pm 2} (τ, t) = \frac{1}{8 N} m, m^{'} \sum n \neq = (m, m^{'}) \prod cos [(D_{nm} + D_{n m^{'}}) t] \cdot [1 - (- 1)^{m - m^{'}}]^{2} J_{m - m^{'}}^{2} (2 D τ) .

M_{2} (τ) = - \frac{1}{G _{2} ( τ )} \frac{d ^{2} F _{\pm 2} ( τ , t )}{d t ^{2}}_{t = 0}

M_{2} (τ) = - \frac{1}{G _{2} ( τ )} \frac{d ^{2} F _{\pm 2} ( τ , t )}{d t ^{2}}_{t = 0}

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Full text

Experimental and theoretical investigations of quantum state transfer and decoherence processes in quasi-one-dimensional systems in multiple-quantum NMR experiments

G. A. Bochkin1, S. I. Doronin1, S. G. Vasil’ev1, A. V. Fedorova1, E. B. Fel’dman1

1 Institute of Problems of Chemical Physics, Chernogolovka, Moscow Region, 142432, Russia

Abstract

Multiple quantum (MQ) NMR methods Baum are applied to the analysis of various problems of quantum information processing. It is shown that the two-spin/two-quantum Hamiltonian Baum describing MQ NMR dynamics is related to the flip-flop Hamiltonian of a one-dimensional spin system in the approximation of the nearest neighbor interactions. As a result, it is possible to organize quantum state transfer along a linear chain. MQ NMR experiments are performed on quasi-one-dimensional chains of \BPChem1̂9F nuclei in calcium fluorapatite \BPChemCa_5(PO_4)_3F. Relaxation of the MQ NMR coherences is considered as the simplest model of decoherence processes. A theory of the dipolar relaxation of the MQ NMR coherences in one-dimensional systems is developed. A good agreement of the theoretical predictions and the experimental data is obtained.

quantum state transfer, many-qubit superposition state, decoherence, fidelity, MQ NMR, dipolar relaxation

I INTRODUCTION

The development of experimental and theoretical methods of quantum information processing (QIP) is an important direction of quantum informatics. NMR methods are the simplest ones among different experimental approaches Nielsen . It is well known that the liquid state NMR methods Jones are widely used for experimental realizations of quantum gates and algorithms on the basis of the pseudo-pure states Cory ; Ger . However, the possibilities of QIP based on the liquid state NMR are limited because the number of correlated qubits is very small (about 10) Warren and quantum correlations in such systems are not strong. In particular, entanglement is almost absent Braun .

At the same time, the potential of the solid state NMR methods Feld1 is not exhausted yet. Multiple quantum (MQ) spectroscopy in solids Baum is an example of such a method. MQ NMR not only creates multi-qubit coherent states but also allows the investigation of their relaxation under the action of the correlated spin reservoir. MQ NMR Baum is an important method for the investigation of various problems of quantum information processing such as the transmission of quantum information Cappellaro and decoherence processes Doronin ; Kaur .

One-dimensional MQ NMR methods are very suitable for solving problems of quantum informatics. The reason is that a consistent quantum-mechanical theory of MQ NMR dynamics has been developed only for one-dimensional systems Feld2 ; Feld3 ; Doronin2 ; Feld4 . That theory is based on the fact that the non-secular two-spin/two-quantum Hamiltonian Baum , describing MQ NMR dynamics, is the XY Hamiltonian Mattis , which can be diagonalized exactly for one-dimensional systems in the approximation of the nearest neighbor interactions Abragam . As a result, MQ NMR dynamics in such systems can be studied analytically. In particular, only MQ NMR coherences of the zeroth and plus/minus second orders arise in a one-dimensional chain initially prepared in a thermodynamic equilibrium state Feld2 ; Feld3 ; Doronin2 ; Feld4 .

The developed theory Feld2 ; Feld3 ; Doronin2 ; Feld4 is based on the model of an isolated spin chain with the nearest neighbor interactions. Such a model is realized in quasi-one-dimensional chains of \BPChem1̂9F nuclei in calcium fluorapatite \BPChemCa_5(PO_4)_3F with a hexagonal system of fluorine nuclei. The distance between neighboring chains is about three times larger than the distance between the nearest fluorine nuclei in the same chain. Taking into account that the dipole-dipole interaction (DDI) of the next nearest spins in the chain is eight times weaker than the nearest neighbor interaction Abragam one can use the model of an isolated spin chain with the nearest neighbor interactions for the interpretation of the experimental data obtained from the MQ NMR experiments Baum .

The information resource of MQ NMR exceeds the resource of conventional NMR Feld4 . However, that resource can be used only if the XY Hamiltonian is created with sufficient accuracy Doronin3 . The XY Hamiltonian is obtained as the averaged two-spin/two-quantum Hamiltonian Baum which is the average Hamiltonian Haeberlen in the MQ NMR experiments.

