Abstract
Modern techniques allow experiments on a single atom or system, with new phenomena and new challenges for the theoretician. We discuss what quantum mechanics has to say about a single system. The quantum jump approach as well as the role of quantum trajectories are outlined and a rather sophisticated example is given.
Ensemble versus Individual System in Quantum Optics^{1}^{1}1Invited lecture at Fundamental Problems in Quantum Theory Workshop, August 37, 1997, University of Maryland Baltimore County
Gerhard C. Hegerfeldt[*]
Institut f r Theoretische Physik, Universität Göttingen, Germany
1. Introduction
Until a decade or so ago only experiments involving many atoms were possible, e.g. atoms in an atomic beam or in a gas. With a beam one would have a repetition of measurements on an ensemble, while experiments on atoms in a gas — dilute and with no cooperative effects — can often be viewed as a simu ltaneous measurement on an ensemble.
This is well adapted to the statistical interpretation of quantum mechanics. For the purpose of the present lecture a quantum mechanical expectation value is understood as a mean value of measurements on systems of an ensemble, i.e. ensemble averages of an observable or mean square deviations and so on.
With the advent of atom traps, in particular the Paul trap, and with laser cooling it became possible to store a single atom (ion) — or two, three or more — in a trap for hours or days and to experiment with it, e.g. study its interaction with light, microwave radiation or with other atoms. For a single system the statistical interpretation of quantum mechanics, based on ensembles, is not so readily applicable as in the case of beam or a gas. The question we want to address here is the following.
“Does quantum mechanics allow statements for a single system ?”
The answer will be: “Yes, to some extent.”
Of course this is trivial if the probability in question is 0 or 1. As a more interesting example of what can happen consider the macroscopic dark periods (‘electron shelving’) of a single threelevel atom as proposed by Dehmelt [1]. The atom is supposed to have a ground state 1 and two excited states 2 and where the former is strongly coupled to 1 and decays rapidly, while the latter is metastable. The transition is strongly driven by a laser, and the transition is weakly driven.
Semiclassically the behavior of such a single atom is easy to understand. The electron makes rapid transitions between levels 1 and 2, accompanied by a stream of spontaneous photon emissions, in the order of . These can be detected (and even seen by the eye; the stimulated emissions are in the direction of the laser). From time to time the weak driving of the transition manages to put the electron into the metastable level where it stays for some time (‘shelving’). During this time the stream of spontaneous photons is interrupted and there is a dark period. Then the electron jumps back to level 1 and a new light period begins. In an ensemble of such atoms (e.g. gas with no cooperative effects) light and dark periods from different atoms will overlap, and consequently one will just see diminished fluorescence. Only light and dark periods from a single or a few atoms are directly observable.
In quantum mechanics, however, the atom will always be in a superposition of the three state , , and and never strictly in state for an extended period, i.e. there will always be a small admixture of . Since decays rapidly the question arises if the dark periods still occur. Experiments [2] and early theoretical treatment [3] have answered this affirmatively. The duration of the dark periods is random and can be seconds or even minutes.
To treat problems like these involving a single system Wilser and
the author [4, 5, 6] developed the quantum jump approach which is
equivalent and simultaneous to the MonteCarlo wave function approach (MCWF)
[7] and to the quantum trajectory approach [8]. Our
approach is based on
standard quantum mechanics and nothing new to the latter is added or
required. In the next section we will give a short exposition of the
quantum jump approach with its associated random (‘quantum’)
trajectories. In Section 3 we discuss the notion of a spectrum in a
light period and of conditional
spectra which can be defined for a single fluorescing system. In this
case the quantum jump approach leads to more general quantum
trajectories.
2. The quantum jump approach. Quantum trajectories
This approach is based on standard quantum mechanics and adds no new assumptions or properties to the latter. In many cases it is just a practical tool for questions concerning a single system and often has technical and conceptual advantages. More details can be found in Refs. [4, 6, 9, 10, 11, 12] and in the recent survey [13].
The underlying idea is that it should make no difference physically
whether or not the photons emitted by an atom are detected and
absorbed once they are sufficiently far away from the atom.
It therefore suggests itself to employ gedanken photon
measurements, over all space and with ideal detectors, at instances
a time apart[14]. For a single driven atom this may look as
in Fig. 1. 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 at the th measurement
a photon will be detected (and absorbed), the next photon at the
th measurement and so on.
Limits on :

Ideally one would like to let to simulate continuous measurements. But this is impossible in the framework of standard quantum mechanics with ideal measurements due to the quantum Zeno effect [15].

