10 June 2002

Physics Letters A 298 (2002) 213–218

www.elsevier.com/locate/pla

Quantum Zeno effect and non-relativistic strongmatter–radiation interaction

Marco Frasca

Via Erasmo Gattamelata, 3, 00176 Roma, Italy

Received 26 February 2002; accepted 4 April 2002

Communicated by P.R. Holland

Abstract

Using an expansion obtained by the dual principle in perturbation theory for the Schrödinger equation Phys. Rev. A 58 (1998)3439, it is proved that, in the case of a strong monochromatic electromagnetic wave interacting with a particle in a Coulombpotential, the non-decay probability at small times satisfies the behavior for the quantum Zeno effect. 2002 Elsevier ScienceB.V. All rights reserved.

PACS: 03.65.Xp; 02.30.Mv

In a series of papers Bender and coworkers [1] pro-posed an approach to do perturbation theory in quan-tum field theory where the kinetic term is consideredas a perturbation. A few years before, extending ananalysis done by Khalfin [2] for the decay of quantumsystems at long times, Misra and Sudarshan [3] provedthat a quantum system that undergoes continuous mea-surement can display the so-called quantum Zeno ef-fect, slowing down the decay by measurement till tostop it completely due to a non-exponential propertyof the decay of quantum systems at short times. Theseresults on quantum decay are non-perturbative and putout the question of explaining the successful experi-mental situation that always favors the exponential lawor to find a way to verify by experiments this peculiarbehavior of quantum mechanics. Indeed, some work

E-mail address: marcofrasca@mclink.it (M. Frasca).

has been done in this direction proving that such aZeno effect is a physical reality [4].

Since the introduction of powerful sources of laserlight, the analysis of the interaction between matterand radiation has generally been done using small per-turbation theory. Anyhow, studies by Russian theo-rists have been accomplished in the sixties attempt-ing to extend the analysis to strong electromagneticfields [5]. An analytical approach has been also real-ized by Keldysh, Faisal and Reiss [6] to obtain closeformulas for the ionization rates of atoms in a strongelectromagnetic field. This field is actually a hot topicin atomic physics due to the appearance of higher or-der harmonics of the laser field that can prove reallyamenable to a technological application (for a reviewsee, e.g., [7]). The peculiarity of this laser sources isthat they give very short pulses of light.

Quite recently we have introduced a perturbationmethod [8] that generalizes the one pioneered by Ben-der and coworkers in quantum field theory. Indeed, the

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214 M. Frasca / Physics Letters A 298 (2002) 213–218

latter can be seen as a particular case. But our aim is tosee if in other situations this approach can be proved tobe more fruitful. Actually, we were able to show a dif-ferent way to interpret the adiabatic approximation togenerate good asymptotic approximations as solutionsto a given Schrödinger equation [9]. From this stand-point, it is easily realized that the radiation–matter in-teraction in the non-relativistic approximation, as ap-proached, e.g., in quantum optics, can be successfullystudied.

Our aim here is to use this dual approach inperturbation theory to prove a quite general resultfor strong radiation–matter interaction in the non-relativistic approximation using the results acquiredin the studies of Misra and Sudarshan. In particular,we will show that a particle in a Coulomb potentialstrongly interacting with a monochromatic wave inthe dipole approximation will decay with a law that,for short times, is non-exponential and agrees withthe requirements to have a quantum Zeno effect. In aintense laser fields the non-exponential behavior is arule rather than the exception but, we will show that,at small times such non-exponential behavior is theproper one for the quantum Zeno effect. The definitionof how strong is the interaction will be definedthrough the perturbation series we analyze till firstorder. Particularly, we reobtain the so-called Keldyshparameter that we will define later by the proof of thetheorem and characterizing the perturbation series wewill use in our proof.

We would like to point out that a recent workin a similar direction has shown an anti-Zeno effectin a three-level atom interacting with an intenseelectromagnetic field [10].

It is interesting to note that the non-decay probabil-ity is obtained through an unitary operator and this isenough to prove that the decay, cannot be exponen-tial [11]. Our aim will be to show that, in a strongelectromagnetic field, the probability to find a parti-cle in the initial state decays as(1− (t2/τ2

Z)) at smalltimes and to determine, in some approximation, theZeno timeτZ . In this way, quantum Zeno effect candevelop also in this case as devised in Ref. [3].

To introduce duality in perturbation theory [8] weconsider a problem with a time-independent Hamil-tonian

(1)(H0 + λV )∣∣ψ(t)

⟩ = i∂

∂t

∣∣ψ(t)⟩,

being λ a parameter, and the original Hamiltonianhas been splitted in two partsH0 andV . There is afreedom in the choice of the way one can split theoriginal Hamiltonian and so our aim is to use thisfreedom to obtain two different perturbation series.Indeed, whenλ → 0 (usually calling V a smallperturbation) one obtains the Dyson series

(2)∣∣ψ(t)

⟩ = e−iH0t T e−iλ∫ t

0 dt ′VI (t ′)∣∣ψ(0)⟩,

beingT the time ordering operator and

(3)VI (t)= eiH0t V e−iH0t .

