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Measurement and Ergodicity in Quantum Mechanics

Mariano Bauer1 and Pier A. Mello1

1Instituto de Fısica, Universidad Nacional Autonoma de Mexico,

Ap. Postal 20-364, 01000 Mexico, D. F., Mexico

(Dated: December 30, 2013)

Abstract

The experimental realization of successive non-demolition measurements on single microscopic

systems brings up the question of ergodicity in Quantum Mechanics (QM). We inquire whether

time averages over one realization of a single system are related to QM averages over an ensemble

of similarly prepared systems. We adopt a generalization of von Neumann model of measurement,

coupling the system to N “probes” –with a strength that is at our disposal– and detecting the

latter. The model parallels the procedure followed in experiments on Quantum Electrodynamic

cavities. The modification of the probability of the observable eigenvalues due to the coupling

to the probes can be computed analytically and the results compare qualitatively well with those

obtained numerically by the experimental groups. We find that the problem is not ergodic, except

in the case of an eigenstate of the observable being studied.

PACS numbers:

1

The question of ergodicity in Quantum Mechanics (QM) has long been studied, a “quan-

tum ergodic theorem” (QET) having been formulated by von Neumann as early as 1929

[1]. The authors of Ref. [2] presented a comprehensive discussion of the investigations on

the subject, from von Neumann’s QET up to recent publications. Following Ref. [2], QM

ergodicity for a macroscopic (more than 1020 particles) quantum system means that

limT→∞

1

T

∫ ∞

0

〈ψ(t)|A|ψ(t)〉dt = 1

d

∑

i∈d

〈ψi|A|ψi〉. (1)

The QM expectation values 〈ψ(t)|A|ψ(t)〉 and 〈ψi|A|ψi〉 appearing in Eq. (1) are interpreted

as averages over an ensemble of similarly prepared systems, to be called a QM ensemble [3];

they exhibit the result of measurements of the dynamical variable A, independently of the

invasive nature of single observations. In addition, a time average appears on the LHS

of Eq. (1) and, on the RHS, an ensemble average over the d eigenstates belonging to the

microcanonical subspace. Nevertheless, Ref. [2] points out that the property described in the

QET “is not precisely analogous to the standard notion of ergodicity as known from classical

mechanics and the mathematical theory of dynamical systems”. Some simple illustrations

of the behavior of the time average appearing in Eq. (1) can be found in Ref. [4].

Here, we pose a question of a different kind, applicable to systems not necessarily macro-

scopic, motivated by theoretical considerations and experimental developments. Out of a

QM ensemble, consider one single system s; suppose that we measure at time t1 the observ-

able A, and we measure A again at t2. If the first measurement is very invasive, we disturb

the system so much that the next measurement at t = t2 no longer reflects a property of

s. We thus introduce the first stage of the measurement, or “pre-measurement”, explicitly

in the QM description, by coupling the system s to a “probe” π1 at time t1: the idea is to

detect, at t+1 (i.e., right after t1), a property of the probe, not the system itself, and through

the entanglement of the two obtain information on the system observable. We may control

the disturbance through the system-probe coupling strength.

Another source of disturbance is the system observable A failing to commute with the

total Hamiltonian H for the system, the probe and their interaction, since a system prepared

in an eigenstate |an〉 of A would be found, in the course of time, in other eigenstates |an′〉with non-zero probability. This can remedied by requiring [A, H ] = 0, so that A, which is

then a constant of motion, is a “quantum non-demolition” (QND) observable [5–7].

Next, we couple the same system s to another probe π2 at t = t2 and detect the probe

2

at t = t+2 , etc. This is an extension of von Neumann’s model (vNM) of measurement [8, 9].

From the detection of the various probes successively coupled to one system we construct

a time average, from which we shall investigate whether we can find information on the

observable of interest A as applicable to a QM ensemble.

Non-demolition measurements distributed in time [10] and ensembles of measurements on

single microscopic systems (few particles or field modes) [11, 12] provide a situation closer

to the classical formulation of the notion of ergodicity [13].

The motivation of the present letter is thus two-fold. On the one hand we develop a

theoretical analysis of an ergodic property –relating time averages on a single system to

QM ensemble averages– which, to the best of our knowledge, has not been done in the

past. On the other, the theory is compared with its actual materialization in true quantum

non-demolition laboratory experiments.

