Hidden PT Symmetry and quantization of coupled-oscillators model of QASER
HHidden PT Symmetry and quantization of coupled-oscillators model of QASER
Lida Zhang ( 张 理 达 ), G. S. Agarwal , W. P. Schleich , , and M. O. Scully Texas A & M University, College Station, Texas 77843, Universit¨at Ulm, D-89081 Ulm, Germany (Dated: September 24, 2018)Using Maxwell-Bloch equations it has been shown how the superradiance can lead to amplificationand gain at a frequency much larger than the pumping frequency. This remarkable effect has beenexamined in terms of a simpler model involving two coupled oscillators with one of them paramet-rically driven. We show that this coupled oscillator model has a hidden parity-time (PT) symmetryfor QASER, we thus bring PT symmetry to the realm of parametrically coupled resonators. More-over, we find that the QASER gain arises from the broken PT symmetry phase. We then quantizethe simplified version of the QASER using quantum Langevin equations. The quantum descriptionenables us to understand how the system starts from quantum fluctuations.
I. INTRODUCTION
Since the advent of lasers at the last 60s, it has notonly pushed science forward in almost every aspect butalso altered humans’ life to an unprecedented level. Aslaser and laser-based technologies have become one ofthe indispensable building blocks of our modern society,it is natural to think of higher-frequency lasers in X-rayregime which can provide untrahigh intensity and preci-sion (e.g., nonoscale) compared to current optical tech-nology, and have great potential in coherent imaging ofmacromolecule structures and tissues which are crucialin medicine, chemistry and biology [1–3]. Unfortunately,owing to the very short wavelength of X-ray which canpenetrate normal mirrors, the X-ray laser can not be sim-ply generated using active cavities as optical lasers. It istherefore vital to find new physical mechanisms to gen-erate X-ray laser in a compact and efficient way.An outstanding and promising example is QuantumAmplification by Superradiant Emission of Radiation(QASER) [4, 5], which utilizes an intense low-frequencylaser to drive the atomic ensemble around its lowercollective frequency (infrared) which matches the fre-quency difference between the two normal modes withmuch higher frequencies (X-ray) in the light-atom inter-acting system, resulting in amplification at the higher-frequency. The essence of QASER is coupling atomicsuperradiance and a combination parametric resonanceto get exponential generation of X-rays. Surprisinglythe QASER shares many aspects of two-coupled oscilla-tors asymmetrically and parametrically driven by a low-frequency field [4, 6]. The asymmetry in the parametriccoupling is especially interesting as in physics typicallywe deal with Hermitian couplings. We show in this paperthat this very asymmetric coupling is associated with thehidden parity-time (PT) symmetry in the model.We note that the study of the PT-symmetric Hamilto-nians [7, 8] has attracting considerable attentions espe-cially after experimental realization of the PT-symmetryin optical systems [9–16]. The most prominent featureof PT symmetry is that PT-symmetric Hamiltonians cangive either real or complex eigenvalues depending on theratio between its real and imaginary parts. If all the eigenvalues of the PT-symmetric Hamiltonian are real,the system is said to be in an unbroken PT-symmetryphase; Otherwise, it refers to a broken PT-symmetryphase of the system when complex eigenvalues start toappear, leading to exponentially growing/decreasing be-havior. Besides its great impact on the foundations ofquantum theory [17–22], PT-symmetry is becoming in-creasingly important in optical physics and is playing asignificant role in many intriguing applications such assingle-mode lasing [23, 24], negative refraction [25, 26],nonreciprocal propagation [12, 13] and so on.We show here that the QASER is indeed linked to abroken PT-symmetry phase. This is illustrated in Sec. IIby firstly simplifying the basic model of QASER for twocoupled oscillators at short evolution time when only thefundamental Floquet components are important, the re-sulting equations of motion are found to be equivalent toa two-mode PT-symmetric system as shown in Sec. III.With the help of the two-mode PT-symmetric system,we are able to get the approximated quantum Langevinequation for the QASER in the short evolution time asgiven in Sec. IV, finding that the fundamental Floquetcomponents of QASER can be exponentially generatedfrom quantum vacuum fluctuations.
