Click-counting distributions in optical transitions

ANNALS

OF PHYSICS

216, 268-290 (1992)

Click-Counting

Distributions

in Optical Transitions

K. W~DKIEWICZ Institute

of Theoretical Warsaw

Physics, Warsaw 00-651, Poland*

University,

AND

J. II. EBERLY University

Department of Physics and Astronomy, of Rochester, Rochester, New York

14627

Received July 31, 1991

We show that it is possible to associate with a quantum average a bivalued “click-counting distribution” analogous to photon-counting distributions. From this distribution it is possible to deduce individual microscopic realizations of the quantum ensemble. We show that these individual realizations of the quantum ensemble. We show that these individual realizations, calted by us microscopic quantum jumps, are described by a random-telegraph master equation. We derive these microscopic quantum jumps and their ~click-counting dist~butions” accompanying the effects of ionization, two-level laser excitation and photon antibunching. We illustrate the nonlocal character of these “click-counting distributions” in the framework of Einstein, Podolsky, and Rosen spin correlations. 0 1992 Academic Press, Inc.

I. INTRODUCTION

Long before a complete formulation of quantum mechanics was established, Bohr [I] and Einstein [2] introduced in their theories the concept of an abrupt transition (better known under the term quantum jump) of an atomic system during the stimulated or spontaneous emission of a light-quantum. Unlike Bohr, Einstein considered that such abrupt changes of the atomic state during a spontaneous emission act are “a weakness of the theory... that it leaves time and direction of elementary processes to chance” [2 J. This fundamental statistical aspect of the elementary quantum jump that made Einstein so uneasy about his theory received further support when Born introduced the probabilistic interpretation of the wave function. In a lecture given to the * Also associated with the Center for Advanced Studies and Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131.

268 ~3~916/92

$9.00

Copyright 0 I992 by Academic Press, Inc. Ail rights of reproduction in any form resewed.

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269

British Association at Oxford in 1926, Born expressed the opinion that “every process consists of elementary processes which we are accustomed to call transitions or jumps; the jump itself seems to defy all attempts to visualize it; and only its result can be ascertained...” [3]. Th is view was strongly supported by Dirac when he wrote in his Quantum Mechanics that during a polarization measurement the photon “has to make a sudden jump” and that the states it will jump into cannot be predicted, but are governed by probability laws [4]. The almost universal acceptance of the Copenhagen interpretation of quantum mechanics, based on ensemble averages of isolated microscopic events, has led to the widespread opinion that a description of a quantum system’s distinct separate quantum events is not provided by the wave function of the system. According to this view only statistical averages (including moments and correlations, of course) can be predicted from the properties of the wave function. The idea that from a continuous wave function one can derive jumplike individual microscopic events is not widely accepted in the traditional interpretation of quantum mechanics. In agreement with Born’s opinion quoted above, an ensemble average will smooth out individual quantum jumps, leaving continuous and well-behaved quantum expectation values. For example, in a radioactive decay an experimental average over many individual events leads to the well-known exponential decay law even if, in the process of collecting the relevant experimental data, a whole series of individual random “clicks” of the detector has been recorded. These individual clicks are, of course, manifestations of individual quantum jumps and, as we argue below, are governed by quantum mechanics. A similar situation occurs in photon counting, where the electric current at the detector is triggered in an instantaneous way, contrary to the classical picture in which an accumulation time is required for the absorption of the electromagnetic energy that causes the photoelectron to be emitted [S]. Evidence of an abrupt transition from an excited to a ground state of an atom has been further provided in the prediction and observations of the photon antibunching effect [6, 73, where the joint probability for the absorption of a second photon vanishes just after the first photon has been observed. A very similar situation occurs if photon polarization is measured in a cascade experiment which represents the optical version of the spin orientation measurements of the Einstein Podolsky Rosen (EPR) correlations [8]. In such experiments photons or spins are detected in a joint measurement involving two photomultipliers or two polarization analyzers [S-lo]. In all these experiments, the probabilities or the correlation functions arise from repeated individual random “clicks” triggered by a prepared ensemble of individual microscopic systems. For example, in the photon antibunching experiments just mentioned, data have been collected from a large number of radiating single atoms, entering and leaving a laser beam. In this case the antibunching joint probability distribution emerges as a result of an ensemble average over many abrupt acts of single-atom spontaneous radiation. In the optical tests of local realism random “clicks” triggered by linear polarizers

