## Abstract

An experimental demonstration of a classical analogue of the quantum Zeno effect for light waves propagating in engineered arrays of tunneling-coupled optical waveguides is reported. Quantitative mapping of the flow of light, based on scanning tunneling optical microscopy, clearly demonstrates that the escape dynamics of light in an optical waveguide side-coupled to a tight-binding continuum is slowed down when projective measurements, mimicked by sequential interruptions of the decay, are performed on the system.

©2008 Optical Society of America

## 1. Introduction

The quantum Zeno and anti-Zeno effects, i.e. the inhibition or acceleration of the irreversible decay of an unstable quantum state into a continuum (reservoir) induced by measurements, are commonly viewed as basic manifestations of the influence of observations on the evolution of a quantum system [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Experiments showing interruptions of Rabi oscillations or analogous forms of nearly-reversible evolution for essentially stable few level systems have been reported as a demonstration of Zeno dynamics (see e.g. [2, 10, 11]), rising a rather broad debate on Zeno dynamics and its connection with the theory of quantum measurements (see e.g. [10, 11, 12, 13]). It was also argued that slow down of the decay may be attained in purely classical systems [14] that can mimic the dynamics of a two-level quantum system. An experimental demonstration of the Zeno effect in a truly decaying system is very hard to be observed in microscopic quantum systems because the required measurement intervals are too short and decay acceleration (anti-Zeno effect) appears to be much more ubiquitous [7]. To date, there is solely one landmark experiment demonstrating both Zeno and anti-Zeno effects in a truly unstable quantum system [15].

Exploitation of quantum-classical analogies, on the other hand, has been used on many occasions to mimic and visualize at a macroscopic classical level some basic quantum dynamical features characteristic of the Schrödinger equation that have difficult or even impossible access in quantum systems (see e.g. [16]). In particular, it was recently theoretically demonstrated [17, 18] that tunneling escape of light waves in an optical waveguide side-coupled to a semi-infinite array (from now on referred to as ‘semi-array’) exactly mimics the decay dynamics of a truly unstable quantum state coupled to a continuum, thus providing a classical realization (‘optical Zeno effect’) of the non-exponential decay behavior in the Friedrichs-Lee model commonly used in theoretical studies of the quantum Zeno effect [6, 7, 8, 9, 19]. In an optical system, projective measurements (corresponding to the collapse of the wave function according to von Neumann’s projection postulate) are simply mimicked by sequential spatial alternations of the semi-array. Each interruption restarts the decay process erasing the memory of the continuum and therefore ideally corresponds to a “wave collapse” in the quantum realm [17]. The spatial length between successive interruptions plays the role of the time interval between successive quantum measurements. The use of an engineered photonic structure to mimic at a classical level the Zeno dynamics, far from being a mere curiosity, offers the rather unique advantage of a direct and precise mapping of the space-time evolution of the Schrödinger probability density which is not fully accessible for quantum tunneling.

We have recently shown [20] that scanning tunneling optical microscopy (STOM) [21] based on a hollow pyramid mounted on a silicon cantilever provides an excellent tool for accurate quantitative tracing of the flow of discretized light in evanescently-coupled optical waveguides [22], allowing for high reproducibility and easy interpretation of the data. In this work we report on the first experimental demonstration, by using the same STOM setup, of the optical Zeno effect for the decay of light waves in an optical waveguide.

## 2. Sample and methods

The structures designed for the experimental demonstration of the optical Zeno effect are shown in Fig. 1 and consist of a single-mode straight channel waveguide (W) which is side-coupled either to a semi-array (S) or to a set of alternating semi-arrays (S1, S2, etc.) of 16 identical waveguides in the geometries shown in Figs. 1(a) and 1(b), respectively.

