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LETTER
Dualism between optical and difference parametric
amplification
To cite this article: Wayne Cheng-Wei Huang and Herman Batelaan 2017 EPL 119 24002
View the article online for updates and enhancements.
- Design criteria for ultrafast optical
parametric amplifiers
C Manzoni and G Cerullo
- Theory of polariton parametric interactions
in semiconductor microcavities
C Ciuti, P Schwendimann and A
Quattropani
- Quantum phase transitions in the driven
dissipative Jaynes-Cummings oscillator:
From the dispersive regime to resonance
Th. K. Mavrogordatos
This content was downloaded from IP address 129.59.95.115 on 28/10/2017 at 13:30
July 2017
EPL, 119 (2017) 24002
doi: 10.1209/0295-5075/119/24002
www.epljournal.org
Dualism between optical and difference parametric amplification
Wayne Cheng-Wei Huang1,2 and Herman Batelaan1
1
2
Department of Physics and Astronomy, University of Nebraska-Lincoln - Lincoln, NE 68588, USA
The Institute for Quantum Science and Engineering, Texas A&M University - College Station, TX 77843, USA
received 30 June 2017; accepted in final form 5 September 2017
published online 5 October 2017
PACS
PACS
PACS
42.65.Yj – Optical parametric oscillators and amplifiers
42.65.Sf – Dynamics of nonlinear optical systems; optical instabilities, optical chaos
and complexity, and optical spatio-temporal dynamics
42.50.Nn – Quantum optical phenomena in absorbing, amplifying, dispersive and
conducting media; cooperative phenomena in quantum optical systems
Abstract – Breaking the symmetry in a coupled wave system can result in unusual amplification
behavior. In the case of difference parametric amplification the resonant pump frequency is equal
to the difference, instead of the sum, frequency of the normal modes. We show that sign reversal
in the symmetry relation of parametric coupling give rise to difference parametric amplification
as a dual of optical parametric amplification. For optical systems, our result can potentially be
used for efficient XUV amplification.
c EPLA, 2017
Copyright Parametric processes are essential to quantum optical applications including frequency conversion, quantum
communication, and nonclassical state generation [1–4].
In particular, the application of squeezed light in precision
measurement has led to enhanced sensitivity for gravitational wave detection [5]. Parametric interaction occurs
when driving a nonlinear dipole with two frequency inputs. In a doubly resonant cavity, two non-degenerate target frequencies, ωe > ωg , can be parametrically coupled
to a pump frequency through a nonlinear medium [6,7].
When the pump frequency ν is equal to the sum-frequency
Σω ≡ ωe + ωg or the difference-frequency Δω ≡ ωe − ωg ,
resonant parametric interaction occurs. A sum-frequency
will facilitate energy transfer from the pump field Ep to the
target fields Ee,g , leading to amplification. A differencefrequency will promote energy exchange between the target fields without changing their total energy [8]. In
the framework of quantum optics, the former corresponds
to the anti–Jaynes-Cummings interaction and the latter
amounts to the Jaynes-Cummings interaction [9,10].
In a recent proposal by Svidzinsky et al. [11], a semiclassical approach was used to show that the JaynesCummings interaction could lead to strong amplification
of light in a super-radiant atomic gas, if such a coupled system is driven with an external difference-frequency pump.
This quickly leads to the conceptual difficulty that energy
conservation is violated. In optical parametric amplification (OPA) energy transfers from the pump field to the
target fields because one sum-frequency photon, having
higher energy, breaks into two target-frequency photons
with smaller energy [10]. In the case that the differencefrequency pump drives the amplification, such a photon
picture cannot apply since the energy of one differencefrequency photon is less than the total energy of two
target-frequency photons. Assuming that this effect exists, what is then the mechanism for energy transfer? To
shed light on this puzzle, we turn to Maxwell equations
where OPA was originally studied [12–14].
In this letter, we show that difference parametric amplification (DPA), i.e., amplification based on a differencefrequency drive, does not violate energy conservation at
the level of classical physics. We illustrate the dualism
between DPA and OPA through the symmetry relation of
parametric coupling. Given that quantum mechanics is a
superior theory with respect to classical mechanics, a corresponding quantum mechanism should exist. We argue
that the combination of DPA and the Jaynes-Cummings
Hamiltonian will lead to non-Hermiticity. This gives rise
to complex-valued expectation values and may explain
why the photon picture does not apply for DPA.
