tunneling time, the hartman eff d s l i lieffect, and

Tunneling Time, the Hartman Eff d S l i liEffect, and Superluminality: Resolving an Old ParadoxResolving an Old Paradox

Herbert G. WinfulHerbert G. WinfulUniversity of Michigan, Ann Arbor

[email protected]

1

Gedanken Experiment to pMeasure Tunneling Time

AR.Y. Chiao, et al

BWhich detector clicks first, A or B?

2

The Orthodox View “On average the tunneling photons arrived before those that traveled through air implying an averagethose that traveled through air implying an average tunneling velocity of about 1.7 times that of light.”[A.M.Steinberg, P.G.Kwiat, and R.Y.Chiao, PRL,1993]“E i t h th t th h t i th h“Experiments show…that the photon passing through that barrier arrives first.” [Introductory Quantum Optics, G.C.Gerry and P.L.Knight,2005]“Twin-photon experiments DO in fact revealthat light that has passed through a barrier can arrive at a detector much sooner ("superluminally") thanat a detector much sooner ( superluminally ) than light that has propagated freely.” [Anonymous PRL referee,2006]

3

The Current Belief

It is widely believed that tunneling photons travel with a group velocity that exceeds the vacuum speed of light.that exceeds the vacuum speed of light.The implication is that electron t li l b l i l*tunneling can also be superluminal*.

[*Superluminal=faster than light]

4

Outline

A brief history of tunneling timeGroup delay and dwell timeThe Hartman EffectThe Hartman Effect“Superluminal” tunneling experimentsMeaning of tunneling group delayMeaning of tunneling group delayOrigin of the Hartman EffectReinterpretation of tunneling experimetsConclusions

5

Conclusions

Tunneling

6

Importance of tunneling

Radioactive alpha decay Biological processes such as photosynthesisphotosynthesisJosephson junctionsEsaki tunnel diodesScanning tunneling microscopyScanning tunneling microscopy…etc

7

Stationary-state tunneling (1)

V

EE

22

2 ( ) 0m E Vψ ψ∇ + − =h

8

2h

Stationary-state tunneling (2)

P(x) ψ(x) 2P(x)= ψ(x)

Barrier transmission probability

T2

: e−2κ L

T : e

2m(V − E)9

κ =( )h2

Barrier transmission versus energy

(E/V)

10

(E/V)

Tunneling: the movie

Re ψ(x)e−iEt /h{ } ψ( ){ }

11

A brief history of tunneling y gtime

Condon (1931): “How long does it take to tunnel through a barrier?”tunnel through a barrier?MacColl (1932): “It takes no appreciable time” [Phys. Rev. 40, 621].time [Phys. Rev. 40, 621].Bohm (1948), Wigner, Eisenbud (1952): energy derivative of transmission phase shift h ld ld d lshould yield time delay.

Hartman (1962): time delay in tunneling is shorter than the “equal” time; can be fastershorter than the “equal” time; can be faster than light! [J. Appl. Phys. 33, 3427]

12

Tunneling Wavepackets

13

Relative Sizes of Wave packets

14

Method of stationary phase y pand group delay

ψ (x,t) = f (E − E0∫ )ψ E (x)e− iEt / hdEIncident wave packet ψ ( , ) f ( 0E∫ )ψ E ( )p

d

ψ(xt)= f(E E)∫ T(E)exp iφ(E)+ikx iEt/h⎡⎣ ⎤⎦dE

Transmitted

ψ(x,t)= f(E−E0)E∫ T(E)exp iφt(E)+ikx−iEt/h⎡⎣ ⎤⎦dE

∂τg = h ∂

∂E(φ + kL)Group delay

15

The Schrödinger equation2 2

( ) ( ) ( )V x x t i x tψ ψ⎡ ⎤∂ ∂

− = −⎢ ⎥h

h (1) 2 ( ) ( , ) ( , )2

V x x t i x tm x t

ψ ψ=⎢ ⎥∂ ∂⎣ ⎦h . (1)

