Sufficiency Criterion for the Validity of the Adiabatic Approximation
Sufficiency Criterion for the Validity of the Adiabatic
Approximation
D. M. Tong1,2 ∗ , K. Singh1, L. C. Kwek1,3, and C. H. Oh1†
1 Department of Physics, National University of Singapore,
10 Kent Ridge Crescent, Singapore 119260, Singapore
2Department of Physics, Shandong Normal University, Jinan 250100, China
3National Institute of Education, Nanyang Technological University,
1 Nanyang Walk, Singapore 639798, Singapore
(Dated: October 29, 2018)
Abstract
We examine the quantitative condition which has been widely used as a criterion for the adiabatic
approximation but was recently found insufficient. Our results indicate that the usual quantitative
condition is sufficient for a special class of quantum mechanical systems. For general systems, it
may not be sufficient, but it along with additional conditions is sufficient. The usual quantitative
condition and the additional conditions constitute a general criterion for the validity of the adiabatic
approximation, which is applicable to allN−dimensional quantum systems. Moreover, we illustrate
the use of the general quantitative criterion in some physical models.
PACS numbers: 03.65.Ta, 03.65.Vf
∗ Electronic address: phytdm@nus.edu.sg
† Electronic address: phyohch@nus.edu.sg
http://arxiv.org/abs/0704.1184v1
The adiabatic theorem reads that if a quantum system with a time-dependent nonde-
generate Hamiltonian H(t) is initially in the n-th instantaneous eigenstate of H(0), and
if H(t) evolves slowly enough, then the state of the system at time t will remain in the
n-th instantaneous eigenstate of H(t) up to a multiplicative phase factor. The theorem is
a useful tool [1, 2, 3, 4, 5, 6, 7] but its practical application relies on the criterion of the
“slowness” required by the theorem. In the literature, the “slowness” is usually encoded by
the quantitative condition,
〈En(t)|Ėm(t)〉
En(t)−Em(t)
≪ 1, m 6= n, t ∈ [0, τ ] (1)
where Em(t) and |Em(t)〉 are the eigenvalues and eigenstates of H(t), and τ is the total
evolution time. This quantitative condition had been deemed to be a sufficiency criterion,
but it was recently found insufficient. In order to resolve the counterexample raised in [8],
we showed that fulfilling only the quantitative condition cannot guarantee the validity of
the adiabatic approximation[9]. While this explains the counterexample in [8], it raises the
obvious question: In what situations would the criterion be sufficient and more importantly
how can it be extended to general cases? There have been some attempts to address different
aspects of this problem[10, 11, 12, 13], but it remains largely unresolved.
In order to resolve the problem, we trace the quantitative condition to its source, and
we note that the condition has never been convincingly proven. It looks likely that the
condition was first derived from some special quantum systems and it was extended to
systems beyond its range of applicability. For instance, in Refs. [3, 4], condition (1) was
obtained by assuming a first order approximation and by requiring both Em(t) − En(t)
and 〈Em(t)|Ėn(t)〉 to be constants. However, such a proof is not convincing because a
first order approximation may be taken as a good approximation of the exact value only
if all the higher order corrections are proven to be much smaller. We note that this is not
the case here. This has also been pointed out in Refs. [10, 11]. Therefore, even if the
condition is sufficient for the special quantum systems, a convincing proof is still necessary.
In any case, the sufficiency criterion for general systems is grossly lacking. It should be
emphasized that the lack of a sufficiency criterion weakens the applicability of the adiabatic
theorem. In the present paper, we address this sufficiency criterion issue. Firstly, we furnish
a new proof to show that the quantitative condition is indeed a sufficiency criterion for
the adiabatic approximation in the quantum systems which satisfy the requirement of both
Em(t)− En(t) and 〈Em(t)|Ėn(t)〉 being constant. Secondly, to extend its validity, we show
that the the quantitative condition along with some additional conditions is sufficiency
for general systems. Thus, the usual quantitative condition and the additional conditions
constitute a general criterion for the adiabatic approximation, which is applicable to all
N−dimensional quantum systems. Moreover, we illustrate the use of the general criterion
in some physical models.
