Frequency super-resolution with quantum environment engineering in a weakly coupled three-nuclear-spin system

Abstract

Optical super-resolution has been widely employed to beat the spatial diffraction limit, which is often stated by Abbe-Rayleigh criterion. Analogously, we propose a frequency super-resolution method, which beats conventional frequency spectral resolution limit often approximated by the full width at half maximum of the spectral peak, $\Gamma$. This method utilizes recently developed quantum environment engineering technique. With numerical simulations and experiments, we demonstrate, as a proof-of-concept, the frequency super-resolution method in a three-nuclear-spin system (Trifluoroiodoethylene) by successfully decomposing a thermal state spectrum of the spin $F_3$ into four peaks of engineered pseudo-pure states of the quantum environment. The ultimate frequency resolution reaches $\sim 0.005 \,\Gamma$. This method is potentially useful in spectral decomposition of weakly coupled small-size nuclear spin systems.

Introduction

Optical spectrum (emission or absorption) is an efficient and powerful tool to probe the microscopic world, such as atomic and molecular structure by infrared and visible light^1,2, chemical and biological structure by microwave and radio frequency³. Fine and hyperfine structures are pursued by pushing very extreme laboratory conditions, e.g. lowering the temperature to nanokelvin⁴, raising the vacuum to ultrahigh level (nanobar)⁵, and increasing the magnetic field to tens of Tesla^6,7. However, the spectrum is ultimately limited by a “diffraction limit”, i.e. two adjacent emission (or absorption) lines are not discernible if their separation is smaller than their line width $\Gamma$, which is due to homogeneous and inhomogeneous line broadening^8,9. To improve the spectral resolution, numerical methods rooted in information science are employed^10,11. For instance, the multiple signal classification technique is widely used to precisely determine the peak positions of multi-peak signals¹². However, these numerical multiple peak fitting methods often rely on the assumptions of peak number and peak profile which are often conjectured without direct and physical justification.

An alternative way to beat the frequency spectral resolution limit may be borrowed from the idea of optical super-resolution^{13,14,15,16,17}. In conventional optics, two point light sources are not spatially resolvable on the image plane if their distance is smaller than the Abbe-Rayleigh criterion (spatial diffraction limit)^18,19. A clever idea in photoactivated localization microscopy (PALM) by switching on and off local fluorescence is employed to beat the diffraction limit and reaches $\sim$10 nm spatial resolution with the light wavelength $\sim$500 nm¹⁷. In fact, such an optical super-resolution is realized by introducing an extra dimension of time where the multiple peaks are stochastically separated but keep the Airy disc unchanged for each point light source without considering numerical method to reduce the Airy disc. In analogy, we may purposefully introduce an extra dimension to break the frequency spectral resolution limit of overlapping spectra and realize frequency super-resolution (FRESURE), by “switching on” a single peak and “turning off” the remaining peaks but keeping the line width untouched (find more details in Sect. 1 of Supplementary Material²⁰).

Our main idea is to realize FRESURE by introducing the extra dimension, quantum environment (QE) state. It is known that a quantum system always couples with its environment, which may be divided into quantum and classical parts. The interaction of the system with its QE often introduces line splitting, e.g. hyperfine splitting by electron and nuclear spin coupling, while the interaction with its classical environment causes line broadening (line width) in the nuclear spin system we consider here²¹. Once the line width is larger than the line splitting (under frequency spectral resolution limit), one would fail to resolve the splitting. With the advancement of quantum technology for recent decades, it now becomes possible to precisely manipulate and engineer a certain QE state^22,23,24,25, which exhibits a single peak with the same line width (similar to PALM but in a definite way). Therefore, FRESURE may be realized by introducing a set of engineered QE states, serving as the additional dimension.

In this paper we put forward a theory of FRESURE for a weakly coupled n-nuclear-spin system, where one spin is considered as the quantum system (emitting radio frequency light) and the other spins the QE²⁶. Numerical simulations and experimental results of nuclear magnetic resonance (NMR) further confirm the validity of the theory. Potentially, super-resolved spectra may find many scientific and practical applications in much broader areas, not only in physics but also in chemistry, biology, medicine, and magnetic resonance imaging.

