Criticality of environmental information obtainable by dynamically controlled quantum probes

A universal approach to decoherence control combined with quantum estimation theory reveals a critical behavior, akin to a phase transition, of the information obtainable by a qubit probe concerning the memory time of environmental fluctuations. This criticality emerges only when the probe is subject to dynamical control. It gives rise to a sharp transition between two dynamical phases characterized by either a short or long memory time compared to the probing time. This phase-transition of the environmental information is a fundamental feature that facilitates the attainment of the highest estimation precision of the environment memory-time and the characterization of probe dynamics.

A simple quantum-probe, such as a qubit, is capable of extracting information on the environment dynamics and its space-time fluctuations through the spectrum of the dephasing noise the probe is subject to [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. This information is the subject of an emerging field of research dubbed environmental quantum-noise spectroscopy [5,6]. Its most straightforward implementation is by monitoring the freeinduction decay of an initially-prepared qubit-probe coherence and inferring the dephasing characteristics from this decay [15,16]. A more promising option is to exert a control (driving) field, whether pulsed or continuouswave (CW), on the qubit-probe and study its dephasing as a function of the control-field characteristics [4][5][6]. Pulsed control of qubit dephasing is commonly described by dynamical decoupling [17][18][19][20][21]. However, for the purpose of environment-noise spectroscopy it is useful to resort to the universal formula for the rate of decoherence under dynamical control [22][23][24][25], which is at the heart of the unified theory of dynamically-controlled open quantum systems [26][27][28]. This universal formula allows the design of control fields or pulse sequences that through the choice of a spectral filter function are optimally tailored to the specific environment-noise spectrum and the task at hand [25,29]: decoherence control [22,23,26,27,[30][31][32][33][34], state-transfer [35,36] or storage [36][37][38] in a fluctuating environment. Here, the filter function will be optimally adapted to the task of probing the environment-noise spectrum by a qubit [39].
For the purpose of characterizing the memory effects of an environment that interacts with a qubit probe, we here put forward an approach based on the aforementioned universal formula for decoherence control [22][23][24][25][26][27][28][29] combined with quantum estimation theory [39,[54][55][56][57]. We show that the information (estimation-precision) concerning the environment-noise fluctuation-spectrum obtained by this approach may exhibit critical behavior as a function of the memory-time parameter. This critical behavior, akin to a phase transition, is only revealed under dynamical control: it defines a sharp boundary between the short-and longtime regimes of the probe decoherence corresponding to long-and short-memory of the environment respectively. By contrast, free-induction decay of the probe coherence undergoes the usual smooth transition between the tworegimes, thus conforming to the gradual change from non-Markovianity to Markovianity that has been previously analyzed [52,53]. The criticality or phase-transition of the environmental information revealed here is a fundamental feature that characterizes dynamical behavior with practical implications on the attainment of the highest estimation precision of the environment memory-time (see Discussion).
Controlled qubit-probe as a sensor of the environmental fluctuations.-We consider a dynamically-controlled qubit-probe experiencing pure dephasing in the weakcoupling probe-environment regime (Fig. 1a), which is characterized by the attenuation (decay) factor J ( x B , t) of the qubit coherence [39] where x B are a set of parameters that describe the environment and the attenuation factor obeys the universal formula [22][23][24][25][26][27][28][29] J Here G( x B , ω) is the coupling spectrum (spectral density) of the environment noise (the Fourier transform of its autocorrelation function). Explicitly, x B = [g, τ c , β], with τ c as the correlation or memory time of the environment noise, i.e. the inverse width of its spectral density, g as the effective probe-environment coupling-strength, and β as a power law exponent that defines the type of stochastic (noise) process. The filter function F t (ω) explicitly depends upon the dynamical control of the probe during time  Figure 1: (a) Estimation of the environmental noise fluctuations by a qubit that probes the environment. A dynamically controlled qubit-probe undergoes pure-dephasing by the environment due to the interaction H SB = gσzB, where σz is the Pauli operator for the probe and B is the environment (bath) operator. The dephasing is revealed by the attenuation (decay) factor J that characterizes the optimal qubit-observable σx( x B , t) = e −J ( x B ,t) , x B being the set (unknown) noise parameters, for an (optimal) initial state -the symmetric superposition of the spin-up/-down states in the basis σz, 1 √ 2 (|↑ + |↓ ) = |+ [39,57,60]. Here we focus in estimating x B = τc. (b) Time dependence of the normalized attenuation factor J of the qubit-state probing an Ornstein-Uhlenbeck process (β = 2) for free evolution (dashed) compared to its counterpart under dynamical control (solid). The latter time dependence exhibits a smooth transition (marked by a circle) between two well-defined dynamical phases (regimes) depending on the ratio t τc .