We note that DPA, if realized, presents potential advantages for delivering efficient XUV amplification. The
state-of-the-art upconversion light sources are based on
either multiphoton excitation or higher-harmonic generation [15–17]. These processes suffer from deteriorating
conversion efficiency as the target frequency gets into the
24002-p1
Wayne Cheng-Wei Huang and Herman Batelaan
Fig. 1: (Color online) Parametric pumping for two coupled
cavities. The transmissivity of the coupling mirrors (dashed
line) determines the strength of normal mode splitting, hence
the difference-frequency between the two cavity modes Δω ≡
ωe − ωg . The nonlinear medium (blue) is assumed to mediate
the parametric interaction between the cavity modes Ee,g and
the pump Ep with nonlinear coupling parameters χe,g .
ultraviolet regime [18–20]. In contrast, DPA remains as
a first-order nonlinear process regardless of how high the
target frequency is. This feature renders DPA a potential
mechanism for efficient amplification in the XUV regime.
To illustrate the concept of DPA and its connection to
OPA, we start with Maxwell equations for waves in a nonlinear medium [9],
(1) ∂ 2 E
4π ∂ 2 PNL
=
,
c2 ∂t2
c2 ∂t2
and rotating-wave frame Ẽ(t) = Ẽ(t)e−iωt to
⎧ 2 d
∗ iω t
∗ iω t
⎪
−iωe t
2 ˜ −iωe t
⎪
e
e
˜
˜
˜
⎪
E
E
e
+
E
e
e
+
E
e
+ω
=
e
e
e
e
⎪
e
⎪
dt2
⎪
⎪
⎪
χg ˜ −iωg t
∗
∗
⎪
⎪
+ E˜g eiωg t E˜p e−iνt + E˜g eiνt ,
Eg e
⎨
2
⎪
d2 ˜ −iωg t ˜ ∗ iωg t ⎪
2
⎪
˜g e−iωg t + E˜g ∗ eiωg t =
⎪
E
+ω
E
e
+
E
e
g
g
g
⎪
⎪
dt2
⎪
⎪
⎪
χe ˜ −iωe t
∗
∗
⎪
⎩
Ee e
+ E˜e eiωe t E˜p e−iνt + E˜g eiνt ,
2
(3)
where E˜p ≡ A0 e−iφ is the pump amplitude. To simplify the above equations, we eliminate the non-resonant
terms with the rotating-wave approximation. Also, we
will use the slow-varying approximation, dEe,g (t)/dt ωe,g Ee,g (t), to reflect the slow-varying envelope Ee,g (t) and
focus only on the fast dynamics at the timescales 1/ωe,g .
Under these assumptions, eq. (3) becomes
⎧
⎪
dE˜e
iχg ˜ ˜ −iΔt
∗
⎪
−iΔs t
˜
˜
⎪
E
,
E
E
=
e
+
E
e
g p
g
p
⎨
dt
4ωe
(4)
˜g
⎪
d
E
iχ
∗ iΔt
∗
e
⎪
−iΔ
t
s
˜
˜
˜
˜
⎪
Ee Ep e
,
=
+ Ee Ep e
⎩
dt
4ωg
where Δ ≡ ν − Δω and Δs ≡ ν − Σω are the pump detunings from the difference-frequency Δω ≡ ωe − ωg and
where (1) represents the linear dielectric response of the the sum-frequency Σω ≡ ωe + ωg , respectively. Later, the
medium which, for simplicity, is assumed to be isotropic validity of the approximations will be shown by the agreedispersionless. The dipole moment of the nonlinear ment between the analytical solution and the numerical
medium PNL acts as a driving source and couples the tar- simulation of eq. (2).