The dynamical vNM of measurement [8] for the system and the probes is used. We

compute the QM average of the time-averaged operator Q defined in Eq. (2) below and

its dispersion over the ensemble, in order to analyze the question of ergodicity. We also

study the reduced density operator of the system proper, to find how it is affected by its

interaction with the N probes. We then analyze the probability distribution (pd) of the

eigenvalues of the observable A conditioned on the N detected probe positions: we give

an analytical treatment of the so-called “decimation” process that has been observed –and

analyzed numerically– in the experiments.

We consider an “extended system” (ES) consisting of the system proper s and the N

probes π1, · · ·πN used in the gedanken experiment described above. We designate by ω one

of theN realizations of the ES, so that the QM ensemble is the collection of such realizations.

For simplicity, each probe πi will be considered one-dimensional, and its canonically

conjugate dynamical variables will be denoted by Qi, Pi in the Schrodinger picture, and by

Qi(t), Pi(t) in the Heisenberg picture defined through an evolution operator associated with

the ES. For the vNM introduced below we show that the various operators Qi(t+i ) commute

among themselves; also, Qi(t+i ) remains constant ∀t > t+i , so that Qi(t

+i ) = Qi(t > t+N) and

thus all the Qi(t+i ) can be evaluated after the last interaction. Therefore, we can define a

“time-averaged” operator associated with the ES as

Q =1

N

N∑

i=1

Qi(t+i ) =

1

N

N∑

i=1

Qi(t+N ). (2)

3

All the Qi(t+N) evaluated at the same time t+N commute among themselves. We can thus

detect the observable Q by detecting, on the realization ω of the ensemble, the various

observables Qi(t+N) (arising from probe πi each) at one fixed time t+N , and subsequently

constructing the “time-averaged” quantity

Q(ω) =1

N

N∑

i=1

Q(ω)i , (3)

where Q(ω)i is the result of measuring Qi in the realization ω. The question we raise is whether

the time average Q(ω) over one realization ω of a single ES depends on that realization, and

whether it has any relation with the QM expectation value 〈A〉t ≡ 〈ψ(t)|A|ψ(t)〉 which is

computed at one time over the QM ensemble. The model Hamiltonian used in this paper

and defined in Eq. (4) ensures stationarity in the sense of the theory of stationary random

processes [13].

QM provides no way of calculating the time average Q(ω) of Eq. (3) on one system.

However, the standard rules of QM allow to compute the statistical properties of Q(ω) across

a QM ensemble.

To illustrate the analysis, we first consider the system proper s coupled to only two probes

π1 and π2, intended to “pre-measure” the system observable A at t = t1, and A again at

t2 (> t1) with the same strength ǫ. We assume the two interactions to be of such a short

duration that their time dependence can be approximated by delta functions centered at

times t1 and t2, respectively. We disregard the intrinsic dynamics of the system and the

probes and write the total time-dependent Hamiltonian as [7, 9, 10]

H(t) = ǫδ(t− t1)AP1 + ǫδ(t− t2)AP2 , 0 < t1 < t2 . (4)

We use for A the spectral representation A =∑

n anPan . The position and momentum

operators of the two probes are designated as Qi and Pi (i = 1, 2). The unitary evolution

operator is given by [9]

U(t) = e−i~ǫθ(t−t2)AP2e−

i~ǫθ(t−t1)AP1 , (5)

where θ(t) is the Heaviside function.

4

The probe operators Qi(t+i ) in the Heisenberg picture are given by

Q1(t+1 ) = U †(t+1 )Q1U(t

+1 ) = e

i~ǫAP1Q1e

− i~ǫAP1 (6a)

Q2(t+2 ) = U †(t+2 )Q2U(t

+2 )

= ei~ǫAP1e

i~ǫAP2Q2e

− i~ǫAP2e−

i~ǫAP1 , (6b)

from which we find that the operators Q1(t+1 ) and Q2(t

+2 ) commute and that

Q1(t+1 ) = Q1(t

+2 ) ; (7)

thus, Q1(t) is identical right after the first interaction as right after the second: indeed,

Q1(t) remains constant ∀t > t1. Thus, the operator Q of Eq. (2) becomes

Q =1

2[Q1(t

+2 ) + Q2(t

+2 )] , (8)

where the two Heisenberg operators Qi(t) are evaluated at the same time, after the last

interaction. Within the present model, the evolution operator being given in (5), A is a

constant of motion, making it a “non-demolition observable”.

