Introduction

Metastability, similar to prethermalization in nonequilibrium physics1, arises when the system relaxes into long-lived states before subsequently decaying to true stationarity much slowly. The concept of metastability is widely used, for example, in the description of the long-lived atomic state of individual atoms2, the magnetic state of single nanoparticle3, and the metastable structure and phase of condensed matter systems4,5. In these systems, metastability is mainly due to the barriers separating local minima in the energy levels of the systems.

In classical stochastic dynamics, metastability is a manifestation of a separation of time scales that arises from the splitting in the spectrum of the dynamical generator. Recently, quantum metastability theory has been extended to Markovian open quantum dynamics with a similar spectral structure in the generator6,7,8, where the manifold of metastable states is argued to be composed of disjoint states, decoherence-free subspaces, and noiseless subsystems. Such metastability theory is physically interesting in its own right and stimulates the prediction of metastability phenomena in various quantum models9,10,11,12,13,14,15,16,17. It is also of practical importance to utilize the preserved information of quantum processes for quantum information processing18,19, especially in the presence of control imperfections or environmental noise.

For the continuous-time dynamics described by the Lindblad master equation, metastability can be observed from the distinct two-step decay of temporal correlations6. However, experimental observations remain scarce, mainly due to the difficulty in measuring temporal second- or higher-order correlations with good resolution20,21. Alternatively, certain signatures of metastability can be observed without measuring the temporal correlations (e.g., through the high-accuracy time-resolved heterodyne detection in the superconducting cavities)22; however, such approaches are often not applicable to other experimental settings.

A recent advance in quantum metastability theory is the generalization of the setting from continuous-time to discrete-time open quantum dynamics23,24. Such a framework describes the more general scenario in which each discrete evolution is induced by a quantum channel (or a completely positive and trace-preserving quantum map)25, which may not be generated by continuous-time master equations26,27. With this framework, metastable open quantum dynamics of a quantum system can be directly observed by only sequentially measuring an ancilla qubit, instead of continuously measuring the temporal correlations. Remarkably, metastability has been predicted in the commonly used Ramsey interferometry measurements (RIMs)28,29,30, which is easy to realize in various quantum platforms. By repeating the RIM on a probe qubit, a nearby bath system can be metastably polarized, which is manifested in statistics of the probe measurement outcomes associated with different quantum trajectories of the bath system.

Here, we report an observation of the metastable open dynamics in diamond based on sequential RIMs, using a nitrogen (14N) nuclear spin as the bath system and the nitrogen-vacancy (NV) electron spin as the probe. Due to the hyperfine coupling between the bath and the probe during free evolution, each round of RIM solely applied on the probe spin induces a quantum channel on the nuclear spin, and sequential RIMs induce discrete-time open quantum dynamics on the nuclear spin (Fig. 1a). By recording the statistical results of the sequential probe measurement, we can continuously monitor the state evolution of the target spin. We directly observe the metastable polarization when varying the number of repetitions m, i.e., the bath spin is first steered to metastable (polarized) states for a finite range of m, and eventually relaxes towards the true stable (maximally mixed) state as m continues to increase.

Fig. 1: Metastability in discrete-time open quantum dynamics and its observation in diamond.

a Schematic of metastability in open quantum dynamics. The states outside the metastable manifold (MM, red dot-dashed line) exhibit a two-step relaxation, i.e., first quickly relaxing into the MM and then slowly decaying into the stationary manifold (SM). The discrete-time axis implies that the open quantum dynamics is described by sequential quantum channels. b Spectrum of a quantum channel, with all eigenvalues located within the unit circle (solid circle) of the complex plane. Metastable points are located in the region with ∣λ∣ ≈ 1 (labeled with blue ring in the spectrum). The metastable region of measurement times is determined by the spectral gap Δ between the decaying region (labeled in orange) and the metastable region in the spectrum. c Quantum circuit of sequential RIMs, where R**ϕ(ϑ) is the rotation operator of the probe qubit, U**α is a unitary operator of the bath conditioned on the probe state ({| \alpha \left.\right\rangle }_{e}) (α = 0, 1), and {a1, …, a**m} (a**i = 0, 1) is the sequence of measurement outcomes. d The experimental system, with the electron spin of a nitrogen-vacancy (NV) center in diamond serving as the probe qubit, and the host 14N nuclear spin as the quantum bath system. e Pulse sequences to implement sequential RIMs in experiments. Laser pulses are used to polarize and read out the NV spin state, and resonant microwave (MW) pulses are used to manipulate the probe spin state.