In OPA, the pump frequency is close to the sumget field E with a pump field Ep through PNL = χ(2) Ep E,
frequency
ν ≈ Σω (|Δ| 0), and eq. (4) can be
(2)
is a dielectric tensor that characterizes the
where χ
further
simplified
by making again the rotating-wave
second-order nonlinear response of the medium. Based on
approximation,
eq. (1), we consider the wave dynamics of two eigenmodes
⎧
∗
⎪ dE˜e
Ee,g of frequencies ωe,g in a doubly resonant cavity (see
⎪
= αs E˜g e−iΔs t ,
⎨
fig. 1). We assume small normal mode splitting compared
dt
(5)
⎪
to the eigenfrequencies, 0 < ωe − ωg ωe,g . Through a
˜
d
⎪
⎩ Eg = βs E˜e ∗ e−iΔs t ,
second-order nonlinear medium, the two target fields Ee,g
dt
are parametrically coupled by an injected pump field Ep .
The coupled wave equations can be derived from eq. (1), where the gain parameters are defined as αs ≡ iχg E˜p /4ωe
and βs ≡ iχe E˜p /4ωg . The target field solutions Ẽe,g (t)
⎧ 2
d
E
⎪
can be derived accordingly,
e
⎪
= −ωe2 Ee + χg Eg Ep ,
⎨
dt2
(2)
Ωs t
Ωs t
iΔs
2
⎪
d
E
Ẽ
Ẽ
(t)
=
(0)
cosh
sinh
+
⎪
g
e
e
⎩
= −ωg2 Eg + χe Ee Ep ,
2
Ωs
2
dt2
Ωs t
2αs
sinh
+ Ẽg∗ (0)
e−i(ωe +Δs /2)t ,
where χg and χe are the nonlinear coupling parameters
Ωs
2
for Eg and Ee , respectively. Conventionally, the nonlinear
∗ ∗ (6)
coupling is symmetric with respect to the target frequenΩs t
Ωs t
iΔs
Ẽg (t) = Ẽg (0) cosh
+ ∗ sinh
cies, χg = χe . However, here we make the distinction and
2
Ωs
2
extend the analysis to the more general case that the two
∗ Ωs t
2βs
coupling parameters can be made different, χg = χe . In
+ Ẽe∗ (0) ∗ sinh
e−i(ωg +Δs /2)t ,
addition, we remark that eq. (2) is the diagonalized repΩs
2
resentation for all parametrically coupled systems, includ
ing the cases discussed in refs. [11,21–23]. Given the pump where the OPA gain rate is Ωs = −Δ2s + 4αs βs∗ . The
field Ep (t) = A0 cos (νt + φ), the coupled equations can be analytical solution to eq. (2) is thus Ee,g (t) = (Ẽe,g (t) +
∗
(t))/2. Seeing from eq. (6), we notice that the
transformed with the complex notation E = (Ẽ + Ẽ ∗ )/2 Ẽe,g
∇2 E −
(1)
24002-p2
Dualism between optical and difference parametric amplification
and the solutions for the target fields Ẽe,g (t) are
Ωt
Ωt
iΔ
Ẽe (t) = Ẽe (0) cosh
sinh
+
2
Ω
2
Ωt
2α
sinh
+ Ẽg (0)
e−i(ωe +Δ/2)t ,
Ω
2
Fig. 2: (Color online) Comparison between analytical solutions
and simulations for amplification via parametric pumping. (a),
(b): provided a negative symmetry relation χe χg < 0, amplification of the target fields Ee,g can only be achieved through
a difference-frequency pump ν = Δω. The upper right insets
give the temporal evolution of the target fields. Here, initial
conditions Ẽe (0) = 1 and Ẽg (0) = 0 are assumed. Fourier
spectra of the fields show a single spectral peak at the respective frequencies ωe /2π = 1460 Hz and ωg /2π = 1240 Hz (bottom panels). The width of the spectral peaks characterizes the
exponential growth rate of the field amplitude (top panels).
A good agreement is found between the analytical solutions
(black and red) and the simulation (blue). (c), (d): provided
a positive symmetry relation χe χg > 0, amplification can only
be attained through a sum-frequency pump ν = Σω. The temporal behavior and spectral property are similar to the case of
a difference-frequency pump. The temporal evolution is plotted at the timescale T ≡ 2π/Ω and Ts ≡ 2π/Ωs for DPA and
OPA, respectively.
