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Nonclassical light generation in quasi-phase-matched parametric self-

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Article  in  Journal of Experimental and Theoretical Physics · January 2004

DOI: 10.1134/1.1842876

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Journal of Experimental and Theoretical Physics, Vol. 99, No. 5, 2004, pp. 947–957.
Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 126, No. 5, 2004, pp. 1089–1100.
Original Russian Text Copyright © 2004 by Novikov, Chirkin.

ATOMS, SPECTRA,
RADIATION

Nonclassical Light Generation

in Quasi-Phase-Matched Parametric
Self-Frequency Conversion
A. A. Novikov and A. S. Chirkin*
Department of Physics, Moscow State University, Moscow, 119992 Russia
*e-mail: chirkin@squeez.phys.msu.su, aschirkin@pisem.net
Received April 30, 2004

Abstract—We present a quantum theory of the parametric self-conversion of the laser radiation frequency in
active nonlinear crystals with a regular domain structure. Such crystals feature simultaneous lasing and quasi-
phase-matched parametric conversion of the laser radiation frequency. These processes are described using the
Heisenberg–Langevin equations in two regimes of the subharmonic generation: super- and subthreshold. The
spectral properties of the quadrature components of the laser frequency and its subharmonic and the photon sta-
tistics have been studied as dependent on the pump power, crystal length, and reflectance of the laser cavity
output mirror. Using the obtained analytical expressions, these characteristics are calculated for a active non-
linear Nd:Mg:LiNbO3 crystal with a regular domain structure. In the subthreshold regime, the maximum
decrease in the spectral density of fluctuations in the subharmonic quadrature component relative to the stan-
dard quantum limit may reach 90%; in the above-threshold regime, these fluctuations are virtually not sup-
pressed. A decrease in the spectral density of fluctuations of the laser frequency quadrature does not exceed
10%. In the subthreshold excitation regime, the subharmonic photons obey a super-Poisson statistics; in the
above-threshold regime, the photon statistics is Poisson-like. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION dopant ions provide for active (lasing) properties, while

In recent years, much attention has been devoted to the crystalline matrix plays the role of a nonlinear
ferroelectric crystals possessing periodically inhomo- medium. This system may feature the phenomenon of
geneous nonlinear properties. In such media, the direc- self-frequency conversion, whereby lasing proceeds
tion of the spontaneous polarization vector changes to simultaneously with nonlinear conversion of the laser
the opposite on passage from one domain to another. frequency. Investigations into the properties of active
This may lead to reversal of the sign of the coefficient nonlinear laser crystals are stimulated by the wide
of quadratic nonlinear susceptibility [1, 2], whereby a potential range of applications of compact and reliable
“nonlinear” lattice is formed in the crystal possessing lasers based on such crystals, generating in the visible
homogeneous linear properties. Nonlinear optical crys- and IR spectral range.
tals with a regular domain structure, called RDS crys- The possibilities of self-frequency conversion in
tals or nonlinear photonic (periodically poled) crystals, homogeneous active nonlinear crystals are limited by
have certain advantages over homogeneous nonlinear their dispersion properties. As was noted above, these
crystals. The main advantage is that, by selecting the limitations can be by-passed in the presence of spatially
period of modulation of the nonlinear susceptibility in modulated nonlinear susceptibility. Recently [8–11], it
an RDS crystal, it is possible to compensate mismatch was demonstrated that RDS crystals combining the
of the interacting light waves (quasi-phase-matched advantages of active nonlinear media with periodically
interaction), thus providing conditions for almost arbi- inhomogeneous nonlinear properties are of interest
trary three-wave mixing. At present, the quasi-phase- from the standpoint of realization of various three-wave
matched interactions between light waves are used for interactions, whereby one of the waves can be
generating coherent radiation in a broad spectral range enhanced due to the active properties of the crystal.
from UV to IR. In addition, the quasi-phase-matched Such processes provide a basis for the creation of min-
interactions are of interest from the standpoint of gen- iature self-frequency-conversion lasers.
erating nonclassical light (squeezed light and entangled This study was devoted to the quantum properties of
photon states) [3–6]. Nonclassical light can be used in light generated during the parametric self-frequency
various high-precision optical measurements and in conversion of laser radiation. In the course of this pro-
optical data transmission and processing systems [7]. cess taking place in a resonator based on a active non-
Active nonlinear RDS crystals open new prospects linear RDS crystal, lasing proceeds simultaneously
in nonlinear optics [8]. In such crystals, rare earth with the parametric conversion of the laser frequency.

1063-7761/04/9905-0947$26.00 © 2004 MAIK “Nauka/Interperiodica”

