Quantum Zeno effect with a superconducting qubit
ABSTRACT Detailed schemes are investigated for experimental verification of Quantum
Zeno effect with a superconducting qubit. A superconducting qubit is affected
by a dephasing noise whose spectrum is 1/f, and so the decay process of a
superconducting qubit shows a naturally non-exponential behavior due to an
infinite correlation time of 1/f noise. Since projective measurements can
easily influence the decay dynamics having such non-exponential feature, a
superconducting qubit is a promising system to observe Quantum Zeno effect. We
have studied how a sequence of projective measurements can change the dephasing
process and also we have suggested experimental ways to observe Quantum Zeno
effect with a superconducting qubit. It would be possible to demonstrate our
prediction in the current technology.
Quantum Zeno effect with a superconducting qubit
Yuichiro Matsuzaki1, ∗and Kouichi Semba1
1NTT Basic Research Laboratories, NTT Corporation, Kanagawa, 243-0198, Japan
Detailed schemes are investigated for experimental verification of Quantum Zeno effect with a superconduct-
ing qubit. A superconducting qubit is affected by a dephasing noise whose spectrum is 1/f, and so the decay
process of a superconducting qubit shows a naturally non-exponential behavior due to an infinite correlation
time of 1/f noise. Since projective measurements can easily influence the decay dynamics having such non-
exponential feature, a superconducting qubit is a promising system to observe Quantum Zeno effect. We have
studied how a sequence of projective measurements can change the dephasing process and also we have sug-
gested experimental ways to observe Quantum Zeno effect with a superconducting qubit. It would be possible
to demonstrate our prediction in the current technology.
Quantum Zeno effect(QZE) is one of fascinating phenom-
ena which quantum mechanics predicts. A sequence of pro-
jective measurements to an unstable system can suppress the
decay process of the state [1–4]. This phenomena will be ob-
served if the time interval of projective measurements is suf-
ficiently small and the decay behavior in the time interval is
quadratic. Although it was proved that an unstable system
shows a quadratic behavior in the initial stage of the decay 
, it is difficult to observe such quadratic decay behavior ex-
perimentally, because the time region to show such quadratic
behavior is usually much shorter than typical time resolution
of an measurement apparatus in the current technology. After
showing the quadratic decay, unstable system shows an expo-
nential decay  and QZE doesn’t occur through projective
measurements to a system which decays exponentially. Due
to such difficulty, in spite of the many effort to observe the
QZE, there was only one experimental demonstration to sup-
press the decay process of an unstable state . It is worth
mentioning that, except this experiment, all previous demon-
stration of QZE didn’t focus on a decoherence process caused
by a coupling with environment but focused on a suppression
of a unitary evolution having a finite Poincare time such as
Rabi oscillation [7–13]. Such approach to change the behav-
ior of the unitary evolution by measurements are experimen-
tally easy to be demonstrated, but is different from the original
suggestion of QZE for the decay process of unstable systems
[1–4] with a decoherence process. Throughout this paper, we
consider only such QZE to change decoherence behavior.
In this paper, we suggest a way to demonstrate Quantum
Zeno effect for the decay process of unstable system exper-
imentally with a superconducting qubit. A superconducting
qubit is one of candidates to realize quantum information pro-
cessing and, for a superconducting qubit, the quadratic de-
cay has been observed in an experiment [14, 15] , which is
necessary condition to observe QZE experimentally. More-
over, a high fidelity single qubit measurement has already
been constructed in the current technology . A super-
conducting qubit has been traditionally measured by super-
conducting quantum interference device(SQUID) . The
state of a SQUID is switched from zero-voltage state to a fi-
nite voltage state for a particular quantum state of a supercon-
ducting qubit, while no switching occurs for the other state of
the qubit. Such switching transition produces a macroscopic
signal and so one can construct a measurement for a supercon-
ducting qubit. Also, entirely-new qubit readout method such
as JBA(:Josephson Bifurcation Amplifier) has been demon-
strated [18, 19]. The JBA has advantages in its readout speed,
high sensitivity, low backaction  and absence of on-chip
dissipative process. It is also studied JBA readout mechanism
 and the projection conditions  of the superposition
state of a qubit. All these properties are prerequisite in ob-
serving the QZE. So a superconducting qubit is a promising
system to verify QZE for an unstable state.
