Related content 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. REFERENCES [1] Dutt A., Luke K., Manipatruni S., Gaeta A. L., Nussenzveig P. and Lipson M., Phys. Rev. Appl., 3 (2015) 044005. [2] Ma X., Herbst T., Scheidl T., Wang D., Kropatschek S., Naylor W., Wittmann B., Mech A., Kofler J., Anisimova E., Makarov V., Jennewein T., Ursin R. and Zeilinger A., Nature, 489 (2012) 269. [3] Meekhof D. M., Monroe C., King B. E., Itano W. M. and Wineland D. J., Phys. Rev. Lett., 76 (1996) 1796. [4] Vlastakis B., Kirchmair G., Leghtas Z., Nigg S. E., Frunzio L., Girvin S. M., Mirrahimi M., Devoret M. H. and Schoelkopf R. J., Science, 342 (2013) 607. [5] The LIGO Scientific Collaboration, Nat. Photon., 7 (2013) 613. [6] Colville F. G., Padgett M. J. and Dunn M. H., Appl. Phys. Lett., 64 (1994) 21. [7] Rivoire K., Buckley S. and Vučković J., Opt. Express, 19 (2011) 22198. [8] Boyd R. W., Nonlinear Optics, 2nd edition (Academic Press, San Diego) 2003. [9] Kippenberg T. J. and Vahala K. J., Science, 321 (2008) 1172. [10] Gerry C. C. and Knight P. L., Introductory Quantum Optics (Cambridge University Press, New York) 2005. [11] Svidzinsky A. A., Yuan L. and Scully M. O., Phys. Rev. X, 3 (2013) 041001. [12] Kroll N. M., Phys. Rev., 127 (1968) 1207. [13] Tien P. K. and Shul H., Proc. IRE, 46 (1958) 700. [14] Giordmaine J. A. and Miller R. C., Phys. Rev., 14 (1965) 973. [15] Zhou B., Shi B., Jin D. and Liu X., Nat. Nanotechnol., 10 (2015) 924. [16] Haase M. and Schäfer H., Angew. Chem., Int. Ed., 50 (2011) 5808. [17] Ghimire S., DiChiara A. D., Sistrunk E., Agostini P., DiMauro L. F. and Reis D. A., Nat. Phys., 7 (2011) 138. [18] Wang J., Deng R., MacDonald M. A., Chen B., Yuan J., Wang F., Chi D., Hor T. S. A., Zheng P., Liu G., Han Y. and Liu X., Nat. Mater., 13 (2014) 157. [19] Ferray M., L’Huillier A., Li X. F., Lompré L. A., Mainfray G. and Manus C., J. Phys. B, 21 (1988) L31. [20] Lein M. and Rost J. R., Phys. Rev. Lett., 91 (2003) 243901. [21] Svidzinsky A. A., Zhang X., Wang L. and Wang J., Coherent Opt. Phenom., 2 (2015) 19. [22] Chen G., Tian J., Bin-Mohsin B., Nessler R., Svidzinsky A. A. and Scully M. O., Phys. Scr., 91 (2016) 073004. [23] Zhang X. and Svidzinsky A. A., Phys. Rev. A, 88 (2013) 033854. [24] Fink J. M., Bianchetti R., Baur M., Göppl M., Steffen L., Filipp S., Leek P. J., Blais A. and Wallraff A., Phys. Rev. Lett., 103 (2009) 083601. 24002-p7

1/--страниц