Intuitively, should be large enough to allow the photons to get away from the atom.

should be short compared to level lifetimes.
This leads to the requirement [16]
To treat these gedanken measurements on a single atom we translate them first into an ensemble description as follows. We consider an ensemble, , of many atoms, each with its own quantized radiation field, of which our individual atom plus field is a member. At time the ensemble is described by the state . Now we imagine that on each member of photon measurements are performed at times . We consider various subensembles of :
all systems of for which at time a photon was detected  
all systems of for which at the times no photon  
was detected (i.e. until time no photon!) 
This is depicted in Fig. 2 where our individual system, atom plus radiation field, is denoted by a dot , and it is a member of for .
Now one can proceed by ordinary quantum mechanics and the von NeumannL ders reduction rule [17]. Let be the projector onto the nophoton subspace,
(1) 
and let be the complete timedevelopment operator, including the laser driving and the interaction of the atom with the quantized radiation field. Then the subensemble is described by
(2) 
and the subensemble by
(3) 
The relative size of the subensemble is the probability to find a member of in and is given by the normsquared of the above expression. Hence
(4)  
To calculate we note that
(5) 
and that the inner expression is a purely atomic operator which is easily obtained by second order perturbation theory. For in the above limits one then obtains, on a coarsegrained time scale (for which is very small), that the timedevelopment of is given by a ‘conditional‘, or ‘reduced‘, nonHermitian Hamiltonian in the atomic Hilbert space where, for an level atom,
(6) 
with the atomic part of the Hamiltonian, including the laser driving, and
(7)  
(8) 
and the dipole operator. consists of generalized damping terms, and we note that
where is the Einstein coefficient for the transition from level to level . Thus, on a coarsegrained time scale, one obtains
(9)  
(10) 
In an obvious extension to density matrices,
(11) 
describes the subensemble with no photon detection until time , with the corresponding nophoton probability given by
(12) 
If one lets in Eq. (3), with kept fixed, then one easily sees by the same calculation that the probability to find no photon until time goes to 1 and that one always stays in the nophoton subspace. This means that for the dynamics is frozen to the atomic subspace, and this is a particular form of the quantum Zeno effect.
A single fluorescent atom as a sample path: Quantum trajectories
Now we can return to the gedanken measurements on our single atom driven by lasers. We can distinguish different steps in its temporal behavior.

Until the detection of the first photon, our atom belongs to the subensembles and hence is described by the (nonnormalized) vector
(13) 
The first photon is detected at some (random) time , according to the probability density
(14) 
From this reset state the time development then continues with , until the detection of the next photon at the (random) time . Then one has to reset (jump), and so on.
In this way one obtains a
stochastic path in the Hilbert space of the atom.
The stochasticity of this path is governed by quantum theory, and the path is called a quantum trajectory. The stochastic process underlying these trajectories is a jump process with values in a Hilbert space. If the reset state is always the same, e.g. the ground state, one has a renewal process. If the reset state depends on one has a Markov process only.
As shown in Ref. [6] the ensemble of all possible trajectories obtained in this way leads to a reduced density matrix for the ensemble of atoms which satisfies the usual optical Bloch equations. This is a nice consistency check [22].
In case of a renewal process the parts of a trajectory between jumps behave like an ensemble created by repetition from a single system at stochastic times. In the general case the reset states can all be different so that this repetitive property is no longer true.
Observables for a single system
Since the individual photon detection times for a single driven atom are random, they cannot be predicted. However, time averages along a trajectory are more promising, e.g. the mean distance between two subsequent photon detections or other correlations.
If the underlying stochastic process is ergodic then
time average over a single trajectory = ensemble average
and this equality allows easy calculation. In many cases, e.g. for a renewal process, ergodicity is easy to see. We believe that it is probably true in general for the quantum trajectories[18].
Hence for observables such as time averages quantum mechanics allows predictions for single systems. As applications we mention macroscopic dark periods [4, 9] and quantum counting processes [6] where the axioms introduced by Davies and Srinivas [19] are not needed.
There is a word of caution, however. For the observable “frequency
spectrum of fluorescent radiation” from a single atom the above
trajectories are not (directly) applicable. This has to do with the
timeenergy uncertainty relation. If all photon detection times were
known by measurements then the spectrum would be broadened and
deformed. This shows that the above quantum trajectories are not
“realistic” and should therefore not be overinterpreted. They are
just a useful quantum mechanical tool in certain situations. There is
also a relation with the consistenthistories approach to quantum
mechanics [20].
3. A surprising example. More general quantum trajectories
Sometimes it is advantageous to carry over the quantum jump approach
to a situation where one asks more general questions about the
temporal behavior of a single system subject to observations. To motivate
this we return to the light and dark periods of the Dehmelt system
mentioned in the Introduction.
The light and dark periods are depicted in Fig. 3. The semiclassical
considerations of the Introduction suggest that the light periods are mainly
due to transitions between levels 1 and 2. Now the frequency
distribution of light emitted by a laserdriven twolevel system is
given by the Mollow spectrum. For weak driving this consists of a
Lorentzian spectrum around the laser frequency
(“incoherent part”) plus
a peak (Rayleigh peak,
“coherent part”) at . For strong driving the incoherent
part consists of three Lorentzian parts (Mollow triplet).
Complete spectrum: Let us now turn to the frequency spectrum emitted by the threelevel Dehmelt system and use some simple arguments to see what to expect. Classically, the or Rayleigh peak of the twolevel system would correspond to the emission of an electromagnetic field with sharp frequency . For the threelevel case the amplitude of this field is zero during the dark periods, and hence classically one would have an amplitudemodulated signal. From radio engineering one knows that classical amplitudemodulated signals contain sidebands. Because the dark periods have random lengths the sidebands should be continuous and should lead to a partial broadening of the or Rayleigh peak. Surprisingly, this is exactly what the fully quantum mechanical calculation predicts [21], and the result is given in Fig. 4. On top of the center of the Mollow spectrum there is an additional narrow Lorentzian peak, and then the Rayleigh peak. This is the spectrum from an ensemble of such atoms, or from a single atom whose light emitted from time zero to infinity is spectrally analyzed. For this spectrum one can show that there is no difference between ensemble and single system, due to ergodicity. Fig. 5 shows the enlarged center of the spectrum for a different set of parameters.
By the classical analogy, more frequent dark periods should mean more
modulation and should thus lead to a wider broadening of the Rayleigh
peak. The quantum mechanical calculation again confirms this
expectation.
Spectrum for single system in a light period: Taking the classical explanation one step further would clearly imply:
spectrum in a light period complete spectrum
Remarks

