Introduction

Quantum Zeno effect1,2 (QZE) is a very interesting phenomenon in quantum physics, in which the evolution of a quantum system can be inhibited by frequent measurements. In 1990, based on Cook’s theoretical proposal3, QZE4 was experimentally demonstrated by controlling the transition between two hyperfine levels of 9Be+ with laser pulses and there was a good agreement between theoretical prediction and experimental results. Since then, much effort has been paid to the research on QZE5,6,7,8, quantum anti-Zeno effect9,10,11,12 and quantum Zeno-like effect (QZLE)13,14. Because of its potential application in slowing down or even freezing the dynamic evolution of a system via repeated frequent measurements, it recently has attracted considerable interest as a tool in the fields of quantum information processing15,16,17,18,19,20,21 and ultrasensitive magnetometer22.

QZE has been successfully demonstrated on various physical systems, such as trapped ions23, superconducting qubits24,25, cavity quantum electrodynamics7,16, nuclear magnetic resonance26,27 and Bose-Einstein condensates28,29,30. On the other hand, in solid-state quantum-information technology, a nitrogen vacancy (NV) center which consists of a nitrogen substituting for a carbon and an adjacent vacancy in diamond has been identified as one of the most promising candidates for qubits31,32,33,34,35,36,37,38 due to its long coherence time at room temperature39,40,41 and convenient manipulation under optical field, microwave field and rf field42,43. Quantum Zeno-like phenomenon was experimentally demonstrated by inhibiting coherent spin dynamics induced by the microwave driving between two ground-state electron-spin levels (ms = 0 and ms = 1) of a single NV center44. Therein, only one measurement is performed to analyze the measurement effect on electron-spin states, i.e., the population variation between the electron-spin states ms = 0 and ms = 1 has been made by this single measurement.

In the conventional QZE, repeated instantaneous perfect measurements performed on the system will freeze the evolution of the initial state. The perfect conventional QZE requires infinite measurements with repetitive frequency approaching infinity, which might be impossible in experiment. However, it was recently discovered that perfect freezing of quantum states can also be achieved by more realistic non-projective measurements performed at a finite frequency13. According to Brouwer’s fixed-point theorem45, there always exist some quantum states which satisfy Φ(ρ0) = ρ0, where Φ represents a quantum dynamical evolution process of a system with its initial state ρ0. After n identical cycle process, the system stays at the state as the same as its inital one, i.e., Φn(ρ0) = ρ0. In this way, a QZLE can be achieved with finite-frequency measurements.

In this paper, inspired by the discovery in ref. 13, we present a proposal to achieve the QZLE in a proximal 13C nuclear spin of an NV center by controlling the electron spin. In our proposal, the electron spin plays the role as a detector while the 13C nuclear spin acts as the target. Furthermore, the conventional QZE is demonstrated by modulating the measurement parameters and the external magnetic field. Here, instead of projective measurements, we apply a microwave pulse to induce the transition between different electronic states, followed by initialization of electron spin. Our numerical calculation properly shows that for suitable parameters there exist the conventional QZE and QZLE in a proximal 13C nuclear spin of the NV center.

Results

The model

An NV center in diamond is composed of a nitrogen atom and a vacancy in an adjacent lattice site. It is a defect with C3v symmetry46,47. For the negatively-charged NV center with electron spin S = 1, the ground state is a spin-triplet state 3A with a zero-field splitting D = 2.87 GHz between spin sublevels ms = 0 and ms = ±148. Around NV centers there are three kinds of nuclear spins49,50, i.e., 13C (I = 1/2), 14N (I = 1) and 15N (I = 1/2). They can be manipulated by microwave and rf fields.