(8)
Ωt
Ωt
iΔ
Ẽg (t) = Ẽg (0) cosh
sinh
−
2
Ω
2
Ωt
2β
sinh
+ Ẽe (0)
e−i(ωg −Δ/2)t ,
Ω
2
where the DPA gain rate is Ω = −Δ2 + 4αβ. The important difference in the gain parameters α and β makes
it possible to attain amplification through a differencefrequency pump and a negative symmetry relation χe χg <
ωg > 0). With a sufficiently
0 (αβ = −χe χg |E˜p |2 /16ωe
strong pump, |E˜p | > 4|Δ| ωe ωg /|χe χg |, eq. (8) implies
that the target field can be exponentially amplified with
a real-valued DPA gain rate, Ω ∈ .
We compare the analytical solutions, eqs. (6) and (8),
to the simulation results of eq. (2), assuming the positive
symmetry relation for ν = Σω and the negative symmetry
relation for ν = Δω. The good agreement in both cases
justifies the use of the rotating-wave approximation and
the slow-varying approximation in the analysis (see fig. 2).
Without loss of generality, the target frequencies are taken
to be ωg /2π = 1240 Hz and ωe /2π = 1460 Hz from acoustic waves. This makes the simulation less stiff as the ratio
between the target frequency and the difference-frequency
Δω/2π = 220 Hz is kept within 10. Generally, the solutions can be applied to any frequency regime.
While the resonant frequencies for OPA (ν = Σω) and
DPA (ν = Δω) are vastly apart, both give rise to amplification of the same target frequencies ωe,g with mutually
exclusive parameter regimes, χe χg > 0 and χe χg < 0 (see
fig. 3). The dualism between OPA and DPA is made clear
when considering the energy flows in the coupled wave system. Using eq. (2), the energy transfer to a field can be
calculated through the driving term χe,g Ee,g Ep ,
dEe,g
2
χe,g Ee,g Ep . (9)
= 0 dt
We,g = 0 Ee,g
dt
dynamic behavior of the coupled wave system is fully
determined by what we call the symmetry relation herein,
the sign of χ
e χg . Assuming a sufficiently strong pump,
|E˜p | > 4|Δs | ωe ωg /|χe χg |, the positive symmetry relation χe χg > 0 implies that αs βs∗ = χe χg |E˜p |2 /16ωe ωg > 0.
This guarantees a real-valued OPA gain rate, Ωs ∈ , and
gives rise to exponential amplification of the target fields
According to eq. (8), a difference-frequency pump ν = Δω
under a sum-frequency pump, as expected for OPA.
with positive symmetry relation χe χg > 0 gives the
When a difference-frequency pump ν ≈ Δω is used in- solution
stead, the coupled equations in eq. (4) become
Ee (t) = Qe (0) cos (ωe t) cos (bt/2),
⎧
˜e
⎪
d
E
−iΔt
⎪
(10)
⎪
,
χe A0 Qe (0)
⎨ dt = αE˜g e
sin (ωg t) sin (bt/2),
Eg (t) =
(7)
2ωg b
⎪ dE˜
⎪
⎪
g
⎩
= β E˜e eiΔt ,
where φ = 0 is assumed. The initial conditions are set
dt
to be Ẽe (0) = Qe (0) and Ẽg (0) = 0. Parameters a and
The use of a difference-frequency pump results in a new set b are defined as the real and imaginary parts of the DPA
∗
of gain parameters α ≡ iχg E˜p /4ωe and β ≡ iχe E˜p /4ωg , gain rate Ω = a + ib. With the positive symmetry relation
24002-p3
Wayne Cheng-Wei Huang and Herman Batelaan
Fig. 3: (Color online) Parametric resonance and Fourier spectra for OPA and DPA. (Field amplitude in logarithmic color
scale.) (a) Pronounced amplification of the two target frequencies ωe /2π = 1460 Hz and ωg /2π = 1240 Hz appears for resonant
pumping at the sum-frequency ν/2π = 2680 Hz (OPA) and the difference-frequency pump at ν/2π = 220 Hz (DPA). (b), (c):
the parameter regimes (χe , χg ) for OPA and DPA are mutually exclusive. Initial conditions Ẽe (0) = 1 and Ẽg (0) = 0 are
assumed.
