= , (10.118)

=

2’R

RT + R

which tends to an upper limit of 50% as R approaches zero.

This quantum e¬ect is called an interaction-free measurement, because the

single photon injected into the interferometer did not interact at all”either by ab-

sorption or by scattering”with the object, and yet we can infer its presence by means

of the absence of any interaction with it. Furthermore, the inference of presence or ab-

sence can be made with complete certainty based on the principle of the indivisibility

of the photon, since the same photon could not both have been absorbed by the ob-

ject and later caused the click in the dark detector. Actually, it is Bohr™s wave“particle

complementarity principle that plays a central role in this kind of measurement. In

the absence of the object, it is the wave-like nature of light that ensures”through

destructive interference”that the photon never exits through the dark port. In the

presence of the object, it is the particle-like nature of the light”more precisely the

indivisibility of the quantum of light”which enforces the mutual exclusivity of a click

at the dark port or absorption by the object.

Thus a null event”here the absence of a click at Obj ”constitutes just as much

of a measurement in quantum mechanics as the observation of a click. This feature of

quantum theory was already emphasized by Renninger (1960) and by Dicke (1981),

but its implementation in quantum interference was ¬rst pointed out by Elitzur and

Vaidman. Note that this e¬ect is nonlocal, since one can determine remotely the pres-

ence or the absence of the unknown object, by means of an arbitrarily remote dark

detector. The fact that the entire interferometer con¬guration must be set up ahead

of time in order to see this nonlocal e¬ect is another example of the general principle

in Bohr™s Delphic remark quoted in Section 10.3.3.

The data in Fig. 10.14 show that the fraction of successful measurements is nearly

50%, in agreement with the theoretical prediction given by eqn (10.118). By techni-

cal re¬nements of the interferometer, the probability of a successful interaction-free

measurement could, in principle, be increased to as close to 100% as desired (Kwiat

et al., 1995a). A success rate of · = 73% has already been demonstrated (Kwiat et al.,

1999a). In the 100% success-rate limit, one could determine the presence or absence

of an object with minimal absorption of photons.

This possibility may have important practical applications. In an extension of this

interaction-free measurement method to 2D imaging, one could use an array of these

devices to map out the silhouette of an unknown object, while restricting the num-

ber of absorbed photons to as small a value as desired. In conjunction with X-ray

interferometers”such as the Bonse“Hart type”this would, for example, allow X-ray

pictures of the bones of a hand to be taken with an arbitrarily low X-ray dosage.

¿ Experiments in linear optics

=

>

Fig. 10.14 (a) Data demonstrating interaction-free measurement. The Michelson beam split-

ter re¬‚ectivity for the upper set of data was 43%. (b) Data and theoretical ¬t for the ¬gure

of merit · as a function of beam splitter re¬‚ectivity. (Reproduced from Kwiat et al. (1995a).)

10.7 Exercises

10.1 Vacuum ¬‚uctuations

(+) (+)

Drop the term E3 (r, t) from the expression (10.7) for Eout (r, t) and evaluate the

(+) (’)

equal-time commutator Eout,i (r, t) , Eout,j (r , t) . Compare this to the correct form

(+)

in eqn (3.17) and show that restoring E3 (r, t) will repair the ¬‚aw.

10.2 Classical model for two-photon interference

Construct a semiclassical model for two-photon interference, along the lines of Section

1.4, by assuming: the down-conversion mechanism produces classical amplitudes ±σn =

√

Iσn exp (iθσn ), where σ = sig, idl is the channel index and the gate windows are

labeled by n = 1, 2, . . .; the phases θσn vary randomly over (0, 2π); the phases and

intensities Iσn are statistically independent; the intensities Iσn for the two channels

have the same average and rms deviation.

Evaluate the coincidence-count probability pcoinc and the singles probabilities psig

and pidl , and thus derive the inequality (10.41).

The HOM dip—

10.3

2

Assume that the function |g (ν)| in eqn (10.69) is a Gaussian:

√

2

|g (ν)| = „2 / π exp ’„2 ν 2 .

2

Evaluate and plot Pcoinc (∆t).

¿

Exercises

HOM by scattering theory—

10.4

(1) Apply eqn (8.76) to eqn (10.71) to derive eqn (10.72).

(2) Use the de¬nition (6.96) to obtain a formal expression for the coincidence-counting

detection amplitude, and then use the rule (9.96) to show that |¦pair will not

contribute to the coincidence-count rate.

