Muga G. Time in Quantum Mechanics

Подождите немного. Документ загружается.

8 Experiments on Quantum Transport of Ultra-Cold Atoms in Optical Potentials 233

Fig. 8.13 Part (a) shows a diagram of the interrupted acceleration sequence. The total tunneling

time is the sum of all the tunneling segments. Part (b) shows a typical integrated spatial distribution

of atoms after ballistic expansion. The peaks can be identiﬁed as in Fig. 8.11. However, the area

containing the tunneled fraction of the atoms is now composed of two peaks. Atoms that left the

well during the ﬁrst tunneling segment are offset in velocity from the ones having left during the

second period of tunneling. The amount of separation is equal to the velocity increase of the well

0.4

0.5

0.6

0.7

0.8

0.9

Survival probability

Tunneling time [µs]

0 1012141618208642

Fig. 8.14 Probability of survival in the accelerated potential as a function of duration of the tun-

neling acceleration. The hollow squares show the non-interrupted sequence, the solid circles show

the sequence with interruptions of 50 μs duration every 1 μs. The error bars denote the error of

the mean. The data have been normalized to unity at t

tunnel

= 0 in order to compare to the quantum

mechanical simulations (solid lines; no adjustable parameters). For these data the parameters were

tunnel

= 15, 000 m/s

, a

interr

= 2000 m/s

, t

interr

= 50 μs, and V

/h = 91 kHz. Figure from [12];

234 M.C. Fischer and M.G. Raizen

of the experimental setup into the analysis of the data. Also indicated as solid

lines are quantum mechanical simulations of the decay by numerically integrating

Schr

odinger’s equation for the experimental sequence and determining the survival

probability numerically. The simulations contained no adjustable parameters and

are in good agreement with the experimental data. We attribute the seemingly larger

decay rate for the Zeno experiment as compared to the simulation to the under-

estimate of the actual tunneling time.

Recently it was predicted that an enhancement of decay can be observed for

slightly longer time delays between successive measurements during the non-

exponential region. In contrast to the suppressed decay for the Zeno effect this

prediction was named the anti-Zeno effect [21, 22, 10]. The shape of the unin-

terrupted decay curve in Fig. 8.14 makes this suggestion fairly obvious. After an

initial period of slow decay the curve shows a steep drop as part of an oscillatory

feature, which for longer time damps away to show the well-known exponential

decay. If the system was interrupted right after the steep drop, one would expect

an overall decay that is faster than the uninterrupted decay [22]. The solid circles

in Fig. 8.15 show such a decay sequence, where after every 5 μs of tunneling the

decay was interrupted by a slow acceleration segment. As in the Zeno case, these

interruption segments force the system to repeat the initial non-exponential decay

behavior after every measurement. Here, however, the tunneling segments between

the measurements are chosen longer in order to include the periods exhibiting fast

010203040

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Survival probability

Tunneling time [µs]

Fig. 8.15 Survival probability as a function of duration of the tunneling acceleration. The hollow

squares show the non-interrupted sequence, the solid circles show the sequence with interruptions

of 40 μs duration every 5 μs. The error bars denote the error of the mean. The experimental

data points have been connected by solid lines for clarity. For these data the parameters were

tunnel

= 15, 000 m/s

, a

interr

= 2800 m/s

, t

interr

= 40 μs, and V

/h = 116 kHz. Figure from [12];

8 Experiments on Quantum Transport of Ultra-Cold Atoms in Optical Potentials 235

decay. The overall decay is much faster than for the uninterrupted case, indicated by

the hollow squares in the same ﬁgure.

