Transition from quantum to classical dynamics in weak measurements and reconstruction of quantum correlation

The ability to track and control the dynamics of a quantum system is the key to quantum technology. Despite its central role, the quantitative reconstruction of the dynamics of a single quantum system from the macroscopic data of the associated observable remains a problem. We consider this problem in the context of weak measurements of a single nuclear carbon spin in a diamond with an electron spin as a meter at room temperature, which is a well-controlled and understandable bipartite quantum system. In this work, based on a detailed theoretical analysis of the model of the experiment, we study the relationship between the statistical properties of the macroscopic readout signal of the spin of a single electron and the quantum dynamics of the spin of a single nucleus, which is characterized by a parameter associated with the strength of the measurement. We determine the parameter of measurement strength in separate experiments and use it to reconstruct the quantum correlation. We control the validity of our approach applying the Leggett-Garg test.

Figure 1

Scheme of the experiment. (a) The sequential weak measurements $M_i$ are composed of a sensor initialization part, dynamical decoupling (KDD-XYn), and readout of the sensor state using conventional optical readout of an NV center electron spin. The KDD-XYn filter function is tuned via pulse spacing $\tau$ to the Larmor precession of the weakly coupled ($A_{zz}\approx 100\mathrm{Hz}$) ${}^{13}\mathrm{C}$ nuclear spin, which results in an effective interaction between nuclear and electron spin. The electron spin state is read out after each interaction, which leads to extraction of information about the nuclear spin and its back-action. (b) Schematic evolution trajectory of an electron spin state in the case the nuclear spin is in the $I_e+I_x$ state (c) Schematic evolution of the nuclear spin during sequential measurements. Initially, the nuclear spin is in the thermally mixed state ${\rho }_0=I_e$, which is then partially polarized by measurements along the $x$ axis with magnitude $sin\alpha$. The free precession between the measurements leads to a rotation around the $z$ axis, which is perpendicular to the figure plane. Subsequent measurements affect the measurement by disturbing both the $x$ and $y$ component of the Bloch vector, conditioned on the measurement outcome (see text).

Figure 2

Externally applied classical field and calibration of fluorescence readout. (a) Experimental protocol for modulation assisted method for determining the fluorescence response of the NV spin readout $n_a$ and $n_b$. Sequential measurements with optical readout of the electron spin yield phase information obtained by the interaction with the external signal. The readout $\pi /2$ pulse phase is sinusoidally modulated with an amplitude of $\pi /2$ and a period of eight measurement cycles. (b) The empirically calculated correlation of the centered photon count number trace. The beating in the correlation originates from the presence of two frequencies. The size of the beating is determined by the relative amplitude of the external signal to phase modulation. The solid curve is the best fit of the analytical model of the correlation function (2.16), which includes phase modulation and the unknown external signal. (c) Measurement protocol for the estimation of the classical signal correlation function. (d) Reconstructed correlation function of the classical sinusoidal signal with a stochastic phase.

Figure 3

Reconstruction of quantum correlation function and calibration of fluorescence readout. (a) Experimental protocol for the modulation-assisted method for determining fluorescence response of NV spin readout $n_a$ and $n_b$. Sequential measurements with optical readout of electron spin results in idle measurements as the DD filter function is detuned from ${}^{13}\mathrm{C}$ Larmor frequency. The readout $\pi /2$ pulse phase is sinusoidally modulated with amplitude $\pi /2$ and a period of eight measurement cycles. (b) Calibration of $n_a$ and $n_b$ for optical spin readout of the NV center electron spin. The dependence of average photon counts on angle ${\varphi }_k$ reveals the $n_a$ and $n_b$ photon counts. The model analytical curve [see (2.19)] shows that the response of the NV center depends on $n_a$ and $n_b$. Measurement scheme [Fig. 3] operates at ${\varphi }_k=90$. (c) Measurement scheme of the nuclear spin correlation function, operating at ${\varphi }_k=90$. (d) Empirically estimated correlation of sensor outputs. The solid curve is the best fit of the analytical model for the correlation function, which includes back-action-induced decay of the initial amplitude accounting for a single unknown parameter $\alpha$. (e) Reconstructed correlation function of the quantum signal.

Figure 4

L-G functions for the classical and quantum processes. (a) L-G function for the reconstructed autocorrelation of the classical r.f. signal. (b) NV2 measurements, L-G function for empirical autocorrelations normalized only by ${sin}^2\alpha$. (c) NV2 measurements, L-G function for empirical autocorrelations normalized by ${sin}^2\alpha$, and dephasing factor. (d) NV2 measurements, the result of averaging L-D functions over five experiments with different control sequences.

Figure 5

The influence of the charge state dynamics on the $I_x$ evolution. (a) The evolution of the $I_x$ has smaller decay, which is shown by the red dots. Panel (b) shows a recovered L-G function using both data sets, including the value of ${\alpha }_{est}$ from the full data set and from the shortened $I_x$ to $N=30$. The cut plays important role, due to the less importance of the actual decay values.

Figure 6

External field amplitude calibration. (a) Absorption profiles of the KDD-XY5 sequence as a function of interpulse spacing $\tau$ for various signal amplitudes. (b) The extracted parameter $\alpha$ of the classical signal as a function of signal power on the r.f. generator.

