Field Propagation in Space-Time

The quantum theory of photodetection is a straightforward application of the quantum theory of time dependent perturbation. The first-order approximation calculates the probability that a photon is annihilated at space-time point (r, t) of the photo-dection event, or the photon counting rate of a photodetector at position r and time f,

The second-order approximation gives the probability that a pair of photons is annihilated jointly at the space-time points (ri,fi) and (Г2,<2)> or the joint-detection counting rate of a pair of photodetectors at positions (rj, Г2), and times (t, t^),

The joint-detection counting rate is also called “coincidence” counting rate, although 11 is not necessarily equal to t^. In these formulas, |Фр ) is the state of the field at to, which is the time when the perturbation V(t) is applied, i.e., at the very beginning of the photodetection event. In typical optical measurements, |Ф^) is often evaluated from the state |Ф(0)) that is prepared at the radiation source. The state |Т(())} of the field that is excited at the radiation source is usually known from early measurements or predictable through the study of the radiation mechanism of the light source.

This is a typical quantum mechanical dynamical problem: we are interested in knowing the expectation value of an operator A at time t

with the knowledge of the quantum state of the system at an early time, such as |Ф(0)) at time t = 0.

Since we have shown that any state of a radiation field can be described by the superposition of the eigenstates of the Hamiltonian, such as Eq. (2.4.38), and proved that an energy eigenstate simply develops a phase factor, from time t = 0 to t. The quantum

state of the field at time t is thus written as

where | E) represents the eigenstate of the Hamiltonian. To obtain the result of Eq. (4.5.2), we have assumed a complete orthogonal set of energy eigenstates

Eq. (4.5.1) is thus rewritten as

We are ready to work in the Heisenberg representation to propagate the field operator from t = 0 at the radiation source to to of the photodetection event, rather then working in the Schrodinger representation to develop the state of | Ф(0)). If we define a time-dependent operator A(t) by

then we can rewrite Ecj. (4.5.3) as

Differentiating Eq. (4.5.4) with respect to time t, we find

if A(t) does not depend explicitly on time t. In Eq. (4.5.6), we have assumed that H commutes with e~>Ht/h. Eq. (4.5.6) is called Heisenberg equation of motion for operators.

It is easy to show that for any product of two or more operators, such as A = BC, we can define a product of two or more time-dependent operators, such as A(t) = B(t)C(t),

where e~lHt/llel"t/,i = has been used. It is also easy to show that B(t) and C(t) obey the Heisenberg equation of motion, respectively, and thus can be propagated, individually,

Based on the above results, in the Heisenberg representation, we have the counting rate of a photodetector, or the probability for a photon to be annihilated at space-time point (M),

and the joint-detection counting rate of a pair of photodetectors, or the probability for a pair of photons to be annihilated jointly at space-time points (r (, t,) and (r^, t2)

with the understanding that the field operators E^(rj,tj) are propagated from (r°,f°) of the radiation source. Notice, we have used a different notation, t°, to distinguish the perturbation time, to, of the photodetection event. To simplify the notation, we usually choose t° = 0 and write the state |Ф(())} as |Ф). If the radiation source can be treated as a point source, we may also choose r° = 0. Care has to be taken in the cases when the finite size of the source has to be taken into account, especially in “near-field” measurements. Eqs. (4.5.9) and (4.5.10) are the basic formalisms for photon counting measurements.

Regarding the propagation of the field operator in space-time, in general, we need to obtain the time evolution of the annihilation and creation operator and to find the solution of the Helmholtz equation subject to the corresponding boundary condition of the field. The mathematical tools for defining the spatial mode function of the field are well developed in classical optics and there is no difference in quantum optics. The time evolution of the annihilation operator, fik(t), and the creation operator, a£(t), can be easily obtained for certain types of Hamiltonians, such as the free field Hamiltonian of Eq. (2.4.25), by solving the Heisenberg equation of motion

with the solution

where we have assigned = 0. We thus write the free field operator in the Heisenberg representation

where u/,.(r) is the solution of the Helmholtz equation of Eq. (2.4.43) subject to the required boundary conditions. In Eq. (4.5.13), E(+)(r, t) and E(-)(r. t) correspond to the physical processes of photon annihilation and creation at the space-time point (r. f). respectively.

