key: cord-0440475-18trr40l
authors: Lu, Yanan; Zhang, Yuran; Liu, Gangqin; Nori, Franco; Fan, Heng; Pan, Xinyu
title: Observing information backflow from controllable non-Markovian multi-channels in diamond
date: 2019-12-02
journal: nan
DOI: nan
sha: 1cf141087cf933701e12c73203d4581817b52579
doc_id: 440475
cord_uid: 18trr40l

Any realistic quantum system is inevitably subject to an external environment. This environment makes the open-system dynamics significant for many quantum tech-nologies, such as entangled-state engineering, quantum simulation, and quantum sensing. The information flow of a system to its environment usually induces a Markovian process, while the backflow of information from the environment exhibits non-Markovianity. The practical environment, usually consisting of a large number of degrees of freedom, is hard to control, despite some attempts on controllable transitions from Markovian to non-Markovian dynamics. Here, we experimentally demonstrate the engineering of multiple dissipative channels by controlling the adjacent nuclear spins of a nitrogen-vacancy centre in diamond. With a controllable non-Markovian dynamics of the open system, we observe that the quantum Fisher information flows to and from the environment using different noisy channels. Our work contributes to the developments of both noisy quantum metrology and quantum open systems from the view points of metrologically useful entanglement.

Any realistic quantum system is inevitably subject to an external environment. This environment makes the open-system dynamics significant for many quantum technologies, such as entangled-state engineering 1-3 , quantum simulation 4 , and quantum sensing 5 . The information flow of a system to its environment usually induces a Markovian process, while the backflow of information from the environment exhibits non-Markovianity 6 . The practical environment, usually consisting of a large number of degrees of freedom, is hard to control, despite some attempts on controllable transitions from Markovian to non-Markovian dynamics [7] [8] [9] [10] [11] [12] . Here, we experimentally demonstrate the engineering of multiple dissipative channels by controlling the adjacent nuclear spins of a nitrogen-vacancy centre in diamond. With a controllable non-Markovian dynamics of the open system, we observe that the quantum Fisher information flows 13 to and from the environment using different noisy channels. Our work contributes to the developments of both noisy quantum metrology [14] [15] [16] and quantum open systems from the viewpoints of metrologically useful entanglement.

The unavoidable interaction of a quantum open system with its environment leads to the dissipation of quantum coherence and correlations, making its dynamical behaviour a key role in many quantum technologies. According to the orientation of the information flow between the system and the environment, the time evolution of the open quantum system can be classified into either a Markovian or a non-Markovian one. A Markovian process assumes memoryless dynamics of a quantum open system, described by a dynamical semigroup with a time-independent Lindblad generator 17 . However, in the presence of memory effects, e.g., for a strong system-environment coupling, the Markovian approximation fails, and the non-Markovian process, deviating from a dynamical semigroup, allows for a revival of quantum features.

Owing to the memory effects and the ability of recovering quantum features, non-Markovian quantum dynamics 18 opens a new perspective for applications in quantum metrology. Quantum metrology 14, 19 , an emerging quantum technology, aims to use quantum resources to yield a higher precision of statistical errors in estimating parameters compared to classical approaches. However, the useful quantum coherence and multipartite entanglement for sub-shot-noise sensitivity, quantified by the quantum Fisher information (QFI) 20, 21 , are very fragile to decoherence. Moreover, quantum metrological studies can be very different when being subject to either Markovian or non-Markovian noises 15 . Thus, it is important to es-tablish an approach for characterising the non-Markovianity of the open-system dynamics by using the QFI flow between the quantum system and its environment 13 . The usual measures of non-Markovianity include: bipartite entanglement 22 , trace distance 23 , and temporal steering 24 . However, the metrological approach based on QFI also works on the information subflows through different dissipative channels for a class of time-local master equations 13 .

