# Zero- to ultralow-field nuclear magnetic resonance and its applications

Min Jiang  Hefei National Laboratory for Physical Sciences at the Microscale
and Department of Modern Physics, University of Science and Technology of
China, Hefei 230026, China  CAS Key Laboratory of Microscale Magnetic
Resonance, University of Science and Technology of China, Hefei 230026, China
Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China  Ji Bian
Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei 230026, China  CAS Key Laboratory of Microscale Magnetic Resonance,
University of Science and Technology of China, Hefei 230026, China  Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei 230026, China  Qing Li  Hefei National
Laboratory for Physical Sciences at the Microscale and Department of Modern
Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and
Technology of China, Hefei 230026, China  Synergetic Innovation Center of
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei 230026, China  Ze Wu  Hefei National Laboratory for Physical
Sciences at the Microscale and Department of Modern Physics, University of
Science and Technology of China, Hefei 230026, China  CAS Key Laboratory of
Microscale Magnetic Resonance, University of Science and Technology of China,
Hefei 230026, China  Synergetic Innovation Center of Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei 230026,
China  Haowen Su  Hefei National Laboratory for Physical Sciences at the
Microscale and Department of Modern Physics, University of Science and
Technology of China, Hefei 230026, China  CAS Key Laboratory of Microscale
Magnetic Resonance, University of Science and Technology of China, Hefei
230026, China  Synergetic Innovation Center of Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
Minxiang Xu  Hefei National Laboratory for Physical Sciences at the Microscale
and Department of Modern Physics, University of Science and Technology of
China, Hefei 230026, China  CAS Key Laboratory of Microscale Magnetic
Resonance, University of Science and Technology of China, Hefei 230026, China
Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China  Yuanhong
Wang  Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei 230026, China  CAS Key Laboratory of Microscale Magnetic Resonance,
University of Science and Technology of China, Hefei 230026, China  Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei 230026, China  Xin Wang  Hefei National
Laboratory for Physical Sciences at the Microscale and Department of Modern
Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and
Technology of China, Hefei 230026, China  Synergetic Innovation Center of
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei 230026, China  Xinhua Peng<EMAIL_ADDRESS>Hefei National
Laboratory for Physical Sciences at the Microscale and Department of Modern
Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and
Technology of China, Hefei 230026, China  Synergetic Innovation Center of
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei 230026, China

###### Abstract

As a complementary analysis tool to conventional high-field NMR, zero- to
ultralow-field (ZULF) NMR detects nuclear magnetization signals in the sub-
microtesla regime. Spin-exchange relaxation-free (SERF) atomic magnetometers
provide a new generation of sensitive detector for ZULF NMR. Due to the
features such as low-cost, high-resolution and potability, ZULF NMR has
recently attracted considerable attention in chemistry, biology, medicine, and
tests of fundamental physics. This review describes the basic principles,
methodology and recent experimental and theoretical development of ZULF NMR,
as well as its applications in spectroscopy, quantum control, imaging, NMR-
based quantum devices, and tests of fundamental physics. The future prospects
of ZULF NMR are also discussed.

###### Contents

  1. I Introduction
  2. II Basic principles
    1. II.1 Procedure of an NMR experiment
    2. II.2 Spin systems at ultralow field
  3. III Experimental apparatus
  4. IV Spectroscopy
    1. IV.1 Zero-field NMR spectroscopy
    2. IV.2 Near-zero-field NMR spectroscopy
    3. IV.3 Two-dimensional NMR spectroscopy
    4. IV.4 Asymmetric NMR spectroscopy
  5. V Quantum control techniques
    1. V.1 Controllability of ZULF NMR systems
    2. V.2 Universal quantum control
    3. V.3 Numerical optimal control
    4. V.4 Time optimal control
    5. V.5 Transition-selective pulses
    6. V.6 Evaluation of quantum control
  6. VI Imaging
  7. VII NMR-based quantum devices
    1. VII.1 Comagnetometer
    2. VII.2 Floquet maser
  8. VIII Tests of fundamental physics
  9. IX Conclusion and perspective

##  I Introduction

Nuclear magnetic resonance (NMR), conventionally operated at high magnetic
fields (on the order of tesla), is a powerful analytical technique in
chemistry, biology, and medicine ernst1987principles ; liang2000principles ;
wuthrich1986nmr ; levitt2013spin . Although high magnetic fields for NMR have
their virtues, high-field NMR spectrometers with superconducting magnets are
expensive (on the order of $\$1$ million), require constant cryogenic
maintenance, and are immobile. Moreover, due to the existence of strong
magnets, high-field NMR can not be used on samples with metallic inclusions or
implanted devices. In magnetic resonance imaging (MRI) liang2000principles ,
the patients with pacemakers are not available to MRI, because the pacemakers
can malfunction at strong magnetic fields. Also, strong magnets often have
severe magnetic field inhomogeneity, limiting the ability to obtain high-
resolution NMR spectra. Thus conventional NMR requires sophisticate magnetic
field shimming systems to produce spatially uniform field.

In contrast to conventional high-field NMR, an opposite trend called “zero- to
ultralow-field (ZULF) NMR” weitekamp1983zero ; blanchard2007zero ;
mcdermott2002liquid ; ledbetter2011near ; bevilacqua2009all ;
tayler2017invited ; liu2013ultralow ; garcon2019constraints  has developed
toward using ultralow, submicrotesla or even no external field at all.

Figure 1: (Color online) Applications of ZULF NMR: (1) Materials science.
Reprinted with permission from ref. ledbetter2008zero , Copyright @ 2008
National Academy of Sciences. (2) Imaging. Reprinted with permission from
refs. xu2006magnetic ; savukov2013magnetic ; savukov2013anatomical , Copyright
@ 2013 American Chemical Society, @ 2006 National Academy of Sciences, @ 2013
Elsevier. (3) Quantum information processing. Reprinted with permission from
ref. jiang2018experimental , Copyright @ 2018 American Association for the
Advancement of Science. (4) NMR-based quantum devices. Reprinted with
permission from ref. jiang2019floquet . (5) Tests of fundamental physics.
Reprinted with permission from ref. garcon2019constraints , Copyright @ 2019
American Association for the Advancement of Science.

As shown in Fig. 1, ZULF NMR has already demonstrated a variety of
applications ledbetter2008zero ; xu2006magnetic ; savukov2013magnetic ;
savukov2013anatomical ; jiang2018experimental ; jiang2019floquet  ranging from
materials science, quantum information processing, imaging, and NMR-based
quantum devices to tests of fundamental physics. Without the need of
superconducting magnets, ZULF NMR spectrometers could be inexpensive and
portable, and can be used to do research in situ without the need to deliver
samples to the laboratory. In addition to the economy and portability, ZULF
NMR has extremely high absolute magnetic field homogeneity and offers improved
spectral resolution ranging from Hz all the way down to tens of mHz
blanchard2013high , allowing one to detect certain low-frequency spin-spin
interactions that would be challenging to observe in conventional high-field
NMR. For comparison, one has to employ sophisticated and meticulously tuned
shim coils to realize linewidths narrower than 0.5 Hz in high-field NMR
appelt2006chemical . Moreover, ZULF NMR opens the door to new applications
otherwise inaccessible to high-field NMR, such as imaging and spectroscopy
inside of metal objects or heterogeneous materials xu2006magnetic ;
savukov2013magnetic ; savukov2013anatomical . At ultralow field, the Zeeman
interactions are negligible or small enough that they can be treated as a
perturbation on the internal spin-spin interactions (for example, J-coupling
and dipole-dipole interactions). This means that the nuclear spins are
strongly coupled together compared with the coupling to the external magnetic
field. Such strong coupled spin systems should be particular interesting for
quantum simulation, such as investigating the topological order and Lee-Yang
zeros of matter luo2018experimentally ; peng2015experimental .

Although ZULF NMR has many advantages compared with high-field NMR, its
detection was challenging at the beginning of its development. Because ZULF
NMR typically operates in low frequency, the traditional Faraday induction
coils are not highly sensitive for low-frequency signals; thus the samples
should be transferred to high field regime for detection at the early age of
ZULF NMR weitekamp1983zero . In recent years, with the advent of new quantum
sensing techniques, ZULF NMR gradually becomes mature and serves as a
complementary tool to high-field NMR. ZULF NMR systems can use superconducting
quantum interference devices (SQUIDs) as magnetic field sensors
mcdermott2002liquid ; greenberg1998application ; zotev2007squid . One drawback
is that SQUIDs must operate at liquid-helium temperature and require constant
cryogenic maintenance. Recent years have seen increased developments in atomic
magnetometers budker2002resonant ; budker2007optical ;
kominis2003subfemtotesla ; li2006parametric ; shah2007subpicotesla ;
fang2012situ ; sheng2013subfemtotesla ; RNGradiometer , both in sensitivity
and portability. A spin-exchange relaxation-free (SERF) atomic magnetometer
allred2002high ; dang2010ultrahigh , with a measurement volume of
$0.45~{}\textrm{cm}^{3}$, has demonstrated sensitivity of
$0.16~{}\textrm{fT}~{}{\rm{Hz}}^{-1/2}$, comparable to the most advanced
SQUIDs. SERF atomic magnetometers have been recently used in ZULF NMR to
detect pure J-coupling spectra at zero-field ledbetter2009optical ;
blanchard2013high ; theis2011parahydrogen , and determine spin-coupling
topology at near-zero magnetic field ledbetter2011near ; jianginterference .

One of the greatest challenges in ZULF NMR is how to selectively manipulate
individual spin species. This is the basis for developing more sophisticated
pulse sequences, such as multidimensional NMR spectroscopy technique,
dynamical decoupling, spin echo, etc. Nuclear spins with different
gyromagnetic ratios all have identical (zero) or near identical Larmor
frequency at zero and ultralow field, and thus individual manipulation of
different spin species presents a challenge and limits the possible
applications of ZULF NMR. Previous examples of coherent isotropic averaging
pulse sequences in zero-field NMR have been limited to the homonuclear case
llor1995coherent ; llor1995coherent2 . More recent pulse sequences for
heteronuclear spins have been demonstrated, but have been limited by the
ability to generate individual arbitrary rotations for different spin species
thayer1986composite ; lee1987theory ; ledbetter2009optical ; blanchard2013high
; ledbetter2011near . We designed and performed a set of composite pulse
sequences that can realize universal quantum control of heteronuclear spin
systems at zero and ultralow field jiang2018experimental ; bian2017universal .
For manipulating homonuclear spin systems, a technique of shaped pulse
sequences based on numerical optimization method was reported in ref.
jiang2018numerical . Tobias et al. sjolander2020two ; sjolander201713c
realized pulse sequences towards two-dimensional zero-field NMR, which enables
one to measure spin coupling networks. To estimate the fidelity of pulse
sequences, several approaches jiang2018experimental ; ji2018time  have
recently been demonstrated, including quantum information-inspired randomized
benchmarking, state tomography, and partial process tomography. High-fidelity
realization of such exquisite and universal pulse sequences in ZULF NMR is
currently a major goal.

A wide range of applications based on ZULF NMR have been demonstrated ranging
from materials science, imaging, quantum information processing, and NMR-based
quantum devices to tests of fundamental physics. For example, Barskiy et al.
investigated the process of chemical exchange from ZULF NMR spectra
barskiy2019zero . Ledbetter et al. RNLiquidstate  and Wu et al.
RNComagnetometer  demonstrated two prototypes of liquid-state comagnetometer
based on ultralow-field NMR systems, which show the potential to strengthen
limits on spin-gravity coupling by several orders of magnitude. More recently
ZULF NMR has been applied to searches for axion and axion-like-particle dark
matter garcon2019constraints ; wu2019search  and provides a promising axion
research tool in the ultralight axion window. In the theoretical researches,
the absence of a large applied magnetic field allows for the measurement of
antisymmetric spin-spin couplings king2017antisymmetric , which are related to
chirality and have been proposed as a means for detecting molecular parity
nonconservation.

In this review, we first focus on the basic principles and methodology used in
ZULF NMR, such as main stages of NMR experiments and atomic magnetometers.
Then, great potential applications of ZULF NMR are presented, such as high-
resolution spectroscopy, quantum control, imaging, and the development of
quantum devices. Finally, we outline the future prospects of ZULF NMR in
materials science, chemical analysis, and fundamental physics, etc.

##  II Basic principles

We now describe the basic principles of ZULF NMR, including the procedure of a
ZULF NMR experiment and spin system at zero and ultralow field.

###  II.1 Procedure of an NMR experiment

Figure 2: (Color online) Three stages of an NMR experiment - polarization,
encoding, detection. In the presence of a strong magnetic field, the
traditional method is that a sample’s nuclear spins are polarized via thermal
equilibration. Information about the structure or spatial distribution of
nuclei is encoded in the form of spin resonances, such as chemical shifts or
an applied magnetic field gradient; and Faraday induction coils are used to
detect the NMR signal. At ultralow field, spin polarization can be achieved by
dynamic nuclear polarization, spin-exchange optical pumping, chemically
induced dynamic nuclear polarization or parahydrogen-induced polarization;
encoding can take the form of J-coupling or dipole-dipole interaction;
superconducting quantum interference devices (SQUIDs), atomic magnetometers or
nitrogen-vacancy center can be used to detect spin resonances. Reprinted with
permission from ref. ledbetter2013nuclear .