We study a general decay process of unstable system. Al-
though a decay behavior of unstable system has been studied
and conditions for quadratic decay have been shown by sev-
and we confirm the conditions for the exponential decay and
the quadratic decay, respectively. Also, from the analytical
solution of the model, we derive a master equation for 1/f
noise. We consider an interaction Hamiltonian to denote a
coupling with an environment such as HI = λf(t)ˆA where
f(t) is a classical normalized Gaussian noise,ˆA is an oper-
ator of the system, and λ denote a coupling constant. Also,
we assume that the noise is not biased and therefore f(t) = 0
is satisfied where this over-line denotes the average over the
ensemble of the noises. In an interaction picture, by solving
the Schrodinger equation, we obtain
where ρ0= |ψ??ψ| is an initial state and ρI(t) is a state in the
interaction picture. Throughout this paper, we restrict ourself
to a case that the system Hamiltonian commutes with the op-
erator of 1/f noise as [Hs,ˆA] = 0. Firstly, we consider a case
that the correlation time of the noise τc≡?∞
tus, which is valid condition for the most of unstable systems.
By taking average over the ensemble of the noises, we obtain
ρI(t) − ρ0=
much shorter than the time resolution of experimental appara-
ρI(t) − ρ0=
arXiv:1006.2133v2 [quant-ph] 14 Jun 2010
Since the correlation time of the noise is short, we obtain
decomposed of a product of two-point correlation f(ti)f(tj),
and so we obtain
0dτ(t−τ)f(τ)f(0) ? tτc. Also,
since the noise f(t) is Gaussian, f(t1)f(t2)···f(tn) can be
where |Aν? is an eigenstate of the operator
denote a degeneracy.
?ψ|eiHstρ(t)e−iHst|ψ?, which denotes a distance between the
mixed state ρ(t) and a pure state e−iHstρ0eiHst, shows an ex-
ponential decay as following.
Secondly, when the correlation time of the noise is much
longer than the time resolution of experimental apparatus such
as a 1/f noise whose correlation time is infinite, we obtain
This leads us to obtain a master equation for 1/f noise as
ˆA and ν
So a dynamical fidelity F
the ensemble of the noises in (2), we obtain
2t2. So, by taking average over
The behavior of the dynamical fidelity becomes quadratic in
the early stage of the decay (t ?1
? 1 − λ2t2
λ) as following.
|A − A?|2|?Aν|ψ?|2|?A?ν?|ψ?|2(7)
These results show that an unstable system has an exponential
decay for t ? τc, while a quadratic decay occurs for t ? τc.
Let us summarize the QZE. Usually, to observe QZE, sur-
vival probability is chosen as a measure for the decay. How-
ever, we use a dynamical fidelity to observe the QZE rather
than a survival probability to take into account of the effect of
a system Hamiltonian. We consider a case that one performs
a sequence of projective measurements e−iHst|ψ??ψ|eiHstto
an unstable state with an interval of τ =
denotes a pure state evolved by only a system Hamiltonian
and N denotes the number of the projective measurements to
be performed during the time span t. If a dynamical fidelity
has exponential decay without projective measurements such
as F = e−Γt, the success probability to project the unsta-
ble state into e−iHst|ψ? becomes P(N) = (e−Γτ)N= e−Γt.
and so the success probability decreases exponentially as the
time increases. On the other hand, if the dynamical fidelity
has a quadratic decay without projective measurements such
as F = e−Γ2t2, we obtain the success probability to project
the unstable state into the state e−iHst|ψ? becomes as follow-
ing. P(N) =1 − Γ2τ2+ O(τ4)
increasing the number of the measurements, the success prob-
ability goes to unity, and this means that the time evolution of
this state is confined into e−iHst|ψ? which is a purely unitary
evolution without noises, and so one can observe the QZE.