The structures are fabricated by the Ag-Na ion exchange technique [23] in an Er:Yb-doped glass substrate (Schott IOG1). Excitation of waveguideWis accomplished by bonding a single-mode optical fiber to the polished input facet of waveguide W, delivering the radiation at *λ* = 980 nm wavelength emitted by a single-frequency laser diode. The bonding system ensures an excellent long-term stability of the excitation set up, avoiding even small fluctuations of the coupled light power that might be detrimental for the demonstration of the optical Zeno effect. The accurate and quantitative mapping of the light flow along the arrays requested for the demonstration of the optical Zeno effect is accomplished by means of STOM imaging with a commercially available microscope setup [24] employing a hollow-pyramid cantilevered tip. The tip is brought to soft contact with the array and converts the evanescent field at the upper surface of the sample into a propagating wave, which is then focused onto a pinhole for background rejection, followed by a single-photon avalanche diode. Each waveguide is single-mode at the wavelength used in this experiment, ensuring that the signal detected by the tip can be directly related to the field energy distribution inside the waveguide [20, 21]. An atomic force microscopy topography map, in which the waveguides appear as buried stripes at the surface, is acquired simultaneously during each STOM image. An example for the surface of the sample in Fig. 1(b) is shown in Fig. 1(c), where the interruption at the boundary between two semi-arrays is clearly visible.

## 3. Model

Light propagation along the paraxial direction *z* of a weakly waveguiding structure is described by a Schrödinger-like equation for the electric field amplitude *Ψ*(*x*, *y*, *z*), which exactly mimics the temporal evolution of a quantum particle in a two-dimensional multiple-well potential *V*(*x*, *y*)≃*n _{s}*-

*n*(

*x*,

*y*), where

*n*(

*x*,

*y*) is the refractive index profile of the waveguide-array system and

*n*the substrate refractive index. By substituting

_{s}*ħ*↦

*λ*/2

*π*,

*m*↦

*n*(

_{s}*m*being the mass of the particle in the Schrödinger equation), and

*t*↦

*z*, the temporal evolution of the particle wave function in the quantum problem is replaced, in our photonic structure, by the spatial propagation of the electric field amplitude

*Ψ*(

*x*,

*y*,

*z*) along

*z*[17]. If the system is initially prepared with the particle in the well W, it can evolve via tunneling into the chain of adjacent wells and the probability

*P*to find the particle in the well W decays with time toward zero in absence of bound surface states [17]. The decay of a discrete level |

*χ*〉 into a tight-binding continuum |

*ω*〉 of states is commonly described in terms of the Friedrichs-Lee Hamiltonian

*H*=

*H*

_{0}+

*H*(see e.g. [7, 8, 9]), where

_{I}is the free Hamiltonian,

is the interaction Hamiltonian, *g*(*ω*) is the discrete-continuum spectral coupling amplitude, 4 *ħ*Δ is the width of the continuum band, and *σ* accounts for a possible detuning between the position of the discrete level and the center of the continuum, as shown in Fig. 1(d). For the system of our interest, the spectral coupling amplitude takes the specific form [18]

where Δ_{0} measures the strength of the discrete-continuum coupling.

In the optical analogue, the tight-binding approach to energy band formation is usually described in terms of the so-called coupled-mode equations [22]. In this frame, indicating by *c _{χ}* (

*z*) the complex amplitude of the modal field trapped in the waveguideW, the decay dynamics of

*c*(

_{χ}*z*) for the structure of Fig. 1(a) is described by the same Friedrichs-Lee Hamiltonian, where Δ

_{0}assumes the meaning of the coupling rate between the boundary waveguide W and the semi-array, Δ is the coupling rate between adjacent waveguides in the semi-array, and

*σ*accounts for a possible propagation constant mismatch of the mode in waveguideW as compared to waveguides in the semi-array. All these quantities are therefore expressed in units of mm

^{−1}in the optical realm. The decay evolution of

*c*(

_{χ}*z*) is obtained by standard spectral analysis and reads (see e.g. [9, 18])

where

is the self-energy. For |*σ*|<2-(Δ_{0}/Δ)^{2}, there are no bound states and *c _{χ}* (

*z*) (and hence the survival probability

*P*(

*z*) = |

*c*(

_{χ}*z*)|

^{2}) asymptotically decays to zero. However, the decay law shows non-exponential features which can be markedly pronounced [17].