948 NOVIKOV, CHIRKIN

ω1 + ω2 = ω3 is devoted to analysis of the subharmonic photon statis-

Pumping ω1 tics in the subthreshold generation regime.
ω2
ω3
1 3
2. LASER
AND NONLINEAR OPTICAL EQUATIONS
2
The process of parametric self-frequency conver-
Fig. 1. Schematic diagram of ring cavity involving three
sion will be analyzed according to the following
mirrors (1–3) and an active nonlinear RDS crystal featuring scheme. First, we will separately describe laser genera-
three-wave interaction. tion in an active crystal and three-wave interactions in
a nonlinear optical crystal. Then, the two descriptions
will be combined to yield the joint system of equations
In the case under consideration, we are speaking of the for the field operators and atoms of the medium, which
second subharmonic generation. Up to now, quantum describes processes in a nonlinear active crystal. The
theory has been well developed separately for the lasing system will be solved using the Heisenberg–Langevin
and the parametric frequency conversion processes method [12–14], which is known to be especially con-
(see, e.g., monographs [12–14]). However, in the case venient for calculating the correlation functions of
of self-frequency conversion, the two processes are cor- operators and, hence, of the spectral densities.
related. This circumstance alters the physics of this
phenomenon and complicates its theoretical analysis. The process of laser and parametric generation is
studied for a nonlinear active RDS crystal situated in a
Previously [15, 16], the quantum theory of paramet- ring cavity (Fig. 1) provided with two mirrors (1 and 2)
ric self-frequency conversion was developed for gas totally reflecting radiation at the excited frequencies
lasers. In these lasers, the photon lifetime T in the laser and with an output mirror (3). The crystal is pumped
cavity is much longer than the characteristic times of through mirror 1; mirrors 2 and 3 are fully transparent
the inverse population relaxation (T||) and the active to pump radiation.
medium polarization (T⊥): T Ⰷ T⊥, T|| . This condition
allows the inverse population and polarization to be
adiabatically excluded from the equations describing 2.1. Laser Generation in an Active Crystal
generation of the laser radiation. However, the afore-
mentioned active nonlinear crystals are characterized The process of laser generation in an active crystal
by a different relation between the characteristic times: will be first considered without allowance for nonlinear
T Ⰶ T⊥ Ⰶ T|| . For this reason, the results of previous optical properties. We use a quantum approach based
theoretical analysis [15, 16] are inapplicable to such on the Heisenberg operator equation, generally follow-
solid-state laser systems. ing the scheme [12]. Excluding operators related to the
thermal reservoir and modeling the laser field interac-
In this study, a quantum analysis of the parametric tion with other physical systems (except atoms) from
self-frequency conversion is performed for an arbitrary the system of operator equations for the field and atoms
relation between the characteristic times. We will con- of the medium, we obtain the system of Heisenberg–
sider the spectrum of fluctuations of the quadrature Langevin equations [12]:
components and the photon statistics of the laser radia-
tion and subharmonic fields. The general formulas will da l
be used in numerical calculations of the parametric ------- = – k l a l – iT C gσ + F l ( t ), (1)
dt
self-frequency conversion in a active nonlinear
Nd:Mg:LiNbO3 crystal with a regular domain structure. dσ 1
------ = ----- ( – σ + igT ⊥ a l N ) + Γ ( t ), (2)
This paper is organized as follows. In Section 2, the dt τ⊥
Heisenberg–Langevin equations are written separately
for the laser generation and the quasi-phase-matched dσ
†
1
--------- = ----- ( – σ ) – i gT ⊥ a l N ) + Γ ( t ) ,
† † †
three-wave interactions. The parametric self-frequency (3)
dt τ⊥
conversion is analyzed in Section 3, where the corre-
sponding Heisenberg–Langevin equations are written
dN 1
------- = ---- [ P – N + i2T || g ( σa l – σ a l ) ] + Γ N ( t ). (4)
and solved by perturbation method. Section 4 presents † †

the results of calculating the spectrum of fluctuations of dt τ ||

the quadrature components of the laser radiation and its
subharmonic in the sub- and above-threshold regimes These equations are written in the interaction repre-
of second subharmonic generation in a active nonlinear sentation using the rotating wave approximation. The
Nd:Mg:LiNbO3 crystal. In Section 5, the main attention laser field frequency ωl is assumed to coincide with the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 99 No. 5 2004

 NONCLASSICAL LIGHT GENERATION 949

frequency ω0 of transitions between the laser levels We also introduce the operators bj and cj related to
(ω0 = ωl). In Eqs. (1)–(4), the modes with the frequencies ωj , which are defined as
follows (see, e.g., [12]):
M M
[ b j ( t ), b k ( t' ) ] = [ c j ( t ), c k ( t' ) ] = δ jk δ ( t – t' ),
† †

∑ ∑
(7)
σ = σν , N = Nν
〈 b j ( t )〉 = 〈 b j ( t )〉 = 〈 c j ( t )〉 = 〈 c j ( t )〉 = 0,
† †
ν=1 ν=1 (8)
〈 b j ( t )b k ( t' )〉 = 〈 b j ( t )b k ( t' )〉
† †
are operators of the polarization and the population dif-
ference for a laser- active medium containing M atoms, (9)
= 〈 c j ( t )c k ( t' )〉 = 〈 c j ( t )c k ( t' )〉 = 0,
† †
†
respectively; a l (al) is the operator of creation (annihi-
lation) of a photon with the frequency ωl; 〈 b j ( t )〉 b k ( t' )〉 = 〈 c j ( t )〉 c k ( t' )〉 = n j ( T )δ jk δ ( t – t' ),(10)
† †

