Experimental sensing quantum atmosphere of a single spin

Abstract

Understanding symmetry-breaking states of materials is a major challenge in the modern physical sciences. Quantum atmosphere proposed recently sheds light on the hidden world of these symmetry broken patterns. Yet, no experiment has been performed to demonstrate its potential. In our experiment, we prepare time-reversal-symmetry conserved and broken quantum atmosphere of a single nuclear spin and successfully observe their symmetry properties. Our work proves in principle that finding symmetry patterns from quantum atmosphere is conceptually viable. It also opens up entirely new possibilities in the potential application of quantum sensing in material diagnosis.

1 Introduction

Symmetry-breaking state of materials is the wellspring of novel and deep physical thoughts [1]. These subtle forms of symmetry breaking often connect with topology and entanglement [2, 3]. They display a rich variety of phenomena from quantum phase transition [4] to topological states of matter [5]. However, to experimentally identify symmetries of a quantum state is extremely non-trivial. Transport measurements can end up with trivial results due to formation of domains in a sample [6]. Standard spectroscopic experiments like photon scattering [7–9], neutron scattering spectroscopy [7, 10, 11], electron diffraction [12, 13] and transverse field muon-spin rotation spectroscopy [14–16] are limited to detecting orders that are able to scatter the probing fields directly [see Sect. 1 in Additional file 1], while there are lots of quantum orders (sometimes called hidden orders) which do not scatter the probing fields directly.

Recently, it is proposed that quantum fluctuation still enable us to see its symmetry in its quantum atmosphere (QA) [17]. Quantum atmosphere is a vicinity zone where quantum fluctuation is not negligible (Fig. 1(A)). In a virtual-photon-exchange process (indirect-scattering), a sensor could interact with the nearby part of a material (Fig. 1(B) and Sect. 2 in Additional file 1) and map out the topography of symmetry in its spectrum, which is not available in either transport measurements or traditional spectroscopic measurements. Yet, no experiment has been performed to demonstrate its potential.

Figure 1

Concepts of quantum atmosphere. (a) Illustration of quantum atmosphere near the material’s surface. The brown box represents domain with hidden symmetry and topology in the material. Due to quantum fluctuation (two-particle-exchange process), local symmetry can be mapped out via the QA near the surface. The circles represent virtue particles created by quantum fluctuation. (b) The physical mechanism of the quantum atmosphere. The upper panel shows a generic Feynman diagram for the QA probe. The spectra of the sensor is modified due to the two-photon-exchange process. The black straight (wavy) line with an arrow represents propagator for sensor spin (photons). The grey circle represents the scattering vertex $\Pi (p\prime -p)$ where the symmetry information in a material is encoded. The lower panel shows that the energy level of the spin-up state and spin-down state will split with an energy difference δϵ when a spin is put in a time-reversal-symmetry broken atmosphere

Here, we report a method to detect the QA with the a NV center in diamond [18, 19] and directly observe the symmetry property. In the experiment, the NV center is used as a quantum sensor to probe the QA of a single nuclear ¹³C, whose time-reversal symmetry can be precisely manipulated. As an excellent object for a demonstrative experiment, a transition from conserved time-reversal symmetry to broken one is observed in its QA. Our unambiguous demonstration of QA sensing turns this purely theoretical proposal into realistic physical observables, opening up entirely new possibilities in fundamental studies in symmetry detection. The method can be further extended to investigate time-reversal symmetry in condense materials and parity symmetry in electric dipole systems, paving the way for potential applications of quantum sensing in material diagnosis.