The key to observing the Zeno and anti-Zeno effects is the ability to measure

the state of the system in order to repeatedly redeﬁne a new initial state. In our

case the measurement is done by separating in momentum space the atoms still

left in the unstable state from the ones that decayed into the reservoir. In order to

distinguish the two classes of atoms, they must have a separation of at least the

size of the momentum distribution of the unstable state, which in our case is the

width of the ﬁrst Brillouin zone of Δp = 2k

. The time it takes for an atom

to be accelerated in velocity by this amount is the Bloch period τ

= 2v

interr

assuming an acceleration of a

interr

. An interruption shorter than this time will not

resolve the tunneled atoms from those still trapped in the potential and therefore

results in an incomplete measurement of the atom number. To investigate the effect

of the interruption duration, we repeated a sequence to measure the anti-Zeno

effect for varying interruption durations while holding all other parameters constant.

Figure 8.16 displays the results of this measurement, interrupting the decay every

5 μs with an acceleration of a

interr

of 2000 m/s

. The hollow squares show the unin-

terrupted decay sequence as a reference. For an interruption duration smaller than

the Bloch period of 30 μs, the measurement of the atom number is incomplete and

has little or no effect. For a duration longer than the Bloch period, the effect saturates

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

no interruption

interr

= 10 µs

= 20 µs

= 30 µs

= 50 µs

= 60 µs

Survival probability

Tunneling time [µs]

0 1015202530

Fig. 8.16 Survival probability as a function of duration of the tunneling acceleration. The hol-

low squares show the non-interrupted sequence, other symbols indicate the sequence with a ﬁnite

interruption duration after every 5 μs of tunneling. The error bars denote the error of the mean. A

further increase of the interruption duration than as indicated does not result in a further change of

the decay behavior. The experimental data points have been connected by solid lines for clarity. For

these data the parameters were a

tunnel

= 15, 000 m/s

, a

interr

= 2000 m/s

,andV

/h = 91 kHz,

leading to a Bloch period of τ

Physical Society

236 M.C. Fischer and M.G. Raizen

and results in a complete restart of the decay behavior after every interruption. Even

though this method of interruption is not an instantaneous measurement of the state

of the unstable system, we can still accomplish the task of redeﬁning the initial state

by ﬁrst switching the system from an unstable to a stable one, then in a ﬁnite time

perform the measurement, and ﬁnally switching the system back to being unstable

again.

8.6 Conclusions

In conclusion, we have completed a detailed study of the onset of irreversibility in

an unstable quantum system and its control by repeated interrogation. We end this

chapter by making some comments about possible future directions. The develop-

ment of new tools to control many-body systems is a promising direction to follow.

In particular, our group has been working toward the experimental realization of

few-body number states [8]. These states of a deﬁnite number of atoms in the ground

state of a well are an ideal starting point for the study of few-body tunneling and

the onset of irreversibility. Future work in our group will focus on this problem.

Finally, it is a great pleasure to acknowledge and thank the many people who have

collaborated with us on this work over the years. The experiments were carried out

together with Kirk Madison, Steven Wilkinson, Cyrus Bharucha, Patrick Morrow,

and Braulio Guti

errez-Medina. Theoretical work was conducted in parallel to our

experiments together with Qian Niu, Roberto Diener, and Bala Sundaram. This work

was supported by the R. A. Welch Foundation, the National Science Foundation, and

the Sid. W. Richardson Foundation.

References

1. M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions (Dover, New York,

1965) 214

2. N.W. Ashcroft, N.D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976) 214, 217

3. C.F. Bharucha, K.W. Madison, P.R. Morrow, S.R. Wilkinson, B. Sundaram, M.G. Raizen,

Phys.Rev.A55, R857 (1997) 227

4. F. Bloch, Z. Phys. 52, 555 (1928) 207

5. C.B. Chiu, E.C.G. Sudarshan, B. Misra, Phys. Rev. D 16, 520 (1977) 231

6. S. Chu, Science 253, 861 (1991) 212

7. S. Chu, Rev. Mod. Phys. 70, 685 (1998) 206

8. C.S. Chuu, F. Schreck, T.P. Meyrath, J.L. Hanssen, G.N. Price, M.G. Raizen, Phys. Rev. Lett.

95, 260403 (2005) 236

9. C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-Photon Interactions (Wiley and

Sons, New York, 1992) 206, 209

10. P. Facchi, H. Nakazato, S. Pascazio, Phys. Rev. Lett. 86, 2699 (2001) 234

11. M.C. Fischer, Atomic Motion in Optical Potentials, Ph.D. thesis, The University of Texas at

Austin (2001) 212, 214

12. M.C. Fischer, B. Guti

errez-Medina, M.G. Raizen, Phys. Rev. Lett. 87, 040402 (2001) 225, 230, 233, 234, 235

