Muga G. Time in Quantum Mechanics

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6 The Quantum Jump Approach and Some of Its Applications 143

From this one obtains an understanding of the light and dark periods as follows.

Let |λ

 be (nonorthogonal) eigenvectors of H

cond

/ (in case of degeneracy one has

to consider limits) and let λ

| be the dual basis, deﬁned by

λ

|λ

=δ

. (6.64)

Then one can write, for initial state |1,

exp{−iH

cond

t/}|1=



j=1

−iλ

|λ

λ

|1 . (6.65)

Since |λ

1,2

are mainly a superposition of |1 and |2 and |λ

is close to |3,thelast

term in the sum is negligible for times of the order A

−1

and Ω

−1

, while for times

much larger than A

−1

and Ω

−1

, the ﬁrst two terms become exponentially small, and

the third term, though small, remains.

As in (6.38) the probability density for the ﬁrst photon is

w(t) =−P



(t)

=1|exp{iH

IL†

cond

t/}



|22|+A

|33|



exp{−iH

cond

t/}|1



2|exp{−iH

cond

t/}|1



+ A



3|exp{−iH

cond

t/}|1



(6.66)

By the preceding argument, the ﬁrst term is close to the two-level expression in

(6.42). It dominates for small times and then becomes exponentially small. For large

times the other, small, term remains which then slowly drops off with a very small

exponential exponent.

For the two-level system the waiting time between two photons is of the order

of A

−1

and Ω

−1

and waiting times much larger than this practically never happen.

One can therefore now deﬁne what one can reasonably understand as a dark period.

One can pick a time T

satisfying

−1

,Ω

−1

 T

 A

−1

,Ω

−1

. (6.67)

Then, if the waiting time between two photons is larger than T

, this is called a dark

period.

For the temporal behavior of a single atom and its associated quantum trajectory

one has now the following situation. One starts in |1, one quickly has a photon, is

back in |1, again a quick photon, and so on. In a very rare event there is no photon

within a time interval of length T

. Once this has happened there is only a very small

probability density for the next photon and the total waiting time is much larger

than T

The probability of no photon within time T

)

∼

||e

−iλ

|λ

λ

|1||

. (6.68)

144 G.C. Hegerfeldt

The exponential is close to 1 while the remainder is small. Thus, after each photon

there is a small probability, i.e., P

), for a dark period. Therefore a dark period

occurs after an average number of 1/P

) photons; the latter form a “light period.”

Note that if P

) is too small a dark period may in practice never occur. This

clearly depends on the parameters chosen.

The probability density, w

(t), for the length of a dark period is the same as the

probability density for the ﬁrst photon to occur after a time interval of length T

Thus one has from (6.66), after normalization,

(t) = w(t)Θ(t − T



∞



w(t



)

∼

2|Imλ

−2|Imλ

Θ(t − T

)/e

−2|Imλ

(6.69)

and this is essentially an exponential distribution. Here Θ(t) is the step function

which vanishes for t < 0 and is 1 for t > 0. The mean duration, T

, of a dark

period is then

= T

2|Imλ

∼

2|Imλ

. (6.70)

Since after each photon from the 1–2 transition the probability for a dark period

is P

), one needs on the average P

)

−1

tries, i.e., photons, before a dark period

occurs. Therefore the mean duration, T

, of a light period is approximately the num-

ber of these needed photons multiplied by the mean waiting time between photons

for the two-level system, i.e., by τ from (6.43). Hence one has

∼

2Ω

+ A

AΩ

)

. (6.71)

For a quantitative treatment one has to calculate the eigenvalues λ

which can, in

principle, be done in a complicated closed form. Simpler approximate expressions

are [49]

1,2

∼

−



4Ω

− A

+Δ

∼

−

−iΩ

+2iΔ

(Ω

−4Δ

− A

)

2(Ω

−4Δ

)

+8A

(6.72)

With this one obtains from (6.70)

={A

(Ω

−4Δ

)

+4A

}

−1

. (6.73)

For the parameters

6 The Quantum Jump Approach and Some of Its Applications 145

= A

= 10

−1

,Ω

= 10

−1

, A

= 1s

−1

, and Δ

= Ω

/2 (6.74)

this gives

= 0.5s. (6.75)

