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The Zeno and anti-Zeno effects on decay in dissipative quantum systems

We point out that the quantum Zeno effect, i.e., inhibition of spontaneous decay by frequent measurements, is observable only in spectrally finite reservoirs, i.e., in cavities and waveguides, using a sequence of evolution-interrupting pulses or randomly-modulated CW fields. By contrast, such measurements can only accelerate decay in free space.


I. INTRODUCTION
The "watchdog" or quantum Zeno effect (QZE) is a basic manifestation of the influence of measurements on the evolution of a quantum system. The original QZE prediction has been that irreversible decay of an excited state into an open-space reservoir can be inhibited [1], by repeated interruption of the system-reservoir coupling, which is associated with measurements (e.g., the interaction of an unstable particle with its environment on its flight through a bubble chamber) [2,3]. However, this prediction has not been experimentally verified as yet! Instead, the interruption of Rabi oscillations and analogous forms of nearlyreversible evolution has been at the focus of interest [4,5,6,7,8,9,10,11]. Tacit assumptions have been made that the QZE is in principle attainable in open space, but is technically difficult.
We have recently demonstrated [12] that the inhibition of nearly-exponential excitedstate decay by the QZE in two-level atoms, in the spirit of the original suggestion [1], is amenable to experimental verification in resonators. Although this task has been widely believed to be very difficult, we have shown, by means of our unified theory of spontaneous emission into arbitrary reservoirs [13], that two-level emitters in cavities or in waveguides are in fact adequate for radiative decay control by the QZE [12]. Condensed media or multi-ion traps are their analogs for vibrational decay control (phonon emission) by the QZE [14]. We have now developed a more comprehensive view of the possibilities of excited-state decay by QZE. Here we wish to demonstrate that QZE is indeed achievable by repeated or continuous measurements of the excited state, but only in reservoirs whose spectral response rises up to a frequency which does not exceed the resonance (transition) frequency. By contrast, in open-space decay, where the reservoir response has a much higher cutoff, non-destructive frequent measurements are much more likely to accelerate decay, causing the anti-Zeno effect.

II. MEASUREMENT SCHEMES
A. Impulsive measurements (Cook's scheme) Consider an initially excited two-level atom coupled to an arbitrary density-of-modes (DOM) spectrum ρ(ω) of the electromagnetic field in the vacuum state. At time τ its evolution is interrupted by a short optical pulse, which serves as an impulsive quantum measurement [4,5,6,7,8,9,10,11]. Its role is to break the evolution coherence, by transferring the populations of the excited state |e to an auxiliary state |u which then decays back to |e incoherently.
The spectral response, i.e., the emission rate into this reservoir at frequency ω, is hg(ω) being the field-atom coupling energy. We cast the excited-state amplitude in the form α e (τ )e −iωaτ , where ω a is the atomic resonance frequency. Restricting ourselves to sufficiently short interruption intervals τ such that α e (τ ) ≃ 1, yet long enough to allow the rotating wave approximation, we obtain where ∆ = ω a − ω s is the detuning of the atomic resonance from the peak (or cutoff) ω s of G(ω).
To first order in the atom-field interaction, the excited state probability after n interruptions (measurements), W (t = nτ ) = |α e (τ )| 2n , can be written as where The QZE obtains if κ decreases with τ for sufficiently short τ . This essentially means that the correlation (or memory) time of the field reservoir is longer (or, equivalently, Φ(t) falls off slower) than the chosen interruption interval τ . Equation (5) can be rewritten as where the interruptions are seen to cause dephasing whose spectral width is

B. Noisy-field dephasing: Random Stark shifts
Instead of disrupting the coherence of the evolution by a sequence of "impulsive" measurements, as above, we can achieve this goal by noisy-field dephasing of α e (t): Random ac-Stark shifts by an off-resonant intensity-fluctuating field result in the replacement of Eq. (6) by ( Fig. 1) Here the spectral response G(∆ + ω a ) is the same as in Eq. (1), whereas L(∆) is the Lorentzian-shaped relaxation function of the coherence element ρ eg (t), which for the common dephasing model decays exponentially. This Lorentzian relaxation spectrum has a HWHM width ν = ∆ω 2 τ c , the product of the mean-square Stark shift and the noisy-field correlation time. The QZE condition is that this width be larger than the width of G s (ω) (Fig. 1). The advantage of this realization is that it does not depend on γ u , and is realizable for any atomic transition. Its importance for molecules is even greater: if we start with a single vibrational level of |e , no additional levels will be populated by this process.

C. CW dephasing
The random ac-Stark shifts described above cause both shifting and broadening of the spectral transition. If we wish to avoid the shifting altogether, we may employ a CW driving field that is nearly resonant with the |e ↔ |u transition [4,5]. If the decay rate of this transition, γ u , is larger than the Rabi frequency Ω of the driving field, then one can show that κ is given again by Eq. (7), where the Lorentzian (dephasing) width is

D. Universal formula
All of the above schemes are seen to yield the same universal formula for the decay rate where F (ω) expresses the relevant measurement-induced dephasing (sinc-or a Lorentzianshaped): its width relative to that of G(ω) determines the QZE behavior.