Results

Metastability in sequential RIMs

We illustrate the principle of metastability by considering a typical class of sequential quantum channels on a quantum bath system, with the channel being induced by a probe qubit under a RIM sequence23. The probe qubit is coupled to the bath through the general pure-dephasing coupling

$$H={\sigma }_{e}^{z}\otimes {B}_{n}+{{\mathbb{I}}}_{e}\otimes {C}_{n},$$

(1)

where the subscripts e and n refer to the probe and bath, respectively, ({\sigma }_{e}^{{i}) is the Pauli-i operator of the qubit (i = x, y, z) with ({\sigma }_{e}}{z}={| 0\left.\right\rangle }_{e}\left\langle \right.0| -{| 1\left.\right\rangle }_{e}\left\langle \right.1|), ({{\mathbb{I}}}_{e}) is the identity operator of the qubit, and B**n(C**n) is the interaction (free) operator of the bath. We denote the probe qubit rotation along an axis in the equatorial plane as ({R}_{\phi }(\vartheta )={e}^{{-i(\cos \phi {\sigma }_{e}}{x}+\sin \phi {\sigma }_{e}^{y})\vartheta /2}), with ϕ denoting the rotation axis and ϑ the rotation angle.

For a single RIM, the probe qubit is first initialized to state ({| 0\left.\right\rangle }_{e}), and then prepared in a superposition state ({| \psi \left.\right\rangle }_{e}=({| 0\left.\right\rangle }_{e}-i{e}^{{i{\phi }_{1}}{| 1\left.\right\rangle }_{e})/\sqrt{2}) by a rotation ({R}_{{\phi }_{1}}(\frac{\pi }{2})). After the composite system evolving with the pure-dephasing coupling [Eq. (1)] for time τ, the probe undergoes another rotation ({R}_{{\phi }_{2}}(\frac{\pi }{2})), and is finally projectively measured in the basis of ({\sigma }_{e}}{z}) with the measurement outcome a ∈ {0, 1}.

The RIM sequence of the probe spin induces a quantum channel Φ on the bath,

$$\Phi ({\rho }_{n})={{{\rm{Tr}}}}_{e}\left[U({\rho }_{e}\otimes {\rho }_{n}){U}^{{{\dagger} }\right]=\sum_{a=0,1}{M}_{a}{\rho }_{n}{M}_{a}}{{\dagger} },$$

(2)

where ({\rho }_{e}={| \psi \left.\right\rangle }_{e}\left\langle \right.\psi |) (ρ**n) is the initial state of the probe (bath) spin, U = e−iHτ is the propagator for the composite system. By partially tracing over the probe spin, we obtain the Kraus operator M**a = [U0 − (−1)aeiΔϕ**U1]/2, where ({U}_{\alpha }={e}^{-i[{(-1)}{\alpha }{B}_{n}+{C}_{n}]\tau }) is a unitary operator of the bath spin conditioned on the probe spin state ({| \alpha \left.\right\rangle }_{e}) (α = 0, 1), and Δ**ϕ = ϕ1 − ϕ2 is the phase difference between the rotation axes of two π/2 pulses in a RIM. With the measurement outcome a, the bath state is steered to ({M}_{a}{\rho }_{n}{M}_{a}^{{{\dagger} }/p(a)) with probability (p(a)=,{\mbox{Tr}},({M}_{a}{\rho }_{n}{M}_{a}}{{\dagger} })). Note that the Kraus operators satisfy ({\sum }_{a}{M}_{a}^{{\dagger} }{M}_{a}={{\mathbb{I}}}_{n}) with ({{\mathbb{I}}}_{n}) being the identity operator on the bath, which ensures that ∑a**p(a) = 1.