There are three main stages of an NMR experiment - polarization, encoding, and
detection liang2000principles ; levitt2013spin ; ledbetter2013nuclear , as
shown in Fig. 2. In conventional high-field NMR, the three main stages occur
in a same spatial region, whereas the three stages are usually spatially
separated. In traditional high-field NMR, a sample’s nuclear spins are
polarized thermally in a superconducting magnet. In contrast, ZULF NMR
experiments usually use permanent magnet arrays (for example, low-cost Halbach
magnets tayler2017invited ) to polarize nuclear spins in samples. Although the
field strengths of permanent magnets (1-2 T) are relatively weaker than that
of superconducting magnets, they are low cost and portable. In particular,
they are not required with high field homogeneity. However, the polarization
obtained by thermal equilibration is on the order of 1 part per million (see
Sec. II.2). Alternatively, large nuclear spin polarizations can be achieved
through hyperpolarization techniques, such as parahydrogen-induced
polarization (PHIP) adams2009reversible ; theis2011parahydrogen , spin-
exchange optical pumping (SEOP) walker1997spin , dynamic nuclear polarization
(DNP) maly2008dynamic , and chemically induced dynamic nuclear polarization
(CIDNP) lawler1967chemically .

Next, information about the samples’ molecular structure is encoded in the
form of spin-resonance frequencies. In high-field NMR, the spin resonance
frequencies depend on the applied magnetic fields according to spin
gyromagnetic ratios (including the effect of chemical shift) and internal
spin-spin interactions. In contrast, the encoding process is accomplished
through internal spin-spin interactions such as J-coupling and dipole-dipole
interactions. The details are present in Sec. II.2.

The detection of NMR signals is traditionally accomplished with Faraday
induction coils in high-field NMR, in which the induced voltage is
proportional to the rate of nuclear spin precession. However, ZULF NMR
typically operates in low frequency (below 1 kHz), thus traditional Faraday
induction coils are not highly sensitive. With the advent of non-inductive
sensors, such as SQUIDs mcdermott2002liquid ; greenberg1998application ;
zotev2007squid  and atomic magnetometers budker2007optical ;
kominis2003subfemtotesla ; budker2002resonant ; shah2007subpicotesla ;
RNGradiometer , ZULF NMR gradually becomes a mature and useful technique. In
particular, atomic magnetometers based on spin-exchange relaxation-free
mechanics have reached at the subfemtotesla level kominis2003subfemtotesla ;
dang2010ultrahigh . In addition to sensitivity, atomic magnetometers work best
between room temperature and 180 0C without the need of cryogenic condition,
and are well suited for detecting ZULF NMR signals. More recently, it has
realized a gradiometric ZULF NMR spectrometer based on an atomic gradiometer
RNGradiometer , which enables one order of magnitude enhancement in the
signal-to-noise ratio (SNR) compared to the single-channel configuration.

###  II.2 Spin systems at ultralow field

A liquid-state n-spin system at an external ultralow magnetic field can be
described by the Hamiltonian:

$H=H_{J}+H_{Z}=\sum\limits_{i,j>i}{2\pi{J_{ij}}}{\textbf{I}_{i}}\cdot{\textbf{I}_{j}}-\sum\limits_{j}{{\gamma_{j}}{\textbf{I}_{j}}\cdot\textbf{B}},$
(1)

where $H_{J}$ and $H_{Z}$ are the scalar spin-spin coupling (so called
J-coupling) and the Zeeman interaction respectively, ${J_{ij}}$ is the
strength of the J-coupling between the $i$th and $j$th spins.
$\mathbf{I}_{i}=(I_{ix},I_{iy},I_{iz})$ is the spin angular momentum operator
of the $i$th spin, and the reduced Planck constant $\hbar$ is set to one. The
effects of dipole-dipole interactions are ignored here, as in an isotropic
liquid, they are averaged out due to the rapid and random flipping motion of
the molecules. In ZULF NMR, the Zeeman interactions are much smaller than the
coupling interaction ($|\gamma_{j}\textbf{B}|\ll|J|$), so that Zeeman
interactions are treated as perturbations. As we show below, the spectra of
samples at a near-zero field usually split into, for example, doublet,
triplet, etc ledbetter2011near ; appelt2010paths ; appelt2007phenomena . Such
near-zero-field splitting patterns provide plentiful information on energy
degeneracies and can be used as a diagnostic tool to determine molecular
structure.

We consider the thermal equilibration in a permanent magnet, which is a
universal approach for preparing initial spin polarization of samples. The
schematic diagram of ultralow-field NMR apparatus (see, e.g., refs.
jiang2018experimental ; ji2018time ; RNGradiometer ) based on an atomic
magnetometer is shown in Fig. 3a. Liquid-state samples are contained in
standard 5-mm NMR tubes, and pneumatically shuttled between a prepolarizing
Halbach magnet (the field in the center is about 1-2 T) and an alkali-metal
vapor cell. During the transfer, a guiding magnetic field is applied along the
transfer direction (along the $z$ axis). The magnitude of the guiding field
($B_{g}\approx 10^{-4}~{}\textrm{T}$) satisfies
$B_{g}\gg|J_{ij}/(\gamma_{i}-\gamma_{j})|$, so the spin state remains the
high-field equilibrium state.

The way that the guiding field is switched off plays a crucial role in
determining the initial state and, in turn, the amplitude and phase of the
oscillating NMR magnetization signal produced jiang2018experimental ;
blanchard2007zero . We consider two cases where the guiding field is switched
off suddenly or adiabatically. When the guiding field is suddenly switched
off, the relevant initial spin state is

$\begin{array}[]{l}\rho_{0}=\mathbbm{1}/2^{n}-\sum_{j}\varepsilon_{j}{I}_{jz},\end{array}$
(2)

where $\varepsilon_{j}=\gamma_{j}B_{p}/k_{\textrm{B}}T\sim 10^{-6}$,
${I}_{jz}$ is the spin angular momentum operator along the $z$ orientation of
the $j$th spin with gyromagnetic ratio $\gamma_{j}$, $k_{\textrm{B}}$ is the
Boltzmann constant, $T$ is the temperature. In this case, $\rho_{0}$ is called
as sudden state. Alternatively, when the guiding field is adiabatically turned
off (with the characteristic time scale defined by the strength of the
J-coupling), the populations at high field are converted to the populations of
the zero- or ultralow-field eigenstates. For example, the initial state of a
two-spin system ($I$ and $S$) is emondts2014long ; jiang2018experimental ;
blanchard2007zero

$\begin{array}[]{l}\rho_{0}=\frac{(\gamma_{I}+\gamma_{S})B_{p}}{8k_{B}T}(I_{z}+S_{z})-\frac{(\gamma_{I}-\gamma_{S})B_{p}}{4k_{B}T}(I_{x}S_{x}+I_{y}S_{y}),\end{array}$
(3)

which is the adiabatic state. One can similarly achieve the form of the
adiabatic state of multi-spin molecules.

The nuclear spins in the sample then evolve at zero or ultralow bias magnetic
field and generate an oscillating magnetization,

$\rho(t)=e^{-iHt}\rho_{0}e^{iHt},$ (4)

where $\rho(t)$ is the time dependent density matrix. The magnetization along
$\eta=x,y,z$ can be calculated by

$M_{\eta}(t)=n\textrm{Tr}[\rho(t)\sum_{j}\gamma_{j}I_{j,\eta}],$ (5)

where $n$ is the molecular density. Expanding Eq. 5 in terms of matrix
elements between the normalized eigenstates $\left\\{\ket{\phi_{k}}\right\\}$
of $H$:

$M_{\eta}(t)=n\sum_{a,b,j}\rho_{ab}\bra{\phi_{b}}\left(\gamma_{j}I_{j\eta}\right)\ket{\phi_{a}}e^{-i2\pi\nu_{ab}t},$
(6)

where $\ket{\phi_{a}}$ and $\ket{\phi_{b}}$ denote different eigenstates,
$\nu_{ab}$ is the transition frequency between the eigenstates
$\ket{\phi_{a}}$ and $\ket{\phi_{b}}$ and
$\rho_{ab}=\bra{\phi_{a}}\rho_{0}\ket{\phi_{b}}$. Further, the nuclear
magnetization can produce magnetic fields on the vapor cell of atomic
magnetometer. Approximating the NMR sample as a sphere, the magnetic moment
due to the sample magnetization is $\mathbf{m}=(4\pi r_{0}^{3}/3)\mathbf{M}$,
where $r_{0}$ is the radius of the sample, and
$\mathbf{M}=[M_{x},M_{y},M_{z}]$ is the nuclear magnetization. The magnetic
field originated from the sample at the location of the atomic magnetometer is
RNGradiometer

$\displaystyle\mathbf{B}_{s}=\frac{\mu_{0}}{4\pi}\frac{3\hat{\mathbf{n}}(\mathbf{m}\cdot\hat{\mathbf{n}})-\mathbf{m}}{r^{3}},$
(7)

where $\mu_{0}$ is the permeability of vacuum, $\hat{\mathbf{n}}$ is the unit
vector pointing from the sample to the magnetometer and $r$ is the distance
between them. After applying a fast Fourier transform to the time-domain
magnetometer signal, frequency-domain ZULF NMR spectra can be achieved. It is
worthy noting that the magnetometer is sensitive to the oscillating magnetic
field along the $z$-axis (see Sec. III), so only $z$-magnetic field component
is considered.

Figure 3: (Color online) (a) Experimental apparatus of ZULF NMR based on an
atomic magnetometer. The NMR sample is contained in a standard 5-mm NMR tube
and pneumatic shuttled between a Halbach magnet and detection region. (b)
Single-shot zero-field NMR signal of 13C-formic acid. The lifetime of the
singlet-triplet coherence is 10.3s, and the FWHM (full width at half maximum)
is 32 mHz. Reprinted with permission from ref. jiang2018experimental ,
Copyright @ 2018 American Association for the Advancement of Science.

##  III Experimental apparatus

The schematic diagram of zero- to ultralow-field NMR apparatus based on an
atomic magnetometer is shown in Fig. 3. Atomic magnetometers usually operate
at zero or ultralow magnetic field and provide an exquisite sensitive approach
for detecting ZULF NMR signals. The atomic magnetometer consists of a warm
alkali-metal vapor cell (for example, 87Rb vapor cell). Nitrogen (N2) with a
pressure of about one atmosphere is added as buffer gas in the vapor cell, and
improve the detection bandwidth (above 100 Hz) of atomic magnetometer. It is
necessary to match the dimensions of the vapor cell and the detected sample.
The typical size of vapor cells is 0.5-1 cm, comparable to the standard 5-mm
NMR tube. We should note that the conventional atomic magnetometers use large
vapor cells, which are usually several centimeter, and thus are not applicable
to ZULF NMR detection. The vapor cell works in a stable temperature 180
${}^{\textrm{o}}$C by resistive heating with using twisted coils and high-
frequency AC current, and is placed inside a five-layer mu-metal magnetic
shield (shielding factor of $10^{6}$). The alkali-metal atoms in the vapor
cell are optically pumped with a circularly polarized laser beam propagating
in the $y$ direction. The pump-laser frequency is tuned close to the center of
the buffer-gas (N2) broadened and shifted D1 line. The magnetic field is
measured via optical rotation of linearly polarized probe laser beam
propagating along the $x$ direction (for example, see ref. allred2002high ).
The frequency of the probe laser is detuned from the D2 transition by about
100 GHz. To suppress the influence of low-frequency noise, the polarization of
the probe laser beam is usually modulated by a photoelastic modulator (PEM)
RNGradiometer  or Faraday modulator fang2012situ , and the signal is
demodulated by a lock-in amplifier.

