quantum jump statatistics in cavity jkps

7/29/2019 Quantum Jump Statatistics in Cavity JKPS

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Photon Absorption and Emission Statistics of a Two-Level Atom

in a Cavity

Chang J. Lee∗

Department of Nanochemistry, Division of Basic Sciences,

Sun Moon University, Asan 336-708, Korea

Abstract

The absorption and emission of photons by an atom involves quantum jumps between states. We

investigate the quantum jump statistics for the system of a two-level atom and a single-mode cavity

field. We use the Jaynes-Cummings model for this problem, and perform Monte Carlo numerical

simulations and give a detailed exact analysis on these simulations. These studies reveal that the

waiting-time distribution (WTD) for photon absorptions(emissions) has a unique novel statistics,

and that the photon absorption(emission) rate is not uniform, but counter-intuitively depends on

the position in the Rabi cycle. The effects of the nonclassical nature of the field on the WTD is

discussed.

PACS numbers: 42.50.Ar, 42.50.Lc, 42.50.Pq

Keywords: quantum jump, photon statistics, absorption, emission, cavity quantum electrodynamics, Jaynes-

Cummings model

∗ [email protected]

1



I. INTRODUCTION

The experimental and theoretical progress made in the last few decades in atomic physics

and quantum optics has enabled researchers to put fundamental quantum mechanical prob-

lems that have eluded concrete answers to the test. The absorption and emission of photons

by matter is one of these fundamental problems. A canonical model for such processes in the

optical and the microwave regimes is a two-level atom interacting with a monochromatic

light. In the standard approach employing the time-dependent Schrodinger equation the

probability for finding the atom in either state oscillates–a phenomena well-known as the

Rabi oscillations [1]. The oscillations are often interpreted as a periodic change between

upper and lower states of an atom by absorption and emission of photons [2]. However,

the oscillations result from an average of processes taking place in an ensemble of similarly

prepared atoms. As long as a single atom is concerned, discrete quantum jumps rather than

the continuous, oscillatory absorptions or emissions take place [3]. In the standard wave

mechanical treatments these quantum jumps are smeared out and cannot be discerned.

An approach that is based on such individual quantum jumps and reproduces the same

ensemble average as in the standard wave mechanics is the Monte Carlo Wave Function

(MCWF) method [4]. The MCWF theories are mostly concerned with spontaneous emis-

sion. In a previous paper [5] we considered the Rabi problem without dissipation and incor-porated quantum jumps into the evolution under the standard time-dependent Schrodinger

equation. We devised a scheme that, while utilizing the concept of wavefunction collapse

during repeated measurements, avoids the quantum Zeno effect [6–8]. This allowed us to

obtain the photon counting statistics associated with the stimulated absorption and emission

processes of the Rabi model–not possible with the traditional wave mechanical approach.

The photon counting did not show unique statistics, but varied depending upon the Rabi

frequency. This left us to wonder under what conditions the statistics is unique, if there is.

Another aspect not considered was the effects of detuning on the statistics.

The goal of this paper is to present results of further investigation on the photon counting

statistics for the cases of both on- and off-resonance excitations and give a detailed analysis

to account for the statistics. We are particularly concerned with finding the conditions un-

der which the statistics are uniquely determined. We recast the semiclassical Rabi problem

into the Jaynes-Cummings model (JCM) [9]. The JCM is one of the most thoroughly stud-

2



ied quantum optics models, and has been extensively generalized. The novel phenomena

of collapses and revivals of the Rabi oscillation [10] is a direct consequence of field quan-

tization. With the advancement of cavity quantum electrodynamics experiments the JCM

has provided a fertile ground for testing fundamental aspects of quantum mechanics such as

entanglement and has implications for quantum computing [11]. We will show in this paper

that the original, nonextended JCM still has other novel phenomena.

This paper is organized as follows: In section II we briefly review the JCM and introduce

physical parameters that will be used throughout this paper. Quantum jumps simulation of

the evolution for the atom-field coupled system will be given in section III. Photon counting

statistics resulting from the simulation and a detailed analysis of the statistics are given in

this section as well. The paper concludes in section IV with a summary of the main results,

and discussions on some experimental aspects and on the effects of field quantization in the

photon absorption(emission) statistics.

