Home Search Collections Journals About Contact us My IOPscience The QASER revisited: insights gleaned from analytical solutions to simple models This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Laser Phys. 24 094014 (http://iopscience.iop.org/1555-6611/24/9/094014) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 136.243.24.42 This content was downloaded on 04/02/2015 at 22:23 Please note that terms and conditions apply. Laser Physics Astro Ltd Laser Phys. 24 (2014) 094014 (6pp) doi:10.1088/1054-660X/24/9/094014 The QASER revisited: insights gleaned from analytical solutions to simple models* Marlan O Scully Princeton University, Princeton, NJ 08544, USA Texas A&M University, College Station, TX 77843, USA Baylor University, Waco, TX 76798, USA E-mail: [email protected] Received 8 March 2014 Accepted for publication 6 April 2014 Published 15 August 2014 Abstract Lasers and masers typically require population inversion. But with phase coherent atoms (phasers), we get lasing without inversion (e.g. 10% of the atoms excited). However, in recent work we found that it is possible to get coherent light emitted with no atoms excited, via Quantum Amplification of Superradiant Emission of Radiation (QASER). In particular, we found that by utilizing collective superradiant emission, we can generate coherent light at high frequency in the UV or x-ray bands by driving the atomic system with lower frequency source. Here, we present a simple analysis based on near-resonant QASER operation and on a multiphoton Hamiltonian obtained by, e.g. a canonical transformation. Keywords: multiphoton Hamiltonian, quantum amplification of superradiant emission of radiation 1. Introduction motivation for our QASER studies was the interesting connection between the single atom maser and single photon superradiance. The equations for the probability of finding an atom excited in these two problems are intriguingly similar, as explained in table 1. The Nphoton which appears in the single atom maser expression is the essence of stimulated emission and is the harbinger of maser-laser Light Amplification by Stimulated Emission of Radiation. On the other hand, the Natom which appears in the single photon entry in table 1 is the essence of superradiant emission. To make a laser, we must add a drive field and go to the many-atom case. One naturally asks: by adding a drive field and going to the many photon limit can we likewise get superradiant amplification? The answer is 'yes' and we proceed by extending the usual Maxwell–Schrödinger equation to treat the problem of many-atom cooperative emission, i.e. driven superradiance. Lasing without inversion (LWI) is a quantum phase sensitive process requiring some population in the excited state and careful attention to the relative atomic decay rates [1–3] between the states. More recently, we have been studying transient LWI [4–6] to overcome the decay rate road block. We have also been investigating a form of Quantum