Muga G. Time in Quantum Mechanics

Подождите немного. Документ загружается.

6 The Quantum Jump Approach and Some of Its Applications 153

occurred until time t. The normalization is chosen in such a way that the trace gives

the relative size of the subensemble. To begin with, in the time interval [t

, t

+Δ],

a photon will have been detected for some of the atoms, i.e., there will have been a

jump on some trajectories, and this subensemble is described by Rρ(t

)Δt, includ-

ing its relative size. The complementary subensemble is described by the difference

of the original ρ(t

+Δt) and Rρ(t

)Δt, so that

+Δt) = ρ(t

+Δt) − Rρ(t

)Δt . (6.108)

This subensemble as a total would evolve until t

+ 2Δt with the Bloch equations,

i.e., as T

+ 2Δt, t

+ Δt) ρ

(t + Δt), and the sub-subensemble with no photon

detected at time t +2Δt is again a difference,

+2Δt) = T

+2Δt, t

+Δt)ρ

+Δt) −Rρ

+Δt)Δt . (6.109)

In the same vain one obtains for general t

(t +Δt) = T

(t +Δt, t) ρ

(t) −Rρ

(t) Δt . (6.110)

If L

denotes the superoperator which generates the time development of the Bloch

equations, then T

(t +Δt, t) = e

Δt

and (6.110) can be written as

(t +Δt) = (1 +L

Δt) ρ

(t) −Rρ

(t) Δt . (6.111)

This ﬁnally yields for the subensemble with no jump (photon detection) until time t

the equation

˙ρ

(t) =



−R



(t) . (6.112)

The difference in the brackets is just the time development given by the conditional

Hamiltonian, as seen from (6.94).

It may seem from this derivation that one might be able to take the limit Δt → 0

without running into the quantum Zeno effect. However, it is well known that the

optical Bloch equations are not valid for arbitrarily small times because also in their

derivation the Markov property is used, and this requires Δt to be larger than the

correlation time. Hence for Δt → 0 one would have difﬁculties with (6.111).

6.5.3 Connection with Continuous Measurements

In order to describe continuous measurements, Davies and Srinivas [47, 48] have

extended the axiomatics of quantum mechanics by postulates for “homogeneous

quantum counting processes.” In particular, their postulates imply the existence

of two superoperators, J and S

, which map trace class operators to trace class

154 G.C. Hegerfeldt

operators and satisfy certain properties. For an individual system of an ensem-

ble described by a density matrix ρ their meaning is as follows. tr(S

ρ)isthe

probability of ﬁnding no counting event in [0, t], and the probability density,

,...,t

;[0, t]), for ﬁnding a counting event exactly at the times t

,...,t

in [0, t] is given by

,...,t

;[0, t]) = tr



t−t

−t

n−1

J ··· JS

−t



. (6.113)

For a particular system J and S

have to be determined phenomenologically or by

intuition.

A comparison with (6.98) shows that this has the same structure as that obtained

by the QJA and that the unknown superoperators J and S

have to coincide with

the R and T

cond

constructed above. In contrast to the axiomatic theory of Davies

and Srinivas, however, the superoperators are explicitly known in the QJA. In

this way one has arrived, without the need of new axioms, at the results of the

continuous measurement theory of [47, 48] for the general N-level atom within

the usual framework of quantum mechanics and with the reduction rule of von

Neumann and L

uders for demolition measurements, except that a temporal coarse

graining has been used. Counting processes analogous to (6.99) have also been

derived in [44].

6.6 How to Replace Density Matrices by Pure States

in Simulations

In many cases the reset state is not a pure state even if one starts in a pure state.

Moreover, the particular reset state in a quantum trajectory may depend on the

conditional state preceding it and thus on the whole history before the resetting.

If in a simulation one would work with density matrices instead of pure states,

one would enormously complicate the numerics for large dimension N of the level

space since density matrices lead to N

instead of N , as pointed out in Sect. 6.4.3.