Anti-HOM—

10.5

Consider the two-photon state given by eqn (10.48), where C (ω, ω ) satis¬es the (’)-

version of eqn (10.51).

(1) Why does C (ω, ω ) = ’C (ω , ω) not violate Bose symmetry?

(2) Assume that C (ω, ω ) satis¬es eqn (10.56) and the (’)-version of eqn (10.51).

Use eqns (10.71)“(10.74) to conclude that the photons in this case behave like

fermions, i.e. the pairing behavior seen in the HOM interferometer is forbidden.

Interaction-free measurements—

10.6

(1) Work out the relation between the lengths of the arms of the Michelson interfer-

ometer required to ensure that a dark fringe occurs at the output port.

(2) Explain why the probabilities P (failure) and P (success), respectively de¬ned by

eqns (10.116) and (10.117), do not sum to one.

11

Coherent interaction of light with

atoms

In Chapter 4 we used perturbation theory to describe the interaction between light and

matter. In addition to the assumption of weak ¬elds”i.e. the interaction energy is small

compared to individual photon energies”perturbation theory is only valid for times

in the interval 1/ω0 t 1/W , where ω0 and W are respectively the unperturbed

frequency and the perturbative transition rate for the system under study. When ω0

is an optical frequency, the lower bound is easily satis¬ed, but the upper bound can

be violated. Let ρ be a stationary density matrix for the ¬eld; then the ¬eld“¬eld

correlation function, for a ¬xed spatial point r but two di¬erent times, will typically

decay exponentially:

(1) (’) (+)

(r, t2 ) ∼ exp (’ |t1 ’ t2 | /Tc ) ,

Gij (r, t1 ; r, t2 ) = Tr ρEi (r, t1 ) Ej (11.1)

where Tc is the coherence time for the state ρ. For some states, e.g. the Planck

distribution, the coherence time is short, in the sense that Tc 1/W . Perturbation

theory is applicable to these states, but there are many situations”in particular for

laser ¬elds”in which Tc > 1/W . Even though the ¬eld is weak, perturbation theory

cannot be used in these cases; therefore, we need to develop nonperturbative methods

that are applicable to weak ¬elds with long coherence times.

11.1 Resonant wave approximation

The phenomenon of resonance is ubiquitous in physics and it plays a central role in

the interaction of light with atoms. Resonance will occur if there is an allowed atomic

transition q ’ p with transition frequency ωqp = (µq ’ µp ) / and a matching optical

frequency ω ≈ ωqp . In Section 4.9.2 we saw that the weak-¬eld condition can be ex-

pressed as „¦ ω0 , where „¦ is the characteristic Rabi frequency de¬ned by eqn (4.147).

In the interaction picture, the state vector satis¬es the Schr¨dinger equation (4.94), in

o

which the full Hamiltonian is replaced by the interaction Hamiltonian; consequently,

‚

|Ψ (t) ∼ „¦ |Ψ (t) .

i (11.2)

‚t

Thus the weak-¬eld condition tells us that the changes in the interaction-picture state

vector occur on the time scale 1/„¦ 1/ω0 . Consequently, the state vector does not

change appreciably over an optical period. This disparity in time scales is the basis

for a nonperturbative approximation scheme. In the interests of clarity, we will ¬rst

develop this method for a simple model called the two-level atom.

¿½

Resonant wave approximation

11.1.1 Two-level atoms

The spectra of real atoms and the corresponding sets of stationary states display a

daunting complexity, but there are situations of theoretical and practical interest in

which this complexity can be ignored. In the simplest case, the atomic state vector is

a superposition of only two of the stationary states. Truncated models of this kind are

called two-level atoms. This simpli¬cation can occur when the atom interacts with

a narrow band of radiation that is only resonant with a transition between two speci¬c

energy levels. In this situation, the two atomic states involved in the transition are the

only dynamically active degrees of freedom, and the probability amplitudes for all the

other stationary states are negligible.

In the semiclassical approximation, the Feynman“Vernon“Hellwarth theorem

(Feynman et al., 1957) shows that the dynamical equations for a two-level atom are

isomorphic to the equations for a spin-1/2 particle in an external magnetic ¬eld. This

provides a geometrical picture which is useful for visualizing the solutions. The general

zeroth-order Hamiltonian for the ¬ctitious spin system is H0 = ’µB · σ, and we will

choose the ¬ctitious B-¬eld as B = ’Bu3 , so that the spin-up state is higher in energy

than the spin-down state.