In order to make sure that these long dark periods do not occur too seldom one

has to make sure that P

) is not too small. To determine P

) one can avoid the

calculation of the eigenvectors by using

|λ

λ

cond

/ − λ

)(H

cond

/ − λ

)

(λ

−λ

)(λ

−λ

)

. (6.76)

This identity and its cyclic variants are easily checked by applying it to the eigen-

vectors |λ

. Neglecting the exponential in (6.68), which is close to 1, one ﬁnds

)

∼

+4Δ

(Ω

−4Δ

)

+4A

. (6.77)

Intuitively one would expect a maximum for the frequent occurrence of dark periods

if the driving is on resonance (Δ

= 0), but contrary to this, when Ω

is large, the

dark periods are most frequent for Δ

close to Ω

. This was already noticed in [15].

For the above parameters the mean duration of a light period is T

= 1.5 s so that

light and dark periods have a similar mean duration.

From (6.65) and (6.76) another curious effect follows. In a dark period the atom

is essentially in the state |λ

 which can be calculated by applying (6.76) to |1

and then normalize the resulting vector to 1. The eigenvector |λ

 has a very small

second component and therefore the next photon after a dark period can also come

from a 2–1 transition, not only from a 3–1 transition as suggested by a rate-equation

approach. For the parameters in (6.74) the probability for this strange effect is an

astonishing 1/2 [49]!

6.4 The General N-Level System and Optical Bloch Equations

6.4.1 The Conditional Hamiltonian

In the general case, the Hamiltonian for the complete system, N-level atom plus

quantized electromagnetic ﬁeld, is again given by an expression as in (6.10), only

with ω

|22| replaced by



ω



. (6.78)

In the rotating-wave approximation the Hamiltonian then becomes

146 G.C. Hegerfeldt

H =



ω



+ H

(−)

·E

(+)

ext

(t) +h.c.

(−)

(+)

+h.c.

≡ H

+ H

(t) + H

, (6.79)

where now

(−)



i> j

|ij| ,

=i|X|j (6.80)

and where i > j means ω

>ω

. The expression for

E is the same as in (6.13). We

introduce the notation ω

= ω

−ω

One can now proceed as in Sect. 6.2. Instead of the single correlation functions

κ(τ ) in (6.23) one can introduce correlation functions of the form

jim

(τ ) ≡



kλ

2ε

V

·ε

kλ

)(ε

kλ

·D

m

) e

−i(ω

−ω

m

)τ

. (6.81)

The correlation function has an effective width of the order of ω

−1

m

around τ = 0,

and for Δt  ω

−1

m

one can, similarly as in the N = 2 case, extend a time integration

in the second-order contribution to inﬁnity [2]. This amounts to the replacement





−t

dτ e

i(ω

m

−ω

)τ

∼

πδ(ω

−ω

m

) + iP

−ω

m

(6.82)

and corresponds to the usual Markov approximation in the derivation of the optical

Bloch equations [38, 43]. Again, the principal-value term corresponding to a level

shift will be omitted.

Then one ﬁnds that the conditional Hamiltonian for an N -level atom with no

photon detection until time t is, on the coarse-grained timescale, given in the

Schr

odinger picture by

cond

(t) = H

+ H

(t) −i

Γ , (6.83)

where the damping operator

Γ is deﬁned by

Γ ≡



ij

i,> j

ijj

|i| (6.84)

and the generalized damping constants Γ

ijk

≡

6πε

c

·D

k

|ω

k

. (6.85)

6 The Quantum Jump Approach and Some of Its Applications 147

Note that, similar to (6.27), for i > j, one has

ijji

= A

/2 , (6.86)

where A

is the Einstein coefﬁcient for the i– j transition.