χe χg > 0, it follows that a = 0 but b = 0. The energy The corresponding energy flows are
2 2 flow in the fields can be subsequently computed with the
A0 Qe (0) ωe
dWe (t)
≈ −0 χe χg
sin (ωe t) sin (ωg t)
slow-varying approximation,
2 2 dt
4a
ωg
A0 Qe (0) ωe
dWe (t)
≈ −0 χe χg
sin (ωe t) sin (ωg t)
× cos (Δωt) sinh (at),
dt
4b
ωg
(13)
2 2 × cos (Δωt) sin (bt),
A
dWg (t)
Q
(0)
e
0
≈ 0 χ2e
cos (ωe t) cos (ωg t)
(11)
dt
4a
2 2 A0 Qe (0)
dWg (t)
× cos (Δωt) sinh (at).
≈ 0 χ2e
cos (ωe t) cos (ωg t)
dt
4b
The negative symmetry relation makes the energy flows
× cos (Δωt) sin (bt).
in both target fields obtain a positive sign, leading to
As shown in fig. 4(a), the energy flows in the two target simultaneous excitation of the two fields (see fig. 4(b)).
fields have similar strength but opposite sign, implying The modulation term cos (Δωt) indicates that energy is
that energy is exchanged between the two fields. When pumped into the target fields at the rate of the differencethe energy of a field is depleted, the sign of its energy frequency. The fast oscillations in the energy flow are out
flow is reversed. The depletion rate is characterized by of phase, suggesting that the fields take turns to draw energy from the pump. The hyperbolic term sinh (at) shows
the imaginary part of the DPA gain rate b.
the
exponential energy growth in the two fields at the rate
In the case of DPA (negative symmetry relation χe χg <
a,
which
is the real part of the DPA gain rate.
0), the field solutions are
Remarkably, the dynamic behaviors of the target fields
Ee (t) = Qe (0) cos (ωe t) cosh (at/2),
in the two parameter regimes (χe χg > 0 and χe χg < 0)
(12) are reversed if the coupled wave system is provided with
χe A0 Qe (0)
sin (ωg t) sinh (at/2).
Eg (t) =
a sum-frequency pump ν = Σω. Under the positive sym2ωg a
metry relation χe χg > 0, as in the scheme of OPA, the
The sinusoidal functions in eq. (10) are now replaced system will undergo the same amplification as described
by hyperbolic functions because a = 0 but b = 0. by eq. (13) with the parameter a replaced by the real part
24002-p4
Dualism between optical and difference parametric amplification
Fig. 4: (Color online) Temporal evolution of energy flow in OPA and DPA. (a) Provided a positive symmetry relation χe χg > 0,
a difference-frequency pump ν = Δω promotes energy exchange between the target fields Ee,g . Energy flows in the two fields
have opposite signs. A positive sign represents energy gain; a negative sign represents energy loss. The negative energy flow
is flipped to positive as the energy of the respective field is depleted (inset). Initial conditions Ẽe (0) = 1 and Ẽg (0) = 0 are
assumed. (b) Under a negative symmetry relation χe χg < 0, a difference-frequency pump ν = Δω can cause amplification for
the two target fields (DPA). The total energy of the fields increases over time. (c) With the same nonlinear coupling parameters
as (a), a sum-frequency pump ν = Σω can cause field amplification as in the case of (b) (OPA). (d) Given the same nonlinear
coupling parameters as (c), a sum-frequency pump ν = Σω will induce energy exchange between the two target fields as in
the case of (a). These four scenarios indicate that the roles of a difference-frequency pump and a sum-frequency pump are
exchanged in the two mutually exclusive parameter regimes χe χg > 0 and χe χg < 0.
of the OPA gain rate, as = Re(Ωs ) (see fig. 4(c)). When
the symmetry relation is negative χe χg < 0, energy is exchanged between the two target fields in a conservative
way, as shown in fig. 4(d). This behavior can be described
by eq. (11) with the parameter b replaced by the imaginary
part of the OPA gain rate, bs = Im(Ωs ).
The four scenarios summarized in fig. 4 illustrate the
dualism between OPA and DPA. The sign reversal in the
symmetry relation switches the roles of a sum-frequency
pump and a difference-frequency pump in the coupled
wave system. While the positive symmetry relation
χe χg > 0 promotes OPA, the negative symmetry relation χe χg < 0 facilitates DPA. The symmetry relation
reflects the symmetry built into the coupling mechanism.