2 Results

Symmetry information can be directly revealed by measuring the proper physical quantities in the atmosphere. In the vicinity of the target spin, one can visualize the magnetic field $B_{\mathrm{eff}}$ or magnetic field fluctuation $\delta B^{2}$ of the spin. These can be understood from atmospheric point of view (see Sect. 3 in Additional file 1). In the presence of the atmosphere, the free energy for the sensor spin nearby is

$$ \mathcal{F}=A_{0}(\vec{I}-P I_{z})^{2}+A_{ij}I_{i} s_{j} . $$

(1)

Here, s⃗ stands for the spin of the quantum sensor and I⃗ represents the target spin. The polarization direction is chosen to be z⃗. P is the polarization of the spin. The coupling constant between the two spins is $A_{ij}$ ($i,j \in \{x,y,z\}$). The stiffness of the target spin’s magnetic field, $A_{0}$, is inverse to the spin orientation fluctuation rate. In the case of totally polarized target spin, $A_{0} \rightarrow \infty$, so fluctuation is forbidden and I⃗ is restricted to $P I_{z}$. In the case of unpolarized target spin, $A_{0}$ is finite, which indicates that fluctuation is very large, and there is no favorable direction of the target spin. The fluctuation part of the target spin can be integrated out to yield the effective free energy of the quantum sensor in the atmosphere of the target spin (see Sect. 3 in Additional file 1). Considering the longitudinal coupling $A_{zz}$ far exceeds other coupling components, we get

$$ \mathcal {F}_{\mathrm{eff}}=A_{zz}PI_{z}s_{z}+ \frac {A_{zz}^{2}}{4A_{0}}s_{z} s_{z} . $$

(2)

The effect of the quantum atmosphere can be viewed as effective longitudinal field. When the system includes time-reversal symmetry broken atmosphere, then the time reversal would change the system’s state. In this case, the target spin is polarized $(P\neq 0)$, the first term is allowed and its mean magnetic field $B_{\mathrm{eff}} \neq 0$. Thereby, the energy level of the spin-state will split with an energy difference δϵ as shown in Fig. 1(b). When time-reversal symmetry is conserved, the reversal of time would not affect the spin alignment relative to the longitudinal field. The unpolarized target spin ($P = 0$) results in only the second term being exist. Based on this effective action, $\langle s \rangle =0$ while $\langle s_{z}s_{z} \rangle \neq 0$, which indicate that $B_{\mathrm{eff}}= 0$ and the magnetic field fluctuation $\delta B^{2} \neq 0$.

An NV-based optically detected magnetic resonance (ODMR) setup (Fig. 2(A) and Sect. 4 in Additional file 1) is constructed to detect ¹³C’s atmosphere. Under the external magnetic field of 515 Gauss, the energy levels of NV’s electron spin $\vert S_{z} = 0 \rangle$ and $\vert S_{z} = +1 \rangle$ are selected to realize an effective two-level quantum sensor (sensor spin), which can be initialized by laser pulses and manipulated by microwave (MW) pulses to capture the magnetic quantities in the target spin’s atmosphere. The state of the target spin can be engineered by radio-frequency (RF) pulses.

Figure 2

Schematic of the setup and experimental method. (a) Nitrogen-vacancy centre in the quantum atmosphere of a nearby ¹³C nuclear spin is illuminated by a focused green laser beam and controlled by microwave and RF pulses. In the magnified figure, red ball with an arrow represents the target spin and the blue one is the sensor spin. (b) Method of preparation: The sensor spin and target spin are first initialized by laser pumping. Radio-frequency pulses are used to transfer the population between nuclear spin’s states. And by varying the rotation angle of RF (θ), we can prepare distinct ¹³C nuclear spin’s quantum atmospheres from time-reversal symmetry conserved type to broken ones. Detection (Ramsey sequence): the sensor spin is prepared in the $( \vert 0\rangle +i \vert +1 \rangle )/\sqrt {2}$ state with a microwave (MW) $\pi /2$ pulse along the x̂ axis. Subsequently, within the sensing process, the spin evolves under the influence of nuclear spin’s quantum atmosphere for duration τ, immediately followed by another microwave $\pi /2$ pulse along the same axis. Then the spin polarization is read out by the laser. Microwave pulses, MW1 and MW2, are used for recording the sensor spin’s evolution in the different ¹³C states

The Hamiltonian of the effective two-level system is

$$ H = DS_{z}+ \gamma _{e} \vec{B}_{\mathrm{ext}}\cdot \vec{ S}+ \gamma _{e} \vec{B}_{\mathrm{eff}}\cdot \vec {S}, $$