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arise from repeated measurements of the photon (spin) polarization. The EPR correlation function for the joint detection emerges as a result of an ensemble average over many abrupt acts of single-photon polarization detection. A new class of experiments dealing with random quantum “clicks” in the fluorescence of a single trapped ion has been reported recently [l l-131. In these experiments a strong fluorescence signal from two resonant levels has been interrupted whenever the population of one of the levels has been “shelved” by excitation with a weak laser to a third level. The resulting “telegraph-signal” behavior of the fluorescent light gives direct evidence for quantum jumps and provides a direct monitoring of the atomic state and its “clicks.” The novelty of these experiments is due to the fact that the data have been recorded from a single ion and as a result the ensemble average has been replaced by the temporal evolution of the individual microscopic events generated by a single quantum system. In this connection, the quantum jumps of a “shelved” three-level atom have attracted considerable theoretical interest recently [ 14211. To the best of our knowledge the problem of the relation of the quantum mechanical wave function itself to individual “clicks” in photon counting, ionization, or radioactivity has not been addressed within the quantum formalism. In this paper we present an approach to the problem of quantum jumps that establishes a direct link between the jumps and the system wave function. We derive a click-counting probability function which is inherently associated with the wave function of the system and the measuring process and which is also intrinsically biualued. We will show that our approach offers one way to translate Born’s and Dirac’s word-picture of quantum dynamics, quoted above, into a concrete mathematical algorithm. Within this approach it is obvious that all elementary processes are described by a bivalued probability distribution function which we shall call “click-counting distribution” (CCD). We show that this bivalued distribution function obeys a master equation which is typical for a jump-like stochastic process. We illustrate our approach in four physically distinct examples of optical transitions. In Section II we investigate the quantum jumps accompanying ionization from a single bound state to a broad-band continuum. We show how the quantum mechanical expectation value for such a process can be explicitly visualized as an ensemble of individual quantum “clicks.” In Sections III and IV we treat two-level atom driven by a strong, short, resonant laser pulse. We show that individual realizations of such a quantum system consist of random telegraph transitions between the two driven atomic levels. We study the transition rate of these quantum jumps in the case of coherent and incoherent excitations. Explicit derivations of the bivalued probability distribution function are presented for both cases. In Section V we discuss the photon antibunching effect and its relation to quantum jumps. This effect is more complicated than the examples discussed in Sections II-IV, because quantum jumps associated with photon antibunching reflect the statistical properties of a quantum correlation function of a higher order (four electric fields).

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In Section VI we discuss quantum jumps in the EPR correlations. These EPR quantum jumps, like the jumps in the photon antibunching case, reflect correlations of higher order because polarization correlations involving two entangled spin-$ systems are discussed. We show that the EPR correlations, if described by a CCD, have many similarities to the photon antibunching CCD. We show that the local realism expressed in terms of Bell’s inequality is violated by these “click-count” jumps because of the nonlocality of these spin correlations. Finally, some conclusions are given in Section VII.

II. QUANTUM JUMPS IN IONIZATION Let us start our discussion with an ionization process, i.e., a transition from a bound state 10) to any of the states IE) forming the atomic continuum (Fig. 1). Ionization of a single electron can be observed with the aid of a broad band detector in the continuum which will signal the ionization with a single “click.” Experience indicates that this “click” means that the atom has been ionized. The broad band character of the detector assures that this electron is absorbed by it and cannot return to the atom. Repetition of this simple experiment will lead to an entire series of random “clicks” and after a while the ensemble of “clicks” can be sensibly compared with quantum mechanical predictions about ionization. This is standard quantum mechanics and this is how the concept of the ensemble average works. But let us return to the single event, i.e., the single “click.” Can we extract the single “clicks” representing ionization events; i.e., can we derive a quantum jumps from IO) to IE), from the wave function of the system? The answer is yes and, in order to show this, we proceed as follows.

FIG. 1. Diagram indicating one-photon ionization. A negative ion such as H- is an example of a quantum system with only one bound state plus a continuum.

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The wave function of the atomic electron has the form

W(t)>= %dt)IO>+ 1 dEa,(t) I-v,

(2.1)

where la,,(t)1 * = exp( -2yt) is the decaying population of the bound state IO) to the continuum 1E) with a rate given by 2~. Broad band detection is described in our case by the projection operator, (2.2)

where we have integrated over all continuum states. The quantum mechanical average of this operator Tr{b(t)@} is interpreted as the probability to find the ionized electron anywhere in the continuum if the state of the system is described by the density operator p(t), where b(t) = 1$(t)) ($(t)l in our case. This quantum expectation value can be written in the equivalent form

(2.3) where

m

t)= ($(t)l w-PI

Ill/(t)>.

(2.4)

This probability density depends both on the wave function of the system 1$(t)) and on the projection operator P of the measuring detector. The quantum mechanical average written in form (2.3) can be interpreted as an ensemble average of individual events x with a probability density given by Eq. (2.4). For the ionization wave function (2.1) an explicit calculation of this probability density is possible using the well-known pole approximation. As a result we obtain P(~,t)=e-~~‘6(~)+(1-e~*~~)6(x-l)

(2.5)

which is the bivalued distribution function (x = 0 or 1) promised in the Introduction. It obviously represents the wave function 1$(t)) and the broad-band detection mechanism. The two possible values of x correspond to the cases when there is nothing in the continuum (x =0) and there is no detector response or when the electron is in the continuum (x = 1) and the detector clicks, The dynamics of such jumps follows from the master equation,

iP(x,r)=

-2yP(x,t)+2yd(x-

1)

(2.6)

with a natural initial condition at t = 0: P(x, 0) = 6(x); i.e., the atom is not ionized

CLICK-COUNTING

FIG. 2.

A typical

273

DISTRIBUTIONS

step-like

jump.

yet. This master equation describes a step-like jump process with an “absorbing” barrier at x = 1. A typical individual event is represented in Fig. 2. In this figure, t, is the random time of a “click” and the probability to jump from 0 to 1; i.e., the probability to “click” in an infinitesimal time interval At is equal to ydt. In a finite time interval the “clicks” exhibit a Poissonian distribution of tC’s, leading to the well-known quantum shot-noise at the detector. Note that the form of P(x, t) and the associated master equation (2.6) are exact consequences of the specific ionization wave function (2.1) and of the broad-band character of the detection mechanism represented by the projection operator (2.2).

