Quantum Brownian motion under generalized position measurements: A converse Zeno scenario

We study the quantum Brownian motion of a harmonic oscillator undergoing a sequence of generalized position measurements. Our exact analytical results capture the interplay of the measurement backaction and dissipation. Here we demonstrate that no freeze-in Zeno effect occurs upon increasing the monitoring frequency. A similar behavior is also found in the presence of generalized momentum measurements.

decay rate, is excluded. Recently, an analysis of the case in which the monitoring time t scales as a power of N , i.e., t ∝ N α , was carried out finding that the Zeno effect occurs for 0 ≤ α < 1/2 [24].
Apart from a few exceptions [25][26][27], the measurements, which are essential for the QZE, have been treated as instantaneous interruptions of the otherwise unitary dynamics. That means that the interaction between the measured system and the measurement apparatus must take place within a time span that is very short compared to all relevant time scales of the unitary evolution.
Barchielli et al. [28,29] found a suppression of the Zeno-typical dynamical freezing for sequences of generalized measurements which can be characterized by a strength that decreases with increasing measurement frequency allowing for continuous measurements with ongoing dynamics. A similar approach, which additionally allows for a finite duration of the measurement, was developed by Ruseckas and Kaulakis [27]. While for both the approaches in Refs. [27,28] the measurements at subsequent times must be performed with identically prepared measurement apparatuses, Gagen et al. [26] proposed a model with a single apparatus permanently coupled to the system on which the measurement is performed. Despite the difference of the physical picture, whether there are as many measurement apparatuses as individual measurements, or just a single apparatus whose pointer moves with the measured observable, the time evolution of the system density matrix is governed by a Markovian master equation of Lindblad type provided that the measurement strength is properly adjusted to the measurement frequency [26,29].
Fearn and Lamb [30] analyzed the effect of repeated position measurements of fixed strength on the dynamics of a particle moving in a double-well potential finding a delocalization rather than a freezing of the dynamics in the well in which the particle was initially prepared. This result was challenged in Ref. [31] claiming that the freezing of the dynamics in either well would result if only sufficiently many measurements within a fixed duration of time were made. Altmüller and Schenzle [32] argued differently saying that a proper and more microscopic description of the measurement process would lead to the Zeno effect. An important aspect distinguishing their treatment from those in Refs. [30] and [1] is that not a series of measurements with independent measurement apparatuses but rather a continuous interaction with the electromagnetic field is considered. Actually, the reduction of the full two-well system to a two-level system is the feature which enforces the appearance of the Zeno effect in Ref. [32]. This is demonstrated by Gagen et al. [26] for a model of continuous measurements of the particle position. Indeed, when all energy levels are taken into account 1 the localization in the initial well persists for a time span that becomes smaller with decreasing energy gap between the first two levels and with increasing measurement strength. At large enough times a delocalization is always observed.
Generally, in a system with a finite-dimensional Hilbert space a sequence of measurements for detecting the presence of the particle in the initial state yields, at large times, a maximally mixed state of uniform population [25,34]. This occurs as long as the time between subsequent measurements is finite. For a vanishing inter-measurement time, however, the Zeno effect manifests itself in its originally proposed form of hindered decay [1,2]: The rate at which the asymptotic uniform state is approached becomes zero. This behavior has also been found in the presence of an environment [35], where the details of the dynamics depend on the specific spectral density of the environment as well as on the strength of the coupling between the system and its environment.
With the present work we consider a quantum harmonic oscillator interacting with a heat bath of mutually independent quantum harmonic oscillators. The resulting dissipative system provides a model of quantum Brownian motion [36][37][38][39]. The onset of the QZE for quantum Brownian motion has been predicted in Ref. [8] within a perturbative treatment starting from an exact timeconvolution-less master equation [40][41][42][43]. In Ref. [8], the central harmonic oscillator is considered to be initially in a Fock state whose decay is monitored, so that the measured observable, namely the excitation numbern, commutes with the oscillator's Hamiltonian. In contrast, here we investigate the case of an oscillator which is instantaneously monitored by a so-called Gaussian meter, which measures its position within a certain width, in the same spirit of Refs. [26,30]. Notably, such repeated position-like measurement on a harmonic oscillator are of experimental relevance, as for example in nanomechanical resonators [44,45].
