Muga G. Time in Quantum Mechanics

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10 Time Scales and Correlations in Non-Markovian Quantum Open Systems 295

10.5.2 Fluorescence in a Structured Environment

We shall focus in this section on the ﬂuorescence spectra of a two-level atom

immersed in a structured environment. We shall use the results of previous sec-

tions, which are valid for an arbitrary system, provided that it is linearly and weakly

coupled to its environment. If the atom is additionally driven by a laser, the coupling

operator L appearing in the Hamiltonian (10.4) has a particular form, which we shall

derive in the following.

Let us ﬁrst consider the interaction Hamiltonian of the atom with a classical laser

ﬁeld, which in the rotating wave approximation can be written as follows [79, 18]:

= ε(σ

−i(ω

t+φ

)

+σ

i(ω

t+φ

)

) , (10.47)

where ω

is the frequency of the laser, ε its Rabi frequency, and φ

its phase.

Because of the magnitude of the laser ﬁeld, the Hamiltonian H

should be consid-

ered as part of the noninteracting Hamiltonian H

, so that now H

= H

The explicit time dependence of the Hamiltonian can be eliminated by applying the

following unitary operator:

= e

[

iω

t+iφ

]



†

+(σ

−σ

)

. (10.48)

This operation transforms the Hamiltonian into the rotating frame of the laser, where

it can be written as H



= H



+ H



. Here,





†

+ε[σ

+σ

] , (10.49)

with Δ

= ω

−ω

, and



= i



(σ

†

−a

) . (10.50)

A new unitary operation V ,

V =



c −s



(10.51)

may transform the system into a dressed picture, where the Hamiltonian

H =

−1



V has the form (10.4). The constants appearing in the transformation matrix

V are c = cos φ and s = sin φ, where the angle, φ, is given by sin

φ =

[1−sgn(Δ



/Δ

) + 1], with Δ

= ω

−ω

. The noninteracting dressed

state Hamiltonian

is equal to

296 D. Alonso and I. de Vega

= Ω R



†

(10.52)

and the interaction Hamiltonian

has the same form as in (10.4) once the interac-

tion operator L is deﬁned as

L = csR

−s

. (10.53)

Here, R

i

j| are the atomic operators deﬁned in the dressed state basis

1, |

2}, and R

= R

− R

. The quantity Ω = [ε

+Δ

/4]

1/2

is the so-called

generalized Rabi frequency.

Once the coupling operator L is known, the two-time correlation L

†

(τ )L(τ



)

can be easily computed with Eq. (10.27). Note that the form of L is such that the

last term of (10.27) is nonzero, and therefore the quantum regression theorem is not

fulﬁlled.

In order to obtain the emission spectra with (10.46), it is necessary to derive the

Laplace transform of the spatial-dependent correlation function, S

∗

(r, r

,ω).

We should then consider the explicit form of the coupling constants appearing in

(10.4) which for a dipolar coupling is

=−i



2ε

kσ

·d

ik·r

, (10.54)

where d

is the atom dipolar moment,

kσ

is the unit vector in the direction of the

polarization σ for a given wave vector k, and ε

is the electric vacuum permittivity.

By using this deﬁnition, and following Eq. (10.45), we shall now write

S(r, r

,τ) = γ (

2π

)





1BZ

k,σ

ω(k)

−iω(k)τ

i(r−r

)·k

, (10.55)

where

is the unitary vector corresponding to the dipolar moment of the detector,

γ = ω

/(2ε

), and the integrals are performed over the ﬁrst Brillouin zones of

the crystal. In order to calculate the integrals appearing in (10.55), it is clear that the

dispersion relation ω(k) is needed.

Analogously to the tight binding model in solid state physics, the dynamics near

the edge of the band is often described through an effective mass approximation.

This approximation is based on an expansion of the full dispersion relation in the

vicinity of the band edge (see for instance [84–87]). Hence, considering the simplest

case of a cubic lattice, the dispersion relation has the parabolic form ω(k) = ω

A(k −k

)

, where k

is the origin of the ﬁrst Brillouin zone of the crystal (which is

the unitary cell in k space) about which we perform the expansion in each direction,

is the frequency of the band edge, and Ais a constant that depends on the speciﬁc

photonic crystal considered.