It will be shown elsewhere [26] that for simulations one can always go over to

quantum trajectories with pure states which yield both the same jump statistics and

the Bloch equations as the original trajectories. In simple situations this procedure

has been used before [4]. For simulations this is of great numerical advantage if N is

large. The general proof of this statement is somewhat involved and it may be more

instructive to see how this works in a speciﬁc example which is simple enough but

still exhibits all salient features.

To this end we consider a simple three-level cascade system as in Fig. 6.7, where

level 2 may decay to level 1 and level 3 to level 2. Only the 1–3 transition is driven

by a laser. To keep things simple we assume the laser to be on resonance, i.e., ω

−ω

. In the Schr

odinger picture the conditional Hamiltonian is then of the form

6 The Quantum Jump Approach and Some of Its Applications 155

cond



ω

|ii|+

Ω



|13|e

iω

+|31|e

−iω



−

iA

|22|−

iA

|33| ,

(6.114)

where A

and A are the Einstein coefﬁcients of level 2 and 3, respectively. By going

over to the interaction picture we get rid of the time dependence and obtain

cond

Ω



|13|+|31|



−

iA

|22|−

iA

|33| .

(6.115)

The reset operation for this system is given in (6.89), and here we write it in the

form

Rρ = A

|11|+Γρ

|12|

+Γ

∗

|21|+Aρ

|22| ,

(6.116)

where we use the abbreviation Γ ≡ Γ

1332

+ Γ

3213

. Note that |Γ |

≤ A

A so that

Rρ is a positive operator.

The operator H

cond

consists of the part −i(A

/2)|22|, responsible for the

decay of level 2, and a remainder operating in the 2D subspace spanned by |1

and |3.Ifλ

denote the eigenvalues of the 2D part, i.e.,

=−

4Ω

− A

(6.117)

then the conditional time development in the interaction picture is easily determined

as in (6.40) to be given by

exp{−iH

cond

t/}=e

−A

t/2

|22|

√

4Ω

− A



−iλ



|13|+|31|

−λ

−

|11|−

−

|33|



−e

−iλ

−



|13|+|31|

−λ

|11|−

|33|



(6.118)

In particular, one ﬁnds

exp{−iH

cond

t/}|1=

√

4Ω

− A



−iλ



−λ

−

|1+

|3



−e

−iλ

−



−λ

|1+

|3



exp{−iH

cond

t/}|2=e

−A

t/2

|2 .

(6.119)

156 G.C. Hegerfeldt

For an initial state |α the probability density, denoted by w

(t, t

= 0; |α), for the

ﬁrst photon is, by (6.38),

(t, t

= 0; |α) =



α|exp{iH

†

cond

t/}



cond

− H

†

cond



exp{−iH

cond

t/}|α .

(6.120)

For a superposition of |1 and |2,

|α=α

|1+α

|2 , (6.121)

one then obtains

(t, t

=0; |α)=|α

−A

+|α

4AΩ



4Ω

− A



−At/2



sin



4Ω

− A





(6.122)

Figure 6.8 shows the behavior of w

(t, t

= 0; |α) for a particular set of parameters

as detailed in the ﬁgure caption. For these parameters the mean time for the ﬁrst

photon is τ = 13/(8A). Figure 6.8 should be compared with the ﬁrst-photon prob-

ability density for the two-level atom in Fig. 6.5. In Fig. 6.8 the probability density

does not vanish for t = 0 since one starts in a superposition of ground state and ﬁrst

excited state so that the ﬁrst photon can come more quickly.

w/A

0.0

0.4

0.2

0.5

0.3

0.1

7.5 10.02.50.0 5.0

Fig. 6.8 (Color online) Probability density, w

(t, t

= 0; |α), for the ﬁrst photon as a function of

time, for initial state |α=(|1+|2)/

√

2, A

= A, Ω = 2 A, |Γ |

= A

A/2;themeantimefor

the ﬁrst photon is τ = 13/(8A)

