Total Quantum Zeno Effect beyond Zeno Time
Total Quantum Zeno Effect beyond Zeno Time
D. Mundarain1, M. Orszag2 and J. Stephany1
Departmento de F́ısica, Universidad Simón Boĺıvar,
Apartado Postal 89000, Caracas 1080A, Venezuela
Facultad de F́ısica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile
In this work we show that is possible to obtain Total Quantum Zeno Effect in an unstable systems
for times larger than the correlation time of the bath. The effect is observed for some particular
systems in which one can chose appropriate observables which frequent measurements freeze the
system into the initial state. For a two level system in a squeezed bath one can show that there are
two bath dependent observables displaying Total Zeno Effect when the system is initialized in some
particular states. We show also that these states are intelligent states of two conjugate observables
associated to the electromagnetic fluctuations of the bath.
I. INTRODUCTION
An interesting consequence of the fact that frequent
measurements can modify the dynamics of quantum sys-
tems is known as Quantum Zeno Effect (QZE) [1, 2, 3,
4, 5] . In general QZE is related to suppression of in-
duced transitions in interacting systems or reduction of
the decay rate in unstable systems. Also, the opposite
effect of enhancing the decay process by frequent mea-
surements has been predicted and is known as Anti-Zeno
Effect (AZE). The experimental observation of QZE in
the early days was restricted to oscillating quantum sys-
tems [6] but recently, both QZE and AZE were success-
fully observed in irreversible decaying processes.[7, 8, 9].
Quantum theory of measurements predicts reduction
of the decay rate in unstable systems when the time be-
tween successive measurements is smaller than the Zeno
Time which is known to be smaller than the correlation
time of the bath. This effect is universal in the sense that
it does not depend on the measured observable whenever
the time between measurements is very small. This ob-
servation does not preclude the manifestation of Zeno
Effect for times larger than the correlation time for some
well selected observables in a particular bath. In this
work we show that is possible for a two-level system inter-
acting with a squeezed bath to select a couple observables
whose measurements beyond the correlation time for ad-
equately prepared systems lead to the total suppression
of transitions, i.e Total Zeno Effect.
This work is organized as follows: In section (II) we
discuss some general facts and review some results ob-
tained in reference [10] which are needed for our discus-
sion. In Section (III) we define the system we deal with
and identify the observables and the corresponding initial
states which are shown to display Total Zeno Effect. In
section (V) we show that the initial states which show To-
tal Zeno Effect are intelligent spin states, i.e states that
saturate the Heisenberg Uncertainty Relation for two fic-
titious spin operators. Finally, we discuss the results in
Section (VI).
II. TOTAL ZENO EFFECT IN UNSTABLE
SYSTEMS
Consider a closed system with Hamiltonian H and an
observable A with discrete spectrum. If the initial state
of the system is the eigenstate |an〉 of A with eigenvalue
an, the probability of survival in a sequence of S measure-
ments, that is the probability that in all measurements
one gets the same result an, is
Pn(∆t, S) =
where
∆2nH = 〈an|H 2|an〉 − 〈an|H |an〉2 (2)
and ∆t is the time between consecutive measurements.
In the limit of continuous monitoring ( S → ∞,∆t → 0
and S∆t → t ), Pn → 1 and the system is freezed in the
initial state.
In an unstable system and for times larger than the
correlation time of the bath, the irreversible evolution
of the system can be described in terms of the Liouville
operator L{ρ} by using the master equation;
= L{ρ} . (3)
In this case the survival probability in a sequence of S
measurements is:
Pn(∆t, S) = (1 + ∆t 〈an|L{|an〉〈an|}|an〉)S (4)
Then, the survival probability in the limit of contin-
uous monitoring is time dependent and is easy to show
that it is given by
Pn(t) = exp {〈an|L{|an〉〈an|}|an〉t} . (5)
In fact for non zero bath correlation time (τD 6= 0) one
cannot take the continous monitoring limit and the equa-
tion (5) is an aproximation since ∆t cannot be strictly
zero and at the same time be larger than τD. In that case
this expression is valid only when the time between con-
secutive measurements is small enough but greater than
http://arxiv.org/abs/0704.1605v1
the correlation time of the bath. For mathematical sim-
plicity in what follows we consider the zero correlation
time limit and then one is allowed to take the limit of
continuous monitoring.
