Zeno and anti-Zeno effects, and interaction-free measurements
Zeno and anti-Zeno effects
There are many fundamental differences between classical and quantum physics concerning
the concept of measurement. One of the most recently discovered is the so-called Zeno effect.
The Zeno effect is the inhibition of the dynamics of an observed system. If the observation is
accurate enough the evolution may be fully stopped. A typical example is that an excited atom
would never emit a photon if we are able to determine with enough precision the moment of
emission of the photon.
unobserved
atom
photon
oberved
atom
In a first approach, the existence of this effect can be demonstrated invoking the collapse of
the wave function associated to quantum measurements. However, this idea of state reduction
is somewhat obscure and controversial so that it can be questioned whether it provides a
suitable explanation. Our approach to the problem focus on the idea of a purely dynamical
explanation of the Zeno effect, avoiding the use of any obscure or debatable aspect. Following
this reasoning, the Zeno effect would be a consequence of the perturbation of the observed
system caused by the measuring procedure. The perturbation and the mechanism enforcing it
can be understood using simple classical concepts.
We have applied this approach to the generation of photon pairs in spontaneous parametric
down-conversion in a nonlinear crystal illuminated by strong laser (not represented in the
above figure).
L=N
The photons are generated in pairs, so that one of them can serve to detect the moment of
emission of the other one, without seemingly interfering the emission process. This can be
done, for instance, by slicing the crystal into N pieces of the same length. For one of the
modes the trajectory in the sliced crystal is fully equivalent to the trajectory in the original
medium. The other mode is interrupted N times in order to detect the emitted photon. This
detection will reveal the particular piece where the emission of the photon pair has taken
place. We have shown that if the modes satisfy perfect phase matching, then the emission
probability decreases as the number of pieces increases (i. e., when the accuracy of the
observation increases). Therefore this is an example of Zeno effect.
In order to explain the effect we have to take into account that the emission of the photon pair
is a completely quantum process, that can be understood as an emission generated or
stimulated by the vacuum fluctuations impinging on the crystal. In the original unsliced
crystal the emissions from every part of the crystal are stimulated by one and the same
vacuum (see the figure). They are coherent and interfere constructively in the case of perfect
phase matching
2
P  1     N  N 2
On the other hand, for the sliced crystal the emissions of different pieces are generated by
different and statistically independent vacuums (see the figure), so that the corresponding
emissions are incoherent and do not interfere
P
2
2
 Pob 
Pob   1     N  N
N
This fully explains the Zeno effect without invoking quantum collapses nor any other special
quantum feature. Note that the appearance of the effect requires no detectiors whatsoever. The
replacement of coherent by incoherent superpositions is the disturbance caused by the
observation of the moment of emission that fully explains the occurrence of the effect.
Zeno effect in parametric down-conversion
A. Luis and J. Peřina, Phys. Rev. Lett. 76, 4340 (1996)
If we recognize the Zeno effect as having a purely dynamical origin (i. e., as a particular case
of quantum evolution) then the inhibition of the dynamics is not a mandatory consequence,
and we might guess the existence of other different consequences of quantum observation.
We have examined this possibility in the same scheme described above, but now focusing on
a pair of modes that are not phase-matched. In this case the interference in the original crystal
has a destructive character. The observation replacing coherent by incoherent superpositions
prevents destructive interference in such a way that the emission can increase
2
 sen kL / 2 
Unobserved emission P(k )  
 P(k  0)
 kL / 2 
(destructive interference)
1  sen kL / 2 N   2

 P(k  0)
N  kL / 2 N 
A simple analysis shows that for some values of N the observed emission becomes larger than
the unobserved one. Therefore, in this case the observation does not hinders the emission, on
the contrary, it favors it. We refer to this effect (the inverse of the Zeno effect) as anti-Zeno
effect.
Observed emission Pob (k ) 
Anti-Zeno effect in parametric down-conversion
A. Luis and L. L. Sánchez-Soto, Phys. Rev. A 57, 781 (1998)
Recently we have confirmed these effects by means of a more complete and general analysis
which takes into account the spectral distribution of the emission. In this multi-mode analysis
the emission of photon pairs can be regarded as the spontaneous and irreversible decay of an
unstable state: the vacuum state impinging on the crystal. This is interesting since there are
few examples where the spontaneous decay can be monitored with enough accuracy.
Moreover, we have demonstrated that the Zeno and anti-Zeno effects are particular cases of
the general evolution of an observed system where the effect of observation is to introduce a
mode filtering. Depending on the particular form of this mode filtering we get Zeno or antiZeno effect.
These analyses demonstrate that complex quantum phenomena may admit very simple
explanations in terms of basic optical concepts, such as interference and coherence.
Zeno and anti-Zeno effect in parametric down-conversion
A. Luis Phys. Rev. A 66, 012101 (2002)
We have studied the Zeno and anti-Zeno effects in the most simple quantum system: a twolevel system. As the most illustrative example we have focus on an atom illuminated by light
beam (either resonant or nonresonant) coupling two of its internal levels causing transitions
between them. The existence of Zeno effect in this system has been well studied. On the other
hand, some authors have stated that there is no place for the anti-Zeno effect given the
reduced dimensionality. Following the same idea applied above to parametric downconversion, we have demonstrated the possibility of anti-Zeno effect in two-level systems
provided that the level coupling is nonresonant. For the unobserved system the lack of
resonance impedes the level transition. On the other hand, the observation favours them so
this is a true anti-Zeno effect. Moreover, we have demonstrated that in the nonresonant case
the observation produces the appearance of the two effects sequentially: Zeno effect in a short
term regime and anti-Zeno effect in the long term.
