Abstract
A system under constant observation is practically freezed to the measurement subspace. If the system driving is a random classical field, the survival probability of the system in the subspace becomes a random variable described by the Stochastic Quantum Zeno Dynamics (SQZD) formalism. Here, we study the time and ensemble average of this random survival probability and demonstrate how time correlations in the noisy environment determine whether the two averages do coincide or not. These environment time correlations can potentially generate nonMarkovian dynamics of the quantum system depending on the structure and energy scale of the system Hamiltonian. We thus propose a way to detect time correlations of the environment by coupling a quantum probe system to it and observing the survival probability of the quantum probe in a measurement subspace. This will further contribute to the development of new schemes for quantum sensing technologies, where nanodevices may be exploited to image external structures or biological molecules via the surface field they generate.
Introduction
The dynamical evolution of a quantum system is always influenced by its environment^{1,2}. Since one is very often only interested on the system dynamics, the environmental degrees of freedom are traced out and, in the Markovian regime (under the assumption of only very shortlived correlations), this leads to the wellknown KossakowskiLindblad master equation^{3}. As a consequence, this approximation does not take into account all the environmentinduced memory effects, which may produce a back flow of information onto the quantum system^{4,5}. However, the environment is usually unknown and very hard to be characterized. Therefore, in the last few years several theoretical proposals for the characterization of the environment according to whether it can generate Markovian or nonMarkovian dynamics of the system to which it is coupled have been put forward, and a full hierarchy of nonMarkovianity^{6} has been introduced. In particular, it has been shown that classical environments exhibiting timecorrelated random fluctuations can lead to nonMarkovian quantum dynamics^{7,8}. The structure of such an environment can be probed by coupling a (typically small, e.g. one qubit) quantum system (probe) of known dynamics to it, and studying the effect of the environment on the system dynamics. Indeed, the very recent idea of the socalled quantum probes is that their fragile properties, as coherence and entanglement, are strongly affected by the environment features and can be used to detect them. Examples of such physical systems are quantum dots, atom chips and nitrogen vacancy centers in diamond where a good control over the system has been proposed and recently achieved^{9,10,11,12,13,14,15,16,17}. They can be used to probe environments like biological molecules or surfaces of solid bodies or amorphous materials. In this regard, a number of nonMarkovianity measures and witnesses has been proposed, such as geometric measures (i.e. measures based on the geometry of the space of quantum maps), quantities based on the Helstrom matrix (i.e. based on the distinguishability of two states under evolution and observation), or witnesses based on the (non)monotonicity of entanglement measures^{6}. Most of them, however, rely on a full state tomography and, thus, are experimentally difficult to be implemented. An experimentally feasible tool for certain systems is based on the state distinguishability and the Loschmidt echo^{18}.
Recently, the scenario of stochastic measurement sequences has been proposed^{19}, and then studied with a particular focus on the probability for the system (survival probability) to remain confined within a given measurement subspace^{20,21,22}. Indeed, when the time interval between two measurements is random, this survival probability becomes a random variable by itself, and it has been shown by large deviation theory^{23,24,25} that it converges to its most probable value, by increasing the number of the measurements performed on the system^{20}. When the measurements become very frequent, the survival probability increases and a stochastic quantum Zeno regime is accessed^{20,26}. It is the stochastic generalization of quantum Zeno dynamics (QZD), where in the limit of infinitely frequent observation the dynamics of a quantum system is freezed to a unidimensional^{27} or multidimensional^{28,29} subspace of the measurement operator. QZD has been experimentally realized first with a rubidium Bose–Einstein condensate in a fivelevel Hilbert space^{30}, and later in a multilevel Rydberg state structure^{31}. Furthermore, a recent theoretical study and experimental demonstration with atomchips has shown also how different statistical samplings of a randomlydistributed sequence of projective measurements coincides in the quantum Zeno regime, proving an ergodicity hypothesis for randomly perturbed quantum systems^{21}. In this regard, the sensitivity of the survival probability to the stochasticity in the time interval between measurements has been properly analyzed by means of the Fisher information^{22}.
In this work, we propose a method based on the Stochastic Quantum Zeno Dynamics (SQZD)^{20,26} to detect time correlations in random classical fields. Indeed, we use the SQZD formalism to study a quantum system (the probe), subjected to a sequence of equally spaced projective measurements, interacting with an environment modelled by a randomly fluctuating field. Then, the random value of the field leads to a random value of the survival probability in the measurement subspace. In this way, a witness for the environment time correlations can be obtained without characterizing the nonMarkovianity of the quantum probe, by properly analyzing the behaviour of the time and ensemble average of this fluctuating survival probability with respect to the amount of noise temporal correlations. As outline, we first introduce our model of a quantum system coupled to the environment. Then, we review and adapt the SQZD formulation, and show how time correlations in the fluctuating field correspond to different statistical sampling of the random measurements. Finally, we demonstrate for random telegraph noise^{32,33} the imprint of the time scale of the correlated noise on the final survival probability after applying the entire measurement sequence.