We now calculate the oscillatory signal of an atomic magnetometer. The precise
description of the interaction between light and atoms requires the use of a
density matrix, but the ground state of alkali-metal atoms can be described by
a Bloch equation for the electron spin polarization P allred2002high ;
RNGradiometer ; jianginterference :

$\frac{d\textbf{P}}{dt}=\frac{1}{q}\left[\gamma_{e}\textbf{B}(t)\times\textbf{P}+R_{op}\left(\hat{\textbf{y}}-\textbf{P}\right)-R_{rel}\textbf{P}\right],$
(8)

where $\textbf{P}=\left\langle\textbf{S}\right\rangle/S$, S is the electron
spin angular momentum, $\gamma_{e}=g_{s}\mu_{B}$, $g_{s}\approx 2$ is the
electron Landé factor, $\mu_{B}$ is the Bohr magneton, $\textbf{B}(t)$ is the
applied magnetic field, $R_{op}$ is the optical pumping rate due to the pump
beam, $\hat{\textbf{y}}$ is the direction of the pump beam, $R_{rel}$ is the
spin relaxation rate in the absence of optical pumping, and $q$ is the nuclear
slowing-down factor. In the present of a slowly changing magnetic field, the
quasi-steady-state solution to Eq. 8 can be found, we introduce
$\beta=\gamma_{e}\textbf{B}/\left(R_{op}+R_{rel}\right)$ and
$P_{0}=R_{op}/\left(R_{op}+R_{rel}\right)$. The solution of above equation is

$\displaystyle\begin{cases}P_{x}&=P_{0}\frac{\beta_{z}+\beta_{x}\beta_{y}}{1+|\beta|^{2}},\\\
P_{y}&=P_{0}\frac{1+\beta_{y}^{2}}{1+|\beta|^{2}},\\\
P_{z}&=P_{0}\frac{-\beta_{x}+\beta_{z}\beta_{y}}{1+|\beta|^{2}}.\\\
\end{cases}$ (9)

Because the propagation direction of the probe light is along $x$, the Faraday
rotation angle of the probe light is proportional to $P_{x}$ allred2002high ;
dang2010ultrahigh . When the magnetic field is very small ($|\beta|\ll 1$),

$P_{x}\approx P_{0}\beta_{z}.$ (10)

Hence, atomic magnetometer is primarily sensitive to magnetic field along $z$
direction. However, in the presence of a bias magnetic field along $y$,

$P_{x}\approx P_{0}\frac{\beta_{z}+\beta_{x}\beta_{y}}{1+\beta_{y}^{2}}.$ (11)

In this situation, atomic magnetometer can be simultaneously sensitive to the
$x$ and $z$-direction magnetic fields. As discussed in Sec. IV.4, an
asymmetric ZULF NMR spectroscopy can be generated due to that atomic
magnetometer is both sensitive to the NMR signals along $x$ and $z$. A
detailed discussion can be found in ref. jianginterference .

Samples ($\sim 200$ $\mu$L) were flame-sealed under vacuum in a standard
$5$-mm glass NMR tube following five freeze-pump-thaw cycles in order to
remove dissolved oxygen, which is otherwise a significant source of
relaxation. Liquid-state samples are pneumatically shuttled between a
prepolarizing Halbach magnet and an alkali-metal vapor cell. The bottom of the
NMR sample tube is $\sim$1 mm above a vapor cell of the atomic magnetometer.
Before detection, it is usually necessary to apply magnetic field pulses on
the samples’ spins (see Sec. V) for optimizing NMR signal amplitude. There are
two sets of orthogonal three dimensional coils placed around the vapor cell.
One set of coils is used to apply uniform bias magnetic field. Another set of
coils is used to apply pulses. To apply pulses, the desired waveform is
written as a list of time/voltage coordinates ($t$,$V_{x}$,$V_{y}$,$V_{z}$),
which are converted into a three-channel analog output voltage. Each output
channel is fed into a linear high-power amplifier, whose output is connected
to one of the coils.

Figure 4: (Color online) (a) Schematics of the gradiometric detection for ZULF
NMR. The detection is based on an atomic magnetic gradiometer. Inset shows
that the probe beam (with diameter $\sim 0.8~{}\textrm{cm}$) is split into top
and bottom channels using a D-shaped mirror. (b) Single-shot ultralow-field
NMR of 13C-formic acid. The 13C-formic acid time-domain signals and their
spectra are observed in top (top), bottom (middle), and gradiometric (bottom)
channels in the presence of a small magnetic field. Reprinted with permission
from ref. RNGradiometer , Copyright @ 2019 American Physical Society.

Current ZULF NMR spectrometers are always equipped with high-quality magnetic
shields to ensure that ambient magnetic field noise does not dwarf the
magnetization signal. An alternative approach is to separate the magnetization
signal from the noise based on their differing spatial profiles, as can be
achieved using a magnetic gradiometer. We have recently realized a
gradiometric ZULF NMR spectrometer RNGradiometer  (see Fig. 4) with a magnetic
gradient noise of 17$~{}\textrm{fT}~{}\textrm{cm}^{-1}~{}\textrm{Hz}^{-1/2}$
in the frequency range of 100-400 Hz, based on a single vapor cell
(0.7$\times$0.7$\times$1.0 $\textrm{cm}^{3}$). The gradiometric spectrometer
achieves 13-fold enhancement in the signal-to-noise ratio (SNR) compared to
the single-channel configuration. By reducing the influence of common-mode
magnetic noise, the gradiometric NMR spectrometer enables the use of compact
and low-cost magnetic shields.

##  IV Spectroscopy

Due to the high absolute magnetic field homogeneity, ZULF NMR can realize
ultrahigh-resolution spectroscopy as a powerful diagnostic tool for non-
destructive structural investigations of matter. This is contrast to high-
field NMR, where magnetic field inhomogeneity is the main factor to cause NMR
line broadening. In addition, the spectral linewidth of ZULF NMR depends on
spin relaxation times. In contrast to high-field case, the absence of certain
relaxation pathways emondts2014long  (for example, chemical shift anisotropy)
allows ZULF NMR to achieve long spin relaxation times. For example, nuclear
spin relaxation time is about $10.3$ s for formic acid (e.g., see Fig. 3b)
jiang2018experimental , $4.7$ s for acetonitrile, $8.8$ s for acetic acid, and
$6.9$ s for fully labeled acetonitrile. Some NMR samples may have short
relaxation times. For example, nuclear spin relaxation time is about $0.8$ s
for formaldehyde. Overall, ZULF NMR offers spectral resolution ranging from Hz
all the way down to tens of mHz. Moreover, the absence of truncation by a
large applied magnetic field means that ZULF NMR is capable of measuring spin-
dependent interactions that do not commute with the Zeeman Hamiltonian
blanchard2015measurement , which is not generally accessible in conventional
high-field NMR experiments. In this section, we review some recent experiments
of ZULF NMR spectroscopy.

Figure 5: (Color online) Experimental ZULF NMR spectra. The initial spin state
is prepared by the guiding field along the $z$ axis. (a, b) Formic acid
(H13COOH, where the acidic proton is negligible due to rapid exchange), (c, d)
Formaldehyde (H313CCOOH), (e, f) Acetonitrile (13CH313C15N). The spectra are
detected in the absence of a magnetic field and the presence of a magnetic
field.

###  IV.1 Zero-field NMR spectroscopy

Zero-field NMR spectra are entirely determined by J-couplings between nuclear
spins, which are thus called “pure J-coupling spectroscopy”. J-couplings,
which are mediated through chemical bonds connecting two spins, can impart
valuable information about chemical bonding and molecular structure.
Typically, the strength of J-couplings is in the range of 100-200 Hz for
proton and carbon nuclei separated by a single covalent bond, and it quickly
decreases as the number of intermediate bonds grows ernst1987principles . The
high-resolution of zero-field NMR spectra could be at the level of 10 mHz
ledbetter2009optical ; blanchard2013high ; ledbetter2013nuclear  and
facilities the measurement of multi-bond J-coupling. Here we summarize the
recent development of zero-field NMR spectroscopy and its applications in
chemical fingerprinting ledbetter2009optical . We consider typical XAn type
molecules, where each “X” spin couples to $n$ equivalent “A” spins with the
same J-coupling strength J${}_{\textrm{AX}}$. The J-coupling Hamiltonian is
written as

$H_{J}=2\pi J_{AX}\mathbf{K_{A}}\cdot\mathbf{S},$ (12)

here $\mathbf{K_{A}}$ is sum of the spin $A$, $\mathbf{S}$ is the angular
momentum of spin $X$, and the total angular momentum
$\mathbf{F_{A}}=\mathbf{K_{A}}+\mathbf{S}$. The eigenstates
$\ket{{f_{A},{m_{f_{A}}}}}$ have energy

$E^{(0)}=J_{AX}/2[f_{A}(f_{A}+1)-k_{A}(k_{A}+1)-s(s+1)],$ (13)

and its degeneracy is $2f_{A}+1$. Because the detectable magnetization is
$M_{z}(t)=n\textrm{Tr}[\rho(t)\sum\limits_{j}\gamma_{j}I_{jz}]$, we obtain
selection rules for observable coherences: $\Delta f_{A}=0,\pm 1$ and $\Delta
m_{f_{A}}=0$. The equivalence of the XAn imposes an additional selection rule,
$\Delta k_{A}=0$ ledbetter2011near ; appelt2010paths . Accordingly, one could
observe $n-2k$ resonance lines with the frequencies
$\nu^{0}=\frac{1}{2}J_{AX}(1+n-2k)$, where $k=n/2-k_{A}$. Under these
circumstances, we can conclude that for odd $n$, an XAn system produces
$(n+1)/2$ lines at frequencies ranging from $J_{AX}$ to $(n+1)/2\times J_{AX}$
in step of $J_{AX}$, and for even $n$ the number of lines is $n/2$ with
frequencies ranging from $3/2J_{AX}$ to $(n+1)/2\times J_{AX}$ in step of
$J_{AX}$. As shown in Fig. 5a, c, the resulting zero-field J-coupling spectra
consist of a single line at J for XA, and two lines, one at J and the other at
2J, for XA3.

A variety of chemical samples have been detected at zero field, and their
zero-field spectra are high-resolution and information-rich spectra that are
well suited for chemical fingerprinting. For XAn system, such as formic acid
(H13COOH), formaldehyde (13CH2O), methanol (13CH3OH), acetonitrile (13CH3CN),
acetone((CH3)13COCH3), trimethyl phosphate (PO(OCH3)3)) blanchard2007zero ;
ledbetter2011near ; mcdermott2002liquid . For (XAn)Bm system, such as methyl
formate(H13COOCH3), formamide (H${}_{2}^{15}$NOH), ethanol-1
(CH${}_{3}^{13}$CH2OH), ethanol-2 (13CH3CH2OH),glycerol-2 (HO13CH(CH2OH)2)
ledbetter2009optical ; blanchard2007zero . For more complex systems: fully
labeled acetonitrile (13CH313C15N) (see Fig. 5e) ledbetter2011near ;
RNGradiometer , acetone doubly labeled (H313CCO13CH3) weitekamp1983zero ,
aromatic compounds blanchard2013high .

###  IV.2 Near-zero-field NMR spectroscopy

Although zero-field NMR spectra can distinguish some chemical groups, it still
leaves some ambiguity in determination of chemical groups, and this ambiguity
can be removed by application of small magnetic fields ledbetter2011near ;
appelt2010paths ; appelt2007phenomena . Here we introduce the investigations
of near-zero-field NMR spectroscopy, where the Zeeman interaction can be
treated as a perturbation to J-couplings. For simplicity, the direction of the
applied bias field is chosen as the quantization axis ($z$) and the sensitive
axis of the atomic magnetoemter is changed to $y$. The presence of very small
magnetic fields results in splitting of the zero-field NMR lines (Fig. 5b, d,
f), imparting considerable additional information to the pure zero-field
spectra. The molecular conformations of XA and XA2 cannot be distinguished
because the zero-field NMR spectra of XA and XA2 both show single NMR peak
(see Sec. IV.1). In contrast, near-zero-field NMR spectra of XA and XA2 are
significantly different. We take XAn spin system as an example to analyse. The
system Hamiltonian is described in Eq. 1. To first order in $B_{z}$,
eigenstates are those of the unperturbed Hamiltonian, and Zeeman shifts of the
eigenvalues can be read from the diagonal matrix elements of the Zeeman
perturbation. One finds blanchard2007zero ; ledbetter2011near ;
appelt2010paths ; appelt2007phenomena

$\begin{split}&\Delta E(f_{A},k_{A},m_{f_{A}})\\\
&=-\bra{{f_{A}m_{f_{A}}}}B_{z}(\gamma_{A}K_{Az}+\gamma_{X}S_{z})\ket{{f_{A}m_{f_{A}}}}\\\
&=-B_{z}\sum\limits_{m_{kA},m_{s}}\left\langle{k_{A}sm_{k_{A}}m_{s}|f_{A}m_{f_{A}}}\right\rangle^{2}(\gamma_{A}m_{k_{A}}+\gamma_{X}m_{s}),\end{split}$
(14)

where $\gamma_{A}$ and $\gamma_{X}$ are the spins A and X gyromagnetic ratios
respectively, and $\left\langle
k_{A}sm_{kA}m_{s}|{f_{A}m_{f_{A}}}\right\rangle$ is the Clebsch-Gordan
coefficient. The signal in our experiment is the $y$ component of the
magnetization $M_{y}(t)=n\textrm{Tr}[\rho(t)\sum\limits_{j}\gamma_{j}I_{jy}]$,
the relevant selection rules are $\Delta f_{A}=0,\pm 1$, $\Delta k_{A}=0$, and
$\Delta m_{f_{A}}=\pm 1$. Accordingly, one could observe $2(n-2k)$ resonance
lines centered at $\nu^{0}$, with the frequencies

$\begin{cases}\nu_{f_{A},m_{f_{A}};k_{A}}^{f_{A}^{\prime},m_{f_{A}}\pm
1;k_{A}}=\nu^{0}+\Delta\nu,\\\ \nu^{0}=\frac{1}{2}J_{AX}(1+n-2k),\\\
\Delta\nu=[\frac{2m_{f_{A}}(-\gamma_{A}+\gamma_{X})}{1+n-2k}\pm\frac{(n-2k)\gamma_{A}+\gamma_{X}}{1+n-2k}]B_{z},\end{cases}$
(15)

where $\nu^{0}$ is frequency of resonance line in zero field (see Sec. IV.1)
and $\Delta\nu$ implies Zeeman splitting. For example of $n=1$, a doublet (see
Fig. 5b) for the transition between states with $f_{A}=1$ and $f_{A}=0$:

$\nu_{0,0}^{1,\pm 1}=J\pm B_{z}(\gamma_{A}+\gamma_{X})/2.$ (16)

More examples can be found in refs. ledbetter2011near ; appelt2010paths . Some
samples have experimentally detected near-zero-field NMR spectra, such as
formic acid (H13COOH), formaldehyde (13CH2O), methanol (13CH3OH), acetonitrile
(13CH3CN), fully labeled acetonitrile (13CH313C15N) ledbetter2011near ;
mcdermott2002liquid ; RNGradiometer .

