# A nuclear magnetic resonance spectrometer for operation around 1 MHz with a sub 10 mK noise temperature based on a two stage dc SQUID

## Abstract

We have developed a nuclear magnetic resonance spectrometer with a series tuned input circuit for measurements on samples at millikelvin temperatures based on an integrated two-stage superconducting quantum interference device current sensor, with an energy sensitivity when operated at 1.4 K. To maximize the sensitivity both the NMR pickup coil and tuning capacitor need to be cooled, and the tank circuit parameters should be chosen to equalize the contributions from circulating current noise and voltage noise in the SQUID. A noise temperature was measured, at a frequency of 0.884 MHz, with the circuit parameters close to optimum.

The extremely high sensitivity of superconducting quantum
interference devices (SQUIDs) to magnetic flux has been exploited in
nuclear magnetic resonance (NMR) spectrometers by a number of groups
for measurements on both cryogenic and room temperature
samples.^{1} In most of these applications the input
circuit is broadband with the NMR pickup coil and SQUID input coil
forming a superconducting flux transformer, however one can obtain
an improved signal-to-noise ratio () with SQUIDs, at the
expense of bandwidth, by tuning the input circuit. Clarke et
al.^{2; 3} considered noise mechanisms and optimization
of dc SQUID circuits for tuned inputs. SQUID amplifiers with tuned
input circuits have been used for nuclear quadrupole
resonance,^{4} NMR^{5; 6} and magnetic
resonance imaging.^{7}

Cooled preamplifiers have been employed to reduce noise in NMR
systems with tuned source impedances. Performance can be
characterized by the noise temperature (the temperature of
the source at which its contribution to the total output noise power
equals that of the amplifier). For a SQUID amplifier with a tuned
input the optimum is proportional to frequency. For our
experiments on NMR samples at millikelvin temperatures we use
frequencies 1 MHz, at which we may obtain a good with
acceptable rf heating. Cooled GaAs MESFETs have been used to detect
NMR signals at these frequencies, achieving 1 K.
^{8} Using a dc SQUID in open loop mode Freeman
et al.^{5} obtained a of 300 mK at 1.9 MHz,
limited by the readout electronics. Improved noise performance is
possible using two-stage SQUID amplification, and this provides the
primary motivation for this work. Two-stage SQUID amplifiers have
been used by Falferi et al.^{9} in a resonant
gravitational wave detector, with an estimated of 15K
at 11 kHz, and by Mück et al.,^{10} who obtained
a around 50 mK at 0.5 GHz with a microstrip SQUID cooled to
20 mK.

In this work we have developed a tuned NMR spectrometer based on an
integrated two-stage low-transition-temperature (low ) dc SQUID
sensor. The spectrometer sensitivity is limited by Johnson noise in
the resistive element of the input circuit and by SQUID amplifier
noise. The goal of this work was to study the limit imposed by the
SQUID noise. To that end we measured the spectrometer noise as a
function of the temperature of the resistive pickup coil, in the
absence of a sample. The pickup coil was placed in a superconducting
shield to suppress external interference. Performance of a similar
SQUID with a broadband superconducting input circuit used for NMR is
reported elsewhere.^{11}

A schematic diagram of the tuned NMR spectrometer is shown in
Fig. 1. The pickup coil forms a series
resonant circuit with a capacitor and the SQUID input coil
inductance . is the resistance in the input circuit. In
our case is mainly associated with the copper pickup coil, but
has contributions from dissipation in the capacitor and from the
presence of the SQUID. The SQUID sensor consists of a single SQUID
first stage, read out by a 16-SQUID series array, integrated onto a
single chip.^{12} The SQUID, mounted at 1.4 K (rather than
4.2 K) for improved noise performance, is connected directly to the
room temperature readout electronics.^{13} This permits
flux-locked loop (FLL) operation without flux modulation, and large
FLL bandwidths of dc up to 6 MHz. The device contains an array of 16
unshunted SQUIDs in the input circuit, which operates as a
-spoiler.^{4; 5} This reduces the tank circuit
quality factor for high signal levels and hence shortens the
recovery time from large current transients following removal of an
NMR transmitter pulse.

The experiment was mounted on a nuclear adiabatic demagnetization
cryostat, capable of achieving 200 K, previously used
for our earlier NMR measurements on at low
millikelvin temperatures using a SQUID spectrometer with a tuned
input.^{6} The NMR pickup coil was wound from copper wire
and heat sunk to a plate whose temperature could be varied from
10 to 1500 mK. Care was taken to avoid thermal noise from dissipative
elements in the tuning capacitor by keeping it close to the coil
temperature.

