re¬‚ecting surfaces de¬ning a resonant cavity for both the ¬rst- and second-harmonic

frequencies. This arrangement enhances the pump intensity inside the crystal.

By contrast, the exit mirror M2 has an extremely high re¬‚ectivity for the second-

harmonic frequency, but only a moderately high re¬‚ectivity at the ¬rst-harmonic fre-

quency. Thus the mirrors M1 and M2 form a resonator”for both the ¬rst- and second-

harmonic frequencies”but at the same time M2 allows the degenerate signal and idler

beams”at the ¬rst-harmonic frequency ω0 ”to escape toward the homodyne detector.

In Fig. 15.2, the left and right ports of the box indicating the homodyne detector

correspond to two ports of a central balanced beam splitter which respectively emit the

signal and local oscillator beams. The output ports of the beam splitter are followed

by two balanced photodetectors, and the detected outputs of the photodetectors are

then subtracted by means of a balanced di¬erential ampli¬er. Finally, the output of

the di¬erential ampli¬er is fed into a spectrum analyzer, as explained in Section 9.3.3.

It is important to emphasize that the extremely high re¬‚ectivity, for frequency ω0 ,

of the entrance mirror, M1, blocks out vacuum ¬‚uctuations from entering the system,

thereby preventing them from contributing unwanted vacuum ¬‚uctuation noise at this

frequency. As explained in Section 15.2-C, the asymmetry in the re¬‚ectivities of the

mirrors M1 and M2 at the ¬rst-harmonic frequency ω0 allows more squeezing of the

light to occur outside than inside the cavity.

The resulting data is shown in Fig. 15.3, where the output noise voltage, V (θ), of

the spectrum analyzer associated with the homodyne detector is plotted versus the

local oscillator phase θ = θLO , for a ¬xed intermediate frequency of 1.8 MHz.

The crucial comparison of this noise output is with the noise from the standard

Nonclassical states of light

8θ

θ θ + π θ + π θ θ + π

θ

Fig. 15.3 Homodyne-detector, spectrum-analyzer output noise voltage (i.e. the rms noise

voltage at an intermediate frequency of 1.8 MHz) versus the local oscillator phase. (Repro-

duced from Wu et al. (1986).)

quantum limit (SQL), which is determined either by blocking the output of the OPO,

or by changing the temperature of the lithium niobate crystal so that the signal and

idler modes are detuned away from the cavity resonance. The SQL level”which repre-

sents the noise from vacuum ¬‚uctuations”is indicated by the dashed line in this ¬gure.

By inspection of these and similar data, the authors concluded that, in the absence

of linear attenuation, the light output from the OPO would have been squeezed by a

factor e2r > 10. This means the semiminor axis of the noise ellipse of the Gaussian

Wigner function in phase space would be more than ten times the semimajor axis.

Strictly speaking, this experiment in squeezed state generation and detection did

not involve exactly degenerate photon pairs, since the detected photons were symmet-

rically displaced from exact degeneracy by 1.8 MHz (within a bandwidth of 100 kHz).

The exact conservation of energy in parametric down-conversion guarantees that the

shifts in the two frequencies are anti-correlated, i.e. ωi = ω0 + ∆ωi and ωs = ω0 + ∆ωs ,

with ∆ωi = ’∆ωs . Thus the beat notes produced by interference of the upper and

lower sidebands with the local oscillator are exactly the same. Both sidebands are de-

tected in the balanced homodyne detector, but their phases are correlated in just such

a way that for one particular phase θLO of the local oscillator”which can be adjusted

by the piezoelectric translator that controls the location of the mirror MLO ”the sen-

sitive spots of homodyne detection coincide with the least noisy quadrature of the

squeezed light. This is true in spite of the fact that the two conjugate photons may

not be exactly degenerate in frequency, as long as they are inside the gain-narrowed

line width of the optical parametric ampli¬cation pro¬le just below threshold. The

noise analysis for this case of slightly nondegenerate parametric down-conversion can

be found in Kimble (1992).

