Magnetic Resonance with Squeezed Microwaves

Vacuum fluctuations of the electromagnetic field set a fundamental limit to the sensitivity of a variety of measurements, including magnetic resonance spectroscopy. We report the use of squeezed microwave fields, which are engineered quantum states of light for which fluctuations in one field quadrature are reduced below the vacuum level, to enhance the detection sensitivity of an ensemble of electronic spins at millikelvin temperatures. By shining a squeezed vacuum state on the input port of a microwave resonator containing the spins, we obtain a 1.2-dB noise reduction at the spectrometer output compared to the case of a vacuum input. This result constitutes a proof of principle of the application of quantum metrology to magnetic resonance spectroscopy.

Popular Summary

Magnetic resonance spectroscopy has a significant impact on our everyday lives, from medical imaging in hospitals to quality control in beer production. The technique uses powerful magnets and radio waves to reveal concentrations of molecules in a substance. Improving the sensitivity—achieved by both boosting the signal measured and reducing the noise—allows for smaller quantities of materials to be measured. The thermal contribution to this noise can be reduced by cooling to low temperatures (less than 1 K), but even in this limit, fluctuations of the electromagnetic field remain (as required by Heisenberg’s uncertainty principle). Recently, the noise in magnetic resonance spectroscopy of electron spins was reduced to the level of these vacuum quantum fluctuations by using a superconducting amplifier operating at 10 mK. One might expect that the noise cannot be reduced below this limit. We have found a way, however, to improve on this performance.

While the Heisenberg uncertainty principle limits the combined uncertainty of two variables, it does not forbid us from engineering states where the variance in one quantity is reduced at the expense of the other. Such states are called squeezed states, and they provide a way to measure data with a signal-to-noise ratio beyond the vacuum fluctuation limit. We use a Josephson parametric amplifier (a type of driven harmonic oscillator that can amplify a signal) to prepare a squeezed state of the electromagnetic microwave field. In this state, fluctuations that are in phase with the signal from the electron spins are reduced, while noise that was out of phase is amplified. We find a 1.2-dB improvement in the signal-to-noise ratio of the spin echo.

Future work will target larger degrees of squeezing and applications to real-world magnetic resonance measurements.

Figure 1

Principle of squeezing-enhanced pulsed magnetic resonance. A squeezed vacuum state is incident on an ESR cavity of frequency ${\omega }_0$. The cavity contains the spins to be detected, which are tuned into resonance at ${\omega }_0$ by a dc magnetic field $B_0$. A Hahn echo microwave pulse sequence ($\pi /2-\tau -\pi -\tau$) is applied to the spins, leading to the emission of an echo in the detection waveguide on the $X$ quadrature. This echo is noiselessly amplified along $X$ before its homodyne demodulation with a local oscillator (LO) phase such that $I(t)$ is proportional to $X(t)$. The traces in the bottom right-hand gray box, which are not real data, depict schematically the expected difference between SQZ off (blue) and SQZ on (red) output quadrature signals when the squeezed quadrature is aligned along the echo emission quadrature $X$; the signal-to-noise ratio is improved on the $I$ quadrature, which contains the entire echo signal.

Figure 2

Experimental setup. (a) The JPAs providing the squeezed vacuum and the noiseless amplification are superconducting $LC$ resonators containing a SQUID array, tuned to a frequency close to the ESR cavity frequency ${\omega }_0$ by the application of a dc flux bias to the SQUID loops. Modulating this flux at twice the resonator frequency by application of a pump tone yields parametric gain at $\omega \simeq {\omega }_0$ for the signals reflected off the JPA. (b) The ESR cavity, whose optical micrograph is shown, is an aluminum lumped-element $LC$ resonator of frequency ${\omega }_0$ patterned on top of a silicon sample containing the spins. A magnetic field $B_0$ is applied parallel to the sample surface and to the resonator inductor (in blue) to tune the spin frequency. Only spins in the immediate vicinity of the inductor are detected. (c) The resonator is enclosed in a copper box holder and is capacitively coupled to the measurement line with a constant ${\kappa }_C$ via an antenna fed through the 3D copper sample holder, thermally anchored at 20 mK. A second port, much less coupled (constant ${\kappa }_A\ll {\kappa }_C$), is used for characterization (see text). Squeezed microwaves at ${\omega }_0$ are generated by a first JPA denoted SQZ, routed onto the resonator via a circulator, and the reflected signal is noiselessly amplified by a second JPA denoted AMP. Both are operated in the degenerate mode and pumped at $2{\omega }_0$ with respective phases ${\varphi }_S$ and ${\varphi }_A$. Further amplification is provided at 4 K by a high electron mobility transistor (HEMT) amplifier and at 300 K. Homodyne demodulation at the signal frequency yields the quadratures $I(t)$ and $Q(t)$. (d) Measured reflection coefficient $|S_{11}^2|$ (red line; blue line is a fit) yielding ${\omega }_0/2\pi =7.3035\mathrm{GHz}$, ${\kappa }_C=1.6\times {10}^6{\mathrm{s}}^{-1}$, and ${\kappa }_A+{\kappa }_L=6\times {10}^4{\mathrm{s}}^{-1}$, ${\kappa }_L$ being the resonator internal loss rate. We determine ${\kappa }_A=3\times {10}^3{\mathrm{s}}^{-1}$ by measuring the full resonator scattering matrix (not shown).