In our experiments, the open system is provided by a nitrogen-vacancy (NV) centre electron spin, its host 14 N nuclear spin, and a proximal 13 C nuclear spin in diamond (see The nitrogen-vacancy (NV) centre, its host 14 N nuclear spin, and a nearby 13 C nuclear spin form a three-qubit system. (b) Energy levels of the three-qubit system. At the excited-state level anti-crossing (ESLAC), the three spin qubits can be polarised by a short laser pulse and manipulated with resonant radio-frequency (RF) pulses (13.284 MHz for the 13 of ∆ ≃ 2.87 GHz between |0 e and | ± 1 e of the ground spin triplet. An external magnetic field along the NV symmetry axis is applied to degenerate | ± 1 e states. Here, the first qubit is encoded on the |0 e and | − 1 e (hereafter labelled as |1 e ) subspace of the NV electron spin. The second and third qubits are encoded on the |↑ n,c and |↓ n,c states of the host 14 N nuclear spin and the nearby 13 C nuclear spin, respectively (see Fig. 1b for the energy levels). Applying the secular approximation and ignoring the weak nuclear-nuclear dipolar interactions, the effective interaction Hamiltonian of the three-qubit system and the spin bath can be written as 25 (we set = 1)Ĥ

whereĤ R is the interaction Hamiltonian between the electron qubit and the spin bath, A n ≃ −2.16 MHz and A c ≃ 12.8 MHz denote the hyperfine coupling strengths between the electron spin and nearby nuclear spins, respectively. Under an external magnetic field of B z = 482 Gauss, the electron spin and two nuclear spins can be simultaneously polarised by a short laser pumping due to level anti-crossing in the excited state (ESLAC) 26 . In Fig. 1c , we show Rabi oscillations of the 13 C and 14 N nuclear spins, driven by 13.284 MHz and 2.929 MHz radio-frequency (RF) pulses, respectively. We attribute the smooth and damping-free oscillations to the superb coherence of the nuclear qubits and the well-controlled driving pulses. In our first experiment, we employ the electron qubit as a noise sensor, whose dynamical behaviour is modulated by initialising the state of the host 14 N spin and the proximal 13 C spin (see Fig. 2a ). Since the interactions between nuclear spins can be ignored in the time scale of our experiments, these two controllable nuclear spins can be regarded as the regulators of two independent dissipative channels, and other weakly coupled nuclear spins act as another uncontrollable dissipative channel (see Methods for details). The quantum circuits and pulse sequences of the first experiment are shown in Figs. 2c and 2e, respectively. By applying a 3 µs laser pulse (532 nm), these three qubits are polarised to an initial state |Ψ i = |0 e ⊗ |↓ n ⊗ |↓ c . Then, by applying the MW, RF 1 and RF 2 pulses as shown in Fig. 2e , the system is prepared in

The time evolution of the electron qubit can be described by the partial trace after the unitary time evolution of the total Hamil-

, where Tr ncR [· · · ] denotes the partial trace over the host 14 N qubit, the 12.8 MHz 13 C qubit and the spin bath degrees of freedom. Given the generatorŜ z e =σ z e /2, the QFI of the electron qubit can be written as QFI of the electron qubit only subject to the 14 N, 13 C, and the spin bath dissipative channels, respectively. Details on Q R (t) can be found in Methods.

The QFI flow, defined as the rate of change of the QFI, I(t) ≡ ∂ t Q(t; φ 1 , φ 2 ), can be explicitly written as a sum of QFI subflows with respect to different dissipative channels 13

where I i (t) ≡ Q(∂ t ln Q i ), with i = n, c, R, and each QFI subflow corresponds to not only the individual separable dissipative channel but also all channels 13 . Moreover, the inward QFI subflow (I i > 0), resulting from the temporary appearance of a negative decay rate 27 of the time-local Lindblad master equation 17 , is an essential feature of non-Markovian behaviours. Furthermore, we focus on the sum of time integrals of all inward QFI subflows

as a measure of non-Markovianity, and the long-time measure is defined as N (φ 1 , φ 2 ) ≡ N (t → ∞, φ 1 , φ 2 ). Different from the time integral of the total QFI flow, the measure of non-Markovianity in Eq. (4) considers the inward subflow from each dissipative channel and can dig out the non-Markovianity even when the total QFI flow is negative.

In our experiments, the dissipative channels of the 14 N and 13 C qubits can be fully controlled by tuning the durations of the RF 1 and RF 2 pulses, i.e., to adjust φ 1 and φ 2 . When both channels are turned off, φ 1 = φ 2 = 0 (see Figs. 3a and 3b for the QFI), the dynamics of the NV electron spin is only affected by the spin bath and behaves Markovian with the QFI flowing out (see Fig. 3d ). For φ 1 = π/4, π/2, and φ 2 = 0, the channel of the 14 N nuclear spin is open, and the revival of the QFI is shown in Fig. 3a , while the positive QFI flows are observed in Figs. 3e and 3f for witnessing non-Markovian dynamics. For φ 1 = 0, and φ 2 = π/4, π/2, the revival of the QFI and the positive QFI flows, subject to the channels of the 13 parameter φ 2 , the measured N (0, φ 2 ) is compared with the numerical simulation in Fig. 3k .