The corrections to the average Hamiltonian Haeberlen spoil the XY Hamiltonian. These corrections are proportional to powers of the parameter $\epsilon=t_{c}\omega_{loc}$ ( $t_{c}$ is the period of the irradiating sequence of the preparation period of the MQ NMR experiment and $\omega_{loc}$ is of the order of the DDI). We have designed a special probe Doronin3 for the generation of high-power ultra-short pulses and decreased the period $t_{c}$ and the parameter $\epsilon$ in MQ NMR experiments. The results Doronin3 show high accuracy of creating the XY Hamiltonian.

It is very important that the XY Hamiltonian (the MQ NMR Hamiltonian) is related to the flip-flop one by a simple unitary transformation Doronin2 for one-dimensional systems in the approximation of the nearest neighbor interactions. That fact allows us to realize the quantum state transfer along a one-dimensional spin chain. It is also significant that the relaxation of the MQ NMR coherences can be considered as a model for studying the decoherence processes.

The paper is organized as follows. An introduction to MQ NMR dynamics is given in section 2. In section 3 we show that MQ NMR can be used for the quantum state transfer in one-dimensional spin chains. Relaxation of the MQ NMR coherences in one-dimensional spin chains is discussed in section 4. We briefly summarize our results in section 5.

II MQ NMR dynamics of one-dimensional spin systems

The MQ NMR experiment Baum consists of four distinct periods of time (Fig. 1): preparation( $\tau$ ), evolution ( $t$ ), mixing( $\tau$ ) and detection. The spin system is irradiated by a periodic sequence of resonance radio-frequency pulses on the preparation period. As a result, the anisotropic DDI Goldman become rapidly oscillating if the inverse period of the irradiating sequence $t_{c}^{-1}$ strongly exceeds the magnitude of the interactions $\omega_{loc}$ ( $\epsilon=t_{c}\omega_{loc}\ll 1$ ). The theory of the average Hamiltonian Haeberlen makes it possible to find the averaged interactions, the Hamiltonian of which $H^{(0)}$ (the averaged non-secular two-spin/two-quantum Hamiltonian Baum ) up to the terms of the order of $\epsilon^{2}$ has the form

[TABLE]

where the non-secular DDI $H^{\pm 2}$ are as follows:

[TABLE]

Here $I^{\pm}_{i}=I_{ix}\pm I_{iy}$ , $I_{i\alpha}$ is the projection of the spin angular momentum of spin $i$ on the $\alpha$ axis ( $\alpha=x,y,z$ ) and $D_{ij}$ is the coupling constant of the DDI between spins $i$ and $j$ . For a linear one-dimensional spin chain $D_{ij}$ is

[TABLE]

where $\gamma$ is the gyromagnetic ratio, $a$ is the distance between nearest neighbors, and $\theta$ is the angle between the vector connecting spins $i$ , $j$ and axis $z$ , which is the direction of the strong external magnetic field.

To perform MQ NMR experiments we used quasi-one-dimensional chains of \BPChem1̂9F nuclei in calcium fluorapatite \BPChemCa_5(PO_4)_3F Doronin3 . Experiments were performed on a Brucker Avance III spectrometer with a static magnetic field $B_{0}=9.4$ T (the corresponding frequency on \BPChem1̂9F nuclei is 376.6 MHz). The main block of the MQ NMR experiment (Fig. 1) is repeated many times with the phase increment of the radio-frequency pulses irradiating the spin system on the preparation period at each repetition. The applied sequence of radio-frequency pulses is built of basic cycles consisting of eight resonance (for \BPChem1̂9F nuclei) $\pi/2$ pulses (Fig. 1). At the preparations period of the MQ NMR experiment, the system is irradiated by several such cycles. At the phase increment of the radio-frequency pulses $\Phi$ , the nonsecular two-spin/two-quantum Hamiltonian (1) has the form (Fig. 1)

[TABLE]

The phase increments of the radio-frequency pulses serve the purpose of the separation of signals from the MQ NMR coherences of different orders on the free evolution period Shykind (Fig. 1).

Since the MQ NMR coherences cannot be observed directly they are transformed into the transverse magnetization (single-quantum coherence) in the mixing period. During the mixing period the spin system is irradiated with the same sequence of the radio-frequency pulses as during the preparation period but the pulse phase is shifted by $\pi/2$ ( $y$ -pulses are applied instead of $x$ -pulses). Formula (4) shows that the sign of the nonsecular two-spin/two-quantum Hamiltonian (1) changes due to such a phase shift. The time reversal conditions Rhim are necessary for the co-phasing of contributions of different pairs of spins into the MQ NMR coherences of different orders Baum .

