— —

oscillator ¬eld EL and in the signal ¬eld Es (t). Thus a large value for |EL | e¬ectively

ampli¬es the contribution of the weak optical signal to the low-frequency heterodyne

—

signal. For instance, doubling the size of EL , doubles the size of the heterodyne signal

for a given signal amplitude Es . Furthermore, the relative phase between the linear

oscillator and the incident signal is faithfully preserved in the heterodyne signal. To

make this point more explicit, ¬rst rewrite eqn (9.103) as Shet (t) = F cos (ωIF t) +

G sin (ωIF t), where the Fourier components are given by

F = 2 Re [r— t EL Es ] , G = 2 Im [r— t EL Es ] .

— —

(9.104)

We use the Stokes relation (8.7), in the form

r— t = |r| |t| e±iπ/2 , (9.105)

to rewrite eqn (9.104) as

— —

F = ±2 |EL Es | |r| |t| sin (θL ’ θs ) , G = ±2 |EL Es | |r| |t| cos (θL ’ θs ) , (9.106)

where θL and θs are respectively the phases of the local oscillator EL and the signal

Es .

The quantities F and G can be separately measured. For example, F and G can

be simultaneously determined by means of the apparatus sketched in Fig. 9.7. Note

that the insertion of a 90—¦ phase shifter into one of the two local-oscillator arms

allows the measurement of both the sine and cosine components of the intermediate-

frequency signals at the two photon detectors. Each box labeled ˜IF mixer™ denotes the

combination of a radio-frequency oscillator”conventionally called a 2nd LO ” that

operates at the IF frequency, with two local radio-frequency diodes that mix the 2nd

LO signal with the two IF signals from the photon detectors. The net result is that

these IF mixers produce two DC output signals proportional to the IF amplitudes F

and G. The ratio of F and G is a direct measure of the phase di¬erence θL ’ θs relative

to the phase of the 2nd LO, since

F

= tan (θL ’ θs ) . (9.107)

G

The heterodyne signal corresponding to F is maximized when θL ’ θs = π/2 and

minimized when θL ’ θs = 0, whereas the heterodyne signal corresponding to G is

¾ Photon detection

Fig. 9.7 Schematic of an apparatus for two-quadrature heterodyne detection. The beam

splitters marked as ˜High trans™ have |t| ≈ 1.

maximized when θL ’ θs = 0 and minimized when θL ’ θs = π/2, where all the phases

are de¬ned relative to the 2nd LO phase. The optical phase information in the signal

waveform is therefore preserved through the entire heterodyne process, and is stored

in the ratio of F to G. This phase information is valuable for the measurement of

small optical time delays corresponding to small di¬erences in the times of arrival

of two optical wavefronts; for example, in the di¬erence in the times of arrival at

two telescopes of the wavefronts emanating from a single star. Such optical phase

information can be used for the measurement of stellar diameters in infrared stellar

interferometry with a carbon-dioxide laser as the local oscillator (Hale et al., 2000).

This is an extension of the technique of radio-astronomical interferometry to the mid-

infrared frequency range.

Examples of important heterodyne systems include: Schottky diode mixers in the

radio and microwave regions; superconductor“insulator“superconductor (SIS) mixers,

for radio astronomy in the millimeter-wave range; and optical heterodyne mixers,

using the carbon-dioxide lasers in combination with semiconductor photoconductors,

employed as square-law detectors in infrared stellar interferometry (Kraus, 1986).

9.3.2 Quantum analysis of heterodyne detection

Since the ¬eld operators are expressed in terms of classical mode functions and their

associated annihilation operators, we can retain the assumptions”i.e. plane waves, S-

polarization, etc.”employed in Section 9.3.1. This allows us to use a simpli¬ed form

of the general expression (8.28) for the in-¬eld operator to replace the classical ¬eld

(9.101) by the Heisenberg-picture operator

Ein (r, t) = ieL aL2 eikL y e’iωL t + ies as1 eiks x e’iωs t + Evac,in (r, t) ,

(+) (+)

(9.108)

where eM = ωM /2 0 V is the vacuum ¬‚uctuation ¬eld strength for a plane wave

with frequency ωM . This is an extension of the method used in Section 9.1.4 to model

