Muga G. Time in Quantum Mechanics

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72 A. Ruschhaupt et al.

The two-dimensional equation (4.6) can be transformed into an effective one-

dimensional one by inserting the ansatz

χ(x, y) =



(1)

(x)e

(2)

(x)e

i(k



Then we get





(1)

(x)

(2)

(x)









0 Ω(x)

Ω(x) −2Δ − iγ



(1)

(x)

(2)

(x)



where

Δ = Δ

−



(2k

)

is an effective detuning which includes Doppler and recoil terms. We have thus

reduced the original problem to a one-dimensional equation with effective

Hamiltonian

=.p

/2m +





−iγ







Ω(x)

−2Δ



. (4.7)

(The x subindex is dropped in the momentum operator since only effective one-

dimensional equations in the x-direction are considered hereafter.) We shall con-

centrate now on the on-resonance (Δ = 0) case that may be achieved by normal

incidence (k

= 0) and by adjusting the laser detuning to cancel the recoil frequency

/(2m).

4.2.1.1 Stationary Scattering Eigenfunctions

Let us ﬁrst solve the eigenvalue equation

= EΦ

, where Φ

(x) ≡



(1)

(x)

(2)

(x)



and E = 

/2m , (4.8)

for waves incident from the left in the ground state (dropping the subindex x in k

too). In the simplest solvable model Ω(x) = Θ(x)Ω, which corresponds to a sharp

laser beam boundary at x = 0 and a semi-inﬁnite ﬁeld extension. If the atoms are

slow enough the result of the experiment should not depend on the beam width

beyond a minimal length necessary for detection.

For x < 0, left-incoming states must be of the form

(x) =

√

2π



ikx

+ R

−ikx

−iqx



, x < 0, k > 0 ,

4 Detector Models for the Quantum Time of Arrival 73

where q satisﬁes

E +iγ/2 = 

/2m ,

with Im q > 0 for boundedness, while R

and R

are the reﬂection amplitudes.

Note that although E is real, the complete wave functions will not be orthogonal, in

accordance with the non-hermiticity of H

To obtain the form of Φ

(x)forx ≥ 0, we denote by |λ

 and |λ

−

 the eigen-

states of the matrix

−iγ

corresponding to the eigenvalues λ

=−

γ ±

−4Ω

|λ

=



2λ

/Ω



Note that |λ

 are not orthogonal and have not been normalized. Nevertheless one

can write Φ

as a superposition of the form

√

2πΦ

(x) = C

|λ

e

−

|λ

−

e

−

, x ≥ 0 .

From the eigenvalue equation (4.8), one obtains for x ≥ 0

= k

−2mλ

/ = k

+iγ m/2 ∓im

−4Ω

/2 ,

with Im k

> 0 for boundedness. From the continuity of Φ

(x) and its derivative at

x = 0 one gets R

, R

, C

and the wave functions explicitly [31].

4.2.1.2 Wavepackets

Wavepackets Ψ =



(1)

(2)



are treated as superpositions of the stationary eigen-

functions. This is easy for an initial ground state wave packet



(1)

(x,t

)



coming in

from the far left side and with t

in the remote past. Indeed, if

ψ(k) denotes the

momentum amplitude the wave packet would have as a freely moving packet at

t = 0, then

Ψ (x, t) =



∞

ψ(k) Φ

(x)e

−ik

t/2m

. (4.9)

The normalization is chosen such that ||Ψ (t

)||

= 1, for t

in the remote past. The

probability, N(t), of no photon detection for a wave packet from t

up to time t is

given by

N(t) =||e

−iH

(t−t

)/

Ψ (t

)||

74 A. Ruschhaupt et al.

and the probability density, Π(t), for the ﬁrst-photon detection by

Π(t) =−



Ψ (t)|H

− H

†

|Ψ (t) .

Since H

− H

†

=−iγ |22|, the ﬁrst-photon probability density is given by

Π(t) = γ



∞

−∞

dx |ψ

(2)

(x, t)|

4.2.1.3 The Reﬂection Problem and the No-Detection Probability

The probability of no photon detection at all is N(t =∞). Only ψ

(1)

contributes to

this and, because Ω(x) = ΩΘ(x), only for x < 0, since, for x > 0, the ground state

part will eventually be pumped by the laser to the excited state. So we get

N(t =∞) = 1 −



∞



Π(t



) =



∞

dk |R

(k)|

ψ(k)|

As a consequence, Π(t) is, in general, not normalized to 1.

and therefore the probability for missing an atom increases with Ω,the

strength of the laser driving. On the other hand, for k →∞reﬂection becomes

negligible. Hence for faster atoms reﬂection does not pose a problem. For later

purposes we also consider increasingly large γ , the other parameters kept ﬁxed.