The detecting pulse is applied on the detection period for the transmission of the magnetization in the plane perpendicular to the external magnetic field. The Fourier transform with respect to the phase increment makes it possible to obtain MQ NMR spectrum consisting of a set of narrow lines corresponding to different coherence orders Shykind .

We emphasize that only MQ NMR coherences of the zeroth and plus/minus second orders emerge in the model of the isolated spin chains in the approximation of the nearest neighbor interactions Feld2 ; Feld3 ; Doronin2 ; Feld4 . Our experimental work Doronin3 demonstrates that more than 97 % of the observed signal corresponds to the MQ NMR coherences of the zeroth and plus/minus second orders. The theoretical intensities of the MQ NMR coherences of the zeroth ( $G_{0}(\tau)$ ) and plus/minus second ( $G_{\pm 2}(\tau)$ orders on the preparation period in infinite spin chains are the following Feld2 ; Feld3 ; Doronin2 ; Feld4 :

[TABLE]

where $J_{0}$ is the zero-order Bessel function and $D=D_{i,i+1}=16.4\cdot 10^{3}$ rad/s is the DDI nearest neighbor coupling constant in the spin chain of \BPChem1̂9F nuclei in calcium fluorapatite when the external magnetic field is directed along the spin chain.

III A quantum state transfer in one-dimensional spin chains

The ideal quantum state transfer is possible only for homogeneous spin chains consisting of up to 3 spins Christandl . However, one can engineer the coupling constants Christandl2 or add external manipulation of the spins at the chain ends Burgarth in order to perform the perfect transfer for chains of arbitrary length. The quantum state transfer in spin chains in the approximation of the nearest neighbors is performed when spin dynamics is governed by the flip-flop Hamiltonian $H_{ff}$ which can be written as

[TABLE]

Unfortunately, such a Hamiltonian is not the Hamiltonian in MQ NMR experiments. However, the Hamiltonian (6) is related to the two-spin/two-quantum Hamiltonian (1) by a very simple unitary transformation Doronin . Indeed, performing the transformation $U$ of the Hamiltonian (1)

[TABLE]

We apply selective $\pi$ pulses which flip the spins in even positions through $180^{\circ}$ about the $x$ -axis of the rotating reference frame. As a result, we obtain

[TABLE]

Although the transformation (7) cannot be realized experimentally it is very useful for the understanding of peculiarities of the quantum state transfer along spin chains.

The system’s initial state $\rho(0)$ , which we consider, has just spin $l$ polarized Cappellaro2

[TABLE]

where the parameter $\beta$ is inversely proportional to the temperature, $N$ is the number of spins in the chain, $Z=2^{N}\mathop{\rm ch}\nolimits{\frac{\beta}{2}}$ is the partition function, $l$ is a positive odd number and $E_{N\neq l}$ is the identity operator in the Hilbert space of all spins besides spin $l$ . In the MQ NMR experiment the density matrix $\rho(t)$ of the system at time $t$ is

[TABLE]

The polarization of spin $N$ at time $t$ is

[TABLE]

In Eq. (11) we suppose that $N$ is also odd. Thus, we have shown that the quantum state transfer in the MQ NMR experiment can be described by the flip-flop Hamiltonian of Eq. (6).

The spin dynamics of the considered system is solved exactly in terms of fermionic operators and their Fourier transform Doronin2 ; Feld2 . If the initial polarization was on spin $l$ (see Eq. (9)) than the polarization of spin $m$ at time $t$ is Feld5

[TABLE]

where the fermion spectrum $\epsilon_{k}$ is

[TABLE]

where $D=D_{i,i+1}$ , $\omega_{0}$ is the Larmor frequency and

[TABLE]

Eq. (12) was obtained Feld5 in the high temperature approximation Abragam . However, it was shown Feld6 that this relationship holds at arbitrary temperatures.

Numerical calculations show that almost all polarization (more than 90 %) can be transmitted to the given chain site for short chains. For long chains the efficiency of transmission gets lower.

The developed approach to the quantum state transfer relies on the approximation of the nearest neighbor interactions. It is known Cappellaro that the DDI of remote spins are not important for the quantum state transfer in short chains.

IV Relaxation of MQ NMR coherences as a model of decoherence processes

The dipolar relaxation of MQ NMR coherences can be studied on the evolution period of the MQ NMR experiment. Relaxation of MQ NMR coherences in one-dimensional systems has been studied earlier Doronin ; Kaur using the second moments of the line shapes of the MQ coherences of the zeroth and second orders. We suggested Bochkin to perform an analogous investigation using the ZZ-model, where only the ZZ-part of the DDI is taken into account.