The behaviors of repetitive quantum channels can be clearly revealed by the spectral decomposition of a single channel (see “Methods”). A quantum channel has at least one fixed point (Fig. 1b), which is the state that remains unchanged after the channel31,32. It has been proved that the fixed points of the channel in Eq. (2) depend on the commutativity of B**n and C**n (see “Methods”)23. If B**n commutes with C**n, i.e., [B**n, C**n] = 0, the fixed points include the simultaneous eigenstates of B**n and C**n, then sequential RIMs of the probe spin will polarize the bath system to one of the eigenstates, also constituting a quantum non-demolition (QND) measurement on the bath spin33,34,35,36,37,38. However, if [B**n, C**n] ≠ 0, the fixed points are the maximally mixed state in the whole space (or subspace) of the bath, then sequential RIMs cannot effectively measure the bath system but only perturb it.

Quantum metastability emerges when [B**n, C**n] ≠ 0, but C**n is a small perturbation of B**n, then the bath will be polarized for a finite range of repetitions of RIMs. We suppose that the channel has r fixed points and q − r (q > r) metastable points, which collectively span a (q − 1)-dimensional metastable manifold (MM). The region of repetition numbers for metastable polarization of the bath spin can be estimated as ({(1-| {\lambda }_{q+1}| )}^{{-1} , \ll , m , \ll , {(1-| {\lambda }_{q}| )}}{-1}) when λ**q is very close to 1, where λ**q is the eigenvalue of metastable point with the smallest eigenvalue. As m increases and enters the metastable region, the bath state evolves into the MM. Beyond this region, the system further relaxes into the stationary manifold (SM), which is spanned by the fixed points. This process, during which the system first reaches the MM and then relaxes to the SM, is referred to as a two-step relaxation (see Fig. 1a).

The metastability behaviors in sequential RIMs can be directly monitored by the measurement statistics of the probe spin23. To see this, we decompose the average dynamics of sequential quantum channels into stochastic trajectories,

$${\Phi }^{{m}({\rho }_{n})=\sum _{{a}_{1},\ldots,{a}_{m}=0}}{1}{{{\mathcal{M}}}}_{{a}_{m}}\cdots {{{\mathcal{M}}}}_{{a}_{1}}({\rho }_{n}),$$

(3)

where ({{{\mathcal{M}}}}_{{a}_{i}}(\cdot )={M}_{{a}_{i}}(\cdot ){M}_{{a}_{i}}^{{{\dagger} }) with a**i ∈ {0, 1} being the measurement outcome of the ith RIM. For a trajectory with the sequence of measurement outcomes {a1, . . . , a**m}, the bath is steered to ({\rho }_{n}}{{\prime} }={{{\mathcal{M}}}}_{{a}_{m}}\cdots {{{\mathcal{M}}}}_{{a}_{1}}({\rho }_{n})/p({a}_{1},...,{a}_{m})) with probability (p({a}_{1},...,{a}_{m})=,{\mbox{Tr}},[{{{\mathcal{M}}}}_{{a}_{m}}\cdots {{{\mathcal{M}}}}_{{a}_{1}}({\rho }_{n})]). With the number of outcome 0(1) in {a1, . . . , a**m} denoted as m0(m1) satisfying m0 + m1 = m, we can define the measurement polarization X = (m0 − m1)/(2m)23,24,39, which can be used to classify different classes of stochastic trajectories that the bath system undergoes. The measurement distribution of X can show several peaks, with each peak corresponding to the quantum trajectories that lead to a fixed point (or metastable state) of the channel (see “Methods”). So in the metastable region, the distribution exhibits up to q peaks, corresponding to the metastable states. However, as m surpasses this region, the number of peaks gradually reduces to r, reflecting the final stationary states of the bath.