Model
Stochastic Schrödinger Equation
We study a quantum system that is coupled to a bath that effectively acts on the system via a time fluctuating classical field Ω(t) as
where H_{0} is the Hamiltonian of the unperturbed system, while H_{noise} describes the coupling of the environment to the system. In other words, we are modelling the classical environment as a random field represented by the stochastic process Ω(t). We assume that Ω(t) takes real values with mean , and ω(t) is the fluctuating part with vanishing mean value. Figure 1 shows an exemplary twolevel system initially prepared in the ground state 0〉. The random Hamiltonian driving term causes a population transfer to the upper level 1〉. This can be probed by measuring the remaining population in 0〉. The system dynamics for a given realization of the random field Ω(t) is described by the standard Schrödinger equation. If we average over the statistics of the field Ω(t), we find the following master equation:
where 〈ω(t)ω(τ)〉 is the secondorder time correlation of the random field, and [·, ·] represents the commutator. The reduced Planck’s constant ħ is set to unity. For white noise the secondorder time correlation is a Dirac delta distribution, i.e. , and we find the LindbladKossakowski master equation^{3}. In general, the memory kernel can lead to nonMarkovian dynamics depending on the structure and time scale of the Hamiltonian, as for example demonstrated for random telegraph noise (RTN) and 1/fnoise^{7,8}. Indeed, also a Markovian stochastic process Ω(t) (as in the case of RTN, where the time correlation is ) can lead to nonMarkovian dynamics of the quantum system it is coupled to. In particular, in ref. 7 the quantum dynamics of the probe system, a qubit subject to RTN, can be Markovian or nonMarkovian depending on the parameters of the nonfluctuating Hamiltonian term H_{0}. This proves that Markovianity or nonMarkovianity are not a feature of just the noise but of the dynamics of the system coupled to the noise. Thus, in this work we want to analyze just the time correlations of the environment, that are independent from the system coupled to it.
We now consider a system under sequential measurement where each measurement (with projection operator П) occurs after a fixed time interval μ. We call q(Ω) the single measurement quantum survival probability that will depend on the value of Ω during this time interval and, thus, be a random variable. We can now generalize the survival probability to the stochastic process as follows
where α = 1, … N labels the realization of a trajectory, j represents the time order of the m measurements, and Ω_{j,}_{α}(t) is the fluctuating field in this corresponding time interval. The single measurement quantum survival probability q(Ω_{j,}_{α}), moreover, is defined as , where is the state of the system. In order to characterize the stochastic process P_{α}(m), two natural quantities arise: the time average and the ensemble average of the survival probability. The timeaverage is defined here as
The idea is that, using the measured value of the survival probability after the jth measurement, one can estimate the expectation value at m by . We can then average this value for j = 1…M and take the limit of a large number of measurements M. This limit potentially depends on the realization α of the fluctuating field, as will be discussed below. The ensembleaverage is instead defined as
where the average of P_{α}(m) is over a large number of realizations N. In the limit of infinite realizations, does not depend on the single realization but on their probability distribution. In the following section, we examine the behaviours of the time and ensemble averages of the survival probability P_{α}(m), and in particular we study how correlations in the field fluctuations influence these averages.
Results
For each realization α of the stochastic process we characterize the fluctuating field in between two measurements by a constant value Ω_{j,α}(t) → Ω_{j,α} distributed according to a random distribution p(Ω). This is a valid formulation also for more complicated fluctuations, e.g. when the unitary dynamics is governed only by the fluctuating field, i.e. H_{0} = 0. In the latter case, the single quantum survival probabilities to survive in the initial state ψ_{0}〉, thus, become
where . Hence, q depends just on the constant with μ being the length of the time interval between two measurements. Note that for simplicity we chose the initial state ρ_{0} = ψ_{0}〉〈ψ_{0} to be pure. However, the following main results depend just on the statistics of q(Ω) and not on the actual dependence of q on Ω. Thus, for nonvanishing H_{0} we treat Ω just as a parameter that describes the statistics of q(Ω) via the probability distribution p(Ω). Indeed, in the remainder of this section we do not specify H_{0}, H_{noise}, П, ρ_{0} or μ but use just p(Ω) and q(Ω).