If the state describing the system plus the two probes at t = 0 is |Ψ〉0 = |ψ〉(0)s |χ〉(0)π1 |χ〉(0)π2 ,

then for t > t2, i.e., after the second interaction, it is given by (f stands for “final”)

|Ψ〉f = e−i~ǫAP2e−

i~ǫAP1|Ψ〉0 (9a)

=∑

n

(

Pan |ψ〉(0)s

)

(

e−i~ǫanP1|χ〉(0)π1

)

×(

e−i~ǫanP2 |χ〉(0)π2

)

. (9b)

The joint probability density (jpd) of the eigenvalues Q1, Q2 of the two position operators

for times t > t2 is

pf(Q1, Q2) =f 〈Ψ|PQ1PQ2|Ψ〉f (10a)

=∑

n

W (0)an

∣

∣χ(0)π1(Q1 − ǫan)

∣

∣

∣

∣χ(0)π2(Q2 − ǫan)

∣

∣

2. (10b)

Eq. (10a) is written in the Schrodinger picture. The quantity

W (0)an

= (0)s 〈ψ|Pan |ψ〉(0)s (11)

is the Born probability for the value an in the original system state; χ(0)π1 (Q1 − ǫan) is the

shifted wave function of the first probe in the position representation, and similarly for π2.

5

For N probes, the jp amplitude for an, Q1, · · · , QN generalizes to

〈an, Q1, · · · , QN |Ψ〉f = 〈an|ψ(0)s 〉 χ(0)

πN(Q1 − ǫan) · · ·χ(0)

π1(Q1 − ǫan) . (12)

The jpd of an, Q1, · · · , QN for Gaussian packets with the same width σ (which represents

the resolution of the probes) for the initial probe states [9, 10] is then

pf(an, Q1, · · · , QN) =W (0)an

N∏

i=1

e−(Qi−ǫan)2

2σ2

√2πσ2

, (13a)

and the jpd of Q1, · · · , QN by

pf (Q1, · · · , QN) =∑

n

pf (an, Q1, · · · , QN ) . (13b)

The Gaussian assumption allows an analytical treatment in what follows.

Use of Eqs. (13) gives

〈Ψ0|Qi(t+i )|Ψ0〉 = 〈Ψf |Qi|Ψf〉 (14a)

=∑

n

W (0)an

· (ǫan) = ǫ〈A〉0 . (14b)

The LHS of Eq. (14a) is expressed in the Heisenberg picture and the RHS in the Schrodinger

picture. With a similar argument we find

〈Ψ0|Q2i (t

+i )|Ψ0〉 = 〈Ψf |Q2

i |Ψf〉 (15a)

=∑

n

W (0)an

· [(ǫan)2 + σ2]

= ǫ2〈A2〉0 + σ2 . (15b)

One can also show, for i 6= j,

〈Ψ0|Qi(t+i )Qj(t

+j )|Ψ0〉 = 〈Ψf |QiQj |Ψf〉 (16a)

=∑

n

W (0)an

· (ǫan)(ǫan)

= ǫ2〈A2〉0 . (16b)

6

The ensemble average over N realizations of the single-realization time average Q(ω) of

Eq. (3) is given by

⟨

Q(ω)⟩

ω= 〈Ψ0|Q|Ψ0〉 (17a)

=1

N

N∑

i=1

〈Ψ0|Qi(t+i )|Ψ0〉 (17b)

1

ǫ〈Q〉 = 〈A〉0 . (17c)

In the last line we made use of Eq. (14b). As a result, the ensemble expectation value of

the time average Q(ω), which is just the QM average of the operator Q, equals, in units of ǫ,

〈A〉0, the QM Born expectation value of the observable A in the original system state.

The crucial question now is the dispersion of Q(ω) over the ensemble, which we calculate

as

varQ

ǫ2=

1

ǫ2N2

N∑

i,j=1

[⟨

Ψ0

∣

∣

∣Qi(t

+i )Qj(t

+j )∣

∣

∣Ψ0

⟩

−⟨

Ψ0

∣

∣

∣Qi(t

+i )∣

∣

∣Ψ0

⟩⟨

Ψ0

∣

∣

∣Qj(t

+j )∣

∣

∣Ψ0

⟩]

(18a)

= (varA)0 +Ncr

N(18b)

≈ (varA)0, if N ≫ Ncr. (18c)

In the sum of Eq. (18a) we separated the terms i = j from i 6= j and made use of Eqs.

(14), (15) and (16). We defined a “critical N” as

Ncr = (σ/ǫ)2 . (19)

As a result, if N ≪ Ncr, varQ/ǫ2 ≫ (varA)0, whereas if N ≫ Ncr, varQ/ǫ

2 & (varA)0.