6.4.2 The Reset Operation

One can again proceed as in Sect. 6.2 to determine the state after a photon detection,

but additional complications can arise unless all optical transition frequencies ω

and ω

m

are far apart or unless two optical transition frequencies are very close so

that |ω

− ω

m

|Δt  1. The latter case can lead to interesting coherence effects

which were discussed for the Λ system in [31]. In general, oscillatory terms can

appear, but it was shown in [34] that for photon counting these can be omitted. The

result of [23, 34] is

Rρ =



ijm

i> j,>m



jim

+Γ

mji



|ji|ρ|m| . (6.87)

In the sum, only transitions from higher to lower atomic levels appear. This is phys-

ically reasonable since, with a classical laser, one is detecting spontaneous photons

only. Anyway, if the laser photons were also treated quantum mechanically then

photons from stimulated emissions would have the same direction as the incident,

stimulating, laser photons and would therefore not be observed.

As in Sect. 6.2.4, tr(Rρ) Δt is the probability to ﬁnd a photon in Δt, and the

analog of (6.54) again gives the probability density for the ﬁrst photon.

6.4.2.1 Examples

(i) Two-level system. Then D

= 0 unless i or j equals 1, the ground state. Then

Rρ = A ρ

|11|, by (6.87). This means that after a photon detection the

atom is in the ground state as expected and as already found in Sect. 6.2.4.

(ii) Λ-system, two ground states | 1 and | 2, excited state | 3 (cf. Fig. 6.6): only

, D

, and D

are nonzero. Then one has, with the Einstein coefﬁcients

and A

and in matrix notation,

Rρ = ρ

⎛

⎝

1332

+Γ

3213

2331

+Γ

3123

000

⎞

⎠

. (6.88)

Therefore, the normalized reset matrix is always the same and given by the

matrix divided by its trace. The off-diagonal terms vanish only if D

·D

= 0.

For levels 1 and 2 sufﬁciently far apart the off-diagonal terms can be neglected

for most questions so that the atom can be taken to be either in state | 1 or | 2

148 G.C. Hegerfeldt

Fig. 6.6 The Λ conﬁguration. The upper level 3 can decay to the lower levels 1 and 2. In addition

there may be driving of the 1–3 and 2–3 transitions

after a photon detection. On the other hand, for levels 1 and 2 close together

the off-diagonal terms lead to interesting coherence effects such as dark peri-

ods [29, 32] and quantum beats in the correlation function g

(2)

(τ ) even under

illumination with incoherent light [30].

(iii) Three-level cascade system. This level system is shown in Fig. 6.7. In matrix

notation one obtains from (6.87)

Rρ =

⎛

⎝

(Γ

1232

+Γ

3212

)ρ

(Γ

2321

+Γ

2123

)ρ

000

⎞

⎠

. (6.89)

Here the normalized reset matrix depends on ρ. For a quantum trajectory the reset

matrix will therefore in general depend on the time of the resetting since matrix

elements of the conditional density matrix elements vary with time if both D

and

are nonzero. The dependence on the state prior to detection, and thus on time, is

easy to understand for large level separation. If, for example, the atom were in state

| 2 there would only be a transition to | 1, while if it were in a superposition of

| 2 and | 3 it could go to | 1 or | 2 or a mixture thereof.

Fig. 6.7 The three-level cascade conﬁguration

6 The Quantum Jump Approach and Some of Its Applications 149

6.4.3 Quantum Trajectories and Recovering the Bloch Equations

It will now be shown that the collection of all quantum trajectories of an N-level

system with the same initial state yields a solution of the usual Bloch equations for

the system.

We consider a large ensemble of systems (atoms) interacting with the quantized

radiation ﬁeld and an external ﬁeld. After taking a partial trace over the photons, the

system alone is described by a density matrix, ρ(t). To each individual atom there

corresponds a quantum trajectory, each starting with the initial state ρ(t

) and with

jumps at different times. Now consider a time t.Ataﬁxedtimet, each trajectory is

in a particular mixed or pure state, which depends on its history. Let the incoherent

weighted sum of these states be denoted by ρ

(traj)

(t). Then in the time interval (t, t +

dt) a particular subensemble of systems will have no jumps (photon detections), and

this subensemble is described, including its relative size, by

cond

(t +dt, t) ρ

(traj)

(t) U

†

cond

(t +dt, t), (6.90)

since the state of each individual trajectory satisﬁes such an equation and (6.90) is

just the weighted incoherent sum of all these. Similarly, the subensemble of systems

with a jump is described, including its relative size, by

R(ρ

(traj)