where j, k = 1, 2, 3 (operators with different subscripts

cបω l refer to the waves of different frequencies), δjk is the
g = -----------------------
- Kronecker delta, and
4I S V T || T ⊥
បω –1
n j ( T ) =  exp ---------j – 1
is the constant of interaction between an atom and the  kT 
electromagnetic field; T|| and T⊥ are the times of relax-
ation of the inverse population and the polarization of is the average number of thermal phonons for the jth
medium, respectively; IS is the saturation intensity of mode at the thermal reservoir temperature T (at room
the active medium; V is the quantization volume deter- temperature, n j (T) Ⰶ 1).
mined by the transverse size of the pump beam and the According to relations (6)–(10),
crystal length; c is the speed of light in vacuum; TC is
[ F l ( t ), F l ( t' ) ] = 2k l δ ( t – t' ),
†
the cavity round trip time by wave; t is the dimension- (11)
less time representing the current time normalized to TC
〈 F l ( t )〉 = 〈 F l ( t )〉 = 0,
†
(t t/TC); τ|| = T||/TC; τ⊥ = T⊥/TC; (12)
2
〈 F l ( t )〉 = 〈 ( F l ( t ) ) 〉 = 0,
2 †
(13)
kl(1 + η )
P = ---------------------
-
〈 F l ( t )F l ( t' )〉 = 2k l n l δ ( t – t' ).
2 †
g T CT ⊥ (14)
The noise operators Γ(t) and ΓN(t) possess the fol-
is the pump parameter; and η is the excess pump power lowing properties [12]:
(absorbed in the active medium) over threshold (the lat-
〈 Γ ( t )F l ( t' )〉 = 0, 〈 Γ ( t )F l ( t' )〉 = 0,
†
ter corresponds to η = 0, and the lasing condition is (15)
η > 0). Operators Γ(t), ΓN (t), and Fl (t) are the operators
〈 Γ N ( t )F l ( t' )〉 = 0, 〈 Γ N ( t )F l ( t' )〉 = 0,
†
of noise related to the polarization, inverse population (16)
of the active medium, and losses, respectively. The
〈 Γ ( t )Γ ( t' )〉 = 0, 〈 Γ ( t )Γ ( t' )〉 = 0,
† †
(17)
appearance of these operators and the terms τ ⊥ σ,
–1

〈 Γ ( t )Γ ( t' )〉
†
τ ⊥ σ†, τ || (P – N), and klal in Eqs. (1)–(4) reflects the
–1 –1

T T (18)
=  -----C- ( M + N 0 ) + ------C- ( P – N 0 ) δ ( t – t' ),
interaction with thermal reservoir. The coefficient kl is
given by the formula  τ⊥ 2τ || 

αl L + 1 – Rl 〈 Γ ( t )Γ ( t' )〉
†
k l = ----------------------------
-, (5)
2 T T (19)
=  -----C- ( M – N 0 ) – ------C- ( P – N 0 ) δ ( t – t' ),
 τ⊥ 2τ || 
where αl are linear losses in the crystal, Rl is the coeffi-
cient of intensity reflection of the cavity output mirror 2T
〈 Γ N ( t )Γ N ( t' )〉 = --------C-  M – ----------0 δ ( t – t' ),
PN
at the frequency ωl , and L is the laser crystal length. τ ||  M 
(20)
The random force operator Fl is defined as [13, 17]
T
〈 Γ ( t )Γ N ( t' )〉 = – -----C-  1 + ----- σ *0 δ ( t – t' ),
† P
(21)
Fl ( t ) = α l Lb l ( t ) + 1 – R l c l ( t ), (6) τ ||  M
T P
where bl and cl are operators related to the losses in the 〈 Γ ( t )Γ N ( t' )〉 = – -----C-  ----- – 1 σ 0 δ ( t – t' ), (22)
crystal and in the cavity output mirror, respectively. τ ||  M 

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 99 No. 5 2004

950 NOVIKOV, CHIRKIN

where N0 and σ0 are the stationary values of the inverse In deriving these equations, we excluded operators
population and the medium polarization operators and related to the thermal reservoir (this is achieved using a
P is the pump power (see formulas (32) below). procedure similar to that used in deriving laser equa-
Using the equations presented in this section, it is tions (1)–(4)). The coefficients kl and the fluctuation
possible to analyze the process of laser generation in an operators Fl(t) are given by formulas (5) and (6),
active crystal with allowance for losses in the crystal respectively. Taking into account these relations and
and in the output mirror. following the conventional procedure (see, e.g., [20]),
one may readily check that commutation relations for
†
the operators a j and aj are retained with time, so that
2.2. Quasi-Phase-Matched Wave Interactions
in a Nonlinear Crystal
[ a j ( t ), a k ( t ) ] = δ jk .
†
(26)
Now let us proceed to the three-wave process (ω3 =
ω1 + ω2) in a nonlinear optical RDS crystal situated in The quantum equations (24) and (25) describe the
process of three-wave nonlinear optical interactions in
a ring cavity. With neglect of losses, the Hamiltonian of
a crystal situated in the laser cavity with losses. The
this interaction is
joint system of equations (1)–(4) describing the laser
H NL = បε ( Qa 3 a 2 a 1 + Q*a 1 a 2 a 3 ),
† † †
(23) generation process and Eqs. (24)–(25) describing wave
interactions in the nonlinear optical medium is a basis
† of the quantum theory of the process of optical self-fre-
where a j (aj) are the operators of creation (annihila- quency conversion.
tion) of a photon with the frequency ωj ,