Amplification by Superradiant Emission of Radiation (QASER) [7], which is free from decay rate constraints and requires no population in the excited state. The physics behind QASER operation is counter-intuitive, and an electronic RLC oscillator analogue has been developed which supports the basic concept and provides insight. A key feature of QASER operation is the frequency upconversion of a lower frequency drive. Here, we present a simple scheme which sheds light on a QASER operation. The model is depicted in figure 1. In particular, we develop a model which embodies multiphoton operation, and makes a QASER operation more transparent by taking advantage of the simplicity of near-resonant operation. In the next few paragraphs, we discuss the physics behind collective many-atom emission. As mentioned in [7], one 2. QASER physics from the Maxwell–Schrödinger equations Starting with the Maxwell equation for a superradiant field of amplitude εs(z, t) with frequency νs and wavevector ks as driven by the polarization P, we write * It is a pleasure to dedicate this paper to prof I Yevseyev—a true laser physics pioneer. Our field is richer and our path is wider because of him. 1054-660X/14/094014+6$33.00 1 © 2014 Astro Ltd Printed in the UK M O Scully Laser Phys. 24 (2014) 094014 Figure 1. (a) Strong low frequency field drives atoms which cooperatively radiate counter propagating field. (b) Atoms driven from ∣b〉 to ∣a〉 by multiphoton absorption while levels ∣a1〉 and ∣a2〉 are coupled nonresonantly, drive and signal frequencies are νd and νs, respectively. The drive field may be a single photon or an effective multi-photon field as outlined in the appendix. Table 1. Table comparing single atom maser and single photon superradiance in which: ℘ is the dipole matrix element, ℏν is the photon energy, ϵ0 is the vacuum permeability and t is the time. Probability of atomic excitation Key variables V and N Single atom maser sin2 ℘ ℏ ℏν ϵ 0V Nph t V = E.M. cavity volume, Nph = number of photon in cavity Single photon superradiance sin2 ℘ ℏ ℏν ϵ 0υ Nat t υ = atomic cloud volume, Nat = number of atoms in cloud ⎛ ∂2 1 ∂2 ⎞ ⎜ 2 − 2 2 ⎟ εs(z , t )e−i(νt − kz ) = μ0 νs2P(z , t ) (1) ⎝ ∂z c ∂t ⎠ Noting that μ0c2 νs = νs/ϵ0 and using 1 4ν 3℘ 2 γ= , (6) 4πε0 3ℏc 3 ⎛∂ 1 ∂ ⎞⎛ ∂ 1 ∂⎞ N ⎜ ⎟⎜ ⎟ εs(z , t )e−i(νt − kz ) = + μ0 νs2℘ ρab − + ⎝ ∂z V c ∂t ⎠⎝ ∂z c ∂t ⎠ (2) allows us to reduce equation (5) to the final working form ⎛∂ ⎡ 3 N⎤ s ∂⎞ ⎜ + c ⎟ Ωs(z , t ) = − i ⎢ γλ2c ⎥ ρab . (7) ⎣ ⎝ ∂t ⎠ 8π V⎦ ∂z where the (complex) macroscopic polarization of the N superradiant ensemble is given by P = ℘ ρab in which ℘ V is the dipole matrix element and the off-diagonal density matrix for the resonant superradiant collective emission is given by And we define the collective many-atom atomic frequency by 3 2 N Ωa2 = γλ c . (8) 8π V To make a connection with the many-atom single photon collective emission of table 1, let us write the single photon ℘ ℏν Nat in terms of the radiative superradiant