13. M.C. Fischer, K.W. Madison, Q. Niu, M.G. Raizen, Phys. Rev. A 58, R2648 (1998) 223

14. L. Fonda, G.C. Ghirardi, G.C. Rimini, Rep. Prog. Phys. 41, 587 (1978) 227, 228

8 Experiments on Quantum Transport of Ultra-Cold Atoms in Optical Potentials 237

15. O.R. Frisch, Z. Phys. 86, 42 (1933) 205

16. R. Graham, M. Schlautmann, P. Zoller, Phys. Rev. A 45, R19 (1992) 207, 208

17. P.T. Greenland, A.M. Lane, Phys. Lett. A 117, 181 (1986) 228

18. K. Grotz, H.V. Klapdor, Phys. Rev. C 30, 2098 (1984) 228

19. W.M. Itano, D.J. Heinzen, J.J. Bollinger, D.J. Wineland, Phys. Rev. A 41, 2295 (1990) 232

20. L.A. Khalﬁn, JETP 6, 1053 (1958) 227

21. A.G. Kofman, G. Kurizki, Phys. Rev. A 54, R3750 (1996) 234

22. A.G. Kofman, G. Kurizki, Nature 405, 546 (2000) 234

23. J.B. Krieger, G.J. Iafrate, Phys. Rev. B 33, 5494 (1986) 220

24. P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. Kasevich, Phys. Rev. Lett. 74, 4763

(1995) 232

25. R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1983) 207, 208, 210

26. K.W. Madison, Quantum Transport in Optical Lattices, Ph.D. thesis, The University of Texas

at Austin (1998) 226

27. K.W. Madison, C.F. Bharucha, P.R. Morrow, S.R. Wilkinson, Q. Niu, B. Sundaram, M.G.

Raizen, Appl. Phys. B 65, 693 (1997) 227

28. K.W. Madison, M.C. Fischer, R.B. Diener, Q. Niu, M.G. Raizen, Phys. Rev. Lett. 81, 5093

(1998) 223

29. K.W. Madison, M.C. Fischer, M.G. Raizen, Phys. Rev. A 60, R1767 (1999) 224

30. M.P. Marder, Condensed Matter Physics (Wiley and Sons, New York, 2000) 214, 217

31. E.E. Mendez, G. Bastard, Phys. Today 46(6), 34 (1993) 207

32. P.W. Milonni, J.H. Eberly, Lasers (Wiley and Sons, New York, 1988) 209

33. B. Misra, E.C.G. Sudarshan, J. Math. Phys. 18, 756 (1977) 231

34. Q. Niu, M.G. Raizen, Phys. Rev. Lett. 80, 3491 (1998) 229

35. Q. Niu, X.G. Zhao, G.A. Georgakis, M.G. Raizen, Phys. Rev. Lett. 76, 4504 (1996) 221, 226

36. E. Peik, M.B. Dahan, I. Bouchoule, Y. Castin, C. Salomon, Phys. Rev. A 55, 2989 (1997) 217

37. E. Raab, M. Prentiss, A. Cable, S. Chu, D. Pritchard, Phys. Rev. Lett. 59, 2631 (1987) 212

38. M.G. Raizen, C. Salomon, Q. Niu, Phys. Today 50(7), 30 (1997) 207

39. J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, New York, 1994) 232

40. D.A. Steck, Sodium D Line Data (2001), available at http://steck.us/alkalidata 209, 210

41. P. Valanju, E.C.G. Sudarshan, C.B. Chiu, Phys. Rev. D 21, 1304 (1980) 231

42. S.R. Wilkinson, C.F. Bharucha, M.C. Fischer, K.W. Madison, P.R. Morrow, Q. Niu, B. Sun-

daram, M.G. Raizen, Nature 387, 575 (1997) 225, 229

43. R.G. Winter, Phys. Rev. 123, 1503 (1961) 227

44. C. Zener, Proc. R. Soc. Lond. A 137, 696 (1932) 226

45. C. Zener, Proc. R. Soc. Lond. A 145, 523 (1934) 226

Chapter 9

Quantum Post-exponential Decay

Joan Martorell, J. Gonzalo Muga, and Donald W.L. Sprung

After a long time the rain let up, but the clouds stayed, and the

lightning kept whimpering, and by and by a ﬂash showed us a

black thing ahead, ﬂoating, and we made for it.

Mark Twain

9.1 Introduction

Exponential decay is a very general phenomenon in the natural sciences. It occurs

when a quantity N decreases at a rate proportional to its value,

=−

N, N = N

−t/τ

. (9.1)

It occurs in practically every ﬁeld of physics, including unstable or excited ele-