The paper is organized as follows. We introduce in Sect. Physical spectral decomposition and FRESURE with QE engineering the Hamiltonian of the n-nuclear-spin system and derive the theory of physical spectral decomposition and FRESURE. In Sect. FRESURE with numerical simulations, we numerically confirm the validity of theoretic predictions. With a NMR simulator of SpinQ Triangulum²⁷, experimental results are shown in Sect. FRESURE in experiments, which agree well with theoretical results. Finally, in Sect. Conclusions and discussions we summarize and discuss further experiments in the future.

Physical spectral decomposition and FRESURE with QE engineering

We consider a typical spin system with n weakly coupled spin-1/2 nuclei $F_i$ ($i=1,2,\cdots ,n$)²⁸. Its Hamiltonian in a strong bias magnetic field $B_0$ along the z axis can be in principle separated into three parts (we set $\hbar =1$)

$$\begin{aligned} \begin{aligned} H&= H_S + H_Q + H_{SQ},\\ H_S&= {1\over 2} (\omega _0+\delta _n+\eta )\sigma _{nz}, \\ H_Q&= {1\over 2}\sum _{i=1}^{n-1}(\omega _0+\delta _i+\eta )\sigma _{iz} + \sum _{i<j}^{n-1}{J_{ij}\over 4} \sigma _i\cdot \sigma _j, \\ H_{SQ}&= \sum _{i=1}^{n-1}{J_{in}\over 4}\sigma _i\cdot \sigma _n, \end{aligned} \end{aligned}$$

(1)

where $H_S, H_Q$ and $H_{SQ}$ are respectively the system Hamiltonian, the QE Hamiltonian, and the system-environment coupling. The Larmor frequency of the spins is $\omega _0=\gamma B_0$ with $\gamma$ the nuclear gyromagnetic ratio, $\delta _i$ is the chemical shift of the ith nuclear spin, $J_{ij}$ characterizes the spin-spin coupling constant, and $\eta =\gamma b_z$ denotes the noise from inhomogeneous stray magnetic field $\textbf{b}$. Typical NMR experiment satisfies $\omega _0 \gg |\delta _i|, |J_{ij}|, |\textbf{b}|$ thus the effects of $b_x$ and $b_y$ are averaged out and neglected. In addition, we assume weak coupling, i.e. $|\delta _i-\delta _j|\gg |J_{ij}|$, so that the Hamiltonian is further simplified to the secular coupling form after neglecting the transverse coupling terms wth $\sigma _{ix}\sigma _{jx}$ and $\sigma _{iy}\sigma _{jy}$

$$\begin{aligned} H_E\approx {1\over 2}\sum _{i=1}^n(\omega _0+\delta _i+\eta )\sigma _{iz}+{1\over 4}\sum _{i<j}^nJ_{ij} \sigma _{iz}\sigma _{jz}. \end{aligned}$$

(2)

The thermal equilibrium state of the spin system is $\rho _T=e^{-H/(k_BT)}/{\text {Tr}}(e^{-H/(k_BT)})$. In the high temperature limit $k_B T \gg \omega _0 \gg |\delta _i|, |J_{ij}|$ for typical NMR spin systems, the thermal equilibrium state becomes $\rho _T = (I/2^n)+(p/2)\sum _{i=1}^n\sigma _{iz}$ where $p \approx -\omega _0/(2^nk_BT)$ and I is the $2^n \times 2^n$ identity matrix. A thermal initial state is prepared by performing a $\pi /2$ rotation of spin $F_n$ along y axis

$$\begin{aligned} \rho _T(0) = \frac{1}{2^n}I+\frac{p}{2}(\sum _{i=1}^{n-1}\sigma _{iz}+\sigma _{nx}). \end{aligned}$$

(3)