frequency ℏ ϵ 0v decay rate γ, which yields s ρab (z , t ) = ρab (z , t )e−i(νst − ksz ). (3) We proceed by making the usual slowly varying approximation so that equation (2) becomes ⎛∂ 1 ∂⎞ N s i(vt − kz ) ⎟ εs(z , t ) = + μ0 νs2℘ ρab 2iks⎜ + e . (4) ⎝ ∂z V c ∂z ⎠ ℘ ℏν 3 2 Nat λ γc Nat = . (9) 4π v ℏϵ 0 ϵ 0v This is the same collective frequency as in equation (8) modulo, the factor of 2 encountered when connecting radiation field amplitude and photon production rates. Note that the working radiation expression equation (7) for the (resonant) field Ωs is governed by the external drive field Ωd as is discussed in detail in the following. A key point Combining terms and using equation (3) yields ⎛∂ μ cνs2 ℘ 2 N s ∂⎞ ⎜ + c ⎟ Ωs(z , t ) = − i 0 ρab , (5) ⎝ ∂t 2ks ℏ V ∂z ⎠ where Ωs = ℘εs/ℏ. 2 M O Scully Laser Phys. 24 (2014) 094014 is that the drive field is detuned from the atomic transition frequency ωab = νs, and is taken as being much stronger than the superradiant field in the present analysis. We proceed to write the density matrix ρab which obeys the usual equation of motion in the conventional notation . ρab = − iωabρab − i [Ωs e−i(νst − ksz ) + Ωd e−i(νdt − kdz )](ρbb − ρaa ). (10) s Hence, we have the equation of motion for ρab .s = − iΔs ρab − i [Ωs + Ωd e−i [(νd − νs )t − (kd − ks )z ]](ρbb − ρaa ) ρab (11) where Δs = ωab − νs . From (8) and (11), we have ∂⎞ ∂⎛∂ s ⎜ + c ⎟ Ωs = − iΩa2Δs ρab ∂t ⎝ ∂t ∂z ⎠ − Ωa2[Ωs + Ωd e−i [(νd − νs )t − (kd − ks )z ]](1 − 2ρaa ) Figure 2. Shape of the superradiant pulse after time t = 100/Ωa obtained by numerical solution for Δ = 2Ωa and Ω3/Ωa = 0.3, 0.5 and 0.7 (solid lines). Dashed line indicates the initial seed pulse (t = 0). Vertical axis has logarithmic scale. (12) where we have used ρaa − ρbb = 1. Since in a phase matched medium ks = kd, the second term in the square brackets represents conventional parametric conversion. Ignoring for the moment the forward parametric term, we get a QASER equation of motion by noting We proceed by differentiating both sides of (16) and use equation (17) to obtain i ⎛∂ ∂⎞ s ρab = 2 ⎜ + c ⎟ Ωs (13) ∂z ⎠ Ωa ⎝ ∂t ( Likewise, we may obtain a polarization equation of motion by differentiating (17) and using (16) to write and use the Rabi result for the drive field 1⎛Ω ⎞ ρaa = ⎜ d ⎟ (1 − cos2μ(t − z / v )) (14) 2⎝ μ ⎠ ( ) s d d s = − Ωa2 ρbb − ρaa ρ¨ab ρab (19) 2 where we have ignored the smaller time derivatives going as d d Ωs ρ˙bb − ρ˙aa . d d We use equation (14) to write ρbb − ρaa = 1 − η cos 2μt 2 where η = (Ωd/μ) , and thus obtain the same gain equation for both atoms and photons, namely ( where μ = Ωd2 + Δd2 /4 with Δd = ω − νd. From (12)–(14) (ignoring the Ωd term in equation (12)), we have the QASER equation of motion ⎛∂ ⎞⎛ ∂ ∂⎞ ⎜ + iΔs ⎟⎜ + c ⎟ Ωs ⎝ ∂t ⎠⎝ ∂t ∂z ⎠ ⎡ ⎛ Ω ⎞2 = − Ωa2⎢1 − ⎜ d ⎟ + ⎝ μ ⎠ ⎢⎣ ) d d Ω¨ s = − Ωa2 ρbb − ρaa Ωs. (18) ) ( ) X¨ = − Ωa2(1 − η cos 2μt )X (20) ⎛ Ωd ⎞2 ⎛ z ⎞⎤ ⎜ ⎟ cos 2μ⎜t − ⎟⎥Ωs. ⎝ v ⎠⎥⎦ ⎝ μ ⎠ s where X = Ωs or ρab . The