Experimental implementation in the NV system

We demonstrate this protocol experimentally on an NV center and a nearby 14N nuclear spin in a high-purity diamond (Fig. 1d). Nuclear spins around NV centers have great potential for building quantum networks40, storing quantum information38, performing quantum simulations41,42,43 and sensing inertial parameters44. Access to the nuclear spins is usually based on their hyperfine interaction with the electron spin and the optical interface of the NV center. It is therefore an interesting and fundamental problem to monitor and understand the dynamics of these nuclear spins under sequential measurements acting on the central electron spin.

Specifically, an NV electron spin is a spin-1 system with a zero-field splitting D = 2.87 GHz between its ({| 0\left.\right\rangle }_{e}) (m**s = 0) and ({| \pm 1\left.\right\rangle }_{e}) (m**s = ±1) state. In our experiment, a moderate external magnetic field of B = 108.4 ± 0.2 G is applied along the NV symmetry axis (z axis) to lift the degeneracy of the ({| \pm 1\left.\right\rangle }_{e}) states and we work in the ({{| 0\left.\right\rangle }_{e},{| -1\left.\right\rangle }_{e}}) subspace. The Hamiltonian in Eq. (1) becomes (in the rotating frame)

$$H={A}_{zz}{S}_{z}{I}_{z}+Q{I}_{z}^{2}-{\gamma }_{n}{{\bf{B}}}\cdot {{\bf{I}}}+{\tilde{H}}_{1},$$

(4)

where S = (S**x, S**y, S**z) is the NV electron spin-1 operator, I = (I**x, I**y, I**z) is the 14N nuclear spin-1 operator, Azz = −2.16 MHz (A⊥ = −2.63 MHz) is the z**z-component (transverse component) of the hyperfine tensor, Q = −4.95 MHz is the quadrupolar splitting, γ**n = 1.071 kHz/G (γ**e = 2.802 MHz/G) being the gyromagnetic ratio of 14N nuclear spin (electron spin), and ({\tilde{H}}_{1}={\sum }_{\alpha }{| \alpha \left.\right\rangle }_{e}\left\langle \right.\alpha | \otimes {H}_{n}^{{\alpha }) is a second-order perturbation term with ({H}_{n}}{\alpha }\approx \frac{{\gamma }_{e}(2-3| \alpha | )}{2D}[-{\gamma }_{e}({B}_{x}^{2}+{B}_{y}{2})+2{A}_{\perp }({B}_{x}{I}_{x}+{B}_{y}{I}_{y})]). To introduce a non-commutativity between B**n = AzzIz and ({C}_{n}=Q{I}_{z}^{2}-{\gamma }_{n}{{\bf{B}}}\cdot {{\bf{I}}}+{\tilde{H}}_{1}), the external magnetic field is slightly misaligned with the NV axis, i.e., there is a small tilt angle θ between the magnetic field and NV axis. Under this circumstance, the perturbation terms −γ**n(BxI**x + ByI**y) and ({\tilde{H}}_{1}) lead to the metastable polarization of 14N nuclear spin.

For each RIM, the resulting quantum state of the NV spin is read out by counting its spin-dependent photon number. As shown in Fig. 1e, a 532-nm laser is used to excite the NV center, and all experiments are performed at room temperature. Due to the low photon emission rate of the NV center and the poor collection efficiency, only 0.05 photons can be detected on average in each measurement, which is not sufficient to distinguish the spin states. Fortunately, the spin relaxation time of the 14N nuclear spin is very long (see “Discussion” below), and many RIMs can be performed before a state jump occurs. Therefore, to obtain a reasonable signal-to-noise ratio (SNR) and to distinguish the nuclear spin states, we simply sum up the counts of the sequential measurements. We expect that the metastable polarization is easier to observe by readout methods with higher SNR, for example, the resonant excitation of the NV electron spin at low temperature45,46.