Let us consider an N -dimensional quantum system with the Hamiltonian H(t). The
instantaneous nondegenerate eigenvalues and orthonormal eigenstates of H(t), denoted as
Em(t) and |Em(t)〉 respectively, are defined by
H(t)|Em(t)〉 = Em(t)|Em(t)〉, m = 1, . . . , N. (2)
|Em(t)〉 is determined by Eq. (2) up to a phase factor. Hereafter, we choose it such that
〈Em(t)|Ėm(t)〉 = 0[14]. If we assume that the system is initially in the n−th eigenstate
|ψ(0)〉 = |En(0)〉, then its state at time t, |ψ(t)〉, is dictated by the Schrödinger equation
|ψ(t)〉 = H(t)|ψ(t)〉. (3)
In the basis {|Em(t)〉}, |ψ(t)〉 can be expanded as
|ψ(t)〉 =
cm(t)e
′)dt′
|Em(t)〉, (4)
where cm(t) = e
′)dt′〈Em(t)|ψ(t)〉 are the time-dependent coefficients. Substituting it
into the Schrödinger equation, we obtain
dcm(t)
l 6=m
〈Em|Ėl〉e
ωmldt
cl(t) = 0, (5)
which leads to
cm(t) = δmn −
l 6=m
〈Em|Ėl〉e
ωmldt
′)dt′, (6)
where Em ≡ Em(t), |Em〉 ≡ |Em(t)〉, ωml ≡ Em(t) − El(t), and m = 1, 2, ..., N . Here,
n in Eq. (6) is the index of the initial state |En(0)〉. We want to ascertain the criterion
under which the adiabatic approximation is valid, i.e., the condition(s) for which the fidelity
F = |〈En(t)|ψ(t)〉| = |cn(t)| ≈ 1.
We first discuss the quantum systems for which both ωml and 〈Em|Ėl〉 are constants[15].
In this case, from Eq. (6), we have, by partial integration,
cn(t) = 1 + i
〈En|Ėm〉
eiωnmtcm(t)−
eiωnmt
ċm(t
′)dt′
. (7)
Substituting ċm(t) from Eq. (5) into the above equation, we obtain
cn(t) = 1 + i
〈En|Ėm〉
eiωnmtcm(t)
l 6=m
〈En|Ėm〉
〈Em|Ėl〉
eiωnlt
′)dt′. (8)
Noting that |cm(t)| ≤ 1, we have from Eq. (8)
1− |cn(t)| ≤
〈En|Ėm〉
l 6=m
∣〈Em|Ėl〉
∣ |Il|
 , (9)
where Il =
iωnlt
′)dt′.
Clearly, if the integral Il is bounded by a finite number, the quantitative condition (1)
can sufficiently guarantee that 1 − |cn(t)| ≪ 1. We now show that this is indeed the case.
To this end, by letting cl(t) = cl(t)e
iωnlt and substituting it into Eq. (5), we have
dcm(t)
− iωnmcm(t) +
l 6=m
〈Em|Ėl〉cl(t) = 0. (10)
Since Eq. (10) is a system of differential equations, the general solution of cm(t) comprise N
special solutions in the form ame
iλt, where am and λ are time-independent constants. They
are determined by the equations,
(ωnm − λ) am + i
l 6=m
〈Em|Ėl〉al = 0, (11)
where m = 1, 2, ..., N. Solving Eq. (11), we may obtain λ = λ1, λ2, ..., λN , where λj
are nonzero real numbers[16]. For each λj, there is a solution cmj(t) = amje
iλjt. All the
N independent solutions lead to the general solution cm(t) =
j=1 pjamje
iλjt, where the
coefficients pj are determined by the initial conditions cm(0) = δmn. Then, we have |Il| =
0 cl(t
′)dt′
pjamj
eiλjt − 1
∣ ≤ 2
pjamj
∣ , where the latter term is a finite
number, independent of time t. This completes the proof that the quantitative condition (1)
is a sufficiency criterion for the quantum systems which satisfy the requirement that both
ωml and 〈Em|Ėl〉 are constants.
We now discuss general quantum systems. Let us return to Eq. (6). We have, by partial
integration,
cn(t) = 1 + i
〈En|Ėm〉
ωnmdt
cm(t)
〈En|Ėm〉
ωnmdt
′)dt′
〈En|Ėm〉
ωnmdt
ċm(t
′)dt′. (12)
Substituting Eq. (5) into Eq. (12), we have
cn(t) = 1 + i
〈En|Ėm〉
ωnmdt
cm(t)
〈En|Ėm〉
ωnmdt
′)dt′
l 6=m
〈En|Ėm〉
〈Em|Ėl〉e
ωnldt
′)dt′. (13)
In the general case, although it is difficult to estimate exactly the values of the integrals in Eq.
