Dynamics revealed by correlations of time-distributed weak measurements of a single spin

We show that the correlations in stochastic outputs of time-distributed weak measurements can be used to study the dynamics of an individual quantum object, with a proof-of-principle setup based on small Faraday rotation caused by a single spin in a quantum dot. In particular, the third order correlation can reveal the"true"spin decoherence, which would otherwise be concealed by the inhomogeneous broadening effect in the second order correlations. The viability of such approaches lies in that (1) in weak measurement the state collapse which would disturb the system dynamics occurs at a very low probability, and (2) a shot of measurement projecting the quantum object to a known basis state serves as a starter or stopper of the evolution without pumping or coherently controlling the system as otherwise required in conventional spin echo.

The standard von Neumann quantum measurement may be generalized in two aspects. One is measurements distributed in time [1,2], continuously or in a discrete sequence, as in the interesting Zeno [1] and anti-Zeno effects [3]. Timedistributed measurements intrinsically interfere with the evolution of the quantum object [2]. Another generalization is weak measurement in which the probability of distinguishing the state of a quantum object by a single shot of measurement is much smaller than one [4,5,6,7,8]. On the one hand, weak measurement has very low information yield rate; on the other hand, it only rarely disturbs the dynamics of a quantum object by state collapse. As a combination of the two generalizations, time-distributed weak measurements have been used to steer the quantum state evolution [9]. In this paper, we show that the statistical analysis of time-distributed weak measurements may be used to study the dynamics of a quantum object [8]. The outputs of time-distributed measurements bear the stochastic nature of quantum measurements, so the standard noise analysis in quantum optics [10] would be a natural method to be applied. Notwithstanding that, we should emphasize that the stochastic output of time-distributed weak measurement is not the noise in the system, but an intrinsic quantum mechanical phenomenon. Revealing quantum dynamics by correlations of time-distributed weak measurements is complementary to the fundamental dissipation-fluctuation theorem which relates correlations of thermal noises to the linear response of a system [11,12,13,14].
To demonstrate the basic idea, we consider the monitoring of coherent Lamor precession and decoherence of a single spin in a quantum dot, which is relevant to exploiting the spin coherence in quantum technologies such as quantum computing [15,16,17,18]. The difficulty of studying the spin decoherence lies in the fact that the "true" decoherence due to quantum entanglement with environments is often concealed by the rapid "phenomenological" dephasing caused by inhomogeneous broadening in ensemble measurements (e.g., in a typical GaAs quantum dot, the spin decoherence time is ∼ 10 −6 sec, but the inhomogeneous broadening dephasing time is ∼ 10 −9 sec [16,17,18,19,20,21]). Note that many single-spin experiments are still ensemble experiments with temporal repetition of measurements. To resolve the spin decoherence excluding the inhomogeneous broadening effect, spin echo [16,19,21,22,23] and mode-locking of spin frequency [18] have been invoked. In this paper, we will show that the spin dynamics can be revealed in correlations of the stochastic outputs of sequential weak probes. In particular, the third order correlation singles out the "true" spin decoherence. Unlike conventional spin echo, the present method involves no explicit pump or control of the spin but uses the state collapse as the starter or stopper of the spin precession.
We design a proof-of-principle setup (see Fig. 1) based on Faraday rotation, which has been used in experiments for spin measurements [18,20,21,24,25]. The probe consists of a sequence of linearly polarized laser pulses evenly spaced in delay time τ. After interaction with a single spin (in a quantum dot, e.g.), the light polarization is rotated by θ or −θ for the spin state parallel or anti-parallel to the light propagation direction (z-axis). The Faraday rotation angle θ by a single electron spin is usually very small (∼ 10 −6 rad in a quantum dot [24,25]), so the two polarization states of the light corre- sponding to the two different spin states are almost identical. Thus a detection of the light polarization is a weak measurement of the spin, as long as the number of photons per pulse is not too large (see discussions following Eq. (3) for details). The light polarization is detected by filtering through a polarized beam splitter (PBS) which is aligned to let the light with polarization rotated by θ fully pass through and the light with orthogonal polarization be fully reflected. The light with Faraday rotation angle −θ is reflected with probability sin 2 (2θ). For a small θ, the average number of reflected photons is much less than one, so in most cases, a single-photon detector set at the reflection arm would be idle with no clicks and one cannot tell which state the spin could be in. The clicks of the detector form a stochastic sequence. The correlations in the