Our two-stage SQUID is designed to have a single-SQUID-like
flux-voltage characteristic,^{12} with a large overall flux
to voltage transfer function . As shown in
Fig. 1, we model the device as a single SQUID of
inductance (equal to that of the first stage SQUID), with a
voltage noise at the output, and a circulating current noise
in the SQUID loop. includes noise from the room
temperature amplifier as well as from the two SQUID stages. The
spectral densities of these noise sources, , and a
correlation , can be considered white above a few tens of
Hz. In this work we write and in terms of a flux noise
in the SQUID as follows:

(1) |

where is the total effective flux noise measured with the
input circuit open, and is a dimensionless parameter. We
define the overall energy sensitivity as . Tesche et al.^{14} calculated
intrinsic noise in a single SQUID. They found to be real,
and to be of the order of unity but dependent on SQUID bias.
We assume this to be the case for our two-stage device, since noise
from the first stage is dominant. A real does not
contribute to noise at the tank circuit resonance frequency.
^{7}

In an NMR experiment, precession of the magnetization induces a signal voltage in the pickup coil. In this case the total flux in the SQUID is

(2) |

Here is the mutual inductance between the input coil and the
SQUID, is the total impedance
in the input circuit, where . can be
influenced by coupling to the SQUID, mainly through a contribution
to , which we neglect in this model. Changes in the SQUID
parameters due to strong coupling to the input circuit are also
neglected, since they are insignificant for the case of a high-
tuned source close to optimum.^{15} represents
the Johnson noise voltage in , which is at temperature . This
noise source is also white at the frequencies of interest. On
resonance the total flux noise is given by

(3) |

Let the signal be close to the tank circuit resonance frequency and measured in a bandwidth . The signal-to-noise ratio can be defined as

(4) |

where from Eq. (3)

(5) |

For a given experiment the sample geometry and resonant frequency are fixed. The choice of quality factor is limited by the desired bandwidth and the requirement of a sufficiently short recovery time following a transmitter pulse. In order to optimize the sensitivity , and can be varied simultaneously within these constraints. Changing the number of turns in the pickup coil varies and such that . Then Eq. (4) can be written as

(6) |

The highest is achieved by making as large as possible within the constraints. For , is close to 1 when is minimized (see Eq. (8)). Therefore the is maximum when is minimum.

The conditions for minimizing are more evident when Eq. (5) is written as

(7) |

Here is the effective coupling constant between the SQUID and the input circuit. Minimizing involves setting , resulting in

(8) |

where the coupling coefficient . For our device H, nH, and pH, corresponding to .

For NMR at 1 MHz a of the order of 100 is reasonable, then for we obtain H and . We expected so we wound a H pickup coil. By driving the pickup coil via the transmitter we measured at kHz, from which we inferred and . Once is known, full optimization could be achieved by addition of a further resistive element at coil temperature . For these studies of noise as a function of temperature this was not done, to eliminate potential errors arising from temperature gradients.

The temperature dependence of the noise in this tuned circuit was measured at five different SQUID working points (two open loop and three FLL), the results of which are summarized in Table 1. The output of the SQUID electronics was fed to an HP 3588A spectrum analyzer with a negligible noise level. In Fig. 2 we show the frequency dependence of the flux noise measured in open loop mode (working point 1) with the pickup coil at 20 mK and the tuning capacitance at 10 mK. For a high- input circuit the off-resonance flux noise is equivalent to that of an open circuit. We infer the open input flux noise at from the off-resonance noise as shown by the dashed line. This corresponds to , where is Planck’s constant. The inset shows noise in the vicinity of resonance as the pickup coil temperature is varied between 20 and 800 mK. The noise power at resonance, normalized by the off-resonance noise power, can be written as

(9) |

This is plotted versus coil temperature in Fig. 3. These data are linear over a wide temperature range. The data deviate from a linear fit at the lowest temperatures possibly due to insufficient thermalization of the pickup coil. We obtain mK from a fit to the linear region. We obtain an almost identical with the SQUID operated in FLL mode (see Table 1), a consequence of the low parasitic coupling between feedback and input coils achieved through careful SQUID design.

Within our model, at working point 1, we estimate from the value of , obtained from the fit in Fig. 3. Eq. (7) gives a predicted of 14 mK for the naïve assumption of , and mK using the measured value of . The slope of the fit in Fig. 3 is less then that obtained using Eq. (9). This could be explained if a noiseless resistance, arising from coupling to the SQUID, accounted for of the total input resistance.

Under optimum conditions tuning the input circuit gives an improvement in approaching a maximum of order if one cools to well below . This can be very significant and becomes easier to achieve as increases.

In summary we have designed and built a tuned NMR spectrometer based on a two-stage dc SQUID coupled to a conventional wire wound pickup coil to study superfluid confined in submicron thick cavities. Open loop and FLL tests of this spectrometer with the pickup coil in a superconducting shield have shown that mK is achievable, close to the prediction based on the measured energy sensitivity. Improvements in energy sensitivity are possible through operating the SQUID at temperatures lower than 1.4 K.

This work was supported by EPSRC grant EP/C522877/1 and in part by EPSRC ARF EP/E054129/1.

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