To provide a physical context for discussion, we devise
a thought experiment where transition between OPA and
DPA is controlled by a single knob. In fig. 5(a), two identical microwave coplanar waveguide cavities are capacitively
coupled. Normal mode splitting makes two close eigenfrequencies ωe,g . Parametric pumping for the cavity modes
is provided by a feedback loop in two steps. First, the
field signal Ee (xr , t) + Eg (xr , t) is taken from a receiver
antenna sitting at xr and mixed with a pump signal
Ep = A0 cos (νt + φ) through an ideal mixer of output
efficiency χ. Second, the output signal from the mixer
is fed to a driver antenna as the pump for the coupled
cavity. Assuming that the driver antenna is sensitive
to the spatial phase of the fields [24], coupling to each
mode will then have the spatial dependence cos (ke xd ) and
cos (kg xd ), where ke,g = ωe,g /c and xd is the position of
the driver antenna. The coupled wave system can be modeled by Maxwell equations,
⎧
2
2
∂
2 ∂
⎪
⎪
−c
Ee (xd , t) = χ cos (ke xd ) (Ee (xr , t)
⎪
⎪
∂t2
∂x2
⎪
⎪
⎪
⎨
+ Eg (xr , t)) Ep (t),
2
⎪
∂
∂2
⎪
⎪
− c2 2 Eg (xd , t) = χ cos (kg xd ) (Ee (xr , t)
⎪
⎪
2
⎪
∂t
∂x
⎪
⎩
+ Eg (xr , t)) Ep (t).
(14)
24002-p5
Wayne Cheng-Wei Huang and Herman Batelaan
Fig. 5: (Color online) Theoretical demonstration of the transition from OPA to DPA with a change in a single physical parameter
xr . (a) A coupled coplanar waveguide cavity is doubly resonant at ωe,g . Parametric pumping is achieved through a feedback
loop in two steps. First, cavity field signals ωe,g are passed from a receiver antenna to an ideal mixer to be mixed with a pump
signal ν. Second, the output signal ωe,g ± ν of the mixer is sent to a driver antenna to pump the coupled cavity. The inset
shows the spatial phase profiles of the cavity modes cos (ke xr ) (red) and cos (kg xr ) (black). When the receiver antenna (blue) is
parked in the regimes between the nodes (pink), parametric coupling via the feedback loop will take a negative symmetry relation
χe χg < 0, which facilitates DPA. (b) Pumping the cavity at the sum-frequency ν = Σω shows amplification in the regimes of
positive symmetry χe χg > 0. (c) Pumping the cavity at the difference-frequency ν = Δω gives rise to field amplification in the
regimes between the nodes, as indicated in the inset of (a). Initial conditions Ẽe (0) = 1 and Ẽg (0) = 0 are assumed.
Substituting in eq. (14) the cavity modes Ee,g (xd , t) =
cos (ke,g xd )Ee,g (t), the equation can be simplified with the
rotating-wave approximation,
⎧ 2
d
⎪
2
⎪
⎪
⎨ dt2 Ee (t) = −ωe Ee (t) + χg Eg (t)Ep (t),
(15)
⎪
2
⎪
d
⎪
⎩
Eg (t) = −ωg2 Eg (t) + χe Ee (t)Ep (t),
dt2
where the effective nonlinear coupling parameters turn out
to be χe,g (xr ) ≡ χ cos (ke,g xr ). Solutions to eq. (15) will
mimic eqs. (6) and (8) because eq. (15) has the same form
as eq. (2). Assuming resonant pumping, the amplification
solutions are
Ee (t) = Ee (0) cos (ωe t) cosh (Ω0 t/2),
where λe,g = 2π/ke,g and m is an integer. In these
regimes, the symmetry relation is negative χe χg =
χ2 cos (ke xr ) cos (kg xr ) < 0. Outside of these regimes,
OPA can occur with a sum-frequency pump (see fig. 5(b)).
Dualism between DPA and OPA is manifested as the position of the receiver antenna changes the symmetry nature
in the coupling.