For stationary states the wave function separates into a time-independent part and a

complex exponential time factor:

( ) ( ) ( / )t iEt h ( , ) ( )exp( / )Ex t x iEtψ ψ= − h ,

with ( )E xψ a solution of the time independent Schr�dinger equation

−h2

2d 2ψ E

d 2 + (V − E )ψ E = 0

16

2m dx 2 E

Group delay τ g

V(x)Time at which transmittedwave packet reaches a peak

ψ (x;k)ikx

at exit, given that incidentpacket peaks at t=0 at x=0.

ψII (x;k)

ψ I = eikx

R ikx+ ψIII =Teikx

T = T eiφt

Re−ikx

0 L ∂

T = T e φt

0 Lτ g = h ∂

∂Eφ t + kL( )

17

Dwell Time dτTime spent in barrier, averaged over allTime spent in barrier, averaged over all

incoming particles.L

2 jinj 2

L

dxψ∫2ψ

tjrj

0d

injτ =

∫inj

0 L

18

0 L

Hartman Effect (1962)

Tunneling time becomes independent of flength for thick barriers. [T.E.Hartman,J.Appl.

Phys. 33,3427 (1962)]If tunneling time becomes independent of length, tunneling velocity must become infinite!Origin of this effect has been a mystery forOrigin of this effect has been a mystery for decades

19

Hartman Effect

Saturation of groupDelay with barrier length

20

How to measure tunneling gtime?

Not n e t k ith ele t onNot an easy task with electrons.

21

Electromagnetic analogy

Time-independent Schrödinger equation and Helmholtz equation are identical

2d 22

2 0ddzψ β ψ+ =

⎧

( )22

2 ( ) Quantum m E Vβ

⎧ −⎪= ⎨⎪

h

( )2 2 Electromagneticck k⎪ −⎩

22

Waveguide with undersized gregion

23

Frustrated total internal reflection

24

1-D Photonic Bandgap Structure

(0 )E t ( )E L(0 , )FE t(0 , )BE t

( , )FE L t

0z = z L=

Grating coupling constant / Bnκ π λ= ΔBragg wavelength λBragg wavelength λ B

Bragg frequency ω B

25

Detuning Ω = ω −ω B

Validity of Classical EM yApproach

The “single-photon” aspect of this experiment only plays a role in the detection processonly plays a role in the detection process.The tunneling part of the experiment is completely describable by the classicalcompletely describable by the classical Maxwell equations since it is a linear process. “Propagation effects are then governed byPropagation effects are then governed by the classical wave equations, and quantization merely affects detectionquantization merely affects detection statistics and higher-order effects.” (R.Y. Chiao and A.M. Steinberg, Progress in Optics)

26

Electromagnetic Barrier gTransmission

0.9

1

0.7

0.8

n

0.4

0.5

0.6

Tra

nsm

issi

on

Ωc Ω =κv

0.2

0.3

0.4T c

Ωc κvh /

0 2 4 6 8 100

0.1

Frequency Ω

where v = c/n

27

Frequency Ω

Group delay gτ

( )( ) ( ) iT T e φ ωω ω= ( )( ) ( ) φ ΩΩ Ω% iT E( ) ( )T T eω ω( )( ) iT e φ ωω( )Ω%E

( )( ) ( ) φ ΩΩ Ω iT E e( )ΩT

0z= z L=φτ =Ωg

ddΩd

(0, )E t( )E L t τ

28gτ

( , )gE L t τ−

Transmission and Group Delay

29Ω

Distortionless Tunneling

Tunneling without 0.7

0.8

0.9

1

Pulse Spectrum

distortion requires that the pulse b d d h b

0.2

0.3

0.4

0.5

0.6

0.7

Transmission

τp=3.0

bandwidth be narrow compared to

b d

0 2 4 6 8 10 120

0.1

0.2

Frequency Detuning Ω

1

stopband.Broadband pulses 0.5

0.6

0.7

0.8

0.9

PulseSpectrum

τp=0.1

have significant spectral content

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

Frequency Detuning

Transmission

30

outside stopband.0 2 4 6 8 10 12

Frequency Detuning Ω

Tunneling is Quasi-staticIn all tunneling time experiments, pulse length greatly exceeds device length.At any instant field distribution in barrier is approximatelyAt any instant, field distribution in barrier is approximately the steady state distribution.