II. THE JCM

The system we consider consists of a two-level atom interacting resonantly or near-

resonantly with a single quantized cavity mode. We assume the atom-cavity interaction

is in the strong coupling regime, so the dissipative processes in both the atom and the cav-ity are ignored. Furthermore, we are concerned with the inner workings of the atom-field

interaction. Thus, the coupling of the field with the outside measuring device [12] is not

considered here. Such a system is described by the JCM Hamiltonian:

H =1

2hω0σz + hω

a†a +

1

2

+ hλ

aσ+ + a†σ−

, (1)

where ω0 is the resonance frequency between the two atomic states. The lower and upper

states of the atom are denoted as |0 and |1, respectively [13]. σz = |11| − |00|, σ+ =|10|, σ− = |01|. ω is the cavity mode frequency, and a and a† are the cavity field

annihilation and creation operators, respectively. The atom-field coupling constant λ is

assumed to be real.

We consider the cavity is in the Fock state with the photon number m. Other field states

can be expressed in terms of these Fock states, so effects due to different field states can

be inferred from the Fock state properties and the photon distribution function. Since the

3



interaction of the atom with the light in Eq. (1) couples only |0, m + 1 state with |1, mstate, the atom-field system can be expressed as a coherent superposition state

|ψ(t) =

j=0,1

c j(t)| j, m − j + 1. (2)

The evolution of the state is governed by the time-dependent Schrodinger equation

d

dt|ψ(t) = − i

hH |ψ(t). (3)

There are several ways to solve Eqs. (1)-(3), and the results can be found elsewhere [15]. If

initially the atom-cavity system is in the state |0, m + 1, the probabilities for finding the

system in the respective states at time t are given by

P 0(t) = |c0(t)|2 = 1 − Ω(m)2

Ω2eff

sin2 Ωeff t2

,

P 1(t) = |c1(t)|2 =Ω(m)2

Ω2eff

sin2 Ωeff t

2, (4)

where Ω(m) = λ√

m + 1 is the m-photon Rabi frequency. (To simplify notations we will use

Ω instead of Ω(m) in the remaining discussions.) Ωeff =√

∆2 + Ω2 with ∆ = ω0 − ω being

the detuning. If the initial state is |1, m instead, one just needs to switch the labels 0 and

1 in Eq. (4).

As it stands, the above results of wave mechanics show that the probability of being at

either the lower or the upper state oscillates continuously. Measurements with an ensemble

of atoms reproduce the oscillation. However, it does not give any information about when

an individual atom makes a transition to the upper state, and vice versa.

III. QUANTUM JUMPS SIMULATION AND ANALYSIS

A. Rabi oscillation simulation

Photon counting involves measurements. Measurements make the wave function collapse,

and we take P j(t), ( j = 0, 1) in Eq. (4) to be the probability for the collapse |ψ(t) →C |ψ(t) = | j, m − j + 1 to occur. Here, C is the collapse operator and is none but the

atom-field interaction term:

C = aσ+ or a†σ−. (5)

4



Since 12

σz + a†a is a constant of motion, j + (m − j + 1) = m + 1 is a conserved quantity.

Thus, atomic and field states are intimately entangled and we will suppress the field state

and concentrate only on the atomic evolution in the rest of the paper.

In order to simulate measurements we divide the total observation time T obs into small

segments, each having ∆t, such that T obs = N ∆t, (N >> 1) [16]. During each time

segment the atom evolves according to the Schrodinger equation, and for each discrete time

tn = n∆t, (n = 1, 2, 3, . . . , N ) a random number rn is generated. The collapse rule is that

|ψ(tn) → σ+|ψ(tn) = |1, P 1(tn) ≥ rn,

|ψ(tn) → σ−|ψ(tn) = |0, P 1(tn) < rn (6)

In the above, we suppressed the operators and the state labels of the field.

A complication of consecutive measurements is the quantum Zeno effect mentioned earlier.

However, it may be evaded with the scheme proposed in [5]. In a nutshell, we consider an

ensemble of similarly prepared atoms, make a measurement and register the state, one

atom at a time: at t = ∆t for the first atom, at t = 2∆t for the second atom, and at

t = 3∆t for the third atom, etc. Such a series of N measurements constitutes a single

quantum trajectory (or measurement history). Since the atoms are prepared similarly, thetrajectory may be looked upon as one of the many possible trajectories that a particular

atom may follow. Thus, we may regard the measurement history as the quantum trajectory

of an atom. An ensemble average of these quantum trajectories should coincide with the

continuous evolution predicted by Eq. (4).