t. The information about the unknown environment parameters x B is encoded by the probabilities p of finding the qubit-probe in the |+ (symmetric) or |− (antisymmetric) superposition of the qubit energy states when measuring σ x . These probabilities obey As a model to describe the memory time scales of the environment, we consider a generalized Ornstein-Uhlenbeck spectral density [39] where A β is a normalization factor depending on the power-law β ≥ 2. In fact, this model is the building block of universal lineshapes: it may characterize the memory time of arbitrary bosonic environments, if one assumes that a chosen harmonic-oscillator mode constitutes an interface between the qubit-probe and the modes of the environment [58]. The combined spectrum of any environment plus the interface mode is then reshaped, or "filtered", according to the chosen oscillator-mode frequency and its coupling strength with the probe, invariably resulting in a skewed-Lorentzian lineshape [58,59]. The power law regime ∝ ω −β of the spectral density, obtained for ωτ c 1, is the spectral range with the strongest dependence on the frequency ω, describing the short time behavior of the probe-qubit dephasing. We define this limit as the Long Memory (LM) regime. In the opposite limit ωτ c 1, associated with long times, the spectral density becomes independent of the frequency, and the probe coherence attenuation factor J ( x B , t) is then given by the Fermi Golden Rule. We dub this limit the Short Memory (SM) regime.
Identifying the dynamical regimes' criticality by dynamically controlled probes.-Under free induction decay, the LM and SM regimes are attained at times t τ c and t τ c , respectively. The respective decay (attenuation) factors are J LM f ree ∝ g 2 t 2 (independent of τ c ) and The transition from the LM to the SM regime is smooth (Fig. 1b) as the ratio t τc is varied and does not depend on g. Invariably, ∂J f ree ∂τc > 0, without sign change.
Consider now the change that may arise in the character of this transition under dynamical control. An example is a decoupling control sequence of N 1 equidistant π-pulses (known as CPMG) [16,61,62]. The filter function F t (ω) [4,[22][23][24][25][26][27][28][29] then converges to a sum of delta functions (narrowband filters) centered at the harmonics of the inverse modulation period, kω ctrl = πkN with k = 1, 2, 3, .. [5,34]. Another suitable control is CW qubit-driving, which has a single frequency component (k = 1). Under such controls, the LM and SM regimes are attained for ω ctrl τ c 1 and ω ctrl τ c 1, respectively. The corresponding decay factors, J ∝ F t (ω ctrl )G(ω ctrl ) in Eq. (2), are (see Fig. 1b) respectively. This reflects the effect of a narrow-band filter F t (ω ctrl ) that may be used to scan the spectral density G(ω) [4][5][6], upon varying the modulating frequency (pulse-rate or Rabi-frequency) ω ctrl of the control field, all the way from the frequency-independent regime G ∝ τ c for ωτ c 1 to the power-law regime G ∝ ω −β τ −(β−1) c for ωτ c 1 (Fig. 2a). In the limit of extremely narrow spectral filters, i.e. N→ ∞, with ω ctrl = πN t , we have An abrupt change (Fig. 2a) is then revealed in the parametric sensitivity of the attenuation-factor derivative ∂J ∂τc through its change of sign: ∂J ∂τc ∝ −(β−1)J LM τc < 0 for LM and ∂J ∂τc ∝ J SM τc > 0 for SM, implying that ∂J ∂τ c ω ctrl τc(β−1) at a value dependent on the control (modulating) frequency ω ctrl . The central result of this paper is that Eq. (8) signifies the vanishing of the quantum Fisher information (QFI) [54], which quantifies the attainable amount of information on τ c that can be extracted from the measured probe-(qubit-) state probabilities p ± obeying Eq. (3). This vanishing becomes apparent upon considering the expression for QFI [39,60] Hence, at the value ω ctrl τ c (β − 1) 1 β ≈ 1 no information can be extracted, F Q ctrl = 0. Since the minimum achievable relative error (per measurement) of the (unbiased) estimation of τ c is related to the QFI through the Cramer-Rao bound as this error diverges as ω ctrl → ω 0 (Eq. (7)):  the optimal time t SM opt (red circles) corresponds to a linear attenuation factor J SM ∝ t (panel (c) and Fig. (1)b), while for √ 2N gτc > 1, the optimal time t LM opt (blue solid circles) belongs to a regime where J LM ∝ t β+1 (panel (d)). At The relative error, ε ctrl F , thus exhibits a sharp transition between the LM and SM dynamical regimes (Fig. 2b), allowing their clear distinction. This critical behavior of the relative error provides a signature of the environmental noise spectral density through the values of τ c and β, provided we apply an appropriate dynamical control that generates a sufficiently narrow spectral filter, so as to scan the sign change of ∂G ∂τc ω∼ω0 at the critical point (Fig. 2). By contrast, such criticality does not arise under free-induction decay, for which the filter function, F f ree t (ω) is a much broader sinc function centered around ω = 0.