sequence will be analyzed to study the spin dynamics, such as the precession under a transverse magnetic field and the decoherence. This proof-of-principle setup, being conceptually simple and adapted from existing experiments, is of course not the only possible implementation. For example, one can use continuous-wave probe instead of pulse sequences, interferometer measurement of the polarization instead of the PBS filtering, polarization-selective absorption instead of the Faraday rotation, and so on. We shall derive from quantum optics description of the spin-light interaction a weak measurement theory in the formalism of positive operator value measure (POVM) [1,26]. Consider a laser pulse in a coherent state |α, H/V creates a photon with linear polarization H or V) and a spin in an arbitrary superposition C + |+ + C − |− in the basis quantized along the z-axis, the initial spin-photon state is After interaction, the state becomes an entangled one as where |α, ±θ ≡ e αa † ±θ −h.c. |0 (with a ±θ ≡ a H cos θ ± a V sin θ) is a photon coherent state with polarization rotated by ±θ. How much the spin is measured is determined by the distinguishability between the two polarization states When the average number of photonsN = |α| 2 ≫ 1 and the Faraday rotation angle θ is not too small, the two coherent states are almost orthogonal and D → 1, thus a detection of the light polarization provides a von Neumann projective measurement of the spin. For a single spin in a quantum dot, the Faraday rotation angle θ is usually very small. For example, in a GaAs fluctuation quantum dot [24], |θ| ∼ 10 −5 rad for light tuned 1 meV below the optical resonance with a focus spot area ∼ 10 µm 2 . The number of photons in a 10 picosecond pulse with power 10 mW isN ∼ 0.5 × 10 6 . In this case, D 4Nθ 2 ∼ 2 × 10 −4 ≪ 1, the spin states are almost indistinguishable by the photon polarization states. After interaction with the spin, the laser pulse is subject to the PBS filtering which transforms the spin-photon state to be where |β t/r denotes a coherent state of the transmitted/reflected mode with amplitude β. Separating the vacuum state |0 r from the reflected mode and keeping terms up to a relative error O θ 2 , we write the state as where |α sin(2θ) ′ r denotes the (normalized) state of the reflected mode but with the vacuum component dropped. With a probability P 1 = D |C − | 2 ≪ 1, an ideal detector at the reflection arm will detect a photon-click and the spin state is known at |− , while in most cases (with probability P 0 = 1 − P 1 ), the detector will be idle and the spin state becomes C + |+ + √ 1 − DC − |− (up to a normalization factor), which is almost undisturbed by the measurement [since the overlap between the state before the measurement and the state after the measurement is |C In the POVM formalism [1,26], the Kraus operators for the click and no-click cases are respectively, which determine the (non-normalized) post-measurement stateM 0/1 |ψ and the probability P 0/1 = ψ M † 0/1M 0/1 ψ . Between two subsequent shots of measurement, the spin precession under a transverse magnetic field (along xdirection) is described by, whereσ x is the Pauli matrix along the x-direction, and ω is the Larmor frequency. Coupled to the environment and subject to dynamically fluctuating local fields, the spin precession is always accompanied by decoherence. For simplicity, we consider an exponential coherence decay characterized by a decoherence time T 2 . In the quantum trajectory picture [5,10], the decoherence can be understood as a result of continuous measurement by the environment along the x-axis, for which the Kraus operators for the quantum jumps with and without phase flip are respectively [26] where γ ≡ 1 − exp (−τ/T 2 ) τ/T 2 is the coherence lost between two subsequent measurements. For a spin state described by a density operatorρ, the decoherence within τ leads the state toÊ [ρ] ≡Ê 0ρÊ † 0 +Ê 1ρÊ †

.
To study the spin dynamics under sequential measurement, we generalize the POVM formalism for a sequence of n measurement. To incorporate the spin decoherence in the density operator evolution, we define the superoperators for the weak measurement and the free evolution asM 0/1 [ρ] =M 0/1ρM † 0/1 , U [ρ] =ÛρÛ † , in addition to the decoherence superoperator E defined above. For a sequence output X ≡ [x 1 x 2 · · · x n ] as a string of binary numbers, the superoperator, transforms an initial density operatorρ toM X [ρ] (not normalized) and determines the probability of the output P X = Tr M X [ρ] . With the POVM formalism, the spin state evolution under sequential measurement and hence the noise correlations discussed below can be readily evaluated.
To illustrate how a real experiment would perform, we have carried out Monte Carlo simulations of the measurement with the following algorithm: (1) We start from a randomly chosen state of the spin |ψ ; (2) The state after a free evolution isÛ|ψ ; (3) Then the decoherence effect is taken into account by applying randomly the Kraus operatorÊ 0 orÊ 1 to the state (with normalization) with probability 1 − γ/2 or γ/2, respectively; (4) The measurement is done by randomly applying the Kraus operatorM 0 orM 1 to the state (with normalization) corresponding to the output 0 or 1 (no-click or click), with probability P 0 or P 1 given by the POVM formalism.