We furthermore characterise the system behaviour when both controllable dissipative channels are open. With φ 1 = φ 2 = π/2, the time evolution of the QFI of the electron qubit and its QFI flow, compared with the ones obtained from the sum of subflows (red cross), are shown in Figs. 3c and 3i. In Fig. 3j , the measure of non-Markovianity N (t, π 2 , π 2 ) from the QFI subflows, I n,c,R , with respect to different dissipative channels (red bar) is compared with the one from the total QFI flow, I (blue bar). We clearly observe that the measure in terms of QFI subflows quantify more non-Markovianity than the total QFI flow, when the system is subject to multiple dissipative channels.

In the second experiment, we consider the open system, consisting of the electron qubit and the proximal 13 C qubit, which is subject to the controllable noisy channel of the host 14 N qubit and the dissipative channel of the spin bath (see Fig. 2b ). Figures 2d and 2f show the quantum circuit and pulse sequences of the second experiment. Starting with the state |Ψ i = |0 e ⊗ |↓ n ⊗ |↓ c , the system is prepared in |Ψ ′ (0) = (|0 e ⊗ |↓ c + |1 e ⊗ |↑ c ) ⊗ |ψ(φ 1 ) n , with the pulse sequences shown in Figs. 2d and 2f. The electron qubit and 13 C nuclear qubit are maximally entangled at this stage. Similarly, assuming ̺ ′ 0 = |Ψ ′ (0) Ψ ′ (0)| ⊗ ρ R , the time evolution of the electron qubit and the 13 C qubit is described as

, where Tr nR [· · · ] denotes the partial trace over the 14 N spin and the spin bath degrees of freedom. If the QFI of a two-qubit state, with (Ŝ z e +Ŝ z c ) being the generator, is larger than 2, i.e., Q ′ (t; φ 1 ) > 2, it characterises the useful entanglement for quantum-enhanced parameter estimation 14 .

The time evolution of the QFI of the maximally entangled state is shown in Fig. 4a , when the controllable 14 N channel is either closed (φ 1 = 0) or open (φ 1 = π/2). At time t = 0 with φ 1 = 0, we obtain the maximum QFI, Q ′ (0; 0) = 3.687, which is useful for sub-shot-noise-limit metrology 14, 28 . With the dissipative channel of 14 N closed (φ 1 = 0), the QFI flow remains negative, except for few time intervals due to the quantum fluctuations (see Fig. 4b ). In Fig. 4c , the positive QFI flow of the maximally entangled state, with φ 1 = π/2, clearly signals the non-Markovian dynamics of the two-qubit open system. Moreover, the metrologically useful entanglement [Q ′ (t; φ 1 ) > 2] survives for a period of time ( 380 ns) with respect to the Markovian noise of the spin bath. However, it decays faster under the impact of non-Markovian noise by

Our experiments clearly demonstrate the engineering of the non-Markovian dynamics of open systems by manipulating the electron spin, the host 14 N, and the neighbouring 13 C nuclear spins of the NV centre in diamond at room temperature. First, the electron qubit, as an open system, is subject to two controllable dissipative channels of nearby nuclear qubits, of which the QFI flow characterises the non-Markovianity and can be decomposed into subflows from individual channels. Second, when the open system, consisting of the electron qubit and the 13 C qubit, is prepared in the maximally entangled state, the controllable non-Markovian behaviour of the decoherence dynamics of the entanglement witnessed by the QFI flow is reported. By using the QFI as a witness for quantum coherence and metrologically useful entanglement, our work will contribute to the developments of both noisy quantum metrology and the non-Markovian dynamics of quantum open systems in solids 28 . Competing Interests The authors declare no competing interests.

QFI flow and non-Markovianity. Consider a mixed state ρ = j λ j |j j|, with i|j = δ ij and a generatorÔ, the QFI of a state ρ(θ) = exp(−iθÔ)ρ exp(−iθÔ) with respect to a parameter θ can be written as 20

For a single qubit, any state can be expressed as

with r = [s x , s y , s z ] being the Bloch vector and r ≡ (s 2 x + s 2 y + s 2 z ) 1 2 being the Bloch length. The QFI with respect to the generatorÔ =σ z /2 can be calculated as Q = r 2 − s 2 z . For a qubit state with s = [1, 0, 0], its QFI can be calculated to be 1. The QFI plays a central role in quantum metrology and multipartite entanglement witness [29] [30] [31] , which is also sufficient for measuring non-Markovianity 13 .