8πបω 1 ω 2 ω 3 3. PARAMETRIC LASER FREQUENCY

ε = πLd eff -----------------------------
2
- SELF-CONVERSION
n1 n2 n3 c V IN AN ACTIVE NONLINEAR CRYSTAL
is the nonlinear wave coupling coefficient (nj being the Let us consider the parametric self-down-conver-
refractive index for the wave with the frequency ωj , and sion of the laser frequency in an active nonlinear crys-
deff being an effective nonlinearity coefficient depen- tal. The process involves simultaneous lasing and
dent on the polarizations of interacting waves), and Q is quasi-phase-matched division of the laser frequency
the factor taking into account the RDS parameters of the (i.e., generation of the subharmonic with the frequency
crystal. The latter quantity is given by the formula [18]: ω/2). In the three-wave interaction, we have ω1 = ω2 =
ω/2 and ω3 = ω, so that the system of equations describ-
∆kL ∆kΛ
Q = exp  – ------------ sin c ---------- tan ----------- ,
i∆kL ing this process is as follows:
 2  2 4 da 1
-------- = – k 1 a 1 + i2εa 1 a 3 + F 1 ( t ),
†
(27)
where L = SΛ is the crystal length, S is the number of dt
RDS periods in the given crystal, Λ is the period of
modulation of the nonlinear susceptibility (RDS da
--------3 = – k 3 a 3 – iT C gσ + iεa 1 + F 3 ( t ),
2
(28)
period), dt
∆k = k ( ω 3 ) – k ( ω 2 ) – k ( ω 1 ) dσ 1
------ = ----- ( – σ + igT ⊥ a 3 N ) + Γ ( t ), (29)
is the phase mismatch, k(ωj) is the wavenumber corre- dt τ⊥
sponding to the frequency ωj . If the modulation period dN 1
------- = ---- ( P – N + i2gT || ( σa 3 – σ a 3 ) ) + Γ N ( t ). (30)
† †
Λ is much shorter than the characteristic nonlinear dt τ ||
interaction length and the quasi-phase-matching condi-
tion is valid (∆k = 2πm/Λ, where m is an odd integer Equations (27)–(30) are obtained by combining the
indicating the phase-matching order), the wave interac- system of nonlinear optical equations (24), (25) with
tion in the RDS crystal proceeds in the same way as in the system of equations (1)–(4) describing laser gener-
a homogeneous medium [2, 19]. In this case, Q = 2/πm. ation. In addition, we take into account that the wave
According to Eq. (23), the system of Heisenberg– with the frequency ω3 = ω is enhanced in the active
Langevin equations for the nonlinear process under medium and involved in the nonlinear interaction. Sub-
consideration with allowance for losses is as follows: script 1 refers to the ω/2 subharmonic wave.
Since the analysis of the nonlinear system of equa-
da 1, 2
- = – k 1, 2 a 1, 2 + iεa 2, 1 a 3 + F 1, 2 ( t ),
† tions (27)–(30) in the general case is impossible, we
----------- (24)
dt will only analyze two typical regimes from the stand-
point of the parametric process under consideration:
da subthreshold and above-threshold generation of the
--------3 = – k 3 a 3 + iεa 1 a 2 + F 3 ( t ). (25)
dt subharmonic mode. Let the pump power exceed the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 99 No. 5 2004

 NONCLASSICAL LIGHT GENERATION 951

laser generation threshold. Then, the atomic and field change in the stationary laser generation amplitude.
operators can be represented as follows: This variation is cased by additional losses of the laser
wave energy for the generation of the subharmonic
a j ( t ) = A j0 + δa j ( t ), σ ( t ) = σ 0 + δσ ( t ), wave.
(31) Now we will analyze the spectra of fluctuations of
N ( t ) = N 0 + δN ( t ),
the laser radiation and its subharmonic in the two limit-
where Aj0 , σ0 , and N0 are the classical quantities corre- ing excitation regimes.
sponding to the stationary solution of Eqs. (27)–(30) in
the absence of random forces, while δaj(t), δσ(t), and 3.1. Subharmonic Generation
δN(t) are the operators taking into account fluctuations. in Above-Threshold Regime
As can be readily seen, the quantities Aj0 , σ0 , and N0 are The above-threshold regime of subharmonic gener-
in fact the stationary average values, because the oper- ation corresponds to stationary solutions of the type
ator forces F1(t) and F3(t) are mutually uncorrelated given by formulas (32). In this case, system (27)–(30)
and are not correlated with the fluctuation operators linearized in the vicinity of the stationary solution leads
Γ(t) and ΓN(t). to the following equations for the fluctuation operators:
A stationary solution of the system (27)–(30) has the
following form: d ( δa 1 )
- = – k 1 δa 1 + i2ε( A 30 δa 1 + A 10 δa 3) + F 1(t), (34)
†
---------------
dt
ik 1 P
A 30 = – ------, N 0 = --------------------------------------
-2 ,
2ε 2 2
1 + k 1 g T || T ⊥ /ε d ( δa 3 )
- = – k 3 δa 3 – iT C gδσ + iε2 A 10 δa 1 + F 3(t), (35)
---------------
dt
k 1 gT ⊥ P
σ 0 = --------------------------------------------, (32) d ( δσ ) 1
2 ( ε + k 1 g T || T ⊥ /ε )
2 2
--------------- = – ----- δσ + iT C g ( N 0 δa 3 + A 30 δN ) + Γ ( t ), (36)
dt τ⊥
1/2
 k 1 k 3 ( η – k 21 g 2 T || T ⊥ /ε 2 ) d ( δN ) 1
A 10 = ±  -----------------------------------------------------
- . --------------- = – ---- δN + i2T C g
 2 ( ε + k 1 g T || T ⊥ ) 
2 2 2 dt τ || (37)
× ( σ 0 δa 3 A 30 δσ – σ 0 δa 3 – A 30 δσ ) + Γ N ( t ).
† † †
+
This solution exists provided that η ≥
2
k 1 g2T||T⊥/ε2,
which corresponds to the pump power exceeding the Let us use the Fourier transform of Eqs. (34)–(37), for
subharmonic generation threshold (A10 ≠ 0). According example,
to formulas (32), the laser wave amplitude A30 in this ∞
above-threshold regime is independent of the pump 1
∫
δa 1 ( Ω ) = ---------- δa 1 ( t )e dt.
iΩt
power P. In other words, the amplitude A30 remains con-
stant when the pump power grows, and all the excess 2π
–∞
supplied power is spent on increasing the amplitude of
the subharmonic wave. As expected, the subharmonic Upon solving Eqs. (34)–(37) for the Fourier compo-
phase in this parametric process is shifted relative to the nents δa1, 3(Ω), we obtain
laser wave phase by π/2 or –π/2 [21].
x *1 ( – Ω )Z ( Ω ) – y 1 ( Ω )Z ( – Ω )
†
If the parameters of the crystal, pump, and cavity are
δa 1 ( Ω ) = -------------------------------------------------------------------------
-, (38)
such that η < k 1 g2T||T⊥/ε2, then the pump power is
2
y 1 ( Ω )y *1 ( – Ω ) – x 1 ( Ω )x *1 ( – Ω )
below the subharmonic generation threshold. For a sub-
threshold regime of subharmonic generation, a station- δa 3 ( Ω ) = i [ ( iΩ – k 1 )δa 1 ( Ω )
ary solution of the system (27)–(30) is as follows: (39)
+ i2ε A 30 δa 1 ( – Ω ) + F 1 ( Ω ) ] ( 2ε A 10 ) ,
† –1