Figure 2 shows in the right upper panel how the classical fluctuating field Ω causes the survival probability P to decrease at a fluctuating rate, such that a stronger average driving strength within one time interval causes a smaller q and a faster decrease of P. Within each time interval between two measurements the decrease of P is quadratic in the time interval and the field strength. While the field fluctuations are random (and independent identically distributed (i.i.d.)), after a few measurements the influence of these fluctuations on P are averaged out and the decay of P behaves similarly for each realization. When the field fluctuations are correlated, however, the decay of the survival probability depends much stronger on the realization because the probability distribution for Ω_{j+1,α} depends on the value of Ω_{j,α} (and potentially also on the previous history). This means that also the convergence of the timeaverage to its limit value is much slower, since a random deviation will influence not only a single time interval but a range of them, corresponding to the relaxation time τ_{c} associated to the time correlations. Now, we consider a simple correlation model inspired by random telegraph noise (RTN)^{32,33}: we choose Ω_{j+1,α} according to the distribution p(Ω) only with a certain probability , and Ω_{j+1,α} = Ω_{j,α} otherwise. This update probability can be associated to a temperature T by (see Fig. 2), and determines how strong the time correlations are. Indeed, the average time between two field switches is , while for RTN this time equals the relaxation time τ_{c}. The physical interpretation of this RTN environment is that the field value changes when, for example, a charge is trapped, and by thermal fluctuations a trapping energy barrier E has to be overcome for the charge to be released, such that the field value is restored to its previous value. In Fig. 2 temperature grows from left to right yielding different types of disorder. For T = 0, one has , i.e. the value of the field Ω is chosen only once randomly and then always remains the same. The relaxation time τ_{c} is infinite and the timeaverage does not always converge to the same value. This scenario simulates the interaction of the system with an environment that exhibits quenched disorder. Depending on the value of Ω of the given realization α, the decay of the survival probability P_{α}(m) can be faster or slower. On the other side, for infinite temperature we have , representing an annealed disorder environment. Between these two extreme regimes, i.e. for finite temperature, we have , hence a mixture of both behaviours. Here, quenched disorder means a scenario with a static noise that depends on the initial random configuration of the environment, whereas annealed disorder means that the environment changes its configuration randomly in time^{34,35,36}.
Time and Ensemble Averages vs. Noise Correlations
For the timeaverage we introduce the expected frequencies mn_{Ω} with which the event Ω occurs in one realization of a stochastic sequence of m measurements. The timeaverage is then given by
where the product is over all possible values of Ω and n_{Ω}. For independent (thus uncorrelated) and identically distributed random variables Ω_{j,α} the expected frequencies correspond to the underlying probability distribution n_{Ω} = p(Ω). For correlated Ω_{j,α} the convergence of the timeaverage might not be unique or not even exist. This is linked to the Markov property and recurrence of the stochastic process^{37}, as explained in more detail below by introducing the theoretical expressions for the timeaverage in different correlated dynamical regimes. Note that a Markovian stochastic process Ω(t) (or Ω_{j}, with Ω_{j} depending just on Ω_{j−1} as in our model) does not imply Markovian quantum dynamics of the system since a Markovian fluctuating field can generate nonMarkovian quantum dynamics through its time correlations^{7}.
The ensemble average of the survival probability is the expectation value of the survival probability, i.e.
where is the probability distribution of the survival probability P_{α}(m) (which is by itself a random variable depending on the field fluctuations), and is the conditional probability for the event Ω_{j} given the process history. In the case of i.i.d. random variables Ω_{j}, it becomes
Now, we consider three different regimes to calculate the time and ensemble average by varying the value of : (1) annealed disorder (), (2) a finite temperature case, with and the number of measurements m such that at least 5–10 jumps occur, and (3) quenched disorder (). We introduce the shorthand notation , that we will use frequently for A = q and A = In q.
In the case of annealed disorder (an), i.e. uncorrelated noise, the theoretical expression for the two averages follows straightforwardly from the definitions, namely
for the timeaverage and
for the ensemble average – see Eq. (8). In the case of quenched disorder (qu), each single realization has constant q(Ω) and survival probability q(Ω)^{m}. The ensemble average is, thus, the arithmetic average of these single possible outcomes, i.e.
Instead, the timeaverage for quenched disorder does not take a single value but splits into several branches with
since the underlying process is not recurrent, in the sense that, given the value of Ω in the first interval, all the other values of the support of p(Ω) cannot be reached anymore within the given realization of the stochastic process.