¿From equation (5) one observes that the Total Zeno
Effect is possible when
〈an|L{|an〉〈an|}|an〉 = 0 . (6)
Then, for times larger than the correlation time, the pos-
sibility of having Total Zeno Effect depends on the dy-
namics of the system ( determined by the interaction
with the baths), on the observable to be measured and
on the particular eigenstate of the observable chosen as
the initial state of the system.
If equation (6) is satisfied, then equation (5) must be
corrected, taking the next non-zero contribution in the
expansion of ρ(∆t). In that case the eq. (4) becomes:
Pn(∆t, s) =
1 + 〈an|L{L{|an〉〈an|}}|an〉∆t2/2
Then the survival probility for continous monitoring is
Pn(t) = exp{
〈an|L{L{|an〉〈an|}}|an〉∆t
t} (8)
In general L is proportional to γ, the decay constant for
vacuum. Then as one can see a decay rate proportional to
γ2∆t appears. and the decay time is ∝ 1
, which is in
general a number much larger than the typical evolution
time of the system since ∆t ≪ γ. This observation is
particularly important for system in which one cannot
take the zero limit in ∆t, i.e when one has a bath with
a non zero correlation time. Notice that as the spectrum
of the bath gets broader, τD becomes smaller, and one is
able to choose a smaller ∆t, approaching in this way the
ideal situation and the Total Zeno Effect.
III. TOTAL ZENO OBSERVABLES
In the interaction picture the Liouville operator for a
two level system in a broadband squeezed vacuum has
the following structure [11],
L{ρ} = 1
γ (N + 1)
2σρσ† − σ†σρ− ρσ†σ
2σ†ρσ − σσ†ρ− ρσσ†
−γMeiφσ†ρσ† − γMe−iφσρσ (9)
where γ is the vacuum decay constant and N,M =
N(N + 1) and ψ are the parameters of the squeezed
bath. Here σ and σ† are the ladder operators for a two
level system,
(σx − iσy) σ† =
(σx + iσy) (10)
with σx, σy and σz the Pauli matrices.
Let us introduce the Bloch representation of the two
level density matrix
(1 + ~ρ · ~σ) (11)
Using this representation and the master equation one
can obtain the following set of differential equation for
the components of the Bloch vector (ρx, ρy, ρz):
ρ̇x = −γ (N + 1/2 +M cos(ψ)) ρx + γM sin(ψ)ρy
ρ̇y = −γ (N + 1/2−M cos(ψ)) ρy + γM sin(ψ)ρx
ρ̇z = −γ (2N + 1)ρz − γ (12)
which has the following solutions:
ρx(t) =
ρx(0) sin
2(ψ/2) + ρy(0) sin(ψ/2) cos(ψ/2)
e−γ(N+1/2−M) t
ρx(0) cos
2(ψ/2)− ρy(0) sin(ψ/2) cos(ψ/2)
e−γ(N+1/2+M) t (13)
ρy(t) =
ρy(0) cos
2(ψ/2) + ρx(0) sin(ψ/2) cos(ψ/2)
e−γ(N+1/2−M) t
ρy(0) sin
2(ψ/2)− ρx(0) sin(φ/2) cos(ψ/2)
e−γ(N+1/2+M) t (14)
ρz(t) = ρz(0)e
−γ(2N+1)t +
2N + 1
e−γ(2N+1)t − 1
These equations describe the behavior of the system
when there are no measurements.