It is known that a superconducting qubit is mainly affected
by two decoherence sources, a dephasing whose spectrum is
1/f and a relaxation whose spectrum is white. The 1/f noise
causes a quadratic decay to the quantum states as we have
shown. Moreover, such quadratic decay has already been ob-
served experimentally [14, 15] . On the other hand, since the
relaxation process from an excited state |0? to a ground state
|1? is governed by white noise where a high frequency is cut
off, the correlation time of the environment is extremely small
and so only an exponential decay can be observed for a relax-
ation process in the current technology. Therefore, when the
dephasing is relevant and the relaxation is negligible, it should
be possible to observe QZE with a superconducting qubit as
? 1 − Γ2 t2
N. So, by
FIG. 1: A schematic layout of a quantum state in a Bloch sphere
to show how QZE is observed with a superconducting qubit. An
initial state is prepared in |+?, and the state has an unknown rota-
tion around z axis due to a dephasing whose spectrum is 1/f. To
construct a projective measurement |+??+|, one performs a rotation
Uy around y axis, performs a projective measurement |0??0|, and
performs a rotation U†
than a dephasing time, this projective measurement |+??+| recovers
a state into the initial state with almost unity success probability even
under the effect of the dephasing.
y. If a measurement interval is much smaller
Firstly, one prepares an initial state |+? =
which is an eigenstate of ˆ σx. Secondly, in a time interval
the number of the measurement performed. Here, we assume
that an effect of a system Hamiltonian is negligible compared
with the dephasing effect. Note that we perform a selective
measurement here to consider only a case to project the state
into |+??+| and, if the state is projected into the other state,
N, one continues to perform projective measurements
2(ˆ1 1+ˆ σx) to the superconducting qubit where N is
we discard the state as a failure case. As a result, due to the
quadratic decay behavior caused by 1/f noise, the success
probability to project the state into a target state N times goes
to a unity as the number of the measurements becomes larger,
and therefore one can observe QZE. It is worth mentioning
that a direct measurement of ˆ σxwith a superconducting qubit
has not been constructed yet experimentally. So, in order to
know a measurement result of ˆ σxin the current technology,
one has to perform a rotation around y axis before and after
performing a projective measurement about ˆ σz. However, re-
cently, a coupling about ˆ σxbetween a superconducting qubit
and a flux bias control line has been demonstrated  , and
this result shows a possibility to realize a direct measurement
of ˆ σxin the near future. Since it is not necessary to perform
preliminary rotations around y axis, this direct measurement
of ˆ σxhas advantage in its readout speed. Moreover, the cou-
pling of ˆ σxcan be realized in an optimal point about environ-
mental noises and so a high fidelity projective measurement
can be performed there.
In the above discussion, the effect of the relaxation and sys-
tem Hamiltonian is not taken into account. Since they are not
always negligible in a superconducting qubit, it is necessary
to investigate whether one can observe QZE or not under the
influence of them. When considering the effect of dephas-
ing and relaxation whose spectrum are 1/f and white respec-
tively, we use a master equation as following
−2ˆ σ−ρI(t)ˆ σ+
where Γ1and Γ2denote a decoherence rate of relaxation and
dephasing respectively. In this master equation, the first part
is a Lindblad type master equation to denote a relaxation, and
the second part denotes a dephasing whose spectrum is 1/f
coming from the fluctuation of ?. Also, we assume that a sys-
tem Hamiltonian is Hs=1
because we have derived a master equation for 1/f noise only
when the system Hamiltonian commutes with the noise oper-
ator of 1/f fluctuation. We find an analytical solution of this
equation, and when the initial state is |ψ? = |+?, we obtain
ρ(t) = e−iH0t?1
It is worth mentioning that, while the 1/f noise causes a
quadratic dephasing, the relaxation effect also causes an ex-
ponential decay of the phase information, which cannot be
suppressed by projective measurements.