10 Time Scales and Correlations in Non-Markovian Quantum Open Systems 297

In order to calculate the spatial-dependent correlation function, S(r, r

, t,τ), we

assume that the quantity

k,σ

ω(k)

is a slowly varying function in the reciprocal

space (particularly in the region nearby the symmetric point k

). In that case, we

can express function (10.55) as

S(r, r

,τ) = γ (

2π

)



,σ

ω(k

)



1BZ

dke

−iω(k)τ

i(r−r

)·k

. (10.56)

Its Laplace transform is then given by

S(r, r

,ω) = γ

2π



,σ

ω(k

)



1BZ

i(r−r

)·k

i(ω(k) −ω)

. (10.57)

In order to perform the last integrals analytically, we consider the parabolic dis-

persion relation and the limit in which the detector and the emitting atom are far

away from each other. In that way, we can make the change of variable q = k −k

so that we have ω(k) = ω

+ Aq

and extend the limits of integration to inﬁnity.

Then

S(r, r

,ω) =−Q

2π



∞

dqq

ikd

−e

−ikd

ω(q) −ω

, (10.58)

here we have assumed that q = q(sin θ cos φ,sin θ sin φ,cos θ) and d = r − r

being parallel to the z-axis. The last consideration is not too restrictive. Any arbitrary

rotation of the detector (i.e., of d) may give rise to the same result of the integral,

which is performed along the whole reciprocal space. The constant

Q = γ

2π

·d

1 −

·u

)

1 −

·u

)

, (10.59)

provided that the d

= d

. Considering that θ and θ

are the angles between k

and u

, respectively, one ﬁnds

Q = γ

2π

sin

θ sin

. (10.60)

Performing the last integral of (10.58), we just obtain

S(r, r

,ω) =

Q2π

idA

−d/l

, (10.61)

where l = 1/



ω−ω

is the so-called localization length. Note that there are two pos-

sibilities: if ω

<ωthen S is proportional to exp(−id/l) and following Eq. (10.46)

the power spectrum decays as d

−2

;ifω

>ω, S is proportional to exp(−d/|l|), and

298 D. Alonso and I. de Vega

the power spectrum decays exponentially. Hence, emitted photons with frequencies

within the gap will not contribute to the power spectra if the detector is placed far

away from the emitter. The reason is that in the long-time limit, all photons within

the gap are emitted in the form of evanescent modes and remain spatially local-

ized nearby the emitting atom (see Fig. 10.2). In addition, increasing the frequency

detuning with respect to the band edge, which we assume placed at ω

= 0, gives

rise to a decreasing localization length l, and therefore to a faster spatial decaying

of the function S.

Fig. 10.2 (Color online) The power spectrum P(ω) times d

is considered for different values of

the detector distance d, which is measured in units of

√

A/ω

. Solid line: d = 0.1, dotted line:

d = 10, and dashed line: d = 100. The Rabi frequency is Ω = 0.55, so that the left sideband of the

Mollow triplet is within the gap, while the right sideband lies within the band. When the detector is

placed sufﬁciently far away from the emitting atom, the left sideband is no longer detected. More

speciﬁcally, no emission is detected within the gap region

10.6 Discussion and Conclusions

In the last few decades the development of new systems, like quantum dots and atom

lasers, and the engineering of new materials, like photonic crystals, have given rise

to phenomena which cannot be described by considering the Markov approxima-

tion. This has produced a growing interest in developing a theory of non-Markovian

quantum open systems, where a ﬁnite relaxation time for the surrounding environ-

ment is considered. Such ﬁnite relaxation time turns out to play an important role,

giving rise to some important memory effects in the evolution of the quantum open

system.

Memory effects are observed in all the system dynamical quantities, particularly

in its expectation values and its ﬂuctuations, which are encoded by multiple time

correlation functions. In this chapter, we have discussed the different methods that

exist to compute such quantities. Using a stochastic Schr

odinger equation approach,

10 Time Scales and Correlations in Non-Markovian Quantum Open Systems 299

and developing the dynamical equations that quantum observables and correlations

of the system satisfy. Both approaches are complementary to each other, and the

choice between them should be made according to the speciﬁc needs we have.