Consider an NV center and a 13C nuclear spin which locates in the first coordination shell around the NV center51, as shown in Fig. 1(a). In other words, this 13C nuclear spin is at the nearest-neighbor lattice site of the NV center. As a result, there is a strong hyperfine coupling between the nuclear and electronic spins. Figure 1(b) shows the simplified energy-level diagram of the ground-state hyperfine structure associated with the nearest-neighbor 13C nuclear spin. To demonstrate the QZE, the electron-spin states (ms = −1, 0) and nuclear-spin states are chosen to code qubits. The target and detector are initially uncorrelated, i.e., they are in a product state. A strong electron-spin polarization into the ms = 0 sublevel can be induced by circulatory optical excitation-emission. This effect results from spin-selective non-radiative intersystem crossing to a metastable state lying between the ground and excited triplet states52,53. Moreover, the nuclear spin could be well isolated from the electron spin, during the optical polarization and measurement of the electronic state42,54. In other words, the state of nuclear spin could be unperturbed when the initialization and measurement are performed on the electronic spin.

Figure 1
figure 1

Scheme for demonstration of the QZE in an NV center.

(a) A 13C nuclear spin is at the nearest-neighbor lattice site of an NV center. (b) The energy-level diagram of the ground state hyperfine structure where a microwave drives the transition between and with Rabi frequency Ω and driving frequency ω.

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Suppose that the electron spin is initially in its ground state and the nuclear spin is in an arbitrary state. First of all, the whole system evolves freely for a time interval Δtf. Afterwards, a microwave driving is used to perform measurement. As Fig. 1(b) shows, the microwave drives the transition between and with Rabi frequency Ω and driving frequency ω. In the process of measurement, the total system evolves under the Hamiltonian HM = HF + HI for a time interval Δtm, where HF is the free Hamiltonian without measurement and HI is the interaction Hamiltonian describing the transition induced by microwave driving. After the measurement, by optical pumping, the electron spin is initialized in its ground state and meanwhile the electron and nuclear spins are decoupled42,54,55. And then the above process is repeated. When the duration of the 532-nm light pulse for optical pumping is appropriate, the 13C nuclear spin could be well isolated from the electron spin and the nuclear spin state can be preserved. In particular, the dephasing of nuclear spin can hardly be observed for light pulses of ~140 ns which are sufficiently long to polarize the electron spin while leave the state of nuclear spin undisturbed42.

The free Hamiltonian HF

The general Hamiltonian of an NV center and a 13C nuclear spin which locates in the first coordination shell around the NV center is56

Here, the first term stands for the zero-field splitting of the electronic ground state. The second term is the Zeeman energy splitting of the electron with γe being the electronic gyromagnetic ratio. The third term denotes the nuclear Zeeman effect where γn is the 13C nuclear spin gyromagnetic ratio. And the last term describes the hyperfine interaction between the electron spin and the nuclear spin of 13C atom.

Using a permanent magnet, an external magnetic field Bz is applied parallel to the NV axis. Hence, and are removed. Under the condition of weak magnetic field strength, the difference between the zero-field splitting D = 2.87 GHz and the electronic Zeeman splitting is much larger than the hyperfine interaction. In this situation, the electron-nuclear-spin flip-flop processes induced by the hyperfine interaction are sufficiently suppressed. Therefore, this allows for the secular approximation56,57,58 and the SxIx and SyIy terms can be neglected. In other words, for the weak external magnetic field along the NV axis only the longitudinal hyperfine interaction needs to be taken into account and the ground-state manifold of the NV center coupled with a proximal 13C nuclear spin is described by the Hamiltonian

where

and σz is the Pauli-z operator.

The Hamiltonian under measurement

A microwave driving is utilized to perform the measurement, as shown in Fig. 1(b). This microwave pulse drives the transition between and . The driving frequency is set to be resonant with the transition between and and meanwhile largely detuned from that between and . In this way, the transition between and can be induced selectively. Thus the system evolves under the whole Hamiltonian HM = HF + HI, where the interaction Hamiltonian is

In order to analytically calculate the quantum dynamics under the influence of HM, we transform to the rotating frame defined by the transformation , where U(t) = exp(−iHFt), and are the wave functions in the static and rotating frames respectively. Therefore, using equation (23), the Hamiltonian under the rotating frame can be obtained, i.e.,

Dynamic Evolution

Now, let us demonstrate the quantum dynamics of the whole system in a full cycle, which includes a free evolution followed by a measurement process. The electron spin is initially prepared in its ground state, i.e.,

and the nuclear spin is in an arbitrary state

which is spanned by the basis . Thus, the initial state of the total system is .