In Fig. 1 we plot quantum trajectories with 1 and 105 atoms with the atom in the

lower state initially. The parameters used in the simulations are: the Rabi frequency Ω =

1000×

2π, ∆ = 0, the time increment ∆t = 10−6, and the number of time segments N = 2000.

Here, both the frequency and time are in arbitrary units such that frequency × time = 2π.

Thus, these parameters give rise to two Rabi cycles as the figure shows.

As expected, the quantum jumps appearing in the single-atom trajectory disappear and

the Rabi oscillations become evident as the number of atoms in the simulation becomes

large enough. The real beauty of this approach, however, is that it can give information on

the statistics of waiting time for photon absorption and emission.

5



0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0

T i m e ( m e a s u r e m e n t s )

0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0

5 0 0 0 0

1 0 0 0 0 0

Q

u

a

n

t

u

m

T

r

a

e

c

t

o

r

e

s

FIG. 1. Simulation of measurement histories with 1 and 105 atoms (from bottom to top). The

single quantum trajectory shows discrete quantum jumps, while the oscillatory structure is evident

when the number of trajectories is large.

B. Waiting-time distribution(WTD) of photon absorption and emission

1. Simulation of WTD

The atom absorbs a photon in the time interval (tn−1, tn] when the measurement of the

atomic state at t = tn−1 shows that it is at the lower state and at t = tn it is at the

upper state (and vice versa for photon emission). Then we can define the waiting time of

photon absorption as the time interval (expressed as a multiple of ∆ t) between two successive

absorptions. The WTD of absorption is easily obtained by grouping into histogram bins the

time intervals between two absorptions in the Rabi oscillation simulation. The WTD of photon emission can be obtained likewise. Figure 2 shows the simulated WTD of photon

absorptions taking place over one Rabi cycle at exact resonance. The number of atoms in the

simulation is 104 and N = 1000, but other parameters are kept the same as in section III A.

The WTD of emission is almost the same as the one for absorption, and hence will not be

discussed henceforth unless otherwise specified. We find from the figure that the distribution

is sharply peaked about 2–3 ∆t, and the width of the distribution is approximately 5.5 ∆t.

6



2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

1 x 1 0

2 x 1 0

N

u

m

b

e

r

o

f

A

b

s

o

r

p

t

o

n

s

W a i t i n g T i m e

2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

1 x 1 0

2 x 1 0

FIG. 2. The waiting-time distribution of photon absorptions over one Rabi cycle with the atom

excited resonantly. The horizontal axis is the waiting time in multiples of ∆t. The inset shows the

entire domain of waiting times, while the magnified view shows the waiting times up to 20.

The product of parameters ΩN ∆t determines the (fractional) number of Rabi cycles.

For a given number of Rabi cycles, ΩN ∆t = 2qπ (q may be nonintegral), the distribution is

uniquely given regardless of the magnitude of Ω, if the number of measurements N is kept

the same and large enough, say N = 1000, and if ∆t is adjusted such that ∆t = 2qπ/(N Ω).

The variation of distributions for nonintegral multiples of the Rabi cycle has been reported

in a previous paper [5]. The reason for the variation is due to unequal excitation probabilities

at different positions in the Rabi cycle as will be discussed in section III B 2.

When detuning is introduced, the distribution behaves differently. As the detuning in-

creases it becomes more difficult for the atom to make transitions to the upper level, so

the waiting time for photon absorption gets longer and consequently the distribution drawsout. Figure 3 demonstrates this behavior. For the simulation the detuning is increased by

a factor of 2 from bottom to top, ∆ = 2k × Ω, (k = 0, 1, . . . , 8). This gives rise to non-

integral multiples of the Rabi cycle that lead to, as mentioned above, different statistics.

Thus, for fair comparison we adjusted the value of the time increment as ∆t = 2π/(N Ωeff )

with N = 1000, so as to keep the number of Rabi cycles to one and isolate the effect of the

detuning. Here, too, the results of simulations are unique since the value of N is sufficiently

7



2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

1 x 1 0

2 x 1 0

3 x 1 0

W a i t i n g T i m e

5 x 1 0

1 x 1 0

N

u

m

b

e

r

o

f

a

b

s

o

r

p

t

o

n

s

5 x 1 0

1 x 1 0

1 0 0 0

FIG. 3. The waiting-time distributions for off-resonant excitations. Detuning values are given by

∆ = 2

k

× Ω, (k = 0, 1, . . . , 8) (from bottom to top).

large. Simulations with N = 100 noticeably deviate from those with N = 1000 or higher.