Under certain experimental conditions, such as far-field measurements, the plane-wave is a good approximation. The positive and the negative field operators in Eq. (4.5.13) can then be written as

It is common to formally write El+)(r, t) and E((r. t) as follows

with t) the propagator, or the Green’s function, propagating the kth mode of the field from the source to (r. t): we have also formally absorbed the constants into the Green’s function as usual.

Quantized Subfield and Effective Wavefunction of Photon

Einstein’s bundle of ray, or subfield, is a quantized microscopic realistic entity of electromagnetic wavepacket propagating in space-time. This concept is still in the frame work of Maxwell electromagnetic wave theory, except Einstein introduced a concept of granularity and abandoned the continuum interpretation of Maxwell. It is interesting to find that an “effective wavefunction”, of the mth photon, or the mth group of identical

photons, among an ensemble of photons, or groups of identical photons, can be defined from the quantum photodetection theory, precisely, from the calculations of GU)(r, t) and G'U) (r i. f |; Г'2. U)- It is also interesting to find that фт has the same mathematical representation and play the same role as that of Einstein’s subfield Etn(r. t). It should be emphasized even before introducing the concept, however, ipm(r,t) represents the probability amplitude for the mth photon, or the mth groip of identical photons, to produce a photodetection event at space-time (r, £), which is different in nature from a quantized realistic entity of electromagnetic wavepacket. In the following, we introduce the "effective wavefunction” of a photon, or a group of identical photons, in terms of the single-photon state representation and the coherent state representation.

(I) Single-photon state representation.

Assuming a natural weak point-like light source and a far-held measurement of a pointlike photodetector. In the light source, a large number of randomly distributed spontaneous atomic transitions excite a large number of subfields. Although the chance to have a spontaneous emission is very small, there is indeed a tiny probability for an atom to create a photon whenever the atom decays from its higher energy level E-2 (ЛЕ2 Ф 0) down to its ground energy state of E. We have approximated the state of the mth subheld, which is excited by the mth atomic transition, as

where |co| ~ 1 is the probability amplitude for no-held-excitation and |ci| = |e| 1 is the

probability amplitude for the creation of a photon from the mth atomic transition. In a 1-D approximation,

here we have intentionally written the complex amplitudes ат{ш)ег<Рт^ш which is determined by the creation and annihilation operators, into the state and assumed that all radiation modes ш excited by the mth atomic transition have a common phase TO(w) = m, so that it can be moved outside of the integral. The generalized state of the radiation field, created from such a large number atomic transitions, can be formally written as

where, again, we have assumed a common complex amplitude for each wavepacket, although it can still take arbitrary phases from transition to transition. Since |c| -C 1, In Equation (4.6.1) we listed the first-order and the second-order approximations on e.

Considering a simple measurement in which a point-like photodetecter at (r, t) is facing the point-like light source, the field operator at (r.t) can be approximated in 1-D:

where gp(to; r, t) is the Green’s function that propagates the w-mode of the pth subfield from the point-like source to space-time coordinate (r. t). We are ready to calculate G1 l^(r. t):

where we have applied the completeness relation

In Eq.(4.6.3) we have defined the effective wavefunction of the mth photon,

where, again, we have assumed a common phase m for the mth atomic transition, we have also absorbed |cm| into am(ui) as usual. In the point-to-point propagation, or the far-field approximation, ipm{r,t) can be written as

where we have chosen the mth atomic transition as the origin of the space-time coordinate system. It is easy to find that the effective wavefunction фт(r, t) has the same mathematical representation and plays the same role as that of Einstein’s subfield Em(r, t).

It is easy to find that (Дп(г, t)) = 0 for thermal state:

when taking into account all possible values of (mn), agreeing with that of Einstein’s picture, which is calculated from the same mathematics of ensemble average.

(II) Coherent state representation.