Figure 3

Characterization of the prepared squeezed vacuum state. (a) Gain of SQZ (open red diamonds) and AMP (open blue circles) as a function of their pump phase (${\varphi }_{S,A}$, respectively) for the chosen pump amplitude settings, leading to a power gain $G_S^2=6\mathrm{dB}$ and $G_A^2=20\mathrm{dB}$ for ${\varphi }_{S,A}=-\pi /4$. Optimal values for the SQZ and AMP pump phases are indicated with arrows. Dashed curves are sinusoidal fits. (b) The variance $\delta I^2$ in the noise is plotted as a function of the difference between the SQZ and AMP pump phases ${\varphi }_{\mathrm{\Delta }}={\varphi }_S-{\varphi }_A$, ${\varphi }_A$ being set at its optimal value. Data with AMP and SQZ both on (red diamonds) are compared to those obtained with AMP on and SQZ off (blue dashed line), and with AMP and SQZ both off (green dashed line). Shaded areas represent the $5\sigma$ measurement uncertainty. Squeezing is obtained for the optimal setting ${\varphi }_{\mathrm{\Delta }}=\pi /2$. (c) Noise histograms obtained using the optimal phases obtained above, for AMP and SQZ both off (green open squares), in which case the noise is determined by the HEMT amplifier, AMP on and SQZ off (blue crosses), in which case the noise is the sum of the HEMT and the amplified vacuum fluctuations, and AMP and SQZ both on (red open symbols), in which case the fluctuations are reduced below the vacuum level. Gaussian fits for each are also shown (curves). (d) Using the optimal phase settings, the squeezing factor (see main text) ${\eta }_S$ is measured as a function of the SQZ power gain (red dots; rectangles represent the $5\sigma$ measurement uncertainty) by varying the SQZ pump power. A linear fit (dashed line) for the low-gain part of the curve indicates the microwave losses between SQZ and AMP to be ${\eta }_{\mathrm{loss}}=0.54$. The black arrow indicates the SQZ gain selected in the experiment.

Figure 4

Limitations induced by JPA saturation. (a) Output quadratures $X_{\mathrm{out}}$ and $Y_{\mathrm{out}}$ measured for weak coherent signals sent to the SQZ with input phases ${\varphi }_S$ spanning the whole interval between 0 and $2\pi$ for 4 different gains $G_S^{-2}$ indicated by the arrows in Fig. 3 and set using different pump powers (black curve corresponds to SQZ pump off). Note that these data are obtained in a separate calibration run in which the ESR cavity is removed [35]. (b) Measured output power and phase of a signal at ${\omega }_0$ as a function of its input power $P_{\mathrm{in}}$ after amplification by the AMP device, in the same conditions as in (a) but with the JPA operated in the nondegenerate mode by detuning the pump by 300 kHz from $2{\omega }_0$. The output power depends linearly on $P_{\mathrm{in}}$ as long as $P_{\mathrm{in}}<-131\mathrm{dBm}$ (blue lines is a linear fit), while the phase shift is zero (blue line) only for $P_{\mathrm{in}}<-137\mathrm{dBm}$. (c) Time trace of the $I$ quadrature of a weak microwave pulse at ${\omega }_0$ sent via the ${\kappa }_A$ port of the ESR cavity, measured with SQZ off (open circles) and SQZ on (black traces) for different input powers indicated by the arrows in (b). The traces have been averaged 4000 times. Above $-136\mathrm{dBm}$, deviations appear between the SQZ on and SQZ off curves due to AMP saturation.

Figure 5

Squeezing-enhanced spin-echo detection. (a) Energy of the 20 levels of bismuth donors in silicon as a function of $B_0$ (gray lines). The transition between the levels indicated in green is used in the experiment. (b) Hahn-echo-detected magnetic field sweep, showing the bismuth donor resonance line. Blue arrow indicates the field chosen in the rest of the experiment, blue Lorentzian curve indicates the fraction of spins that are within the cavity resonance. (c) Echo signals observed with SQZ off (blue) and on (red) for a single shot (lines) and averaged over 2500 traces (symbols) confirm the signal intensity is identical. SQZ was switched on only during a short $\mathrm{\Delta }t=200\text{−}\mu \mathrm{s}$ window around the echo emission time (dashed rectangle in the pulse sequence). The excitation pulse angle is chosen to be $\approx \pi /3$ in order to avoid saturation effects (see main text). (d) Histograms of the noise around the average signals of (c) measured with 2500 single-shot traces acquired on a $70\text{−}\mu \mathrm{s}$ time window centered on the echo, with SQZ off (blue) and on (red) (see Supplemental Material [34]), and corresponding Gaussian fits (curves). Standard deviations are $0.0858\pm 2\times {10}^{-4}\mathrm{V}$ for SQZ off and $0.0748\pm 2\times {10}^{-4}\mathrm{V}$ for SQZ on, confirming a reduction in the noise accompanying the spin-echo signal when SQZ is on.

Figure 6

Influence of squeezing on the spin coherence times. (a) Coherence time $T_2$ measured with a Hahn echo sequence for SQZ off (blue circles) and on (red squares). Contrary to the experiments in Fig. 5, SQZ is now switched on or off for the entire experimental sequence. The integrated echo signal is plotted as a function of the delay $\tau$ between the $\pi /2$ and $\pi$ pulses. Exponential fits (solid lines) yield $T_{2,\mathrm{off}}=2.5\pm 0.2\mathrm{ms}$ and $T_{2,\mathrm{on}}=2.6\pm 0.4\mathrm{ms}$. (b) Energy relaxation time $T_1$ with SQZ on (blue circles) and off (red circles). Exponential fits (solid lines) yield $T_1=900\pm 50\mathrm{ms}$ with SQZ off and $T_1=450\pm 40\mathrm{ms}$ with SQZ off. Both $T_1$ and $T_2$ curves [(a) and (b)] have their amplitude reduced by $\approx 0.5$ with SQZ on, indicating reduced spin polarization in the steady state when the SQZ is continuously switched on.