6 The Quantum Jump Approach and Some of Its Applications 157

In a simulation of a quantum trajectory with initial pure state |α, one determines

a time, t

say, for the ﬁrst photon according to the probability density w

(t, t

0; |α) by means of random number operators in a standard way. Then, after this

ﬁrst photon, the state has to be reset by the reset operation R in (6.116). For the

present example this can be written in matrix form as

Rρ

cond

⎛

⎝

cond

Γρ

cond

Γρ

cond

Aρ

cond

000

⎞

⎠

, (6.123)

where ρ

cond

is given by the original state |α conditionally time-developed until

time t

and written as a density matrix,

cond

) = exp{−iH

cond

/}|αα|exp{iH

†

cond

/} . (6.124)

Note that

trρ

cond

= w(t

) , (6.125)

the probability density for the ﬁrst photon, as it should. The reset matrix normalized

to trace 1 is denoted by ˆρ,

ˆρ ≡ Rρ

cond

/tr(Rρ

cond

) . (6.126)

If the initial state is either |1, |2,or|3,theresetmatrix ˆρ after the ﬁrst photon

corresponds to a pure state, namely |2, |1,or|2, respectively. But for the initial

state |α from (6.121) one has

cond

) =|α

−A

cond

) =4|α

√

4Ω

− A

sin

4Ω

− A

/4 ,

cond

) =2i ¯α

√

4Ω

− A

−A

/2−At

sin

4Ω

− A

(6.127)

and then the reset matrix need not be pure state. Indeed, for it to be pure, the nonzero

2 × 2 submatrix has to have determinant 0, i.e.,

cond

Aρ

cond

−|Γρ

cond

= 0 .

(6.128)

Since with the initial state |α from (6.121) one has |ρ

cond

= ρ

cond

this

requires |Γ |

= A

A to have a pure reset state.

In case the latter condition is not fulﬁlled the reset matrix can correspond to a

mixed state. To see how large the deviation from a pure state is if |Γ |

< A

A we

diagonalize ˆρ. Its eigenvalues are 0 with eigenvector |3, p

and p

−

= 1 − p

where

158 G.C. Hegerfeldt



1 + 4

|Γ |

|ρ

cond

− A

Aρ

cond

+ Aρ

cond

)

, (6.129)

with corresponding normalized eigenvectors

=

⎛

⎝

Γρ

cond

+ Aρ

cond

− A

cond

⎞

⎠

/norm . (6.130)

When ρ

cond

) vanishes ˆρ is diagonal and the eigenvectors are |1 and |2, respec-

tively.

Fig. 6.9 (Color online) The eigenvalue p

of the reset matrix right after the ﬁrst photon at time t

as a function of t

, for the same initial state and parameters as in Fig. 6.8. In the center of the time

interval, p

differs appreciatively from 1 and thus ˆρ considerably deviates from a pure state

In Fig. 6.9 we have plotted p

as a function of the time t

of the ﬁrst photon,

for the same initial state and parameters as in Fig. 6.8. At the initial time one has

the initial pure state and hence p

= 1. Whenever p

< 1, ˆρ is a mixture, and the

deviation from a pure state is the more pronounced the closer p

is to 1/2.

Figure 6.10 shows that the eigenvectors of the reset matrix ˆρ can deviate signif-

icantly from |1 and |2. In this ﬁgure we have plotted the absolute value, R,ofthe

ratio of the ﬁrst and second components of |p

 as a function of the time t

of the

ﬁrst photon, for the same initial state and parameters as in Fig. 6.8. In the center

of the depicted time interval the ﬁrst and second components of |p

 are of similar

6 The Quantum Jump Approach and Some of Its Applications 159

Fig. 6.10 (Color online) Absolute value, R, of the ratio of the ﬁrst and second components of the

eigenvector |p

 of the reset matrix, as a function of the time t

, for the same initial state and

parameters as in Fig. 6.8. In the center of the time interval the ﬁrst and second components of |p



are of similar magnitude, while at the boundary times it is predominantly in |1

magnitude, while at times close to the boundary of the ﬁgure |p

is close to |1and

−

 close to |2.