In this work we consider the following protocol. We start from the canonical thermal state of the full interacting oscillator-bath system. A first selective Gaussian measurement at t = t 0 then prepares the oscillator in a state centered at some position x 0 with a width σ. The system subsequently evolves, undergoing a sequence of n nonselective measurements; i.e., the measurements leave the system state in a probabilistic mixture of its possible outcomes [46,47]. This intermediate monitoring is performed by Gaussian instruments of width σ acting at equally spaced times t j = t j−1 + τ , where j = 1, . . . , n. A final selective measurement at time t F < t n + τ , performed with a Gaussian meter, again of width σ, and centered at x F , provides the two-point probabil- . A scheme of this protocol is provided in Fig. 1 below. We exactly solve the monitored dynamics capturing the interplay between measurements and unitary dynamics of the oscillator interacting with a heat-bath. In agreement with the finding of Fearn and Lamb [30], no Zeno effect occurs for the oscillator. Instead, its position distribution initially spreads with increasing number of measurements. In the absence of friction, i.e. for an isolated oscillator, the spreading continues and the position undergoes a diffusion process. Since this diffusion takes place in the confining oscillator potential also the energy grows steadily whereby the position measurements fuel this process: Each position measurement suddenly squeezes the position to a narrow range; the concomitant spreading in momentum space subsequently leads to a spreading in position space beyond the width of the antecedent measurement. In clear contrast, in the presence of friction the supply of energy by the measurement can now be balanced by the amount of dissipation. Consequently, the asymptotic position distribution is characterized, after a sufficiently large number of measurements, by a finite width. In neither case a freezing of the dynamics, which is the essential feature of the Zeno effect, occurs. This is because the energy spectrum of the (isolated) oscillator is unbounded from above. Put differently, no lower limit exists in the inter-measurement time below which the unitary dynamics of the total system cannot take place between measurements.
In a recent treatment of the same model [48] a survival probability was found that depicts the Zeno effect. However, the corresponding analysis was based on the iteration of the transition probability for a single pair of measurements thereby neglecting the quantum coherences which build up during the sequence of Gaussian nonselective position measurements.
In the following we describe the model and detail the measurement protocol. Then, we derive the main result, namely the probability distribution for the final measurement conditioned on the result of the first preparing measurement. Finally, we illustrate and discuss the obtained results.
The Hamiltonian (1) provides a model for the quantum Brownian motion of a particle in a harmonic potential. The Heisenberg equation for the position operatorq of the central oscillator has the form of the following generalized Langevin equation [38,39,49] Here,ξ(t) is the quantum Brownian force operator which reads explicitlŷ with t i denoting the time origin. The damping kernel γ(t) is given by γ(t) = 2(M π) −1 ∞ 0 dω[J(ω)/ω] cos(ωt), where J(ω) is the spectral density function defined by J(ω) := π k c 2 k δ(ω − ω k )/(2m k ω k ). In the following, if not stated otherwise, we consider a strictly Ohmic heat bath, a bath with spectral density of environmental coupling whose continuum limit is linear in the frequency, i.e., The above strictly Ohmic case yields γ(t) = 2γδ(t), where the damping parameter γ provides an overall measure of the strength of the coupling with the bath modes.
where ρ qq (t) = q|ρ(t)|q ,q|q = q|q , and where x indicates the center position of the meter.
Here, f (q) denotes a Gaussian slit operator of width σ, reading explicitly with the identity operator 1 B acting in the bath Hilbert space. This Gaussian measurement setting is elucidated in greater detail in A. In the limit σ → 0, the measurement action becomes projective, i.e., lim f (q − x) = |x x|. Note that for a finite slit width σ the coherences with respect to the position basis are not totally obliterated by the generalized measurement described by Eq. (5).
Starting out at a time origin t i = 0 with the initial density operator of the total system ρ(0), one obtains for the probability density W (x 0 , t 0 ) to find the result x 0 in a first position measurement at some later time t 0 > 0 the expression where U (t) = exp(−iĤt/ ) is the time evolution operator of the full system, withĤ given by Eq. (1). In Eq. (7),q(t 0 ) = U † (t 0 )qU (t 0 ) denotes the position operator in the Heisenberg picture.
Similarly, one obtains for the joint probability density W (x 0 , t 0 ; . . . ; x F , t F ) of finding the central oscillator at the positions x 0 , x 1 , . . . , x n , x F in n + 2 measurements at the subsequent times In the following we assume that the full system is initially prepared at time t = 0 in the canonical equilibrium state at temperature T , i.e., ρ(0) = ρ th , where withĤ the Hamiltonian in Eq. (1). Then, the brackets in Eq. (8) (and in the following) denote the canonical thermal expectation value. It is convenient to introduce the corresponding (n + 2)-point characteristic function φ(k 0 , . . . , k F ) defined as the Fourier transform with respect to all positions For the quantum Brownian motion, the characteristic function can be conveniently cast into the where and s F = 0. Details of this derivation can be found in Ref. [50].  Fig. 1. The resulting two-point probability distribution, with initial and final measurements at times t 0 and t F separated by the total time

measurements. A scheme of this protocol is shown in
being a multivariate Gaussian distribution with zero mean, x 0 = x F = 0, and with covariance determined by the set of quantities The antisymmetrized correlation function in the last line of Eq. (15) (13) is contained in the expression ζ 2 (t) − ζ 2 0 , as given by the sum in Eq. (15). In the absence of intermediate measurements, this sum reduces to a single term and the wave packet spreading induced by a position measurement (as discussed in Ref. [50]) is recovered.