To connect this model to the two-level atom, let the two resonantly connected

atomic states be |µ1 and |µ2 , with µ1 < µ2 . The atomic Hilbert space is e¬ectively

truncated to the two-dimensional space spanned by |µ1 and |µ2 , so the atomic Hamil-

tonian and the atomic dipole operator d are represented by 2 — 2 matrices. Every 2 — 2

matrix can be expressed in terms of the standard Pauli matrices; in particular, the

truncated atomic Hamiltonian is

µ2 + µ1 ω21

µ 0

Hat = 2 = I2 + σz , (11.3)

0 µ1 2 2

where I2 is the 2 — 2 identity matrix and ω21 = µ2 ’ µ1 . The term proportional to I2

can be eliminated by choosing the zero of energy so that µ2 + µ1 = 0. This enforces

the relation µB ” ω21 /2 between the two-level atom and the ¬ctitious spin.

When the very small e¬ects of weak interactions are ignored, atomic states have

de¬nite parity; therefore, the odd-parity operator d has no diagonal matrix elements.

For the two-level atom, this implies d = d— σ’ + d σ+ , where d = µ2 d µ1 , σ+ is the

spin-raising operator, and σ’ is the spin-lowering operator. Combining this with the

decomposition E = E(+) + E(’) and the plane-wave expansion (3.69) for E(+) leads

to

(r) (ar)

Hint = Hint + Hint , (11.4)

Hint = ’d · E(+) σ+ ’ d— · E(’) σ’

(r)

ωk d · eks

= ’i aks σ+ + HC , (11.5)

2 0V

ks

Hint = ’d · E(’) σ+ ’ d— · E(+) σ’

(ar)

ωk d— · eks

= ’i aks σ’ + HC . (11.6)

2 0V

ks

¿¾ Coherent interaction of light with atoms

In Hint the annihilation (creation) operator aks a† is paired with the energy-raising

(r)

ks

(ar)

(-lowering) operator σ+ (σ’ ), while Hint has the opposite pairings. In the perturba-

tive calculations of Section 4.9.3 the emission (absorption) of a photon is associated

with lowering (raising) the energy of the atom, subject to the resonance condition

(r) (ar)

ωk = ω21 , so Hint and Hint are respectively called the resonant and antiresonant

Hamiltonians.

The full Hamiltonian in the Schr¨dinger picture is

o

H = H0 + Hint , (11.7)

where

ω21

ωk a† aks +

H0 = σz . (11.8)

ks

2

ks

In the interaction picture, the operators satisfy the uncoupled equations of motion

‚

aks (t) = [aks (t) , H0 ] = ωk aks (t) , (11.9)

i

‚t

‚

i

σz (t) = [σz (t) , H0 ] = 0 , (11.10)

‚t

‚ ω21

σ± (t) = [σ± (t) , H0 ] = “

i σ± (t) , (11.11)

‚t 2

with the solution

aks (t) = aks e’iωk t , σz (t) = σz , σ± (t) = e±iω21 t σ± , (11.12)

where aks , σz , and σ± are the Schr¨dinger-picture operators. Thus the time depen-

o

dence of the operators is explicitly expressed in terms of the atomic transition fre-

quency ω21 and the optical frequencies ωk . This is a great advantage for the calcula-

tions to follow.

The interaction-picture state vector |˜ (t) satis¬es the Schr¨dinger equation

o

‚

|˜ (t) = Hint (t) |˜ (t) ,

i (11.13)

‚t

where

(r) (ar)

Hint (t) = Hint (t) + Hint (t) , (11.14)

ωk d · eks i(ω21 ’ωk )t

(r)

Hint (t) = ’i e aks σ+ + HC (11.15)

2 0V

ks

and

ωk d— · eks ’i(ω21 +ωk )t

(ar)

(t) = ’i

Hint e aks σ’ + HC (11.16)

2 0V

ks

are obtained by replacing the operators in eqns (11.5) and (11.6) by the explicit solu-

tions in eqn (11.12).