We have assumed a weak thermal radiation in the above calculation, in which only the lower order of Eq.(4.6.1) contributes. In the case of arbitrary brightness, especially bright

light condition, we may need to keep all orders of the expansion. In this case, quantum coherent state representation is a better choice. In the quantum coherent state representation, the state of a thermal radiation field can be written as

In this model, the radiation field contains a large number of randomly created and randomly distributed subfields, or wavepackets, each is created from an atomic transition, or a group of identical atomic transitions, or other creation mechanisms, and each subfield contains a large number of radiation modes that can be represented by coherent state. In Eq. (4.6.7), m labels the mth subfield, and к labels the kth wavevector. |om(k)) is an eigenstate of the annihilation operator with an eigenvalue om(k),

Thus, we have,

Considering the same simple measurement with the same field operators of Eq. (4.6.2), we have

Following Glauber’s theory, the probability to produce a photoelectron event at (r, t) is proportional to G^^(r, t) = ({E(_)(r, t)Ei+^(r. <))qm)Eii- Substitute the quantum state and the field operators into GC)(r, t), we have

where we have defined фт{r.t) the effective wavefunction of the mth suhfield:

We have shown earlier (Дп(г, t)) = 0 when taking into account all possible values of (m(fin) in the ensemble average.

It is interesting to find that the quantum mechanical concept of effective wavefunction has the same mathematical representation and plays the same role as Einstein’s concept of subfield. However, the physical means of the two are different in nature: Einstein’s subfield Em(r,t) is a classical real entity of electromagnetic field. The quantum mechanical effective wavefunction фт (r. t) is the probability amplitude for the mth subfield to produce a photodetection event at space-time (r.t).

If there exists two different yet indistinguishable alternatives, such as in the Young’s double-slit interferometer, to produce a photodetection event at space-time coordinate (r.t), the effective wavefunction is easily calculated to be the following linear superposition

which is, again, the same superposition as that in Einstein’s picture, except the electromagnetic subfield .ETO(r, t) is replaced with the effective wavefunction ipm(i ). However, the “interpretation” of the superposition between фтл{r, t) and фтв{r, t) is different from that of Einstein’s picture. In quantum theory, ^тд(г, t) and фтв{r, t.) represent two different yet indistinguishable probability amplitudes for a photon or a group of identical photons to produce a photodetection event at (r, t). It is a principle of quantum mechanics: the probability of producing a photodetection event at (r, t) is proportional to |'0тд(г. t) + фтв{г,t)|2.

Joint Measurement of Composite Radiation Systems

Following Glauber’s theory, the probability to produce a joint photodetection event at space-time coordinates (ri,ti) and (Г2Т2) from thermal radiation field in single-photon state representation is calculated as follows:

where we have defined the effective wavefunction of the m-nth pair of randomly created and randomly distributed photons

with

G(2)(ri>*i;r2,<2) is also named the second-order coherence function, which is the result of a superposition between two different yet indistinguishable two-photon amplitudes: (1) the mth wavepacket is annihilated at (lq, t) and the nth wavepacket is annihilated at (r2,f2); and (2) the mth wavepacket is annihilated at (r2, t2) and the nth wavepacket is annihilated at We name this superposition two-photon interference: two randomly

created and randomly paired wavepackets, or photons in thermal state interfering with the pair itself.

Examine the cross-interference term of Eq.(14.5.9) which contributes to the photon number fluctuation correlation measurement,

we may find that this term is the same as these we have calculated in previous sections, except Einstein’s subfield is replaced by the effective wavefunction.

A similar effective wavefunction of a pair of randomly created and randomly distributed photons can be also calculated from coherent state representation

We will have more discussions on the effective wavefunctions of photons and photon pairs in Chapters 6 and 7.

SUMMARY

In this chapter, we discuss the measurement of quantized light. We start from the measurement of Einstein’s bundle of ray

and then introduced the Glauber photodetection theory in first-order photon counting and in second-order coincidence photon counting based on the time dependent perturbation theory of quantum mechanics,

and

During the calculations of GD)(r, t) and G(2)(r], tx: r2. t,2), we introduced the concept of effective wavefunction for a photon and the concept of effective wavefunction for a pair of photons in the single-photon state representation and in the coherent state representation.