II. BASIC MODEL OF QASER FOR TWOCOUPLED OSCILLATORS
We start from the basic model of QASER for two cou-pled oscillators [4, 6]¨ φ + ω φ − Ω φ = 0 , (1a)¨ φ + ω φ − Ω (1 + δ cos ν d t ) φ = 0 . (1b)Here ω is the frequency of the oscillators, ω (cid:29) Ωwhere Ω is the coupling constant between the two os-cillators, and δ (cid:28) ω = (cid:112) ω − Ω and ω = (cid:112) ω + Ω . Under the modulation of the low-frequency driving field ν d (cid:28) ω , the coupled system a r X i v : . [ qu a n t - ph ] M a y shows strong amplification for φ and φ with gain coeffi-cient G = Ω δ/ (8 ω ω ) when the modulation frequenciesmatches the frequency difference between the two nor-mal modes, i.e., ν d = ω − ω as shown in Fig. 1. Thecoupled-oscillators model may find realizations in cou-pled PT symmetric microresonators [12–15] or optome-chanical systems [16]. By introducing two new variables X = φ + φ , (2a) Y = φ − φ , (2b)Eqs. (1) are transformed into¨ X + ω X −
12 Ω δ cos( ν d t )( X + Y ) = 0 , (3a)¨ Y + ω Y + 12 Ω δ cos( ν d t )( X + Y ) = 0 , (3b)In the above equations, all the variables like φ , φ , X and Y are real functions of t . Now we will introducetwo complex coordinates to further simplify the coupledequations α = ( X + iω ˙ X ) e iω t , (4a) β = ( Y + iω ˙ Y ) e iω t , (4b)one can then obtain (see Appendix A for more detail)˙ α = i Ω δ ω ( αe − iω t + βe − iω t + c.c. ) cos( ν d t ) e iω t , (5a)˙ β = − i Ω δ ω ( αe − iω t + βe − iω t + c.c. ) cos( ν d t ) e iω t , (5b)where “c.c.” represents the counter-rotating terms andcan be neglected. At the resonance condition ω − ω = ν d , according to the Floquet theory one can write thesolution as α = (cid:88) n α n e inν d t , (6a) β = (cid:88) n β n e inν d t . (6b)The full Floquet analysis is given in Ref. [6]. When con-sidering the evolution of the system for a short time orfor small coupling constant δ , only the base component( n = 0) is important, resulting in (see Appendix A formore detail) ˙ α = i Ω δ ω β , (7a)˙ β = − i Ω δ ω α , (7b) ω G a i n C o e ffi c i e n t ( G ) FIG. 1. Gain coefficient G as a function of ω for the twocoupled oscillators. Parameters are chosen as before: Ω = ω / , δ = 0 . , ν d = 2. (Adopted from Ref. [6], Fig. 2). Eqs. (7) would also now serve as the starting point thatwe discuss the relation between the QASER and PT sym-metric system in the next section. Furthermore, we canalso find from Eqs. (7) that¨ α = Ω δ ω ω α . (8)Thus the gain coefficient for α is calculated as G = Ω δ √ ω ω . (9)Choosing the parameters as given in Fig. 1 and ω (cid:39) G (cid:39) . III. HIDDEN PT SYMMETRY OF QASEREQUATIONS
In this section, we try to show the connection betweenthe QASER and PT symmetric system. In order to il-lustrate this, we first introduce two transformations asfollows a = α + iβ √ , b = α − iβ √ , (10a)Ω δ ω = J + g , Ω δ ω = g − J , (10b)Thus Eqs. (7) reduce to˙ a = ga − Jb , (11a)˙ b = Ja − gb . (11b)It can be seen that a has linear gain g ( δ >
0) and b experiences balanced linear loss simultaneously, and theparameter J gives the coupling between the two modes a and b . This equation can be rewritten in matrix form˙ v = i ˆ H eff v , (12)where v = { a, b } T is the vector of variables, and ˆ H eff canbe considered as the effective Hamiltonian of the systemwhose form is ˆ H eff = (cid:34) − ig iJ − iJ ig (cid:35) . (13)It can be easily seen that ˆ H eff is not Hermitian sinceˆ H eff = ˆ H † eff unless g = 0. One can prove that ˆ H eff isinvariant under the combination of time reversal ˆ T : i →− i and parity reflection ˆ P which is defined asˆ P = (cid:34) (cid:35) , (14)which means ˆ H eff is PT-symmetric. The eigenvalues ofˆ H eff can be found as λ ± = ± iλ , (15)where λ = (cid:112) g − J = G > a and b , andthus α and β .We have seen that the QASER is equivalent to a two-mode PT-symmetric system when only considering thefundamental Floquet components, a natural question onemight ask would be whether or not this relation still holdswhen extending to higher Floquet modes? A short an-swer is probably no. This can be seen from the coupledequations for the Floquet components after the rotating-wave approximation as given by Eqs. (A.7), where the n − n + 2 terms (i.e., α n − and β n +2 ) are asym-metrically coupled in the sense they only appear in oneof the n -th equations (either in a n or in b n ), and thusintroduce additional terms after the transformations asdefined in Eqs. (10) for all n components, the result-ing effective Hamiltonian when including higher Floquetmodes is then not PT-symmetric. IV. QUANTIZATION