Figure 7

(a) Sequential weak measurements of target spins with the prior polarization scheme. The sequence contains the Pulse-Pol routine as polarization, radio frequency (r.f.) $\pi /2$ pulse for initiating free precession of ${}^{13}\mathrm{C}$ target spin, and sequence of weak measurements including the KDD-XY3 sequence for NV2 and ${}^{14}\mathrm{N}$ polarization. (b) Average electron spin $S_z$ in the series of measurements $M_0,\cdots ,M_{N-1},M_N$. Orange uncertainty area is a confidence interval of the fitting model. The uncertainty of the initial phase adds additional systematic parameter for calibration in this case, which is not present in the nonpolarized case. (c) The L-G expression of recovered $I_x$ component of the nuclear spin. Note that in this case we took $\langle I_x\rangle$ as a K function of the L-G functional. (d) The L-G functional from the $I_x$ signal divided by the exponential decay with $exp(-{\alpha }^{2}N/4)$. The error bars are also divided by exponential decay and express how well our model of an effective Hamiltonian describes the obtained signal. This view shows that the signal behaves according to the model for several intervals where inequality is violated. The inset shows the integral points of L-G inequality violation corresponding to the violation points on the main chart (first four points), which gives stronger confidence in violation. (e) Depiction of the initial phase with respect to the rotation frame of the nuclear spin drive on the Bloch sphere.

Figure 8

Histogram of the photon count differences $\mathrm{\Delta }n$ in single-shot readout with referenced probing using two microwave $\pi$ pulses, conditioned to $m_i=0$ and $m_i!=0$. In such a case the threshold is positioned on the $\mathrm{\Delta }n=0$.

Figure 9

Result of five series of main experiments on NV2 with KDD-XY5 and $N_{RR}=200$ repetitive readouts using ${}^{14}\mathrm{N}$ as an ancillary memory.

Figure 10

Choice of NV-${}^{13}\mathrm{C}$ pair. (a) Decay rate of the ${}^{13}\mathrm{C}$ correlation signal in a sequence presented in Fig. 7 vs total sequence time $t_s$ normalized by ${\alpha }^2$ (measurement strength). Various ${}^{13}\mathrm{C}$ possess a plateau, which was fitted and extracted as a natural ${\mathrm{\Gamma }}_2$ rate for each ${}^{13}\mathrm{C}$ at corresponding NV. (b). Dependence of the ${}^{13}\mathrm{C}$ decay rate as a function of the $A_{zz}$ coupling of the corresponding ${}^{13}\mathrm{C}$. (c), (d) Map of ${}^{13}\mathrm{C}$ relative positions to their NV center. Note NV2 is located close to the magic-angle latitude with close to zero $A_{zz}$. (e), (f) The $S_z$ correlation results for two NV11 and NV2, respectively. The signals show a clear difference in signal-to-noise ratio. The reason for that is overall better spin readout contrast of fluorescence for NV2, since the ${}^{14}\mathrm{N}$ is polarized as well as reduced decay of the correlation signal due to vanishing $A_{zz}$ coupling. Nonpolarized ${}^{14}\mathrm{N}$ also leads to a shift in the readout phase of the KDD sequence, which ultimately reduces the contrast and even could change the underlying dynamics of the target spin.

Figure 11

Experimental setup scheme. A stands for a wedged mirror, B is a pinhole, C is a long pass filter 650 nm, D is the 3D piezo nanopositioner.

Figure 12

Scheme of the diamond sample and NV center.

Figure 13

Pulsed-ODMR spectra of the single NV center for $m_s=0$ to $m_s=-1$ electron spin transition. ${}^{14}\mathrm{N}$ nuclear spin hyperfine splitting of $2.2\mathrm{MHz}$ is visible. No visible hyperfine coupling for ${}^{13}\mathrm{C}$ within $T_2^{*}$; $\pi$ pulse is $8\mu \mathrm{s}$.

Figure 14

ENDOR spectroscopy of two NV centers. NV5 is an example with detectable nuclear spin coupling for total interrogation time of $100\mu \mathrm{s}$. NV2 is the NV center used in the current research without detectable $A_{zz}$ coupled nuclear spin within the $100\mu \mathrm{s}$ total interrogation time.

Figure 15

High-order KDD-XY30 (totally $600\pi$ pulses) (maximum contrast) spectroscopy of the nuclear spin bath. No trace of $A_{zz}$ coupling, though there is clear indication of negative contrast, caused by coherent interaction via the $A_{zx}$ terms.

Figure 16

Zoom into the 21st resonance of the KDD-XY30 sequence for the NV2. The resonance reveals no $A_{zz}$ coupled nuclear spins and allows for determination of Larmor frequency with higher precision.

Figure 17

DD resonance position in microseconds as a function of its order.

Figure 18

Demonstration of coherent coupling in NV2. Dependence of the evolution on number of pulses in the KDD-N sequence shows the coherent oscillations for NV2, while for other NVs without a signal in the ENDOR sequence, the decay results in loss of coherence, meaning that the $A_{zx}$ coupling is also weak.

Figure 19

2D image of KDD numbers vs the interpulse time. The image shows a coherent evolution of the single weakly coupled nuclear spin. The resonance to the bath is also visible, resulting in dissipative dynamics.

Figure 20

Precise determination of resonant $\tau$ of the DD sequence. Left: dependence of the decay of the correlation function on the interpulse time $\tau$. Right: the no-decay correlation function in the region of the phase-matching condition.

Figure 21

Free precession angle sweeping, and phase-locking effect (left). The decay of the autocorrelation vanishes as the phase acquired by the target spin between successive measurements approaches $\pi$.

Figure 22

Decay of the correlation function for KDD1, 2, 3, 5.

Figure 23

Decay of the correlation function as a function of the measurement strength.