Consider now the hermitian operator σµ associated to
the fictitious spin component in the direction of the uni-
tary vector µ̂ = (cos(φ) sin(θ), sin(φ) sin(θ), cos(θ)) de-
fined by the angles θ and φ,
σµ = ~σ · µ̂ = σx cos(φ) sin(θ)+σy sin(φ) sin(θ)+σz cos(θ)
The eigenstates of σµ are,
|+〉µ = cos(θ/2) |+〉+ sin(θ/2) exp (iφ) |−〉 (17)
|−〉µ = − sin(θ/2) |+〉+ cos(θ/2) exp (iφ) |−〉 (18)
If the system is initialized in the state |+〉µ the survival
probability at time t is
P+µ (t) = exp {F (θ, φ) t } (19)
where
F (θ, φ) = µ〈+| L { |+〉µ µ〈+| } |+〉µ . (20)
In this case the function F (θ, φ) has the structure
F (θ, φ) = −1
γ (N + 1)
ρz(0) + ρ
z(0) +
ρ2x(0) +
ρ2y(0)
(ρz(0)− ρ2z(0)−
ρ2x(0)−
ρ2y(0)
γMρx(0)(cos(ψ)ρx(0)− sin(ψ)ρy(0))
γMρy(0)(sin(ψ)ρx(0) + cos(ψ)ρy(0)) (21)
where now ~ρ(0) = µ̂ is a function of the angles..
In figure (1) we show F (φ, θ) for N = 1 and ψ = 0 as
function of φ and θ. The maxima correspond to F (φ, θ) =
0. For arbitrary values of N and ψ there are two maxima
corresponding to the following angles:
φM1 =
π − ψ
and cos(θM ) = − 1
2 (N +M + 1/2)
φM2 =
π − ψ
+ π and cos(θM ) = − 1
2 (N +M + 1/2)
Theta
FIG. 1: F (φ, θ) for N = 1 and ψ = 0
These preferential directions given by the vectors µ̂1 =
(cos(φM1 ) sin(θ
M ), sin(φM1 ) sin(θ
M ), cos(θM )) and µ̂2 =
(cos(φM2 ) sin(θ
M ), sin(φM2 ) sin(θ
M ), cos(θM ))) define the
operators σµ1 and σµ2 which show Total Zeno Effect if
the initial state of the system is the eigenstate |+〉µ1 or
respectively |+〉µ2 , then each preferential observable has
only one eigenstate displaying Total Zeno Effect. These
eigenstates are:
|+〉µ1 =
|+〉+ i
exp{−iψ
}|−〉 (24)
|+〉µ2 =
|+〉 − i
exp{−iψ
}|−〉 (25)
The other eigenstates of the observables do not dis-
play Total Zeno Effect. As final remark is important to
observe that in the previous calculations we have ever
chosen the state |+〉µ in order to optimize the function
F (φ, θ). In fact one can select the state |−〉µ but the
final observables displaying Total Zeno Effect will be in
the same preferential directions indicated above.
IV. MASTER EQUATION AND
MEASUREMENTS
Besides of the Total Zeno effect obtained in the cases
specified previously it is also very interesting to discuss
the effect of measurements for other choices of the initial
state, the states which do not display Total Zeno Effect.