Number states

15.4 Number states

We have seen in Section 2.1.2 that the number states provide a natural basis for the

Fock space of a single mode of the radiation ¬eld. Any state, whether pure or mixed,

can be expressed in terms of number states. By de¬nition, the variance of the number

operator vanishes for a number state |n ; so evaluating eqn (9.58) for the Mandel

Q-parameter of the number state |n gives

V (N ) ’ N

Q (|n ) ≡ = ’1 , (15.132)

N

where X = n |X| n . Thus the number states saturate the general inequality

Q (|Ψ ) ’1. Furthermore, every state with negative Q is nonclassical; consequently,

a pure number state is as nonclassical as it can be. Since this is true no matter how

large n is, the classical limit cannot be identi¬ed with the large-n limit. Further ev-

idence of the nonclassical nature of number states is provided by eqn (5.153), which

shows that the Wigner distribution W (±) for the single-photon number state |1 is

negative in a neighborhood of the origin in phase space.

15.4.1 Single-photon wave packets from SDC

States containing exactly one photon in a classical traveling-wave mode, e.g. a Gaussian

wave packet, are of particular interest in contemporary quantum optics. In the approx-

imate sense discussed in Section 7.8 the photon is localized within the wave packet.

With almost complete certainty, such a single-photon wave packet state would register

a single click when it falls on an ideal photodetector with unit quantum e¬ciency.

The ¬rst experiment demonstrating the existence of single-photon wave packet

states was performed by Hong and Mandel (1986). The single-photon state is formed

by one of the pair of photons emitted in spontaneous down-conversion, using the appa-

ratus shown in Fig. 15.4. An argon-ion UV laser beam at a wavelength of » = 351 nm

enters a crystal”potassium dihydrogen phosphate (KDP)”with a χ(2) nonlinearity.

Conjugate down-converted photon pairs are generated on opposite sides of the UV

Amp. & Counter

disc.

PMT

A

Interference

filter

Gated PDP

UV

counter 11/23+

filter

MCP

Argon-ion UV laser KDP

B Amp. & Counter

disc.

Lens Field Interference

lens filter

Fig. 15.4 Schematic of Hong and Mandel™s experiment to generate and detect single-photon

wave packets. (Reproduced from Hong and Mandel (1986).)

Nonclassical states of light

beam wavelength at the signal and idler wavelengths of 746 nm and 659 nm, respec-

tively, and enter the photon counters A and B. Counter B is gated by the pulse derived

from counter A, for a counting time interval of 20 ns.

Whenever a click is registered by counter A”and the less-than-unity quantum

e¬ciency of counter B is accounted for”there is one and only one click at counter B.

This is shown in Fig. 15.5, in which the derived probability p(n) for a count at counter

B”conditioned on the detection of a signal photon at counter A”is plotted versus

the photon number n.

The data show that within small uncertainties (indicated by the cross-hatched

regions),

p(n) = δn,1 ; (15.133)

that is, the idler photons detected by B have been prepared in the single-photon

number state |n = 1 . In other words, the moment that the click goes o¬ in counter A,

one can, with almost complete certainty, predict that there is one and only one photon

within a well-de¬ned wave packet propagating in the idler channel. The Mandel Q-

parameter derived from these data, Q = ’1.06 ± 0.11, indicates that this state of light

is maximally nonclassical, as expected for a number state.

15.4.2 Single photons on demand

The spontaneous down-conversion events that yield the single-photon wave packet

states occur randomly, so there is no way to control the time of emission of the wave

packet from the nonlinear crystal. Recently, work has been done on a controlled pro-

duction process in which the time of emission of a well-de¬ned single-photon wave

packet is closely determined. Such a deterministic emission process for an individual

photon wave packet is called single photons on demand or a photon gun. One such

method involves quantum dots placed inside a high-Q cavity. When a single electron is

controllably injected into the quantum dot”via the Coulomb blockade mechanism”

the resonant enhancement of the rate of spontaneous emission by the high-Q cavity

produces an almost immediate emission of a single photon. Deterministic production

of single-photon states can be useful for quantum information processing and quantum

computation, since often the photons must be synchronized with the computer cycles

in a controllable manner.

p(n)

1.0

Fig. 15.5 The derived probability p(n) for

the detection of n idler photons conditioned

0.5

on the detection of a single signal photon in

the 1986 experiment of Hong and Mandel. The

cross-hatched regions indicate the uncertain-

ties of p(n). (Reproduced from Hong and Man-

n

0 1 2 3 4

del (1986).)