In section III B 2 below we will analyze how these distributions come about.

8



( n - 1 ) ( n - 2 )

Q

u

a

n

t

u

m

S

t

a

t

e

T i m e i n t

( n - 3 )

FIG. 4. A representative trajectory that gives rise to the waiting time 2∆t. Here the measurement

time is in units of ∆t and n varies from 3 to N .

2. Analysis of WTD

The analysis to account for the statistics can be done by scrutinizing each quantum

trajectory. For example, let us consider the waiting time 2∆t. Of course, this is the smallest

waiting time for two consecutive absorptions. A representative trajectory that contributes

to this waiting time is shown in Fig. 4. In the figure n varies from 3 to N . If | j; tn, ( j = 0, 1)

denotes the state of the atom at t = tn, the quantum trajectory can be written as

· · · |0; tn−3 → |1; tn−2 → |0; tn−1 → |1; tn · · · , (7)

where · · · denotes that the atom may be at an arbitrary state, and from Eq. (4) the proba-

bility for the atom to absorb a photon in the interval (tn−1, tn] is

Π(2)n ∆t = P 0(tn−3) · P 1(tn−2) · P 0(tn−1) · P 1(tn)∆t

=Ω4

Ω4eff

1 − Ω2

Ω2eff

sin2 Ωeff tn−3

2

sin2 Ωeff tn−2

2

×

1 − Ω2

Ω2eff

sin2 Ωeff tn−1

2

sin2 Ωeff tn

2∆t. (8)

9



The sum of probabilities over all trajectories ranging from n = 3 to n = N = 1000

gives the distribution for the waiting time 2∆t. For the case of on-resonance excitation1000n=3 Π(2)

n = 23.4369, in agreement with the simulated value 234034 divided by the number

of atoms, 104. The summation of Eq. (8) is thus 2.34369

×10−5. For the off-resonance

case ∆ = Ω the sum is1000

n=3 Π(2)n = 32.7145, which again agrees well with the simulated

value 327564 divided by the number of atoms, 104. When multiplied by ∆t = 2π/(N Ωeff ) =

10−6/√

2, it becomes 2.3133 × 10−5.

The analytical expression, Eq. (8) allows us to calculate how much error is introduced

by setting N = 1000. In the limit N → ∞ (and hence ∆t → 0) the sum of Π(2)n ∆t can be

replaced with an integral. The integration is straightforward and in the case of ∆ = 0 the

integration over one Rabi cycle ΩN ∆t = 2π yields

2πΩ

0cos4

Ωt

2sin4 Ωt

2dt =

3π

64Ω, (9)

which is 2.34375 × 10−5 for Ω = 2π × 1000. The relative error due to the finite sum is about

2 × 10−3%, and hence, the error due to limiting N to 1000 measurements is negligible.

Other waiting times can be analyzed in the same manner. For example, for the waiting

time 3∆t the pertinent sum over trajectories

1000n=4 Π

(3)

n =

1000n=4 P 0(tn−4) · P 1(tn−3) · P 0(tn−1) · P 1(tn) (10)

gives 23.4338 for ∆ = 0 and 32.7135 for ∆ = Ω. The simulated values are 234367 and

326328, respectively. And the sum for the waiting time 4∆t,

1000n=5

Π(4)n =

1000n=5

P 0(tn−5) · P 1(tn−4)[P 1(tn−3)

+P 0(tn−3) · P 0(tn−2)] · P 0(tn−1) · P 1(tn), (11)

gives 18.5477 for ∆ = 0 and 25.3731 for ∆ = Ω. Again, they agree with the simulated values

186202 and 253394, respectively. We did the analysis for several other waiting times and

the results for ∆ = 0 are listed in Table I. They all agree with the simulated values, so the

simulated photon distribution can be completely accounted for.

We now turn to the problem of how nonintegral multiples of the Rabi cycle give rise

to different WTDs. To this end let us consider how much contribution a trajectory makes

10



TABLE I. Comparison of WTDs between exact analytical calculations and simulations.