¾

Heterodyne and homodyne detection

imperfect detectors. The annihilation operators aL2 and as1 respectively represent the

local oscillator ¬eld, entering through port 2, and the signal ¬eld, entering through

port 1; and, we have again assumed that the bandwidths of the signal and local os-

cillator ¬elds are small compared to ωIF . If this assumption has to be relaxed, then

the Schr¨dinger-picture annihilation operators must be replaced by slowly-varying en-

o

(+)

velope operators aL2 (t) and as1 (t). In principle, the operator Evac,in (r, t) includes

all modes other than the signal and local oscillator, but most of these terms will

not contribute in the subsequent calculations. According to the discussion in Section

8.4.1, each physical input ¬eld is necessarily paired with vacuum ¬‚uctuations of the

same frequency”indicated by the dashed arrows in Fig. 9.6”entering through the

(+)

other input port. Thus Evac,in (r, t) must include the operators aL1 and as2 describing

vacuum ¬‚uctuations with frequencies ωL and ωs entering through ports 1 and 2 respec-

tively. It should also include any other vacuum ¬‚uctuations that could combine with

the local oscillator to yield terms at the intermediate frequency, i.e. modes satisfying

ωM = ωL ±ωIF . The +-choice yields the signal frequency ωs , which is already included,

so the only remaining possibility is ωM = ωL ’ ωIF . Again borrowing terminology from

radio engineering, we refer to this mode as the image band, and set M = IB and

(+)

ωIB = ωL ’ ωIF . The relevant terms in Ein (r, t) are thus

Ein (r, t) = ieL aL2 eikL y e’iωL t + ieL aL1 eikL x e’iωL t

(+)

+ ies as1 eiks x e’iωs t + ies as2 eiks y e’iωs t

+ ieIB aIB2 eikIB y e’iωIB t

+ ieIB aIB1 eikIB x e’iωIB t . (9.109)

A The heterodyne signal

(+)

The scattered ¬eld operator Eout (r, t) is split into two parts, which respectively de-

scribe propagation along the 2 ’ 2 arm and the 1 ’ 1 arm in Fig. 9.6. The latter

(+)

part”which we will call Eout,D (r, t)”is the one driving the detector. The spatial

(+)

modes in Eout,D (r, t) are all of the form exp (ikx), for various values of k. Since we

only need to evaluate the ¬eld at the detector location xD , the calculation is simpli¬ed

by choosing the coordinates so that xD = 0. In this way we ¬nd the expression

Eout,D (t) = ieL aL1 e’iωL t + ies as1 e’iωs t + ieIB aIB1 e’iωIB t .

(+)

(9.110)

The scattered annihilation operators are obtained by applying the beam-splitter scat-

tering matrix in eqn (8.63) to the incident annihilation operators. This simply amounts

to working out how each incident classical mode is scattered into the 1 ’ 1 arm, with

the results

as1 = t as1 + r as2 , aL1 = t aL1 + r aL2 , aIB1 = t aIB1 + r aIB2 . (9.111)

The ¬nite e¬ciency of the detector can be taken into account by using the technique

(+)

discussed in Section 9.1.4 to modify Eout,D (t).

¾ Photon detection

Applying eqn (9.33), for the total single-photon counting rate, to this case gives

(’) (+)

w(1) (t) ∝ Eout,D (t) Eout,D (t) , (9.112)

and the intermediate frequency part of this signal comes from the beat-note terms

between the local oscillator part of eqn (9.110)”or rather its conjugate”and the

signal and image band parts. This procedure leads to the operator expression

(’) (+)

Shet = Eout,D (t) Eout,D (t) = F cos (ωIF t) + G sin (ωIF t) , (9.113)

IF

where the operators F and G”which correspond to the classical quantities F and G

respectively”have contributions from both the signal and the image band, i.e.

F = Fs + FIB , G = Gs + GIB , (9.114)

where

†

Fs = eL es aL1 as1 + HC , (9.115)

†

FIB = eL eIB aL1 aIB1 + HC , (9.116)

†

Gs = ’ieL es aL1 as1 ’ HC , (9.117)

and

†

GIB = ’ieL eIB aL1 aIB1 ’ HC . (9.118)

By assumption, the density operator ρin describing the state of the incident light

is the vacuum for all annihilation operators other than aL2 and as1 , i.e.

aΛ ρin = ρin a† = 0 , Λ = s2, L1, IB1, IB2 . (9.119)

Λ

These conditions immediately yield

†

aL1 aIB1 = 0 , (9.120)

and

†

a† as1 .

aL1 as1 = r— t (9.121)