III. QUANTUM JUMPS IN AN UNDAMPED

TWO-LEVEL

ATOM

Let us now investigate a different physical system that consists of a two-level atom (with levels 1+ ) and I- )) driven at exact resonance by a coherent and short (compared to the atomic lifetime) laser pulse characterized by an area d(t)=j’ds L?(s ), w h ere Q(t) is the instantaneous Rabi frequency of the electric field laser pulse [22]. The physical quantity of interest is the atomic inversion operator 8, = ) + )( + 1 - I- )( - I. Following our discussion of the ionization case, we can write the quantum expectation value of the inversion operator in the form

(6~)=pxxP(x.r),

(3.1)

where now nx, 1) = (ti(t)l

w - 83) Ill/(t) >

(3.2)

and l+(t)) is the time-dependent state of the two-level atom excited from its ground state 1- ) at t = 0 by coherent resonant light with the pulse area given by e(t). Simple algebra involving the explicit form of It&t)) leads to P(x, t) = $( 1 + cos e(t)) 6(x + 1) + &(1 -cos 0(t)) 6(x-

1).

(3.3)

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Again the atomic wave function and the atomic dynamical variable are associated with a bivalued distribution function, this time associated with the values x = f 1. The two possible values of the random variable x correspond to the inversion jumping from + 1 to - 1, from the ( + ) state to the 1- ) state, and vice versa. In time, the random variable performs abrupt random-telegraph jumps from x + -x. The dynamics of these quantum jumps, associated with the wave function 1$(t)) and the atomic inversion, is described by the master equation

;P(,, t)= --

1

P(x, t) + -

2dt)

1

P(--? f),

Wt)

where l/z(t) = e(t) tan O(t). This master equation describes a random telegraph process with a time dependent jumping frequency given by l/z(t). It is well known that the pulse area is often a more physical parameter than time for providing insight into the two-level excitation process. We can easily convert our random telegraph to a 8 basis. The atomic inversion as a function of the pulse area is given by the expectation value (63) = -cos 6,

(3.5)

where we have assumed that the atom initially is in the ground state. This quantum mechanical expectation value is a smooth function of the exciting pulse area and we can write it in the following form: @,)=jdxxP(x,B).

(3.6)

Then Eqs. (3.4) and (3.5) give (3.7)

where y(8) = tan 8. This master equation describes a random-telegraph process (a click-counting process) with a pulse area-dependent jumping rate y(B). A typical realization of such a telegraph process (as a function of the laser pulse area 0) is shown in Fig. 3.

212 FIG. function

3.

A representation of pulse area.

of the quantum-jumping

E

of a coherently

pumped

two-level

atom,

as a

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DISTRIBUTIONS

The quantum jumps of the driven two-level atom occur with infinite frequency (as a function of 0) for any pulse with area equal to 8 = 7c/2. It is clear that in these cases we can have 6’= 7c/2 and 4 #O at the same time. In addition, the atom experiences its maximum dispersion of the inversion operator da: = 1 at 0 = 7r/2. These maximum atomic fluctuations are exhibited in the individual realizations of the jumping process by an infinite jumping rate. Now what about the Rabi oscillations? How do they affect the individual realizations of the quantum jumps of the atom? In the individual realization representing the wave function I+(t)) and the inversion operator 8,, Rabi oscillations occur only in the temporal behavior of the jump frequency l/t(t). Smooth Rabi oscillations of the atomic inversion result only from an ensemble average of many individual realizations of the atomic quantum jumps represented by Fig. 3.

IV. QUANTUM

JUMPS IN AN INCOHERENTLY

PUMPED TWO-LEVEL

ATOM

Let us now see how the atomic quantum jumps look if the excitation is incoherent. Intuitively it should be clear that some kind of average over the Rabi frequency should occur and that the oscillations should be replaced by damping. Consider, for example, the case of a noisy laser pulse whose field envelope is given as a Gaussian stochastic process. Then the Rabi frequency can be assigned the correlation functions (Q(t)>inc=o

and

(Q(t) Q(t)),,,

= 2rs(t

- t’),

(4.1)

where f is the diffusion constant of this incoherent radiation. For such incoherent radiation the stochastic average of the Rabi oscillations gives (4.2) and, as a result, the following probability distribution inversion driven by an incoherent pump:

is associated with the atomic

P(~,t)=$l+e-~‘)6(x+l)++(l-e~~‘)6(x-l). From this solution we obtain that the dynamics of these quantum described by the master equation

-gP(x,r)=+qx, t)+$(-x, t),

(4.3) jumps is

(4.4)

where the jumping rate of the random telegraph process now is given by the constant r. A typical realization of such a telegraph process is shown in Fig. 4. Indeed, Rabi oscillations have been removed by the incoherence of the laser pump

276

W6DKIEWICZ

FIG.

4.

A representation

AND

of the quantum-jumping

EBERLY

of an incoherently

pumped

two-level

atom.

and a simple random telegraph process with a constant switching frequency r in the atomic population inversion represents the quantum jumps in such an atomic system. It is clear that in the limit of t -+ co, the atomic system reaches an equilibrium state with the incoherent laser pump field. In this limit the atomic inversion is equal to zero and the corresponding random telegraph stochastic process has a probability of 5 to be in the excited or in the ground state. It is known that the fluorescence intensity of a radiating two-level atom is proportional to the atomic population of the excited level, I+ ) ( + 1= $( 1 + 83). The random variable x which corresponds to the atomic inversion can be replaced by a random variable Z= i( 1 +x), which corresponds to the atomic population of the excited level. The fluorescence intensity in this case is given by the expression (I)

= j” dZZP(Z, t),

(4.5)

where

P(ZTt) = .