Finally, for the finite temperature ( fT ) regime the problem is more difficult: the timeaverage is the same as for the annealed disorder, namely
The reason is that, despite of the time correlations, after each field update event the field is chosen according to p(Ω) independently from the history of the process. Also is independent of the current value of the field and for long times the frequencies of each q(Ω) still converge to their expected values n_{Ω} = p(Ω). Instead, in order to calculate the ensemble average we have to take into account all the time correlations examining the occurrence of sequences of Ω(t) that are constant over several time intervals and the updates of the probability according to . If the length of such a sequence is labelled by k, this length k is distributed by the Poisson distribution with the inverse of the probability for an update of Ω. The survival probability for this sequence of constant field is:
The frequency of the updates is also Poisson distributed, with expectation value . The joint survival probability is then
Figure 3 shows the above calculated ensemble averages together with numerical values from the realization of N = 1000 stochastic processes for different values of . In all cases for p(Ω) we have used a bimodal distribution with p_{1} = 0.8, p_{2} = 0.2 and corresponding single measurement quantum survival probabilities q_{1} = 0.999, q_{2} = 0.9. If we decrease (increase) q_{1} and q_{2}, the decay becomes faster (slower). The same happens for an increase (decrease) of p_{2}, that is the probability associated with q_{2} < q_{1}. Note that the probabilities p_{1}, p_{2}, q_{1} and q_{2} along with do fully define the above listed averages and we do not have to specify the Hamiltonian H(t). Their values in Fig. 3 have been chosen to model a system in the Zeno regime (q_{1}, q_{2} close to 1) and with having in mind a bimodal noise field leading to two different possible values of q.
Variance of the Survival probability
The variance and standard deviation of the the distribution of the survival probability are defined as
In order to calculate the variance, we still need to calculate the second moment of the probability distribution of the survival probability. In the special case of infinite temperature or annealed disorder, it is given by
The normalized variance, thus, reads
For finite temperature, we first consider again a sequence of constant Ω, where the square of the survival probability is as follows:
The frequency of the updates of the fluctuation field is again Poisson distributed, with expectation value . Then, the joint squared survival probability is
and the normalized variance reads
Finally, for quenched disorder one has
with the normalized variance
Figure 4 shows the standard deviations (i.e. the square root of the above calculated variances, but without normalization) together with numerical values from the realization of 1000 stochastic processes for each chosen value of . The underlying distribution p(Ω) is as in Fig. 3. We find that the larger is the timecorrelation (the smaller ), the larger is the standard deviation of the survival probability , i.e. the more the outcome depends on the single realization. To average out the nonmonotonic behaviour of , we consider the accumulated standard deviation
i.e. we sum up the standard deviation values for every measurement j = 1…m. The result is shown in Fig. 5. Indeed, for relatively large values of m (>300) does monotonically increase with the amount of noise temporal correlations related to the quantity .
Discussion
If we compare the timeaverage with the ensemble one for different temperatures (i.e. ), we find that the convergence of the timeaverage (quantified by the standard deviation) as well as the expected values for the ensemble average depend on . If we consider N realizations with m measurements each, then for large numbers m and N (i.e. many measurements and many realizations) the frequency of each event q(Ω) is mNp(Ω) independently of . To calculate the timeaverage (for ) we will thus have mp(Ω) events and the value of the timeaverage is then , as shown in Eq. (14). For the ensemble average, instead, we have to average over many realizations, where each time the exponent of q(Ω) will deviate from mp(Ω) according to the (possibly timecorrelated) statistics. Because of these increasing deviations, the ensemble average for annealed disorder is larger than the timeaverage quantified in the Supplementary Information (SI). If we include time correlations, the ensemble average will grow until it takes the maximum for the quenched disorder limit, i.e. the arithmetic average of the quantity q(Ω)^{m}. As a consequence, we find that
For annealed disorder the time and ensemble averages practically coincide: we refer to this equality as an ergodic property of the systemenvironment interaction^{21}. However, the more the q(Ω_{I,α}) are correlated, the more the ensemble average moves away from the time average and the ergodicity is broken. This can be seen in Fig. 6, where time and ensemble averages are simulated for a bimodal distribution p(Ω) for quenched and annealed disorder, and for two values of finite temperature. Also for this simulation, as well as for Figs 3, 4 and 5, we have used a bimodal distribution with p_{1} = 0.8, p_{2} = 0.2 and corresponding single measurement quantum survival probabilities q_{1} = 0.999, q_{2} = 0.9. As shown in the SI, the nonergodic behavior depends essentially on the second and fourth moment of p(Ω). In other terms, for a similar average value, this effect will decrease if we choose p_{1} ≈ p_{2} or q_{1} ≈ q_{2}. The same happens if we change the bimodal distribution into a multimodal or continuous distribution. From an application point of view, this allows us to detect correlations in a fluctuating field by measuring and comparing to each other the time and ensemble averages of the survival probability. Furthermore, by changing the time interval μ between two measurements, we can explore the time scale on which these correlations occur.