Exercises

15.4.3 Number states in a micromaser

Number states have been produced in a standing-wave mode inside a cavity, as opposed

to the traveling-wave packet described above. In the microwave region, number states

inside a microwave cavity have been produced by means of the micromaser described

in Section 12.3. This is accomplished by two methods described below.

In the ¬rst method, a completed measurement of the ¬nal state of the atom after it

exits the cavity allows the experimenter to know”with certainty”whether the atom

has made a downwards transition inside the cavity. Combining this knowledge with

the conservation of energy determines”again with certainty”the number state of the

cavity ¬eld.

In the second method, an exact integer number of photons is maintained inside

the cavity by means of a trapping state (Walther, 2003). According to eqn (12.21),

the e¬ective Rabi frequency for an on-resonance, n-photon state is „¦n = 2g (n + 1),

where g is the coupling constant of the two-level atom with the cavity mode. The

√

Rabi period is therefore Tn = 2π/„¦n = π/ g n + 1 . If the interaction time Tint of

the atom with the ¬eld satis¬es Tint = kTn , where k is an integer, then an atom that

enters the cavity in an excited state will leave in an excited state. Thus the number of

photons in the cavity will be unchanged”i.e. trapped ”if the condition

√

n + 1gTint = kπ (15.134)

is satis¬ed.

Trapping states are characterized by the number of photons remaining in the cav-

ity, and the number of Rabi cycles occurring during the passage of an atom through the

cavity. Thus the trapping state (n, k) = (1, 1) denotes a state in which a one-photon,

one-Rabi-oscillation trapped ¬eld state is maintained by a continuous stream of Ry-

dberg atoms prepared in the upper level. Experiments show that, under steady-state

excitation conditions, the one-photon cavity state is stable. Although this technique

produces number states of microwave photons in a beautifully simple and clean way, it

is di¬cult to extract them from the high-Q superconducting cavity for use in external

experiments.

15.5 Exercises

15.1 Quadrature variances

(1) Use eqn (15.14) and the canonical commutation relations to calculate : X 2 : and

to derive eqns (15.17) and (15.18).

(2) Are the conditions (15.19) and (15.20) su¬cient, as well as necessary? If not, what

are the su¬cient conditions?

(3) Explain why number states and coherent states are not squeezed states.

(4) Is the state |ψ = cos θ |0 + sin θ |1 squeezed for any value of θ? In other words,

for a given θ, is there a quadrature X with VN (X) < 0?

Nonclassical states of light

15.2 Squeezed number state

Number states are not squeezed, but it is possible to squeeze a number state. Consider

|ζ, n = S (ζ) |n .

(1) Evaluate the Mandel Q-parameter for this state and comment on the result.

(2) What quadrature exhibits maximum squeezing?

Displaced squeezed states and squeezed coherent states—

15.3

Use the properties of S (ζ) and D (±) to derive the relations (15.52)“(15.54).

Photon statistics for the displaced squeezed state—

15.4

Carry out the integral in eqn (15.67) using polar coordinates and combine this with

the other results to get eqn (15.69).

Squeezing of emitted light—

15.5

(1) Carry out the calculations required to derive eqns (15.125) and (15.126).

(2) Use these results to derive eqn (15.128).

16

Linear optical ampli¬ers—

Generally speaking, an optical ampli¬er is any device that converts a set of input modes

into a set of output modes with increased intensity. The only absolutely necessary

condition is that the creation and annihilation operators for the two sets of modes

must be connected by a unitary transformation. Paradoxically, this level of generality

makes it impossible to draw any general conclusions; consequently, further progress

requires some restriction on the family of ampli¬ers to be studied.

To this end, we consider the special class of unitary input“output transformations

that can be expressed as follows. The annihilation operator for each output mode

is a linear combination, with c-number coe¬cients, of the creation and annihilation

operators for the input modes. Devices of this kind are called linear ampli¬ers. We

note in passing that linear ampli¬ers are quite di¬erent from laser oscillator-ampli¬ers,

which typically display the highly nonlinear phenomenon of saturation (Siegman, 1986,

Sec. 4.5).