(3)

where S is Pauli spin-1/2 operator of the two-level subspace spanned by the spin states $\vert S_{z} = 0 \rangle$ and $\vert S_{z} = +1 \rangle$, $B_{\mathrm{ext}}$ is the external magnetic field applied along NV’s axis, D is the zero filed splitting between sensor spin’s $\vert S_{z} = 0 \rangle$ and $\vert S_{z} = +1 \rangle$ states, which is measured as 2.87 GHz, $\gamma _{e}$ is electron’s gyromagnetic ratio being $2.803~\mathrm {MHz/Gauss}$, $B_{\mathrm{eff}}$ is the magnetic field the sensor detects. In the rotating frame, the first two terms in the Eq. (3) are omitted, and the remaining magnetic field term to be measured is the longitudinal coupling between the ¹³C nuclear spin and the sensor, $A_{zz}I_{z} {S_{z}}$, which corresponds to the first term in Eq. (2). The $A_{zz}=13.56$ MHz is the coupling strength in our experiment (see Sect. 5 in Additional file 1). The second term in Eq. (2) is induced by the time-reversal symmetry conserved quantum atmosphere of the target spin. The strength of this term is proportional to the magnetic-fluctuation strength, which can be obtained from the repetitive measurements of the magnetic field. Other components of the coupling can be neglected because they are much weaker than the longitudinal component. Therefore, the magnetic field is measured as $B_{\mathrm{eff}}=\langle A_{zz}PI_{z}\rangle /\gamma _{e}$ and magnetic fluctuation as $\delta B^{2} = \langle (A_{zz}(I_{z} - PI_{z}))^{2}\rangle /\gamma _{e}^{2}$. The bracket means the average of repetitive measurements.

To probe the quantum atmosphere of the nuclear spin, we monitor the sensor spin’s evolution under the influence of ¹³C which has different polarization along the +ẑ direction. The sensor spin is first initialized by laser pumping. In the meanwhile, excited-state level-anticrossing near 510 Gauss allows electron-nuclear-spin flip-flops to occur near resonantly [20]. Since ¹³C preferentially relaxes into $\vert m_{I}=\uparrow \rangle$ during the process, high degree of initial polarization is prepared. RF pulses are then applied to transfer the population between the $\vert m_{I}=\uparrow \rangle$ and $\vert m_{I}=\downarrow \rangle$ states (Fig. 2(B) and Sect. 6 in Additional file 1) [21]. Different ¹³C nuclear spin states corresponding to distinct quantum atmospheres can be prepared by controlling the rotation angle (θ) of RF pulses. It follows with a Ramsey sequence [22] to record magnetic field information from ¹³C nuclear spin. The sensor spin is initially rotated to $( \vert 0\rangle +i \vert +1 \rangle )/\sqrt {2}$ state with a microwave $\pi /2$ pulse along the x̂ axis. During the sensing process, the sensor spin evolves under the influence of quantum atmosphere for a duration τ. Subsequently, another microwave $\pi /2$ pulse along the same axis is applied on the sensor spin followed by the final read-out of the spin polarization by the laser. Limited by the power of the microwave (pulse excitation bandwidth 6.5 MHz), the whole magnetic field information is unavailable in a single Ramsey sequence. Therefore, we adopt the strategy to split the sensing process into two steps to record the sensor spin’s evolution in the different ¹³C states. In the first step, MW1 is detuned from the sensor spin’s resonance frequency between $\vert m_{s}=0, m_{I}=\uparrow \rangle$ and $\vert m_{s}=+1, m_{I}=\uparrow \rangle$ by +1 MHz (4321.0 MHz). The large off-resonance of the sensor spin in the $\vert m_{I}=\downarrow \rangle$ subspace leaves only the information of $\vert m_{I}=\uparrow \rangle$ state observable. Complementarily, the procedure above is repeated but with MW2 detuned from the resonance frequency between $\vert m_{s}=0, m_{I}=\downarrow \rangle$ and $\vert m_{s}=+1, m_{I}=\downarrow \rangle$ by −1 MHz (4305.5 MHz). The experiment has been repeated for $6\times 10^{5}$ times to build good statistics.