The success probability P(N) to project the state into a
state e−iHst|+??+|eiHstis calculated as
P(N) = (1
where T1 = (Γ1)−1and T2 = (Γ2)−1denote a relaxation
time and a dephasing time respectively, and so we obtain
ˆ σ+ˆ σ−ρI(t) + ρI(t)ˆ σ+ˆ σ−
2(Γ2)2t[ˆ σz,[ˆ σz,ρI(t)]]
2?ˆ σzfor ? ? ∆,
2Γ1t−Γ2t2|1??0| + (1 −1
P(N) ? (1−
the T1is much smaller than T2, one can observe that success
probability goes to almost a unity as one increases the number
of the projective measurements. It is worth mentioning that
T2? 1. So, as long as
FIG. 2: A success probability to perform projective measurements
into a target state under the effect of dephasing and relaxation is
plotted. The horizontal axis and the vertical axis denote the success
probability and the number of measurements respectively The lowest
line(blue) is the case of t = 60(ns), and the other lines denote a case
of time t = 50,40,30(ns), respectively. As one increases the num-
ber of the qubits, the success probability increases. Here, we assume
a relaxation time T1 = 100ns and a dephasing time T2 = 40 ns,
we assume a Hamiltonian as Hs ?
the optimal point for a superconducting qubit, and so a co-
herence time T2of this qubit becomes as small as tens of ns.
In the current technology, it takes hundreds of ns to perform
JBA and so one has to use a switching measurement to uti-
lize a SQUID which can be performed in a few second. The
state of a SQUID remains a zero-voltage when the state of a
qubit is |0?, while a SQUID makes a transition to a finite volt-
age state to produce a macroscopic signal for a state of |1?.
One of the problems of a measurement by a SQUID is that,
once a transition to a finite voltage state occurs, this switching
destroys quantum states of the qubit and following measure-
ments cannot be performed after the transition. However, as
long as the state of a qubit is |0?, the state of a SQUID remains
a zero-voltage state and so sequential measurements becomes
possible. Since one postselects a case that all of measurement
results are |0? while one discards the other case as a failure,
the SQUID can be utilized to observe QZE with the selective
measurements. Importantly, it is also possible to observe QZE
at the optimal point where T2becomes as large as µs. A recent
demonstration of coupling about ˆ σxbetween a superconduct-
ing qubit and a flux bias control line shows a possibility to
have a relevant 1/f noise caused by a fluctuation of ∆ due to
a replacement of a Josephson junction with a SQUID  .
Then, the noise operator from the 1/f fluctuation becomes ˆ σx
which commutes a system Hamiltonian at the optimal point
as H = ∆ˆ σx. So, by replacing the notation from ˆ σz(|0?) to
ˆ σx(|+?), it is possible to apply our analysis in this paper to a
case observing QZE at the optimal point. Moreover, since T2
at the optimal point is much longer than a necessary time to
2?ˆ σzwhich is far from
perform JBA, a sequence of measurements are possible for all
measurement results. This motivates us to study a verification
of QZE without postselection as following.
Finally, we discuss how to observe QZE without postselec-
tion of measurement results, which can be realized by JBA.
Such non-selective measurements to a single qubit is modeled
asˆE(ρ) = |φ+??φ+|ρ|φ+??φ+|+|φ−??φ−|ρ|φ−??φ−| where
|φ+? = e−iHst|+? and |φ−? = e−iHst|−? are orthogonal
with each other. So, when performing this non-selective mea-
surement with a time interval τ =
dephasing and relaxation, we obtain
Nunder the influence of
ρ(N,t) = e−iHst?1
where we use a result in (9). Note that a non-diagonal term is
decayed by the white noise and 1/f noise, and only the decay
caused from 1/f noise is suppressed by the measurements.
FIG. 3: A decay behavior of a phase term |?0|ρ|1?| under the effect of
denote the time(ns) and the number of measurements, respectively.
The decay is suppressed by the frequent measurements and so QZE
is observed. Here, we assume a relaxation time T1 = 1 µs and a
dephasing time T2 = 400 ns, respectively.
A possible experimental way to remove out the effect of the
white noise is measuring ?0|ρ(N,t)|1? and ?0|ρ(1,t)|1? sep-
arately by performing a tomography, and plotting the value
of ?0|ρ(N,t)|1?/?0|ρ(1,t)|1? = e−
t. As a result one can observe the suppression of the de-
phasing caused by 1/f noise through frequent measurements.