(4.6)

From Eq. (3.3) and the definition (4.6) we conclude that the CCD (4.6) is bivalued with Z= 0 or Z= 1. The quantum mechanical average of the fluorescence intensity corresponds to an ensemble average of these random “on” and “off” quantum clicks and, as a result, we obtain for coherent excitation (3.5), (I) = sin2 O/2,

(4.7a)

and for the incoherent excitation modelled by (4.2), (I)={(1

-exp(-Z?)).

(4.7b)

The function (4.6) represents a single-fold distribution function of a stochastic process corresponding to a CCD of quantum jumps. This function should appear as a marginal distribution from a higher jump-like distribution which describes an optical process in which photons are correlated. We shall discuss such a process in the following section.

CLICK-COUNTING

V. QUANTUM

JUMPSAND

277

DISTRIBUTIONS PHOTON ANT~BU~CHING

Photon antibunching in resonance fluorescence [6] (first reported in 1977 by Kimble et al. [7]) strongly supports the concept of an abrupt atomic transition during spontaneous emission. Compared to the examples discussed in the previous sections, photon antibunching involves a higher correlation function of the electromagnetic field. The joint probability to detect a photon at time t fz, if a photon has been detected at time t, is proportional to the four-point coherence function [6] G(t+z;t)=(d+(t)b+(t+t)ci(t+t)o’(t)),

(5.1)

where G+ and cr are the raising and lowering atomic dipole operators. This formula represents the quantum version of the classical coherence function (Z( t + t) Z(t)) with the instantaneous intensity being being identified, for the sake of simplicity, with the upper state population operator Ix 6’(t) d(t). The quantum nature of the detection mechanism is reflected in the fact that G may be written G(t+T;t)={:~(~(t+~)~(r)):),

(5.2)

i.e., as a normal and chronological product of the atomic transition operators. In order to extract the jump character of the intensity we shall rewrite Eq. (5.2) in the form

i.e., in the form of a statistical average over random intensities Z and Z, with a distribution function P(Z, t + z; IO, t). The quantum nature of the statistical process is reflected in the definition of this distribution function. Following the procedures from the previous sections we derive that P(Z, t+z;z,,

t)= (: ~[~(z-~+(t+?)~(t+~))x~(z~-~+(r)~(r))]

:).

(5.4)

In agreement with the rules of classical stochastic theory we can rewrite this joint probability distribution in the form of a product of a conditional probability P(Z, tt z/ I,, t) and a one-fold distribution P(Z,, t): P(I, t + t; I,, t) = P(Z, t + 5 /z,, t) P(Z*, t).

(5.5)

For the intensity operator i=e’a we have the trivial identity emiAi= (1 - 1) + ie-“. Thus all the orderings and algebraic calculations in formula (5.4) can be carried out exactly. As a result we can write the conditional probability dist~bution in the form P(z,t+zIZ~,t)=P(0j0)6(1}6(z~)+P(1~1)6(1-1)6(1~-1) -tP(OI l)fJ(Z)S(Z,-

l)+P(l10)6(Z-

1)&Z,),

(5.6)

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where all the functions P( i /j) = P( i, t + T /j, t) (i, j = 0, 1) can be calculated explicitly for the strong resonance fluorescence problem. Clearly the probability distribution function (5.6) indicates a bivalued dimensionless intensity, with the only possible values for the random variables being 0 or 1. For the value 0 no radiation is detected, and for the value 1 radiation has been detected. This gives a physical interpretation of the four functions entering into the definition (5.6). For example, P(0, t -t r j 0, t) is a conditional probability that, if at a given time t no radiation has been detected, then at a later time t + r, still no radiation will be detected. Because of this, P(O]O) gives a temporal correlation between “dark” periods at the detector. For the same reason P(0 11) correlates dark with bright periods, P( 1 11) correlates bright with bright, and, finally, P( l/O) correlates dark with bright. For these functions we have the sum rules PfOl l)+P(l I l)= 1 and P(1 [O)+P(O/O)= 1. This means, for example, that the sum of the two probabilities to start in a bright period and end up either in a dark or in a bright period is, of course, a certainty. Formula (5.3) exhibits the fact that one can visualize the coherence function (5.2) as an ensemble average over abrupt jumps between all possible combinations of dark and bright periods. The term “abrupt” applies because the intensity is assigned one of only two values 0 and 1. Note that the marginal averages of the “click-counting distribution” given by Eq. (5.6) reproduce the statistical properties of fluorescence discussed in the previous section. For strong-field resonance fluorescence these functions can be calculated exactly. At resonance, in the steady state limit (t + co) and when the Rabi frequency $2 is much larger than the Einstein A-coefficient (Q $- A), these formulas are particularly simple. In the steady state limit the P’s are stationary and their only dependence is on the time r: (5.7a) and ~(O,~~l)=~(l,~~O)=~[I+exp(-3~~/4)cos~~]. In Fig. 5 we have plotted

the probability

for dark-dark

(5.7b) and bright-bright

0.8

I

I

I

I

I

0123456 AT FIG. 5. The probability for dark-dark and bright-bright laser excitation: D = 3A.

transitions as a function of Az, for a strong

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DISTRIBUTIONS

transitions (5.7a) as a function of A7. In Fig. 6 we have plotted the probability for dark-bright and bright-dark transitions (5.7b) as a function of A7. The equivalence of dark-dark and bright-bright correlations as well as dark-bright and bright-dark functions is due to the equal population of the two levels when 51% A. In this case Eq. (5.6) can be written in a simpler form, P(I,t+7/f,,r)=~fl-e-3~“‘4COSSZ7)6(1-I,) + $1 +r-‘--3.4r!4cos Q7) 6(1+ zo - l),