Given σ, the resolution of the probe, the intensity ǫ of the interaction defines how quickly

the value Ncr is attained.

We have thus found that the time average Q(ω), measured in units of ǫ, has, in general,

a finite dispersion over the ensemble of realizations ω, even when N/Ncr → ∞. This means

that we do not have ergodicity, and that we cannot verify the ensemble predictions of QM by

means of a sequence of measurements on a single system. An exception to the last statement

is when the original QM state is an eigenstate of the observable A, because then (varA)0 = 0

and varQ/ǫ2 → 0 as N → ∞.

7

To clarify to what extent has the system proper been altered due to its interaction with

the N probes, we calculate the final reduced density operator of the system. Tracing over

π1, · · ·πN the total density operator |Ψ〉f f〈Ψ| from Eq. (9) generalized toN Gaussian probes,

we find

ρ(f)s =∑

nn′

e−N

8Ncr(an−an′ )2

(

Pan |ψ〉(0)s(0)s 〈ψ|Pan′

)

. (20)

We have also used the definition (19) for Ncr. The non-demolition character of the process is

again apparent in this result: the diagonal matrix elements of the reduced density operator

are unchanged by the interaction with the probes, whereas the off-diagonal ones are changed,

depending on the interaction strength ǫ.

We have thus found that for N probes such that N ≪ Ncr, ρ(f)s ≈ |ψ〉(0)s

(0)s 〈ψ|, and the

system state is not altered appreciably by the detections. In contrast, for N ≫ Ncr,

ρ(f)s ≈ Pan |ψ〉(0)s(0)s 〈ψ|Pan′

, (21)

and the final state is a mixture like the one found after a non-selective projective measure-

ment on the original pure state [14]. This result is eventually attained as N increases, no

matter how small is ǫ/σ, as long as it is fixed. However, the final system state can be kept

close to the original one for N as large as we please, by choosing, in our experiment, Ncr

large enough.

In the experiments referred to above [11, 12], the authors analyze the change suffered by

the pd of the photon number in the cavity, conditioned on the detection of the N probes. We

now carry out this study within the theoretical model we have developed. For this purpose,

we compute the pd of the an’s conditioned on the detected N probe positions Q1, · · · , QN .

The starting point is Eq. (13a). First, integrating (13a) over Q1, · · · , QN , we find that the

final marginal distribution of an coincides with the original one, i.e.,

W (f)an

= W (0)an

. (22)

From what we mentioned right after Eq. (20), this is a consequence of the non-demolition

character of the vNM Hamiltonian we are using. From Eq. (13a) we obtain the conditional

8

pd of an as

pf (an|Q1, · · · , QN)

=W (0)an

e−(an−QN/ǫ)2

2N (σǫ )

2

∑

n′ W(0)an′

e−(an′−QN/ǫ)

2

2N (σǫ )

2

, (23)

where

QN =1

N

N∑

i=1

Qi (24)

is the same quantity defined in Eq. (3) for one realization ω. The use of Gaussian func-

tions for the probes results in the dependence of pf (an|Q1, · · · , QN) on the probe positions

occurring only through the arithmetic average QN of Eq. (24).

The result is that the pd of the system an’s, which was originallyW(0)an , after its interaction

with the probes and conditioned on a specific realization ω of theN -tuple of values Q1, · · ·QN

of the N probe positions, has become modulated by the second factor in Eq. (23), whose

effect is to “disect” it into a slice centered at an ∼ QN/ǫ, which is thinner the larger is N .

Notice that the N -tuple of probe positions Q1, ..., QN appearing in the above equations

pertains to one realization ω, and so does the averaged quantity QN . This is the “decimation

process” referred to in Refs. [11, 12], exhibited very clearly, e.g., in Fig. 2 of Ref. [12].

In these references, which use non-Gaussian functions for the probes that depend on the

experimental setup, the decimation process is exhibited numerically, while the present model

allows an analytical treatment.

We now examine the following question. Suppose we add 1 probe. In a similar way as

in Eq. (23), pf (an|Q1, · · · , QN , QN+1) depends only on the average QN+1. From Eq. (23)

we see that the new slice is centered at QN+1/ǫ and its width is smaller than for N probes.

The question is whether QN+1 has any relation to QN : the pd of QN+1 conditioned on the

values Q1, · · · , QN of the preceding N probes is found from Eq. (13b) to be

pf (QN+1|QN , · · · , Q1) = pf(QN+1|QN)

=e−

N(N+1)

2σ2 (QN+1−QN )2

√

2π(

σN+1

)2.