mentary particles, nuclei, atoms, molecules, or quantum dots. Typical examples of

microscopic decay processes are nuclear alpha-decay, atomic autoionization, spon-

taneous emission of a photon from an excited state, or escape from a potential trap.

In the quantum world, however, the exponential law is not an obvious consequence

of the basic dynamical law, which is Markovian (i.e., memoryless) and applies to

amplitudes, not probabilities. Accordingly, the derivation of exponential decay from

ﬁrst principles has been the subject of much scrutiny and debate.

Quantum decay was ﬁrst examined in nuclear physics: the decay of natural

radioactive nuclei has been studied for more than a century [116]. Gamow [41]

developed the ﬁrst quantal theory for alpha-decay. He explained, at least quali-

tatively, the vast range of alpha-decay lifetimes as a tunneling phenomenon by

J. Martorell (B)

Departament d’Estructura i Constituents de la Materia, Universitat de Barcelona, 08028 Barcelona,

Spain, martorell@ecm.ub.es

J.G. Muga

Departamento de Qu

ımica-F

ısica, UPV-EHU, Apdo. 644, 48080 Bilbao, Spain, jg.muga@ehu.es

D.W.L. Sprung

Department of Physics and Astronomy, McMaster University, Hamilton, Ontario,

Canada L8S 4M1, dwsprung@mcmaster.ca

Martorell, J. et al.: Quantum Post-exponential Decay. Lect. Notes Phys. 789, 239–275 (2009)

DOI 10.1007/978-3-642-03174-8

 Springer-Verlag Berlin Heidelberg 2009

240 J. Martorell et al.

imposing “outgoing wave boundary conditions” on the wave function at the outer

edge of the conﬁning barrier. Such an outgoing condition can be satisﬁed, in general,

only for a discrete set of complex energies. The corresponding “Gamow states” are

usually associated with “resonances,” a concept that almost always arises in dis-

cussions of decay, especially exponential decay. Gamow states decay exponentially

in time, with twice the imaginary part of the complex energy, divided by ,giving

the inverse lifetime. However, they are unphysical states because the wave function

grows exponentially outside the interaction region in coordinate space. They can,

however, be used successfully for performing “resonant state” or “pole” expansions

of the physical wave function [45]. A simple explanation of why the Gamow state

provides the correct decay rate, using a more conventional expansion in (real energy)

stationary eigenstates, is given in [10].

Weisskopf and Wigner, in another inﬂuential paper [139, 140], worked out a more

elaborate theory for spontaneous emission from an excited atom interacting with the

quantized radiation ﬁeld, as introduced by Dirac [23]. This approximate theory also

predicts exponential decay and sets a standard paradigm for treating general decay

problems in which an initial discrete bound state (or more generally a set of them) is

coupled to (“embedded into”) a perturbing continuum of states that destabilizes the

discrete state, making it a decaying resonance. The relation between Fermi’s golden

rule and exponential decay was explored at length by Hillery [57] for radiative decay

of a discrete atomic state, either in free space or in a cavity. The 2p →1s decay in

hydrogen was examined in detail and estimates were made for the deviations at long

and short times.