Starting from the thermal equilibrium state, a pseudo pure state (PPS) is prepared with time averaging method^27,29,30, $\rho _{PPS}=(1-q)I/2^n+q|00\cdots 00\rangle \langle 00\cdots 00|$, about the latter of which there are n bits located in $|0\rangle$. Since the identity matrix I has no contribution to the free induction decay (FID) signal, the PPS acts like a pure state³¹. All initial PPSs may be prepared by flipping the different spin $F_i$ except $F_n$ with $\pi$ pulses and rotating spin $F_n$ with a $\pi /2$ pulse,

$$\begin{aligned} \begin{aligned} \rho _1(0)&=\frac{1-q}{2^n}I+q|00\cdots 0\rangle \langle 00\cdots 0|\otimes (I_2+\sigma _{nx})/2, \\ \rho _2(0)&=\frac{1-q}{2^n}I+q|00\cdots 1\rangle \langle 00\cdots 1|\otimes (I_2+\sigma _{nx})/2, \\ \vdots \\ \rho _{2^{n-1}}(0)&=\frac{1-q}{2^n}I+q|11\cdots 1\rangle \langle 11\cdots 1|\otimes (I_2+\sigma _{nx})/2. \end{aligned} \end{aligned}$$

(4)

where $I_2$ is a $2 \times 2$ identity matrix of spin $F_n$.

Starting from an initial spin state $\rho (0)$, the system evolves under the effective Hamiltonian $H_E$, Eq. (2). At time t, the FID signal of spin $F_n$ is $s(t)\equiv \langle \sigma _{nx}\rangle =\text {Tr}(U_E\rho (0)U_E^\dagger \sigma _{nx})$, where $U_E=e^{-itH_E}$. It is straightforward to calculate the FID signal for all initial PPSs

$$\begin{aligned} \begin{aligned} s_1(t)=&\text {Tr}(U_E\rho _1(0)U_E^\dagger \sigma _{nx})=q\cos (\omega _1 t), \\ s_2(t)=&\text {Tr}(U_E\rho _2(0)U_E^\dagger \sigma _{nx})=q\cos (\omega _2 t), \\ \vdots \\ s_{2^{n-1}}(t)=&\text {Tr}(U_E\rho _{2^{n-1}}(0)U_E^\dagger \sigma _{nx})=q\cos (\omega _{2^{n-1}} t). \end{aligned} \end{aligned}$$

(5)

where $\omega _1 = \alpha +\sum _{i=1}^{n-1}{J_{in}/2}, \omega _2 = \alpha +\sum _{i=1}^{n-2}{J_{in}/2}-J_{(n-1)n}/2, \cdots$ and $\omega _{2^{n-1}} = \alpha -\sum _{i=1}^{n-1}{J_{in}/2}$ with $\alpha =\omega _0+\delta _n+\eta$. Obviously, the FID signal starting from each initial PPS exhibits different and single oscillation frequency. If the initial state is the thermal initial state, the FID is $s_T(t)=\text {Tr}(U_E\rho _T(0)U_E^\dagger \sigma _{nx}) =p[\sum _{m=1}^{2^{n-1}}\cos (\omega _m t)]$. It is obvious that $s_T(t)$ consists of $2^{n-1}$ oscillations, each of which represents a peak after Fourier transform. Interestingly, one immediately finds

$$\begin{aligned} s_T(t)=\frac{p}{q}\sum _{m=1}^{2^{n-1}}s_m(t). \end{aligned}$$

(6)

Such a relation indicates a known NMR result that one can decompose the spectrum from the thermal initial state into the spectra from all different initial PPSs which are actually prepared by QE engineering on spins $F_i (i=1,2,\cdots ,n-1)$.

Different from usual numerical multiple-peak-fitting methods¹¹, such a spectral decomposition is physical and independent of the distribution of magnetic noise $\eta$, i.e. the profile of the peaks. It is especially useful if the line widths of adjacent peaks are larger than the peak splitting, which is nothing but FRESURE (similar to optical super-resolution). Of course, the weak coupling condition $|\delta _i-\delta _j| \gg |J_{ij}|$ is required in principle in order for the physical spectral decomposition to be valid. In Sect. 4 of Supplementary Material, we specifically obtain when $|J_{ij}|/|\delta _i-\delta _j|\sim 0.2$, the weak coupling approximation can work well, which is specifically showed in Fig. 4 of Supplementary Material²⁰.