¿¿

Resonant wave approximation

11.1.2 Time averaging

The slow and fast time scales can be separated explicitly by means of a temporal

¬ltering operation, like the one introduced in Section 9.1.2-C to describe narrowband

detection. We use an averaging function, (t), satisfying eqns (9.35)“(9.37), to de¬ne

running averages by

∞ ∞

f (t) ≡ (t ’ t ) f (t ) =

dt dt (t ) f (t + t ) . (11.17)

’∞ ’∞

The temporal width ∆T de¬ned by eqn (9.37) will now be renamed the memory

interval Tmem . The idea behind this new language is that the temporally coarse-

grained picture imposed by averaging over the time scale Tmem causes amnesia, i.e.

averaged operators at time t will not be correlated with averaged operators at an

earlier time, t < t ’ Tmem . The average in eqn (11.17) washes out oscillations with

periods smaller than Tmem , and the average of the derivative is the derivative of the

average:

df d

(t) = f (t) . (11.18)

dt dt

The separation of the two time scales is enforced by imposing the condition

1 1

Tmem (11.19)

ω21 „¦

on Tmem . A function g (t) that varies on the time scale 1/„¦ is essentially constant over

the averaging interval, so that

∞

g (t) ≡ (t ’ t ) g (t ) ≈ g (t) .

dt (11.20)

’∞

The combination of this feature with the normalization condition (9.36) leads to the

following rule:

(t ’ t ) ≈ δ (t ’ t ) when applied to slowly-varying functions . (11.21)

It is also instructive to describe the averaging procedure in the frequency domain.

We would normally denote the Fourier transform of (t) by (ω), but this particular

function plays such an important role in the theory that we will honor it with a special

name: ∞

(t) eiωt .

K (ω) = dt (11.22)

’∞

The properties of (t) guarantee that K (ω) is real and even, K — (ω) = K (’ω) =

K (ω), and that it has a ¬nite width, wK , related to the averaging interval by wK ∼

1/Tmem. The frequency-domain conditions corresponding to eqn (11.19) are

„¦ wK ω21 , (11.23)

and the time-domain normalization condition (9.36) implies K (0) = 1. Performing

the Fourier transform of eqn (11.17) gives the frequency-domain description of the

¿ Coherent interaction of light with atoms

averaging procedure as f (ω) = K (ω) f (ω). Thus for small frequencies, ω wK , the

original function f (ω) is essentially unchanged, but frequencies larger than the width

wK are strongly suppressed. For this reason K (ω) is called the cut-o¬ function.1

11.1.3 Time-averaged Schr¨dinger equation

o

Since |˜ (t) only varies on the slow time scale, the rule (11.21) tells us that it is

e¬ectively unchanged by the running average, i.e. |˜ (t) ≈ |˜ (t) . Consequently,

averaging the Schr¨dinger equation (11.13), with the help of eqn (11.18), yields the

o

approximate equation

‚

|˜ (t) = H int (t) |˜ (t) .

i (11.24)

‚t

(ar)

According to eqn (11.16), all terms in Hint (t) are rapidly oscillating; therefore, we

(ar)

expect that H int (t) ≈ 0. This expectation is justi¬ed by the explicit calculation in

(ar)

Exercise 11.1, which shows that the cut-o¬ function in each term of H int (t) is evalu-

ated with its argument on the optical scale. In the resonant wave approximation

(RWA), the antiresonant part is discarded, i.e. the full interaction Hamiltonian Hint (t)

(r)

is replaced by the resonant part H int (t). The traditional name, rotating wave approx-

imation, is suggested by the mathematical similarity between the two-level atom and

a spin-1/2 particle precessing in a magnetic ¬eld (Yariv, 1989, Chap. 15).

(r)

Turning next to the expression (11.15) for H int (t), we see that the exponentials

involve the detuning ∆k = ωk ’ ω21 which will be small near resonance; therefore, the

(r)

average of Hint (t) will not vanish. The explicit calculation gives

(r)

gks e’i∆k t σ+ aks + HC ,

Hrwa (t) ≡ H int (t) = ’i (11.25)

ks

where

ωk d · eks

gks = K (∆k ) , (11.26)

2 0V

and we have introduced the new notation Hrwa (t) as a reminder of the approximation

in use. The cut-o¬ function in the de¬nition of the coupling constant guarantees that

only terms satisfying the resonance condition |ω21 ’ ωk | < wK will contribute to

Hrwa .

With the resonant wave approximation in force, we can transform to the Schr¨- o

dinger picture by the simple expedient of omitting the time-dependent exponentials

in eqn (11.25). Thus the RWA Hamiltonian in the Schr¨dinger picture is

o

Hrwa = H0 ’ d · E(+) σ+ ’ d— · E(’) σ’

g— a† σ’ ,

= H0 ’ i gks aks σ+ + i (11.27)

ks ks

ks ks

where H0 is given by eqn (11.8). This observation provides the following general scheme

for de¬ning the resonant wave approximation directly in the Schr¨dinger picture.

o

1 This is physics jargon. An engineer would probably call K (ω) a low-pass ¬lter.

¿

Resonant wave approximation