For typical applications of linear, optical ampli¬ers”e.g. optical communication or

the generation of nonclassical states of light”it is desirable to minimize the noise added

to the input signal by the ampli¬er. The ¬rst source of noise is the imperfect coupling

of the incident signal into the ampli¬er. Some part of the incident radiation will be

scattered or absorbed, and this loss inevitably adds partition noise to the transmitted

signal. In the literature, this is called insertion-loss noise, and it is gathered together

with other e¬ects”such as noise in the associated electronic circuits”into the category

of technical noise. Since these e¬ects vary from device to device, we will concentrate

on the intrinsic quantum noise associated with the act of ampli¬cation itself.

In the present chapter we ¬rst discuss the general properties of linear ampli¬ers

and then describe several illustrative examples. In the ¬nal sections we present a

simpli¬ed version of a general theory of linear ampli¬ers due to Caves (1982), which

is an extension of the earlier work of Haus and Mullen (1962).

16.1 General properties of linear ampli¬ers

The degenerate optical parametric ampli¬er (OPA) studied in Section 15.2 is a linear

device, by virtue of the assumption that depletion of the pump ¬eld can be neglected. In

the application to squeezing, the input consists of vacuum ¬‚uctuations”represented

by b2,in (t)”entering the mirror M2, and the corresponding output is the squeezed

state”represented by b2,out (t)”emitted from M2. Both the input and the output

have the carrier frequency ω0 . Rather than extending this model to a general theory

of linear ampli¬ers that allows for multiple inputs and outputs and frequency shifts

Linear optical ampli¬ers—

¼¼

between them, we choose to explain the basic ideas in the simplest possible context:

linear ampli¬ers with a single input ¬eld and a single output ¬eld”denoted by bin (t)

and bout (t) respectively”having a common carrier frequency.

We will also assume that the characteristic response frequency of the ampli¬er and

the bandwidth of the input ¬eld are both small compared to the carrier frequency. This

narrowband assumption justi¬es the use of the slowly-varying amplitude operators

introduced in Chapter 14, but it should be remembered that both the input and the

output are reservoir modes that do not have sharply de¬ned frequencies. Just as in the

calculation of the squeezing of the emitted light in Section 15.2, the input and output

are described by continuum modes.

All other modes involved in the analysis are called internal modes of the ampli¬er.

In the sample“reservoir language, the internal modes consist of the sample modes and

any reservoir modes other than the input and output. A peculiarity of this jargon is

that some of the ˜internal™ modes are ¬eld modes, e.g. vacuum ¬‚uctuations, that exist

in the space outside the physical ampli¬er.

The de¬nition of the ampli¬er is completed by specifying the Heisenberg-picture

density operator ρ that describes the state of both the input ¬eld and the internal

modes of the ampli¬er. This is the same thing as specifying the initial value of the

Schr¨dinger-picture density operator. Since we want to use the ampli¬er for a broad

o

range of input ¬elds, it is natural to require that the operating state of the ampli¬er is

independent of the incident ¬eld state. This condition is imposed by the factorizability

assumption

ρ = ρin ρamp , (16.1)

where ρin and ρamp respectively describe the states of the input ¬eld and the ampli¬er.

In the generic states of interest for communications, the expectation value of the

input ¬eld does not vanish identically:

bin (t) = Tr [ρin bin (t)] = 0 . (16.2)

Situations for which bin (t) = 0 for all t”e.g. injecting the vacuum state or a

squeezed-vacuum state into the ampli¬er”are to be treated as special cases.

The identi¬cation of the measured values of the input and output ¬elds with the

expectation values bin (t) and bout (t) runs into the apparent di¬culty that the

annihilation operators bin (t) and bout (t) do not represent measurable quantities. To

see why this is not really a problem, we recall the discussion in Section 9.3, which

showed that both heterodyne and homodyne detection schemes e¬ectively measure a

hermitian quadrature operator. For example, it is possible to measure one member of

the conjugate pair (Xβ,in (t) , Yβ,in (t)), where

1 ’iβ †

e bin (t) + eiβ bin (t) ,

Xβ,in (t) =

2 (16.3)

1 ’iβ †

e bin (t) ’ eiβ bin (t) .