The result in Eqs. (13 -15) holds for any bath spectral density function. Specifically, for the Ohmic bath considered here, the symmetrized and antisymmetrized position correlation functions read [36,39,49] where ω r = ω 2 0 − γ 2 /4 denotes the effective frequency of the damped oscillator, β = 1/k B T the inverse temperature, and ν n = 2πnk B T / the Matsubara frequencies. In the limit of vanishing friction γ → 0 one recovers the results for the free harmonic oscillator prepared in the canonical thermal state.

IV. RESULTS
We are interested in the conditional probability density that a measurement taken at time t F yields the result x F , given that the system was initially prepared by a measurement taken at time t 0 with outcome x 0 , in the presence of n nonselective measurements between t 0 and t F , according to the scheme presented in Fig. 1. This conditional probability is defined as 9 The numerator in the rhs. of Eq. (17) is given in Eq. (13) while for the denominator we obtain with ζ 2 0 as defined in Eq. (15) above. Combining Eqs. (13) and (18) one finds from Eq. (17) which is a Gaussian probability density function with mean valuē and variance given by Here we made use of the definitions in Eqs. (15) and (16).
Equations (19)(20)(21)  Finally, it is interesting to study how the width Σ 2 τ (t) of the distribution (19) evolves in time and how it becomes influenced by the monitoring rate µ = 1/τ as well as by the coupling γ to the environment. In this spirit, the interesting issue to investigate is whether the traditional Zenophenomenon eventually emerges for vanishing τ . For this purpose, assume that the final selective measurement is performed after a time τ past the last nonselective measurement of the sequence, so thatt = (n + 1)τ . Then, considering that A[t − (n + 1)τ ] = A(0) = 0, the series in Eq. (21) can be approximated by the following time integral in the small-τ limit It follows that, in this small-τ limit, the width of the conditional probability distribution (19) emerges as This shows that the width of the distribution diverges as τ → 0, as it also does for σ → 0. In the latter limit a projective measurement of position is attained which in turn entails the injection of an infinite amount of energy upon measuring.
From Eq. (23) two interesting limits of the variance can be taken at fixed, small but finite τ .
The first is the frictionless limit at finite time t The second is the long-time limit for γ = 0 .
The Further insight into the behavior of the conditional probability density P (n) (x F , t F |x 0 , t 0 ) shown with Figs. 2 and 3 can be obtained by visualizing the time evolution of its width Σ 2 τ (t). In Fig. 4 this quantity is plotted by using the exact expression (21) for different measurement rates and for the three dissipation strengths γ used in Figs. 2 and 3. The time evolution and stationary values of the curves at the higher monitoring rates µ are qualitatively accounted for by the small-τ limit (23). In particular, Fig. 4 depicts the linear increase of Σ 2 τ (t) with increasing time at γ = 0 (see Eq. 24). In contrast, for finite dissipation, we observe a saturating, stationary behavior, in accordance with the analytic expression in Eq. (25).
We conclude this section with two remarks. First, the results presented are substantially unaf-

V. DISCUSSION AND CONCLUSIONS
With this work we studied the quantum Brownian motion of a dissipative oscillator undergoing a sequence of position-type generalized measurements by so-termed Gaussian slit instruments. The latter are characterized by a finite width σ around a specified position x and yield projective measurements in the limit σ → 0. The time evolution of the quantum Brownian particle subject to such repeated, instantaneous measurements was studied through the exact two-point quantum probability distribution with intermediate nonselective measurements. This intermediate monitoring was accounted for by suitably modifying the formalism described in Ref. [50]. We found that an increase of the monitoring rate enhances the position spreading after a first measurement at a and of the measurement rate µ = τ −1 (in units of ω0/2π). The symmetrized correlation function S(t) was numerically determined by truncating the sum in Eq. (16) to the first 2000 terms. The behavior at large µ is accounted for by the analytical small-τ expression (23). In particular, at γ = 0, the width Σ 2 τ (t) increases linearly with time, with a coefficient proportional to µ = τ −1 [see Eq. (24)]. On the contrary, for γ = 0, a steady state is reached where the width gets smaller as γ is increased with fixed µ. This is because the larger γ the more effectively the energy input from the measurements is dissipated. On the other hand, at fixed γ, the width increases with µ, as the rate of energy These results are in contrast to what occurs for projective measurements on systems with bounded Hamiltonians. Under these conditions, the conventional Zeno effect follows rigorously [51]. For a harmonic oscillator under the influence of generalized position measurements both conditions are clearly violated leaving room for a dynamical evolution of the system under permanent observation.