To be specific let us consider measurements of the ob-
servable σµ = ~σ · µ̂. The modified master equation with
the measurement of σµ is given by [10]:
= Pµ L {ρ} Pµ + (1− Pµ) L {ρ} (1− Pµ) (26)
where
Pµ = |+〉µ µ〈+| (27)
and L{ρ} is given by (9). This equation can be solved
using the Bloch representation of the density matrix. In
this case we can write the density operator in terms of
a second set of rotated Pauli matrices that includes the
Pauli observable which we are measuring :
(1 + ρµσµ + ρασα + ρβσβ) (28)
where σα and σβ are two Pauli matrices projected in two
orthogonal direction to the vector µ̂. During the process
of measurement one obtains always eigenvectors of σµ
observable, these eigenvectors have the property of being
zero valued for the other two observables. Then during
the measurement process the quantities ρα and ρβ are
equal to zero because these quantities coprrespond to the
mean values of the respectives observables. Then in this
case the density matrix can be written in term of one
parameter which corresponds to the mean value of the
observable that is bein measured;
(1 + ρµσµ) (29)
ρµ = 〈σµ〉 = Tr {ρσµ} (30)
Then the master equation is reduced to the following
differential equation:
ρ̇µ = Tr {ρ̇σµ}
= Tr {(Pµ L {ρ} Pµ + (1− Pµ) L {ρ} (1− Pµ)) σµ}
= Tr {L {ρ}σµ} (31)
This equation could induced to think that the evolution
with and withouth measurements are equal, but we must
remember that the density matrix in the right hand side
of (??) is the density with measurements. Substituting
the form of the density matrix during the measuring pro-
cess one can obtain a real differential equation for ρµ:
ρ̇µ = α+ βρµ (32)
where
Tr {L {1}σµ} (33)
Tr {L {σµ}σµ} (34)
In our case and measuring σµ1 one obtains
α = 2 γ (N −M + 1/2) (35)
β = −α = −2 γ (N −M + 1/2) (36)
The solution to the differential equation is
ρµ(t) = 1 + (ρµ(o) − 1) e−αt (37)
one can observe the Total Zeno Effect when ρµ(o) = 1
which correspond to having as initial state |+〉µ1 .
In figure (2) we show the evolution of 〈σµ1〉, that is the
mean value of observable σµ1 , when the system is initial-
ized in the state |+〉µ1 . without measurements (master
equation (9)) and with frequent monitoring of σµ1 (mas-
ter equation (26)). Consistently with our discussion of
frequent measurements, the system is freezed in the state
|+〉µ1 (Total Zeno Effect).
In figure (3) we show the time evolution of 〈σµ1〉 when
the initial state is |−〉µ1 without measurements and with
measurements of the same observable as in previous case.
One observes that with measurements the system evolves
from |−〉µ1 to |+〉µ1 . In general for any initial state the
system under frequent measurements evolves to |+〉µ1
which is the stationary state of Eq. ( 26) whenever we
do measurements in σµ1 . Analogous effects are observed
if one measures σµ2 . In contrast, for measurements in
other directions different from those defined by µ̂1 or µ̂2
, the system evolves to states which are not eigenstates
of the measured observables.
< σ 1 > (t)
FIG. 2: 〈σµ1(t)〉 for N = 1 and ψ = 0. Solid circles: no mea-
surements. Empty circles: with measurements. One measures
σµ1 and the initial state is |+〉µ1
.5  0
.5  1
< σ 1 > (t)
.5  0
.5  1
< σ 1 > (t)
FIG. 3: 〈σµ1(t)〉 for N = 1 y ψ = 0. Solid circles: no mea-
surements. Empty circles: with measurements. One measures
σµ1 and the initial state is |−〉µ1
V. INTELLIGENT STATES
Aragone et al [12] considered well defined angular mo-
mentum states that satisfy the equality (∆Jx∆Jy)
| 〈Jz〉 |2 in the uncertainty relation. They are called
Intelligent States in the literature. The difference with
the coherent or squeezed states, associated to harmonic
oscillators, is that these Intelligent States are not Mini-
mum Uncertainty States (MUS), since the uncertainty is
a function of the state itself.
In this section we show that the states |+〉µ1 and |+〉µ2
are intelligent states of two observables associated to the
bath fluctuations. The master equation (9) can be writ-
ten in an explicit Lindblad form
2SρS† − ρS†S − S†Sρ
using only one Lindblad operator S,
N + 1σ −
N exp {iψ}σ† (39)
S = cosh(r)σ − sinh(r) exp {iψ}σ† (40)
Obviously any eigenstate of S satisfies the condi-
tion (6). It is very easy to show that the S opera-
tor has two eigenvectors |λ±〉 with eigenvalues λ± =
M exp{iψ/2}. It is also easy to observe that these
two states are exactly the same states founded in the
previous section, |λ+〉 = |+〉µ1 and |λ−〉 = |+〉µ2 .
