arXiv:quant-ph/0101037v1 8 Jan 2001
Zeno dynamics yields ordinary constraints
P. Facchi,1S. Pascazio,2,3A. Scardicchio2and L. S. Schulman4
1Atominstitut der¨Osterreichischen Universit¨ aten, Stadionallee 2, A-1020, Wien, Austria
2Dipartimento di Fisica, Universit` a di Bari I-70126 Bari, Italy
3Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy
4Physics Department, Clarkson University, Potsdam, NY 13699-5820, USA
(February 1, 2008)
The dynamics of a quantum system undergoing frequent measurements (quantum Zeno effect)
is investigated. Using asymptotic analysis, the system is found to evolve unitarily in a proper
subspace of the total Hilbert space. For spatial projections, the generator of the “Zeno dynamics”
is the Hamiltonian with Dirichlet boundary conditions.
PACS numbers: 03.65.Bz; 03.65.Db; 02.30.Mv
Frequent measurement can slow the time evolution of a quantum system, hindering transitions to states different
from the initial one [1,2]. This phenomenon, known as the quantum Zeno effect (QZE), follows from general features
of the Schr¨ odinger equation that yield quadratic behavior of the survival probability at short times .
However, a series of measurements does not necessarily freeze everything. On the contrary, for a projection onto
a multi-dimensional subspace, the system can evolve away from its initial state, although it remains in the subspace
defined by the “measurement.” This continuing time evolution within the projected subspace we call quantum Zeno
dynamics. It is often overlooked, although it is readily understandable in terms of a theorem on the QZE  that we
will recall below.
The aim of this article is to show that Zeno dynamics yields ordinary constraints. Under general conditions, the
evolution of a system undergoing frequent measurements takes place in a proper subspace of the total Hilbert space
and the wave function satisfies Dirichlet boundary conditions on the domain defined by the measurement process.
Moreover, the evolution is generated by a self-adjoint Hamiltonian and remains reversible within the Zeno subspace.
This shows that the irreversibility is not compulsory, as noted in .
The QZE has been tested on oscillating systems  and there has been a recent observation of non-exponential
decay (leakage through a confining potential) at short times . Although these experiments have invigorated studies
on this issue, they deal with one-dimensional projectors (and therefore one-dimensional Zeno subspaces): the system is
forced to remain in its initial state. This is also true for interesting quantum optical applications . The present work
therefore enters an experimentally uncharted area, although the property of being a multidimensional measurement
is not at all exotic, and in particular applies to the most basic quantum measurement: position. The latter is the
paradigm for the present work.
We introduce notation. Consider a quantum system, Q, whose states belong to the Hilbert space H and whose
evolution is described by the unitary operator U(t) = exp(−iHt), where H is a time-independent semi-bounded
Hamiltonian. Let E be a projection operator that does not commute with the Hamiltonian, [E,H] ?= 0, and EH = HE
the subspace defined by it. The initial density matrix ρ0of system Q is taken to belong to HE:
ρ0= Eρ0E, Trρ0= 1. (1)
The state of Q after a series of E-observations at times tj= jT/N (j = 1,···,N) is
ρ(N)(T) = VN(T)ρ0V†
N(T),VN(T) ≡ [EU(T/N)E]N
and the probability to find the system in HE(“survival probability”) is
P(N)(T) = Tr
Our attention is focused on the limiting operator
V(T) ≡ lim
Misra and Sudarshan  proved that if the limit exists, then the operators V(T) form a one-parameter semigroup,
and the final state is
ρ(T) = lim
N→∞ρN(T) = V(T)ρ0V†(T). (5)
The probability to find the system in HEis
P(T) ≡ lim
N→∞P(N)(T) = 1. (6)
This is the QZE. If the particle is constantly checked for whether it has remained in HE, it never makes a transition
A few comments are in order. First, the final state ρ(T) depends on the characteristics of the model investigated and
on the measurement performed (the specific forms of VNand V depend on E). Moreover, the physical mechanism that
ensures the conservation of probabilities within the relevant subspace hinges on the short time behavior of the survival
probability: probability leaks out of the subspace HE like t2for short times. Since the infinite-N limit suppresses
this loss, one can inquire under what circumstances V(T) actually forms a group, yielding reversible dynamics within
the Zeno subspace.
