Decoherence Control of Nitrogen-Vacancy Centers

Quantum mechanical systems lose coherence through interacting with external environments—a process known as decoherence. Although decoherence is detrimental for most of the tasks in quantum information processing, a substantial degree of decoherence is crucial for boosting the efficiency of quantum processes, for example, in quantum biology and other open systems. The key to the success in simulating those open quantum systems is therefore the ability of controlling decoherence, instead of eliminating it. Motivated by simulating quantum open systems with Nitrogen-Vacancy centers, which has become an increasingly important platform for quantum information processing tasks, we developed a new set of steering pulse sequences for controlling various coherence times of Nitrogen-Vacancy centers; our method is based on a hybrid approach that exploits ingredients in both digital and analog quantum simulations to dynamically couple or decouple the system with the physical environment. Our numerical simulations, based on experimentally-feasible parameters, indicate that decoherence of Nitrogen-Vacancy centers can be controlled externally to a very large extend.

simulating open quantum systems through an extension of the idea of dynamical decoupling and apply it to NV centers. Dynamical decoupling has been applied to NV center for eliminating the noise from environment and thus prolong the coherence time of electron spin 40,54,55 . In this paper, we developed a new set of decoupling pulse sequences that can control the decoherence of the off-diagonal matrix elements of the system density matrix.
The pulse sequences of interest in this work are different from those in dynamical decoupling [30][31][32] , whose primary goal is to decouple the system from the noise of the environment. In other words, the goal of dynamical decoupling was to maintain the purity of the quantum system. Here we aim to control the decoherence by exploiting the existing environment, so that we can simulate the dynamics of an open quantum system without the need to maintain the purity of the system, instead, environment of the quantum simulator can be employed to be as part of open quantum system. Consequently, we can avoid the need of including extra ancilla qubits as in other digital approaches of simulating open quantum systems.

Results
Full system controllability. A NV center can be viewed as a 3-dimensional qudit with a Hamiltonian H NV of the following form 40,54,55 : where S z is the 3-dimensional spin operator for the spin-1 particle {m = 1,0,−1}, D is the zero-field splitting, γ = gμ B is the gyromagnetic ratio with μ B the Bohr magneton, g is the g-factor of electron, and B z is the static magnetic field applied along z direction ([111] axis). Before we demonstrate how to control decoherence, we first show how to simulate a general 3-dimensional system with a NV center in rotating frame.
For any given Hamiltonian Ĥ and quantum state Ψ , and a unitary operator = −Û e iAt associated with a self-adjoint operator Â (where we used the natural units that set =1), the corresponding quantum state in the rotating frame is given by, We can obtain an effective Hamiltonian H rot in the rotating frame as follows: In our case, we include two sets of microwave pulses to a NV center. In the laboratory frame, the Hamiltonian in Eq. ((1)) becomes: where H NV is the Hamiltonian shown as in Eq. ((1)), S x is a 3 × 3 spin operator, B 1/2 is the magnetic field amplitude of the applied microwave along the x axis, and ω 1 and ω 2 are the frequencies of the applied microwave. The target Hamiltonian H S can be obtained by choosing a rotating frame reference where , which gives the following simulated Hamiltonian: If the microwaves are turned off, then B 1 = B 2 = 0, while in the resonance conditions: ω 1 = D + γB z and ω 2 = D − γB z , the system Hamiltonian is H S = 0.