If at all, the notion of spectrum in a light period can only be meaningful for a single system!

The timefrequency uncertainty relation will of course introduce a broadening. The light period under consideration should therefore be so long that this broadening is negligible.

How does one know one is in a light period? By photon counting. But this disturbs the spectrum, by the timeenergy uncertainty relation, as explained in Section 2! So how to measure the spectrum without disturbing it?
To overcome the last objection to a proper quantum mechanical notion
of the spectrum in a light period, we suggested in Ref. [21]
the setup depicted in Fig. 6. A laserdriven
atom emits radiation. In the right halfspace an ideal broadband
photodetector registers all photons and triggers a
spectrometer (spectral analyzer) in the following way. A light period
in the right halfspace is defined as a sequence of photon detections
whose temporal distance in less than some prescribed time . A
dark period is a time interval which is longer than and
with no detection occurring.
Now the broadband counter in the right halfspace opens the spectral
analyzer in the left halfspace at the beginning of a light period and
closes it at the end (after an additional time , to be precise).
All data for light
periods in the right halfspace of length less than some prescribed
are discarded. In this way one obtains a sequence of spectral data for
the left halfspace referring to light periods in the right halfspace
of length at least . The spectral data for individual light periods
may depend on the actual detection pattern for the photons in the
right halfspace and one can average over these. The notion of spectrum
outlined above may be properly called a conditioned or conditional
spectrum because one performs a selection of spectral
data based on prescribed conditions.
Generalized quantum trajectories: To calculate this conditional spectrum in a light period we have generalized the above quantum jump approach and its quantum trajectories in a natural way, adapted to the problem at hand [21]. At a nophoton detection in the right halfspace the state is not projected with the projector of Eq. (1) onto the global nophoton subspace, but rather onto those nophoton states belonging to modes with momenta in the right halfspace. The resulting projected state will contain, in addition to an atomic part, also photon modes with momenta in the left halfspace. This leads to a more complicated time development between detections. Once a photon has been detected and absorbed in the right halfspace one has to reset the state (a “jump”). The jump in general leads to a density matrix involving the atoms and photons from the left halfspace. In this way one obtains a quantum trajectory consisting of density matrices where the latter simultaneously describe the atom together with a subset of modes of the radiation field.
This generalized quantum jump approach has been used to calculate the conditional spectrum in a light period [21]. As expected the broadening of the Rayleigh peak disappears. For the same parameters as in Fig. 5 the dotted curve in Fig. 7 shows the corresponding spectrum in a light period which is long enough so that the broadening of the Rayleigh peak due to the timefrequency uncertainty relation is negligible. Of course, with increasing length such light periods become very rare. To define a light period we have chosen , and so classically a light phase can still have some amplitude modulation. This explains the small bump in the line center.
The above example is in several respects amazing. First of all, quite
elementary classical arguments about radiation from driven atoms
turn out to be qualitatively correct (to a large extent also
quantitatively so, cf. Ref. [21]), although the quantum
solution is much more complicated. Furthermore, it shows that there
are interesting questions concerning a single system which at first
sight seem contradictory but still allow a quantum theoretical
treatment.
4. Conclusions
The usual statistical interpretation of quantum mechanics uses the ensemble point of view. In this view a state vector or density matrix describes not a single system but an ensemble of identically prepared systems. Nevertheless, it is sometimes useful to ascribe a state or density matrix also to a single system. By this we mean that the system is prepared by the same apparatus as that for the corresponding ensemble.
In this lecture we have tried to convey several points.