Figure 6: (Color online) Experimental two-dimensional zero-field NMR spectrum
of ethanol mixture using ($\pi_{x/y}-t_{1}-\pi_{x/y}$) pulse sequence. The
entire spectral window (both positive and negative frequencies) is displayed
in F1, whereas only the positive part of the F2 axis is shown. The spectrum
contains cross-peaks between transitions belonging to the same spin-state
manifolds. Meanwhile, there is no coherence transfer, and therefore no cross-
peaks, between either the two isotopomers or peaks corresponding to the same
isotopomer but different spin-manifolds. Reprinted with permission from ref.
sjolander2020two .

###  IV.3 Two-dimensional NMR spectroscopy

Spectral complexity increases rapidly with spin-system size. The development
of two-dimensional spectroscopy is a major reason behind the analytical power
of NMR, allowing the resolution of more crowded spectra. Additionally, many
pulse sequences exist that enable the mapping of coupling networks, the
simplification of spectral assignment, and structure elucidation. In liquid-
state analytical chemistry, multiple-quantum coherence filters combined with
two-dimensional detection techniques provide one of the standard ways to map
coupling networks ernst1987principles . Recently, Sjolander et al.
sjolander2020two ; sjolander201713c  introduced a technique of two-dimensional
correlation and single- and multiple-quantum experiments in the context of
liquid-state zero-field J-spectroscopy. For example, at zero field the
spectrum of ethanol appears as a mixture of 13C isotopomers, and correlation
spectroscopy is useful in separating the two composite spectra, as shown in
Fig. 6. Two-dimensional spectroscopy further improves the high resolution
attained in zero-field NMR since selection rules on the coherence-transfer
pathways allow for the separation of otherwise overlapping resonances into
distinct cross-peaks sjolander2020two .

###  IV.4 Asymmetric NMR spectroscopy

It has been extensively demonstrated that the ultralow-field NMR signals
recorded with atomic magnetometers uncommonly differ from the expected values
with a few tens of percent distortion ledbetter2011near ; RNComagnetometer ;
RNLiquidstate ; RNGradiometer ; garcon2019constraints ; wu2019search .
Ultralow-field NMR spectra of even simple samples like formic acid (containing
13C-1H spin pairs) suffer from severe asymmetric amplitudes (Fig. 7),
differing greatly from those predicted by the standard NMR theory
appelt2010paths ; appelt2007phenomena ; ledbetter2011near . Figure 7 shows the
experimentally observed NMR spectra of some samples, clearly exhibiting
asymmetric characteristics on their spectral amplitude. For example, the
relative NMR peak amplitudes for acetic acid are theoretically predicted to be
1:3:6:6:3:1, but in practical experiments the ratio is measured to be
1:4.6:12.3:16.4:6.2:1.5 (as shown in Fig. 7c).

Figure 7: (Color online) Experimental asymmetric ultralow-field NMR spectra.
(a) Formic acid, (b) Formaldehyde, (c) Acetic acid, (d) Fully labeled
acetonitrile. These are partial spectra of the samples. A small magnetic field
($\approx$$72.9$ nT) is applied along $x$ in the experiments. Results show
that the amplitudes of the NMR peaks with yellow color are different from the
corresponding peaks with red color. (e) Basics of the quantum sensor resulting
in the interference effect: The magnetic field components along the $x$\- and
$y$-axes both affect the sensor but the observed signal is given only by the
sum of their contributions, which have different sensitivities (amplitudes
$\textrm{A}_{x}$, $\textrm{A}_{y}$) and phases ($\Phi_{x}$, $\Phi_{y}$).
Reprinted with permission from ref. jianginterference , Copyright @ 2020
Wiley-VCH GmbH.

Recently, we uncovered an interference effect in atomic magnetometers
jianginterference , which causes an important systematic effect to greatly
deteriorate the accuracy of measuring magnetic fields. Based on the reported
interference effect, good agreement is found between the prediction and the
asymmetric amplitudes of resonant lines in ultralow-field NMR spectra. Figure
7e schematically shows the basic interference effect resulting from the
response of atomic magnetometer. ${\textrm{A}_{\xi}}(\nu,{B_{z}})$ and
${\Phi_{\xi}}(\nu,{B_{z}})$ ($\xi=x,y$) are, respectively, the amplitude and
phase response functions. Specifically, the output signal of the atomic
magnetometer $s(t)$ is an oscillating signal. The amplitude of the oscillating
$s(t)$ signal is

$\displaystyle S_{tot}^{2}$ $\displaystyle=$
$\displaystyle(S_{x}^{2}+S_{y}^{2})(1+\chi\cos\Delta\phi),$ (17)

where $S_{x}=S_{x}(\nu,{B_{z}})=\alpha\textrm{A}_{x}(\nu,{B_{z}})B_{x0}$ and
$S_{y}=S_{y}(\nu,{B_{z}})=\alpha\textrm{A}_{y}(\nu,{B_{z}})B_{y0}$
respectively denote the amplitude of the output oscillating signal when there
only exists the $x$\- or $y$-component in the input magnetic field
$\textbf{B}(t)=[B_{x0}\cos(2\pi\nu t+\theta_{x0}),B_{y0}\cos(2\pi\nu
t+\theta_{y0}),B_{z}]$. The proportionality constant $\alpha$ summarises, e.g.
amplifier gains and conversion factors of detectors. The term
“$\chi\cos\Delta\phi$” represents the interference effect when the $x$\- and
$y$-components are both nonzero. Here the interference phase
$\Delta\phi=\Delta\Phi+(\theta_{x0}-\theta_{y0})$ and the interference
contrast $\chi\equiv 2S_{x}S_{y}/(S_{x}^{2}+S_{y}^{2})$. Based on this
interference model, the asymmetric amplitudes of ZULF NMR spectra can be well
explained jianginterference . In addition, a standard approach is presented to
detecting and characterizing the interference effect in, but not limited to,
atomic magnetometers, enabling a higher confidence for gaining precise
knowledge from NMR spectra jianginterference .

##  V Quantum control techniques

In a conventional high-field NMR experiment, the strong static magnetic field
ensures the wide separation of resonance frequencies for non-zero-spin nuclei
(for example, 1H, 13C, 15N, and 19F) allowing each to be addressed selectively
using AC magnetic field pulses vandersypen2005nmr . In contrast, as the
resonance frequencies are not well separated (or even equal to zero) in ZULF
NMR, the selection of individual spin species is not straightforward
tayler2016nuclear , and thus presents a challenge and limits the possible
applications of ZULF NMR jiang2018experimental . Nevertheless, quantum control
methods are developed that could overcome the difficulties and achieve
sophisticated manipulations of spins in ZULF NMR bian2017universal ;
jiang2018numerical ; ji2018time ; tayler2016nuclear ; sjolander2016transition
; llor1995coherent ; lee1987theory ; llor1991scaling . These include zero-
field spin echoes llor1991scaling , rank-selective decoupling sequences
llor1995coherent ; llor1995coherent2 , universal quantum control based on
composite pulses bian2017universal  and numerical optimization methods
jiang2018numerical , time-optimal control ji2018time , and transition-
selective pulses sjolander2016transition . Moreover, experiments based on
these methods are performed tayler2016nuclear ; sjolander2016transition ;
jiang2018experimental ; ji2018time . Experimental techniques to characterize
the performance of quantum controls are also demonstrated
jiang2018experimental ; ji2018time . In this section, we give a review of
these quantum control methods and experiments in ZULF NMR.

###  V.1 Controllability of ZULF NMR systems

Figure 8: (Color online) Graphs representing the network of a $4$-spin system.
Nuclear spins (colored balls and different colors representing different
nuclear species) are labeled with different numbers. Graph (a) satisfies the
controllability condition (i) and (b) satisfies condition (ii), so they can be
completely controlled. Graphs (c), (d) and (e) satisfy neither condition (i)
nor (ii). Reprinted with permission from ref. jiang2018numerical , Copyright @
2018 American Physical Society.

The internal and control Hamiltonian of a liquid state NMR system at zero and
ultralow field are given in Sec. II.2. Before presenting how to realize
quantum control in such systems, one important issue is whether, or under what
conditions, it is possible to control a quantum system using any physically
permitted operations, i.e., the controllability problem. In a closed quantum
system, complete controllability refers to the ability of realizing any
element in $\textrm{SU}(2^{n})$ dong2010quantum . The complete controllability
of a spin system at zero and ultralow field can be described in terms of the
graph albertini2002lie : the spin network is represented by a graph, as shown
in Fig. 8. Each node representing a spin is labeled by a number; the edge
connecting $i$th and $j$th spins corresponds to the J-coupling. From ref.
albertini2002lie , the complete controllability of the system is related to
the connectivity of the graph. Mathematically, a graph is connected when there
is a path between every pair of nodes. Then the above system is complete
controllable if one of the following conditions is satisfied:

(i) The graph representing the spin system is connected, and gyromagnetic
ratios of the spins are all different.

(ii) The graph representing the spin system is connected. The $m$ ($m\geq 1$)
spins with unique $\gamma$ (each of them has different gyromagnetic ratio from
all the others in the system) are defined in a set $S$. Define the rest spins
with the same $\gamma$ to form sets $S_{1}$, $S_{2}$, …, $S_{p}$,
respectively. All the elements in $S_{1}$ to $S_{p}$ are transferred to $S$ by
the following disintegration algorithm: 1. Define set
$\Gamma=\\{S_{1},S_{2},...,S_{p}\\}$. 2. If at least two spins $j$ and $k$ in
$S_{i}$ and a spin $l$ in $S$ satisfy $J_{jl}\neq J_{kl}$, then transfer spin
$j$ and $k$ in set $S_{i}$ to set $T$. If there is no element in $S$ and no
set in $\Gamma$ satisfying this property, stop and put all the elements of set
$T$ into set $S$. 3. Go back to step $2$ and repeat the process.

Note that condition (i) is a necessary and sufficient condition of complete
controllability, while condition (ii) is a sufficient condition.

###  V.2 Universal quantum control

Figure 9: (Color online) Quantum control methods in ZULF NMR. (a) Composite
pulse sequence realizing $\pi/2$ gate on 13C in a 13C-1H system. Reprinted
with permission from ref. bian2017universal , Copyright @ 2017 American
Physical Society. (b) Optimal control pulses for realizing 1H $\pi$ rotation
along $z$ in 1H-31P system. Reprinted with permission from ref.
jiang2018numerical , Copyright @ 2018 American Physical Society. (c) Time-
optimal control fields and corresponding trajectories on the Bloch sphere for
realizing $R^{H}_{y}(\pi)$. Reprinted with permission from ref. ji2018time ,
Copyright @ 2018 American Physical Society. (d) Bloch sphere representation
showing how the density matrix evolves in the interaction frame during the
application of the AC pulse sjolander2016transition . Here the magnitude of
the detected signal is proportional to the projection of the final state onto
the $x$-$y$ plane. Reprinted with permission from ref. sjolander2016transition
, Copyright @ 2016 American Chemical Society.