The exact solution Feld2 ; Feld3 ; Doronin2 ; Feld4 for MQ NMR dynamics in one-dimensional systems allows us to obtain the density matrix $\sigma(\tau)$ on the preparation period of the MQ NMR experiment Baum in the approximation of the nearest neighbor interactions Abragam ; Goldman and to write it as follows

[TABLE]

where $\sigma_{i}(\tau)$ ( $i=0,2,-2$ ) describes the MQ NMR coherence of order $i$ . If the number of the spins $N\gg 1$ the contributions $\sigma_{0}(\tau)$ , $\sigma_{2}(\tau)$ , $\sigma_{-2}(\tau)$ are Feld2 ; Feld3

[TABLE]

where $k=\frac{2\pi n}{N}$ , ( $n=-N/2,\,-N/2+1,\,\dots N/2-1$ ) and $a_{k}^{+}$ , $a_{k}$ are the fermion operators Feld2 ; Feld3 .

We study the dipolar relaxation of the MQ NMR coherences Bochkin on the evolution period of the MQ NMR experiments Baum . This period begins immediately after the preparation period Baum and the density matrix (15) can be used as the initial state for the relaxation. The relaxation of MQ NMR coherences is caused by the secular (with respect to the external magnetic field directed along the $z$ -axis) DDI

[TABLE]

An investigation of the relaxation process with the Hamiltonian (19) is too complicated. However, the problem is substantially simplified Bochkin , if we restrict ourselves to the ZZ-part of $H_{dz}$ only and consider the Hamiltonian

[TABLE]

The problem with the Hamiltonian $H_{ZZ}$ was called Bochkin the ZZ-model.

This approximation is widely used Abragam in spin physics (for example, see Feld7 ). The intensity $F_{0}(\tau,t)$ of the MQ NMR coherence of the zeroth order in the course of the evolution period at the time moment $t$ is

[TABLE]

In Eq. (21) $I_{z}=\sum_{i=1}^{N}I_{iz}$ and $F_{0}(\tau,0)=G_{0}(\tau)$ (see Eq. (5)).

Analogously, the intensities of the MQ NMR coherences of the plus/minus second order on the evolution period are given by

[TABLE]

Our calculations demonstrate Bochkin that the MQ NMR coherence of the zeroth order is not subject to the dipolar relaxation in the ZZ-model. However, the experimental investigation Bochkin performed using quasi-one-dimensional chains of \BPChem1̂9F nuclei in calcium fluorapatite \BPChemCa_5(PO_4)_3F shows the dipolar relaxation of the MQ NMR coherence of the zeroth order due to the flip-flop part of the Hamiltonian (19). The experimental data Bochkin also demonstrate that relaxation does not lead to the full disappearance of the MQ NMR of the zeroth order. Relaxation ends with the stationary intensity of that MQ coherence. The point is that the density matrix of Eq. (16) contains a part which is proportional to the operator $I_{z}$ , commuting with the DDI Hamiltonian (19). As a result, one can obtain Bochkin the stationary intensity of the MQ NMR coherence of the zeroth order $F_{o}^{st}$ which is

[TABLE]

The experimental dependence of the stationary intensity of the MQ NMR coherence of the zeroth order during the preparation period is presented in Fig. 2. The experimental data are in a close agreement with Eq. (21).

Calculations of the relaxation of the MQ NMR coherences of the plus/minus second order with Eq. (21) yield Bochkin

[TABLE]

We have shown Bochkin that the decay of the MQ NMR coherences of the second order conforms reasonably well to a Gaussian function $S(t)=\exp\{-\frac{M_{2}(\tau)t^{2}}{2}\}$ , where the second moment $M_{2}(\tau)$ of the line shape of this coherence is

[TABLE]

In Fig. 3 the dependencies of the experimental and theoretical (at $N=150$ ) dipolar relaxation times on the duration of the preparation period are shown. One can see that the theoretical predictions satisfactorily describe the experimental data.

V Conclusions

We considered the methods of MQ NMR spectroscopy of one-dimensional spin systems and their application for solving problems of quantum informatics. The quantum state transfer in linear spin chains can be realized with high fidelity using MQ NMR dynamics. Decoherence of many-qubit quantum state created on the preparation period of the MQ NMR experiment Baum can be studied on the evolution period of that experiment. The dipolar relaxation of the MQ NMR coherences is considered as a model of the decoherence process. The theory of the dipolar relaxation of the MQ NMR coherences is developed. The performed MQ NMR experiments on the quasi-one-dimensional system of \BPChem1̂9F nuclei in calcium fluorapatite \BPChemCa_5(PO_4)_3F are in a good agreement with the theoretical predictions.

Acknowledgements.

The work is supported by the Russian Foundation for Basic Research (Grants 16-03-00056 and 16-33-00867) and the Program of RAS ”Element base of quantum computers” (Grant No 0089-2015-0220).

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