Waiting time Exact integral Numerical value of column 2 Simulation

(in units of ∆t) with Ω = 2π × 1000

2 3π64Ω 2.34375 × 10−5 2.34034 × 10−5

3 3π64Ω 2.34375 × 10−5 2.34367 × 10−5

4 19π512Ω 1.85547 × 10−5 1.86202 × 10−5

5 7π256Ω 1.36719 × 10−5 1.36756 × 10−5

6 323π16384Ω 9.85718 × 10−6 9.8078 × 10−6

7 233π16384Ω 7.1106 × 10−6 7.0979 × 10−6

8 1361π131072Ω 5.1918 × 10−6 5.1965 × 10−6

9 253π32768Ω 3.86047 × 10−6 3.8758 × 10−6

10 12295π2097152Ω 2.93136 × 10−6 2.9140 × 10−6

11 9539π2097152Ω 2.27427 × 10−6 2.2332 × 10−6

12 60443π16777216Ω 1.80134 × 10−6 1.7850 × 10−6

2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

0 . 0 0

0 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

0 . 0 5

0 . 0 6

0 . 0 7

= 2

= 0 . 5

P

r

o

b

a

b

t

y

= 0

FIG. 5. The probability density Π(2)n versus n for the atom to follow the trajectory shown in Fig. 4

for some detuning values. Here n varies from 3 to 1000.

11



to the WTD. Again, we take up the case of the waiting time 2∆t. In Fig. 5 we plot the

probability Π(2)n given in Eq. (8) versus n, (n = 3, . . . , 1000) for some detuning values. It

shows that for ∆ = 0 the biggest contributions come from n values at about N/4 = 250

and 3N/4 = 750. However, as the detuning increases the n value that makes the largest

contribution shifts toward N/2 = 500. In any case, the contributions to Π(2)n across different

n values and hence different trajectories in a Rabi cycle are not equal. Likewise, different

trajectories contribute differently to other waiting times. In fractional multiples of the Rabi

cycle a trajectory may be missing or included more than once and, threfore, can give rise to

different WTDs.

Finally, let us find out where in the Rabi cycle the photon absorption(emission) rate is the

highest. If one examines the one-atom quantum trajectory shown in Fig. 1, one can find that

the photon absorptions (and emissions) are concentrated in the region where P 0(t) ≈ P 1(t).

The probability for a photon absorption(emission) to take place during the time interval

(tn−1, tn], regardless of what happens before t = tn−1 and after t = tn, is given by

P 0(tn−1) · P 1(tn) ∆t

= 1

−Ω2

Ω2eff

sin2 Ωeff tn−1

2

Ω2

Ω2eff

sin2 Ωeff tn

2

∆t,

P 1(tn−1) · P 0(tn) ∆t

=Ω2

Ω2eff

sin2 Ωeff tn−1

2

1 − Ω2

Ω2eff

sin2 Ωeff tn2

∆t. (12)

Both probabilities become equal to each other (as should be) if ∆t → 0. Especially, for

∆ = 0 they approach 14 sin2 Ωtn ∆t. We show in Fig. 6 the simulated absorption rate and it

agrees with the analytical expression given by Eq. (12) divided by ∆t. Note that for ∆ = 0,although the transition probability P 1(t) for the atom initially at the ground state to the

upper state is the highest in the middle of the Rabi cycle t = π/Ω, it is where both absorption

and emission rates are the lowest . The highest rates are located at about t = π/(2Ω) and

t = 3π/(2Ω), where P 0(t) ≈ P 1(t) and this confirms the aforementioned qualitative remark

based on the observation of one-atom trajectory. However, the position shifts to t = π/Ω as

∆ increases.

12



0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

1 0 0 0

2 0 0 0

3 0 0 0

T i m e i n m u l t i p l e s o f t

1 0 0 0

2 0 0 0

3 0 0 0

= 0 . 5

1 0 0 0

2 0 0 0

3 0 0 0

A

b

s

o

r

p

t

o

n

r

a

t

e

1 0 0 0

2 0 0 0

3 0 0 0

= 1 0

FIG. 6. The absorption rate (simulated) as a function of time. Time is given in units of ∆t. The

location of the maxima shifts as the detuning increases from bottom to top.