In the free evolution, the total system evolves under its free Hamiltonian HF for a time interval Δtf, which is described by the evolution operator

Apparently, without the driving, the evolution operator of the total system is diagonal. At the end of free evolution, the state of the total system becomes . Afterwards, a microwave pulse is used to drive the transition between and . The total system evolves under the Hamiltonian for a time interval Δtm. Having transformed to the rotating frame, a time-independent Hamiltonian is obtained and the corresponding evolution operator is

After a cycle with duration τ = Δtf   + Δtm, the state of the whole system is

Utilizing equation (24), the final state of the whole system in the static frame reads

By partially tracing over the degree of the electron spin, the final state of the nuclear spin reads

On the other hand, the initial state of the nuclear spin can be decomposed into its eigenbasis as13

where C0 = 1/2, C1 = β, C2 = β*, C3 = (2α − 1)/2, I is the identity operator, and are the raising and lowering operators respectively. After the first cycle, the nuclear spin is in the state

where the eigenvalues are

After N cycles, the nuclear-spin qubit evolves into the state

Here, λ0 = λ3 = 1 are related to fixed points45 independent of all parameters. However, λ1 and λ2 are modulated by the parameters (Ω, Bz, Δtm, Δtf). By adjusting these parameters, the quantum Zeno and Zeno-like effects can be observed.

Hereafter, by analyzing the dependence of the eigenvalues on the parameters, we demonstrate the existence of quantum Zeno and Zeno-like effects.

Quantum Zeno-like effect

In equation (15), the eigenvalues λ0 = λ3 = 1 mean that are the fixed points independent of the combination of parameters (Ω, Bz, Δtm, Δtf) after repeated measurements. To be specific, if the initial state of the nuclear spin is of diagonal form, the state will not be changed and thus is preserved. This is a QZLE on the nuclear spin, similar to that in ref. 13.

On the other hand, the σ+ and σ− components of the initial nuclear-spin state are exponentially suppressed, when with appropriate parameters (Ω, Bz, Δtm, Δtf). Therefore, the following process is achieved by sufficiently-many measurements.

Furthermore, the existence of λ3 = 1 preserves the polarization of the nuclear spin. If α is 1 or 0, or can be obtained. Thus, the polarization of the 13C nuclear spin near the NV center is frozen. It may be a potential way to preserve the polarization of 13C nuclear spin against its hyperfine interaction with electron spin.

Quantum Zeno effect

For the eigenvalues λ1 and λ2, we consider both the ideal situation when Δtf, Δtm→0 and the realistic situation of finite Δtf and Δtm.

In the ideal situation, equation (17) is simplified as

From equation (18), we learn that the eigenvectors σ+ and σ− contribute to the off-diagonal part of ρ(T)(Nτ). When Δtf and Δtm are small enough, the eigenvalues and can be very close to unity. Furthermore, γnBzNτ = 2nπ is assumed, where Nτ = T is the fixed total evolution time and n is an integer. Because the eigenvalues and approach unity quadratically in the high-measurement-frequency limit, an arbitrary nuclear-spin state is exactly preserved by infinitely-frequent measurements. Here, the conventional QZE is recovered.

On the other hand, consider the realistic condition in an NV center, e.g. non-vanishing Δtm due to a finite pulse width. When the Rabi frequency and the pulse width are chosen to meet the following requirement

with n1 being positive integer, λ1 = λ2 = 1 can be obtained. In this case, arbitrary initial state is the Zeno-like fixed point which depends on the appropriate choice of parameters (Ω, Bz, Δtm, Δtf). In other words, under certain measurement conditions, the QZLE is observed. Figure 2 shows the locations where QZLE will occur in the parameter space of (Ω, Δtm). In the case of , after repeated measurements the elements of σ+ and σ− in ρ(T)(T) may disappear due to accumulated loss. However, by tuning parameters to the vicinity of the points (Ω, Δtm) given in equation (20), equation (17) can be expanded to the second order of Δtm as

Figure 2
figure 2

Measurement conditions for the QZLE.

The QZLE will occur at the specified locations in the parameter space (Ω, Δtm). The points are determined by equation (20) with n1 = 1, 2, 3.