Figure 2 presents the metastable dynamics of the target nuclear spin under sequential RIMs acting on the NV electron spin. Before the measurement, the 14N nuclear spin is in the thermal state, which means that its population is evenly distributed among the set of states (\left.{{{| 0\left.\right\rangle }_{n},| \pm 1\left.\right\rangle }}_{n}\right}) (m**I = 0, ±1). For a small number of RIMs (m < 5 k), the shot noise of the photon counts exceeds the count difference of the measured states, so that no efficient signal can be detected in this range. As the number of RIMs increases (m > 5 k), the measurement results begin to concentrate on two regions, which is a clear signature of the metastable polarization of the bath spin, as can be seen in the middle slice of Fig. 2a, b. We classify the trajectories based on the average photon counts with a threshold value of 0.05. If the average photon counts are less than 0.05, it is classified as a dar**k state; otherwise as a brigh**t state. The experimentally resolved bright state corresponds to the maximally mixed state of the subspace spanned by ({{| 0\left.\right\rangle }_{n},{| -1\left.\right\rangle }_{n}}), i.e., ({\rho }_{{{\rm{B}}}}=({| 0\left.\right\rangle }_{n}{\langle 0|+| -1\rangle }_{n}\left\langle \right.-1| )/2). This bright state, together with the dark state ({\rho }_{{{\rm{D}}}}={| 1\left.\right\rangle }_{n}\left\langle \right.1|), are the extremely metastable states (EMSs) of the one-dimensional MM spanned by the fixed point and the metastable point with higher eigenvalue, as shown in Fig. 2d. Beyond the metastable region (m > 600 k), the two peaks mentioned above gradually disappear and a single peak appears, corresponding to the stable state of the system, i.e., the maximally mixed state in the whole space ({{\mathbb{I}}}_{n}/3). More data and simulations can be found in Figs. S4–S6.

Fig. 2: Metastable dynamics of a 14N nuclear spin induced by sequential RIMs of a nearby NV electron spin.

a, b Typical PL time trace of an NV center under sequential RIMs and the corresponding distribution of the measurement results. The total and average photon counts are shown on the left and right axes, respectively. The duration of free evolution, τ = 374 ns. The number of measurement repetition m is 5 k, 60 k, 250 k, 600 k from top to bottom. When m = 5 k, it is difficult to distinguish the nuclear spin quantum states. When m = 60 k, the jump signal is concentrated in two distinct photon count intervals. As m increases further to 250 k, the overlap between the two distribution peaks becomes more pronounced. When m exceeds the metastable region, the photon counting peaks gradually merge. The blue envelopes in (b) indicate the normalized numerical simulation results. c Numerical results. The evolution of the nuclear spin state fidelity under sequential RIMs. The initial thermal state of the nuclear spin is polarized to either the dark state ({| 1\left.\right\rangle }_{n}\left\langle \right.1|) or the bright state (({| 0\left.\right\rangle }_{n}\left\langle \right.0|+{| -1\left.\right\rangle }_{n}\left\langle \right.-1| )/2). The solid lines are fits to the simulation results. The fidelity FD,B quantifies how close the quantum states represented by the trajectories are to the ideal dark or bright states, respectively. d The channel spectrum, with one fixed point, two metastable points and six decaying points. The right pane is a magnified view of the spectrum near the metastable points, and the deviations of their eigenvalues from 1 (1 − ∣λ∣) are shown accordingly. The metastable region and the underlying channel spectrum gap are marked with blue shades in (c) and (d), respectively. Monte Carlo simulations are conducted with a tilt angle θ = 8.8° and 3000 samples.

To quantitatively understand the metastable dynamics of the nuclear spin, we perform numerical simulations that take into account the parameters of hyperfine coupling, the strength and orientation of the external magnetic field, and the readout efficiency of the NV electron spin. The metastable states can be characterized by comparing them with the ideal polarized states, i.e., by the fidelity ({F}_{{{\rm{D(B)}}}}=F({\rho }_{{{\rm{D(B)}}}},{\rho }_{n}^{{{\prime} })), where (F(\rho,\sigma )=,{\mbox{Tr}},\sqrt{{\rho }}{1/2}\sigma {\rho }^{{1/2}}) and ({\rho }_{n}}{{\prime} }) is the actual state. As shown in Fig. 2c, the experimentally observed metastable dynamics of the 14N can be well reproduced with a tilt angle of the magnetic field of θ = 8.8°.