(5.8)

where I or I, can be equal to 0 and 1 and the one-fold stationary distribution given by the following formula: P( IO) = 5 S(Z,) c 4 6(Z, - 1).

is

(5.9)

These formulas obviously describe a random-telegraph signal with jumps between 0 and 1. From Eq. (5.8) we obtain the random-telegraph master equation for the quantum jumps in strong field resonance fluorescence,

where the time-dependent

dwell frequency of the telegraph is given by i=tA+RtanQr

(5.11)

and the transition matrix f(Zl I’) is off-diagonal. This formula is quite similar to (3.4), but differs by the constant rate associated with spontaneous emission in the presence of strong laser light. To an excellent approximation it can be assumed that the process of spontaneous emission in the presence of a laser field is Markovian, and as a result of this approximation the stochastic process associated with the telegraph master equation

FIG. 6. The probability for dark-bright laser excitation: Q=3A.

and bright-dark

transitions as a function of At, for a strong

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(5.10) is fully described by a two-point joint distribution function with the marginal (4.5), already discussed in the previous sections. In fact it is quite easy to generate the one-fold distributions presented in the previous section considering, for example, some quantum mechanical quasi-distribution functions (for example, the diagonal P-representation or the Wigner distribution function). What is impossible to obtain from such quasi-distributions is a meaningful (i.e., a positive) joint distribution function with the jump-like interpretation. We believe that the CCD presented in this section offers a very attractive and physically appealing interpretation. The photon antibunching property can be clearly seen in Eqs. (5.7), where for T = 0 the correlation functions of dark-dark and bright-bright periods vanish and we have, accordingly: P(0, T = 0 11) = P( 1, T = 0 IO) = 1. This is precisely what photon antibunching is, a perfect correlation of bright and dark periods with no correlation at all between dark-dark and bright-bright periods. Let us discuss for completeness the case of weak excitation by the laser (Sz
+2ePA’-2ePA”2)

(512a)

P( 1 11) = (Q2/A2)( 1 - ePAr/2)2

(5.12b)

P(110)=(Q2/A2)(1+2e-A’-2ePA’/2)

(5.12~)

P(OI 1) = 1 - (Q2/A2)(l -e-A7’2)2,

(512d)

where for notational reasons we have omitted the time argument T in P(iIj). Note that the strong-field symmetry between different periods is now gone. The dark-dark and the bright-bright correlations are not equal to each other any more. This is, of course, due to the weak-excitation limit in which the system is most likely to stay in a state with no outgoing radiation. For T = 0 the dark-dark periods are correlated with probability P(0, T = 0 IO) = 1 - Q2/A2, and the bright-bright because P( 1, T = 0 11) = 0. In addition we have periods are “anticorrelated” P(~,T=OIO)=Q~/A~ and P(O,~=o(l)=l. This means that, as in the case of

0

0

2

4

6

8

10

AT

FIG.

A=3Q.

7.

The

probabolity

for dark-bright

transition

as a function

of

AT, for a weak

laser excitation:

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DISTRIBUTIONS

0

0

2

4

6

8

IO

AT

FIG. 8. The probability A=3D.

for bright-bright

transition as a function of As, for a weak laser excitation:

strong-meld resonance fluorescence, the bright-bright periods are “antibunched,” because P( 1, r = 0 / 1) = 0. For weak excitation we have a dark-dark jump which is much more probable than the dark-bright transition. This is, of course, due to the fact that L? 6 A and the atomic transitions are not saturated. In Fig. 7 we have plotted the probability for dark-bright transition (5.12~) as a function of Ar. In Fig. 8 we have plotted the bright-bright (512b) transition as a function of At. The corresponding dark-dark (512a) and bright-dark (512d) transitions can be obtained by a subtraction of these expressions from unity. These positive CCDs lead to a simple statistical interpretation of photon correlations in terms of random numbers 1 and 0 for the variables I and I,. The quantum mechanical average in this case is represented by an ensemble average of sequences of random numbers 1 and 0. This random character of these variables can be applied in the description of the photon correlations detected by two photomultipliers (see Fig. 9). In this case to each photomultiplier there corresponds a sequence of random variables denoted by I and I,. These are the only possible outcomes of the experiment (with perfectly efficient detectors). On each single photomultiplier the outcomes are completely random and the bright and dark outcomes occur with equal probability ($ in this case). The quantum nature of these correlations shows up in the fact that these two perfectly random sequences (on the first and on the second photomultipliers) are correlated and the correlations are given by Eqs. (5.6). These formulas predict that FLUORESCENCE I+>

PHOTOMULTIPLIERS

/

l = (1,0,0,1,0,1,0,0,1,...) D-l--

Time delay

I->

=

f-)1.

I ,, = (o.r.r,o,o,~,o,l.l,...)

FIG. 9. Fluorescence photons emitted by a two-level atom are detected and correlated with a time delay T by two photomultipiers I and I,.