Heteronuclear spin systems satisfying condition (i) are completely
controllable. This means that universal quantum control can be implemented in
such systems nielsen2002quantum . Universal quantum control in ZULF NMR might
find applications in several areas including correlation J-spectroscopy
sjolander2020two , chirality measurement king2017antisymmetric , and Cosmic
Axion Spin Precession Experiment (CASPEr) garcon2019constraints ;
budker2014proposal . A practical way to achieve universal control is to
realize a set of universal logic gates, e.g., arbitrary single-qubit gates and
two-qubit controlled-NOT gates (CNOT) nielsen2002quantum . Here a “qubit” is
represented by a “spin-$1/2$” particle. Implementing the above gates in
heteronuclear spin systems satisfying condition (i) by composite pulses is
studied in ref. bian2017universal . Denote an arbitrary single-qubit gate on
spin $i$

$U_{\mathbf{n}}^{i}(\theta)=e^{-i\mathbf{n}\cdot\mathbf{I}_{i}\theta},$ (18)

where $\mathbf{n}$ is the unit vector and $\theta$ is the rotation angle.
According to ref. bian2017universal , arbitrary single-qubit gates in such
systems are realized by the following pulse sequence

$U_{\mathbf{n}}^{1}(\theta)=U_{\mathbf{n}_{\perp}}^{2\sim
n}(\pi)e^{-iH_{\textrm{dc}}(-{\mathbf{B_{n}}})t/2}U_{\mathbf{n}_{\perp}}^{2\sim
n}{}^{\dagger}(\pi)e^{-iH_{\textrm{dc}}(-{\mathbf{B_{n}}})t/2},$ (19)

where $\theta=\gamma_{1}Bt$ is the desired rotation angle,
$|\mathbf{B_{n}}|=B$ is the control field strength,
$U_{\mathbf{n}_{\perp}}^{2\sim
n}(\pi)=U_{\mathbf{n}_{\perp}}^{2}(\pi)U_{\mathbf{n}_{\perp}}^{3}(\pi)...U_{\mathbf{n}_{\perp}}^{n}(\pi)$,
and
$H_{\textrm{dc}}(\textbf{B})=-\sum\limits_{j}{{\gamma_{j}}{\textbf{I}_{j}}\cdot\textbf{B}}$.
${\mathbf{n}_{\perp}}$ denotes an arbitrary direction that is perpendicular to
$\mathbf{n}$. Spin $1$ is rotated by the desired angle, while the changes in
other spins are refocused at the end of the sequence. Note that J-couplings
are omitted during DC pulses. A schematic diagram of implementing the above
method is demonstrated in Fig. 9a. The sequence relies on single-qubit $\pi$
pulses on rest of the qubits. Given that the gyromagnetic ratios of different
qubits are different, one could construct the $\pi$ pulses through applying a
constant control field along the desired axis for some fine-tuned time $\tau$.
This leads to high fidelity $\pi$ pulses in typical small spin systems
bian2017universal . Here the fidelity is defined by

$F=|\textrm{Tr}(U^{\dagger}_{\textrm{ideal}}U)|/2^{n},$ (20)

which describes the accuracy of a realized unitary operation $U$ with respect
to the ideal one $U_{\textrm{id}}$. For example, in a two-qubit 13C-1H system,
as $\gamma_{\textrm{H}}/\gamma_{\textrm{C}}\approx 4$, one could choose
$\tau=\pi/(B\gamma_{C})$ such that 13C undergoes $\pi$ rotation, while 1H
undergoes approximately $4\pi$ rotation, and a $\pi$ pulse on 13C is thus
realized with $F\approx 0.9994$. In general, $\tau$ could be found numerically
through maximizing $F$ in a given range of pulse duration. For example, in a
C-H-F system, $\pi$ pulse on both 1H and 19F simultaneously (leaving $13$C
alone) is obtained with $F=0.9933$ through the numerical method. For
complicated spin systems, the above method may fail bian2017universal . In
that case, one could apply numerical optimal control method as explained in
Sec. V.3.

One still need CNOT gates between arbitrary spin $i$ and $j$ to achieve
universal control. Its matrix form in $I_{z}$ basis
$\\{|0\rangle_{i}|0\rangle_{j},|0\rangle_{i}|1\rangle_{j},|1\rangle_{i}|0\rangle_{j},|1\rangle_{i}|1\rangle_{j}\\}$
with $I_{z}|0\rangle=\frac{1}{2}|0\rangle$ and
$I_{z}|1\rangle=-\frac{1}{2}|1\rangle$ reads:

$\textrm{C}\textrm{NOT}_{ij}=\left({\begin{array}[]{*{20}{c}}1&0&0&0\\\
0&1&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{array}}\right),$ (21)

where spin $i$ is the control spin, and spin $j$ is the target spin. This
operation flips spin $j$ (target spin) when spin $i$ (control spin) is in
state $|1\rangle$ and does nothing when spin $i$ is in state $|0\rangle$. In a
two-qubit system, this operation could be further decomposed into
vandersypen2005nmr

$\textrm{C}\textrm{NOT}_{ij}=\sqrt{i}U_{z}^{i}(\frac{\pi}{2})U_{z}^{j}{}^{\dagger}(\frac{\pi}{2})U_{x}^{j}(\frac{\pi}{2})U^{(i,j)}_{\text{zz}}(\frac{\pi}{2})U_{y}^{j}(\frac{\pi}{2}),$
(22)

where

$U^{(i,j)}_{\text{zz}}(\phi)=e^{-iH^{(i,j)}_{0}t}U_{z}^{j}(\pi)e^{-iH^{(i,j)}_{0}t}U_{z}^{j}{}^{\dagger}(\pi),$
(23)

$H^{(i,j)}_{0}={{2\pi J_{ij}}}\mathbf{I}_{i}\cdot\mathbf{I}_{j}$ and
$\phi\equiv 2\pi J_{ij}t$ ($J_{ij}>0$). For spin systems with $n$ spins
($n>2$), the main barrier to implementing the CNOT gate is the implementation
of $U^{(i,j)}_{\text{zz}}(\theta)$ in a large coupled spin network, where only
the coupling $H^{(i,j)}_{0}$ is active. To achieve this, one needs to turn off
the undesired couplings, as achieved by refocusing vandersypen2005nmr  in
high-field NMR. This can be achieved in zero-field NMR via a concatenated
scheme by recursively building on a basic sequence bian2017universal .

The above universal quantum control scheme was experimentally demonstrated in
ref. jiang2018experimental . Single-qubit control for 13C and 1H and two-qubit
control via a CNOT gate in 13C-formic acid was realized on a home-built ZULF
NMR spectrometer jiang2018experimental . The experimental apparatus and
procedure were outlined in Sec. III. The average fidelity for 13C single-qubit
control was $f_{\textrm{avg}}=0.9960(2)$. This was obtained by a quantum
information-inspired randomized benchmarking method that will be explained in
Sec. V.6. In this experiment, it was the unitary errors which originated from
pulse imperfections that principally limited the single-spin control fidelity.
The CNOT${}_{\textrm{HC}}$ gate was also realized in ref.
jiang2018experimental , and the fidelity was around $F=0.9877$.

###  V.3 Numerical optimal control

In ref. jiang2018numerical , numerical optimization method based on the
gradient ascent pulse engineering (GRAPE) algorithm khaneja2005optimal  is
applied to manipulate nuclear spins in zero-field NMR. In particular, it is
shown that this method is capable of designing control pulses for homonuclear
spin systems satisfying condition (ii). This is not achievable via the
composite pulse scheme presented in Sec. V.2. Thus the numerical optimization
method greatly improves the ability to manipulate spin systems at zero field
jiang2018numerical .

GRAPE is a gradient-based numerical optimization method khaneja2005optimal .
It promises to achieve precise and robust quantum control in a wide range of
quantum systems, and is widely used in the magnetic resonance community and
quantum information science community glaser2015training ; tovsner2009optimal
. As an example, $\pi$ pulse on 1H in 1H-31P system (i.e., Phosphorous acid)
is obtained by the GRAPE method jiang2018numerical . The optimal control
fields are illustrated in Fig. 9b. The total duration $T=0.5$ ms (i.e., number
of pieces: $20$ and piecewise duration: $25$ $\mu$s) and the fidelity
$F=0.9997$. It was further demonstrated in ref. jiang2018numerical  that
universal control could also be achieved by this method. In particular, it is
generally considered to be formidable for manipulating homonuclear spins at
zero field due to the same gyromagnetic ratios. However, homonuclear spins in
systems satisfying condition (ii) in Sec. V.2 can be distinguished utilizing
the information on different J-couplings. For instance, in 1H-13C1-13C2 system
(i.e., Trichloroethylene) composed of two 13C spins, the system has different
J-couplings between 13C and 1H, i.e., J${}_{\text{H}\text{C}_{1}}=202.3$ Hz,
J${}_{\text{H}\text{C}_{2}}=13.4$ Hz. This system satisfies condition (ii) and
hence it is universally controllable. By using the GRAPE method, a $\pi$
$z$-pulse on 13C1 is achieved with $T=90$ ms (i.e., number of pieces: 450 and
piecewise duration: 200 $\mu\text{s}$) and $F=0.9999$.

Moreover, the requirement of robustness against pulse imperfections could be
incorporated into the GRAPE method. According to Sec. V.2, the main errors
come from pulse imperfections that principally limit the performance of
control. In ref. jiang2018numerical , the robustness tests (against pulse
imperfections) for single-qubit and two-qubit gates in 1H-31P system are
demonstrated. The result indicates that the optimal control pulses are more
robust compared with the composite pulse scheme presented in Sec. V.2.
Therefore, the GRAPE method allows for decreasing requirement of hardware
stability and calibration accuracy in practical experiments. Note that this
method could be straightforwardly generalized to ultralow-field NMR. Together
with its high fidelity and capability of designing control fields in
homonuclear spin systems, future experiments based on this method is worth
exploring.

###  V.4 Time optimal control

Time-optimal control (TOC) problems in quantum systems are ubiquitous and
important in multiple applications glaser2015training ; khaneja2001time ;
carlini2006time ; d2007introduction . Because the inevitable noise from the
environment degrades quantum states and operations over time, inducing quantum
dynamics in minimal time utilizing TOC becomes a preferable choice geng2016 .
The analytic knowledge of the TOC is useful even in cases where such a control
is not the one physically implemented. It gives information about the inherent
time limitations of the system, therefore indicating a benchmark for the time
of any control law ji2018time . However, analytic solutions are rare in
optimal control and theoretical calculations have to be complemented by
difficult numerical simulations glaser2015training . Previous works mainly
considered time optimization with controls which address spin individually
khaneja2001time ; geng2016 . However, this is not the case in zero-field NMR,
where all spins are simultaneously affected by the control fields. In ref.
ji2018time , TOC for arbitrary single-qubit rotation in a two-qubit system
under simultaneous control is analytically obtained and experimentally
implemented in a zero-field NMR experiment. This further improves the ability
to manipulate spins in ZULF NMR.

To obtain the TOC, the following procedure is adopted in ji2018time . The
Pontryagin’s maximum principle (PMP) was first applied to reduce the number of
unknown parameters to $7$. Then, a newly developed symmetry reduction
technique is applied to further reduce the number of unknown parameters to
$3$. Finally, the TOC and optimal time are obtained by solving a set of
equations. For example, the control magnetic fields
$\mathbf{B_{n}}=(B_{x},B_{y},B_{z})$ are analytically obtained to realize
$R_{y}^{H}(\theta)$ in a two-qubit 1H-13C system, as illustrated in Fig. 9c.
There is a $70\%\sim 80\%$ reduction in the time cost of $R_{y}^{H}(\theta)$
compared to the composite pulse scheme introduced in Sec. V.2. The experiments
were performed in the 13C-formic acid sample using a home-built zero-field NMR
spectrometer. The average fidelity for 1H single-qubit TOC control was
$f_{avg}=0.99$ ji2018time .

###  V.5 Transition-selective pulses

Apart from the above universal control methods, there are alternative methods
adding an increased capability for spin control in ZULF NMR
sjolander2016transition ; tayler2016nuclear . For example, the use of high-
field selectivity in ZULF NMR was reported in ref. tayler2016nuclear , where a
magnetic field was temporarily imposed on the sample, allowing one to apply AC
pulses that individually address different spin species. Moreover, many NMR
techniques rely on selectivity beyond differentiating spin species, to the
extent of addressing individual transitions freeman1991selective . In ref.
sjolander2016transition , frequency selective pulses in ultralow-field regime
were demonstrated with a typical excitation bandwidth of $0.5$-$5$ Hz. The
resonant AC pulses were utilized to drive spin populations directly between
the zero-field eigenstates and therefore allowed greater control over which
transitions are excited. In ref. sjolander2016transition , the evolution of
the system under the AC magnetic field, J-coupling and a small static field
were obtained analytically in an interaction frame. This is illustrated in
Fig. 9d. The result is that the total Hamiltonian can be block diagonalized to
a time-independent Hamiltonian acting on a two level system.

Figure 10: (Color online) (a) Clifford-based randomized benchmarking. The
signals acquired using four different independent pulse sequences (readout
operations), $M\in\\{\textrm{No
operation},\pi^{z}_{S},\pi^{x}_{I},\pi^{z}_{S}-\pi^{x}_{I}\\}$, are averaged
together. Random sequences with $P=e^{\pm i\pi A}$ and $C=e^{\pm i(\pi/2)B}$
are applied for each sequence length $m$, where the Clifford gates are
realized by combined operations $PC$. Here,
$A\in\\{\mathbf{1},S_{x},S_{y},S_{z}\\}$ and $B\in\\{S_{x},S_{y},S_{z}\\}$.
The recovery operation $R$ is chosen to return the system to the initial state
in the absence of control error. (b) Randomized benchmarking results for 13C
single-spin control. Each point is an average over $32$ random sequences of
$m$ Clifford gates, and the error bars indicate the standard error of the
mean. A single exponential decay shown with a solid line is used to fit the
fidelity decay. The randomized benchmarking results yield an average error per
Clifford gate $\epsilon_{g}=0.0040(2)$, and an imperfection of the state
initialization and readout $d_{if}=0.0141$. (c) Pulse sequences for
implementing the CNOT gate, as explained in Sec. V.2. (d) Output of the CNOT
gate applied to the sudden state. (e) Reconstructed CNOT gate in the
computational basis. The fidelity of the CNOT gate is $0.9877(2)$. Reprinted
with permission from ref. jiang2018experimental , Copyright @ 2018 American
Association for the Advancement of Science.