Consider now the standard fictitious angular momen-
tum operators for the two level system are {Jx =
σx/2, Jy = σy/2, Jz = σz/2} and also two rotated op-
erators J1 and J2 which are consistent with the electro-
magnetic bath fluctuations in phase space (see fig. 2 in
ref [10]) and which satisfy the same Heisenberg uncer-
tainty relation that Jx and Jy . They are,
J1 = exp{iψ/2Jz}Jx exp{−iψ/2Jz}
= cos(ψ/2)Jx − sin(ψ/2)Jy (41)
J2 = exp{iψ/2Jz}Jy exp{−iψ/2Jz}
= sin(ψ/2)Jx + cos(ψ/2)Jy (42)
These two operators are associated respectively with the
major and minor axes of the ellipse which represents the
fluctuations of bath.
In terms of J1 y J2 we have
J− = σ = (Jx − iJy) = exp{iψ/2}(J1 − iJ2) , (43)
J+ = σ
† = (Jx + iJy) = exp{−iψ/2}(J1 + iJ2) . (44)
Then S can be written in the following form:
S = exp{iψ/2} (cosh(r) − sinh(r)) (J1 − iαJ2) (45)
cosh(r) + sinh(r)
cosh(r) − sinh(r)
= exp{2r} (46)
Following Rashid et al ( [13]) we define a non hermitian
operator J−(α)
J−(α) =
(J1 − iαJ2)
(1 − α2)1/2
so that
S = exp{iψ/2} (cosh(r) − sinh(r)) (1− α2)1/2 J−(α)
After some algebra one obtains that
S = 2λ+ J−(α) (49)
¿From this equation one can observe that the eigen-
states of S are also eigenstates of J−(α) with eigenvalues
±1/2. It is known that the eigenstates of J−(α) are in-
telligent states of J1 and J2, i.e they satisfy the equality
condition in the Heisenberg uncertainty relation for these
observables:
∆2J1∆
2J2 =
|〈Jz〉|2
VI. DISCUSSION
We have shown that Total Zeno Effect is obtained for
two particular observables σµ1 or σµ2 , for which the az-
imuthal phases in the fictitious spin representation de-
pend on the phase of the squeezing parameter of the bath
and the polar phases depend on the squeeze amplitude.
In this sense, the parameters of the squeezed bath specify
some definite atomic directions.
When performing frequent measurements on σµ1 ,
starting from the initial state |+〉µ1 , the system freezes
at the initial state as opposed to the usual decay when
no measurements are done. On the other hand, if the
system is initially prepared in the state |−〉µ1 , the fre-
quent measurements on σµ1 will makes it evolve from
the state |−〉µ1 to |+〉µ1 . More generally, when perform-
ing the measurements on σµ1 , any initial state evolves
to the same state |+〉µ1 which is the steady state of the
master equation (26) in this situation.
The above discussion could appear at a first sight sur-
prising. However, taking a more familiar case of a two-
level atom in contact with a thermal bath at zero temper-
ature, if one starts from any initial state, the atom will
necessarily decay to the ground state. This is because
the time evolution of 〈σz〉 is the same with or without
measurements of σz . In both cases the system goes to
the ground state, which is an eigenstate of the measured
observable σz. In the limit N,M → 0, σµ1 → −σz , and
the state |+〉µ1 → |−〉z, which agrees with the known
results.
Finally, we found that the two eigentates of the two
preferential observables displaying QZE are also eigen-
states of S operator and consequently intelligent states
of J1, J2 which are rotated versions of Jx, Jy obsevables.
A. Acknowledgements
Two of the authors(D.M. and J.S.) were supported
by Did-Usb Grant Gid-30 and by Fonacit Grant No G-
2001000712.
M.O was supported by Fondecyt # 1051062 and Nu-
cleo Milenio ICM(P02-049)
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