monitored system during the time δt of the system-meter interaction, whose coupling strength g has dimensions of a frequency. The full Hamiltonian iŝ withĤ defined in Eq. (1).
We assume that the interaction starts at t 0 = 0. For δt sufficiently small and g not too large, by using the Baker-Campbell-Hausdorff formula, the time evolution operator can be factorized as where Note that the operator e ixP M is a displacement operator for the meter, namely e ixP M |Q = |Q + x .
Assuming the initial system-meter factorized initial state ρ tot = ρ(0) ⊗ σ M (0) ≡ ρ 0 ⊗ σ 0 , and that after the time δt the meter state is projected into the state |Q [26], the system state after the measurement reads where |q(t) = e −iĤt/ |q . Now, by neglecting B, which is proportional to δt 2 , we get An additional assumption, which simplifies things further, is that the state of the system under the free evolution induced byĤ alone does not change appreciably during the time interval δt, i.e., |q(δt) |q . Whether this assumption is sensible depends on the state of the system previous to the measurement. Within the above approximations, the Gaussian measurement is attained by the following choice for the preparation of the density matrix of the meter at the initial time where µ(ξ, ξ ) = 1 We then get for the probability density to read from the meter the result x 0 This result amounts to taking the trace of the last line of Eq. (5), provided that Q 2 = σ 2 [cf.
Eq. (7)]. Finally, we note that, especially for the non-dissipative case, as the monitoring proceeds and the oscillator is excited to higher energies, the assumption of instantaneous measurement may break down.

Appendix B: Momentum measurements
The formalism developed in Sections II and III does not rely on the choice of the oscillator's positionq as the observable being measured. Indeed it applies as it is also to the case in which the measurements are Gaussian momentum measurements with operators The conditional probability density P (n) (p F , t F |p 0 , t 0 ) retains the same structure as the one given in Eq. (19) for P (n) (x F , t F |x 0 , t 0 ), with the difference is that in Eqs. (20) and (21) one has to use the momentum symmetrized and antisymmetrized correlation functions S pp (t) and A pp (t). These quantities combine to yield the momentum correlation function C pp (t) = S pp (t) + iA pp (t), so that C pp (0) = p 2 = S pp (0). In turn C pp (t) is given by the second time derivative of C(t), namely C pp (t) = −M 2 d 2 /dt 2 C(t) [49]. As can be seen by inspection of Eq. (16), the second time derivative of the symmetrized equilibrium position correlation function S(t) diverges (logarithmically) at t = 0 in the strictly Ohmic case [36,53]. The physically motivated introduction of a high-frequency cutoff regularizes this divergent behavior. A simple case is the Drude regularization [36,54,55] for which the spectral density function assumes the algebraically decaying form J(ω) = M γω(1 + ω 2 /ω 2 D ) [cf. Eq. (4)], where the Drude cutoff is ω D γ, ω 0 . Starting from the expression in Ref. [54], after some manipulations, S(t) reads (t ≥ 0) The parameters α,η, and δ are implicitly defied by the relations: 2α + δ = ω D , α 2 + η 2 = ω 2 0 ω D /δ, and α 2 + η 2 + 2αδ = ω 2 0 + γω D . Up to the first order in γ/ω D , the parameters in Eq. (B2) read By inspection of Eq. (B2) one sees that the strict Ohmic case is recovered in the limit ω D → ∞.
Indeed, in this limit, α → γ/2, η → ω r , and δ → ω D = ∞. As a result, Eq. (B2) reduces to the corresponding Ohmic expression in Eq. (16). On the other hand, for temperatures such that ν n → δ, the corrsponding n-th coefficient in the sum on the last line of Eq. (B2) vanishes.
The additional contributions to S(t) brought by the Drude regularization [cf. Eq. (16)] are at most of the order ω 0 /ω 2 D . Nevertheless, the introduction of the cutoff yields for the second derivative of S(t) a non-divergent behavior, which in turn entails the finiteness of S pp (0) = p 2 (see below).