For a simulation one now has two quite different situations. When one starts in

the atomic state |1, the atom is pumped to a superposition of |3 and |1.Afterthe

ﬁrst photon it jumps to |2. After the second photon it jumps from |2 to |1,so

after two photons one is back to the original starting state |1 and the simulation

continues as before. When one starts in the atomic state |2, the atom decays to |1,

and after that the simulation continues as before.

However, if one starts from a superposition of |1and |2, which can be prepared,

e.g., by a pulse from a second laser, the situation is completely different. In this case

the atom will in general be reset to a nonpure state right after the ﬁrst photon. In

a simulation one would then have to continue the conditional time development

until the next photon with a density matrix. For a higher dimensional atomic state

space a simulation with density matrices is much more time consuming than with

pure states. As noted at the beginning of this section, one can bypass this compli-

cation and go over to a sequence of (simulated) pure states, without changing the

overall jump statistics in a trajectory and still generating the Bloch equations for an

ensemble of such trajectories [26]. This tremendously reduces numerical effort in

more complicated cases. We will demonstrate this procedure in the simple cascade

160 G.C. Hegerfeldt

model outlined above, although in this case the numerical simpliﬁcations are not so

pronounced.

Replacing density matrices by pure states in simulations: Onestartsinthestate

|α, develops with U

cond

(t, 0), calculates the probability density w

(t, t

= 0; |α)

for the ﬁrst photon as in (6.122), and generates the ﬁrst photon time, t

, by a random

number generator according to this probability density. Then one determines the

corresponding reset matrix ˆρ ≡ ˆρ(t

) as in (6.123). Instead of resetting the atomic

state to ˆρ(t

) one determines its eigenvalues p

) and eigenstates |p

) and

resets the atom to the pure state |p

) with probability p

)orto|p

−

) with

probability p

−

). Then one applies the conditional time development U

cond

(t, t

)

to the pure state chosen and calculates the probability density w

(t, t

; |p

))for

the second photon. With this probability density one generates the time, t

, for this

second photon and determines the reset matrix at time t

, which obviously depends

on the pure reset state chosen at time t

. Again, instead of resetting the atom to this

reset matrix, one resets to one of its eigenvectors with the probability given by the

corresponding eigenvalue. This procedure is depicted in Fig. 6.11. Continuing in

this way, one generates a pseudo-quantum trajectory with pure states which does

not correspond to an actual physical trajectory. Nevertheless, the jump statistics

obtained by time averaging over such a trajectory agrees with the photon statistics

of the original physical process [26].

Fig. 6.11 Simulation of a trajectory consisting of pure states. One starts in |α, develops with U

cond

generates the ﬁrst jump time t

, and determines the reset state for U

cond

, 0)|αand its eigenvalues

) and eigenvectors |p

).Att

one resets to |p

) with probability p

). Then one

develops the chosen reset state with U

cond

(t, t

), generates the next jump time t

, and determines

the reset state for U

cond

, t

)|p

) and its eigenvalues p

) and eigenvectors |p

).Att

one resets to |p

) with probability p

),andsoon

Recovering the Bloch equations: Now one can repeatedly generate a large ensem-

ble of such trajectories, always starting with the same initial state |α. At a ﬁxed time

t, each trajectory is in a particular pure state, which depends on its history. Let the

incoherent weighted sum of these states be denoted by ρ

(sim)

|α

(t). It can then be shown

[26] that this density matrix satisﬁes the optical Bloch equations of the original

problem with the same initial condition, i.e., |αα|, and hence renders a solution of

the Bloch equations of the original problem. In general, for a large number of levels,

this can be numerically extremely advantageous.