Barrier

τ κ

τ κ

Ω

⇒ >>cRequirement: Pulse bandwidth 1 / << = v

1 /p

pv

31

κ τ >>Since ~1, this implies pL v L

Tunneling time experimentsEnders and Nimtz (1992): Electromagnetic measurements show “zero time” tunnelingmeasurements show “zero time” tunnelingSteinberg, Kwiat, and Chiao, (1993): Measurements show superluminal groupMeasurements show superluminal group velocities in photon tunneling (~1.7c)Spielmann et al (1994): PhotonicSpielmann, et al (1994): Photonic experiments confirm Hartman’s predictionsLonghi et al (2002): Photonic experimentsLonghi, et al (2002): Photonic experiments show no reshaping of tunneled pulses.

32

Experimental confirmation of pHartman effect

Balcou andBalcou and Dutriaux,PRL, 78, 852

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are needed to see this picture.

PRL, 78, 852 (1997)

33

The SKC Experiment

34

Steinberg, Kwiat, and Chiao, g, , ,PRL 71, 708 (1993)

F d l 3 6 fFree space delay=3.6 fsMeasured group delay=2.13 fsInferred group velocity=1.7c1.1L mμ=

35

Count rate ~

2( / )[1 ]px ce τ− Δ−[1 ]e

36

Results of Steinberg, Kwiat, g, ,Chiao (SKC) Experiment

Measured group delay of tunneling single photon ~ 2 13 fs The delay is less than onephoton ~ 2.13 fs. The delay is less than one optical cycle (2.34 fs) at 702 nm!From this delay a “group velocity” of 1.7c wasFrom this delay a group velocity of 1.7c was inferred.Reasonable agreement with group delay

d b d ll’prediction based on Maxwell’s equations.Conclusion is that tunneling is superluminal and that single photons tunnel faster thanand that single photons tunnel faster than light.

37

A th E i t l R ltAnother Experimental Result(S. Longhi, et al, Phys. Rev. E, 2001)

Reference

Tunneled

38

Frustrated total internal reflection

New Scientist, August 17, 2007“Nimtz and Stahlhofen said the reflected photons and the tunneled photons both arrived at their respective photodetectors at the same time, leading them to

l d h f hconclude that some of the microwaves traveled faster than the speed of light. They also found th t th t li ti did 'tthat the tunneling time didn't change on a distance of up to three feet.”

39

The ExperimentThe ExperimentIncident Microwave PulseIncident Microwave Pulse

16 ns wide Gap d~5 cm

Wavelength=3.28 cm(*The pulsesare muchlonger thanlonger thanshown here.)

TransmittedDelay 100 ps

Reflected

40

Delay=100 psDelay=100 ps

Klein-Gordon Equation forKlein Gordon Equation for Tunneling Pulses

2 221E E∂ ∂ 2

2 2 2

1F FF

E E Ez v t

κ∂ ∂− =

∂ ∂

)cos(~ tKzEF Ω−2 2 2 2Dispersion relation: /K v κ=Ω −

F

cCutoff κνΩ =41

c

Evanescent Waves (1)

For frequencies below cutoff frequency, we have exponentially attenuating standing “waves”standing waves

)cos()exp(~ tzE F Ω−γ22 2Attenuation constan ( /t )γ = Ω −Ω v

)cos()exp( tzE F Ωγ

Attenuation constan ( /t )γ = Ω −Ωc v

42

Evanescent Waves (2)

Evanescent waves do not go anywhere. They merely stand and wave, every part in phase.