For λ → ∞ one has a strong perturbation. Werescale time asτ = λt so that Eq. (1) becomes

(4)

(1

λH0 + V

)∣∣ψ(τ)⟩ = i

∂

∂τ

∣∣ψ(τ)⟩.

Then, by this rescaling, we have reverted the role ofH0 andV giving a Dyson series with the developmentparameter 1/λ

(5)∣∣ψ(t)

⟩ = e−iV τ T e−i 1λ

∫ τ0 dτ ′ H0F (τ

′)∣∣ψ(0)⟩,

being now

(6)H0F (τ )= eiV τH0e−iV τ .

This perturbation series is dual to the series in Eq. (2)being its development parameter 1/λ that is the in-verse of the one of the series Eq. (2). But this hasbeen obtained by a symmetry of the original Hamil-tonian where one part or the other can be chosen ar-bitrarily: this is duality in perturbation theory. Indeed,a series can be obtained from the other by the inter-changeH0 ↔ V , settingλ = 1. This approach can bestraightforwardly extended to time-dependent pertur-bations as done in Ref. [8]. With this method one canrecover the result of Bender et al. [1] by taking forV

the kinetic term of aλφ4 theory.Our starting point is the Hamiltonian

(7)H =H0 + e

mA(t) · p + e2

2mA2(t),

being

(8)H0 = p2

2m+U(x),

the Hamiltonian without the electromagnetic field andA(t) the potential vector of the electromagnetic field

M. Frasca / Physics Letters A 298 (2002) 213–218 215

in the dipole approximation. We have also taken theinteracting particle (as, e.g., the electron of a hydrogenatom) as having a negative charge and the unitsh̄ =c = 1. All we require at this stage is that the potentialvector has a Fourier series

(9)A(t) =+∞∑

n=−∞Ane

−inωt ,

having set

(10)An = ω

2π

2π/ω∫0

A(t)einωt dt.

As required by duality in perturbation theory [8], weremove by an unitary transformation the part wherethe fieldA(t) appears in Hamiltonian (7) given by

(11)

UF (t) = exp

[−i

e

m

t∫0

dt ′ A(t ′) · p

− ie2

2m

t∫0

dt ′ A2(t ′)],

and we arrive at the so-called Kramers–HennebergerHamiltonian [12]

(12)HKH = p2

2m+U

(x − α(t)

),

with

(13)α(t) = e

m

t∫0

dt ′ A(t ′),

having recognized that the unitary transformation (11)defines a translation operator in space. Indeed, thistransformation was introduced in [12] and defines theso-called Kramers–Henneberger frame.

In this Letter we are interested to a fieldA(t)

having a Fourier series, giving for Hamiltonian (12)the expression

(14)HKH = p2

2m+ V0(x)+

∑n=0

Vn(x)e−inωt ,

being

(15)Vn(x)= ω

2π

2π/ω∫0

U(x − α(t)

)einωt dt.

This form is very useful to analyze the interactionof a particle in a potential with a strong periodicalelectromagnetic field. At this stage, we can apply theinteraction picture by choosing as an unperturbed termthe Hamiltonian

(16)H ′0 = p2

2m+ V0(x),

so that, we are left with the Schrödinger equation tosolve

(17)eiH′0t

∑n=0

Vn(x)e−inωt e−iH ′0tUD(t) = i

∂

∂tUD(t),

and one can formally do perturbation theory at thisstage by deriving a Dyson series from the aboveSchrödinger equation. It is important to point outthat the initial state of the system is generally aneigenstate ofH0 and there is no need to diagonalizethe HamiltonianH ′

0. Anyhow, in our proof we alsoconsider the eigenstates ofH ′

0.So, we have the Dyson series

UD(t)= I − i

t∫0

dt1 eiH ′

0t1∑n=0

Vn(x)e−inωt1e−iH ′0t1

(18)+ · · · ,that is the essential point on which the main result ofthe Letter is obtained. The series (18) gives us for thestate of the interacting system

(19)∣∣ψ(t)

⟩ =UF (t)e−iH ′

0tUD(t)∣∣ψ(0)

⟩.

Already at this stage we can conclude that the decaycannot be exponential [3] as the unitary evolution isnot generally compatible with an exponential decay as

(20)d

dt

⟨ψ(0)

∣∣UF (t)e−iH ′

0tUD(t)∣∣ψ(0)

⟩∣∣t=0 = 0,

and this does not agree with an exponential formof the decay law at small times. This result is non-perturbative and absolutely general in quantum me-chanics and here we have shown as also it applies forstrong laser–atom interaction in a quite general way.What we want to do is to prove that the quantum Zenoeffect can be realized in the interaction of matter withstrong electromagnetic fields.

From the Dyson series above and the unitarytransformations we have done, one realizes the leading

216 M. Frasca / Physics Letters A 298 (2002) 213–218

order solution given by

(21)∣∣ψ(t)

⟩ ≈UF (t)e−iH ′

0t∣∣ψ(0)

⟩,

and so, at small times one has the proper decaybehavior for the quantum Zeno effect. To supportthis conclusion, we need to prove that the higherorder corrections can indeed be neglected when theelectromagnetic field is really strong and to determinethe Zeno time in this approximation. As we will see,by “strong” we mean that a small Keldysh parameteris the key development parameter for the perturbationseries.