In this case the state vector for x > 0 becomes simply the plane wave with wave

number k in the ground state. This means that for increasing γ there is less and less

reﬂection, but also less and less absorption, i.e., photon detection, so that the effect

of the laser on the atom decreases. Moreover, if both γ and Ω go to inﬁnity with

γ/Ω kept ﬁxed, then R

→−1 and all atoms are reﬂected without having been

detected.

Depending on the parameters, the delay and reﬂection problem may be

either very relevant or negligible [31].

4.2.2 The Connection to an Idealized Arrival Time Distribution

We have just seen that avoiding reﬂection by weak driving, Ω/γ  1, would cause

a severe delay problem since the laser would take more time to pump the atom to

the excited state, see Fig. 4.3.

Should it not be possible to somehow compensate the delay in Π(t) by that of

the atom at rest and thus arrive, in some limit, at a delay-free ideal distribution? To

More precisely, if Ω and γ are of the same order of magnitude and γ 

k

k or, alternatively, if

Ω/γ  1andΩ

/γ 

k

k,thenR

≈−1. Physically, both conditions mean that the distance

traveled by the atom in the time it takes for a photon to be scattered is much less than the de Broglie

wavelength.

4 Detector Models for the Quantum Time of Arrival 75

Fig. 4.3 Time-of-arrival distributions: Flux J (solid line, here indistinguishable from Kijowski’s

)andΠ (ﬁrst photon, dots). Note the delay in Π. The initial state is a minimum-uncertainty-

product Gaussian for the center-of-mass motion of a single Cesium atom in the ground state with

v=9.0297 cm/s, x=−1.85 μm, and Δx = 0.26 μm; Ω = 0.0999γ ; all ﬁgures are for the

transition 6

3/2

–6

1/2

of Cesium with γ = 33.3MHz

achieve this, we assume the “experimental” arrival time distribution Π (t)tobethe

convolution of a hypothetical ideal distribution, Π

, with the detection probability

density W (t) for an atom at rest [56], which is put in the laser ﬁeld in the ground

state,

Π = Π

∗ W,

W(t) =

γΩ

4|S|

−γ t/2

St/2

−e

−St/2

Θ(t) ,

where

S =

(γ

−4Ω

)

1/2

The delay in Π is then contained in W and the hypothetical ideal distribution Π

obtained by deconvolution from

Π/

W, where the tildes denote the Fourier

transforms, e.g.,

Π(ν) =

dte

−iνt

Π(t). One ﬁnally obtains, using (4.9) and taking

the γ →∞limit [31],

(t) → J(0, t) ,

which is the ﬂux for the free wave function ψ

(x, t)atx = 0, i.e., the one evolving

without laser, see also Fig. 4.4. This is an extremely interesting result since J is a

76 A. Ruschhaupt et al.

Fig. 4.4 Excellent agreement between Π

(ﬁlled circles)andJ (solid line); deviations from Π

(dotted line)andΠ (dot-dashed line). The initial wave packet is a coherent combination ψ =

−1/2

(ψ

+ ψ

) of two Gaussian states for the center-of-mass motion of a single Cesium atom

that become separately minimal uncertainty packets (with Δx

= Δx

= 0.021 μm and average

velocities v

= 18.96 cm/s, v

= 5.42 cm/s) at x = 0andt = 2 μs; Ω = 0.37γ

natural candidate for the arrival time distribution. We note that J is normalized to 1

for a wave function which has only positive-momentum components.

4.2.3 Generalizations of the Atom-Laser Model

Different generalizations of the atom-laser model have been worked out:

In [32] the effects of a ﬁnite laser width (in contrast to the semi-inﬁnite laser

described before) are examined. In that setting, the ﬂux can again be obtained from

the probability density for the arrival of the ﬁrst photon in a given limit.

In [45] the three-dimensional formulation of the atom-laser model is investigated

in detail. It is shown that within typical conditions for optical transitions the results

of the simple one-dimensional version are generally valid. Differences that may

occur are consequences of Doppler and momentum-transfer effects. Ways to mini-

mize these are discussed.

In [82] the idea of the atom-laser model is transferred to an ionization model. A

method to achieve perfect detection of ultra-cold atoms at ﬁxed incident velocity is

proposed.

4.3 Complex Potentials

Allcock [4–6] modeled the arrival detector by a local and energy-independent com-

plex potential, i.e., he put forward a simpliﬁed operational approach for arrival

times. The operational (unnormalized) time-of-arrival distribution Π is then identi-

ﬁed as the absorption rate

4 Detector Models for the Quantum Time of Arrival 77

Π =−dN/dt ,

where N is the decreasing norm of the quantum wave packet for the undetected

particle. Later, Muga and coworkers used more complicated functional forms for

the complex potential, including a real part, to show that the reﬂections induced

by the step model could be partially avoided [65, 75]. Imaginary potentials have

also been used in the phenomenological event-enhanced quantum theory (EEQT)

of Blanchard and Jadczyk [18, 20, 19] to simulate time-of-arrival and traversal time

measurements [21, 75, 78–80], and Halliwell [44] derived the imaginary step poten-

tial used by Allcock from an abstract two-level detector model.