IV. CONCLUSIONS AND DISCUSSIONS

In this paper we applied the quantum-jump approach to the JCM to gain further insights

on the stimulated absorption and emission processes. This approach enabled us to obtain

photon counting statistics for both on- and off-resonance excitations. It shows that for on-

13



or near-resonant excitations an atom absorbs or emits a photon mostly in less than 10 mea-

surements. As the detuning increases, however, the waiting time to observe an absorption

or emission spreads out. Thus, the observed WTDs for both absorbed and emitted photons

tend to bunch less as the detuning increases. The statistics was accounted for by analyti-

cally examining quantum trajectories that contribute to each waiting time. The resulting

statistics is a novel one, not following the well-known distributions such as the Poisson dis-

tribution. It was also shown how the photon absorption or emission rate varies with the

position in a Rabi cycle and how the rate variation depends on the detuning. We found

that the maximum absorption(emission) rate is located not where the absorption(emission)

probability is the highest, but where the absorption and the emission probabilities are the

same, instead.

Time-resolved measurements of absorption and emission of photons experimentally would

be prohibitive, while they are readily simulated numerically. Fortunately, the photon count-

ing statistics is not sensitive to measurement frequency as long as it is large enough, say

1000 measurements per Rabi cycle. This is because the absorption(emission) probability is

a slowly varying function of time such as shown in Fig. 5, so the probability changes only

slightly during ∆t. Thus, one does not have to worry about how many photon absorp-

tions and emissions take place in the time interval between measurements to get the photon

statistics right. In other words, one does not have to measure each and every photon ab-sorption(emission) to obtain the statistics. Another interesting fact is that, if ∆t is adjusted

subject to ΩN ∆t = constant, the WTD is uniquely given regardless of the magnitude of the

Rabi frequency. Thus, measurement interval ∆t for the initially vacuum field can be 10 times

longer than that of the field with 100 photons. Consequently, extracting statistics from the

vacuum Rabi oscillation may be experimentally more feasible than high-field experiments

as long as measurement time is concerned.

We considered only the number state of the field in this paper. For the Rabi oscilla-tion phenomena the results with the number state closely resemble the semiclassical ones

except for the vacuum field. Thus, for the non-vacuum fields the quantization of the cavity

field is not required to obtain the WTDs. The granularity of the field emerges, however,

when one considers other initial photon states, expressed in terms of the number states:

|β =

m cm(β )|m. The probability for the atom to be in the upper state is obtained by

the weighted sum

m |cm(β )|2P 1(t), where P 1(t) is given by Eq. (4). The initial photon dis-

14



tribution gives a corresponding distribution of Rabi frequencies and hence the distribution

of Rabi cycles, Ω(m)N ∆t = 2q mπ. Here q m is a fractional multiple of the Rabi cycle for the

number state |m. Consequently, such a distribution uniquely determines the WTD as it

does the collapses and revivals of the Rabi oscillations [10]. A possible application of this

result is the inverse problem of getting the initial photon distribution in the cavity from

WTD measurements. A further investigation needs to be done to see if such an applica-

tion is feasible. Another avenue to investigate is to apply the approach discussed here to

multimode, multiphoton extensions of the JCM [14] as well as multilevel atoms.

ACKNOWLEDGMENTS

This work was supported by a 2010 Research Grant from Sun Moon University.

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15



(1980). (b) G. Rempe, H. Walther, and N. Klein, Phys Rev Lett. 58, 353 (1987).

[11] For a review, see S. Haroche and J-M. Raimond, Exploring the Quantum , (Oxford University

Press, Oxford, 2006).

[12] T. Hayrynena et al., Eur. Phys. J. D56, 113 (2010).

[13] This choice of the atomic labels stems from the isomorphism between a two-level atom and a

spin-1/2 particle, a fermion. The atomic state labels then can be regarded as the occupation

numbers of the fermion. For a detailed discussion see [14].

[14] C. J. Lee, Phys. Rev. A42, 1601 (1990).

[15] See, for example, W. H. Louisell, Quantum Statistical Properties of Radiation , (Wiley, New

York, 1973), Chap. 5.

[16] The time a quantum jump takes may be infinitesimally small. In this paper we do not worry

about how short the jump time is. We can always change the value of N as needed. In

practice we choose some large N value that gives results virtually identical to the ones when

N → ∞. For a discussion on the time related to quantum jumps see L. S. Schulman in,

Time in Quantum Mechanics, Edited by J. G. Muga, R. Sala Mayato, and I. L. Egusquiza,

(Springer-Verlag, Berlin, 2002) p. 99, and references therein.

quantum jump statatistics in cavity jkps

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