Muga G. Time in Quantum Mechanics

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122 J. Mu

noz et al.

5.6.4 Approximations

While the previous results bring dwell-time information closer to experimental real-

ization, the difﬁculty is translated onto the measurement of the ffcf, not necessarily

an easy task. A simple approximation is to substitute the expectation of the prod-

uct of two ﬂux operators by the product of their expectation values, which are the

current densities. Using the wave packet of Eq. (5.86), we have compared the times

obtained with the full expression (5.76) and with this approximation in Fig. 5.7. The

two results approach each other as Δ

→ 0, also by increasing L and/or k

3 4 5

1/Δ

Relative Error (%)

Fig. 5.7 (Color online) Comparison of the relative error of 

 using the approximation C

(τ )

instead of C(τ ) for free motion. α = 0.5,

 = m = 1, and L = 100

The exact result can be approached systematically, still making use of ordinary

current densities, as follows: First decompose

J(x

, t +τ )

J(x

, t) by means of

1 =

P +

Q , (5.90)

P =|ψψ| , (5.91)

so that

J(x

, t +τ )

J(x

, t) =

J(x

, t +τ )(|ψψ|+

J(x

, t) . (5.92)

It is useful to decompose

Q further in terms of a basis of states orthogonal to |ψ

and to each other, {|ψ

},

Q =



|ψ

ψ

| , (5.93)

5 Dwell-Time Distributions in Quantum Mechanics 123

that could be generated by means of a Gram–Schmidt orthogonalization procedure.

Now we can split Eq. (5.75):

C(τ ) =

(τ ) +

(τ ) , (5.94)

where

(τ ) has the structure of

C, but with P inserted between the two ﬂux oper-

ators in each of the four terms. Similarly

(τ ) has Q inserted and can be itself

decomposed using Eq. (5.93).

We deﬁne C(τ ) = C

(τ )+C

(τ ) by taking the real part of ψ|

(τ )+

(τ )|ψ.

is the zeroth-order approximation discussed before and only involves ordinary,

measurable current densities [9]. The nondiagonal terms from C

, ψ|

J(x

, t)|ψ



ψ

J(x

, t +τ)|ψ, can also be related to diagonal elements of

J by means of the

auxiliary states

|ψ

=|ψ+|ψ

 ,

|ψ

=|ψ+i|ψ

 ,

|ψ

=|ψ−i|ψ

 , (5.95)

since

ψ|

J(x, t)|ψ

=

ψ

J(x, t)|ψ

−

ψ

J(x, t)|ψ



−

ψ

J(x, t)|ψ

+

ψ

J(x, t)|ψ



−

ψ

J(x, t)|ψ

 . (5.96)

5.7 Final Comments

The quantum dwell-time distribution of a particle in a spatial region and its second

moment present nonclassical features even for the simplest case of a freely moving

particle, such as bimodality (due to two different eigenvalues for the same energy),

with the strongest deviations from classical behavior occurring for de Broglie wave-

lengths of the order or larger than the region width. Progress in ultra-cold atom

manipulation makes plausible the observation of these effects and motivates further

effort to achieve an elusive direct measurement of the dwell times or to link the

distribution and its moments to other observables. We have in this regard pointed

out that the ﬂux–ﬂux correlation function provides access to the second moment.

The potential impact on cold atom time–frequency metrology [42, 26] and other

ﬁelds in which the dwell time plays a prominent role (such as conductivity [7] or

chaos) remains an open question.

124 J. Mu

noz et al.

Acknowledgments We acknowledge discussions with M. B

uttiker, J. A. Damborenea, and

B. Navarro. This work has been supported by Ministerio de Educaci

on y Ciencia (FIS2006-10268-

C03-01), the Basque Country University (UPV-EHU, GIU07/40), EU Integrated Project QAP, and

the EPSRC QIP-IRC (GR/S82176/0).

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Chapter 6

The Quantum Jump Approach and Some

of Its Applications

Gerhard C. Hegerfeldt

6.1 Introduction

Modern techniques allow experiments on a single-driven atom or a single system.