A. Finite reservoirs: A Lorentzian line
The simplest application of the above analysis is to the case of a two-level atom coupled to a near-resonant Lorentzian line centered at ω s , characterizing a high-Q cavity mode [12]. In this case, where g s is the resonant coupling strength and Γ s is the linewidth (Fig. 2). Here G s (ω) stands for the sharply-varying (nearly-singular) part of the DOM distribution, associated with narrow cavity-mode lines or with the frequency cutoff in waveguides or photonic band edges. The broad portion of the DOM distribution G b (ω) (the "background" modes), always coincides with the free-space DOM ρ(ω) ∼ ω 2 at frequencies well above the sharp spectral features. In an open cavity, G b (ω) represents the atom coupling to the unconfined freespace modes. This gives rise to an exponential decay factor in the excited state probability, regardless of how short τ is, i.e., where κ s is the contribution to κ from the sharply-varying modes and γ b = 2πG b (ω a ) is the effective rate of spontaneous emission into the background modes. In most structures γ b is comparable to the free-space decay rate γ f . In the short-time approximation, taking into account that the Fourier transform of the Lorentzian G s (ω) is Φ s (t) = g 2 s e −Γst , Eq. (2) yields (without the background-modes contribution) The QZE condition is then On resonance, when ∆ = 0, Eqs. (5) and (12) yield Thus the background-DOM effect cannot be modified by QZE. Only the sharply-varying DOM contribution κ s may allow for QZE. Only the κ s term decreases with τ , indicating the QZE inhibition of the nearly-exponential decay into the Lorentzian field reservoir as τ → 0. Since Γ s has dropped out of Eq. (14), the decay rate κ is the same for both strong-coupling (g s > Γ s ) and weak-coupling (g s ≪ Γ s ) regimes. Physically, this comes about since for τ ≪ g −1 s the energy uncertainty of the emitted photon is too large to distinguish between reversible and irreversible evolutions.
The evolution inhibition, however, has rather different meaning for the two regimes. In the weak-coupling regime, where, in the absence of the external control, the excited-state population decays nearly exponentially at the rate g 2 s /Γ s + γ b (at ∆ = 0), one can speak about the inhibition of irreversible decay, in the spirit of the original QZE prediction [1]. By contrast, in the strong-coupling regime in the absence of interruptions (measurements), the excited-state population undergoes damped Rabi oscillations at the frequency 2g s . In this case, the QZE slows down the evolution during the first Rabi half-cycle (0 ≤ t ≤ π/2g −1 s ), the evolution on the whole becoming irreversible. A possible realization of this scheme is as follows. Within an open cavity the atoms repeatedly interact with a pump laser, which is resonant with the |e → |u transition frequency. The resulting |e → |g fluorescence rate is collected and monitored as a function of the pulse repetition rate 1/τ . Each short, intense pump pulse of duration t p and Rabi frequency Ω p is followed by spontaneous decay from |u back to |e , at a rate γ u , so as to destroy the coherence of the system evolution, on the one hand, and reshuffle the entire population from |e to |u and back, on the other hand (Fig. 3). The demand that the interval between measurements significantly exceed the measurement time, yields the inequality τ ≫ t p . The above inequality can be reduced to the requirement τ ≫ γ −1 u if the "measurements" are performed with π pulses: Ω p t p = π, t p ≪ γ −1 u . This calls for choosing a |u → |e transition with a much shorter radiative lifetime than that of |e → |g .
where R is the geometric-mean reflectivity of the two mirrors, f is the fractional solid angle (normalized to 4π) subtended by the confocal cavity, and L is the cavity length. It shows, that the population of |e decays nearly-exponentially well within interruption intervals τ , but when those intervals become too short, there is significant inhibition of the decay.

B. Open-space reservoirs
The spectral response for hydrogenic-atom radiative decay via the p · A free-space interaction is given by [15] G(ω) = αω where α is the effective field-atom coupling constant and the cutoff frequency is Using measurement control that produces Lorentzian broadening [Eq. (7)] we then obtain where In the range we obtain from Eq. (17) the anti-Zeno effect of accelerated decay. This comes about due to the rising of the spectral response G(ω) ≈ αω as a function of frequency (for ω ≪ ω c ). The Zeno effect can hypothetically occur only for ν > ∼ ω c ∼ 10 19 s −1 . But this range is well beyond the limit of validity of the present analysis, since ∆E ∼hν > ∼h ω c may then induce other decay channels ("destruction") of |e , in addition to spontaneous transitions to |g .

IV. CONCLUSIONS
Our unified analysis of two-level system coupling to field reservoirs has revealed the general optimal conditions for observing the QZE in various structures (cavities, waveguides, phonon reservoirs, and photonic band structures) as opposed to open space. We note that the wavefunction collapse notion is not involved here, since the measurement is explicitly described as an act of dephasing (coherence-breaking). This analysis also clarifies that QZE cannot combat the open-space decay. Rather, impulsive or continuous dephasing are much more likely to accelerate decay by the inverse (anti-) Zeno effect.