In order to effectively test our method for a real quantum system, we consider the twolevel Hamiltonian
with the Pauli matrices σ_{x}, σ_{z}, a fluctuating driving Ω(t) of the system (e.g. an unstable classical light field) and a detuning term Δ. We set Δ = 2π × 5 MHz and Ω ∈ 2π × {1,5} MHz as a fluctuating RTN field with equal probability for both values. We initially prepare the system in the ground state 0〉 and perform projective measurements in this state spaced by intervals of constant length μ = 100 ns. Such scheme may be implemented on many different experimental platforms and, very recently, has been realized in the stochastic quantum Zeno context with a BoseEinstein condensate on an atomchip^{20} under similar conditions. Note that the secondorder time correlation function for the RTN is exponential in time, i.e. , where the relaxation time τ_{c} is equal to the average time between two field switches. The left panel of Fig. 7 shows how the convergence of the timeaverage for M = 2000 depends on the correlation time τ_{c}. The error bars indicate the 20th and 80th percentiles and, thus, contain 30 final values for a total of 50 realizations of the time average calculated for M = 2000. The right panel, instead, shows how the accumulated standard deviation , Eq. (25), depends on the correlation time τ_{c}. We plot the value of obtained from 1000 realizations of the stochastic process. Finally, Fig. 8 shows the time and ensemble averages along with the standard deviation, for an average time between the fluctuating field switches such as 10,10^{3},10^{5},10^{7} ns. It can be clearly seen how a noise correlation time longer than the time interval μ generates a growing standard deviation of the survival probability, which can then be exploited as a witness of timecorrelated noise.
Conclusions
By studying SQZD in timecorrelated environments we have shown how an ergodicity property quantitatively depends on the time scale of the noise correlations. By doing so, we propose a new (quantum Zenobased) way to detect time correlations in random classical fields coupled to a quantum probing system. The time correlations in the noise field determine whether and how fast the survival probability converges to its statistical mean, hence how the standard deviation of the survival probability over many experimental realizations will reveal information on the noise field. Then, we further improve this dependence by summing the standard deviation over the whole measurement series. This way we obtain a witness of time correlations in the fluctuating field coupled to the quantum probe. Let us stress that this approach can be generalized by applying different measurement operators. Indeed, it has been demonstrated that quantum Zeno dynamics allows to confine the dynamics within decoherencefree subspaces^{38,39}. By turning around that point of view, one can realize different initial states and measurement operators and thus also probe the effect of the environment on different subspaces of the system. Each of them might experience different time correlations of the noise as they predominantly couple to a different bandwidth of the noise spectrum. In conclusion we have introduced a novel method to examine time correlations and spectra of an environment acting on a quantum probing system as a random Hamiltonian term. This method does not rely on quantum state and process tomography but on a simple Zenobased measurement scheme. It is also platform independent and thus can be used in very different implementations of the physical system and frequency/time scale ranges of the noisy environment field. Therefore, these results are expected to move further steps towards novel technologies for quantum sensing, where the fragile properties of quantum systems, as coherence, and especially here Zeno phenomena are exploited to probe an environmental fluctuating field and indirectly the presence of external artificial and biological molecules that are difficult to image otherwise.
Additional Information
How to cite this article: Müller, M. M. et al. Stochastic quantum Zenobased detection of noise correlations. Sci. Rep. 6, 38650; doi: 10.1038/srep38650 (2016).
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Acknowledgements
The authors gratefully acknowledge S. Ruffo, S. Gupta, A. Smerzi, and F.S. Cataliotti for useful discussions. This work was financially supported from the Ente Cassa di Risparmio di Firenze through the project QBIOSCAN.
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M.M. and S.G. carried out the analytic calculations. M.M. performed the numerical simulations. F.C. led the theory and supervised the work. All authors contributed to the discussion, analysis of the results and the writing of the manuscript.
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Müller, M., Gherardini, S. & Caruso, F. Stochastic quantum Zenobased detection of noise correlations. Sci Rep 6, 38650 (2016). https://doi.org/10.1038/srep38650
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