For the waveguide structure manufactured for our experiment, the boundary waveguide W and the waveguides in the semi-arrays have approximately the same refractive index profile, i.e. we may assume *σ*/Δ≃0. For the waveguide separations of the fabricated structure (*a* = 9.5 *µ*m and *a*
_{0} = 11 *µ*m), the values of the coupling rates between the waveguides turn out to be Δ≃0.435 mm^{−1} and Δ_{0}≃0.223 mm^{−1}, as measured using two reference directional couplers (see e.g. [25]). Solid line in Fig. 2 shows, correspondingly, the predicted decay law for *P*(*z*) in the waveguide W of Fig. 1(a), as obtained by Eq. (4). The inset in Fig. 2 shows the corresponding behavior of the effective decay rate *γ*
_{eff}(*z*)=-(1/*z*)ln*P*(*z*), compared to the ‘natural’ constant decay rate
${\gamma}_{0}=\frac{2{\left(\frac{{\Delta}_{0}}{\Delta}\right)}^{2}}{\sqrt{1-{\left(\frac{{\Delta}_{0}}{\Delta}\right)}^{2}}}$
, obtained in the frame of Gamow’s approach to quantum tunneling decay [9, 17], which would give an exponential decay for *P*(*z*). The spatial length *τ* = 3.1mm between successive interruptions in the sample of Fig. 1(b) plays the role of the time interval between successive measurements, and for the observation of the Zeno effect it should be chosen smaller than a characteristic ‘Zeno time’ *τ*
_{Z}, where *τ*
_{Z} is the smallest root of the equation *γ*
_{eff}(*τ*
_{Z})=*γ*
_{0} [9]. In our case, one can estimate *τ*
_{Z} ≃ 30 mm, i.e. much longer than the sample length (~12 mm), and anti-Zeno effect can be a priori excluded. Dashed line in Fig. 2 shows the same decay behavior simulated by means of a fully numerical integration of the paraxial wave equation for *Ψ*(*x*, *y*, *z*) using a beam-propagation software [26]. *P*(*z*) is calculated by projecting, at each propagation distance *z*, the envelope *Ψ*(*x*, *y*, *z*) over the fundamental mode of waveguide W. The good agreement between the two simulations further supports the use of a tight-binding approach (coupled-mode equations) to tackle the optical problem.

## 4. Results and discussion

The detailed experimental spatial maps for the light intensity distribution |*Ψ*(*x*,0, *z*)|^{2} in the (*x*, *z*) plane of the sample are shown in Figs. 3(a) and 3(b) for the samples of Figs. 1(a) and 1(b), respectively.

The maps are obtained by taking successive STOM scans along the propagation *z* direction with steps of Δ*z* = 500*µ*m. Each scan covers a sample area of size 170*µ*m×20*µ*m in the (*x*, *z*) plane, thus comprising the main waveguideW and all the 16 waveguides in the semi-arrays S, S1, S2, etc. From the acquired STOM images, the integrated optical signal along the x direction is normalized in order to take absorption and internal losses into account. In this way the decay law for the light power trapped in waveguide W can be calculated, as fully described in [20]. In Fig. 3(c) we show the experimental decay curves for the samples of Figs. 3(a) (squares) and 3(b) (circles). The difference in the decay behavior between the two curves clearly shows that, as anticipated before, slow down of the decay due to frequent projective measurements is obtained when the light trapped in waveguideWis alternately coupled to the right and left semi-arrays S1, S2, etc. (optical Zeno effect). Each interruption of the semi-arrays restarts the decay process in the waveguide W and thus mimics a projective (von Neumann) measurement in the corresponding quantum mechanical problem [17]. The spatial distance *τ* between successive interruptions is large enough to ensure that light trapped in the semi-array after each interruption is scattered out into the substrate in a short distance (~600 *µ*m). In this way, the ‘memory’ in the continuum is erased after each interruption, as shown in the inset of Fig. 3(c), where the normalized power decay law in each of the semi-arrays is plotted and superimposed to the theoretical prediction.

## 5. Conclusion

In conclusion, we reported on the experimental demonstration of a classical analogue of the quantum Zeno effect for light waves [17] in an engineered photonic structure. Escape of light waves via optical tunneling in a waveguide side-coupled to a semi-array has been accurately measured by STOM imaging using a cantilevered hollow pyramid tip, showing that periodic interruptions of the decay result in a deceleration of the decay process.

We gratefully acknowledge A. Bassi, G. Cerullo, V. Foglietti, M. Lobino, M. Marangoni, and M. Savoini for fruitful discussions and for their support during the experimental sessions.

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