L2

Furthermore, the independently generated signal and local oscillator ¬elds are uncor-

related, so the total density operator can be written as a product

ρin = ρL ρs , (9.122)

where ρL and ρs are respectively the density operators for the local oscillator and the

signal. This leads to the further simpli¬cation

a† as1 = a† as1 . (9.123)

L2 L2 s

L

¾

Heterodyne and homodyne detection

From eqn (9.120) we see that the expectation values of the operators F and G are

completely determined by Fs and Gs , and eqn (9.123) allows the ¬nal result to be

written as

F = eL es 2 Re r— t a† as1 s , (9.124)

L2

L

a†

G = eL es 2 Im r— t as1 , (9.125)

L2 s

L

which suggests de¬ning e¬ective ¬eld amplitudes

EL = eL aL2 , Es = es as1 . (9.126)

L s

With this notation, the expectation values of the operators F and G have the same

form as the classical quantities F and G:

—

F = ±2 |EL Es | |r| |t| sin (θL ’ θs ) ,

(9.127)

—

G = ±2 |EL Es | |r| |t| cos (θL ’ θs ) .

This formal similarity becomes an identity, if both the signal and the local oscillator

are described by coherent states, i.e. aL2 ρin = ±L ρin and as1 ρin = ±s ρin .

The result (9.127) is valid for any state, ρin , that satis¬es the factorization rule

(9.122). Let us apply this to the extreme quantum situation of the pure number state

ρs = |ns ns |. In this case Es = es as1 s = 0, and the heterodyne signal vanishes. This

re¬‚ects the fact that pure number states have no well-de¬ned phase. The same result

holds for any density operator, ρs , that is diagonal in the number-state basis. On the

other hand, for a superposition of number states, e.g.

|ψ = C0 |0 + C1 |1s , (9.128)

the e¬ective ¬eld strength for the signal is

—

Es = es ψ |as1 | ψ = es C0 C1 . (9.129)

Consequently, a nonvanishing heterodyne signal can be measured even for superposi-

tions of states containing at most one photon.

B Noise in heterodyne detection

In the previous section, we carefully included all the relevant vacuum ¬‚uctuation terms,

only to reach the eminently sensible conclusion that none of them makes any contribu-

tion to the average signal. This was not a wasted e¬ort, since we saw in Section 8.4.2

that vacuum ¬‚uctuations will add to the noise in the measured signal. We will next

investigate the e¬ect of vacuum ¬‚uctuations in heterodyne detection by evaluating the

variance,

2

V (F ) = F 2 ’ F , (9.130)

of the operator F in eqn (9.114).

¾ Photon detection

Since the calculation of ¬‚uctuations is substantially more complicated than the

calculation of averages, it is a good idea to exploit any simpli¬cations that may turn

up. We begin by using eqn (9.114) to write F 2 as

F 2 = Fs + Fs FIB + FIB Fs + FIB .

2 2

(9.131)

The image band vacuum ¬‚uctuations and the signal are completely independent, so

there should be no correlations between them, i.e. one should ¬nd

Fs FIB = Fs FIB = FIB Fs . (9.132)

Since the density operator is the vacuum for the image band modes, the absence of

correlation further implies

Fs FIB = FIB Fs = 0 . (9.133)

This result can be veri¬ed by a straightforward calculation using eqn (9.119) and the

commutativity of operators for di¬erent modes.

At this point we have the exact result

2

V (F ) = Fs + FIB ’ Fs

2 2

= V (Fs ) + V (FIB ) , (9.134)

where we have used FIB = 0 again to get the ¬nal form. A glance at eqns (9.115)

and (9.116) shows that this is still rather complicated, but any further simpli¬cations

must be paid for with approximations. Since the strong local oscillator ¬eld is typically

generated by a laser, it is reasonable to model ρL as a coherent state,

aL2 ρL = ±L ρL , ρL a† = ±— ρL , (9.135)

L

L2

with

±L = |±L | eiθL . (9.136)

The variance V (FIB ) can be obtained from V (Fs ) by the simple expedient of replacing

the signal quantities {as1 , as2 , es } by the image band equivalents {aIB1 , aIB2 , eIB }, so we

begin by using eqns (9.111), (9.119), and (9.135) to evaluate V (Fs ). After a substantial

amount of algebra”see Exercise 9.5”one ¬nds

e’2iθL V (as1 ) + CC

2 2

V (Fs ) = ’e2 |r t| |EL |

s

a† as1 ’ | as1 |

2 2 2

+ 2e2 |r t| |EL |

s s1

a† as1 ,

2 2 2 2

+ e2 |r| |EL | + (eL es ) |t| (9.137)

s s1

where |EL | = eL |±L | is the laser amplitude. We may not appear to be achieving very

much in the way of simpli¬cation, but it is too soon to give up hope.

