# Quantum Zeno and anti-Zeno effects in an asymmetric nonlinear optical coupler

###### Abstract

Quantum Zeno and anti-Zeno effects in an asymmetric nonlinear optical coupler are studied. The asymmetric nonlinear optical coupler is composed of a linear waveguide () and a nonlinear waveguide () interacting with each other through the evanescent waves. The nonlinear waveguide has quadratic nonlinearity and it operates under second harmonic generation. A completely quantum mechanical description is used to describe the system. The closed form analytic solutions of Heisenberg’s equations of motion for the different field modes are obtained using Sen-Mandal perturbative approach. In the coupler, the linear waveguide acts as a probe on the system (nonlinear waveguide). The effect of the presence of the probe (linear waveguide) on the photon statistics of the second harmonic mode of the system is considered as quantum Zeno and anti-Zeno effects. Further, it is also shown that in the stimulated case, it is easy to switch between quantum Zeno and anti-Zeno effects just by controlling the phase of the second harmonic mode of the asymmetric coupler.

Quantum Zeno and anti-Zeno effects in an asymmetric nonlinear optical coupler

Kishore Thapliyal*, Anirban Pathak

Jaypee Institute of Information Technology, A-10, Sector-62, Noida, UP-201307, India

^{0}

^{0}footnotetext: *tkishore36@yahoo.com

Keywords: Quantum Zeno effect, quantum anti-Zeno effect, optical coupler, waveguide.

## 1 Introduction

Zeno’s paradoxes have been in discussion since fifth century. In 1977,
Mishra and Sudarshan [1] introduced a quantum analogue
of Zeno’s paradox, which is later termed as quantum Zeno effect. Quantum
Zeno effect in the original formulation refers to the inhibition of
the temporal evolution of a system on continuous measurement [1, 3, 2],
while quantum anti-Zeno or inverse Zeno effect refers to the enhancement
of the evolution instead of the inhibition [4] (see
[5, 6] for reviews on Zeno
and anti-Zeno effect). In the last four decades, quantum Zeno effect
has been studied in different physical systems, such as two coupled
nonlinear optical processes[7], parametric down-conversion[8]
and cascaded parametric down-conversion with losses[9].
Similarly, quantum anti-Zeno effect is also reported in various physical
systems. Specifically, quantum anti-Zeno effect is observed in parametric
down-conversion[10] and in radioactive decay
processes, where the measurement causes the system to disintegrate
[11]. In Ref. [12], both
the effects (quantum Zeno and anti-Zeno) are reported in two-level
systems. Further, a geometrical criterion for transition between the
Zeno and anti-Zeno effects has also been discussed in Ref. [13].
Quantum Zeno effect using environment-induced decoherence theory has
also been discussed in the past[14]. Agarwal and
Tewari proposed a scheme for an all optical realization of quantum
Zeno effect using an arrangement of beam splitters [15].
In addition to the above mentioned theoretical studies, quantum Zeno
effect has also been experimentally realized in trapped beryllium
ions [16] and with the help of rotators and polarizers
[17].

Recently, the interest on quantum Zeno effect has increased by manifold
as it has found its applications in counterfactual quantum computation[18],
where computation is accomplished using the computer in superposition
of running and not running states and later to infer the solution
from it; counterfactual quantum communication, in which information
is sent without sending the information encoded particles through
the communication channel[19]; and quantum
Zeno tomography, which essentially uses interaction free measurement[20].
Interestingly, detection of an absorbing object without any interaction
with light, using quantum Zeno effect with higher efficiency has already
been demonstrated [21]. These facts
motivated the study of Zeno effect in macroscopic systems[22],
too. Quantum Zeno and anti-Zeno effects in the nonlinear optical couplers
have been studied in the recent past[23, 24, 25, 26]
by considering that one of the mode in the nonlinear waveguide is
coupled with the auxiliary mode in a linear waveguide, and the auxiliary
linear mode acts as a probe (continuous observation) on the evolution
of the system (nonlinear waveguide) and changes the photon statistics
of the other modes of the nonlinear waveguide, which are not coupled
with the probe. This kind of continuous interaction with an external
system is equivalent to the original inhibition or enhancement of
evolution of the system on continuous measurement as measurement is
strong coupling with a measuring device[27, 28, 29].

An important example of nonlinear optical coupler is an asymmetric
nonlinear optical coupler, consisting of a nonlinear waveguide with
nonlinearity operating under second harmonic generation
coupled with a linear waveguide. This system is studied earlier and
the nonclassical properties (such as squeezing, antibunching and entanglement)
have been reported in this system for both codirectional and contradirectional
propagation of the field in the linear waveguide ([30, 31, 32, 33]
and references therein). We will consider here the codirectional propagation
of the fields as in contradirectional propagation the solution was
only valid at both the ends of the coupler, i.e., not valid for ,
where is the interaction length of the coupler [32, 33].
It has already been established that the nonclassical effects can
be observed in the asymmetric nonlinear coupler[30, 31, 32, 33].
The presence of quantum Zeno and anti-Zeno effects reported in this
work further establishes the existence of nonclassicality in this
asymmetric coupler.

To study the quantum Zeno and anti-Zeno effects closed form analytic
expressions for different field operators are used here. These expressions
were obtained earlier by using Sen-Mandal perturbative approach[30, 31]
and a completely quantum mechanical description of the system. The
solutions of Heisenberg’s equations of motion used here are better
than the conventional short-length solutions as these solutions are
not restricted by length[33]. In what follows,
we use the solutions reported in Refs. [30, 31]
to establish the existence of quantum Zeno and anti-Zeno effects in
the asymmetric nonlinear optical coupler.