In summary, exponential decay is ubiquitous and a standard topic in quantum

mechanics books where it is derived in ﬁrst-order time-dependent perturbation the-

ory using Fermi’s golden rule. What is less well known, at least at textbook level, is

that at both short and long times, quantum mechanics predicts deviations from the

exponential law. The Zeno effect, a subject which has attracted much attention in

recent years [11] and is treated in both the previous and present volumes of Time in

Quantum Mechanics, is associated with the short time deviation. The deviations at

long times are less often discussed and constitute the central topic of this chapter.

Figure 9.1 shows the survival probability for a particle initially conﬁned inside a

very simple spherically symmetric potential (s-wave decay). (Details of the potential

will be given in Sect. 9.3.3.) Up to t  50 the decrease is exponential. After a

transition near t  100, beyond t  150 the decay is again smooth and clearly

follows a different law. This is the post-exponential decay at issue. We will discuss

how it is determined by the corresponding Hamiltonian.

From another perspective, Figs. 9.2 and 9.3 show the absolute value of the

radial s-wave function determined from a numerical solution of the time-dependent

Schr

odinger equation. The leftmost part of the x-axis corresponds to the inner well

(see inset of Fig. 9.1). Figure 9.2 covers the range of times corresponding to expo-

nential decay. As time progresses the amplitude in the well decreases exponentially

and an outgoing wave packet expands beyond the barrier. The line for the earliest

time shows the emission of several humps corresponding to higher energy wave

packets. These are evidence that the chosen initial state is a superposition of several

9 Quantum Post-exponential Decay 241

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

0 100 200 300 400 500 600

P(t)

time

–0.5

0.5

1.5

01234567

v (r)

Fig. 9.1 Survival probability, P(t), for an s-wave-conﬁning potential consisting of a square well,

a surrounding square barrier, and an outer inverse square potential. Units such that

 = 2m = 1.

Continuous line: exact solution. Dashed line: prediction for asymptotic decay, Eq. (9.47). Inset:

shape of the potential

0.05

0.1

0.15

0.2

0.25

0.3

0 10203040506070

|u(r,t)|

Fig. 9.2 Snapshots of |u(r, t)| ,0< r < 72, for times corresponding to exponential decay. From

top to bottom at r = 30, the continuous lines correspond to times t = 15, 30, 45, 60, and 75

quasibound states. Figure 9.3 corresponds to post-exponential times and shows a

much more regular trend: the wave function moves steadily outward and the part

inside the inner well is a smooth continuation inward of the main outer part. It is

this residual interior amplitude that shows the power law decrease.

From a theoretical viewpoint, this is an interesting regime, where the classical

prediction of Eq. (9.1) differs from the quantal prediction. Quantum mechanics

allows a very general argument leading to elegant results. On the experimental side,

access to the post-exponential regime has proven to be difﬁcult. There have been

many attempts to provide convincing experimental evidence of post-exponential

242 J. Martorell et al.

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 10203040506070

|u(r,t)|

Fig. 9.3 Snapshots of |u(r, t)| ,0< r < 72, for times corresponding to intermediate and post-

exponential decay. From top to bottom at the left,thecontinuous lines correspond to times t =

75, 90, 105, 120, 135, and 150

decay, with scant success [103, 101, 48, 104, 96]. Consequently, various ideas have

been put forth to explain the lack of success. It has been argued that repetitive

measurements on the same system, and the ensuing repeated reduction of the wave

packet, or simply the interaction with the environment would lead to persistence of

the exponential regime to times well beyond those expected in an isolated system

[38, 71, 112]. A recent measurement of post-exponential decay in organic molecules

in solution [121] has thus come as a surprise, triggering renewed interest in the

subject.