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a photon correlation measurement can be understood as a correlation between completely random sequences Z,, = (1, 0, 0, 1, 1, 0, 0, ...) and I= (0, 0, 1, 1, 0, ...). These correlations will change with the time delay z between the two photomuitipliers according to Eqs. (5.6~(5.121, Because of this we shall call these correlations nonlocal. This terminology is borrowed from the theory of spin correlations for the EPR wave function (see next section). The nonlocality means here just the fact that by changing the time delay z between the two photomultipliers we can change the correlations between compIetely random sequences of numbers. Note that photon antibuching for z = 0 in this case is equivalent to a correlation of two random sequences I, = (1, 1, 0, 1, 0, 0, ...) and I= (0, 0, 1, 0, 1, 1, ...) with a probability one to have 1 and 0 corresponding to 0 and 1 in the other sequence. The nonlocality of these correlations follows from the fact that whenever Z0 = 1 or 0 on the photomuliplier (Z,), we must have, with probability one, I= 0 or 1 on the second (even if it is remote) photomultiplier (I). A local and a positive single-fold distribution function will never reproduce antibunching because we have: (Z2)/(Z)2 - 1 = j dZP(Z)(Z- (Z))2/(Z)2 >O. For different values of the time delay T, this correlation probability between the two sequences will be modified according to the formula (5.6). We shall discuss the similarities of this result to the EPR correlations in the next section. It is important to keep in mind iome facts about Markov stochastic processes (spontaneous emission to an excellent approximation is Markovian). In order to describe the statistical properties of the intensity one needs TWOprobability distributions: (1) the one-fold P(Z, t) which is rather trivial and (2) the conditional distribution function P(Z, t 1I,) which is the key to the Markovian statistical interpretation in terms of “click-counting” quantum jumps. Correlations of Z and I0 are calculated with the help of the joint distribution PfZ, t; I,) = P(Z, tlZO) P(Z,) which relates the conditional and the one-fold distributions. Our CCD derived in this and in the previous section are of such a type.

VI. EPR JUMPS Let us start our discussion of this section with a brief description of the EPR correlations for a singlet state of two spin-i particles [S]. Such correlations are equivalent to those obtained in photon cascade experiments [9-lo]. The EPR state of the two spins is given by the density matrix fi = MEPR)

(6.1)

where, following Bohm, the singlet state of the two spins I~EPIo=~(l+;-)-I-;+)) Jz

(6.2)

is the EPR wave function applied to a spin state. We have denoted’ here by I+ >

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283

and 1- ) the individual spin up and down states of the two particles. If the spins are measured along some axes a and b, the observable associated with such measurements are represented by the projection operators B(a) = $( 1 + S(a)),

f(b) = $( 1 + c?(h)),

(6.3)

where we have understood ii(a) = 8. a and cF(b)= 6. b. The quantum mechanical correlations between measurements of the spin a and of the spin b by linear polarizers is given by the expectation value of the projection operators, p(a; b) = (&a)

P(b)) = a( 1 - cos 01),

(6.4)

where a is the relative angle between the two unit vectors a and b. These are the standard quantum mechanical predictions for the spin-correlated state given by Eq. (6.2). Following our approach developed in the previous sections we can rewrite the quantum correlation function (6.4) in the form

where the CCD function is given by the formula:

With the help of this CCD we have rewritten the quantum mechanical correlation function in a form which has remarkable similarities to the photon antibunching correlation function given by Eq. (5.3). Because the projection operators can have their eigenvalues equal to 1 or 0, i.e., can represent only “yes” or “no” answers, the values R, and & can take only values equal to 1 and 0. The bivalued distribution given by Eq. (6.6) is positive everywhere, but depends on the polarization directions a and b. The distribution function which depends on the orientation a of the first analyzer and on the orientation b of the second (possibly even remote) analyzer is nonlocal (see also the previous section for the discussion of the nonlocality in photon antibuching). In the framework of EPR correlations it is customary to call an analyzerdependent distribution function a nonlocal distribution function. The nonlocality of this distribution function makes the Bell’s inequality [22] void, because in order to obtain this inequality the existence of a universal, local (polarization independent) distribution in the parameters A, and A, (hidden parameters in this case) is essential. Quantum mechanics tells us that if we insist on a “click-counting distribution” of the form given by Eq. (6.6), we can do it but only under the condition that the statistical distribution of the parameters A, and A,, is nonlocal. If we insist on a local

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distribution we shall end up with a quantum mechanical quasi-distribution which is not positive definite [24-251. In order to elucidate this point further, let us calculate the explicit form of the nonlocal distribution function (6.6) for the spin singlet state. From Eq. (6.6) and the form of the singlet state we obtain P(~,;I,)=P(O;O)6(;1,)6(~,)+P(l; + P(0; 1) &I,)

1)&A,-

l)&&-

1)

42, - 1) + P(0; 1) &A, - 1) &A,),

(6.7)

where P(O;O)=P(l;

l)=$[l-coscr]

(6.8a)

P(0; 1)=P(1;0)=~[1+c0scr].

(6.8b)

and

For notational convenience we have omitted the angle c( argument in the P(i;j). There is a striking similarity between these expressions and the photon antibunching CCD (5.6). In fact, mathematically these two expressions are identical and represent a bivalued distribution of a random-telegraph signal. This result shows that one can regard the spin measurements of the EPR wave function as an ensemble of random-telegraph signals at each spin polarization analyzer. Only after a statistical average of these quantum jumps is performed do we obtain smooth quantum mechanical probabilities. The EPR correlations are just correlations of two sequences of random numbers jumping between values 0 and 1 (“no” and “yes” answers) for polarization measurements performed with linear analyzers (see Fig. 10). Following the procedure developed in Section V we can construct a conditional probability distribution for “yes” and “no” answers represented by the distribution (6.6). This conditional distribution is defined by the following relation (see also Eq. (5.5)) fv,; The distribution

FIG. 10. polarization

P(A, 1lb) is the conditional

Iz;

2,) = R&I I &I fv,).