(13) as we did in the above special case, it is still possible to obtain bounds on the integrals,
which will lead to the sufficiency criterion. Noting that |cm(t)| ≤ 1 and
ωnmdt
= 1, we
〈En|Ėm〉
ωnmdt
′)dt′
〈En|Ėm〉
dt′, (14)
〈En|Ėm〉
〈Em|Ėl〉e
ωnldt
′)dt′
〈En|Ėm〉
∣〈Em|Ėl〉
∣ dt′. (15)
From Eqs. (13), (14), and (15), we obtain
1− |cn(t)| ≤
〈En|Ėm〉
〈En|Ėm〉
l 6=m
〈En|Ėm〉
∣〈Em|Ėl〉
∣ dt′. (16)
Since the sums on the right hand side of expression (16) are finite terms, the approximation
1 − |cn(t)| ≪ 1 is guaranteed if each of the terms is small. This is met if the following
conditions
〈En(t)|Ėm(t)〉
En(t)−Em(t)
≪ 1, t ∈ [0, τ ], (17)
〈En(t)|Ėm(t)〉
En(t)− Em(t)
dt≪ 1, (18)
〈En(t)|Ėm(t)〉
En(t)−Em(t)
∣〈Em(t)|Ėl(t)〉
∣ dt≪ 1, (19)
are satisfied, where m 6= n and τ is the total evolution time for which the adiabatic ap-
proximation is valid, t ∈ [0, τ ]. Expression (A) is just the well-known quantitative condition
(1), and expressions (B) and (C) are two additional conditions, which set a bound on the
total evolution time. Physically, if a quantum system satisfying condition (A) is initially
in its n−th eigenstate, it will remain close to its n−th instantaneous eigenstate during the
initial short time but it may deviate from its instantaneous eigenstate at a later time. The
additional conditions provide a time scale, τ , for which the state remains close to the instan-
taneous eigenstate. τ can be obtained after calculating the integrals in Eqs. (18) and (19).
In some special cases, the integrals may be easily evaluated. For instance, if En(t)−Em(t) is
a monotonic function of t, the integral in Eq. (18) can be carried out. In some other cases,
we may not be able to evaluate the integrals analytically. In such instance, we may simplify
the conditions by appealing to estimations. In any case, the stronger expressions
〈En(t)|Ėm(t)〉
En(t)−Em(t)
τ ≪ 1, (20)
〈En(t)|Ėm(t)〉
En(t)− Em(t)
∣〈Em|Ėl〉
τ ≪ 1, (21)
can always cover conditions (B) and (C) respectively, where |f(t)|M means the maximal
modulus of f(t) for t ∈ [0, τ ]. As conditions (b) and (c) are stronger than (B) and (C), the
latter should be preferentially used when possible.
The quantitative condition (A) (i.e.(1)) and the additional conditions (B) and (C) con-
stitute a general quantitative criterion for the adiabatic approximation. The general cri-
terion can sufficiently guarantee the validity of the approximation and it is applicable to
all N−dimensional quantum systems. That is, if a quantum system, initially in the eigen-
state |En(0)〉, fulfills the general criterion, it will remain, with high probability, in the n-th
instantaneous eigenstate |En(t)〉 up to a phase factor. The fidelity between the approx-
imate state and the exact state may be estimated by Eq. (16). It is interesting to use
this criterion to reexamine the counterexample furnished with two related Hamiltonians
Ha(t) = iU̇(t)U+(t) and Hb(t) = iU̇+(t)U(t) in [8, 9]. Suppose the eigenstate |Eim〉 of
H i(t) (i = a, b) has been properly chosen such as 〈Eim|Ė
m〉 = 0. We may have the relation,
〈Ebn|Ė
m〉 = e
(Ean−E
〈Ean|Ė
m〉, which result in that condition (B) cannot be satisfied for
Hb(t), in general, if it is satisfied for Ha(t). Hence, the counterexample is ruled out from
the adiabatic systems. We now apply the general criterion to some quantum systems.
Firstly, we specialize the general criterion for the quantum systems in which En(t)−Em(t)
is a monotonic function of t. Many interesting adiabatic systems may belong to this class.
In this case, we have
〈En|Ėm〉
〈En|Ėm〉
ωnm(τ)
ωnm(0)
〈En|Ėm〉
τ. (22)
Since | lnωnm(τ)/ωnm(0)| is a finite number and hence the first term is small under condition
(A), the additional condition (B) can be written as
〈En|Ėm〉
τ ≪ 1. (23)
As we cannot simplify the integral in (19) here, we use the stronger expression (c). Therefore,
conditions (A), (B1) and (c) constitute the sufficiency criterion. Since in real physical
experiments the total evolution time τ is usually finite, condition (c) is automatically ensured
by condition (A). Hence, in this case, we may take (A) and (B1) as the adiabatic criterion.