The remaining part of this paper is organized as follows. In Section 2, we briefly describe the momentum operator that provides a completely quantum mechanical description of the system and also note the analytic expressions of the field operators required for the present study. In Section 3, we show the existence of quantum Zeno and anti-Zeno effects in the asymmetric nonlinear optical coupler and show the spatial evolution of the Zeno parameter. The variation of the Zeno parameter has also been studied with the phase of the coherent input in the second harmonic mode of the nonlinear waveguide, phase mismatch between fundamental and second harmonic modes in the nonlinear waveguide, and linear coupling between probe and the system. Finally, the paper is concluded in Section 4.

## 2 The System and Solution

The asymmetric nonlinear optical coupler, schematically shown in Fig. 1, is prepared by combining a linear waveguide with a nonlinear waveguide of nonlinearity. As medium can produce second harmonic generation, we may say that the codirectional asymmetric nonlinear optical coupler studied here for the investigation of possibility of observing quantum Zeno and anti-Zeno effects is operated under second harmonic generation. The momentum operator for this specific coupler in interaction picture is[30]

(1) |

where stands for the Hermitian conjugate, and
is the phase mismatch between the fundamental and second harmonic
beams, and () being the linear (nonlinear) coupling
constant is proportional to the linear (nonlinear) susceptibility
. The value of is considerably
smaller than (typically
which leads to , unless an extremely strong pump is
present in the nonlinear waveguide.

In Refs. [30, 31] closed form analytic expressions for the evolution of the field operators corresponding to the Hamiltonian (1) were obtained using Sen-Mandal perturbative approach valid up to the linear power of the nonlinear coupling coefficient . The field opeartors reported there are:

(2) |

where

(3) |

where

The number operator for the second harmonic field mode in the nonlinear waveguide, i.e., mode is given by

(4) |

The initial state being the multimode coherent state with all three modes and the eigen kets of annihilation operators , and , respectively. For example, after the operation of the field operator on such a multimode coherent state we would obtain

(5) |

where are the initial number of photons in the field modes , and , respectively. Further, the coupler discussed here can operate under two conditions: spontaneous and stimulated. In the spontaneous case, and , whereas in the stimulated case, and .

## 3 Quantum Zeno and Anti-Zeno Effects

The number of photons in the second harmonic mode for the initial multimode coherent state (5) is given by

(6) |

In the absence of the probe mode, i.e., and , we have

(7) |

where . The effect of the presence of the probe mode can be given as , where is the Zeno parameter. The non-zero value of the Zeno parameter implies that the presence of the probe affects the evolution of the photon statistics of the system, i.e., the positive (negative) value of the Zeno parameter means the photon generation is increased (decreased) due to the continuous measurement of the probe on the linear mode of the nonlinear waveguide. In the present case, using Eqs. (6) and (7), we may obtain the analytic expression for the Zeno parameter as

(8) |

When the Zeno parameter becomes negative (positive), it implies the existence of quantum Zeno (anti-Zeno) effect. Here, the expression for the Zeno parameter is of the form , where we have considered . So, the Zeno parameter becomes zero in the spontaneous case, and neither the quantum Zeno effect nor the anti-Zeno effect can be observed. In the stimulated case, the functional form of the Zeno parameter suggests that we can control the quantum Zeno and anti-Zeno effects in the asymmetric coupler just by controlling the phase of the input coherent beam of the mode, i.e., changing . As the phase change of in changes the quantum Zeno effect into quantum anti-Zeno effect, and vice versa. For example, in Fig. 2, we show the spatial evolution of the Zeno parameter for (smooth blue line) and (dashed red line). While the choice illustrates the existence of quantum Zeno effect, is found to illustrate the existence of quantum anti-Zeno effect.

Further, the effect of change of the other parameters on the Zeno parameter can also be observed. Specifically, Fig. 3 shows that the transition between quantum Zeno and anti-Zeno effects can be observed with change in phase mismatch between fundamental and second harmonic modes in the nonlinear waveguide. Similarly, Fig. 4 illustrates the variation of quantum Zeno effect, as the Zeno parameter remains negative, with linear coupling between probe and the system. A similar behaviour can also be observed for quantum anti-Zeno effect. However, the variation of the Zeno parameter with change in nonlinear coupling constant is observed to be linear in nature and decreasing with nonlinear coupling constant (corresponding plot is not included in this paper).

## 4 Conclusion

Existence of quantum Zeno and anti-Zeno effects are reported in an
asymmetric nonlinear optical coupler prepared by combing a linear
and a nonlinear waveguide of nonlinearity
(cf. Fig. 2). Further, it is also shown that in the
stimulated case, it is easy to switch between quantum Zeno and anti-Zeno
effects just by controlling the phase of the second harmonic mode
in the asymmetric coupler discussed here. This flexibility to switch
between the quantum Zeno and anti-Zeno effects was not observed in
earlier studies on quantum Zeno and anti-Zeno effects in optical couplers[23, 24, 26].
Further, the effects of change in linear coupling and phase mismatch
on the spatial variation of the Zeno parameter are also illustrated.

The approach adopted here and in Ref. [33] is quite general and same may be used for the similar studies on other nonlinear optical couplers. Further, the theoretical results reported here seem to be easily realizable in experiment as the coupler used here is commercially available and the photon number statistics required to study quantum Zeno and anti-Zeno effects can be obtained using high efficiency photon number resolving detectors.

## Acknowledgments

K. T. and A. P. thank Department of Science and Technology (DST), India for support provided through the DST project No. SR/S2/LOP-0012/2010. They also thank B. Sen, A. Venugopalan and J. Peina for some useful discussions.

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