The quantum jump approach was originally developed for the description of the

temporal evolution of such a driven system and was later extended to more general

situations like a moving particle coupled to a spatially conﬁned laser beam or to

spin-boson baths. In this chapter the underlying ideas are presented and illustrated

by simple examples. Applications include the spectacular macroscopic light and

dark periods in the ﬂuorescence of a single atom, quantum counting processes, and

arrival times.

The quantum jump approach

takes its name from the jump-like change of state

when a measurement is performed. A simple example for such a jump-like change

occurs in a two-level atom interacting with the quantized radiation ﬁeld. If ini-

tially the atom is in the excited state and the radiation ﬁeld in the no-photon state

(vacuum), then the complete system evolves into a superposition of atomic and pho-

ton states. If at a speciﬁc time one detects a photon by absorption then, by standard

theory, right thereafter the atom is in the ground state.

A more complicated example would be a single atom in a trap irradiated by

a classical laser. Again, the complete system (atom plus radiation ﬁeld) evolves

under the joint time development into a complicated superposition of atomic and

photon states, and again one could detect the ﬂuorescence photons (outside the laser

direction, laser light is not detected). Right after each photon detection the state

of the atom is reset to the ground state for the two-level case and in general to

G.C. Hegerfeldt (B)

Institut f

ur Theoretische Physik, Universit

at G

ottingen, Friedrich–Hund–Platz 1,

D-37077 G

ottingen, Germany, hegerf@theorie.physik.uni-goettingen.de

The quantum jump approach was developed in [35, 49, 23]. It is equivalent and simultaneous to

the Monte Carlo wave function approach (MCWF) [17] and to the quantum trajectory approach

[14].

Hegerfeldt, G.C.: The Quantum Jump Approach and Some of Its Applications. Lect. Notes

Phys. 789, 127–174 (2009)

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

 Springer-Verlag Berlin Heidelberg 2009

128 G.C. Hegerfeldt

more complicated states and even to density matrices. Moreover, the reset state may

depend on the state prior to the detection and thus on the time a photon is detected.

Of course, recoil can also change the atomic center-of-mass momentum.

The quantum jump approach (QJA) has been developed to study the temporal

behavior of a single system coupled to a “bath,” such as an atom coupled to the

radiation ﬁeld, driven by an external ﬁeld like a laser. A spectacular example for a

phenomenon that can be observed for a single system but not for an ensemble is the

famous macroscopic quantum jumps of a three-level atom, predicted by Dehmelt

[19]. He considered an atom with a ground state 1 and two excited states 2 and 3

where the former is strongly coupled to 1 and decays rapidly, while the latter is

metastable (cf. Fig. 6.1). The 1–2 transition is strongly driven by a laser and the 1–3

transition is weakly driven.

Fig. 6.1 (Color online) System with macroscopic light and dark periods. Three-level system in a

V conﬁguration. Level 2 decays rapidly while level 3 is metastable. The 1–2 transition is strongly

driven by a laser, and the 1–3 transition is weakly driven

Semi-classically one would expect the electron to make rapid transitions between

levels 1 and 2, accompanied by a stream of spontaneous photon emissions, in the

order of 10

photons per second. These would form a visible light period. From time

to time the weak driving of the 1–3 transition, however, would manage to put the

electron into the metastable level 3 where it would stay for some time (“shelving”).

During this time the stream of spontaneous photons would be interrupted, leading to

a dark period. Then the electron would jump back to level 1, with a new light period

beginning. This is depicted in Fig. 6.2, where each short line denotes the emission

of a spontaneous photon.

Fig. 6.2 Each line represents an individual photon. Groups of photons represent a light period,

with a dark period in between two such groups

Experimentally, the individual photons are not resolved, and in the ﬂuorescence

signal one just observes light and dark periods [10–13]. The light and dark periods

can last from milli-seconds to minutes and their durations are random. In an ensem-

ble of such atoms (e.g., gas with no cooperative effects) light and dark periods from

different atoms would overlap, and consequently one would just see diminished

ﬂuorescence. Only light and dark periods from a single or a few atoms are directly

observable.