(t)) dt . (6.91)

Hence, with

cond

(t +dt, t) = 1 −



cond

dt (6.92)

one has

(traj)

(t +dt) = ρ

(traj)

(t) −



cond

(traj)

−ρ

(traj)

†

cond

}dt +R(ρ

(traj)

(t)) dt (6.93)

and this gives

˙ρ

(traj)

=−



cond

(traj)

−ρ

(traj)

†

cond

}+Rρ

(traj)

. (6.94)

This coincides with the quantum optical Bloch equations (which are usually written

in a somewhat different way [38, 43]). Since the initial condition is the same, one

has that ρ

(traj)

(t) yields a solution of the optical Bloch equations.

Therefore, the QJA lends itself to a numerical solutions of the Bloch equations

by simulations of quantum trajectories. If R always resets to pure states, one can

work in dimension N instead of N

for the Bloch equations. For large N this is a

tremendous numerical advantage. On the other hand, if R resets to a density matrix

150 G.C. Hegerfeldt

this advantage would be lost; however, as indicated in Sect. 6.6 one can always reset

to (auxiliary) pure states.

There are quite efﬁcient ways to derive master equations like the Bloch equations.

Hence, for given Bloch equations, one can read off from (6.94) either H

cond

or Rρ

if the other is known, or one can make an educated guess for both from the form of

the Bloch equations.

If one is only interested in simulating solutions of the Bloch equations there

are alternative schemes to achieve this, e.g., trajectories of diffusion processes

[22, 46]. Still another type of trajectories has been used in [3], see Chap. 10 to

study photonic crystals and band gap materials where the Markov property does not

hold.

6.5 Quantum Counting Processes

The QJA will now be used to determine the statistics of broadband photon counting

for a general N-level atom and, more generally, for systems with jumps (with reset

operation R) and a conditional time development between jumps. To be completely

general we use density matrices.

6.5.1 Counting Statistics

For a density matrix the conditional time development is given by a superoperator,

cond

, deﬁned by

cond

(t, t

) ρ(t

) ≡ U

cond

(t, t

) ρ(t

) U

†

cond

(t, t

) . (6.95)

If ρ(t

) is normalized, its trace gives the probability P

(t) ≡ P

(t; ρ(t

)) of ﬁnding

no photon (jump) between t

and t.

The subensemble with (i) no photon found between t

and t and (ii) a photon

at t + Δt is described by RT

cond

(t, t

) ρ(t

) since R performs the resetting. The

size of the subensemble relative to the original ensemble, and thus the probability

(t, t

; ρ(t

))Δt to ﬁnd the ﬁrst photon after t

at t +Δt, is the trace of this times

Δt , i.e.,

(t, t

; ρ(t

)) = tr



cond

(t, t

) ρ(t

)



, (6.96)

which is the analog of (6.54).

Analogously, the subensemble for which photons are found in the intervals

, t

+Δt],...,[t

, t

+Δt] but none in between and none between t

+Δt and t

is described by

cond

(t, t

) RT

cond

, t

n−1

) ··· RT

cond

, t

) ρ(t

) , (6.97)

6 The Quantum Jump Approach and Some of Its Applications 151

where T

cond

, t

i−1

+ Δt) has been approximated by T

cond

, t

i−1

). The size of this

subensemble relative to the original one is the trace times (Δt)

, and it gives the

probability for this event. Going over to the coarse-grained timescale we have that

the probability density w(t

,...,t

;[t

, t]) for ﬁnding exactly n photons at times

< t

···< t

in the interval [t

, t] is given by

w(t

,...t

;[t

, t]) = tr



cond

(t, t

) RT

cond

, t

n−1

) ··· RT

cond

, t

) ρ(t

)



(6.98)

Putting t = t

the superoperator T

cond

(t, t

) drops out and one obtains the prob-

ability density, w(t

, ···, t

), for ﬁnding n photons at t

< ··· < t

and none in

between,

w(t

,...t

) = tr



cond

, t

n−1

) ··· RT

cond

, t

) ρ(t

)