During the sensing process, the sensor spin state undergoes Larmor precession at different frequencies due to the influence of ¹³C nuclear spin, which is in a superposition state of $\vert m_{I}=\downarrow \rangle$ and $\vert m_{I}=\uparrow \rangle$. The two eigenstates exert different magnetic fields on the sensor, shifting its original Larmor frequency $f_{0}$ to $f_{0}- A_{zz}/2$ and $f_{0}+A_{zz}/2$. By varying the duration τ, the evolution of the sensor spin can be recorded in the time domain. To obtain the effective magnetic field, we transform the signal from time domain to frequency domain and divide it by electron’s gyromagnetic ratio.

Figure 3(A) depicts the observed phase transition of ¹³C’s quantum atmosphere between time-reversal symmetry conserved type and broken ones at different levels. These states are prepared to different polarization, P, ranging from $-P_{0}$ to $P_{0}$. The nuclear spin polarization is $P=P_{0} \cos(\theta )$, where θ is the rotation angle of RF pulse and $P_{0}$, measured to be $0.91(1)$, is the natural polarization due to optical pumping. The experimentally prepared polarization of target nuclear spin may deviates from the expected value due to several factors, e.g., laser-induced depolarization [23] and temperature fluctuation. Figure 3(B)–(D) shows three representative QAs in Fig. 3(A) and their magnetic field distributions. For nuclear spin with time-reversal symmetry broken atmosphere, its magnetic field is highly concentrated in the positive (Fig. 3(B) with $P=0.91(1)$) or negative (Fig. 3(D) with $P=-0.91(1)$) part centered near $A_{zz}/(2\gamma _{e})=\pm 2.419$ Gauss. While for the time-reversal symmetry conserved one (Fig. 3(C) with $P= 0.00(1)$), its magnetic field is evenly distributed on both sides.

Figure 3

Spectrum of quantum atmosphere with varying symmetry property. (a) The whole spectrum of fifteen states polarized to different quantum atmospheres with different RF rotation angle. The brighter the color is, the more likely the magnetic field will distribute at that field strength. (b) three typical quantum atmospheres. The nuclear spin polarization is $P=P_{0} \cos(\theta )$. Left: The nuclear spin is polarized to its $\vert m_{I}=\uparrow \rangle$ state corresponding to a time-reversal symmetry broken atmosphere, it applies a positive magnetic field to the sensor spin. Middle: The nuclear spin is fully unpolarized within the margin of error, no magnetic field is applied to the sensor corresponding to a time-reversal symmetry conversed atmosphere. Right: the nuclear spin is polarized to its $\vert m_{I}=\downarrow \rangle$ state, it applies a negative magnetic field to the sensor corresponding to a time-reversal symmetry broken atmosphere. The scattered dots are the experimental results while the real lines are our simulant results. Every point is averaged over $6 \times 10^{5}$ repetitive measurements

To quantitatively determine the symmetry of the ¹³C nuclear spin’s atmosphere, we first measure the mean magnetic field $B_{\mathrm{eff}}$ and magnetic fluctuation $\delta B^{2}$ of each polarization state (see Sect. 7 in Additional file 1). The Symmetry Indicator Γ is defined as the ratio of the magnetic fluctuation to the square of magnetic field, $\Gamma = {\delta B^{2}}/{B_{\mathrm{eff}}^{2}}$, which provides a method to quantitatively measure the symmetry-breaking level of quantum atmosphere. When Γ equals 0, the broken level of time-reversal-symmetry reaches maximum (i.e. the spin is fully polarized) and as the parameter would grow larger, the symmetry breaking level gets lesser. Notably, Γ comes to divergence when time-reversal symmetry is conserved.