∑

nW(0)an e−

N+12σ2 (QN+1−ǫan)2

∑

nW(0)an′

e−

(QN−ǫan′ )

2

2σ2N

.

(25)

9

The dependence on QN , · · · , Q1 occurs only through the average QN associated with of the

preceding N probes.

In the first exponential in Eq. (25), the variable QN+1 is centered at QN and has a

variance σ2/N(N + 1). The numerator of the second factor is also a function of QN+1,

but very wide, of the order of the spectrum width of the observable A. We thus have,

approximately

E(QN+1|QN , · · · , Q1) ≈ QN (26a)

var(QN+1|QN , · · · , Q1) ≈ σ2

N(N + 1). (26b)

Thus the resulting pd of QN+1, conditioned on QN , · · · , Q1, depends only on the average

QN and is approximately centered at QN , with a variance σ2/N(N + 1). The conclusion is

that the addition of probes will eventually make the conditioned pd narrow, no matter how

wide the initial pd of Q1 is.

We go back to Eq. (23), which gives the pd of an conditioned on a given set of N probe

positions Q1, · · ·QN , i.e., for one realization ω. The “disecting factor” for this realization

is centered at an ∼ QN/ǫ; a different realization ω′ of QN would give a disecting factor

centered at some other value of QN/ǫ: through the ensemble of realizations for fixed N , the

pd of QN/ǫ is given by

pf(

QN/ǫ)

=∑

n

W (0)an

e−(QN/ǫ−an)

2

2 σ2Nǫ

√

2π σ2

Nǫ

, (27)

which is shown schematically in Fig. 1. The various peaks occurring in pf(

QN/ǫ)

are

__

Q| | /|a1 aa n

. . .

. . .n+1

. . .

. . .ε

σ / N1/2

N

__

p( )QN

FIG. 1: Schematic illustration of the pd of QN of Eq. (27).

centered at an; if N is large enough, the peaks do not to overlap and the area under the

curve of the peak centered at an gives the original value W(0)an . As we run through the

ensemble of realizations, the disections shown in Eq. (23) appear with the frequency of

occurrence of QN , Eq. (27); if we then integrate over QN , we recover W(0)an , just as observed

in Ref. [12], in their Fig. 3.

10

From Eq. (27) and Fig. 1 we now see very clearly the effect of not having ergodicity. From

one realization to another, QN jumps at random from one very narrow peak to another,W(0)an

being the fraction of realizations whose QN/ǫ ≈ an. When the original state is an eigenstate

of A with eigenvalue an0, only one peak around an0 occurs and we eventually attain ergodicity

in the limit N → ∞.

In summary, in this paper we investigated whether the notion of ergodicity, defined in the

theory of stationary random processes [13], is realized in QM, i.e., whether the time average

over one realization of a single system coincides with the average over a QM ensemble.

To control the disturbance produced by the measurement, i) we required the observable A

in question to be a non-demolition one; ii) we introduced a succession of N probes with

resolution σ which interact instantaneously with the system at times ti, with a coupling

strength ǫ that is at our disposal, and we detect the probes. This “gedanken” scheme has

been actually materialized in real experiments dealing with QED cavities, in which the

probes are atoms that traverse the cavity at successive times and are eventually detected.

The answer we found is that, in general, the system is not ergodic, as the resulting time

average has a non-zero dispersion over a QM ensemble. Thus, from the time average over one

realization of the system proper plus N probes, we cannot infer the QM ensemble average.

The reduced density operator for the system after the process is not appreciably altered

by the N detections if N ≪ Ncr, where Ncr = (σ/ǫ)2 and σ is the resolution of the probes.

If N ≫ Ncr, an initially pure state eventually becomes a complete mixture.

The probability of occurrence of the eigenvalues an, conditioned on the measured values

of the N probes, is just the original Born probability modulated by a factor that depends

on the average QN of such measured values. This is the “decimation process” observed in

the experiments. Gaussian functions for the probes provide an analytical treatment of these

results.

The statistical distribution over the QM ensemble of the average quantity QN consists

of a series of peaks centered at the eigenvalues an, the area under each one giving the

corresponding original Born probability. This is exactly what has been observed in the

QED experiments. The presence of more than one peak is precisely a consequence of not

having ergodicity in the problem.

In our analytical treatment we used a simplified Hamiltonian for the system proper plus

the N probes. A generalization to a more complete and realistic Hamiltonian is under study.

11

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12