Yβ,in (t) =

2i

The quadrature angle β is determined by the relative phase between the input signal

and the local oscillator employed in the detection scheme. The operational signi¬cance

¼½

General properties of linear ampli¬ers

of the complex expectation value bin (t) is established by carrying out measurements

of Xβ,in (t) for several quadrature angles and using the relation

1 ’iβ †

= Re e’iβ bin (t)

bin (t) + eiβ bin (t)

Xβ,in (t) = e . (16.4)

2

With this reassuring thought in mind, we are free to use the algebraically simpler

approach based on the annihilation operators. An important example is provided by

the phase transformation,

bin (t) ’ bin (t) = e’iθ bin (t) , (16.5)

of the annihilation operator. The corresponding transformation for the quadratures,

Xβ,in (t) ’ Xin (t) = Xβ,in (t) cos θ + Yβ,in (t) sin θ , (16.6)

Yβ,in (t) ’ Yin (t) = Yβ,in (t) cos θ ’ Xβ,in (t) sin θ , (16.7)

represents a rotation through the angle θ in the (X, Y )-plane. As explained in Section

8.1, these transformations are experimentally realized by the use of phase shifters.

16.1.1 Phase properties of linear ampli¬ers

From Section 14.1.1-C, we know that the noise properties of the input/output ¬elds are

described by the correlation functions of the ¬‚uctuation operators, δbγ (ω) ≡ bγ (ω) ’

bγ (ω) , where γ = in, out. Thus the input/output noise correlation functions are

de¬ned by

1 † †

Kγ (ω, ω ) = δbγ (ω) δbγ (ω ) + δbγ (ω ) δbγ (ω) (γ = in, out) . (16.8)

2

The de¬nitions (14.98) and (14.107) relating the input/output ¬elds to the reservoir

operators allow us to apply the conditions (14.27) and (14.34) for phase-insensitive

noise. The input/output noise reservoir is phase insensitive if the following conditions

are satis¬ed.

(1) The noise in di¬erent frequencies is uncorrelated, i.e.

Kγ (ω, ω ) = Nγ (ω) 2πδ (ω ’ ω ) , (16.9)

where

1

†

Nγ (ω) = δbγ (ω) δbγ (ω) + (16.10)

2

is the noise strength.

(2) The phases of the ¬‚uctuation operators are randomly distributed, so that

δbγ (ω) δbγ (ω ) = 0 . (16.11)

With this preparation, we are now ready to introduce an important division of the

family of linear ampli¬ers into two classes. A phase-insensitive ampli¬er is de¬ned

by the following conditions.

Linear optical ampli¬ers—

¼¾

2

(i) The output ¬eld strength, bout (ω) , is invariant under phase transformations

of the input ¬eld.

(ii) If the input noise is phase insensitive, so is the output noise.

Condition (i) means that the only e¬ect of a phase shift in the input ¬eld”i.e. a rota-

tion of the quadratures”is to produce a corresponding phase shift in the output ¬eld.

Condition (ii) means that the noise added by the ampli¬er is randomly distributed in

phase. An ampli¬er is said to be phase sensitive if it fails to satisfy either one of

these conditions.

In addition to the categories of phase sensitive and phase-insensitive, ampli¬ers

can also be classi¬ed according to their physical con¬guration. In the degenerate OPA

the gain medium is enclosed in a resonant cavity, and the input ¬eld is coupled into

one of the cavity modes. The cavity mode in turn couples to an output mode to

produce the ampli¬ed signal. This con¬guration is called a regenerative ampli¬er,

which is yet another term borrowed from radio engineering. One way to understand

the regenerative ampli¬er is to visualize the cavity mode as a traveling wave bouncing

back and forth between the two mirrors. These waves make many passes through the

gain medium before exiting through the output port.

The advantage of greater overall gain, due to multiple passes through the gain

medium, is balanced by the disadvantage that the useful gain bandwidth is restricted

to the bandwidth of the cavity. This restriction on the bandwidth can be avoided

by the simple expedient of removing the mirrors. In this con¬guration, there are no