In this article we show that Zeno dynamics for a position measurement yields a particular kind of dynamics within
the subspace defined by that measurement, namely unitary evolution with the restricted Hamiltonian, and with the
domain of that (self adjoint) operator defined by Dirichlet boundary conditions. This elucidates the reversible features
of the evolution for a wide class of physical models. As a spinoff, our proof provides a rigorous regularization of the
example considered in .
We start with the simplest spatial projection. Q is a free particle of mass m on the real line, and the measurement
is a determination of whether or not it is in the interval A = [0,L] ⊂ R. The Hamiltonian and the corresponding
evolution operator are
U(t) = exp(−itH). (7)
H is a positive-definite self-adjoint operator on L2(R) and U(t) is unitary. We study the evolution of the particle
when it undergoes frequent measurements defined by the projector
dx χA(x)|x??x|, (8)
where χAis the characteristic function
?1 for x ∈ A = [0,L]
Thus EAis the multiplication operator by the function χA. We study the following process. We prepare a particle
in a state with support in A, let it evolve under the action of its Hamiltonian, perform frequent EAmeasurements
during the time interval [0,T], and study the evolution of the system within the subspace HEA= EAH. We will show
that the dynamics in HEAis governed by the evolution operator
V(T) = exp(−iTHZ)EA,
2m+ VA(x),VA(x) =
?0 for x ∈ A
This is the operator obtained in the limit (4). In other words, the system behaves as if it were confined in A by rigid
walls, inducing the wave function to vanish on the boundaries x = 0,L (Dirichlet boundary conditions).
We now prove our assertion. Let the particle be initially (t = 0) in A. We recall the propagator in the position
G(x,t;y) ≡ ?x|EAU(t)EA|y? = χA(x)?x|U(t)|y?χA(y)
?im(x − y)2
where t = T/N is the time when the first measurement is carried out and the particle found in EA. To study the
properties of G we choose a complete basis in L2(A)
un(x) = ?x|un? =
(n = 1,2,...) .(13)
When these functions define the eigenbasis of H, H is self adjoint and
H|un? = En|un? ,En=?2n2π2
so that H has Dirichlet boundary conditions. The matrix elements of G are
Gmn(t) ≡ ?um|EAU(t)EA|un? =
?im(x − y)2
un(y) . (15)
Let r = x − y, R = (x + y)/2 and λ = m/2?t, so that
dr um(R + r/2)un(R − r/2)exp?iλr2?. (16)
where r0(R) = L − |L − 2R|. We now use the asymptotic expansion
dx f(x)eiλx2= gstat(λ) + gbound(λ),(17)
gstat(λ) = f(0) +
4λf′′(0) + O(λ−2), (18)
2ia√iπλ[f(a) + f(−a)] + O(λ−3/2) (19)
are the contributions of the stationary point x = 0 and of the boundary, respectively. By expanding the inner integral
in (16) as in (17)–(19) one gets
dr um(R + r/2)un(R − r/2)exp?iλr2?
= um(R)un(R) +
dr2[um(R + r/2)un(R − r/2)]r=0+ O(λ−3/2). (20)
(Note that the contribution of the boundary vanishes identically.) Using this result, we integrate by parts and after
a straightforward calculation obtain
= ?um|un? −it
2m|un? + O(t3/2)
+ O(t3/2). (21)
With this formula we can carry out the limit required in Eq. (4). At time T, in the energy representation, the
Gmn(T) = ?m|V(T)|n?
This is precisely the propagator of a particle in a square well with Dirichlet boundary conditions and proves (10)–(11).