Since the 14N nuclear spin is a three-level system, three-state jumps are expected in the PL time trace. The fact that only two stages appear indicates that two of them cannot be distinguished under the current experimental conditions (B = 108.4 G and θ = 8.8°). Interestingly, when a smaller angle of the external magnetic field (θ < 2°) is chosen, three-state jumps are observed in the experiment, as shown in Fig. 3a, b. The physical picture behind this is that the external magnetic field can influence the metastability behavior by changing the value of [B**n, C**n]. Through numerical simulations we find that under the magnetic field with θ = 8.8°, three peaks emerge at around 5 k measurements and two of them merge into one before 60 k measurements, so that only two-state jumps are observed in our measured region (Fig. 2a). Figure 3c shows the calculated dependence of the two-state jump and three-state jump regions as a function of the orientation of the magnetic field. For a smaller tilt angle of the external magnetic field (e.g., θ = 2°), the metastable region of the three-state jump can survive under a large number of RIM measurements, which allows the experimental observation of the three-peak signals (see Supplementary Note II for more details).

Fig. 3: Magnetic field dependence of the metastable region.

a Three-state jump signal of the 14N nuclear spin. The signal is observed when the magnetic field is close to the NV axis (θ = 2°), with a RIM duration τ = 374 ns and the number of repetitions m = 60 k. The red fitting line is extracted from the hidden Markov model. b The first 100 points of of three-state jump signal. The three plateau levels correspond to the states ({|+1\left.\right\rangle }_{n}), ({| 0\left.\right\rangle }_{n}), and ({| -1\left.\right\rangle }_{n}) of the 14N nuclear spin. c Phase diagram for metastability. The metastable region for the number of measurements m depends on the orientation of the magnetic field. The dashed line marks the case of m = 60 k, with the two black triangles indicating experimental conditions used. The left point with θ = 2° gives the three-state jump signals explicitly shown in (a) and (b). The corresponding metastable region, quantified by the deviations of their eigenvalues from 1, is shown in the inset. The right point with θ = 8.8° corresponds to the two-state jump signal shown in Fig. 2a.

Single-shot readout of a single nuclear spin

The observed metastability provides a simple and efficient way to realize high-fidelity single-shot readout of the nuclear spin. Under a magnetic field tilt angle θ = 7° ± 2° and a RIM interval of τ = 374 ns, the quantum jumps of the nuclear spin are clearly observed in the range of measurement numbers from 5 k to 250 k, as shown in Fig. 2a. We then characterize the readout fidelity of the nuclear spin using the method developed in ref. 36,47. By pre-selecting bright-state and dark-state photon count thresholds, we filter quantum jump signals across different measurement repetitions. This results in photon number distributions for bright and dark states with initialization fidelity higher than 99%. We then quantify the state-dependent readout fidelities, as summarized in Fig. 4a and Fig. S9b, c. It is worth noting that both the external magnetic field and the RIM interval affect the occurrence of the metastable states, so as to the optimal readout region. With a repetition number of m = 80 k (total measurement time 1.695 s), a readout fidelity of 97% ± 2% with a threshold count value of 2950 is achieved.

Fig. 4: Single-shot readout of the nuclear spin state.

a Readout fidelity of the 14N nuclear spin as a function of the measurement repetition number. The data points with θ = 7° ± 2°, τ = 374 ns are shown in red squares, while the data points with θ = 7° ± 2°, τ = 370 ns are shown in blue circles. b 14N nuclear spin relaxation time (bright state in orange and dark state in blue) as a function of the duty cycle of the excitation laser pulses. The inset illustrates the experimental pulse sequence. The duty cycle is defined as the ratio of the measurement time to the waiting time in the sequence. In (a) and (b), the vertical error bars display the statistical error (s.d.) from the fit. c 14N nuclear spin NMR, obtained by varying the radio-frequency (RF) pulse frequency at a fixed pulse duration of 30 μs. d 14N nuclear spin Rabi oscillations. A resonant RF pulse of a specific length is applied between every two sequential RIMs to flip the nuclear spin between the ({| 0\left.\right\rangle }_{n}) and ({|+1\left.\right\rangle }_{n}) states, while the electron is at the ({| 0\left.\right\rangle }_{n}) state.