solution to equation (20) can be written in the form (15) X (t ) = x+eiωt + x−e−iωt (21) which upon assuming ω = Ωa and neglecting the small x¨± terms yields Equation (15) is essentially the QASER equation of motion expressed in the form of a Mathieu equation. The backward propagation aspect of QASER operation as depicted in figure 2 is given in [8]. ηΩ 2 x˙+ = − i a x− (22) 4ω ηΩ 2 x˙− = i a x+ (23) 4ω 3. QASER gain To gain insight into QASER dynamics, we write equation (7) for the case of no z-dependence and equation (11) for the case of Δs = 0 while ignoring the parametric conversion term going as Ωd. Then, we have the simple coupled field-matter equations and therefore x¨± = Gx± , (24) where the gain G is given by s Ω˙ s = − iΩa2ρab (16) ( η G = Ωa , (25) 4 ) s d d ρ˙ab = − iΩs ρbb − ρaa (17) and as before we have assumed Ωa = ω. It is important to note that both the field oscillator and the collective atom oscillator experience gain. where we have indicated, by the superscript ‘d’, that the populations ρaa(ρbb) are governed by the driving field. 3 M O Scully Laser Phys. 24 (2014) 094014 Acknowledgments (b) (a) The National Science Foundation Grant has supported this work PHY-1241032 (INSPIRE CREATIV) and the R A Welch Foundation (Awards A-1261). It is a pleasure to thank O Kocharovskaya, Y Rostovtsev, W Schleich, G Shchedrin, A Svidzinsky, D Wang and L Yuan for useful discussions. Figure A1. (a) Odd number of drive photons couple, for example, the 1S and 2P levels of hydrogen. (b) Even photon number couples for example 1S and 2S. Appendix. Multiphoton Hamiltonian analysis Using the Gibson approach [9], the Hamiltonian in the interaction picture is given by eiΘ = ∣b⟩⟨b ∣+ cos θ (∣1⟩⟨1 + 2⟩⟨2∣) Hint (t ) = V12(t ) + Vs(t ) + Vd (t ) (A.1) V12(t ) = ℏα12[∣2⟩⟨1∣ + adj.] sin (νd t − k dz ) (A.2a) The essential operator product as it appears in equations (A.2b) and (A.2c) then reads Vs(t ) = ℏα v∣1⟩⟨b∣eiωt aŝ e−i(νst − ksz ) + adj. (A.2b) + i sin θ (∣1⟩⟨2 + 2⟩⟨1∣). (A.9) ̂ ̂ (A.10) eiΘ ∣1⟩⟨b∣e−iΘ = cos θ∣1⟩⟨b ∣+ i sin θ∣2⟩⟨b∣, i ωt Vd (t ) = ℏαd∣1⟩⟨b∣e sin (νd t − k dz ) + adj. (A.2c) and so from equations (2), (7) and (8), we have where α12 is the Rabi frequency of the drive field coupling ∣1〉, ∣2〉 and ℏαv is the vacuum Rabi coupling between ∣a1〉, ∣b〉; note that we usually use the notation ∣a〉, ∣b〉 for the laser levels [9]. Here, the upper level ∣a〉 is replaced by ∣a1〉 and ∣a2〉, but we will call the upper levels ∣1〉 and ∣2〉 for convenience. The other symbols have their usual meaning, e.g. α v = ℘abE 0/ ℏ where ℘ab is the dipole matrix element and E 0 = ℏωab / ϵ 0V is the electric field per photon of the weak signal field which is resonant with the atomic frequency ωab and aŝ is the signal field annihilation operator. The strong drive field of frequency νd and strength Ed is treated classically and the corresponding Rabi frequency is αd. We proceed by transforming V1,2 away such that the Schrödinger equation vs ̂ = ℏα v (cos θ∣1⟩⟨b ∣+ isin θ∣2⟩⟨b∣) as†̂ eiωt e−i(νst − ksz ) + adj. (A.11a) vd̂ = ℏαd (cos θ∣1⟩⟨b ∣+ i sin θ∣2⟩⟨b∣) eiωt sin (νdt − k dz ) + adj. (A.11b) Recalling that θ = x cos φ (with x = − Ω12/ν) and using the Bessel generating functions ∞ cos( x cos φ) = J0(x ) + 2 ∑ (−)ℓ J2ℓ(x ) cos 2ℓφ, (A.12a) ℓ= 1 ∞ sin (x cos φ ) = 2 ∑ (−)ℓ J2ℓ+ 1(x ) cos[(2ℓ + 1)φ] (A.12b) i Ψ˙ = − Hint Ψ (A.3) ℏ ℓ= 0 yields is replaced by cos θ 1⟩⟨b + i sin θ 2⟩⟨b∣ ∞ ⎧ ⎫ = ⎨J0(x ) + 2 ∑ ( − )ℓ J2ℓ(x )cos 2ℓ(νd t − k dz )⎬∣1⟩⟨b∣ ⎩ ⎭ ℓ= 0 i Φ˙ = − (vd̂ (t ) + vs(̂ t ))Φ, (A.4) ℏ where ̂ (A.5) Ψ = e−iΘ (t )Φ, with (A.6) Θ (̂ t ) = θ (t )(∣1⟩⟨2∣ + adj.), (A.7) vd̂ (t ) + vs(̂ t ) = eiΘ (t )(Vd (t ) + Vs(t ))e−iΘ (t ). ⎪ ⎧ ∞ ⎫ + ⎨2 ∑ ( − )ℓ J2ℓ+ 1(x )cos (2ℓ + 1)(νd t − k dz )⎬∣2⟩⟨b∣. ⎩ ℓ= 0 ⎭ (A.13) ⎪ ⎪ ⎪ ⎪ ∞ ⎧ ⎫ vs(̂ t ) = ℏα v ⎨J0(x ) + 2 ∑ (−)ℓ J2ℓ(x )cos 2ℓ(νd t − k dz )⎬ ∣ 1⟩ ⎩ ⎭ ℓ= 1 We expand the exponential operator in equation (5) ̂ ⎪ ⎪ From equations (13) and (A.11a, A.11b), we distinguish the small processes involving an odd and even number of photons as depicted in figure A1 and listed in equations (A.14a, A.14b) and (A.15a, A.15b) as odd where θ(t) = (α12/νd)cos(νdt − kdz). In such a case, the transformed drive and signal field interaction Hamiltonian are given by eiΘ = 1 + iθ (t )(∣1⟩⟨2 + 2⟩⟨1∣)− ⎪ θ (t ) (∣1⟩⟨1 + 2⟩⟨2∣) + .... 2 2 (A.8) and by using 1 = ∣b〉〈b∣ + ∣1〉〈1∣ + ∣2〉〈2∣ we write this as ⎪ ⎪ ⎪ ⎪ ⟨b ∣ eiωt aŝ e−i(νst − ksz ) + adj . (A.14a) 4 M O Scully Laser Phys. 24 (2014) 094014 ∞ ⎧ ⎫ ⎨ vd̂ (t ) = ℏΩd J0(x ) + 2 ∑ (−)ℓ J2ℓ(x )cos 2ℓ(νd t − k dz )⎬ ∣ 1⟩ ⎩ ⎭ ℓ= 1 ⎪ ⎪ ⎪ ⎪ where p = 2m + 1 as it appears in equation (20) with Δp = ω − pνd, and the effective coupling frequency for odd p is 2p Ωp = αd Jp(x ), odd. (A.22) x ⟨b ∣ eiωt sin (νd t − k dz ) + adj . (A.14b) even Likewise, for an even number of drive photons we take from equation (A.15b) terms having ℓ = m which go as 2mνd ≈ ω. Such a term is proportional to ⎧ ∞ ⎫ vs(̂ t ) = ℏα v ⎨2 i ∑ (−)ℓ J2ℓ+ 1(x )cos (2ℓ + 1)(νd t − k dz )⎬ ∣ 2⟩ ⎩ ℓ= 0 ⎭ ⎪ ⎪ ⎪ ⎪ eiνt − kz iωt e , 2i(−)m J2m + 1e−i(2m + 1)(νt − kz ) (A.23) 2i ⟨b ∣ eiωt aŝ e−i(νst − ksz ) + adj . (A.15a) and from the term in the series for which ℓ = m − 1, we have another term going as 2mνd ⎧ ∞ ⎫ vd̂ (t ) = ℏΩd ⎨2 i ∑ (−)ℓ J2ℓ+ 1(x )cos (2ℓ + 1)(νd t − k dz )⎬ ∣ 2⟩ ⎩ ℓ= 0 ⎭ ⎪ ⎪ ⎪ ⎪ e−i(νt − kz ) iωt e , 2i(−)m − 1J2m − 1e−i(2m − 1)(νt − kz ) (A.24) −2i ⟨b ∣ eiωt sin (νd t − k dz ) + adj . (A.15b) so that equation (A.15b) now reduces to Consider the case of a hydrogen atom driven by an odd number of photons to an excited Rydberg state. The associated atomic frequency ω is multiphoton resonant with two terms in the sum of equation (A.14b). For example, let us take the term in equation (A.14b) with ℓ = m which goes as Hence from equations (A.15b), (14) and (A.25), we have vd̂ (t ) = (−)m [J2m + 1(x ) + J2m − 1(x )]ei(ω − 2mν )t ei2mkz . (A.25) ℏαd 4m vd̂ (t ) J2m(x )eiΔ2m t ei2mkz ∣ 2⟩⟨b ∣ +adj . , = (−)m x ℏ α d 2(−)m J2mcos 2m(νd t − k dz )sin (νd t − k dz )eiωt ≈ 2(−)m