Apart from the theoretical interest, the importance of multiple time correlation

functions relies on the fact that many measurable quantities are related to them. For

instance, an atom interacting with the electromagnetic ﬁeld emits some radiation

that can be detected in the laboratory. In many experiments, the measured quantity

is the number of photons, a quantity that turns out to be proportional to a certain two-

time correlation function of system observables. Henceforth, predicting this exper-

imental result requires a theory to describe not only quantum mean values but also

multiple time correlation functions. For the Markov case there are well-developed

tools to compute multiple time correlation functions, the most relevant being the

so-called quantum regression theorem. For the non-Markovian case, the structure

of the evolution equations of system ﬂuctuations is much more complicated, since

some memory effects should be taken into account. In general, non-Markovian ﬂuc-

tuations do not follow the predictions of the quantum regression theorem, an issue

that has been widely treated in the literature and we have discussed in this chapter.

Acknowledgments We would like to thank J.G. Muga for his kind invitation to contribute to this

volume and its invaluable support. We thank H. Carmichael, G.C. Hegerfeldt, A. Ru

ız, and L.S.

Schulman for their comments at different stages of this work and G. Nicolis, P. Gaspard, J.I. Cirac,

and W.T. Strunz for support and encouragement. This work has been supported by Ministerio de

Ciencia y Tecnolog

ıa of Spain (FIS2007-64018) and by the EU projects CONQUEST and SCALA.

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Chapter 11

Double-Slit Experiments in the Time Domain

Gerhard G. Paulus and Dieter Bauer

11.1 Introduction

Writing about “double-slits in time” bears the risk of entering mined area. It is thus

advisable to ﬁrst set the stage by describing the kind of experiments, either real ones

or gedankenexperiments, we are going to discuss.

Let us consider some massive particle(s) fulﬁlling a nonrelativistic, time-depen-

dent Schr

odinger equation. Initially, the particle(s) are conﬁned inside some region,

e.g., a particle beam impinging on a shutter, atoms in a magneto-optical trap (MOT),

or electrons in an atom. At later times particle(s) are allowed to leave the region upon

passing through externally controlled “time slits.” Such time slits can be realized

by chopping or modulating a particle beam [62, 4, 23, 29], using laser light as an

ultrafast atomic mirror [63, 3], or ionizing atoms using intense laser pulses [41],

respectively.

Moshinsky studied in the seminal 1952 paper entitled “Diffraction in Time” [49]

the case of a perfectly absorbing shutter at z = 0, blocking a particle beam of wave

vector k

along the z-axis z|Ψ (t = 0)=exp[ik

z]Θ(−z) (with Θ the Heaviside

step function) for times t < 0 but removed suddenly at t = 0. Moshinsky showed

that for t > 0

|z|Ψ (t)|





+C(ξ )





+ S(ξ )





(11.1)

holds, where ξ = (k

t − z)/(

√

πt) and C and S are the Fresnel integrals (units

are used where  and the mass of the particle are unity). Plotting the probability

G.G. Paulus (B)

Institute of Optics and Quantum Electronics, Friedrich Schiller University of Jena, Jena 07743,

Germany; Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA,

ggpaulus@ioq.uni-jena.de

D. Bauer

Max-Planck-Institut f

ur Kernphysik, Postfach 103980, Heidelberg 69029, Germany,

dbauer@mpi-k.de

Paulus, G.G., Bauer, D.: Double-Slit Experiments in the Time Domain. Lect. Notes Phys. 789,

303–339 (2009)

DOI 10.1007/978-3-642-03174-8

 Springer-Verlag Berlin Heidelberg 2009

304 G.G. Paulus and D. Bauer

density (11.1) at a given spatial point z



as a function of time one obtains the diffrac-

tion pattern well known from spatial diffraction at a straight edge (see, e.g., [8]).