In conclusion, we have studied detailed schemes for experi-
mental verification of QZE to a decay process with a super-
conducting qubit. Since a superconducting qubit is affected
by a dephasing whose spectrum is 1/f, the dephasing pro-
cess shows a quadratic decay which is suitable for an exper-
imental demonstration for QZE, while the relaxation process
in a superconducting qubit has an exponential decay which
causes unwanted dephasing effect to observe QZE. We have
suggested a way to observe QZE experimentally even under
N(T2)2for a fixed time
an influence of relaxation. Our prediction should be feasible
even in the current technology. Authors thank H. Nakano for a
useful discussion. This work was done during Y. Matsuzaki’s
short stay at NTT corperation, and also this work was sup-
ported in part by Grant-in-Aid for Scientific Research of Spe-
cially Promoted Research # 18001002 by MEXT, and Grant-
in-Aid for Scientific Research (A) # 22241025 by JSPS.
∗Electronic address: matsuzaki@ASone.c.u-tokyo.ac.jp
 B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 3491
 A. Kofman and G. Kurizki(London, Nature 405, 546 (2000).
 R. J. Cook, Phys. SCR. T21, 49 (1988).
 P. Facchi, H. Nakazato, and S. Pascazio, Phys. Rev. Lett. 86,
 H. Nakazato, M. Namiki, and S. Pascazio, Int. J. Mod. B 10,
 M. C. Fischer, B. Gutierrez, and M. Raizen, Phys. Rev. Lett. 87,
 C. Balzer,T. Hannemann,
W. Neuhauser, and P. E. Toschek, Opt. Commun 211, 235
 C. Balzer, R. Huesmann, W. Neuhauser, and P. E. Toschek, Opt.
Commun 180, 115 (2000).
 K. Molhave and M. Drewsen, Phys. Lett. A 268, 45 (2000).
 B. Nagels, L. Hermans, and P. Chapovsky, Phys. Rev. Lett. 79,
 E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley,
W. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. 97, 260402
 P. Toschek and C. Wunderlich, New J. Phys. 14, 387 (2001).
 L. Xiao and J. Jones, Phys. Lett. A 359, 424 (2006).
 F. Yoshihara, K. Harrabi, A. Niskanen, and Y. Nakamura, Phys.
Rev. Lett. 97, 167001 (2006).
 K. Kakuyanagi, T. Meno, S. Saito, H. Nakano, K. Semba,
H. Takayanagi, F. Deppe, and A. Shnirman, Phys. Rev. Lett.
98, 047004 (2007).
 J. Clarke and F. Wilhelm, Nature 453, 1031 (2007).
 J. Clarke and A. Braginski, The SQUID handbook. Vol. 1,
Fundamentals and technology of SQUIDs and SQUID systems
 I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe,
C. Rigetti, L. Frunzio, and M. H. Devoret, Phys. Rev. Lett. 93,
 A. Lupascu, S. Saito, T. Picot, P. C. de Groot, C. J. P. M. Har-
mans, and J. E. Mooij, Nature Physics 3, 119 (2007).
 H. Nakano, S. Saito, K. Semba, and H. Takayanagi, Phys. Rev.
Lett. 93, 207002 (2004).
 K. Kakuyanagi, H. Nakano, S. Kagei, R. Koibuchi, S. Saito,
A. Lupascu, and K. Semba (2010), arXiv:1004.0182v2.
 Y. Makhlin and A. Shnirman, Phys. Rev. Lett. 92, 178301
 J. M. Martinis, S. Nam, J. Aumentado, , and K. M. Lang, Phys.
Rev. B 67, 094510 (2003).
 L. S. Schulman, J. Phys. A 30, L293 (1997).
 A. Fedorov, A. Feofanov, P. Macha, P. Forn-D´ ıaz, C. Harmans,
and J. Mooij, arXiv:1004.1560 (2010).
D. Reis,C. Wunderlich,