Noises in Nitrogen-Vacancy (N-V) centers.
In the process of diamond growth, impurities such as isotope 13 C and nitrogen sneak inside. The most possible result of substituting Nitrogen atoms is to generate P1 centers(while NV centers is a P1 centers that captured a vacancy). Normally, a NV center is subject to a local environment dominated by the surrounding nuclear spins of 13 C's and electron spins of P1 centers, which effectively produce a random magnetic field b(t) to the NV center 54,40,56 , i.e., which is also the system-environment interaction Hamiltonain of the simulated quantum system, since in rotating frame H SB remains the same as in laboratory frame. For the case where the environment is dominated by the nuclear spins, the random fluctuation of the magnetic field can be regarded as stationary, i.e., b(t) = b, and is usually approximated as Markovian and Gaussian 54 , i.e., with a probability distribution , where σ b is the variance of the random magnetic field from the spin bath. For the case where the noise come from electron spins instead, the random process of b(t) can be approximated by the Ornstein-Uhlenbeck process 40 , with a correlation function C(t) given by the follow: , where l describes the characteristic strength of the coupling of the NV center to the bath, and R = 1/τ c is the transition rate, with τ c being the correlation time of the spin bath 40 .
Strengthening decoherence. We note that the evolution operator e −iHt of the total system can be divided by many small time slices, Δt ≡ t/n, in which H B is the Hamiltonian of the environment which can be treated as interacting spin system 56 . One way to strengthen decoherence, assisted by the environment, can be achieved as follows: first, turn off the system Hamiltonian momentarily, for a time period, λΔt, where λ > 0, the evolution of the total system is then: , which can be realized by seting where ω is the frequency of the microwave and ε is the diagonal element of simulated quantum system, here the factor λ + 1 1 appears as the diagonal elements do not disappear in the same rotating frame when turning off the microwaves. Then we allow the total system to evolve freely for a SCIeNtIfIC REPORts | 7: 11937 | DOI:10.1038/s41598-017-12280-z time period of Δt. The pattern is then repeated for n times, i.e., , which can be appoximately rewritten as if Δt is small enough. Therefore, in the large-n limit, we obtain an effective Hamiltonian as follow:

S S B B eff
which contains an interaction term H SB amplified by a factor of (1 + λ). The side product is that the environment Hamiltonian is also amplified, that there also appear a factor λ before the environment Hamiltonian H B . However, for the spin environments of NV centers, the effect can be ignored.

Controlling noises from nuclear and electron spin baths. In the following we consider combining
Trotter expansion with decoupling pulses to control the decoherence in NV centers. For NV centers in ultra-pure diamonds(in which there is no measurable nitrogen impurity), the dominant decoherence source comes from the surrounding 13 C spin bath 54 , which is random but stationary within the time-scale of system dynamics. Consequently, for a qubit initialized in a pure state, ψ α β = + 0 1 0 , and H S = 0, the off-diagonal matrix element (or coherence), , decays as follows: which implies that the effective coherence time T 2 can be controlled by the parameter λ, The results are shown in Fig. 1, from Fig. 1(a), it is clear that tuning the parameter λ can strengthen the decoherence, i.e., we can destroy the coherence via increasing the value of λ. The coherence time can be extract from above and shown in Fig. 1(b).
In diamonds with nitrogen impurities, such as Type I, the electron spins also contribute to the decoherence of the system, which means that both nuclear and electron noise should be included. We investigate the decoherence of a two-level system numerically with experimental parameters of NV center, in which both nuclear and electron spin noise are included. The simulated results are shown in Fig. 1(c) and (d), in which only one microwave is applied. The parameters are as follow: The static magnetic field is B z = 100 Gauss, the zero-field splitting is D = 2.87GHz, the frequency of applied microwave is a little away from the resonance frequency(which is 3.15 GHz) of NV center at the given static magnetic field with a detuning 1.9 × 10 6 /(1 + λ) Hz and with a amplitude of 1.717 Gauss, λ is varied from zero (i.e. the ⁎ T 2 ) to 3. the coherence time vs λ is shown in Fig. 1(d), it shows that the coherence time of NV center decrease when λ increases, which shows a effective control of decoherence caused by system-environment interaction.
Weakening decoherence. Let us consider a two-level system and a swap gate defined by: , after which the two state applied undergo an opposite evolution under the influence of environment noise and thus decouple the quantum system from the environment(i.e. the microwave is turned off) after two swap gates are inserted, shown as in Fig. 2(a). During the decoupling part of Trotter decomposition,the evolution is then: where we set t 1 = (λ − μ)Δt and t 2 = μΔt. The overall evolution becomes

S S B S B
, which implies the following effective Hamiltonian:

S S B eff
Supposing a qubit initialized in a pure state, ψ α β = + 0 1 0 , and H S = 0 and the environment noise is quasi-static, for the case in NV center, it corresponds to the case that the noise come from the nuclear spin, the coherence vs evolution time under Eq. ((9)) becomes: Figure 2. (a) Pulse sequence for two-level system, where x means the microwave is applied along x axis, the red bar is the idea π pulse (i.e. the swap gate), blue rectangles mean the evolution of H S , which in the numerical calculations we always set it to be 0 without losing generalization. (b) Coherence vs evolution time with different distance between two swap gate, the variance of the random magnetic field is set to be 0.2 Gauss. (c) Coherence time vs τ, which is defined in µ τλ This is shown in Fig. 2(b) and the coherence time is: which is shown in Fig. 2(c). Here we set the distance between two swap gate as μΔt = τλΔt/2, where τ is the variable shown in Fig. 2 . When τ = 0, it is equal to the case with no decoupling pulse, but when τ = 1, the distance between the two swap gate is half of the λΔt, which is the same as the CPMG pulse 57,58 .
In the presence of electron spin noise, the coherence factor of a two-level system subject to dynamical decoupling is given 59 is the random noise, the function f(t;t′) depends on the pulse sequence as: is the total evolution time. Furthermore, the spectral density of the noise C(ω) is given 40  where δ = λΔt, δ 1 = (λ − μ)Δt and δ 2 = μΔt, Eq. (14) is the spectral dependence of decoupling pulses and is known as "Filter function". The values of ω | |  f t ( , ) 2 are shown in Fig. 2(d) with different values of τ, which is defined in the following relation: μΔt = τλΔt/2. We obtained the coherence factor shown as in Fig. 2(e), after applying a large cut-off frequency. It is obviously that when τ becomes larger, the coherence time becomes longer.
Fine-tuning decoherence for qudit. For general multi-level systems, i.e., qudit, we have an extra tool to fine-tuning the decoherence for different off-diagonal elements in the density matrix. Here we consider only the stationary noise from nuclear spin in three-level system. Let us consider that the dynamical decoupling pulses, = + + − − u 1 0 0 1 1 1 12 , are applied on only one channel(shown as in Fig. 3(a)), between = m 1 and = m 0 , then the relevant part in evolution operator becomes: iH t i H t ib t L t t 12 12 ( ) ( , ) , which gives λ + = ∆ t t t 1 2 shown as in Fig. 3(a). Therefore, when we choose < t t 1 2 , we effectively make state 1 experience less dephasing then state 0 , and vice versa. For any given initial state, ψ , then the off diagonal elements of the associated density matrix decay as follows: here the lower label 1, 2, 3 represent the state = m 1 , = m 0 and = − m 1 , also this can be got after averaging of environment noise which is stationary Gaussian. In other words, the coherence times of the off-diagonal elements are given by: The dependence of the coherence times with the parameter μ is shown in Fig. 3(c). We see that the coherence time T 2 12 is more sensitive to the change of μ, compared with the other coherence times. Furthermore, additional dynamical decoupling pulses, e.g., (shown in Fig. 3 . It gives Figure 3. (a) Pulse sequences for the case applying the decoupling pulses in only one channel in three-level system, labels have the same meaning as Fig. 1. (b) Pulse sequences of the case that applying the decoupling pulses in two channel of the three-level system, where MW1 and MW2 mean the first and second microwave with different frequencies. (c)The coherence time vs τ(which is defined in µ τλ ∆ = ∆ t t /2), as τ increases, the coherence time T 2 12 increases and the other two coherence time remains almost the same, decrease or increase slightly. (d-f) show the coherence time between the three levels, with (d) is the T 2 12 , (e) the T 2 13 and (f)T 2