J2me−i2mνdt e−iνdt iωt i(2m + 1) kdz e e , −2i even (A.26) so for 2m = n drive photons, we have the even number effective Rabi frequency (A.16) in which (2m + 1)νd ≈ ω. Likewise there is a term in (A.14b) with ℓ = m + 1 which is nearly resonant, that is 2n Ωn = αdi n Jn(x ), even. (A.27) x eiνdt iωt i(2m + 1) kdz e e . −2i (A.17) A couple of points should be made: x = Ω12/νd is much larger than αd = Ω1b/ω since the Rydberg matrix elements 〈1∣r∣2〉 are larger than the direct matrix elements 〈1∣r∣b〉. Furthermore, νd (e.g. IR) is substantially smaller that ω (e.g. UV). The Hamiltonian for the near-resonant (ω ≅ νs) interaction from equations (A.11a) and (A.12a) is given by vd̂ (t ) (−)m ≅ [J2m(x ) + J2m + 2(x )]ei(ω − 2(m + 1)ν )t ei(2m + 1) kdz ∣ 1⟩ ℏαd i (A.18) vs ̂ = ℏΩs∣1⟩⟨b∣eiωt e−i(νst − ksz ) + adj. , (A.28) 2( − )m + 1J2m + 2 cos 2(m + 1)(νd t − k dz ) sin(νd t − k dz )eiωt ≈ 2( − )m + 1J2m + 2e−i2(m + 1) νdt Thus, equation (A.14b) reduces to ⟨b ∣ +adj . where Ωs = J0(x ) α v n (t ) with n (t ) being the average number of photons in the superradiant pulse at time t. Thus far, the effort has focused on one three-level atom driven by a strong drive field having Rabi frequency αd. We may apply this to the many-atom QASER, e.g. for the case of odd (e.g. three) photon absorption, we find from equation (22) with p = 3. We have Recalling the recursion relation for Bessel functions 2k Jk − 1(x ) + Jk + 1(x ) = Jk(x ), (A.19) x and taking 2m = k − 1 in equation (18), we have vd̂ (t ) (−)m 2(2m + 1) ≅ J2m + 1(x )ei [ω − (2m + 1) νd ]t ei(2m + 1) kdz ∣ 1⟩ x ℏαd ⟨b ∣ +adj . (A.20) 6 Ωd = αd J3(x ). (A.29) x In the limit of small x, Jp(x) ∼ (x/2)p/p! and x2 Ωd ~αd . (A.30) 8 where we have neglected overall phase factors. Thus, the effective Hamiltonian for the absorption of p drive photons may be written as Recalling that x = Ω12/νd and considering the case 1S → nP 3 1 and using ⟨nP∣r∣1S⟩ ≅ n2a 0 and ⟨2P∣r∣1S⟩ = a 0, then 2 2 vd̂ = ℏΩp eiΔp t eipkdz∣1⟩⟨b∣ + adj. , odd (A.21) 5 M O Scully Laser Phys. 24 (2014) 094014 Ω12 ∼ 3n αd. Hence, if αd ∼ 10 and νd ∼ 10 , then for n ∼ 10 we have x ≲ 1 and Ω ∼ 1012. 2 13 [4] Kilin S Y, Kapale K T and Scully M O 2008 Phys. Rev. 100 173601 [5] Svidzinsky A A, Yuan L and Scully M O 2013 New J. Phys. 15 053044 [6] Yuan L, Wang D, Svidzinsky A A, Xia H, Kocharovskaya O A, Sokolov A, Welch G R, Suckewer S and Scully M O 2014 Phys. Rev. A 89 013814 [7] Svidzinsky A A, Yuan L and Scully M O 2013 Phys. Rev. X 3 041001 [8] Yuan L, Svidzinsky A A and Scully M O 2014 QASING in the backward direction via nearly resonant multiphoton drive (in preparation) [9] Gibson G 2003 Phys. Rev. A 67 043401 14 References [1] Nikonov D E, Scully M O, Lukin M D, Fry E S, Hollberg L N, Padmbandu G G, Welch G R and Zibrov A S 1996 Proc. SPIE 2798 342–50 [2] Lee H and Scully M O 1998 Found. Phys. 28 585–600 [3] Scully M and Zubairy S 1997 Quantum Optics (Cambridge: Cambridge University Press) 6

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