Note that the t3/2contribution in (21) drops out in the N → ∞ limit since it appears as N ×O(1/N3/2). It is worth
emphasizing that although this result has been proved using the basis (13), the information obtained is a property of
the propagator, and therefore holds true in general. Our choice was a matter of convenience. With a different basis
and nonvanishing boundary conditions, the dominant contribution of order λ−1/2in gbound(λ) would have given a
nondiagonal term in (20)–(22), showing that the chosen basis is not the right eigenbasis of HZ(i.e., for the limiting
This result can be generalized to a wide class of systems. Let
2m+ V,U(t) = exp(−itH), (23)
where V is a regular potential. (It may be unbounded from below, for example V (x) = Fx, although within A the
total Hamiltonian H should be lower bounded.) The measurement performed is again application of the projector
(8) and we study the short-time propagator
G(x,t;y) = χA(x)
?im(x − y)2
−it(V (x) + V (y))
The basis to be used for representing the propagator is again that of the Hamiltonian with Dirichlet boundary
conditions in [0,L]
|un? = En|un?,un(x)|x=0,L= 0. (25)
As before (r = x − y, R = (x + y)/2, λ = m/2?t)
Using the asymptotic expansion (17)–(19), a calculation identical to the previous one yields
dR2+ V (R)
= ?un|um? −it
|um? + O(t3/2)
and the limiting propagator at time T again reads
Gmn(T) = δnme−iTEn/?.(28)
Again, the simplicity of the proof is due to the choice of the basis (25), satisfying Dirichlet boundary conditions.
We have also obtained an improvement with respect to earlier approaches to this problem. The aforementioned
theorem by Misra and Sudarshan  requires that the Hamiltonian be lower bounded from the outset. However, we
need only require that the Hamiltonian be lower bounded in the Zeno subspace. Despite the fact that for unbounded
potentials (like V = Fx) H may not be lower bounded on the real line, the evolution in the Zeno subspace is governed
by the Hamiltonian
2m+ VA(x),VA(x) =
for x ∈ A
that can be lower bounded in A, yielding a bona fide group for the evolution operators.
The above calculation and conclusions can readily be generalized to higher dimensions, so long as the measurement
projects onto a set in Rnwith a smooth boundary (except at most a finite number of points). We again take
x,y ∈ Rnand let the measurement-projection be defined by A ⊂ Rn, which is not necessarily bounded. Again setting
r = x − y, R = (x + y)/2, Eq. (26) becomes
dr un(R + r/2)e−itV (R+r/2)/2?un(R − r/2)e−itV (R−r/2)/2?eiλr2, (30)
where D(R) is the transformed integration domain for r. The n-dimensional asymptotic expansions (17)–(19) read
gstat(λ) = f(0) +
4λ△f(0) + O(λ−2),(31)
gbound(λ) = O(λ−1/2) × f(boundary) + O(λ−3/2) (32)
and the theorem follows again because f vanishes on the boundary (Dirichlet). The proof is readily generalized
to non-convex and/or multiply-connected projection domains, the only difficulty being that the integration domain
in (30) must be broken up. It is interesting to notice that at those points at which the boundary fails to have a
continuously turning tangent plane, the asymptotic contribution of the discontinuity in the boundary in (32) would
be of yet higher order in λ.
In conclusion, for traditional position measurements, namely projections onto spatial regions, we have shown that
Zeno dynamics uniquely determines the boundary conditions, and that they turn out to be of Dirichlet type. This
is also relevant for problems related to the consistent histories approach [11–13], where different boundary conditions
were proposed. For us, the frequent imposition of a projection, the traditional idealization of a measurement, provides
all the decohering of interfering alternatives that is needed. On the other hand, in the works just cited one seeks
a restricted propagator (using the path decomposition expansion ) and such interference can occur. A second
issue discussed in these works (especially ) is the validity of the Trotter product formula in certain cases. Again,
our implicit use of this formula (in Eq. (12), etc.) is nothing more than its use for a free particle (or a particle
in an ordinary potential). This is because the propagator of Eq. (12) provides time evolution under a sequence of
operations: the particle evolves freely (on the entire line) for a time t, and then one applies the projection (left and
right multiplication by the operator EA).