η
1/2
k3
A 30 = ------------------------------
1/2
-, N 0 = -----------------
2
-, where
( 4g T || T ⊥ )
2
g T CT ⊥
x 1 ( Ω ) = – 4ε A 10 – i2εy A *30
2
(33)
1/2 3
iη k
σ 0 = -------------------------------------
1/2
-, A 10 = 0. – ( iΩ – k 1 ) ( iΩ – k 3 – gT C L ⊥ N 0 + x ),
( 4g T C T || T ⊥ )
4 2

y 1(Ω) = y(iΩ – k 1) – i2ε A 30(iΩ – k 3 – gT C L ⊥ N 0 + x),

A comparison of Eqs. (32) and (33) shows that exceed-
ing the subharmonic generation threshold leads to a Z ( Ω ) = ( k 3 – iΩ + gT C N 0 L ⊥ – x )F 1 ( Ω ) + y ( Ω )

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 99 No. 5 2004

952 NOVIKOV, CHIRKIN

× F 1 ( – Ω ) + i2ε A 10 [ F 3 ( Ω ) + i ( 1 – 2γ A *30 )L ⊥ Γ ( Ω )
† given by formulas (33). In this case, equations for the
fluctuation operators are as follows:
– i2 A 30 γ L ⊥ Γ ( – Ω ) + γ Γ N ( Ω ) ],
†
d ( δa 1 )
---------------- = – k 1 δa 1 + i2ε A 30 δa 1 + F 1 ( t ),
†
(45)
2 –1
dt
γ = γ ( Ω ) = 2gT C A 30 L ⊥ ( iΩ – τ •∃ + 4gT C L ⊥ A 30 ) ,
–1

d ( δa 3 )
γ ∑ˆ<dmM ( – Ω ) - = – k 3 δa 3 – iT C gδσ + F 3 ( t ),
--------------- (46)
dt
x = x(Ω) = 2gT C γ ( A *30 N 0 L ⊥ – iσ 0 ), x* = x* ( – Ω ), d ( δσ ) 1
--------------- = – ----- δσ + iT C g ( N 0 δa 3 + A 30 δN ) + Γ ( t ), (47)
y = y(Ω) = 2gT C γ ( A 30 N 0 L ⊥ + iσ 0 ), y* = y* ( – Ω ), dt τ⊥
d ( δN ) 1
gT C --------------- = – ---- δN
L ⊥ = L ⊥ ( Ω ) = ------------------
-. dt τ || (48)
iΩ – τ ⊥
–1

i2T C g ( σ 0 δa 3 + A 30 δσ – σ *0 δa 3 – A 30 δσ ) + Γ N ( t ).
† †
+
We are interested in statistical properties of the
quadrature components of the laser frequency and its Solving these equations, we obtain the following
subharmonic at the laser cavity output, which are expressions for the Fourier spectra of field operators:
measured using the balance homodyne detection tech-
nique [22]. Let us introduce the quadrature Fourier δa 1, out ( Ω ) = 1 – R1
components,
( iΩ – k 1 )F 1 ( Ω ) – i2ε A 30 F 1 ( – Ω )
† (49)
× ------------------------------------------------------------------------------- – c 1 ( Ω ),
4ε A 30 – ( iΩ – k 1 )
2 2 2
X j ( Ω ) = δa j, out ( Ω ) + δa j, out ( Ω ),
†
(40)

1 – R3
Y j ( Ω ) = i ( δa j, out ( Ω ) – δa j, out ( Ω ) ),
†
(41) δa 3, out ( Ω ) = -----------------------
-
yy* – xx*
where δaj, out is the Fourier component of the field oper-
× [ zF 3 ( Ω ) – yF 3 ( – Ω ) + ( zx 2 – i A 30 L ⊥ γy )Γ ( Ω )
†
ator with the frequency ωj at the cavity output deter- (50)
mined by the following boundary condition at the out- + ( – i A 30 L ⊥ γz – yx *2 )Γ ( – Ω )
†
put mirror [17]:
+ ( zγ – yγ * )Γ N ( – Ω ) ] – c 3 ( Ω ),
δa j, out ( Ω ) = 1 – R j δa j ( Ω ) – c j ( Ω ). (42)
where
According to relations (40) and (41), the quadrature x 2 = x 2 ( Ω ) = i ( 1 – 2γ A 30
* )L ⊥ , x 2* = x 2* ( – Ω ),
components are δ-correlated:
z = z ( Ω ) = iΩ – k 3 – gT C N 0 L ⊥
〈 X j ( Ω ) X j ( Ω' )〉 = S X, j ( Ω )δ ( Ω – Ω' ),
†
(43) + 2gT C γ * ( A 30 N 0 L ⊥ + iσ 0 ).
The quadrature components of the laser frequency
〈 Y j ( Ω ) X j ( Ω' )〉 = S Y, j ( Ω )δ ( Ω – Ω' ),
†
(44) are also δ-correlated and their spectral densities are
described by relations (43) and (44). Using formulas (49)
where SX, j (Ω) and SY, j (Ω) are the spectral densities of and (50), it is possible to obtain analytical expressions
fluctuations of the quadrature components. Using for the spectral densities SX, j(Ω) and SY, j(Ω). The plots
Eqs. (38)–(44), it is possible to derive analytical of these spectral densities are also presented below in
expressions for the spectral densities SX, j (Ω) and Section 4.
SY, j (Ω), but the resulting formulas are rather cumber- Thus, the proposed theory allowed us to obtain ana-
some and are not presented here. Below (see Section 4) lytical expressions for the spectrum of fluctuations of
we present the plots of the spectral densities SX, j (Ω) the quadrature components of the laser frequency and
and SY, j (Ω) versus various parameters of the problem its subharmonic excited simultaneously in the same
under consideration. crystal in two limiting regimes of the parametric self-
frequency conversion. The spectral densities of both
quadrature components depend on various parameters
3.2. Subharmonic Generation of the system under consideration. For this reason,
in Subthreshold Regime these dependences will be considered for a particular
In the subthreshold regime of subharmonic genera- case of the active nonlinear Nd:Mg:LiNbO3 crystal, in
tion, the stationary solution of system (27)–(30) is which one of the self-frequency conversion processes