Times t < z



, i.e., times less than the “time of ﬂight” to the chosen position z



correspond to the “shadow” region where the probability that a particle yet arrived

is rather small (although nonzero). However, for times t > z



, the probability

density does not simply assume the constant value it has for z < 0 but oscillates

around this value, exactly as the light intensity does in the illuminated region behind

a straight edge. This diffraction in time has been revisited several times in the lit-

erature. In [45] it is analyzed in terms of Wigner distributions and tomographic

probabilities. The effect of the aperture function on the temporal diffraction pattern

was studied in [19], and in the very recent reference [20], Moshinsky’s shutter was

replaced by a moving mirror, which offers the possibility to slow down or accelerate

atom beams and may be useful for atom interferometry in the time domain. A valu-

able review on exact solutions for time-dependent phenomena of such kind includ-

ing concise introductions into the relevant theoretical methods can be found in [39].

While spatial diffraction also occurs for massive particles, diffraction in time

does not occur for light. This – perhaps at ﬁrst sight puzzling – asymmetry is due to

the fact that only for stationary solutions both the Schr

odinger and the wave equa-

tions reduce to the Helmholtz equation and thus have formally equivalent solutions.

For time-dependent phenomena the different dispersion of matter and light waves

yields different dynamics

[10, 11].

When talking or writing about “interference in time” a common objection one

encounters is that this cannot be because time is not an operator, at least in “main-

stream” quantum theory [30, 31]. As a consequence, the “occurrence time” of phys-

ical events is not an observable and thus does not fulﬁll an uncertainty relation

with the energy (i.e., usually the Hamiltonian) of the system. However, aside from

the fact that interference per se does not require operators, uncertainty relations

between times and energies emerge for reasons different from noncommuting oper-

ators in many quantum mechanical calculations. In fact, in the theoretical analysis

of Sects. 11.2 and 11.4, we shall also meet (at least) two variants of the time–energy

uncertainty relation: peaks in energy spectra become narrower if the interaction time

increases and interference structures in energy depend on the time delay which

characterizes the preparation of the quantum state, in our case the delay between

two or more “slits” in time. An excellent overview of the multifaceted aspects of the

time–energy uncertainty relation, in general, is given in Chap. 3 of the ﬁrst volume

of this book series [12] (see also [28], the classic papers [2, 44], and, e.g., [50, 38]).

11.2 Wave Packet Interference in Position and Momentum Space

The theoretical analysis in this section and later on in Sect. 11.4 is performed for

electronic wave packets using atomic units. The generalization to any kind of matter

waves is straightforward (see also [11]).

More mathematically, the ﬁrst time derivative in the Schr

odinger equation and the second time

derivative in the wave equation yield different dynamics.

11 Double-Slit Experiments in the Time Domain 305

The wave function of a free electronic Gaussian wave packet fulﬁlling the time-

dependent Schr

odinger equation (TDSE)

∂

∂t

|ψ

(t)=

|ψ

(t) (11.2)

reads in position space

r|ψ

(t)=



1 + it/a



3/2

exp



−

−2ia

k · r +ia



1 + it/a





(11.3)

and in momentum space

p|ψ

(t)=Aa

exp



−

(k − p)

−



. (11.4)

Here, A is a normalization constant, k is the center-of-mass momentum of the wave

packet (i.e., equal to the group velocity in atomic units), and a determines the width

of the wave packet.

11.2.1 Double-Slit in Space

Let us ﬁrst analyze the “standard” double-slit in space with the help of Gaussian

wave packets. We assume that two wave packets of equal width and amplitudes

start at t = 0 with equal center-of-mass momentum k = k

, k

> 0, at the two

spatial positions r =±de

. The wave function at a later time t > 0 reads in position

space, according to (11.3),

r|ψ

(t)=



1 + it/a



3/2

exp



−

+ z

−2ia

z + ia



1 + it/a







exp



−

(x −d)



1 + it/a





+exp



−

(x +d)



1 + it/a





(11.5)

and in momentum space, according to (11.4),

For an explicit Fourier transformation the integral

√

2π



∞

−∞

dx exp[−ip

x] exp[−αx

+βx − γ ] =

√

2α

exp[−γ ]exp



(β −i p

)

4α



(for Re α>0) is useful.

Muga G. Time in Quantum Mechanics - Vol. 2

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