We then consider a quantum process, described by a timelocal master equation 17, 27 ∂ρ ∂t

whereĤ is the Hamiltonian for the open system without coupling to the bath, γ j (t) is the time-dependent decay rate, and A j (t) is the time-dependent Lindblad operator. The QFI flow of the quantum open system can be divided into different subchannels as 13 I ≡ ∂Q/∂t = j I j , with I j = γ j (t)J j , and J j ≤ 0. Therefore, the existence of any positive QFI subflow characterises a quantum non-Markovian process based on the completely positive divisibility 17 .

Measure of non-Markovianity via QFI subflows. From Eq. (4), the non-Markovianity can be witnessed by the sum of time integrals of all inward QFI subflows. When φ 1 = φ 2 = π/2, QFI subflows I n (t), I c (t) and I R (t) can be calculated from QFI with different parameters: Q R ≡ Q(t, 0, 0), Q nR ≡ Q(t, 0, π 2 ), and Q cR ≡ Q(t, 0, π 2 ), which can be expressed, withQ ≡ ∂Q/∂t, as

Setup and sample. Our experiments are performed under ambient conditions on a high-purity bulk diamond (Element Six, with N concentration < 5 p.p.b., and natural abundance of 13 C isotopes). The NV centre is located 10 µm below the diamond surface. To enhance the photon collection efficiency of the NV centre, solid immersion lenses (SILs) are etched on the diamond surface 32 . The photon detection rate is 450 kcps, when the laser (532 nm) excitation power is 240 µW.

We use the ODMR to detect nuclear spins, which are strongly coupled to the NV electron spin.

From the continuous-wave ODMR spectrum under a small magnetic field B ≃40 Gauss (see Extended Data Figs. 1a and 1b), the host 14 N nuclear spin and the nearby 13 C nuclear spin with 12.8 MHz coupling strength are identified 33 . To identify the nuclear spin with a weaker coupling strength, the external magnetic field is tuned to 482 Gauss (along the quantisation axis of the NV centre), and both the host 14 N and the 12.8 MHz 13 C nuclear spins can be polarised by a short laser pulse. We then measure the pulsed-ODMR spectra of the NV centre (see Extended Data Fig. 1c) . By reducing the MW power, two other 13 C nuclear spins are resolved (see Extended Data Fig. 1d) . The hyperfine splittings caused by these two 13 C nuclear spins are 0.9 MHz and 0.4 MHz, respectively. These two nuclear spins are partially polarised by the laser pumping (see Extended Data Fig. 1d) , which is consistent with the literature results 34 . In our experiments, the 12.8 MHz 13 C nuclear spin and the 14 N nuclear spin are taken as fully controllable decoherence channels, while the other nearby weakly-coupled 13 C nuclear spins behave as uncontrollable decoherence channels.

Coherent manipulation of the NV centre. The electron spin and nuclear spins are manipulated by the resonant MW and RF pulses. Extended Data Fig. 1e shows a typical Rabi oscillation signal of the NV electron spin, with Rabi frequency 23.8 MHz. Extended Data Fig. 1f shows the Ramsey oscillation from a deliberate MW detuning (2 MHz) and the beating between different transitions, according to the state of the 0.4 MHz 13 C nuclear spin. By fitting the Ramsey signal, we obtain the coherence time of this NV centre as T ⋆ 2 ≈ 2.9 µs.

State tomography. The sequence to realise the single-qubit state tomography can be divided into three parts: (i) Directly collect fluorescence photon counts L z ; (ii) prepare the state, apply a π 2 MW pulse along the x-axis, and then collect the fluorescence photon counts L y ; (iii) prepare the state, apply a π 2 MW pulse along the y-axis, and then collect the fluorescence photon counts L x . In addition, we collect the fluores-cence photon counts L 0 in the bright state (m s = 0), and L 1 in the dark state (m s = −1), as references to normalise the florescence signal. With the signals on different bases, we can reconstruct the state by calculating the Bloch vector

The two-qubit state tomography technique is based on the single-qubit state tomography method. The key point is to apply one or two additional transfer pulses (RF/MW pulses, see concrete sequences in Ref. 35 ) to divide the full 4×4 density matrix of two qubits into several 2×2 reduced density matrices. Then, we can in turn obtain the real and imaginary parts of each element of the density matrix by implementing the single-qubit state tomography in each working transition (see Refs. 35, 36 ).