FRESURE with numerical simulations

Under the weak coupling assumption, we have simplified the original Heisenberg spin coupling to the Ising coupling, as shown in Eq. (2). However, the ratio between the coupling constant and the chemical shift difference $|J_{ij}/(\delta _i-\delta _j)| \sim 0.1$ is not small enough in our experimental case of Trifluoroiodoethylene, $C_2F_3I$, as shown in Fig. 1. Therefore, we employ numerical simulations to check the validity of the physical spectral decomposition. In our numerics, we adopt the Hamiltonian Eq. (1) with $n=3$ and the Zeeman splitting of the strong bias magnetic field $\omega _0 = 39.638$ MHz. We assume $\eta$ follows a probability distribution with a Lorentzian profile $f(\eta ) = (\Gamma /2\pi ) /[\eta ^2 + (\Gamma /2)^2],$ where $\Gamma = 40$ Hz. Other parameters we use are given in Fig. 1b.

Fig. 1

(a) Structure of molecule $C_2F_3I$. The nuclear spins of three fluorine atoms are spin-1/2. (b) Chemical shifts (diagonal) and coupling constants (off-diagonal) for the fluorine spins roughly obtained by the FFT spectrum in Fig. 1 of Supplementary Material²⁰ with the unit of Hz. We set the second peak of spin $F_1$ as 0 Hz (see Sect. 2 in Supplementary Material²⁰ for details).

With these specific parameters and a given initial density matrix of three spins, we calculate the FID signal at different evolution time t under the Hamiltonian Eq. (1). The FID signal is actually averaged over $10^5$ runs of random $\eta$ values that obey the Lorentzian probability distribution function. A spectrum is then obtained by performing a fast Fourier transform (FFT) of the averaged FID signal. We plot in Fig. 2 the spectra with five initial states, the thermal initial state and four initial PPSs (A, B, C, and D). As a comparison, the sum spectrum of four initial PPSs is also presented. Due to their independence of each other, four PPS spectra may be also plotted in an additional dimension denoted by the QE state.

Fig. 2

Numerical results of physical spectral decomposition. Four dashed lines denote the spectra obtained from FID signal $s_A, s_B, s_C, s_D$ of initial PPSs. The sum (purple solid line) of four dashed lines agrees well with the spectrum from the thermal initial state $s_T$ (blue circles). The stray magnetic-field noise $\eta$ follows a Lorentzian distribution with a full width at half maximum (FWHM) $\Gamma = 40$ Hz. Four PPS spectra are also plotted in an additional dimension denoted by the QE state. The separation of B and C is 19.5 Hz. Each spectrum is averaged $10^5$ times.

From Fig. 2, we see that the sum spectrum almost coincides with the thermal spectrum, indicating that the thermal spectrum can be decomposed into four PPS spectra with equal weight in the weak coupling regime we consider here. Moreover, the peak positions of the four PPSs’ spectra (1091.3, 1040.0, 1020.5, 974.1) agree well with the analytical results (1088.5, 1039.5, 1020.5, 971.5) obtained by Eq. (5) with $n=3$, manifesting the fact that small deviation between the original Heisenberg coupling and the effective Ising coupling is negligible.

Different from the numerical fitting method “dmfit” (see Fig. 2 in Supplementary Material²⁰ for fitting results), the decomposition into four PPSs is essentially physical (shown in Fig. 2) and there is no need to carefully adjust any parameter, while a close fitting can also be obtained by adjusting the parameters in “dmfit”. More importantly, by engineering the QE state as an additional dimension, we are able to definitely distinguish two close peaks B and C whose distance is smaller than their line width $\Gamma$. In the spirit of optical super-resolution^14,32,33, we can thus realize FRESURE by QE engineering.

FRESURE in experiments

To confirm the above analytical and numerical results on the spectral decomposition and the realization of FRESURE, we carry out NMR experiments with a real physical system composed of three $^{19}$F nuclear spins of Trifluoroiodoethylene ($C_2F_3I$). The structure of the molecule is shown in Fig. 1a. The natural Hamiltonian of the system is written as Eq. (1) with $n=3$ and parameters given in Fig. 1b, the exact values of which are not strictly known³⁴. The experiments are performed at a fixed temperature 35$^{\circ }$C, using a SpinQ Triangulum ($B_0= 0.97$ Tesla)²⁷.