The agreement between the deconvolution performed by Allcock with a weak

imaginary step potential and the deconvolution of Sect. 4.2.2, in the short lifetime

(weak driving) limit, suggests that there must be a connection. Indeed, the laser-

atom model can be reduced to a local and energy-independent complex potential

model in the weak driving regime. To see this ﬁrst at a heuristic and intuitive level,

let us write the stationary Schr

odinger equation for the effective Hamiltonian (4.7)

in components, putting Δ = 0,

Eφ

(1)

(x) =

(1)

(x) +



Ω(x)φ

(2)

(x) , (4.10)

Eφ

(2)

(x) =

(2)

(x) +



Ω(x)φ

(1)

(x) −iγ/2 φ

(2)

(x) . (4.11)

If γ is much larger than the rest of energies in (4.11), we can approximate φ

(2)

from this equation as

(2)

(x) ≈

Ω(x)

iγ

(1)

(x) .

Putting this into Eq. (4.10) we get

Eφ

(1)

(x) ≈

(1)

(x) −iV

(x)φ

(1)

(x) , (4.12)

where

(x) =

Ω(x)

2γ

, (4.13)

i.e., there results a local, purely imaginary, and energy-independent potential for

state 1.

For a wavepacket with stationary components satisfying (4.12), ψ

(1)

(t) satisﬁes

the one-dimensional Schr

odinger equation

78 A. Ruschhaupt et al.

i

(1)

(t) ≈

/2m − iV

(.x)

(1)

(t) . (4.14)

Since ψ

(2)

(t) → 0, one has

N(t) ≈||ψ

(1)

(t)||

and so

Π(t) =−

≈





∞

−∞

dxV

(x)



(1)

(x, t)



Therefore, in the simple semi-inﬁnite laser case Ω(x) = ΩΘ(x), we get

Π(t) ≈





∞

dx |ψ

(1)

(x, t)|

, (4.15)

where V

= Ω

/2γ is now a constant. Equations (4.14) and (4.15) relate the

atom-laser model to the complex potential model of Allcock, see also [44].

An interesting case is the limit of the delta potential V

(x) =−iV

δ(x), for which

the signal Π(t) provides the probability density,

Π(t) ≈



(1)

(0, t)



However, this is not the probability density for the freely moving wave unless the

additional limit V

→ 0 is taken.

4.3.1 Complex Potential from Feshbach’s Theory

A more rigorous and systematic derivation of the complex potential follows from

using Feshbach’s projector techniques [37, 38, 61]. They let us obtain an exact

generalized, non-local and energy-dependent, complex potential treatment for laser

and atom parameters that cannot be described with the simple local and energy-

independent approach and also set its conditions of validity.

We shall now include detuning, as in (4.7),

= K1 + V =









Ω(x)

−(2Δ + iγ )



where K = .p

/2m is the kinetic energy operator and 1 =|11|+|22|

the unit operator in the internal state subspace. We want to solve the eigenvalue

4 Detector Models for the Quantum Time of Arrival 79

equation (4.8). Therefore, we may apply the complementary projectors

P =|11|, Q =|22| ,

to write, using the standard manipulations of the partitioning technique [37, 38, 61],

an exact integro-differential equation for the ground state amplitude φ

(1)

Eφ

(1)

(x) = K φ

(1)

(x) +



∞

−∞



x, 1|V

opt

(E)|x



, 1φ

(1)



) ,

where the exact complex non-local and energy-dependent “optical” potential is

opt

(E) = PV

opt

(E)P = PV P + PVQ(E + i0 − QHQ)

−1

QV P ,

and its coordinate representation is given by

x, 1|V

opt

(E)|x



, 1=−

i|x−x



Ω(x)Ω(x



) ,

with q =

√

2mE



(1 + μ)

1/2

,Imq ≥ 0, and μ =

2Δ+iγ

2E/

. The exact equation for the

excited state is

(2)

(x) =−

2



∞

−∞



i|x−x



Ω(x



)φ

(1)



) .

If we set Ω(x) = Ω

ω(x), where Ω

is the maximum of |Ω(x)|, we get by partial

integration



∞

−∞



i|x−x



ω(x



) f (x



) =

i

ω(x) f (x) +O





, (4.16)

for some function f (x) and 2E/ being constant.