Next, we investigate the relaxation mechanism of the 14N nuclear spin under repeated measurements. Previous studies have shown that the flip-flop process in the excited state of the NV center is the dominant factor, and usually a large external magnetic field (>2000 G) is applied to suppress this term so that single-shot readout of the nuclear spin can be achieved36,37. Alternatively, our results show that the nuclear spin relaxation can be largely suppressed by simply shortening the duration of the excitation laser. Moreover, by inserting additional waiting intervals between the sequential RIMs, a longer nuclear spin T1 is observed, as summarized in Fig. 4b. The longest T1 measured in our experiment is 15 ± 3.5 s. To our knowledge, this is the first time that a single-shot readout of the 14N nuclear spin has been achieved at room temperature and under a small magnetic field.

Finally, we perform nuclear magnetic resonance (NMR) and Rabi measurements to demonstrate coherent control of the 14N nuclear spin. The nuclear spin state is first prepared by a RIM measurement sequence (either in the bright or dark state), then an RF pulse is applied to flip the nuclear spin, and a second RIM measurement is performed to determine the nuclear spin state. The RF pulse has either a fixed duration (30 μs for NMR) or a fixed frequency (at the resonance frequency of NMR). The sequence is repeated 80 k times, and the probability of nuclear spin flip is recorded. Figure 4c shows the NMR signal at 4973.0 kHz, which corresponds to the 14N nuclear spin being in ({|+1\left.\right\rangle }_{n}) state. Figure 4d shows the Rabi oscillation of the 14N nuclear spin between the states ({|+1\left.\right\rangle }_{n}) and ({| 0\left.\right\rangle }_{n}). These results further confirm that the observed quantum jump signal originates from the 14N nuclear spin.

Discussion

In conclusion, we have directly observed metastability in the discrete-time open quantum dynamics of a single nuclear spin in diamond, induced by sequential RIMs of a probe NV electron spin. The observed metastable polarization of the nuclear spin demonstrates that QND measurements of an arbitrary quantum system can be realized in a much broader range of conditions, and go far beyond the QND experiments where the ancilla and the measured system are tuned to maximally entangled states36,37,38. Such schemes can also be easily extended to other physical platforms, including trapped ions, superconducting circuits, and other semiconductor single-spin systems.

The framework for constructing QND measurements can be generalized to implement dynamical-decoupling-based quantum metrology in noisy environments, which enables us to learn the probe-bath and bath-bath coupling strengths48,49, the magnetic field46,50,51 and bath spin polarization52,53. On the other hand, our results point out a potential issue in constructing quantum networks with NV centers or other single-spin systems, since sequential measurements of an ancilla can cause its nearby memory to equilibrate towards a maximally mixed state, leading to the loss of information about the stored state.

The generality of our protocols also sets a blueprint for future studies of nonequilibrium phenomena in open quantum systems. It will be interesting future work to realize more general channels in both theory and experiments, whose fixed points may include decoherence-free subspaces or noise-free subsystems, and to explore the metastability phenomena for protected coding and logical gates. Such phenomena can be directly relevant to quantum information processing, since the fixed points of a quantum channel with preserved information can become metastable when control errors or environmental noise are taken into account.

Methods

Experimental setup and sample information

The experiments are performed with a [100]-oriented, type IIa single-crystal diamond produced by Element Six, with a natural 13C isotopic abundance of 1.1%. The NV centers used in this study are generated through electron irradiation and subsequently annealed under vacuum conditions. To improve photon collection efficiency, a solid immersion lens (SIL) is etched onto the diamond surface. Under saturation conditions, the photon counting rate of the NV center is about 300 kcps. The diamond was mounted on a custom-built confocal microscope, and a 532-nm laser was used to initialize and read out the NV spin states. The laser pulses are timing-controlled with an acoustic optical modulator (AOM, Gooch & Housego 3350-199) and focused on the diamond sample through an objective (NA = 0.9). The fluorescence emitted from the NV center is collected with the same objective, passed through a dichroic mirror and a 650-nm long-pass filter, then coupled into a multimode fiber and detected with a single photon detector (SPD). The fluorescence photons are counted with an data acquisition card (NI-6343).