The experiments are divided into three steps in general: thermal equilibrium state preparation after long enough relaxing time, initial state preparation by applying a $\pi /2$ pulse to spin $F_3$ (and $\pi$ pulses to spin $F_{1,2}$ after preparation of a PPS), and measurement of the FID signal of $F_3$. Typical FID and spectrum of the three spins are presented in Fig. 1 of Supplementary Material²⁰. To clearly distinguish the FIDs from the large background noise, we average the FID signals over 70 times before performing the FFT to obtain the final spectrum. We deliberately configure the system off its sweet spot so that $\Gamma =60$Hz in order to demonstrate the advantages of physical spectral decomposition and FRESURE.

Fig. 3

Fitting results (FWHM is 130.6Hz) of the thermal spectrum (blue circles) with four Lorentzian peaks (black dashed lines) by assuming the same width and height. The separation of 1 and 4 is 117.6 Hz. The sum (purple solid line) shows only a single fat peak.

Experimental results for the thermal initial state are shown in Fig. 3, where $\tilde{s}_T(\omega ) = FFT(s_T(t))$. We roughly observe three small peaks on a high plateau. According to the conventional method, we decompose numerically the thermal initial state results into four Lorentzian peaks with the constraints of the same height and width, by inputting the system-parameter-simulated $\omega _i$ as initial value and further requiring the fitting parameters within the range $[\omega _i -10, \omega _i +10]$ Hz. The fitting results are shown in Fig. 3 and in Table 1 (see Fig. 3 and Table 1 of Supplementary Material²⁰ for other fitting results and parameters with program “dmfit”¹¹). Not surprisingly, as shown in Fig. 3, the sum of the decomposed four peaks exhibits only a single fat peak, not close to the experimental results at all. Clearly, numerical fitting methods are difficult to correctly decompose the thermal experimental data.

To demonstrate the physical spectral decomposition, we plot experimental results from the thermal initial state and four PPSs in Fig. 4. For each PPS, we observe a single peak, $\tilde{s}_{A, B, C, D}(\omega ) = FFT(s_{A, B, C, D}(t))$. The peak positions with their average value and the widths of four PPSs are listed in Table 1. The former one indicates that the four peaks are indeed four single peaks, while the latter one suggests that we are unable to discern B and C in the thermal initial state. Additionally, experimental peak positions are around 10Hz larger than analytical peak positions (1088.5, 1039.5, 1020.5, 971.5), which is reasonable with considering the value of $\eta$. To check the validity of physical spectral decomposition, we multiply the sum of the four PPSs’ spectrum by a constant $\lambda$. An optimal $\lambda$ is searched by the least square fit, i.e. minimizing the value of $\int _{\omega _-}^{\omega _+} (\tilde{s}_T- \lambda \tilde{s}_S)^2 d\omega$ where $\omega _- = 850$ Hz, $\omega _+ = 1200$ Hz, and $\tilde{s}_S = \tilde{s}_A + \tilde{s}_B + \tilde{s}_C + \tilde{s}_D$. We find $\lambda =1.5$, bigger than 1 compared with Fig. 2, which may depend on the experimental imperfections, such as PPS purity with specific preparation process of PPSs²⁷, the pulse strength error, non-resonance pulse and other potential errors. To theoretically predict $\lambda$, we make numerical simulation considering three possible errors above and obtain $\lambda \sim 1.32$, which indicates other factors not considered have contribution to the imperfections.

Fig. 4

Experimental results of physical spectral decomposition. The lines and symbols are the same as those in Fig. 2. Clearly, the three peaks of the thermal state can be decomposed into four PPSs. The separation of B and C, $d_{B-C}$, is 24.4 Hz. $\Gamma$ is around 60 Hz. Each spectrum is averaged 70 times.

Obviously, the sum spectrum agrees well with the thermal spectrum in Fig. 4, indicating that the physical spectrum decomposition is valid for the $C_2F_3I$ spin system. More importantly, the spectral distance between peaks B and C is much smaller than their widths. It is rather challenging to discern the peaks B and C with conventional methods. However, these peaks are completely separated by introducing the extra dimension of the QE state, as shown in Fig. 4. Therefore, we demonstrate the FRESURE experimentally by the QE state engineering.

Table 1 Peak positions $\omega _i$ with their standard deviation and FWHM $\Gamma _i$ ($\Gamma _i^L$) with their standard deviation of the spectrum of PPSs, extracted from experimental results (as shown in Fig. 4) and the fitting results (as shown in Fig. 3).