Zeno and anti-Zeno effects in two-level systems
A. Luis, Phys. Rev. A 67, 062113 (2003)
The Zeno effect was introduced originally as the inhibition of the spontaneous irreversible
decay of an unstable state, caused by a frequent enough observation. However, the accurate
monitoring of the decay of natural unstable systems presents serious technical problems
(leaving aside the valuable example of the parametric down-conversion discussed above).
Because of this some authors have introduced the idea of inducing decay processes amenable
to observation. This can be done by coupling an stable atomic level to a truly unstable one.
This causes an induced unstability of the first level whose properties can be controlled at will
by varying the characteristics of the coupling. This is also known as reservoir engineering.
We have analyzed these schemes examining the concurrence of the so called coherent and
incoherent Zeno effects (the halting of reversible and irreversible dynamics respectively). We
have demonstrated that these schemes can be regarded as the competition of two Zeno effects
taking place simultaneously. This is because the decay of the unstable level provides an
effective observation of the transition from the unstable to the stable levels. Their relative
strength allows us to explain the characteristics of these induced-observed decays. We have
demonstrated also the appearance of the anti-Zeno effect.
Zeno effect in spontaneous decay induced by coupling to an unstable level
A. Luis Phys. Rev. A 64, 032104 (2001)
The existence of the Zeno and anti-Zeno effects reveals the possibility of controlling the
dynamical properties of physical systems by means of rather subtle and sophisticated
mechanisms of purely quantum origin. This opens the possibility of new applications of the
quantum theory that are certainly worth investigating.
In this sense we have applied the Zeno effect to the preparation and control of the internal
state of an atom. Moreover, we have shown that this effect can serve to protect the state
against external perturbations.
As a further application we have used this idea to construct a two beam interferometer
(specifically a Mach-Zehnder) where the two beams have rather dissimilar origins: one is an
electromagnetic wave while the other one is the quantum wave associated to the vibrations of
a material system, such as atom or molecule. In other words, we have constructed a photonphonon interferometer.
Quantum-state preparation and control via the Zeno effect
A. Luis, Phys. Rev. 63, 052112 (2001)
Construction of a matter-light interferometer via the Zeno effect
A. Luis, J. Opt. B: Quantum Semiclass. Opt 3, 238 (2001)
Interaction-free measurements
A quantum phenomenon closely related to the Zeno effect is the so-called interaction-free
measurement. This can be illustrated with a simple example.
The objective of the measurement is to detect the presence or absence of a given object i a
given place. The object is assumed to be a perfect absorber and it can be situated in one of the
internal paths of a Mach-Zehnder interferometer illuminated by a single photon. The
interferometer is prepared so that when the object is absent the photon leaves the
interferometer at a given output port with certainty (bright port). The presence of the object
destroys the interference and there is some probability that the photon leaves the
interferometer by the other output port (dark port). When this occurs we have full certainty
about the presence of the object. Moreover the photon never impinged on the object
(otherwise it would have disappeared). Therefore, the detection of the presence of the object
has occurred seemingly without any interaction between the object and the apparatus (the
photon).
out
object in
in
object
out
Leaving aside the theoretical interest of this phenomenon we have possible practical
applications derived from the possibility of optically examining photosensitive specimens that
otherwise might be destroyed or seriously damaged by light.
This phenomenon seemingly defy basic believes concerning quantum detection such as the
every measurement disturbs the observed system. Moreover it seems natural to consider that
any alteration must be caused by some kind of interaction. Our objective was then to look for
such an unavoidable disturbance.
We have been able to demonstrate that in this kind of detection seemingly without interaction
there is always an effective interaction at work that produces the disturbance of the observed
system. This effective interaction can be only revealed by a fully quantum treatment of the
problem, including the observed system. This detail was not taken into account by previous
approaches. Such an interaction does not involve energy or momentum exchange. It is
revealed by the increase of the fluctuations of the relative phase of the coherent superposition
object  p in  ei (1  p) out
We have demonstrated that this interaction can be observed and measured in an
interferometric arrangement where this phase can be recast as a phase difference. The
randomization of the phase will imply the degradation of the visibility of then interference.
Dynamical analysis of seemingly interaction-free measurements
A. Luis y L. L. Sánchez-Soto, Phys. Rev. A 58, 836 (1998)
Our interest in understanding the mechanism behind this kind of measurements lead us to
study them at the most basic level: the resonant interaction between light and a two-level
atom. In this case the purpose of the measurement is to detect whether the atom is in the
excited or in the ground level. We have demonstrated that this can be done without
absorption-emission of photons, so this can be regarded as in interaction-free measurement.
Also in this case we have demonstrated that there is an effective interaction that derives
naturally from the atom-field joint evolution equations. In this case the variable experiencing
the disturbance caused by the interaction is the phase of the atomic dipole.
In order to clearly disclose these facts we have examined how this measuring arrangement
affects the dynamics of the atom interacting with a classical field. Among the consequences
of observation we have found the frequency narrowing of the excitation probability. It may
also occur that the resonance splits into two resonances. On the other hand, there can be
complete population inversion. The atom can be also transparent at resonance reaching an
extremely large dispersion.
All these points are incompatible with the presumed lack of interaction. Moreover, they
clearly demonstrate the existence of an effective interaction in the so-called interaction-free
measurements.
Atom-field resonant interaction without exchange of photons
A Luis y L. L. Sánchez-Soto, Phys. Lett. A 252, 130 (1999)
Dynamics of a two level atom observed via an interaction-free measurement
A. Luis y L. L. Sánchez-Soto, Phys. Rev. A 60, 56 (1999)