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Here, the conventional QZE happens in the neighbourhood of the QZLE points. In other words, the QZE occurs for a series of parameter combinations corresponding to finite-frequency measurements with finite coupling strengths. Since the eigenvalues λ1 and λ2 approach unity quadratically under the repeated measurements, arbitrary nuclear-spin state is exactly preserved by finitely-frequent measurements, even though it is affected by the free evolution.

Discussion

The conventional QZE and QZLE are demonstrated in a 13C nuclear spin around an NV center by controlling the electron spin. Both of the QZE and QZLE can be observed by modulating the Rabi frequency and the magnetic field and the free-evolution time and the pulse width. Our numerical calculation properly shows that for suitable parameters there exist both the QZE and QZLE in an NV-center system under the experimental condition. Consequently, the conventional QZE and QZLE are obtained with finite-frequency imperfect measurements.

In order to put our experimental proposal into practice, the secular approximation should be valid, i.e. the applied magnetic field strength should not be too strong56. As a consequence, the magnetic field strength Bz could be less than 200 G. Additionally, due to the resonance condition, the driving frequency ω equals to the level spacing between and , i.e., . At the same time, the level spacing ω1 between and is . To selectively only induce the transition between and , the large-detuning condition should be fulfilled. Since the hyperfine coupling between the electron spin and a 13C nuclear spin in the first coordination shell is known to be 130 MHz51,59, the Rabi frequency Ω can be no more than 10 MHz. Furthermore, the initialization of the NV center will take approximately ~140 ns42 and Δtm and Δtf are on a time scale about 2 μs. Thus, a single cycle process will take about 5 μs. The intrinsic dephasing time of the 13C nuclear spin T2n was observed as around one second41. To ignore the decoherence effect induced by the environment, we restrict the total experiment time as , i.e., the total experiment time T is chosen as 100 ms. In this case, we can demonstrate the quantum Zeno and Zeno-like effects for roughly 2 × 104 cycles.

On the other hand, due to the presence of the nitrogen nucleus, 14N (I = 1) or 15N (I = 1/2) and the 13C nuclear spin bath, the dephasing time of the electron spin is 58 μs39. The duration τ = Δtf + Δtm of a cycle is smaller than the dephasing time of the electron spin by one order. Because the hyperfine coupling between the electron spin and the nitrogen nucleus AN < 4 MHz50 is much smaller than Azz, the dephasing effect induced by the nitrogen nucleus can be neglected. The dipole-dipole interactions between the electron spin and the other 13C nuclear spins are too weak. Therefore, the dephasing effect induced by all nuclear spins can also be neglected.

Last but not the least, the nuclear spin bath induces the dephasing of the 13C nuclear spin. Because the total experiment time is sufficiently short and the magnetic dipole-dipole interactions between the 13C nuclear spin and the other nuclear spins are weak enough60, the dephasing effect induced by the spin bath can be neglected. Meanwhile, after every measurement, we decouple the electron and 13C nuclear spins and initialize the electron spin in its ground state 42,54 without perturbing the 13C nuclear spin. In this process, the nuclear spin is supposed to be completely isolated from the environment42,54. Therefore, the nuclear spin hardly evolves during this process.

In conclusion, as shown in Fig. 2, our numerical calculation properly indicates that under practical conditions we can demonstrate the conventional QZE and QZLE in 13C nuclear spin around the NV center with finite-frequency imperfect measurements.

Methods

The rotating frame

Since the original Hamiltonian in the measurement process HM is time-dependent, the whole system is transformed to a rotating frame defined by the transformation , where , and are respectively the wave functions in the static and rotating frames. Now, we derive the relationship between the Hamiltonian in the static frame HM and the Hamiltonian in the rotating frame . Because the time evolution of still fulfills the Schrodinger equation in the rotating frame, i.e.

the effective Hamiltonian in the rotating frame reads

Correspondingly, the relationship between the density matrix in the static frame ρ(t) and the density matrix in the rotating frame ρR(t) is

Additional Information

How to cite this article: Qiu, J. et al. Quantum Zeno and Zeno-like effects in nitrogen vacancy centers. Sci. Rep. 5, 17615; doi: 10.1038/srep17615 (2015).