Critical behavior of the maximal estimation precision of τ c .-The critical behavior shown above is also manifest, under the same control on the probe, for the maximized estimation precision, i.e. the smallest possible minimal rela- Min. Rel. error Relative correlation time g CPMG, N max =20 Hahn Figure 4: Minimal relative error per measurement in the estimation of τc as a function of gτc for a Lorentzian environmental spectrum under dynamical control. Practical limitations on the number of pulses of a CPMG sequence, such that N ≤ Nmax, may prevent to attainment of the ultimate bound (dashed line) [39]. Then, a sudden change of the dynamical control strategy as a function of gτc may help: For gτc lower than the critical value, the highest precision is achieved by the single-pulse Hahn echo (N = 1). However, if gτc is larger than the critical value CPMG sequence with N = Nmax is optimal. This dynamical control strategy under practical limitations reduces the minimal error represented by the shaded area, which is determined by the optimal control on each side of the intersection (critical value) of the Hahn and the CPMG curves.
tive error in the estimation of τ c in Eq. (10), The error minimization is the outcome of selecting the optimal time t opt at which the measurement (cf. Eqs. (1), (2)) is performed on the probe, following its dephasing under the control we have applied. Figure 3 shows the critical behavior of the maximum precision per measurement, ε ctrl F (τ c , t opt ) for the Lorentzian spectrum (β = 2) following CW control of the qubit-probe as a function of separates two regions characterized by different scalinglaws of the minimal relative-error as a function of √ 2N gτ c (Fig. 3a). These scaling-laws are dictated by the different dynamical regimes for the attenuation (decay) factor shown in Figs. 3c,3d. The optimal probing (measurement) and control time t opt is also shown to undergo a sudden transition at the critical point (13), as shown in Fig. 3b. This optimal time corresponds to the best tradeoff between a signal amplitude contrast, e −2J 1−e −2J , and the parametric sensitivity of the signal attenuation factor, ∂J ∂τc 2 . The optimal tradeoff occurs (Fig. 3b) at either a long time, t SM opt (red circle) corresponding to a linear attenuation factor J SM ∝ t (Fig.  3c) or at a short time, t LM opt (blue circle) corresponding to J LM ∝ t β+1 (Figs. 3d). These optimal control times in the two regimes are situated on both sides of the critical value (Fig. 3b) The criticality of the relative error as a function of ω ctrl described by Eq. (11), defines two local, unequal minimum values of ε ctrl F , located on either side of the critical point ( Fig. 2b and insets Fig. 3a). What determines the global minima of ε ctrl F is the optimal time t opt obtained from Eq. (12). The critical behavior emerges when this global minimum jumps between the two local minima as a function of the parameter √ 2N gτ c (Fig. 3b). At the critical point, both local minima are equal, as displayed in Fig. 2b.
Discussion.-We have demonstrated a critical behavior of information (estimation-precision) on the environment fluctuation (noise) spectrum of generalized Ornstein-Uhlenbeck process extracted by a probe subject to appropriate dynamical control as a function of the environment memory-time gτ c . This finding applies to any bosonic environment, provided the probe and the environment are suitably interfaced by a chosen oscillator mode [58,59] (cf. discussion following Eq. (4)).
We have shown that similar critical behavior is manifest for the maximal estimation precision of τ c . At the critical point there is a massive loss of information on τ c . Near this point, the optimal time for measuring and controlling the quantum-probe is either very short, corresponding to little parametric sensitivity, or very long, corresponding to significant decay of the signal.
The critical behavior of the maximal estimation precision of τ c has paramount practical implications: (i) Complete dynamical behavior characterization: Rather than mapping out the long-and short-memory probe dynamics regimes by varying the probing time, the critical behavior demonstrated here, allows one to characterize the complete dynamics as consisting of two distinct dynamical phases (regimes) according to the maximal information they yield near the critical point (13).
(ii) Sudden change of the optimal dynamical control sequence: The critical point depends on the control scheme: thus, for CPMG control [16,61,62] gτ c ≈ 1 √ 2N when probing a Ornstein-Uhlenbeck process (Lorentzian spectra). This fact highlights the importance of optimizing the number of pulses N so as to improve the estimation precision, if N is bounded by N max due to practical limitations. Under these conditions, the ultimate bound on the estimation precision found in Ref. [39] 48, may not be attained for gτ c ≈ 1 √ 2Nmax . Then, a sudden change of N should be undertaken as a function of gτ c in order to optimize the estimation: For gτ c lower than a certain critical value shown in Fig. 4, the best precision is achieved by the single-pulse Hahn echo (N = 1). However, if gτ c is larger than this critical value, then the CPMG sequence with N = N max is optimal. Qualitatively similar considerations apply for generalized Ornstein-Uhlenbeck processes.
To sum up, the critical behavior of the environmental information revealed here is a fundamental feature that facilitates the attainment of the highest estimation precision of the environment memory-time. It represents an alternative characterization of the probe-qubit dynamics under suitable control or observation that leads to a phase transition on the dynamical behavior [44,[63][64][65][66][67]. Intriguingly, the absence of information has been shown to provide a distinctive signature of the environmental noise estimation.