Step (2)-(4) are repeated for many times. The output is a random sequence of clicks, as shown in Fig. 2 (d).
To study the correlation of the stochastic output, we first consider the interval distribution K(n), defined as the probability of having two clicks separated by n − 1 no-clicks [10], where 0 n−1 means a string of n − 1 zeros. By a straightforward calculation, up to O γD 2 and O nD 3 , for γ, D ≪ ωτ < π. A successful measurement at the beginning of an interval projects the spin to the basis state |− along the optical (z) axis. Then, the spin precesses under the external magnetic field about the x-axis. The interval is terminated by a second successful measurement among the periodic attempts after a time lapse of nτ.
The decay of the oscillation is due to the spin decoherence. The overall decay e −nD/2 is due to decreasing of the probability of unsuccessful measurement with increasing time. The measurement also induces a little phaseshift to the oscillation. Obviously, the smaller the distinguishability D, the less the spin dynamics is disturbed by the measurement. In experiments, often the photon coincidence correlation instead of the interval distribution is measured. The second order correlation g (2) (nτ) is the probability of having two clicks separated by n−1 measurements [10], regardless of the results in between, By Fourier transformation and summation in the frequency domain, The spin precession, the decoherence, and the measurementinduced decay are all seen in the second order correlation function [see Fig. 2]. Note that the overall decay of the interval distribution manifests itself as a measurement-induced dephasing of the oscillating signal in the correlation function. The Monte Carlo simulation shows that 10 10 shots of measurement would yield a rather smooth profile of the spin dynamics, which requires a time span of about 3 seconds for the parameters used in Fig. 2.
In addition to the decoherence due to the dynamical fluctuation of the local field, there is also phenomenological dephasing due to static or slow fluctuations, i.e., inhomogeneous broadening which exists even for a single spin since the sequential measurement contains many shots which form an ensemble. The inhomogeneous broadening is modeled by a Gaussian distribution of ω with mean value ω 0 and width σ. With the inhomogeneous broadening included, the ensemble-  t 2 ), with parameters the same as for Fig. 2 (c). The insets (a) and (b) show the oscillation details of G(t 1 , t 2 ) in the range 0 ns≤ t 1,2 ≤ 30 ns and 90 ns≤ t 1,2 ≤ 120 ns, respectively. averaged correlation function becomes Since usually σ ≫ T −1 2 , the decay of the 2nd order correlation is dominated by the inhomogeneous broadening effect [see Fig. 2 (c)].
To separate the spin decoherence from the inhomogeneous broadening, we resort to the 3rd order correlation g (3) (n 1 τ, n 2 τ), the probability of having three clicks separated by n 1 − 1 and n 2 − 1 measurements. The idea can be understood in a post-measurement selection picture: After the first click, the second click has the peak probability appearing at an integer multiple of the spin precession period, so the coincidence of the two earlier clicks serves as filtering of the spin frequency and the third click would have a peak probability appearing at n 2 τ = n 1 τ, similar to the spin echo. The 3rd order correlation in the absence of inhomogeneous broadening is g (3) (t 1 , t 2 ) ∝ g (2) (t 1 )g (2) (t 2 ). The ensemble-average leads to e −(T −1 2 +τ −1 D/4)(t 1 +t 2 ) e −σ 2 (t 1 −t 2 ) 2 /2 cos ω 0 (t 1 − t 2 ) . (15) Fig. 3 plots G (3) (t 1 , t 2 ) ≡ g (3) (t 1 , t 2 ) − g (2) (t 1 ) g (2) (t 2 ) to exclude the trivial background. Along the direction t 1 = −t 2 , the 3rd order correlation oscillates and decays rapidly (with timescale σ −1 ). But the oscillation amplitude decays slowly (with timescale T 2 ) along the direction t 1 = t 2 , as expected from the last term of Eq. (15).
In conclusion, we have given a statistical treatment of sequential weak measurements of a single spin. The character-istics of the weak measurement consist in the negligible perturbation of the spin state except for the projective state collapse when the measurement is successful in identifying the spin state. We show that the third order correlation reveals the spin decoherence from the inhomogeneous broadening. The theory presented here for sequential pulse measurement can be straightforwardly generalized to continuous weak measurement by letting the pulse separation τ → 0 while keeping the average power of the light unchanged (i.e., D/τ = constant). In the proof-of-principle setup based on Faraday rotation, all optical elements have been assumed ideal for conceptual simplicity. An investigation of the defects, e.g., in the PBS and in the photon detector, shows that they do not change the essential results presented here but only reduce the visibility of the features. Details will be published elsewhere.