Secondly, we consider a quantum system defined by the parameterized Hamiltonian H(s),
where s = t/T, t ∈ [0, T ]. The well-known proofs of adiabatic theorem given in Refs. [5, 6]
were carried out by using such a Hamiltonian. We now apply the general criterion to the
system. By substituting t = Ts into conditions (A), (B), and (C), we obtain
〈En|Ėm〉
En −Em
〈En(s)|Ėm(s)〉
En(s)− Em(s)
〈En|Ėm〉
En − Em
〈En(s)|Ėm(s)〉
En(s)− Em(s)
(En(s)−Em(s))
En(s)− Em(s)
〈En(s)|Ėm(s)〉
En(s)− Em(s)
〈En|Ėm〉
En − Em
∣〈Em|Ėl〉
∣ dt ≤
〈En(s)|Ėm(s)〉
En(s)−Em(s)
∣〈Em(s)|Ėl(s)〉
. (24)
Since all the terms on the right of the above expressions can be arbitrarily small as T
becomes large, conditions (A), (B), and (C) are met if T is large enough. We then arrive
at the conclusion that the adiabatic approximation is always valid for quantum systems
defined by Hamiltonians of the forms H( t
) with t ∈ [0, T ], as long as T is large enough.
This conclusion agrees with the results in [5, 6]. What is special about such Hamiltonians is
that the criterion expressed by (A), (B), and (C) can always be satisfied by choosing values
of the parameter T . However, once the parameter T is chosen, there is also a bound on
total evolution time, τ = T . The adiabatic approximation is valid if t ∈ [0, T ], and it may
be invalid if t is larger than T .
Finally, we apply the general criterion to a concrete model to understand how the time
constraint functions. Consider a spin-half particle in a rotating magnetic field, H(t) =
(σx sin θ cosωt + σy sin θ sinωt + σz cos θ). For this model, we have E1 − E2 = ω0,
〈E1|Ė2〉 = −
sin θeiωt cos θ. Substituting them into (A), (b), and (c), we have ω sin θ/ω0 ≪ 1
, (ω sin θ/ω0) · ωτ cos θ ≪ 1, and (ω sin θ/ω0) · ωτ sin θ ≪ 1. It shows that, besides the usual
condition, there is a time constraint, t ∈
0, k 1
, where k is a certain number. The time
constrain limits the total evolution time to a finite number of rotating periods of the magnetic
field. Such a time limit is acceptable in physics, and it is also consistent with the geometric
phase consideration. In Ref. [17], it was shown that the geometric phase calculated by using
the adiabatic approximation may differ appreciably from its exact value if the evolution time
is too large. The difference reads δγ ≃ −ωτ sin θ · ω sin θ
2(ω0+2ω cos θ)
. This implies that, in order
to guarantee δγ ≪ 1, ωτ must be finite. So, the application of the adiabatic approximation
on geometric phase indicates that the time constraint is reasonable.
In summary, we have examined the quantitative condition (1). Our results indicate that
the usual quantitative condition (1) itself is a sufficiency criterion for the adiabatic approx-
imation when it is applied to the quantum systems in which (Em − El) and 〈Em|Ėl〉 are
constants. It may not be a sufficiency criterion when it is applied to a general quantum
system. To extend its validity to general systems, we have shown that the usual condi-
tion (1) and the additional conditions (18) and (19) constitute a general criterion for the
adiabatic approximation. The general criterion can sufficiently guarantee the validity of
the adiabatic approximation and it is applicable to all N−dimensional quantum systems.
We have examined a few examples to illustrate its use. It should be noted that when the
sufficiency conditions are used, the eigenstates |Em〉 need to be properly chosen such that
〈Em|Ėm〉 = 0.
D. M. Tong acknowledges the useful discussions with B. C. Sanders, K.-P. Marzlin, and
L.-A. Wu. This work was supported by NUS Research Grant No. R-144-000-189-305 and
NSF of China No. 10675076. Tong also acknowledges the financial support from IQIS at
University of Calgary when he visited there.
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[14] If |Em〉 does not satisfy 〈Em|Ėm〉 = 0, then it is replaced by e
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|Em〉.
[15] One example of such systems is a spin-half particle in a rotating magnetic field with θ = π/2.
See the last page.
[16] Since λ can be taken as the eigenvalues of the hermitian matrix defined by Mml = ωnmδml +
i〈Em(t)|Ėl(t)〉, they must be real numbers. Besides, λ = 0 is not a solution in general, other-
wise, it leads to a trivial proof.
http://arxiv.org/abs/quant-ph/0607118
http://arxiv.org/abs/quant-ph/0603175
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