(6.9)

of the event A, (“yes” or “no”) to occur

(0,1,1,0.0,1....)

EPR correlations of two sequences of random numbers jumping measurements performed with two linear analyzers a and b.

between

values

0 and 1 for

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DISTRIBUTIONS

under the condition that 3”,+(“yes” or “no”) has occurred. Because of the full symmetry between the two detectors we have P(I,l&) = P(J.,I ,I,). From Eqs. (6.1)-(6.9) we calculate that these conditional probabilities for the particular transitions are P(O]O)=P(l/l)=sin’01/2,

(6.10a)

P( 1 IO) = P(0 / 1) = cos2 cr/2*

(6.lOb)

These positive and nonlocal distributions lead to a simple statistical interpretation of the spin transitions and of the violation of Bell’s inequality in terms of random numbers 1 and 0 for the variables R, and A,. The quantum mechanical average in this case is represented by an ensemble average of two sequences of random numbers 1 and 0. This random character of these variables can be applied in the description of the EPR correlations measured by two polarizers. To each polarizer there corresponds a sequence of random variables denoted by 1;, and /I,,. These are the only possible outcomes of the experiment. On each single polarizer the outcomes are completely random and the “yes” and “no” answers occur with equal probability ($ in this case). The nonlo~ality of the EPR correlations shows up in the fact that these two perfectly random sequences (on the first and the second polarizers) are correlated and the correlations are given by Eqs. (6.10). These formulas predict that the EPR wave function can be understood as a nonlocal correlation between two random sequences 1, = (1, 0, 0, 1, LO, 0, ...) and &, = (0, 0, 1, 1, 0, ...). The nonlocality of these correlations follows from the fact that whenever I, = 1 or 0 on the polarizer a, we must have ,I, = 1 or 0 on the polarizer b with the probability sin* ~x/2, i.e., the outcomes on b (possibly even a remote analyzer) are determined by the outcomes on the analyzer a. Let us illustrate, using these random sequences, the violation of Bell’s inequality, Let us assume that in the first series of experiments we set CI= n/4. According to the formulas (6.10) we have P(OlO)=P(l

(I)=-

1 2i

1 -;

>

x0.15....

(6.11)

This means that we have 15% confidence that the outcome on b will be the same as the outcome on a. Let as assume that in the second series of experiments we set TV= 37r/4. For this angle we have P(O)O)=P(l[

l)=f

z 0.85..

(6.12)

This means that we now have 85% confidence that the outcomes on b will be the same as the outcomes on a. The joint probabilities for c1= 7r/4 and 51= 37r,l4 can be obtained by a multiplication of the conditional probabilities by 0.5. This follows from the fact that the probabilities to have 0 or 1 in any of these experiments is OS,

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because the individual spins behave as perfectly incoherent mixtures of spin up and spin down states. From Eqs. (6.1 l)-(6.12) we obtain that the joint probability for these two orientations of the polarizers have to satisfy the following condition: fi P(3J4)-P(n/4)=~=0.35....

(6.13)

This is precisely what the Bell’s inequality violation is. It is well known that for a coplanar geometry it is possible to reduce Bell’s inequality to the inequality [S], jP(3n/4)-P(n/4)1

GO.25,

(6.14)

i.e., a clear contradiction with the result (6.13) given by the nonlocal “clickcounting distributions.” The probability of reproducing the same sequence on the polarizer a if the sequence on the polarizer b is known is nonlocal, because the “click-counting distribution” depends on the relative orientation c1of the polarizers. This is how the EPR quantum jumps violate local realism.

VII.

DISCUSSION

The examples discussed in this paper show in a few cases how one can associate a bivalued distribution function with the observables and the wave function I+) of a system. We have shown that with a quantum average one can associate a “clickcounting distribution” analogous to photon counting distributions. From this “click-counting distribution” one can deduce the statistical nature of the individual realizations involved in the examples discussed. We have shown that these individual realizations have the form of a random-telegraph signal, whose distribution function is determined by the Schrijdinger wave equation and, in addition, are consistent with the views quoted from Born or Dirac. We have shown that all of these randomly quantum-jumping systems can be described by a random-telegraph master equation with appropriate boundary conditions and switching times. Our representation of quantum mechanical expectation values as ensemble averages of randomly jumping variables allows for a clear picture of the measuring process in quantum mechanics as follows. Let us assume that we have a system in a state described by a density matrix /? We probe the system and ask a certain elementary question represented by a projection operator p. The probability for a positive answer is given by P=Tr{fip}.