6 The Quantum Jump Approach and Some of Its Applications 161

6.7 Inclusion of Center-of-Mass Motion and Recoil

6.7.1 Conditional Hamiltonian and Reset Operation

Until now, the N-level system was assumed to be at rest. For atoms or ions in a trap

this is often a good approximation. In general, however, the motion of the system

should be taken into account, in particular when the laser intensity varies in space

and when the center-of-mass (cm) motion should be treated quantum mechanically,

as for lower velocities.

As an example we consider a two-level system with quantized cm motion, inter-

acting with a classical laser and with the quantized radiation ﬁeld. In the Schr

odinger

picture the corresponding Hamiltonian is of the form

H =

P/2m +ω

|22|+



ω

†

kλ

D ·E(

X, 0) +e

D ·E

(

X, t) , (6.131)

where

X and

P denote cm position and momentum operator and

D the dipole oper-

ator as in (6.11). The classical laser ﬁeld is of the form

(x, t) = Re E

(x)e

−iω

, (6.132)

where we allow for a position-dependent laser amplitude. We put

kλ

≡ ie



2ε

V

kλ

·D

and go over to a “laser-adapted cm ﬁeld interaction picture” by means of the

operator

/2m + ω

|22|+H

(6.133)

and denote the resulting Hamiltonian by H

Lcm

. With the rotating wave approxima-

tion one then has in this interaction picture

Lcm

=−Δ|22|

+|12|



kλ

k,λ

exp{−i(ω

−ω

)t}exp{ik · (

X +

Pt/m)}+h.c.



|21|Ω(

X +

Pt/m) +h.c., (6.134)

where Δ = ω

−ω

is the detuning and the position-dependent Rabi frequency is

Ω(x) = eD

·E

(x) .

As is (6.7) we consider the zero-photon matrix element of the time-development

operator from t to t +Δt. In the Dyson series of the time-development operator we

162 G.C. Hegerfeldt

keep only terms proportional to Δt. Then only the laser contributes to the ﬁrst order,

which is

−





t+Δt





|21|Ω(

X +



/m) +h.c.

The second-order term in the time-development operator becomes



−





0



t+Δt









Lcm



Lcm



) |0

=

−



|22|



t+Δt











kλ

exp{i(ω

−ω

)(t



−t



)}

exp{ik ·(

X +



/m)}exp{−ik ·(

X +



/m)} , (6.135)

where again, as in Sect. 6.2.1, the laser terms have been omitted since they are

proportional to (Δt)

We note that

X(t



) ≡

X +



/m =

X +

Pt/m +

P(t



−t)

is the time development of

X in the free Heisenberg picture of the cm motion of the

system. With Δt ≈ 10

−12

s as in (6.1) and the atomic velocity around 1 m/s one has

Δx ≡ νΔt = 10

−12

m. With optical wavelengths one has k ∼ 2 ×10

−1

so that

kΔx ∼ 2 ×10

−6

and therefore the

P(t



− t) part can be neglected in the exponent.

It can similarly be neglected in Ω(

X +



/m) of the ﬁrst-order term if Ω(x) does

not vary signiﬁcantly over a distance of Δx. Hence, under these conditions one has

X(t



) ≈

X(t



) ≈

X(t) . (6.136)

Then the last two exponentials in (6.135) cancel. As in Sect. 6.2.1 for the two-

level case without cm motion this leads to the damping term −

A|22| and one

obtains as conditional Hamiltonian in the laser-adapted cm interaction picture

Lcm

cond

=−Δ|22|−

A|22|+



Ω(

X(t))|21|+h.c.. (6.137)

Reversing the interaction picture with respect to the free cm Hamiltonian, one

obtains for the conditional Hamiltonian in the laser-adapted interaction picture,

denoted as in (6.36) by H

cond

/2m − Δ|22|−

A|22|+



Ω(

X)|21|+h.c.. (6.138)

In the Schr

odinger picture this becomes

Muga G. Time in Quantum Mechanics - Vol. 2

Подождите немного. Документ загружается.