Compared with the numerical results presented in Fig. 2, the three peaks of the thermal initial state in Fig. 4 are not distinct. Two side peaks are not symmetric with respect to the central one. The mechanism behind such an asymmetry of the side peaks is unclear and worth exploring in the future. In addition, the experimental results for the PPSs are not exactly in Lorentzian shape because of their long and flat tails. To be more specific, we list in Table 1 the Lorentzian width $\Gamma _i^L$, which is defined as the fitting width of each PPS experimental data above 70% of its height (to avoid their exceptional long tail), assuming a Lorentzian line shape. The validity of physical decomposition with non-Lorentzian spectral profile of the PPSs may indicate independence of the decomposition method from the line shape.

To explore the FRESURE limit, we calculate the Allan deviation of the peak positions of four PPSs with the Eq.(4) from³⁵. The results are presented in Fig. 5. Clearly, as the measurements are repeated many times, the Allan deviation decreases rapidly and reaches its minimum which is in general below 1 Hz. Finally, the Allan deviation increases as the total measurement time becomes too long. Apparently, the FRESURE limits, 1 Hz in the worst case and 0.3 Hz ($\sim 0.005\,\Gamma$) in the best case, are well below the peak width $\Gamma _i \sim 60$ Hz.

Fig. 5

Dependence of Allan deviation ($\sigma$) of four PPSs on the number of averaged measurements data (M) in one group. Total sample number is 70 and the interval between two measurement is 3 min.

We demonstrate the method of FRESURE in a rather small and simple nuclear spin system. With more advanced quantum control techniques and stronger bias magnetic fields, it is possible to explore the FRESURE method in more complicated but practical systems. A strategy for a complicated large system is divide-and-conquer, i.e. first treating the close and weakly coupled spins, then the remote spins with weaker coupling strength. We remark that quantum resources to prepare the PPSs increase exponentially with the number of nuclear spins.

Conclusions and discussions

To summarize, we propose a physical method to decompose a multiple-peak spectrum into several single peaks by the QE state engineering technique. Numerical simulations (under the original Hamiltonian Eq. (1)) and experimental results in $C_2F_3I$ nuclear spin system (with SpinQ Triangulum) confirm the validity of the physical method in the weak coupling regime. With this method at hand, it is possible to realize spectral frequency super-resolution. The limit of the frequency super-resolution reaches $\sim 0.005\; \Gamma$ in experiments, with $\Gamma$ the FWHM of the single peaks. Such a frequency super-resolution realized by QE engineering method may be useful to measure the spin coupling constants in noisy and low-field NMR systems. For example, the spin coupling strength $J_{13}$ and $J_{23}$ can be calculated from the more accurate values of four PPSs’ peak positions in Fig. 4.

Although the physical spectral decomposition theory assumes a weak coupling between the system and its QE, our numerical simulations and experimental results indicate that this theory is still valid with the original Hamiltonian Eq. (1) at the not-so-small value $|J_{ij}/(\delta _i-\delta _j)| \sim 0.1$ (see Sect. 4 of Supplementary Material for more information). Additionally, it is worth exploring in the future the asymmetry and the long tail properties of the spectra of four PPSs, though the $\Gamma _i$ and $\Gamma _i^L$ in Table 1 are quite close.

Compared with spectral line narrowing, the line width of a single peak in FRESURE keeps untouched. As shown in Table 1, the line width in experiment after FRESURE is still around 60 Hz and much larger than the distance between peaks B and C ($\sim 24$ Hz). While in line narrowing methods, such as dynamical decoupling (including spin echo), dynamic nuclear polarization, or nuclear spin state narrowing, the line width of each single peak is strongly reduced, so that the peak distance eventually becomes larger than the reduced line width. Additionally, the limit of line width by line narrowing, such as dynamical decoupling or environmental state narrowing, is in the order of $T_1^{-1}$ with $T_1$ the longitudinal spin relaxation time^36,37. For the $C_2 F_3 I$ nuclear spin system, $T_1\sim 10$ s and the narrowest line width is about 0.1 Hz^34,38. Obviously, the resolution with FRESURE ($\sim 0.3$ Hz) is close to the limit of line narrowing techniques, but the experimental condition is significantly relaxed, e.g. lower bias field and noisier stray field fluctuation.