6 The Quantum Jump Approach and Some of Its Applications 129

The above semi-classical description is adequate when the atom is irradiated by

incoherent light, with the level separations in the optical domain, since in this case

one can use rate equations. However, when the atom is irradiated by coherent light

as by a laser, quantum coherences can build up. Quantum mechanically the atom

will then in general be in a superposition of the three states |1, |2, and |3 but

never strictly in the state |3 for an extended period, i.e., there will always be a

small admixture of |2. These coherences can change the situation considerably and

can lead to quantitatively different predictions for the frequency and length of the

light and dark periods. The small admixture of |2could, in principle, even have the

strange consequence that the ﬁrst photon after a dark period could come from a 2–1

transition instead of a 3–1 transition as predicted by the semi-classical argument,

although an experimental veriﬁcation of this effect is presently beyond reach.

The QJA was speciﬁcally developed to treat problems like these. The QJA has

turned out to be a practical tool for questions concerning a single system and often

has technical and conceptual advantages. An extensive review [42] appeared some-

time ago, and therefore this chapter will be more intent on trying to explain the

basics and illustrate the approach by simple, but interesting, examples. In particular,

the above light and dark periods will be treated in some detail and it will be shown

that, with suitably chosen parameters, the ﬁrst photon after a dark period can indeed

come in about half of all cases from the 2–1 transition. More details as well as

applications can be found in [35, 23, 29, 30, 5, 31, 8, 32, 33, 6, 34, 9, 7] and in the

brief survey [24]. Also included are some more recent extensions of the approach

to incorporate recoil (cf. e.g. [25]) and the possibility of replacing density matrices

by pure states [26] as well as applications to arrival times and to spin-boson models

which have been discussed in [27, 28].

6.2 Repeated Measurements on a Single System: Conditional

Time Development, Reset Operation, and Quantum

Trajectories

In this section the basic ideas underlying the QJA will be explained by means of an

atomic system which is driven by a laser and radiates photons. For the example of

a two-level system the resulting conditional Hamiltonian for the time development

between photon detections will be calculated explicitly in Sect. 6.2.1. From the con-

ditional Hamiltonian the probability density for the ﬁrst photon is derived in Sect.

6.2.2. The reset operation which yields the atomic state right after a photon detection

will also be explicitly determined for this example in Sect. 6.2.4. Although the result

for the two-level system is obviously just the ground state, the explicit calculation

for this simple example facilitates the understanding of the general case in Sect. 6.4.

In Sect. 6.2.5 the notion of quantum trajectories will be explained by means of this

example.

It is intuitively reasonable that it should make no difference physically whether

or not the photons radiated by an atom are detected and absorbed once they are

130 G.C. Hegerfeldt

sufﬁciently far away. It therefore suggests itself to employ gedanken photon mea-

surements, over all space and with ideal detectors, at instances a time Δt apart.

For a single-driven atom this may look as in Fig. 6.3. Starting in some initial state

with no photons (the laser ﬁeld is considered as classical), at ﬁrst one will detect no

emitted photon in space and then, say at the n

th measurement, a ﬁrst photon will

be detected (and absorbed), the next photon at the n

th measurement, and so on.

Fig. 6.3 Repeated photon measurements at times Δt apart. The ﬁrst are null measurements where

no photon is found

Ideally one would like to let Δt → 0 to simulate continuous measurements. But

this is impossible in the framework of quantum mechanics with ideal measurements

due to the quantum Zeno effect [39, 36]. To get an idea how to choose Δt, intuitively

it should be large enough to allow the photons to get away from the atom. On the

other hand, Δt should also be short compared to level lifetimes so that these can be

resolved in time. This leads to the requirement

Δt

∼

−13

−10

s . (6.1)

For a similar requirement compare the book by Dirac [20]. For consistency one has

of course to check that all results do not depend on the particular value of Δt when

chosen in this interval. Also, for systems coupled to other baths, another range for

Δt may be appropriate.