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 NONCLASSICAL LIGHT GENERATION 953

(namely, the quasi-phase-matched parametric self-fre- Sω

quency-doubling) was observed [23]. 2.0
1
4. QUADRATURE-SQUEEZED LIGHT 1.8
GENERATION AT THE LASER FREQUENCY
AND ITS SUBHARMONIC 1.6
IN Nd:Mg:LiNbO3 RDS CRYSTAL 2
Now we will calculate the spectra of quadrature 1.4
components of the laser frequency and its subharmonic
for the process of parametric self-frequency conversion
in a active nonlinear Nd:Mg:LiNbO3 RDS crystal. The 1.2
crystal can be pumped by radiation with a wavelength
of 0.81 µm, for example, from a diode laser [23]. Laser 1.0
3
generation is observed at a wavelength of 1.084 µm, so
that the subharmonic wavelength is 2.168 µm. In para-
metric interaction of the ee–e type, the laser and sub- 0.8
0 0.05 0.10 0.15 0.20
harmonic waves possess extraordinary polarization.
ΩTC
The condition of quasi-phase-matched interaction is
obeyed for a nonlinear susceptibility modulation period Fig. 2. The spectra of fluctuations in a laser field quadrature
of Λ = 22 µm [23]. The other parameters of the in a above-threshold regime of subharmonic generation,
calculated for various ratios of the pump power to the
Nd:Mg:LiNbO3 RDS crystal were as follows: effective threshold power Ppump/Pth = 10 (1), 15 (2), and 20 (3).
nonlinearity coefficient, deff = 34.4 pm/V; linear losses Other parameters: Rω/2 = 1; Rω = 0.9; L = 0.5 cm.
at the frequencies ω/2 and ω, α1 = α3 = 0.08 cm–1; sat-
uration intensity, IS = 104 W/cm2; number of active Sω/2
dopant atoms, M = 1018; longitudinal and transverse 30
relaxation times, T|| = 10–4 s and T⊥ = 6.7 × 10–10 s, respec-
tively; typical cavity round trip time, TC = 3 × 10–11 s. 25

4.1. Spectral Characteristics 20

for Subharmonic Generation
in Above-Threshold Regime 15 1
The characteristic spectral densities of quadrature
components calculated using formulas (38)–(44) are 10
presented in Figs. 2 and 3, where the unit spectral
density corresponds to the standard quantum limit. Fig- 2
5
ure 2 shows the spectra of fluctuations in one of the
laser radiation quadratures at various pump powers. As 3
can be seen, the more the pump power exceeds the
0 0.02 0.04 0.06 0.08 0.10
threshold level, the stronger the suppression of quadra-
ture fluctuations relative to the quantum limit in a cer- ΩTC
tain spectral region. In other words, a quadrature- Fig. 3. The spectra of fluctuations in a subharmonic field
squeezed light is generated in this spectral region. quadrature in a above-threshold regime of generation, cal-
According to Fig. 2, the maximum efficiency of fluctu- culated for various coefficients of reflection of the cavity
ation suppression below the standard quantum limit in output mirror Rω/2 = 0.95 (1), 0.97 (2), 0.99 (3). Other
the quadrature component for the laser frequency can parameters: Rω = 1; Ppump/Pth = 15; L = 0.5 cm.
amount to 10%. These results refer to the X-quadrature
field components; as for fluctuations of the Y-quadra-
ture components, these must increase in accordance exhibits rather insignificant suppression of laser radia-
with the uncertainty relation, which was confirmed by tion fluctuations in the regime of above-threshold sub-
the results of our calculations. harmonic generation.
Figure 3 shows the spectra of fluctuations in one of
the subharmonic field quadratures for various reflec-
4.2. Spectral Characteristics
tion coefficients of the cavity output mirror. As can be
for Subharmonic Generation in Subthreshold Regime
seen, fluctuations in this quadrature component are
virtually not suppressed. Thus, the active nonlinear Figres 4–6 show the spectra of fluctuations in one of
Nd:Mg:LiNbO3 crystal with the above parameters the quadrature components of the subharmonic field,

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 99 No. 5 2004

954 NOVIKOV, CHIRKIN

20
Sω/2(Ω)
18 1.000
1.000
16
0.9000

14 0.8000
0.7000
12
P pump/Pth

0.6000
10 0.5000
0.4000
8
0.3000
6 0.2000
0.1000
4

2
0 0.02 0.04 0.06 0.08 0.10
ΩTC
Fig. 4. Variation of the spectrum of fluctuations in a subharmonic field quadrature in a subthreshold regime of generation depending
on the ratio of the pump power to the threshold power Ppump/Pth . Other parameters: L = 0.5 cm; Rω = 1; Rω/2 = 0.8. The inset shows
the scale of the spectral density levels, in which a darker color corresponds to a lower level of fluctuations.