Experiments with a single system (atom, ion) show phenomena which are absent for an ensemble (e.g. in a gas with no cooperative effects).

A convenient tool for a description of such experiments with a single system is often given by the quantum jump approach with its (random) quantum trajectories. This approach is based on standard quantum mechanics and does not go beyond it.

The quantum jump approach is useful for questions related to the statistics of photons emitted from a single system. More general problems, like the frequency spectrum in a light period, require more general trajectories.

The set of quantum trajectories for a driven atom give the reduced density matrix equations (optical Bloch equations) for the ensemble of atoms.

Since quantum trajectories are adapted to the particular problem under consideration a quantum trajectory is not a “realistic” property of a single system. Rather, the complete information is contained in the state vector (or density matrix) of the system plus radiation field.
An interesting question relating to ergodicity was touched upon in the text. Are time averages independent of the particular quantum trajectory? Or could they vary for different individual systems? That they do not is usually assumed in experiments. Is this always true or does one need additional assumptions?
References
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 [12] G.C. Hegerfeldt and D.G. Sondermann, Quantum Semiclass. Opt. 8, 121 (1996)
 [13] M.B. Plenio and P.L. Knight, preprint, quantph/9702007
 [14] Alternatively one can use the Markov assumption. Cf., e.g., R. Reibold, J. Phys. A 26, 179 (1993). See also M. Porrati and S. Putterman, Phys. Rev. A 39, 3010 (1989).
 [15] As will be pointed out further below, the calculation of the photon emission probabilities will automatically give zero for the present case when . This is a special manifestation of the quantum Zeno effect. Extensive references on this effect can be found in A. Beige and G.C. Hegerfeldt, Phys. Rev. A 53 (1996) and in A. Beige, G.C. Hegerfeldt, and D.G. Sondermann, Found. Physics (Dec. 1997, in press).
 [16] For a similar requirement cf. P.A.M. Dirac, The Principles of Quantum Mechanics, 4th Ed., Clarendon Press (Oxford 1959), p. 181.
 [17] The projection postulate as commonly used nowadays is due to G. Lüders, Ann. Phys. 8, 323 (1951). For observables with degenerate eigenvalues his formulation differs from that of J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer (Berlin 1932) (English translation: Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955), Chapter V.1. The projection postulate intends to describe the effects of an ideal measurement on the state of a system, and it has been widely regarded as a useful tool. Cf. also the remark in P.A.M. Dirac, The Principles of Quantum Mechanics, 1st Ed., Clarendon Press (Oxford 1930), p. 49, about measurements causing minimal disturbance. In later editions this passage has been omitted.
 [18] It would indeed be very interesting if there were nonergodic quantum trajectories. Then there could be different time averages for different trajectories.
 [19] M.D. Srinivas and E.B Davies, Optica Acta 28, 981 (1981)
 [20] R.B Griffiths, this conference.
 [21] G.C. Hegerfeldt and M.B. Plenio, Phys. Rev. A 52, 3333 (1995); G.C. Hegerfeldt and M.B. Plenio, Phys. Rev. A 53, 1178 (1996); M.B. Plenio, Doctoral Dissertation, University of Göttingen, 1994
 [22] The quantum trajectories can therefore be used to obtain numerical solutions of the Bloch equations by simulations. This was stressed in Ref. [7]. For the simulation of the Bloch equations also other trajectories can be used which do not have the same physical meaning as in the above approach. Cf. e.g. N. Gisin, Phys. Rev. Lett.52, 1657 (1984); N. Gisin and I.C. Percival, J. Phys. A: Math. Gen. 25, 5677 (1992) and the survey by D.G. Sondermann, in: Nonlinear, deformed and irreversible quantum systems, edited by H.D. Doebner, V.K. Dobrev, and P. Nattermann, World Scientific 1995; p. 273.
Figure Captions
Fig. 1: Repeated photon measurements.
Fig. 2: Ensemble and subensembles. The dot denotes our single
system.
Fig. 3: The light periods consist of rapid sequences of photon
emissions.
Fig. 4: Frequency spectrum with an additional narrow Lorentzian
peak due to dark periods ().
Fig. 5: Enlarged center of spectrum for a different set of
parameters ().
Fig. 6: The broadband photodetector in the right halfspace triggers
the spectrometer in the left halfspace during a long light
period.
Fig. 7: Dotted line: The narrow Lorentzian peak of Fig. 5 is absent in a long light period ().