To control the NV electron spin and the 14N nuclear spins, we use an arbitrary waveform generator (AWG, Techtronix 5024C) to generate transistor-transistor logic (TTL) signals and low-frequency analog signals. The synchronized TTL signals are used to control the AOM, RF switches and counters. For electron spin manipulation, the microwave (MW) carrier signal from a MW source (Rohde & Schwarz SMIQ03B) is combined with two analog signals using an IQ-mixer (Marki Microwave, IQ1545LMP). An additional analog channel is used to generate the RF signal for nuclear spin manipulation. Both the MW and RF signals are amplified by amplifiers (Mini Circuits, ZHL-16W-43-S+ and ZHL-32A-S+). The MW and RF signals are combined by a customer-designed duplexer (DC-200 MHz, 1200 MHz-4000 MHz) and transmitted to the NV position via a gold co-planar waveguide deposited on the diamond surface.

As can be seen in the ODMR spectrum of the NV center, there is no splitting due to hyperfine interaction with strongly coupled 13C nuclear spin, and a static magnetic field B = 108.4 ± 0.2 G is applied close to the NV symmetry axis. The transition between the ({| 0\left.\right\rangle }_{e}\leftrightarrow {| -1\left.\right\rangle }_{e}) NV spin states is addressed via resonant microwave pulses (at the m**I = 0 state of the 14N nuclear spin). To compensate for the temperature drift of the experimental system, we re-measured the ODMR spectra and adjusted the microwaves accordingly before each of the sequential RIMs. Figure S8b shows the free-induction decay (FID) signal of the measured NV center, which yields ({T}_{2e}^{*}=1.15,\mu {{\rm{s}}}).

For the 14N nuclear spin, the measurement strength oscillates as a function of the RIM duration τ. Considering the nuclear spin relaxation and the NV electron spin dephasing time, τ = 374 ns is chosen to perform the experiments. Under these circumstances, the dynamics induced by sequential RIMs precedes the 14N nuclear spin T1 process. For each RIM, an optical pulse of 220 ns is used to read out the population of the NV electron spin. After the first measurement, the nuclear spin randomly collapses to ({| 0\left.\right\rangle }_{n}) or ({| \pm 1\left.\right\rangle }_{n}), and in the following measurement, it remains at the same state until a quantum jump occurs. Results with more τ value can be found in the Supplementary Information Note III.

Quantum channels and their representations

Quantum channel, also called completely positive and trace-preserving (CPTP) map, describes the most general (closed or open) quantum dynamics that a quantum system can undergo. Each quantum channel has four different representations: the Kraus representation, the Stinespring representation, the natural representation, and the Choi representation. The Kraus representation is the most commonly used one and characterizes a channel by a set of Kraus operators ({{{M}_{a}}}_{a=1}^{{r}), which satisfy ({\sum }_{a=1}r{M}_{a}{{\dagger} }{M}_{a}={\mathbb{I}}), such that (\Phi (\cdot )={\sum }_{a=1}}{r}{M}_{a}(\cdot ){M}_{a}^{{{\dagger} }=\mathop{\sum }_{a=1}}{r}{{{\mathcal{M}}}}_{a}(\cdot )). The Stinespring representation is a dilation of a quantum channel, which is realized by coupling the bath system to a probe system, subjecting the composite system to a unitary evolution and then tracing over the probe system, i.e., Φ(ρ**n) = Tre[U(ρ**e ⊗ ρ**n)U†], where ({{{\rm{Tr}}}}_{e}[\cdot ]) denotes the partial trace over the probe qubit.

Since a quantum channel acts on the operator space of the bath system and can be considered as a superoperator, it is more convenient to use its natural represen