These measurements on a single atom can be translated into an ensemble descrip-

tion as follows. Let E be an ensemble of many atoms, each with its own quantized

radiation ﬁeld, of which the considered atom plus ﬁeld is a member. At time t

= 0

the ensemble is described by the state |0

|ψ

. Now one imagines that on each

member of E photon measurements are performed at times Δt, ..., nΔt, ... .Now

we denote, for n = 1, 2, ..., by E

(nΔt)

the subensemble which consists of all systems

of E for which at the times Δt, ..., nΔt no photon was detected. This is depicted in

Fig. 6.4 where the individual system under consideration, atom plus radiation ﬁeld,

is denoted by a dot “·” and it is a member of E

(nΔt)

Now one can proceed by ordinary quantum mechanics and the von Neumann–

uders reduction rule [1]. Let P

be the projector onto the no-photon subspace,

≡|0

1

0

| , (6.2)

and let U(t, t

) be the complete time-development operator, including the laser

driving and the interaction of the atom with the quantized radiation ﬁeld. Then the

subensemble E

(Δt)

is described by

6 The Quantum Jump Approach and Some of Its Applications 131

Fig. 6.4 Ensemble E and subensembles. E

(nΔt)

denotes the subensemble with no photons found

until, and including, the nth measurement. The dot denotes the single system actually under con-

sideration for which it is assumed that no photon was found until the nth measurement so that it

lies in E

(nΔt)

U(Δt, 0)|0

|ψ

(0) (6.3)

and the subensemble E

(nΔt)

U(nΔt, (n − 1)Δt)P

...P

U(Δt, 0)|0

|ψ

(0)≡|0

|ψ

cond

(t) , (6.4)

where we have put t ≡ Δt. The relative size of the subensemble E

(nΔt)

is the prob-

ability to ﬁnd a member of E in E

(nΔt)

and is given by the norm square of the above

expression. Hence

(t) ≡|ψ

cond

(t)

(6.5)

is the probability to ﬁnd no photon until time t = nΔt. Since the decrease of P

(t)

in the time interval Δt is the probability, w

(t)Δt, of ﬁnding a photon in Δt, one

has

(t)Δt =−(P

(t +Δt) − P

(t)) . (6.6)

This relation also results from the reset operation, as explained in Sect. 6.2.4.

To calculate |ψ

cond

(t) we note that

U(t



+Δt, t



=|0

0

|U(t



+Δt, t



) |0

0

| (6.7)

and that the inner expression is a purely atomic operator which is easily obtained by

second-order perturbation theory.

The time development of |ψ

cond

(t) occurs under the condition that no photon is

observed until time t, i.e., it is a conditional time development, and therefore the

norm of |ψ

cond

(t) decreases. It will be shown in the following that, on a coarse-

grained timescale, |ψ

cond

(t) can be written by means of a nonunitary conditional

132 G.C. Hegerfeldt

time-development operator U

cond

(t, 0) as

|ψ

cond

(t)=U

cond

(t, 0)|ψ

(0) , (6.8)

where U

cond

(t, 0) satisﬁes the Schr

odinger equation

cond

(t, 0) =−



cond

(t)U

cond

(t, 0) (6.9)

with a complex conditional Hamiltonian H

cond

. The nonunitarity of U

cond

(t, 0) cor-

responds to the decrease in time of the probability P

(t) of ﬁnding no photon until

time t. This will be illustrated now for the example of a two-level system. The

general case will be treated in Sect. 6.4.

6.2.1 The Conditional Hamiltonian for a Two-Level System

In this section the atom will be assumed to be at rest and located at the origin. Its

center-of-mass motion will be taken into account in Sect. 6.7. For optical wave-

lengths the atom can be treated as point-like. Let E

ext

(0, t) denote an external ﬁeld

at the origin. Then the Hamiltonian for the complete system, atom plus quantized

electromagnetic ﬁeld

E, is given in the Schr

odinger picture by [37, 16]

H = ω

|22|+



ω

†

kλ

D ·

E(0, 0) +e

D · E

ext

(0, t) , (6.10)

where

D is the electronic dipole operator,

D =



i|

|j|ij|≡



|ij| . (6.11)

In the usual rotating-wave approximation the Hamiltonian becomes

H = ω

22



ω

†

kλ

(−)

·E

(+)

ext

(t) +h.c.

(−)

(+)

+h.c.

≡ H

+ H

(t) + H

, (6.12)

where

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