0.9
Sω/2(Ω)
0.8 1.000
1.000
0.7 0.9000
0.8000
0.6
0.7000
0.5 0.6000
Rω/2

0.5000
0.4
0.4000
0.3 0.3000
0.2000
0.2
0.1000
0.1

0 0.02 0.04 0.06 0.08 0.10 0.12 0.14

ΩTC
Fig. 5. Variation of the spectrum of fluctuations in a subharmonic field quadrature in a subthreshold regime of generation depending
on the coefficient of reflection Rω/2 of the cavity output mirror at the subharmonic frequency ω/2. Other parameters: Ppump/Pth =
10; Rω = 1; L = 0.5 cm. The inset shows the scale of the spectral density levels, in which a darker color corresponds to a lower level
of fluctuations.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 99 No. 5 2004

 NONCLASSICAL LIGHT GENERATION 955

0.6
Sω/2(Ω)
1.000
0.5 1.000
0.9000

0.4 0.8000
0.7000
0.6000
L, cm

0.3
0.5000
0.4000
0.2
0.3000
0.2000
0.1 0.1000

0 0.01 0.02 0.03 0.04 0.05

ΩTC
Fig. 6. Variation of the spectrum of fluctuations in a subharmonic field quadrature in a subthreshold regime of generation depending
on the length L of the active nonlinear crystal. Other parameters: Ppump/Pth = 10; Rω = 1; Rω/2 = 0.9. The inset shows the scale of
the spectral density levels, in which darker color corresponds to a lower level of fluctuations.

which were calculated using formulas (40)–(44), (49), where Ppump = 0.5 W, Pth = 1.25k3 W, k3 is the dimen-
and (50). The calculations were performed for various sionless quantity given by formula (5), and αp =
pump powers, reflection coefficients of the cavity out- 2ln2 cm–1. The maximum efficiency of suppression of
put mirror, and crystal lengths. The parameters of the the subharmonic wave quadrature fluctuations for the
Nd:Mg:LiNbO3 crystal were the same as those used indicated parameters is about 90% (for the X-quadra-
above for the above-threshold regime of subharmonic ture field). As for the laser frequency generated in this
generation; other parameters are indicated in the figure regime, our calculations showed that virtually no sup-
captions. In Figs. 4–6, the level of the standard quantum pression of fluctuations take place.
limit also corresponds to the unit spectral density.
Analysis of the data presented in Figs. 4–6 shows
that the maximum suppression (corresponding to the 5. PHOTON STATISTICS
darkest area) of the quadrature field fluctuations takes IN PARAMETRIC SELF-FREQUENCY
place at a nonzero frequency, in contrast to the case of CONVERSION
the above-threshold regime of subharmonic generation.
Figure 4 shows that the higher the pump power, the In the general case, calculations of the photons dis-
stronger the suppression of fluctuations in the subhar- tribution functions for the laser frequency and subhar-
monic quadrature. However, this is accompanied by monic frequency encounter considerable difficulties.
narrowing of the spectral band in which the fluctuations For this reason, we will restrict the consideration to
are effectively suppressed. According to Fig. 5, an analysis of the statistical properties of photons within
increase in the reflectance of the output mirror for the the framework of the second-order moments and calcu-
subharmonic wave leads to a significant growth in the late the average photon number and its dispersion. In
efficiency of suppressing fluctuations in the corre- order to simplify calculations, we consider the average
photon number 〈nj 〉 = 〈 a j aj 〉 and the Fano factor
†
sponding quadrature. An analogous behavior is
observed in response to a change in the length of the
active nonlinear crystal as depicted in Fig. 6, which was 〈 n j 〉 – 〈 n j〉
2 2
calculated for F j = ----------------------------
-
〈 n j〉
–α L
P pump ( 1 – e p )
η + 1 = -------------------------------------
-, at the subharmonic frequency (j = 1) and laser fre-
P th quency (j = 3) inside the cavity.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 99 No. 5 2004

956 NOVIKOV, CHIRKIN

In the vicinity of the stationary solution of we obtain the relation

Eqs. (27)–(30), according to expressions (31),
ε η
2
k 1 – ( 2ε A 30 ) = k 1 – ----------------
2 2 2
〈 n j〉 = A j0 + 〈 δa j δa j〉 .
2 † -.
(51) 2
g T || T ⊥
For the dispersion of the photon number, to within the
same accuracy we have Thus, as expected, the average number of photons at the
subharmonic frequency, the dispersion of this number,
σ j = 〈 n j 〉 – 〈 n j〉 = A j0 ( 1 + 2 〈 δa j δa j〉 )
2 2 2 2 †
and the corresponding Fano factor sharply increase
(52) when the pump power approaches the subharmonic
+ A j0 ( 〈 δa j δa j〉 + 〈 δa j δa j 〉 ).
2 † †
generation threshold (η ≈ k 1 g2T||T⊥/ε2).
2