Finally, we address the problem of photon conservation
in DPA. The standard Hamiltonian for second-order nonlinear interaction is Ĥs = Ĥ0 + h̄χ(2) (â†e â†g Ẽp + âe âg Ẽp∗ +
â†e âg Ẽp +âe â†g Ẽp∗ ) [11], where χ(2) is a real-valued parameter. The two terms â†e â†g Ẽp and âe âg Ẽp∗ describe the anti–
Jaynes-Cummings interaction that supports OPA. The
photon picture for OPA is that one sum-frequency photon breaks into two lower energy photons at the target
(16)
χ cos (ke xr )A0 Ee (0)
frequencies. The other two terms â†e âg Ẽp and âe â†g Ẽp∗ are
sin (ωg t) sinh (Ω0 t/2),
Eg (t) =
2ωg Ω0
the Jaynes-Cummings interaction which promotes energy
exchange between the target fields. If we generalize the
2
2
where Ω0 =
±χ cos (ke xr ) cos (kg xr )A0 /4ωe ωg , and
three-body Hamiltonian with nonlinear coupling parameinitial conditions are Ẽe (0) = Ee (0) and Ẽg (0) = 0. The
ters χe and χg ,
± sign in Ω0 corresponds to the sum-frequency pump ν =
Σω(+) and the difference-frequency pump ν = Δω(−).
Ĥg = Ĥ0 + h̄(χg â†e â†g Ẽp + χe âe âg Ẽp∗
As the wavelengths of the two cavity modes are slightly
+ χg â†e âg Ẽp + χe âe â†g Ẽp∗ ),
(18)
off, the nonlinear coupling can be either symmetric (same
sign) or asymmetric (opposite sign) depending on the po- the generalized Hamiltonian Ĥ will yield a set of quang
sition of the receiver antenna xr . For the symmetric case tum Heisenberg equations that resemble eq. (4). Note
we say that the symmetry relation is positive (χe χg > 0), that the standard Hamiltonian Ĥ is resumed by chooss
and for the asymmetric case the symmetry relation is neg- ing χ = χ = χ(2) in Ĥ . The generalized Hamiltonian
e
g
g
ative (χe χg < 0). In the example of fig. 5(c), the cavity
Ĥg in eq. (18) leads to solutions for âe (t) and âg (t) simis pumped with the difference frequency ν = Δω. As the
ilar to eqs. (6) and (8). In particular, when the symmereceiver antenna moves across the spatial phase profiles
try relation is negative χe χg < 0, a difference-frequency
of the cavity modes, amplification (DPA) occurs in the
pump can give rise to an amplification solution. Howregimes
ever, while the quantum solutions are similar to those
(17) from the classical analysis, expectation values of operators
xr ∈ ((2m + 1)λe /4, (2m + 1)λg /4),
24002-p6
Dualism between optical and difference parametric amplification
do not agree with corresponding classical observables. In
the case of χe = χg , or χe = χg = iχ(2) , the generalized Hamiltonian Ĥg becomes non-Hermitian and the expectation value of total energy Hg is complex-valued,
which is undesirable. Although the difficulty of applying
the photon picture for DPA does not exclude the possibility of realizing DPA, as is predicted by a classical analysis,
the incompatibility does suggest that the quantization of
DPA is not straightforward in the framework of conventional nonlinear optics and further theoretical works are
needed.
In conclusion, we derive from Maxwell equations the
classical solutions for DPA as an alternative pathway
of parametric amplification. In contrast to OPA, amplification in DPA requires a difference-frequency pump
and a negative symmetry relation of parametric coupling. We illustrate the dualism between OPA and DPA
by showing their corresponding roles in mutually exclusive
parameter regimes. As the DPA gain rate Ω =
−Δ2 − χe χg |E˜p |2 /4ωe ωg scales weakly with increasing
target frequencies ωe,g , DPA could be suitable for efficient
X-ray amplification.
∗∗∗
The authors thank Da-Wei Wang, Luojia Wang,
Xiwen Zhang, Anatoly Svidzinsky, Wolfgang
Schleich and Marlan Scully for advice. WH wishes
to give special thanks to Steve Payne and William
Seward for helpful discussions. This work utilized highperformance computing resources from the Holland Computing Center of the University of Nebraska. Funding
for this work comes from NSF EPS-1430519 and NSF
PHY-1602755.
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