Having realized the PT-symmetric nature of theQASER equations, we proceed to quantize them in orderto study the quantum nature of the system. Let’s firsthave a closer look at the definitions of α and β given inEqs. (4) and (6), it is apparent that α and β are reminis-cent of multi-mode annihilation operators, and α and β are similar to single-mode operators, as well as a and b .We may then replace a and b by ˆ a and ˆ b respectively inorder to quantize Eqs. (11). However, this is not enoughto get the quantum equations owing to the linear gainand loss which would introduce quantum noises [27–30].After including the random noise terms, one can obtainthe quantum Langevin equations for ˆ a and ˆ bd ˆ adt = g ˆ a − J ˆ b + ˆ f a , (16a) d ˆ bdt = J ˆ a − g ˆ b + ˆ f b , (16b)where ˆ f a and ˆ f b represent the quantum noise for the twomodes respectively, and satisfy the correlations (cid:104) ˆ f † a ( t ) ˆ f a ( t (cid:48) ) (cid:105) = 2 gδ ( t − t (cid:48) ) , (cid:104) ˆ f a ( t ) ˆ f † a ( t (cid:48) ) (cid:105) = 0 , (17a) (cid:104) ˆ f b ( t ) ˆ f † b ( t (cid:48) ) (cid:105) = 2 gδ ( t − t (cid:48) ) , (cid:104) ˆ f † b ( t ) ˆ f b ( t (cid:48) ) (cid:105) = 0 . (17b)An exceptional feature of Eqs. (16) is that spontaneousgeneration grows exponentially from vacuum for PT-symmetry broken phase. This can be shown by calculat-ing (cid:104) ˆ a † ˆ a (cid:105) and (cid:104) ˆ b † ˆ b (cid:105) when the input states for both fieldsare vacuum (cid:104) ˆ a † ˆ a (cid:105) = J gλ (cid:20) gJ (cosh(2 λt ) −
1) + λ + g λJ sinh(2 λt ) − t (cid:21) , (18a) (cid:104) ˆ b † ˆ b (cid:105) = J gλ (cid:18) sinh(2 λt )2 λ − t (cid:19) . (18b)For the values J and g defined in Eqs. (10), it has beenshown in the previous section that PT symmetry is bro-ken in the system, and exponential growth of sponta-neous generation appears for ˆ a and ˆ b as shown above.One can then obtain the quantized form of Eqs. (7) bytransforming ˆ a and ˆ b back to ˆ α and ˆ β d ˆ α dt = i Ω δ ω ˆ β + ˆ f α , (19a) d ˆ β dt = − i Ω δ ω ˆ α + ˆ f β , (19b)and the noise terms are given byˆ f α = ˆ f a + ˆ f b √ , (20a)ˆ f β = ˆ f a − ˆ f b √ i . (20b)Then the nonzero correlations between the quantum fluc-tuations can be derived (cid:104) ˆ f † α ( t ) ˆ f α ( t (cid:48) ) (cid:105) = (cid:104) ˆ f † β ( t ) ˆ f β ( t (cid:48) ) (cid:105) = ( ω + ω )Ω δ ω ω δ ( t − t (cid:48) ) , (21a) (cid:104) ˆ f α ( t ) ˆ f † α ( t (cid:48) ) (cid:105) = (cid:104) ˆ f β ( t ) ˆ f † β ( t (cid:48) ) (cid:105) = ( ω + ω )Ω δ ω ω δ ( t − t (cid:48) ) , (21b) (cid:104) ˆ f † α ( t ) ˆ f β ( t (cid:48) ) (cid:105) = (cid:104) ˆ f α ( t ) ˆ f † β ( t (cid:48) ) (cid:105) = − i ( ω + ω )Ω δ ω ω δ ( t − t (cid:48) ) , (21c) (cid:104) ˆ f † β ( t ) ˆ f α ( t (cid:48) ) (cid:105) = (cid:104) ˆ f β ( t ) ˆ f † α ( t (cid:48) ) (cid:105) = i ( ω + ω )Ω δ ω ω δ ( t − t (cid:48) ) . (21d)One can then calculate the spontaneous generation forˆ α and ˆ β when the input state for both fields is vacuum S α = ω + ω ω ω (cid:2) ( ω − ω )Ω δt + 8 ω ω (cosh(2 λt ) − √ ω ω ( ω + ω ) sinh(2 λt )] , (22a) S β = ω + ω ω ω (cid:2) ( ω − ω )Ω δt + 8 ω ω (cosh(2 λt ) − √ ω ω ( ω + ω ) sinh(2 λt )] . (22b)where S O = (cid:104) ˆ O † ˆ O (cid:105) with O ∈ { α, β } . For illustration,we have plot S α and S β against evolution time t as shownin Fig. 2. Similar to ˆ a and ˆ b , ˆ α and ˆ β show exponen-tial growing behavior against time t even when both ofthe input fields are vacuum. The quantum generationstarting from vacuum is different from semiclassical