Let us now examine the limit μ →∞keeping 2E/ and Ω

constant. To

apply perturbation theory, we assume an asymptotic expansion of φ

(1)

in powers of

1/μ and compare equal powers of 1/μ on both sides taking (4.16) into account. It

follows that

Eφ

(1)

= Kφ

(1)

(x) +W (x)φ

(1)

(x) +O





where

W(x) =

Ω(x)



2(2Δ + iγ )

, (4.17)

80 A. Ruschhaupt et al.

i.e., there results a local and energy-independent approximation W (x) to the exact

optical potential. For the excited state, we get

(2)

(x) =

2Δ + iγ

Ω(x)φ

(1)

(x) +O





, (4.18)

i.e., the population of the excited state is very small compared to the ground state

population when the local potential applies.

Note that (4.17) includes real and imaginary parts that can be modiﬁed by modu-

lating the laser intensity and/or detuning. This may be used to control and optimize

the atom dynamics and detection with quantum optical means [71].

Another interesting limit which yields the same effective potential is μ →∞

while 2E/ and Ω

/μ remain constant [81].

The time-dependent two-component state vector for the undetected atom Ψ is

obtained by solving the time-dependent Schr

odinger equation corresponding to H.

For the atom incident from the left in the ground state, Ψ is given by a linear super-

position of the stationary waves. If the local and energy-independent potential W(x)

is a good approximation for all stationary waves composing the wavepacket, then

the ground state component satisﬁes a closed time-dependent Schr

odinger equation

i

∂

∂t

(1)

(x, t) ≈

(1)

(x, t) +W (x)ψ

(1)

(x, t) ,

whereas the excited state component becomes

(2)

(x, t) ≈

2Δ + iγ

Ω(x)ψ

(1)

(x, t) .

As a consequence the photon detection rate, or operational arrival time distribution,

is given by

Π =−

≈−

∂

∂t



∞

−∞



1 +

(x)

(2Δ + iγ )



|ψ

(1)

(x, t)|

≈−

∂

∂t



∞

−∞

dx|ψ

(1)

(x, t)|

For Δ = 0 this result corresponds to the phenomenological one-channel models that

make use of an imaginary potential [4–6, 21, 78–80], which can now be physically

interpreted as W

Δ=0

(x) =−iV

(x) where V

(x) is given by Eq. (4.13). The phys-

ical signiﬁcance of these approximations may also be understood by applying the

Markovian limit in the time-dependent version of the partitioning technique due to

Zwanzig [91].

4 Detector Models for the Quantum Time of Arrival 81

4.3.2 Multiple Photon Case

In general the atom will emit many photons in consecutive ﬂuorescence cycles and

the distribution of the photon detection times may be of interest. The multiple pho-

ton case can also be studied by the quantum jumps approach. In this section we shall

assume that the following conditions are satisﬁed: Δ = 0, γ  Ω

and the energy

is assumed to be small, i.e., E  γ/2, but large with respect to the recoil energy.

This simpliﬁes the resetting of the wave function after a photon has been detected.

With the abbreviation g(x) =

Ω(x)

√

the effective ground state potential becomes

W(x) =−i



g(x)

and, using the above assumptions, the quantum jump algorithm

takes the following form:

(1) Start with an atom in the ground state.

(2) In each time step (δt γ,Ω

) evaluate the detection probability δ P between

t and t +δt,

δ P = γ

(2)

(t)

δt =

g · ψ

(1)

(t)

δt .

(3) Choose uniformly a random number r ∈ [0, 1].

(4a) If r >δP then no detection occurs and the wave function of the atom at

time t +δt is given by

(1)

(x, t + δt) =

√

1 − δP



1 −



δt



−i



g(x)



(1)

(x, t) ,

(2)

(x, t + δt) =−

√

g(x)ψ

(1)

(x, t + δt) .

(4b) If r <δP then a photon is detected and the wave function collapses and is

given at time t + δt by

(1)

(x, t + δt) =

(2)

(t)

(2)

(x, t) =−i

g · ψ

(1)

(t)

g(x)ψ

(1)

(x, t) ,

(2)

(x, t + δt) = 0 .

(For a more general treatment including the effect of recoil see [72].)

(5) Let t + δt → t and go to (2) for possible further detections until a certain

maximum propagation time is reached. This would complete one “trajectory” with

a record of photon emissions at a sequence of time steps. For obtaining averaged

statistical results many trajectories starting at step (1) must be run.

It is remarkable that only the ground state ψ

(1)

has to be taken into account for

this algorithm, i.e., for dealing with the multiple-photon case, under these approxi-

mations.

Note that this algorithm is also essentially coincident with the “PDP algorithm”

of phenomenological “event-enhanced quantum theory” (EEQT) applied for arrival

times using one complex potential detector (see, for example, [78, 20]). While the

Muga G. Time in Quantum Mechanics - Vol. 2

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