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the second-harmonic, pump light to enter the OPO, while also serving as one of the
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


θ θ + π θ + π θ θ + π

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)

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
Gated PDP
counter 11/23+

Argon-ion UV laser KDP
B Amp. & Counter
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
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.



Fig. 15.5 The derived probability p(n) for
the detection of n idler photons conditioned
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-
0 1 2 3 4
del (1986).)

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

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

Photon statistics for the displaced squeezed state—
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—
(1) Carry out the calculations required to derive eqns (15.125) and (15.126).
(2) Use these results to derive eqn (15.128).
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—

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
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
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) =
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
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)
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)
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)


Nγ (ω) = δbγ (ω) δbγ (ω) + (16.10)
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—

(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

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