0.9

1

0.7

0.8

0.9

t=0 ; π

0.4

0.5

0.6

INTE

NSITY

t=π/4Ω 2 2 2cos ( )zFE e tγ−= Ω

0.1

0.2

0.3t=π/3Ω

430 0.1 0.2 0.3 0.4 0.50

0.1

DISTANCE(z/L)

t=π/2Ω

Tunneling simulation: 3pτ =

0.9

1

0.7

0.8

0.9

INCIDENT

TUNNELED

(Normalized)

0.4

0.5

0.6

INTE

NSIT

Y

INCIDENT

REFERENCE

=3

0.2

0.3

0.4IN τp=3

5 10 15 20 250

0.1

0.2

44

5 10 15 20 25TIME(vt/L)

Long Pulse Tunneling: 3Pτ =

100

101

κL=5 τp=3

10−1

10

16 17

18 19

10−3

10−2

Y D

ENSI

TY

10 11

12

19 20

10−4

10

ENER

GY 12

13

14

15

0 0.2 0.4 0.6 0.8 1

10−5

45

0 0.2 0.4 0.6 0.8 1DISTANCE(z/L)

Tunneling long pulse

EnergyDensity

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0 L

46Distance along Barrier

0 L

Short pulse tunneling: 0.1pτ =

0.9

1TransmittedIncidentReference

0.7

0.8

0.9 IncidentReference

0.5

0.6

NTEN

SITY

KL=5 TP=0.1

0.3

0.4INT KL=5 TP=0.1

0

0.1

0.2

47

0 0.5 1 1.5 2 2.5 3 3.5 40

TIME(vt/L)

Short pulse tunneling: 0.1pτ =

1.4

1

1.2

Y

T=0.5

0.6

0.8

GY

DENS

ITY

0.7

0.8

κL=5 τp=0.1

0.4

0.6

ENER

GY

0.8

0.9

1.0 1.1

0.2

1.1

48

0 0.2 0.4 0.6 0.8 10

DISTANCE(z/L)

“Tunneling” short pulse

EnergyDensityDensity

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Distance along Barrier0 L

49

Distance along Barrier

Group delay from transmissionGroup delay from transmission phase shift

Phase shift: ( )1tan / tanhφ γ γ−= Ω⎡ ⎤⎣ ⎦v L

/τ φ= Ωg d dGroup delay :

22⎡ ⎤⎛ ⎞

φgp y

222 2

2

tanh sech os1 cgL L

vL

vκ γ γ φγ γ

τγ

⎡ ⎤⎛ ⎞Ω−⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦=

v vγ γ γ⎝ ⎠⎢ ⎥⎣ ⎦

50

the quantity in bNot luee that = τgv

Dwell Time dτDwell time=Stored Energy/Input PowerA property of an entire wave function with reflected and transmitted components.Does not differentiate betweenDoes not differentiate between reflection and transmission channels.It is not a transit time if pulse is mostlyIt is not a transit time if pulse is mostly reflected.

51

Time-average stored energy1 ( )U dvε μ∗ ∗= +∫ E E H H( )4 vol

μ∫22 t h1⎛ ⎞⎡ ⎤⎛ ⎞ΩL2

0

22 2

2

tanh sech cos12

κ γ γ φγ γ γ

ε⎛ ⎞=⎜⎡ ⎤⎛ ⎞Ω

−⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

⎟⎝ ⎠

L L Lv

U E A⎣ ⎦

Dwell time: lifetime of stored energy escapingthrough both ends d

i

UP

τ =

52

ginP

Dwell time is identical to group delay!