QFI envelope induced by the spin bath. To quantitatively describe the influence of the spin bath on the dynamics of the quantum open system, we measure the long-term QFI of the NV electron spin, when all the controllable channels (12.8 MHz 13 C and 2.16 MHz 14 N) are turned off after polarising by a short laser (see Extended Data Fig. 2b) . In Extended Data Figs. 2a and 2b, we find that the coherence and QFI decay non-monotonously, with a 0.4 MHz oscillation. We attribute this oscillation to the effects of the partially polarised 13 C nuclear spin, of which the hyperfine coupling strength is 0.4 MHz, and the quantisation axis is not the same as the NV electron spin's 34 . Therefore, we can only regard the 0.4 MHz 13 C nuclear spin as an uncontrollable quantum channel and fit the experimental results using the formula Q R (t) = exp[−(t/T * 2 ) α ][1 − sin 2 φ 0 sin 2 (A c0 t/2 + ϕ 0 /2)], with α = 0.89, T * 2 ≃ 1026 ns, φ 0 = 0.37π, A c0 = 0.4 MHz, and ϕ 0 = 0.21π. However, within the time scale of our experiments (0-600 ns), the dissipative channel of the 0.4 MHz 13 C nuclear spin behaves Markovian, which can be included in the Markovian channel of the spin bath (see Extended Data  Fig. 2b) .

Data processing. The measurements in our experiments are repeated at least 4×10 5 times to obtain a good signal-to-noise ratio. In the first experiment, each data point of the QFI is calculated from the measured photon counts (L x , L y , L z , L 0 , and L 1 ) using Eqs. (5) (6) (7) . After calculating the QFI, we smooth the data by averaging 5 data points around each specific point (the adjacent average smoothing method) to calculate the QFI flow. In the second experiment, each data point of the QFI is obtained from the full 4×4 density matrix of two qubits. The final density matrix is extracted from the measured photon counts, according to the two-qubit state tomography technique, and optimised by a maximum likelihood estimation (MLE) to reduce measurement errors. 

A ten-qubit solid-state spin register with quantum memory up to one minute

Generation and manipulation of Schrödinger cat states in Rydberg atom arrays

Generation of multicomponent atomic Schrödinger cat states of up to 20 qubits

Quantum simulation

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The theory of open quantum systems

Decoherence of quantum superpositions through coupling to engineered reservoirs

Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems

Local detection of quantum correlations with a single trapped ion

Controllable non-Markovianity for a spin qubit in diamond

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Detecting non-Markovianity via quantified coherence: Theory and experiments

Quantum Fisher information flow and non-Markovian processes of open systems

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Quantum metrology in non-Markovian environments

Metrology with PTsymmetric cavities: Enhanced sensitivity near the PT -phase transition

Colloquium: Non-Markovian dynamics in open quantum systems

General non-Markovian dynamics of open quantum systems

Quantum metrology with nonclassical states of atomic ensembles

Statistical distance and the geometry of quantum states

Quantum spin squeezing

Entanglement and non-Markovianity of quantum evolutions

Measure for the degree of non-Markovian behavior of quantum processes in open systems

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Dynamic polarization of single nuclear spins by optical pumping of nitrogen-vacancy color centers in diamond at room temperature

Quantum non-Markovianity: Characterization, quantification and detection

Demonstration of entanglement-enhanced phase estimation in solid

Multipartite entanglement and high-precision metrology

Entanglement, nonlinear dynamics, and the Heisenberg limit

Characterization of topological states via dual multipartite entanglement

Nanofabricated solid immersion lenses registered to single emitters in diamond

13 C hyperfine interactions in the nitrogen-vacancy centre in diamond

High-resolution spectroscopy of single NV defects coupled with nearby 13 C nuclear spins in diamond

Noiseresilient quantum evolution steered by dynamical decoupling

Decoherence-protected quantum gates for a hybrid solid-state spin register

Acknowledgements We would like to acknowledge Zi-Yong Ge and Qing Ai for helpful discussions, and Da Wei Lu for the codes of maximum likelihood estimation.