Using this method, one is able to discriminate between signals belonging to
manifolds of different total proton angular momentum in the zero-field
spectrum of [15N, 13C2]acetonitrile. It is further shown that the sense of
rotation of pulsed fields may select between positive and negative changes in
angular-momentum projection. Using two orthogonal pulsing coils each
generating an AC field, a rotating magnetic field is generated. It allows one
to select transitions with $\Delta m_{F}=+1$ or $\Delta m_{F}=-1$. A weak DC
field is applied to the 13C-formic acid, and the resulting spectrum contains
two observable transitions centered about the J-coupling frequency. By
applying the above rotating magnetic field, both transitions could be
addressed selectively, sidestepping the usual limit on frequency selectivity.
These techniques should facilitate ZULF NMR spectroscopy of larger, more
demanding spin systems or mixtures and open a way to adapting a suit of
established high-field experiments to ultralow field sjolander2016transition ;
tayler2016nuclear .

###  V.6 Evaluation of quantum control

Being able to quantify the performance of coherent quantum control is
important for the development of quantum architectures vandersypen2005nmr .
Full characterization of any quantum process, and then calculation of the
fidelity of control, is possible through a procedure known as quantum process
tomography (QPT) vandersypen2005nmr . However, QPT requires an exponential
number of experiments, making it experimentally prohibitive for quantum
systems larger than a few qubits. To overcome this, in ref.
knill2008randomized , it was reported the randomized benchmarking (RB) method
to determine the error probability per gate in computational contexts. In a
recent work jiang2018experimental , Clifford-based RB was applied to estimate
the fidelity of single-qubit controls in the 13C-1H system in ZULF NMR. The
randomized benchmarking pulse sequences are shown in Fig. 10a. The sudden
state is selected as the initial state. To measure the coefficient of $S_{z}$
independently, the same temporal averaging method as the state tomography is
adopted, as explained in Fig. 10a. Measuring the decay of the coefficient of
$S_{z}$ with respect to the number ($m$) of randomized Clifford gates in the
benchmarking sequence yields the average fidelity for 13C single-spin control.
By averaging the coefficients of $S_{z}$ over $k$ different randomized
benchmarking sequences with the same length $m$, and normalizing this averaged
value to that of $m=0$, the normalized signal $\bar{F}$ can be written as
$\bar{F}=(1-d_{if})(1-2\epsilon_{g})^{m}$, where $d_{if}$ is due to the
imperfection of the state initialization and readout, and $\epsilon_{g}$ is
the average error per Clifford gate jiang2018experimental . As explained in
Fig. 10b, the average fidelity for 13C single-spin control is estimated to be
$f_{\textrm{avg}}=1-0.0040(2)=0.9960(2)$, which is resilient to the state
preparation and measurement errors.

Figure 11: (Color online) (a) Ultralow-field MRI detection apparatus. (b)
Ultralow-field MRI of human brain. These images reveal some obvious brain
features and boundaries. Reprinted with permission from ref.
savukov2013magnetic , Copyright @ 2013 American Chemical Society. (c)
Ultralow-field MRI of two fingers. Reprinted with permission from ref.
savukov2013anatomical , Copyright @ 2013 Elsevier. (d) and (e) Schematic of
the experimental setup and MRI of two parallel cylindrical channels. Reprinted
with permission from ref. xu2006magnetic , Copyright @ 2006 National Academy
of Sciences. Figure 12: (Color online) (a) Pentane-hexafluorobenzene
comagnetometer based on ZULF NMR. Ultralow-field NMR signals from a mixture of
1H-pentane (C5H12) and 19F-hexafluorobenzene (C6F6). The time-domain signal is
well described by a sum of two exponentially decaying sinusoids. As shown in
the inset, the spectrum shows two peaks at the Larmor precession frequencies
of 1H and 19F nuclei ($\nu_{h}$ and $\nu_{f}$). $\nu_{h}$ and $\nu_{f}$ are
both proportional to external magnetic field, but $\nu_{f}/\nu_{h}$ is
insensitive to magnetic field and can be used to construct as a
comagnetometer. (b) The ratio $\nu_{f}/\nu_{h}$ versus external magnetic field
modulation. The modulation on the ratio is not visible, which is in good
agreement with theoretical prediction. (c) Effects of magnetic field gradients
on the pentane-hexafluorobenzene comagnetometer. Reprinted with permission
from ref. RNLiquidstate , Copyright @ 2012 American Physical Society. (d)
Comagnetometer based on a liquid of identical molecules. Experimentally
measured J-coupling spectrum of acetonitrile-$2$-13C (13CH3CN) at $J_{CH}$ and
2$J_{CH}$ in a 100 nT bias field along $z$. The splittings $\Delta\nu_{1}$ and
$\Delta\nu_{2}$ are both proportional to external magnetic field, and can be
employed to realize a comagnetometer based on an ensemble of identical
molecules. (e) The ratio $\Delta\nu_{2}/\Delta\nu_{1}$ remains constant
although an oscillating magnetic field is applied. (f) In the presence of a
constant bias magnetic field $B_{z}=100$ nT, the normalized frequency ratio as
a function of the gradient $dB_{z}/dy$. Reprinted with permission from ref.
RNComagnetometer , Copyright @ 2018 American Physical Society.

The performance of the CNOT gate in the 13C (target spin)-1H (control spin)
system is also evaluated in ref. jiang2018experimental . The CNOT gate is
implemented according to the method presented in Fig. 10c and Sec. V.2. As the
experimental error is dominated by unitary error, a partial quantum process
tomography (see ref. jiang2018experimental ) is implemented to characterize
the CNOT gate. It requires fewer measurements than standard QPT. It takes two
independent initial states as the input states and measure the corresponding
output states after applying the CNOT gate. The CNOT gate is then
reconstructed by using a numerical minimization technique
jiang2018experimental . One of the output state is illustrated in Fig. 10d.
The reconstructed CNOT gate is illustrated in Fig. 10e, and then the fidelity
is calculated to be $0.9877(2)$.

##  VI Imaging

MRI is a widely used non-invasive technique for materials research and medical
diagnosis liang2000principles . Because the image quality improves with the
magnetic field strength, conventional clinical scanners use fields as high as
1-3 T. However, traditional MRI instruments are immobile and expensive, in
particular they have long been off-limits to patients with pacemakers. Also,
high-field MRI is not applicable to metallic materials because of the short
penetration depth of radiofrequency radiation. To overcome the difficulties
mentioned above, one solution is to perform MRI at ultralow field on the order
of microtesla xu2006magnetic ; savukov2013magnetic ; savukov2013anatomical .
In particular, ultralow-field MRI instruments can be constructed at much lower
cost, can be more portable, and can have many other advantages including
higher contrast to anomalies and absence of susceptibility artifacts. Starting
with the early work with SQUIDs as the low-field MRI detector
zotev2008microtesla , atomic magnetometers have surpassed the sensitivity of
SQUIDs and could be a promising detector for MRI allred2002high ;
dang2010ultrahigh . References savukov2013magnetic ; savukov2013anatomical
reported ultralow-field MRI systems for human brain and fingers, as shown in
Fig. 11a, b and c. In this MRI system, a $\sim$80 mT prepolarization field
first prepolarizes spins and then is switched off before the MRI sequence
begins, and then spins evolve and are detected under 4 mT field through a
pick-up coil transferring to an atomic magnetometer. Moreover, it is also
possible to perform MRI of metallic materials, which are of extensive interest
because metallic materials broadly used in filtration, catalysis, and
biomedicine. Xu et al. xu2006magnetic  reported remote-detection imaging (see
Fig. 11d, e) where water was first prepolarized through a permanent magnet and
flowed into a spatially separated encoding region with 31 G bias field, and
detection was performed with an atomic magnetometer at ultralow field.

Although the demonstrated MRI image quality is much lower than that typical in
high-field MRI, there are still potential to improve the current MRI quality
such as improving the prepolarization field and optimizing the sensitivity of
atomic magnetometers. The ultralow-field MRI can find practical applications
where high-field MRI is restricted. For example, the ultralow-field MRI
removes the off-limits to patients with pacemakers and provides a safe way of
imaging for such patients. In addition, such a MRI system could be very
portable and thus can quickly move to patients without the need of relocating
them to MRI facility. This should be particular important because the
relocation is dangerous, especially when life support system needs to be
connected savukov2013magnetic .

##  VII NMR-based quantum devices

ZULF NMR operates at zero or ultralow fields, which is particularly
appropriate for precise measurements without electromagnetic interference and
operate as sensitive quantum devices. In this section, we review recent
developments of comagnetometer and Floquet maser based on ZULF NMR.

###  VII.1 Comagnetometer

Liquid-state comagnetometers based on ZULF NMR were demonstrated in refs.
RNLiquidstate ; RNComagnetometer . As shown in Fig. 12a, the comagnetometer is
based on ultralow-field NMR of binary mixtures of mutually miscible solvents,
for example a mixture of pentane and hexafluorobenzene, which is similar to
that of noble gas comagnetometer. As shown in the inset, the spectrum
demonstrated two peaks at the Larmor precession frequencies of 1H and 19F
nuclei ($\nu_{h}$ and $\nu_{f}$, respectively). Because $\nu_{h}$ and
$\nu_{f}$ are both proportional to external magnetic field, $\nu_{f}/\nu_{h}$
is insensitive on external magnetic fields (Fig. 12b). Therefore, they can be
employed to realize a comagnetometer, which can resist magnetic field
interference but retains sensitivity to Zeeman-like nonmagnetic spin
interactions wu2019search ; garcon2019constraints . In such a comagnetometer,
however, magnetic field gradients are one of the major sources of systematic
errors (see Fig. 12c) because there exists some spatial separation between the
ensemble-averaged position of different spin species.

To overcome this difficulty, reference RNComagnetometer  reported a single
species comagnetometer, in which different nuclear spins are probed within the
same molecules. Specifically, under the influence of a bias magnetic field,
the J-coupling resonance lines at different frequencies split into separate
peaks, as shown in Fig. 12d. The frequency separation between the split peaks
for each J-coupling resonance has distinct linear coefficients with respect to
the magnetic field, i.e., $\Delta\nu_{1}=(\gamma_{h}+\gamma_{c})B_{z}$ and
$\Delta\nu_{2}=\frac{1}{2}(\gamma_{h}+3\gamma_{c})B_{z}$. As shown in Fig.
12e, the ratio $\Delta\nu_{2}/\Delta\nu_{1}$ is insensitive to external
magnetic field changes. Thus, measurements of these splittings can be employed
as a comagnetometer. In particular, because the probed spins in a same
molecule interact with same magnetic field, such a comagnetometer can greatly
resist the perturbation of magnetic field gradients (Fig. 12f).

Liquid-state comagnetometers based on ZULF NMR are suitable for precision
measurements, such as sensitive gyroscopes and to search for spin-gravity,
permanent electric dipole moments, axionlike dark matter and other fields in
fundamental physics wu2019search ; garcon2019constraints .

###  VII.2 Floquet maser

The invention of the maser stimulated many revolutionary technologies such as
lasers and atomic clocks gordon1955maser ; oxborrow2012room . Despite enormous
progress, the realizations of masers are still confined to a limited variety
of systems, in particular, the physics of masers remains unexplored in
periodically driven (Floquet) systems, which are generally defined by time-
periodic Hamiltonians and enable to observe many exotic phenomena such as time
crystals. Recently we reported the first theoretical and experimental
demonstration of a Floquet based maser jiang2019floquet  comprised of
periodically driven 129Xe spins in an vapor cell. In contrast to liquid-state
ZULF NMR, We used noble gas atoms 129Xe with nuclear spin $I=1/2$ in a setup
depicted in Fig. 13a. A 0.5 cm3 cubic vapor cell made from pyrex glass
contains 5 torr 129Xe, 250 torr N2, and a droplet of enriched 87Rb. 129Xe
spins are polarized by spin-exchange collisions with optically-pumped 87Rb
atoms in a bias magnetic field $B_{0}$ ($\approx 750$ nT) along the polarized
direction ($z$ axis). Similar to a microwave cavity in conventional masers
gordon1955maser ; oxborrow2012room , the 129Xe spins in experiments are
embedded in a feedback circuit, as shown in Fig. 13a, which employs an atomic
magnetometer as a sensitive detector of 129Xe spins and simultaneously
supplies the real-time output audio-frequency signal of the magnetometer to
the spins.