2⎡ ⎤222 2

2

tanh sec h1 cosdL L

vL

vκ γ γ φγ γ γ

τ⎡ ⎤⎛ ⎞Ω

−⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

=

g

v vγ γ γ

τ

⎝ ⎠⎢ ⎥⎣ ⎦=

And both are proportional to stored energy in the barrier.Winful, Optics Express, 2002; PRL 2003

53

For PBG’s dwell time is identical to group delay!

tanhU L Lτφτ κ∂= ⎡ ⎤= ⎢ ⎥= =d g

inP Lvτ

κτ = = ⎢ ⎥⎣

=∂Ω ⎦

=

And both are proportional to stored energy in the barrier.

54

Origin of the Hartman effect…or h d ti t t dwhy does time appear to stand

still?still?

55

Saturation of stored energy gywith barrier length

20

1lim / /iU E A P vε κ κ= =

0.9

1

02lim / /inL

U E A P vε κ κ→∞

0.6

0.7

0.8

EN

SIT

Y

ENERGY DENSITY lim 1/g vτ κ=⇒0.3

0.4

0.5

EN

ER

GY

DE gL→∞

Group delay saturates because stored

0 0.2 0.4 0.6 0.8 10

0.1

0.2

DISTANCE

p yEnergy saturates.

56

DISTANCE

Origin of Hartman Effect

0.9

1

0.7

0.8

0.9

ergy

0.4

0.5

0.6

mali

zed

Ener

g

SATURATION OF STORED ENERGY WITH BARRIER WIDTH

0.2

0.3

0.4

Norm

a ENERGY WITH BARRIER WIDTH

0 1 2 3 4 5 6 7 80

0.1

0.2

57

0 1 2 3 4 5 6 7 8

Barrier Width

Stored Energy and Group gy pDelay

Barrier

Free space

58

Ω

Origin of “superluminality”

For the same input power, the less energy stored, the smaller the delay

UinPiP

1a<

U a UinP

59

Energy Density in Air

60

Energy Density in Barrier

61

Limiting group delayAt midband ( =0) group delay is justΩthe inverse of cutoff frequency (filter

bandwidth)

1bandwidth)

1lim 1/gLvτ κ

→∞= =

ΩLc

→ Ω

62

Relation between group delay g p yand cutoff frequency

0.9

1

0.7

0.8

n 10.4

0.5

0.6

Tra

nsm

issi

on

Ωc

1gτ =

0.2

0.3

0.4T c

gcΩ

0 2 4 6 8 100

0.1

Frequency Ω

c

63

Frequency Ω

Test of the Limiting Group g pDelay

Full width of stop band 250 nmλΔ =Center wavelength 702 nmλ =

SKC experiment

20/λ λΔ = Δf c

Predicted: τg∞ =1/πΔf =2.1 fs

Center wavelength 0 702 nmλ =

g∞

Measured group delay=2.13 fs

Bandwidth 9.2 GHzBW=

Longhi, et al experiment

Limiting group delay 1/ 34.6 psg vτ κ∞ = =

Measured group delay=34 ps

64

Meaning of Group Delay

Lifetime of stored Uτenergy escaping

through both endsP P P

ginP

τ =

in t rP P P= +

1U tPrP 1 in t r

g

P P PU U Uτ

= = +

1 1 1g

= +

65

g t rτ τ τ

Group delay = lifetime of stored energy

P P=Stored EnergyinP out inP P=

1

0 z=L

20 0

12inP cE Aε=

20 0 0

12

U E ALε=

z=0 z=L

0Free space group delayU Lτ = =

66

Free space group delay ginP c

τ

Group delay = time to empty tank from both sides simultaneously

y

For symmetricbarrier delay is the

H.G. Winful, NJP, 2006

Ene

rgy barrier, delay is the

same whether monitored on

Sto

red reflection or

transmission side

S

ReflectionTransmission

67

Reflection

Classical Notion of Transit TimeClassical Notion of Transit Timet 0t=0

A BA B

68

Classical Notion of Transit Time

0

Classical Notion of Transit Time

t=0

A BA B

69

Not a propagation delay!!!It has been assumed for years that thefor years that the group delay in tunneling is atunneling is a propagation delay.Based on that LBased on that assumption, superluminal group

gvτ

=superluminal group velocities have been inferred.

gτ

70

Tunneled Intensity Negligible y g gCompared to Reference

Even at its peak thetunneled pulse istunneled pulse is orders of magnitude smaller than the reference pulse.reference pulse.