To achieve our aim, we consider the case where theFourier series of the electromagnetic potential keepsjust the first two terms giving

(22)A(t) = εEω

cos(ωt),

having neglected the 0th term being a constant andhaving defined the amplitude of the potential by theelectric fieldE and the frequency of the electromag-netic waveω. The vectorε is the polarization vec-tor. This is often taken to characterize a laser field. Inthis case is quite easy to write down the coefficientsof the Fourier series of the potential in the Kramers–Henneberger Hamiltonian (14). They are given by

(23)Vn(x)= 1

π

1∫−1

Tn(y)U(x − λLεy)dy√

1− y2,

whereTn(y) are the Chebyshev polynomials of or-dern. Besides, forn < 0 one has to put(−1)|n|T|n|(y).We have set

(24)λL = eEmω2

,

the length representing the greatest amplitude of themotion of a free particle of chargee in an electric fieldvarying in time as sin(ωt). In a Coulomb field we canwrite, taking the polarization vectorε oriented alongthex axis,

(25)

Vn(x)= −Ze2

π

1∫−1

Tn(x′)√

(x − λLx ′)2 + y2 + z2

× dx ′√

1− x ′2 .

So, we can compute the first order correction to thesolution for the particle in a Coulomb potential as

iZe2

π

t∫0

dt1 eiH ′

0t1

(26)

×∑n=0

1∫−1

Tn(x′)√

(x − λLx ′)2 + y2 + z2

dx ′√

1− x ′2

× e−inωt1e−iH ′0t1

∣∣ψ(0)⟩.

Now, we do not know how to diagonalize the Hamil-tonian

H ′0 = p2

2m

− Ze2

π

1∫−1

1√(x − λLx ′)2 + y2 + z2

dx ′√

1− x ′2 ,

(27)

but numerical studies have been accomplished [13].We put

(28)H ′0|α〉 =Eα|α〉,

with α containing all the quantum numbers of thegiven eigenstate and substitute

(29)∑α

|α〉〈α| = 1,

into the expression (26) giving

Ze2

π

∑α,β

|α〉⟨β∣∣ψ(0)⟩∑n=0

ei(Eα−Eβ−nω)t − 1

Eα −Eβ − nωA(n)αβ ,

(30)

where

(31)

A(n)αβ = 〈α|

1∫−1

Tn(x′)√

(x − λLx ′)2 + y2 + z2

× dx ′√

1− x ′2 |β〉.We want to analyze Eq. (30) at small times. Assumingto neglect higher order term (we justify this assump-tion below) one has

(32)

γ∑α,β

∑n=0

A(n)αβ |α〉⟨β∣∣ψ(0)

⟩

×[iωt − (ωt)2

2

(Eα −Eβ

ω− n

)],

M. Frasca / Physics Letters A 298 (2002) 213–218 217

where appears the constant

(33)γ = Ze2

λLω,

known in literature as the Keldysh parameter thatcharacterizes the regime of interaction of a particlein a Coulomb potential with a strong radiation field.The regime of interest is then given byγ � 1. In thiscase we can neglect the above term and the othersone beingO(γ 2) that give a small contribution intothe evaluation of the Zeno time. It is also interestingto note that, for a time such thatωt � 1, the choiceto neglect the above term is enforced. It shouldbe emphasized that the smallness of the Keldyshparameter gives meaning to the perturbation series weare adopting.

To evaluate the Zeno time we have to compute theprobability of permanence of the particle in the state∣∣ψ(0)

⟩. We have, taking for the potential vector the

Eq. (22), after a straightforward algebra

P|ψ(0)〉 = ⟨ψ(0)

∣∣UF (t)e−iH ′

0t∣∣ψ(0)

⟩× ⟨

ψ(0)∣∣eiH ′

0tUF (t)†∣∣ψ(0)

⟩ ≈ 1− t2

τ2Z

,

(34)

being

1

τ2Z

= ⟨ψ(0)

∣∣(H ′0 + eE

mωε · p

)2∣∣ψ(0)⟩

(35)−(⟨ψ(0)

∣∣H ′0 + eE

mωε · p

∣∣ψ(0)⟩)2

,

that is, the Zeno time is given by the variance of thestatic part of the Kramers–Henneberger HamiltonianH ′

0 with added the field termeEmω

ε · p. This parabolicbehavior of the decay probability can give rise tothe quantum Zeno effect. This same behavior can beobtained for weak laser fields but the effect appearsreally difficult to observe [14]. In Ref. [15] deviationsat large times were also find in the weak field limit: itwould be interesting to extend this analysis to the casewe discussed in this Letter.

The existence of a Zeno time proves that thequantum Zeno effect, that is, the slowing down of thedecay of the initial state by measurement, can happenalso when a strong laser field is applied to a particlein a Coulomb potential. As a further result, we have

obtained a perturbation series that could be useful insome way, at least at the leading order to obtain someresults in the strong laser–atom interaction.

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