Figure 13: (Color online) (a) Schematic of the 129Xe nuclear-spin maser. The
129Xe nuclei are polarized and detected via spin-exchange collisions with
optically pumped 87Rb. (b) Floquet states of a periodically driven Floquet
system. (c) Spectrum based on Floquet 129Xe nuclear-spin maser. Reprinted from
ref. jiang2019floquet .

Unlike the common masers that exploit inherent transitions, the realized maser
called ”Floquet maser” oscillates at the frequencies of transitions between
Floquet states shown in Fig. 13b. Using the Floquet maser technique,
ultrahigh-resolution spectra of the Floquet system are observed with two
orders of magnitude better resolution compared to that limited by decoherence
(Fig. 13c). As the spectral resolution is greatly increased, a different
regime emerges where high-order Floquet sidebands become significant and
complex spectra are expected, enabling accurate measurement of physical
parameters, e.g., atomic scalar and tensor polarizabilities, magnetic fields,
ultralight bosonic exotic fields, and multiphoton coherences. In particular,
the Floquet maser could be extremely sensitive detector of low-frequency
magnetic fields, which is significantly better than state-of-the-art
magnetometers.

##  VIII Tests of fundamental physics

The applications of ZULF NMR are not limited to applied science, but also can
be applicable to tests of fundamental physics. In particular, ZULF NMR-based
quantum devices have extensively studied to measure magnetic fields or pseudo-
magnetic fields with high sensitivity, which are well suited to test of
fundamental physics including the searches for spin-gravity coupling
RNComagnetometer ; RNLiquidstate , spin-axion coupling garcon2019constraints ;
wu2019search ; jiang2019floquet , molecular parity nonconservation
king2017antisymmetric . The proposal of Cosmic Axion Spin Precession
Experiment proposed the use of NMR techniques to search for dark-matter-driven
spin precession budker2014proposal . Based on this, the potential couplings
between nuclear spins and axions are similar to the Zeeman interactions,
enabling their direct detection via high-resolution ZULF NMR spectra.
Reference garcon2019constraints  reported that as nuclear spins move through
the galactic dark-matter halo, they couple to dark matter and behave as if
they were in an oscillating magnetic field, generating a dark-matter-driven
NMR signal. No dark matter signal was detected above background, establishing
new experimental bounds for dark matter bosons with masses ranging from
$1.8\times 10^{-16}$ to $7.8\times 10^{-14}$ eV (corresponding to Compton
frequencies ranging from $\sim$45 mHz to 19 Hz). To explore the window of
lower Compton frequencies, low-frequency magnetic field may cause
electromagnetic interference because ZULF NMR spectra are also sensitive to
real magnetic fields. To overcome this, reference wu2019search  reported the
search of ultralight axion dark matter via a liquid-state comagnetoemter,
ranging from $10^{-22}$ to $1.3\times 10^{-7}$ eV. In addition to the spin-
axion searches, a comagnetometry scheme based on ZULF NMR is capable of
measuring the hypothetical spin-dependent gravitational energy of nuclei at
the $10^{-17}$ eV level RNComagnetometer ; RNLiquidstate .

##  IX Conclusion and perspective

ZULF NMR provides an alternative magnetic resonance modality which is
performed in the absence of an applied magnetic field or at extremely small
magnetic field. Over the years, ZULF NMR based on atomic magnetometers has
gradually become mature and served as a complementary technique to
conventional high-field NMR. This new NMR modality is particularly attractive
because of inexpensive, portability, and high absolute magnetic field
homogeneity. In addition to the advantages mentioned above, ZULF NMR opens the
door to applications in, such as imaging and spectroscopy inside of metal
objects or in metallic materials, which are inaccessible to traditional high-
field NMR. In this review, we review the recent advances of ZULF NMR
demonstrated in spectroscopy, quantum control, imaging, NMR-based quantum
sensing, and tests of fundamental physics.

ZULF NMR is still a relatively young field and its current applications are
mainly in laboratories. The applications of ZULF NMR in industry are still
lacking, for example, in the characterization of oil in well logging and food
safety testing. In future, to become a more powerful and versatile technique
in research and industry, a plenty of efforts on theories and experiments are
still necessary. Firstly, the detection sensitivity of ZULF NMR is still lower
than that of the state-of-the-art high-field NMR spectrometers. This can be
overcome by several techniques as follows. The current atomic magnetometers
are still far from reaching the quantum-noise-limited sensitivity
budker2007optical ; several orders of magnitude improvement are possible in
the sensitivity, such as optimizing the magnetic field sensitivity of atomic
magnetometers or atomic magnetic gradiometer to subfemtotesla
kominis2003subfemtotesla ; dang2010ultrahigh , or even attotesla level. In
addition to the atomic magnetometer sensitivity, it is also possible to
improve the nuclear spin polarization through the techniques of parahydrogen-
induced polarization (PHIP) adams2009reversible ; theis2011parahydrogen  and
dynamic nuclear polarization (DNP) barskiy2019zero . Secondly, the size of
ZULF NMR spectrometer is usually at the level of one meter, which is mainly
limited by lasers, magnetic shield, etc. Combining with micro-nanotechnology
of atomic devices shah2007subpicotesla  or anisotropic magnetoresistive sensor
verpillat2008remote  and microfluidic chip ledbetter2008zero , it is promising
to realize a much smaller and lighter “NMR-on-a-chip” devices, which could be
well suited for real-time chemical analysis in situ without the need to
deliver sample to the laboratory. With the development of nanoscale magnetic
sensors, ZULF NMR will promisingly reach single-molecule level
budker2019extreme , which is useful for metabolomics at a single-cell level.
Thirdly, the quantum control techniques for ZULF NMR should be developed to
analyse the exceedingly complex spectra of small molecules and even
macromolecules. For example, multidimensional ZULF NMR techniques are at the
early stage and could be developed as a powerful diagnostic tool for analysing
molecular structure.

We believe that the ZULF NMR will progress dramatically over the coming years
and shed new lights in applications ranging from materials science, imaging,
and quantum information processing to fundamental physics.

Acknowledgements

We thank Dmitry Budker for valuable discussions. This work was supported by
National Key Research and Development Program of China (Grant No.
2018YFA0306600), National Natural Science Foundation of China (Grants Nos.
11661161018, 11927811), Anhui Initiative in Quantum Information Technologies
(Grant No. AHY050000), and USTC Research Funds of the Double First-Class
Initiative (Grant No. YD3540002002).

Keywords

zero- to ultralow-field NMR; atomic magnetometer; spectroscopy; quantum
control; imaging; NMR-based quantum devices; tests of fundamental physics.

Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * [1] Richard R Ernst, Geoffrey Bodenhausen, and Alexander Wokaun.  Principles of nuclear magnetic resonance in one and two dimensions.  Number BOOK. 1987.
  * [2] Zhi-Pei Liang and Paul C Lauterbur.  Principles of magnetic resonance imaging: a signal processing perspective.  SPIE Optical Engineering Press, 2000.
  * [3] Kurt Wüthrich.  Nmr with proteins and nucleic acids.  Europhysics News, 17(1):11–13, 1986.
  * [4] Malcolm H Levitt.  Spin dynamics: basics of nuclear magnetic resonance.  John Wiley & Sons, 2013.
  * [5] DP Weitekamp, A Bielecki, D Zax, K Zilm, and A Pines.  Zero-field nuclear magnetic resonance.  Physical review letters, 50(22):1807, 1983.
  * [6] John W Blanchard and Dmitry Budker.  Zero-to ultralow-field nmr.  eMagRes, pages 1395–1410, 2016.
  * [7] Robert McDermott, Andreas H Trabesinger, Michael Mück, Erwin L Hahn, Alexander Pines, and John Clarke.  Liquid-state nmr and scalar couplings in microtesla magnetic fields.  Science, 295(5563):2247–2249, 2002.
  * [8] MP Ledbetter, Thomas Theis, JW Blanchard, Hattie Ring, Paul Ganssle, Stephan Appelt, Bernhard Blümich, Alex Pines, and Dmitry Budker.  Near-zero-field nuclear magnetic resonance.  Physical review letters, 107(10):107601, 2011.
  * [9] Giuseppe Bevilacqua, Valerio Biancalana, Yordanka Dancheva, and Luigi Moi.  All-optical magnetometry for nmr detection in a micro-tesla field and unshielded environment.  Journal of Magnetic Resonance, 201(2):222–229, 2009.
  * [10] Michael CD Tayler, Thomas Theis, Tobias F Sjolander, John W Blanchard, Arne Kentner, Szymon Pustelny, Alexander Pines, and Dmitry Budker.  Invited review article: Instrumentation for nuclear magnetic resonance in zero and ultralow magnetic field.  Review of Scientific Instruments, 88(9):091101, 2017.
  * [11] Guobin Liu, Xiaofeng Li, Xianping Sun, Jiwen Feng, Chaohui Ye, and Xin Zhou.  Ultralow field nmr spectrometer with an atomic magnetometer near room temperature.  Journal of Magnetic Resonance, 237:158–163, 2013.
  * [12] Antoine Garcon, John W Blanchard, Gary P Centers, Nataniel L Figueroa, Peter W Graham, Derek F Jackson Kimball, Surjeet Rajendran, Alexander O Sushkov, Yevgeny V Stadnik, Arne Wickenbrock, et al.  Constraints on bosonic dark matter from ultralow-field nuclear magnetic resonance.  Science advances, 5(10):eaax4539, 2019.
  * [13] MP Ledbetter, IM Savukov, D Budker, V Shah, S Knappe, J Kitching, DJ Michalak, S Xu, and A Pines.  Zero-field remote detection of nmr with a microfabricated atomic magnetometer.  Proceedings of the National Academy of Sciences, 105(7):2286–2290, 2008.
  * [14] Shoujun Xu, Valeriy V Yashchuk, Marcus H Donaldson, Simon M Rochester, Dmitry Budker, and Alexander Pines.  Magnetic resonance imaging with an optical atomic magnetometer.  Proceedings of the National Academy of Sciences, 103(34):12668–12671, 2006.
  * [15] I Savukov and T Karaulanov.  Magnetic-resonance imaging of the human brain with an atomic magnetometer.  Applied physics letters, 103(4):043703, 2013.
  * [16] I Savukov and T Karaulanov.  Anatomical mri with an atomic magnetometer.  Journal of Magnetic Resonance, 231:39–45, 2013.
  * [17] Min Jiang, Teng Wu, John W Blanchard, Guanru Feng, Xinhua Peng, and Dmitry Budker.  Experimental benchmarking of quantum control in zero-field nuclear magnetic resonance.  Science advances, 4(6):eaar6327, 2018.
  * [18] Min Jiang, Hui Li, Zhennan Zhu, Xinhua Peng, and Dmitry Budker.  Floquet maser.  arXiv preprint arXiv:1901.00970, 2019.
  * [19] John W Blanchard, Micah P Ledbetter, Thomas Theis, Mark C Butler, Dmitry Budker, and Alexander Pines.  High-resolution zero-field nmr j-spectroscopy of aromatic compounds.  Journal of the American Chemical Society, 135(9):3607–3612, 2013\.
  * [20] Stephan Appelt, Holger Kühn, F Wolfgang Häsing, and Bernhard Blümich.  Chemical analysis by ultrahigh-resolution nuclear magnetic resonance in the earth’s magnetic field.  Nature Physics, 2(2):105–109, 2006.
  * [21] Zhihuang Luo, Jun Li, Zhaokai Li, Ling-Yan Hung, Yidun Wan, Xinhua Peng, and Jiangfeng Du.  Experimentally probing topological order and its breakdown through modular matrices.  Nature Physics, 14(2):160–165, 2018.
  * [22] Xinhua Peng, Hui Zhou, Bo-Bo Wei, Jiangyu Cui, Jiangfeng Du, and Ren-Bao Liu.  Experimental observation of lee-yang zeros.  Physical review letters, 114(1):010601, 2015.
  * [23] Ya S Greenberg.  Application of superconducting quantum interference devices to nuclear magnetic resonance.  Reviews of Modern Physics, 70(1):175, 1998.
  * [24] Vadim S Zotev, Andrei N Matlashov, Petr L Volegov, Algis V Urbaitis, Michelle A Espy, and Robert H Kraus Jr.  Squid-based instrumentation for ultralow-field mri.  Superconductor Science and Technology, 20(11):S367, 2007.
  * [25] Dmitry Budker, W Gawlik, DF Kimball, SM Rochester, VV Yashchuk, and A Weis.  Resonant nonlinear magneto-optical effects in atoms.  Reviews of modern physics, 74(4):1153, 2002.
  * [26] Dmitry Budker and Michael Romalis.  Optical magnetometry.  Nature physics, 3(4):227–234, 2007.
  * [27] IK Kominis, TW Kornack, JC Allred, and Michael V Romalis.  A subfemtotesla multichannel atomic magnetometer.  Nature, 422(6932):596–599, 2003.
  * [28] Zhimin Li, Ronald T Wakai, and Thad G Walker.  Parametric modulation of an atomic magnetometer.  Applied physics letters, 89(13):134105, 2006.
  * [29] Vishal Shah, Svenja Knappe, Peter DD Schwindt, and John Kitching.  Subpicotesla atomic magnetometry with a microfabricated vapour cell.  Nature Photonics, 1(11):649–652, 2007.
  * [30] Jiancheng Fang, Tao Wang, Wei Quan, Heng Yuan, Hong Zhang, Yang Li, and Sheng Zou.  In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect.  Review of Scientific Instruments, 85(6):063108, 2014.
  * [31] Dong Sheng, S Li, Nezih Dural, and Michael V Romalis.  Subfemtotesla scalar atomic magnetometry using multipass cells.  Physical review letters, 110(16):160802, 2013.
  * [32] Min Jiang, Román Picazo Frutos, Teng Wu, John W. Blanchard, Xinhua Peng, and Dmitry Budker.  Magnetic gradiometer for the detection of zero- to ultralow-field nuclear magnetic resonance.  Phys. Rev. Applied, 11(2):024005, 2019.
  * [33] JC Allred, RN Lyman, TW Kornack, and Michael V Romalis.  High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation.  Physical review letters, 89(13):130801, 2002.
  * [34] HB Dang, Adam C Maloof, and Michael V Romalis.  Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer.  Applied Physics Letters, 97(15):151110, 2010.
  * [35] MP Ledbetter, CW Crawford, A Pines, DE Wemmer, S Knappe, J Kitching, and D Budker.  Optical detection of nmr j-spectra at zero magnetic field.  Journal of magnetic resonance, 199(1):25–29, 2009.
  * [36] Thomas Theis, Paul Ganssle, Gwendal Kervern, Svenja Knappe, John Kitching, MP Ledbetter, Dmitry Budker, and Alex Pines.  Parahydrogen-enhanced zero-field nuclear magnetic resonance.  Nature Physics, 7(7):571–575, 2011.
  * [37] Min Jiang, Wenjie Xu, Qing Li, Ze Wu, Dieter Suter, and Xinhua Peng.  Interference in atomic magnetometry.  Advanced Quantum Technologies, page 2000078.
  * [38] A Llor, Z Olejniczak, and A Pines.  Coherent isotropic averaging in zero‐field nuclear magnetic resonance. i. general theory and icosahedral sequences.  Journal of Chemical Physics, 103(10):3966–3981, 1995.
  * [39] A Llor, Z Olejniczak, and A Pines.  Coherent isotropic averaging in zero-field nuclear magnetic resonance. ii. cubic sequences and time-reversal of spin couplings.  Journal of Chemical Physics, 103(10):3982–3997, 1995.
  * [40] AM Thayer and A Pines.  Composite pulses in zero-field nmr.  Journal of Magnetic Resonance (1969), 70(3):518–522, 1986.
  * [41] CJ Lee, D Suter, and ANDA Pines.  Theory of multiple-pulse nmr at low and zero fields.  Journal of Magnetic Resonance (1969), 75(1):110–124, 1987.
  * [42] Ji Bian, Min Jiang, Jiangyu Cui, Xiaomei Liu, Botao Chen, Yunlan Ji, Bo Zhang, John Blanchard, Xinhua Peng, and Jiangfeng Du.  Universal quantum control in zero-field nuclear magnetic resonance.  Physical Review A, 95(5):052342, 2017.
  * [43] Min Jiang, Ji Bian, Xiaomei Liu, Hengyan Wang, Yunlan Ji, Bo Zhang, Xinhua Peng, and Jiangfeng Du.  Numerical optimal control of spin systems at zero magnetic field.  Physical Review A, 97(6):062118, 2018.
  * [44] Tobias F Sjolander, John W Blanchard, Dmitry Budker, and Alexander Pines.  Two-dimensional single-and multiple-quantum correlation spectroscopy in zero-field nuclear magnetic resonance.  arXiv preprint arXiv:2004.04456, 2020.
  * [45] Tobias F Sjolander, Michael CD Tayler, Arne Kentner, Dmitry Budker, and Alexander Pines.  13c-decoupled j-coupling spectroscopy using two-dimensional nuclear magnetic resonance at zero-field.  The journal of physical chemistry letters, 8(7):1512–1516, 2017\.
  * [46] Yunlan Ji, Ji Bian, Min Jiang, Domenico D’Alessandro, and Xinhua Peng.  Time-optimal control of independent spin-1/2 systems under simultaneous control.  Physical Review A, 98(6):062108, 2018.
  * [47] Danila A Barskiy, Michael CD Tayler, Irene Marco-Rius, John Kurhanewicz, Daniel B Vigneron, Sevil Cikrikci, Ayca Aydogdu, Moritz Reh, Andrey N Pravdivtsev, Jan-Bernd Hövener, et al.  Zero-field nuclear magnetic resonance of chemically exchanging systems.  Nature communications, 10(1):1–9, 2019.
  * [48] M. P. Ledbetter, S. Pustelny, D. Budker, M. V. Romalis, J. W. Blanchard, and A. Pines.  Liquid-state nuclear spin comagnetometers.  Phys. Rev. Lett., 108(24):243001, 2012.
  * [49] T. Wu, J. W. Blanchard, D. F. Jackson Kimball, M. Jiang, and D. Budker.  Nuclear-spin comagnetometer based on a liquid of identical molecules.  Phys. Rev. Lett., 121(2):023202, 2018.
  * [50] Teng Wu, John W Blanchard, Gary P Centers, Nataniel L Figueroa, Antoine Garcon, Peter W Graham, Derek F Jackson Kimball, Surjeet Rajendran, Yevgeny V Stadnik, Alexander O Sushkov, et al.  Search for axionlike dark matter with a liquid-state nuclear spin comagnetometer.  Physical review letters, 122(19):191302, 2019.
  * [51] Jonathan P King, Tobias F Sjolander, and John W Blanchard.  Antisymmetric couplings enable direct observation of chirality in nuclear magnetic resonance spectroscopy.  The Journal of Physical Chemistry Letters, 8(4):710–714, 2017.
  * [52] Micah P Ledbetter and Dmitry Budker.  nuclear magnetic resonance.  Phys. Today, 66(4):44, 2013.
  * [53] Ralph W Adams, Juan A Aguilar, Kevin D Atkinson, Michael J Cowley, Paul IP Elliott, Simon B Duckett, Gary GR Green, Iman G Khazal, Joaquín López-Serrano, and David C Williamson.  Reversible interactions with para-hydrogen enhance nmr sensitivity by polarization transfer.  Science, 323(5922):1708–1711, 2009.
  * [54] Thad G Walker and William Happer.  Spin-exchange optical pumping of noble-gas nuclei.  Reviews of modern physics, 69(2):629, 1997.
  * [55] Thorsten Maly, Galia T Debelouchina, Vikram S Bajaj, Kan-Nian Hu, Chan-Gyu Joo, Melody L Mak-Jurkauskas, Jagadishwar R Sirigiri, Patrick CA van der Wel, Judith Herzfeld, Richard J Temkin, et al.  Dynamic nuclear polarization at high magnetic fields.  The Journal of chemical physics, 128(5):02B611, 2008.
  * [56] Ronald G Lawler.  Chemically induced dynamic nuclear polarization.  Journal of the American Chemical Society, 89(21):5519–5521, 1967\.
  * [57] S Appelt, FW Häsing, U Sieling, A Gordji-Nejad, S Glöggler, and B Blümich.  Paths from weak to strong coupling in nmr.  Physical Review A, 81(2):023420, 2010.
  * [58] Stephan Appelt, F Wolfgang Häsing, Holger Kühn, and Bernhard Blümich.  Phenomena in j-coupled nuclear magnetic resonance spectroscopy in low magnetic fields.  Physical Review A, 76(2):023420, 2007.
  * [59] Meike Emondts, Micah P Ledbetter, Szymon Pustelny, Thomas Theis, Brian Patton, John W Blanchard, Mark C Butler, Dmitry Budker, and Alexander Pines.  Long-lived heteronuclear spin-singlet states in liquids at a zero magnetic field.  Physical review letters, 112(7):077601, 2014.
  * [60] John W Blanchard, Tobias F Sjolander, Jonathan P King, Micah P Ledbetter, Emma H Levine, Vikram S Bajaj, Dmitry Budker, and Alexander Pines.  Measurement of untruncated nuclear spin interactions via zero-to ultralow-field nuclear magnetic resonance.  Physical Review B, 92(22):220202, 2015.
  * [61] Lieven MK Vandersypen and Isaac L Chuang.  Nmr techniques for quantum control and computation.  Reviews of modern physics, 76(4):1037, 2005.
  * [62] Michael CD Tayler, Tobias F Sjolander, Alexander Pines, and Dmitry Budker.  Nuclear magnetic resonance at millitesla fields using a zero-field spectrometer.  Journal of Magnetic Resonance, 270:35–39, 2016.
  * [63] Tobias F Sjolander, Michael CD Tayler, Jonathan P King, Dmitry Budker, and Alexander Pines.  Transition-selective pulses in zero-field nuclear magnetic resonance.  The Journal of Physical Chemistry A, 120(25):4343–4348, 2016.
  * [64] A Llor, Z Olejniczak, Joseph R Sachleben, and A Pines.  Scaling and time reversal of spin couplings in zero-field nmr.  Physical Review Letters, 67(15):1989–1992, 1991.
  * [65] Daoyi Dong and Ian R Petersen.  Quantum control theory and applications: a survey.  Iet Control Theory and Applications, 4(12):2651–2671, 2010.
  * [66] Francesca Albertini and Domenico D’Alessandro.  The lie algebra structure and controllability of spin systems.  Linear algebra and its applications, 350(1-3):213–235, 2002.
  * [67] Michael A Nielsen and Isaac Chuang.  Quantum computation and quantum information.  American Association of Physics Teachers, 2002.
  * [68] Dmitry Budker, Peter W Graham, Micah Ledbetter, Surjeet Rajendran, and Alexander O Sushkov.  Proposal for a cosmic axion spin precession experiment (casper).  Physical Review X, 4(2):021030, 2014.
  * [69] Navin Khaneja, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen, and Steffen J Glaser.  Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms.  Journal of magnetic resonance, 172(2):296–305, 2005.
  * [70] Steffen J Glaser, Ugo Boscain, Tommaso Calarco, Christiane P Koch, Walter Köckenberger, Ronnie Kosloff, Ilya Kuprov, Burkhard Luy, Sophie Schirmer, Thomas Schulte-Herbrüggen, et al.  Training schrödinger’s cat: quantum optimal control.  The European Physical Journal D, 69(12):1–24, 2015.
  * [71] Zdeněk Tošner, Thomas Vosegaard, Cindie Kehlet, Navin Khaneja, Steffen J Glaser, and Niels Chr Nielsen.  Optimal control in nmr spectroscopy: Numerical implementation in simpson.  Journal of Magnetic Resonance, 197(2):120–134, 2009.
  * [72] Navin Khaneja, Roger Brockett, and Steffen J Glaser.  Time optimal control in spin systems.  Physical Review A, 63(3):032308, 2001.
  * [73] Alberto Carlini, Akio Hosoya, Tatsuhiko Koike, and Yosuke Okudaira.  Time-optimal quantum evolution.  Physical review letters, 96(6):060503, 2006.
  * [74] Domenico d’Alessandro.  Introduction to quantum control and dynamics.  CRC press, 2007.
  * [75] J Geng, Y Wu, and X et al. Wang.  Experimental time-optimal universal control of spin qubits in solids.  Physical review letters, 117(17):170501, 2016.
  * [76] Ray Freeman.  Selective excitation in high-resolution nmr.  Chemical reviews, 91(7):1397–1412, 1991.
  * [77] Emanuel Knill, D Leibfried, R Reichle, J Britton, R B Blakestad, J D Jost, C Langer, Roee Ozeri, S Seidelin, and D J Wineland.  Randomized benchmarking of quantum gates.  Physical Review A, 77(1):012307, 2008.
  * [78] Vadim S Zotev, Andrei N Matlashov, Petr L Volegov, Igor M Savukov, Michelle A Espy, John C Mosher, John J Gomez, and Robert H Kraus Jr.  Microtesla mri of the human brain combined with meg.  Journal of Magnetic Resonance, 194(1):115–120, 2008.
  * [79] James P Gordon, Herbert J Zeiger, and Charles H Townes.  The maser—new type of microwave amplifier, frequency standard, and spectrometer.  Physical review, 99(4):1264, 1955.
  * [80] Mark Oxborrow, Jonathan D Breeze, and Neil M Alford.  Room-temperature solid-state maser.  Nature, 488(7411):353–356, 2012.
  * [81] F Verpillat, MP Ledbetter, S Xu, DJ Michalak, C Hilty, L-S Bouchard, S Antonijevic, D Budker, and A Pines.  Remote detection of nuclear magnetic resonance with an anisotropic magnetoresistive sensor.  Proceedings of the National Academy of Sciences, 105(7):2271–2273, 2008.
  * [82] Dmitry Budker.  Extreme nuclear magnetic resonance: Zero field, single spins, dark matter….  Journal of Magnetic Resonance, 306:66–68, 2019.