Figure 4(A) exhibits some realistic physical observables of quantum atmosphere with different time-reversal symmetry broken level. The error bars are obtained by dividing the data into three groups and calculating the variance from their deviations from the mean value. The upper panel shows the magnetic field of QA and the lower panel shows their fluctuation. When the target spin is not polarized, there is no favorable direction. As a result, no magnetic field is exerted by the ¹³C spin but maximum fluctuation is presented. When it is polarized, the spin shows certain orientation. In this case, even the slightest fingerprint of nonzero energy shift (effective magnetic field) could be sensed due to the exquisite sensitivity of NV center [24]. The measured magnetic fluctuation is slightly lifted from the theoretical value (see Sect. 7 in Additional file 1), which is mainly contributed by the inevitable background noise. Figure 4(B) shows Γ as a function of ¹³C spin polarization P. Γ sees a huge spike around $P=0$ and, within the rage of error permitting, Γ goes through the divergent point. As P deviates from this point, Γ sharply decreases. Judging from the property of the symmetry indicator, we associate this point with the time-reversal-invariant quantum atmosphere and other cases as time-reversal symmetry broken classical atmosphere [17].

Figure 4

Phase diagram of different quantum atmosphere. The red dots denote the experiment data, the blue curves denote the theoretical predictions, which we find agreement. (a)The physical observables in the experiment are the mean magnetic field $B_{\mathrm{eff}}$ and the magnetic fluctuation $\delta B^{2}$ of the ¹³C spin. The upper panel shows the field of ¹³C’s QA with different polarization P. The lower panel shows their magnetic field fluctuation of ¹³C’s QA. Notably, when time-reversal-symmetry is conserved, there is no favorable direction of the ¹³C spin, which results in to zero net magnetic field and maximum magnetic field fluctuation. (b) shows symmetry indicator Γ as a function of P. Γ sees a huge spike around $P=0$ and sharply decreases as P deviates from the that point. It is quite clear from the diagram that time reversal symmetry is broken except the state that nuclear spin is unpolarized. Error bars are mainly due to the photon statistics

3 Discussion

Although in this experiment a strongly coupled nuclear spin is employed, it is important to note that the physical picture and general methods are applicable to any target spins irrelevant to its coupling with the sensor [25]. For example, ¹³C spin with a coupling weaker by several orders of magnitude ($A_{zz} \sim$ kHz) has been experimentally accessed and manipulated with the method of high order dynamical decoupling sequence [26]. Adopting specific polarization and readout schemes [27], target spins, weakly and strongly coupled, could be sensed in its quantum atmosphere to reveal their symmetry.

With the proof-in-principle implementation of QA sensing of a single spin, we turn this purely theoretical proposal into realistic physical observables. Standard techniques (i.e. superconducting quantum interference device (SQUID) magnetometry, neutron scattering and muon-spin rotation) used for bulk crystals, large-area thin films and nanoparticles in solution are challenging for QA detection because the quantum fluctuation effect is greatly suppressed [28]. Optical techniques (for example, magneto-circular dichroism (MCD)) also face the obstacle that fluctuation in the material’s atmosphere will induce a broad background [29].

NV centers provide exquisite sensitivity to magnetic [26] and electrical signals [30] and nanoscale spatial resolution [31], making it a promising candidate for detecting and diagnose the symmetry of a lot of exotic materials. For example, it is suggested that quantum fluctuation involving Chern–Simons interaction will produce a sort of parity and time-reversal symmetry violating atmosphere above a topological insulator, which induces an effective Zeeman field on the quantum sensor nearby [17]. Its strength is within NV centers sensitivity of magnetometry. Furthermore, chiral superconductivity may also be directly identified by sensing its time-reversal symmetry violated QA.

It is worth noting that QA based probing is different from stray field sensing or electric fields sensing. Most of the previous experiments measures nonzero Electric(E) or magnetic(B) fields [30, 32]. The major difference is that if the mean value of E or B field around the material is zero, the symmetry of a quantum state can still be revealed directly with QA based sensing. Therefore, it provides a new approach to diagnosing condense matter systems, which will help to extend the field of quantum sensing based on NV centers. In the meanwhile, our result shows that QA based probing is very concise, making it more accessible for the above experimental measurements.

4 Conclusion

In conclusion, we demonstrate the feasibility of sensing quantum atmosphere using NV-based quantum sensing techniques and turn this purely theoretical concept into realistic physical observables. In the future, our method can be extended to reveal subtle forms of symmetry breaking in materials, opening up entirely new possibilities in the searching and the study of hidden symmetries. The present observation also raises intriguing possibility in diagnosis more complex materials [17], e.g., topological insulators and superconductors [5, 33].