. (6.99)

This can be rewritten as follows. We denote by ˆρ

the normalized reset matrix at

time t

. Then one has, by (6.96),

cond

, t

) ρ(t

) =w

, t

; ρ(t

)) ˆρ

cond

, t

)RT

cond

, t

) ρ(t

) =w

, t

; ρ(t

)) RT

cond

, t

)ˆρ

, t

; ρ(t

)) w

, t

;ˆρ

)ˆρ

(6.100)

and so on. In this way one obtains

cond

, t

n−1

) ···RT

cond

, t

) ρ(t

) = w

, t

; ρ(t

)) ···w

, t

n−1

;ˆρ

n−1

)ˆρ

(6.101)

and therefore

w(t

,...t

) = w

, t

; ρ(t

)) w

, t

;ˆρ

) ··· w

, t

n−1

;ˆρ

n−1

) .

(6.102)

This result can also be obtained directly by considering quantum trajectories and the

probability densities after the jumps.

For the two-level system, the V system and the Λ system, in example (ii) of Sect.

6.4 the reset matrix is always the same. In general, if after each jump the system is

always reset to the same normalized state or density matrix, ˆρ

say, i.e., if for any ρ,

one has

Rρ = λ

ˆρ

= tr(Rρ)ˆρ

, (6.103)

then the memory is lost after each detection. In this case (6.99) factorizes into single-

photon probabilities:

w(t

,...,t

) = tr



cond

, t

n−1

)ˆρ



···tr



cond

, t

)ρ(t

)



= w

, t

n−1

;ˆρ

) ··· w

, t

;ˆρ

) w

, t

; ρ(t

)) .

(6.104)

152 G.C. Hegerfeldt

In this case the analysis becomes particularly easy and many quantities can be cal-

culated by Laplace transform as in [49] for the case of a single ground state.

It is also of practical interest to consider the probability density, denoted by

p(t

,...,t

), associated with trajectories for which jumps (i.e., photons from the

corresponding atom) are found in the intervals [t

, t

+ Δt],...,[t

, t

+ Δt] and

arbitrarily many jumps in between. To describe this subensemble, denote the time-

development superoperator for the optical Bloch equations by T

(t, t

), i.e.,

ρ(t) = T

(t, t

) ρ(t

) .

(6.105)

Then starting at time t

with ρ(t

), the time development goes with T

(t, t

) until

t = t

, then there is the reset operation R to those trajectories with a jump, after

that the time development with T

, t

), reset, and so on. The subensemble of

trajectories for which jumps are found in the intervals [t

, t

+Δt],...,[t

, t

+Δt]

and arbitrarily many jumps in between and later is therefore described at time t > t

by the density matrix

(t, t

n−1

) RT

, t

n−1

) ··· RT

, t

) ρ(t

) , (6.106)

and the relative size of the subensemble is given by the trace of this times (Δt)

Therefore the probability density, p(t

,...,t

), for ﬁnding jumps (photons from the

atom) at times t

< ···< t

, with arbitrarily many jumps in between and at later

times, is given by

p(t

,...,t

) = tr



, t

n−1

) ··· RT

, t

) ρ(t

)



. (6.107)

The superoperator T

(t, t

n−1

) in (6.106) has dropped out since it conserves the trace.

The above probability densities determine a classical stochastic process whose

sample paths are given by the photon detection times of a single radiating atom.

Without external pumping these paths terminate. Ergodicity allows one to replace

time averages over a single trajectory by ensemble averages. In many cases the latter

can be computed analytically.

6.5.2 Converse: From Bloch Equations to the Conditional Time

Development

In this section it will be shown that one can derive the conditional time develop-

ment directly from the Bloch equations and the reset operation, without using the

projection machinery from Sects. 6.2.1 and 6.4.1.

We start at time t

and consider an ensemble of trajectories of radiating atoms

whose incoherent sum of states at time t is described by a density matrix ρ(t). It is

assumed that ρ(t) obeys the Bloch equations. We denote by ρ

(t) the density matrix

for the subensemble of trajectories for which no jump (no photon detection) has

Muga G. Time in Quantum Mechanics - Vol. 2

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