¾

Heterodyne and homodyne detection

The ¬rst promising sign comes from the simple result

2 2

V (FIB ) = e2 |r| |EL | . (9.138)

IB

This represents the ampli¬cation”by beating with the local oscillator”of the vacuum

¬‚uctuation noise at the image band frequency. With our normalization conventions,

the energy density in these vacuum ¬‚uctuations is

ωIB

uIB = 2 0 e2 = . (9.139)

IB

V

In Section 1.1.1 we used equipartition of energy to argue that the mean thermal energy

for each radiation oscillator is kB T , so the thermal energy density would be uT =

kB T /V . Equating the two energy densities de¬nes an e¬ective noise temperature

ωIB ωL

≈

Tnoise = . (9.140)

kB kB

This e¬ect will occur for any of the phase-insensitive linear ampli¬ers studied in Chap-

ter 16, including masers and parametric ampli¬ers (Shimoda et al., 1957; Caves, 1982).

With this encouragement, we begin to simplify the expression for V (Fs ) by intro-

ducing the new creation and annihilation operators

b† (θL ) = eiθL a† , bs (θL ) = e’iθL as1 . (9.141)

s s1

This eliminates the explicit dependence on θL from eqn (9.137), but the new oper-

ators are still non-hermitian. The next step is to consider the observable quantities

represented by the hermitian quadrature operators

e’iθL as1 + eiθL a†

bs (θL ) + b† (θL )

s s1

X (θL ) = = (9.142)

2 2

and

bs (θL ) ’ b† (θL ) e’iθL as1 ’ eiθL as1

s

Y (θL ) = = . (9.143)

2i 2i

These operators are the hermitian and anti-hermitian parts of the annihilation oper-

ator:

bs (θL ) = X (θL ) + iY (θL ) , (9.144)

and the canonical commutation relations imply

i

[X (θL ) , Y (θL )] = . (9.145)

2

By writing the de¬ning equations (9.142) and (9.143) as

X (θL ) = X (0) cos θL + Y (0) sin θL ,

(9.146)

Y (θL ) = X (0) sin θL ’ Y (0) cos θL ,

the quadrature operators can be interpreted as a rotation of the phase plane through

the angle θL , given by the phase of the local oscillator ¬eld. In the calculations to

follow we will shorten the notation by X (θL ) ’ X, etc.

¿¼¼ Photon detection

After substituting eqns (9.141) and (9.144), into eqn (9.134), we arrive at

1

+ |r| |EL | e2 + e2 + |t| e2 a† as1 e2 .

2 2 2 2 2

V (F ) = 4 |r t| |EL | e2 V (Y ) ’

s s IB s L

s1

4

(9.147)

The combination V (Y ) ’ 1/4 vanishes for any coherent state, in particular for the

vacuum, so it represents the excess noise in the signal. It is important to realize that

the excess noise can be either positive or negative, as we will see in the discussion of

squeezed states in Section 15.1.2. The ¬rst term on the right of eqn (9.147) represents

the ampli¬cation of the excess signal noise by beating with the strong local oscillator

¬eld. The second term represents the ampli¬cation of the vacuum noise at the signal

and the image band frequencies. Finally, the third term describes ampli¬cation”by

beating against the signal”of the vacuum noise at the local oscillator frequency. The

2 2 2

strong local oscillator assumption can be stated as |r| |±L | |t| , so the third term is

negligible. Neglecting it allows us to treat the local oscillator as an e¬ectively classical

¬eld.

The noise terms discussed above are fundamental, in the sense that they arise

directly from the uncertainty principle for the radiation oscillators. In practice, exper-

imentalists must also deal with additional noise sources, which are called technical in

order to distinguish them from fundamental noise. In the present context the primary

technical noise arises from various disturbances”e.g. thermal ¬‚uctuations in the laser

cavity dimensions, Johnson noise in the electronics, etc.”a¬ecting the laser providing

the local oscillator ¬eld. By contrast to the fundamental vacuum noise, the technical

noise is”at least to some degree”subject to experimental control. Standard practice

is therefore to drive the local oscillator by a master oscillator which is as well controlled

as possible.