Aside from the challenge of understanding and extending these experimental

results, the motives for studying post-exponential decay are manifold. Winter, for

example, argued that hidden variable theories could produce observable effects in

samples that have decayed for many lifetimes [142]. More recently Krauss and Dent

have suggested that late-time behavior of unstable systems may have important

cosmological implications [69]. Decay at long times is quite sensitive to delicate

measurement or environmental effects, becoming a testing ground for theories of

these processes. At a fundamental level, the detailed form of long time deviations

may help to distinguish between standard, Hermitian quantum mechanics [26], and

modiﬁcations proposed to incorporate a microscopic arrow of time [98, 97]. From a

practical point of view, Norman pointed out that post-exponential decay could set a

limit to the validity of radioactive dating schemes [104]. In a recent paper we have

also argued that the deviation could be a diagnostic tool for characteristics of certain

cold atom traps [80]. Indeed, due to technological advances in lasers, semiconduc-

tors, nanoscience, and cold atoms, microscopic interactions are now relatively easy

to manipulate, making decay controllable and post-exponential decay more acces-

sible to experimental scrutiny and/or applications. For example, under appropriate

conditions it could become the dominant regime and be used to implement an anti-

Zeno effect, speeding up decay [73].

9 Quantum Post-exponential Decay 243

Finally, let us also mention that recent experiments on propagation of electric

ﬁelds in periodic waveguide arrays provide a classical analog of a simple quantum

system with exponential decay [74, 19]. These experiments may allow one to reach

the post-exponential region in a particularly simple case, as we will discuss below.

9.1.1 What We Shall Not Discuss

This review focuses, as emphasized in the title chosen for the chapter, on sys-

tems whose decay is exponential over several lifetimes, i.e., systems where an iso-

lated resonance is dominant, or whose decay is characteristic of a post-exponential

regime, as for resonances very close to threshold [63, 42]. There are many phenom-

ena where decay is not at all exponential: A trivial example is the survival prob-

ability of a freely moving quantum particle represented, say, by a Gaussian wave

function. Overlapping resonances in nuclear physics can also make the decay quite

complex [24], even for two degenerate discrete levels [115]. The level structure of

polyatomic molecules leads to complicated coherence effects of interaction between

levels and regular or irregular quantum beats [82]. Nonexponential decay signals

may also result from the incoherent addition of purely exponential contributions

from different members of the ensemble [27, 134, 102] or from temporal ﬂuctua-

tions of the environment on a timescale larger than the lifetime. Power law decay in

the photoluminescence of conjugated polymers has also been reported but its mech-

anism involves incoherent processes [100, 16]. Another interesting phenomenon

is the suppression of decay of dressed bound states, in or out of the continuum,

formed by the interaction of the original discrete state(s) with the continuum states

[127, 128, 78]. For other examples of nonexponential decay see [119, 126, 31].

Interesting as they are, these cases and systems are beyond the scope of this review,

although we may brieﬂy comment on some of them.

9.1.2 Previous Reviews

A number of monographs and review articles on the decay process in quantum

mechanics, including post-exponential decay, exist. Among the textbooks, Sakurai

[124] gives an excellent summary of the basic theory. The most frequently cited

review is that of Fonda et al. [38]. It includes a section on the interpretation of

repetitive experimental measurements. The monograph of Goldberger and Watson

[47] uses the resolvent method to relate the asymptotic decay law to the threshold

behavior of the energy density. Cohen-Tannoudji et al. [12] considered decay in the

context of an initial discrete state coupled to a continuum of ﬁnal states. In the same

context Nakazato et al. [93] and Facchi et al. [32, 33] include a detailed discussion

of Laplace transforms to extract asymptotic contributions to the survival probabil-

ity. See also Longhi [76] for a recent extension to the more complex problem of

coupling to a cavity with intrinsic gains and losses.

Muga G. Time in Quantum Mechanics - Vol. 2

Подождите немного. Документ загружается.