23
, in which τ 1 and τ 2 are defined in µ τ λ ∆ = ∆ t t /2 1(2) 1 (2) , the blue region means the coherence time is small and the red ones is the case the coherence time increases sharply.
SCIeNtIfIC REPORts | 7: 11937 | DOI:10.1038/s41598-017-12280-z λ + + = ∆ t t t t 1 2 3 . Consequently, for any initial state, ψ α β γ = + + − 1 0 1 0 , the time-dependent off-diagonal elements of the density matrix are given by: which means that the coherence times of the off-diagonal elements are given by: The coherence times with parameters µ 1(2) are shown in Fig. 3(d-f)(in which the case of τ τ ≤ ≤ 2 2 1 is also included, where τ 1 and τ 2 are defined in µ τ λ ). In the plots, the orange region means the coherence time is small and the blue-white one is the case the coherence time increases sharply. It shows that the coherence between different levels can be tuned though changing the insert time of the two decoupling time, which is similar with the case of only one channel is applied with decoupling pulse. For the case that decoupling pulses are applied on both channels, the coherence time T 2 13 increases when µ 1 and µ 2 are close to λ (shown in Fig. 3(e)), while the other two coherence time remains essentially the same as the one before applying the decoupling pulses, and here the coherence time of other two state without decoupling pulses applied increase at a range that is different from the other channels, this shows that with decoupling pulses applied only two channels, we can get the coherence time between all the states tuned.

Discussions
Trotter decomposition combined with decoupling pulses is shown to be able to control the decoherence in NV centers, although this method is numerically tested only for NV centers with two and three energy levels, further generalization of our method to d-dimensional ( ≥ d 4) systems is possible. Following the previous results, the decoherence of different off-diagonal elements can be controlled by the following sequence: where t ij 's are the adjustable waiting time before the swap gate between i and j level is applied after the next swap where t 0 is included as the waiting time before the first swap gate applied.
In conclusions, in this work, we have presented a new method that can control the decoherence in NV center by combining the Trotter decomposition and decoupling pulses, which have potential applications in simulating open quantum system with NV center. The scheme exploits the intrinsic decoherence from the environment, and contains the benefits of the university of digital quantum simulation and also the efficiency of analog quantum simulation. Our results indicate that such a scheme is experimentally feasible. This method could also be used as a test bed for non-Markovian open quantum systems 60,61 , without obvious roadblock. And also the method we proposed here can be applied to quantum simulation of various open quantum system, if it has the same type of environment of the quantum simulator, a simplest classification is Markovian or non-Markovian for example.

Methods
Model of environment in Nitrogen-Vancancy center. The environment we take into consideration contains both electron spins (mainly come from P1 center, i.e. the neutral charge state of substitutional Nitrogen) and nuclear spins ( 13 C), but mainly nuclear spins, which is more similar to the exact bath of deep (about 10 μm) NV centers in diamond with a nitrogen content whose concentration is small than 1 ppb. The model used in this paper come from the literatures 40,54 , for the noise of nuclear spin, it was treated to be stationary compared with the evolution of the NV center, while for the electron spin noise, it is decribed as the Ornstein-Uhlenbeck process.
Numerical calculations. The algorithm adopted here for Fig. 1(c) is RK4 (Runge-Kutta method of the 4-th order), which is a well-known numerical method of ordinary differential equations. In the calculations we used ideal π pulse (i.e. the swap gates), while in laboratory frame, finite-time length of microwave is used to inverse the population of the corresponding two level if needed.
Simulating a open quantum system in NV center. The simulation of the quantum system is realized in a rotating frame. The process is as follow: 1, Initialize the NV center to the | 〉 0 state, from which the population can be read with laser, all of the other states can be read by transfering the population to this state; 2, Apply the microwave which is used to realize the simulating of the target quantum system, evolution for a time duration δt, during SCIeNtIfIC REPORts | 7: 11937 | DOI:10.1038/s41598-017-12280-z which both the quantum system and the environment are included; 3, Turn off the microwave for a duration of λδt, then only the environment is included in the evolution; 4, Repeat this process for n times, and finally with Trotter theorem, the strengthening of the system-environment interaction can be realized. To weaken the decoherence of the simulated quantum system, two π pulses are applied to decouple the system from the environment, and the two pulses are inserted during the same period of evolution when the simulated quantum system is turned off.