71

There is no Pulse Shortening!!!

72

Looking inside a barrier

73

Input pulse



74

Output pulse

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81

Train Analogy to Hartman EffectTrain Analogy to Hartman EffectEach successive car carries half the number of passengers

N N/2 N/4 N/8 N/16

Each successive car carries half the number of passengersin previous car:

N N/2 N/4 N/8 N/16…

The maximum number of passengers is 2N no matter how longThe maximum number of passengers is 2N no matter how longthe train is.Group delay = time it takes to empty the train, proportional to p y p y , p pnumber of passengers on train.If the number of passengers becomes independent of length,

d th d l82

so does the group delay.

Stored energy in cavity (tank)Stored energy in cavity (tank)

gy

Incident

red

Ene

rS

tor

Reflected

d

83

Effect of barrier:

Reduces stored energy below free space value.Creates partial standing wave in barrierCreates partial standing wave in barrier.Reduces accumulated phase below the propagation phase shift.Delays output peak by lifetime of storedDelays output peak by lifetime of stored energy.

84

Free space delay /L vτ =Free space delay 0 /L vτ =

Add propagationAdd propagationdelay to equalizepaths

Barrier delay 0(tanh ) /g L Lτ τ κ κ=

LΔ

85

Barrier delay 0( )g

Absence of DispersionNo dispersion in the middle of stop band.

Phase delay equals group delay if U does not depend on Ωy q g p y p

in in

U UP P

φ φ∂= ⇒ = Ω

∂Ω

Phase shift and phase delay simply proportional to stored energy.

86

gy

Reinterpretation of SKR pExperiment

Since measured delay is less than one optical cycle, best to describe as phase shiftbest to describe as phase shift.Signal and idler photons are simply independent modes of the electromagnetic field.gBarrier reduces energy stored in region occupied by signal photon.Phase shift of signal mode reduced because of reduced stored energy.To equalize paths need to add propagation delay toTo equalize paths, need to add propagation delay to signal path.

87

Reinterpretation of Tunneling p gExperiments (1) Longhi et al

Parameters: L=2cm, 2.8Lκ =tanh 0.35g U L

U Lτ κ= = =

0 0U Lτ κHence barrier stores 35% of the energy stored in barrier-free region.Group delay due to barrier should be 35% of barrier value.

0.35 34 psgτ τ= =

2 cm L=0 / 97 psL vτ = =

Experimental group delay: 97 ps – 63 ps=34 ps

8882.065 10 m/sv= ×

Relation top Quantum p QTunneling

Hartman effect due to saturation of number of particles under barrier.Group delay is lifetime of transientGroup delay is lifetime of transient state.

89

ConclusionsGroup delay in tunneling is not a transit time but the lifetime of stored energy leaking outbut the lifetime of stored energy leaking out of both ends. [New J. Phys., June 2006]Group delay due to barrier is shorter than forGroup delay due to barrier is shorter than for equal length of free space because barrier stores less energy for same input power.

ff d f dHartman effect is due to saturation of stored energy with barrier length.Short group delay does not implyShort group delay does not imply superluminal velocity.

90

References

H.G. Winful, Opt. Express, 10, 1491 (2002) H G Winful Phys Rev Lett 90H.G. Winful, Phys. Rev. Lett., 90,032901 (2003)H.G. Winful, IEEE JSTQE, 9, 17 (2003)H.G. Winful, Nature, 424, 638 (2003)H.G. Winful, Nature, 424, 638 (2003)H.G. Winful, Phys. Rep. 436, 1 (2006)

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tunneling time, the hartman eff d s l i lieffect, and

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