(7.1)

But by using the procedure developed in this paper we can rewrite Eq. (7.1) in the form p=prxP(x),

(7.2)

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287

where P(x)=Tr{b@x-p)}. Because of the properties of the projection expression

(7.3)

operator P, we obtain from (6.3) the

P(x)=(l-p)6(x)+p&x-l),

(7.4)

i.e., a yes and no distribution function (x = 0, 1) with the associated probabilities (1 -p) and p. Before the measurement, the system can be regarded as jumping between the two answers and, during the measurement, this probability “collapses” by jumping into one of the answers. This is rather standard language when discussing quantum mechanical measurements. More than this, we have discussed specific physical examples for which the random-telegraph master equation involving one-fold and two-fold distribution functions can be worked out exactly. Due to the spectral theorem we believe that our approach based on yes and no questions is quite general and holds for any projection measure associated with an arbitrary selfadjoint observable [26]. It is possible, of course, to ask more complicated questions than just yes or no regarding state occupation, and it is possible to generalize our approach accordingly. For example, a many-level system permits simple yes-no questions to be asked about a given level. But questions that take account of all of the levels can also be formulated, and a many-step telegraph framework [27] would be natural for their treatment. We remarked in Section II that it is possible to derive a quantum jump from the wave function of a system. We gave an elementary example of this. It may be useful to say what this means in different language. From the beginning of quantum mechanics it has been a goal to formulate quantum theory in terms of directly observable quantities. The density matrix formalism does this, even for pure-state evolution, since the elements of the density matrix correspond in well-known ways to measurable “populations” and “coherences.” However, it has been felt that the density matrix and its equation of motion do not provide the deepest possible connection between quantum uncertainty and measurable reality because the density matrix exhibits quantum jumps no more directly than the wave function or its probability amplitudes do. That is, solutions of the von Neumann-Liouville equation for the quantum mechanical density matrix are always smooth deterministic functions of time. As we say, the probability of an event changes smoothly as mandated by the Schrodinger or von Neumann-Liouville equations, and this in no way contradicts the fact that the event itself happens abruptly when it happens. Since any quantum event (whatever it is) under consideration will occur abruptly, it would be best to have a stochastic rather than deterministic equation to describe it. However, it is well known that there has never been found a

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stochastic equation dealing just with observable quantities that is adequate to replace Schrodinger’s equation in describing probabilistic quantum reality. We have, nevertheless, found in Sections II-V different stochastic master equations associated with a random-telegraph process. These stochastic equations are not inferred from one or another statistical ansatz but follow from Schrodinger’s equation and thereby represent the quantum jumping associated with the observable events of ionization, atom inversion, or photon antibunching. For the ionization problem this master equation is given by (2.6). It has been possible to find it by slightly relining the traditional search for such equations. We have decided not to ask about the quantum evolution of our atom undergoing ionization, except to ask only whether or not it has become ionized. This is a yes-no question and is represented in quantum theory by a simple projection operator. the stochastic variable x in our treatment, which has all of the attributes of a purely classical stochastic process, is simply the eigenvalue of this projection operator. That is, what we have done is to show that quantum mechanics provides the rules, through Schrodinger’s equation, that govern the classically stochastic evolution of the result of an observation. We must point out explicitly that although x behaves exactly like a rather ordinary classical stochastic variable, it is nevertheless quantum mechanical (depends on Planck’s constant at the least). Of course, one’s ability to solve Schriidinger’s equation in a given circumstance determines whether one can actually obtain the desired classical stochastic equation for x. In the simple case of Section II, for example, the stochastic equation gives an excellent approximation under most conditions of ionization to the behavior of the yes-no variable x, simply because the solution depends on the validity of the pole approximation in a well-known way. A different choice of yes-no question would have led to an entirely different challenge to find the equation for x. For example, if we had asked whether the atom was, or was not, in a band of continuum states narrow enough to make the pole approximation dubious we would not have found anything as simple as (2.5) which is the beginning of the quantitative analysis in Section II. In fact, our analyses in Sections III-V yield different forms of the stochastic equation, just because a different type of yes-no question has been posed to the same two-level atom. For the photon antibunching effect, the stochastic equation provides two sequences I and I, of random bright (yes) and dark (no) questions associated with the two photon counters. This means that at each detector we can observe a random sequence of 0 and 1 corresponding to no arrival of a photon or to an arrival of a photon. The master equation (5.10) gives dynamical correlations between these two random events, resulting from the interaction of the two-level atom with strong laser light. These correlations can be exhibited with the help of quantum jumps between the dark and bright periods at the two detectors, i.e., random telegraph jumps between the two random sequences I and 1,. These jumps are fully described by quantum mechanics and exhibit the nonclassical effect of photon antibunching. For the EPR correlations, the stochastic equation provides two sequences A, and i,, of random “yes” and “no” questions associated with the two polarization

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analyzers. This means that at each analyzer we can observe a random sequence of O’s and l’s corresponding to polarization spin measurements. The random-telegraph master equation is fully consistent with EPR correlations between these two random events. These jumps or random clicks are fully described by quantum mechanics and violate Bell’s inequality based on local realism. A different question arises about the possibility of actually observing these individual quantum jumps accompanying a measurement, due to finite detector resolution. Independent of detector resolution, there is no doubt that probabilities or correlations between dark and bright periods of the detector in ionization, resonance fluorescence, or photon antibunching can be measured and are described accurately by the expressions derived in this paper. These expressions give a probabilistic interpretation of various clicks monitored in an experiment. Another issue is the transformation, in a detector, of quantum properties into a classical signal. A measurement is basically always associated with a macroscopic device which adds extra ingredients associated with the measuring process [28,29]. The macroscopic “clicks” at the detector are nothing other than the more or less smoothed microscopic jumps derived in this paper.

ACKNOWLEDGMENTS The authors thank I. Biafynicki-Birula, R. J. Cook, J. Javanainen, P. L. Knight, P. Meystre, M. 0. Scully, and H. Walther for discussions about quantum jumps. We thank G. S. Agarwal, F. Haake, and A. Schenzle for their open criticism of our views about the microscopic quantum jumps. This work was supported by the U.S. National Science Foundation Physics Division and Division of International Programs, and by the U.S. Office of Naval Research.

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