In the above-threshold regime of subharmonic genera-

If the number of thermal phonons is n 1 (T) Ⰶ 1, for-
tion, we have |Aj0|2 Ⰷ 〈δ a j δaj〉 and, hence,
†
mulas (55) an (56) yield for the Fano factor
〈 n j〉 ≈ A j0 , 〈 n j 〉 – 〈 n j〉 ≈ A j0 .
2 2 2 2
(53) F 1 = 1.5 + 2 〈 n 1〉 . (57)
This implies that the Fano factor is Fj = σ j /〈nj〉 ≈ 1
2
In the other limiting case ( n 1 (T) Ⰷ 1), the Fano factor
(j = 1, 3) and, therefore, the photon statistics in the laser is independent of proximity to the subharmonic gener-
cavity excited at these frequencies is Poisson-like. ation threshold, being determined by the average num-
In the subthreshold regime of subharmonic genera- ber of thermal phonons: F1 = n 1 (T).
tion, the photon statistics at the laser frequency remains
Poisson-like as well (|A10| = 0). The photon statistics at It should be noted that, far away from the subhar-
the subharmonic frequency depends on the average val- monic generation threshold at 〈n1〉 Ⰶ 1, the Fano factor
for the excited biphotons, F1 = 1.5, differs from the
ues 〈δ a 1 δa1〉 and 〈(δ a 1 δa1)2〉. The time variation of the
† †
value (F = 2) for the biphotons generated as a result of
operator δ a 1 (δa1) under the action of the random force
†
the spontaneous parametric scattering [17, 22]. We
can be readily determined using Eq. (45) and the Her- believe that a decrease in the Fano factor is related to
mitian-conjugated relation. As a result, we obtain the inertial character of the response of the system
under consideration to a random action (i.e., to the pres-
t
–k 1 θ
ence of a term describing losses in Eq. (45)).
δa 1 ( t ) ∫e ( F 1 ( t – θ ) cosh ( 2ε A 30 θ )
† †
=
(54)
0
6. CONCLUSIONS
+ iF 1 ( t – θ ) sinh ( 2ε A 30 θ ) ) dθ.
We have developed a theory describing the genera-
Using this solution, taking into account the statistical tion of nonclassical light during laser self-frequency
†
properties of fluctuations related to the operators F 1 (t) conversion in an active nonlinear crystal. Using this
description, based on the Heisenberg–Langevin equa-
and F1(t) according to Eqs. (12)–(14), and considering tions, we have thoroughly analyzed the process of self-
the stationary subthreshold regime of subharmonic down-conversion (halving) of the laser frequency. The
generation, we obtain expressions for the average pho- calculations were performed for an arbitrary relation
ton number, between the photon lifetime in the laser cavity and the
characteristic times of the inverse population relaxation
( 2ε A 30 ) + 2k 1 n 1 ( T )
2 2
〈 n 1〉 = -------------------------------------------------
-, (55) and the active medium polarization. We considered the
2 ( k 1 – ( 2ε A 30 ) )
2 2
process of subharmonic generation in the sub- and
above-threshold regimes. It was established that there
and for the dispersion of this number, are optimum parameters of the crystal, pumping, and
2 cavity favoring the maximum efficiency of suppressing
k1
σ1
2
= ---------------------------------------
- fluctuations in the quadrature components of the laser
2 ( k 1 – ( 2ε A 30 ) )
2 2
frequency and subharmonic fields below the level of the
standard quantum limit. We have also considered the
× [ 3 ( 2ε A 30 ) + 8 ( 2ε A 30 ) n 1 ( T )
2 2
(56) photon statistics for the generated light fields.
The results of our theoretical analysis show that the
+ 4 ( k 1 – ( 2ε A 30 ) )n 1 ( T ) ] .
2 2 2
active nonlinear RDS crystals can be used as effective
Taking into account that, according to (33), sources of nonclassical light. Further expansion of the
possibilities of such crystals with respect to self-fre-
η
1/2 quency conversion due to the presence of RDS opens
A 30 = ------------------------------
1/2
-, good prospects for the creation of small-size sources of
( 4g T || T ⊥ )
2
nonclassical radiation in various wavelength ranges.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 99 No. 5 2004

 NONCLASSICAL LIGHT GENERATION 957

RDS crystals can feature, besides the traditional 8. G. D. Laptev, A. A. Novikov, and A. S. Chirkin, J. Russ.
nonlinear optical interactions, consecutive three-wave Laser Res. 23, 183 (2002).
interactions of the optical modes having the common 9. K. S. Abedin, T. Tsuritani, M. Sato, and H. Ito, Appl.
pump wave. In this context, it is important to develop a Phys. Lett. 70, 10 (1997).
quantum theory of such processes in active nonlinear 10. J. Capmany, D. Callejo, V. Bermudez, et al., Appl. Phys.
RDS crystals. In such systems, the laser cavity features Lett. 79, 293 (2001).
three coupled processes: lasing and two nonlinear wave 11. J. Capmany, Appl. Phys. Lett. 78, 144 (2001).
interactions. The theory of self-frequency conversion 12. H. Haken, Light 2. Lasers Light Dynamics (North-Hol-
during consecutive interactions in RDS crystals can be land, Amsterdam, 1985; Mir, Moscow, 1988).
developed through generalization of the approach pre- 13. D. F. Walls and G. J. Milburn, Quantum Optics
sented in this study. (Springer, Berlin, 1995).
14. M. O. Skully and M. S. Zubairy, Quantum Optics (Cam-
ACKNOWLEDGMENTS bridge University Press, Cambridge, 1997).
15. V. N. Gorbachev and E. S. Polzik, Zh. Éksp. Teor. Fiz.
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fruitful discussions. 43, 6303 (1991).
This study was supported in part by the INTAS 17. C. W. Gardiner, Quantum Noise (Springer, Berlin, 1991).
Foundation, grant no. 01-2097. 18. G. D. Laptev and A. A. Novikov, Kvantovaya Élektron.
(Moscow) 31, 981 (2001).
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