The present work has implications for the notion of “hard wall,” as used for example in elementary quantum
mechanics. Everyone would agree (we expect) that this notion is an idealization. However, in many cases where this
idealization is useful the “wall” is dynamic rather than static, the result of some fluctuating atomic presence. In
this article we have a sufficient condition for the validity of this notion in a dynamic situation. Moreover, there is
a quantitative framework (arising from our asymptotic analysis and finite-time-interval QZE effects) for gauging the
effects of less than perfect hard walls.
This work is supported in part by the TMR-Network of the European Union “Perfect Crystal Neutron Optics”
ERB-FMRX-CT96-0057 and by the United States NSF under grant PHY 97 21459.
 A. Beskow and J. Nilsson, Arkiv f¨ ur Fysik 34, 561 (1967); L. A. Khalfin, Zh. Eksp. Teor. Fiz. Pis. Red. 8, 106 (1968)
[JETP Lett. 8, 65 (1968)].
 B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
 For a review, see H. Nakazato, M. Namiki and S. Pascazio, Int. J. Mod. Phys. B10, 247 (1996); D. Home and M. A. B.
Whitaker, Ann. Phys. 258, 237 (1997).
 P. Facchi, V. Gorini, G. Marmo, S. Pascazio and E.C.G. Sudarshan, Phys. Lett. A275, 12 (2000).
 R.J. Cook, Phys. Scr. T21, 49 (1988); W.H. Itano, D.J. Heinzen, J.J. Bollinger and D.J. Wineland, Phys. Rev. A41, 2295
(1990); T. Petrosky, S. Tasaki and I. Prigogine, Phys. Lett. A151, 109 (1990); Physica A170, 306 (1991); A. Peres and A.
Ron, Phys. Rev. A42, 5720 (1990); S. Pascazio, M. Namiki, G. Badurek and H. Rauch, Phys. Lett. A179 (1993) 155; T.P.
Altenm¨ uller and A. Schenzle, Phys. Rev. A49, 2016 (1994); S. Pascazio and M. Namiki, Phys. Rev. A50 (1994) 4582; L.S.
A. Beige and G. Hegerfeldt, Phys. Rev. A53, 53 (1996); A.G. Kofman and G. Kurizki, Phys. Rev. A54, R3750 (1996);
L.S. Schulman, Phys. Rev. A57, 1509 (1998).
 S.R. Wilkinson et al., Nature 387 (1997) 575.
 A. Luis and J. Periˇ na, Phys. Rev. Lett. 76, 4340 (1996); M.B. Plenio, P.L. Knight and R.C. Thompson, Opt. Comm. 123,
278 (1996); A. Luis and L. L. S´ anchez–Soto, Phys. Rev. A 57, 781 (1998); K. Thun and J. Peˇ rina, Phys. Lett. A 249, 363
(1998); J.ˇReh´ aˇ cek et al, Phys. Rev. A62, 013804 (2000).
 R.P. Feynman and G. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
 L. S. Schulman, Techniques and Applications of Path Integration (John Wiley, New York, 1981; reissued in paperback Download full-text
 N. Bleistein and R. A. Handelsman Asymptotic Expansions of Integrals (Dover Publications, New York, 1986).
 N. Yamada and S. Takagi, Prog. Theor. Phys. 85, 985 (1991); 86, 599 (1991); 87, 77 (1992); N. Yamada, Phys. Rev. A54,
 J.B. Hartle, Phys. Rev. D44, 3173 (1991).
 I.L. Egusquiza and J.G. Muga, Phys. Rev. A62, 032103 (2000).
 A. Auerbach, S. Kivelson and D. Nicole, Phys. Rev. Lett. 53, 411 (1984).