cal-culations where one concentrates on gain by assuming asmall seed field.Although we adopted an open system approach to un-derstand gain in the system; it is possible to develop anHamiltonian framework [31] for the oscillator system de-scribed by Eq. (3). In this work by Giese et al both the os-cillator have time dependent frequency modulation in ad-dition to a coupling which depends on the low frequencydrive. The most important aspect of the Hamiltoniandescription is that one of the oscillators has a negativemass and inverted potential. The gain is then interpretedas arising from the oscillator with inverted energy spec-trum. Giese et al also argue by considering a simplercoupled oscillator Hamiltonian that gain can only arise ifone of the oscillators has inverted spectrum. V. CONCLUSIONS
In conclusion, we have found that the coupled-oscillators model of QASER for short evolution time or S α S β FIG. 2. Exponential spontaneous generation from quantumvacuum fluctuations for the fundamental Floquet componentsof QASER. Parameters are chosen as before: Ω = ω / , δ =0 . , ω = 8. weak parametric coupling when higher Floquet compo-nents are negligible is equivalent to a two-mode parity-time (PT) symmetric system. In the context of PTsymmetry, we interpret the exponential growth in theQASER is associated to the PT symmetry breaking ofthe system which leads to imaginary eigenvalues. More-over, by investigating the quantum nature of the two-mode PT symmetric system, we have derived the quan-tum Langevin equations for the QASER and demonstratethat the exponential gain originates from the quantumvacuum fluctuations.G. S. A. thanks colleagues, especially Anatoly Svidzin-sky for discussions and the Texas A & M University forsupport. We acknowledge the support of Office of NavalResearch Grant N00014-16-1-3054 and Robert A. WelchFoundation Award A1261. Appendix A: Simplifications of the coupledequations
Here we give the detailed steps on the derivation ofEq. (7). We first introduce two complex variables as z = X + iω ˙ X , (A.1a) z = Y + iω ˙ Y , (A.1b)then one can find that˙ z + iω z = iω ( ¨ X + ω X ) , (A.2a)˙ z + iω z = iω ( ¨ Y + ω Y ) . (A.2b)Inserting these expressions into Eqs. (3) leads to˙ z + iω z − i Ω δ ω cos( ν d t )( z + z ∗ + z + z ∗ ) = 0 , (A.3a)˙ z + iω z + i Ω δ ω cos( ν d t )( z + z ∗ + z + z ∗ ) = 0 . (A.3b)Now we define α = z e iω t , (A.4a) β = z e iω t , (A.4b)here α and β represent the slow-varying parts of z and z , respectively. Then one obtains˙ α = i Ω δ ω ( αe − iω t + βe − iω t + c.c. ) cos( ν d t ) e iω t , (A.5a)˙ β = − i Ω δ ω ( αe − iω t + βe − iω t + c.c. ) cos( ν d t ) e iω t , (A.5b)where “c.c.” denote the counter-rotating terms which wewill neglect in the following. Furthermore we consider the resonant case when ω − ω = ν d , physically corre-sponding to the combination parametric resonance [4, 6].Eqs. (A.5) are a Floquet system whose solution can bewritten as α = (cid:88) n α n e inν d t , (A.6a) β = (cid:88) n β n e inν d t , (A.6b)Eqs. (A.5) can be transformed into˙ α n = − inν d α n + i Ω δ ω ( α n − + α n +1 + β n + β n +2 ) , (A.7a)˙ β n = − inν d β n − i Ω δ ω ( α n − + α n + β n − + β n +1 ) . (A.7b)We now consider the evolution of the system for a shorttime where the higher components in α n and β n for n ≥ α = i Ω δ ω β , (A.8a)˙ β = − i Ω δ ω α . (A.8b) [1] C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel,A. Schropp, A. Schwab, S. Stephan, M. Burghammer,S. Sch¨oder, and C. Riekel, Phys. Rev. Lett. , 090801(2008).[2] I. Robinson and R. Harder, Nat Mater , 291 (2009).[3] H. N. Chapman and K. A. Nugent, Nat Photon , 833(2010).[4] A. A. Svidzinsky, L. Yuan, and M. O. Scully, Phys. Rev.X , 041001 (2013).[5] M. O. Scully, Laser Physics , 094014 (2014).[6] G. Chen, J. Tian, B. Bin-Mohsin, R. Nessler, A. Svidzin-sky, and M. O. Scully, Physica Scripta , 073004 (2016).[7] C. M. Bender and S. Boettcher, Phys. Rev. Lett. , 5243(1998).[8] C. M. Bender, Rep. Prog. Phys. , 947 (2007).[9] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti,M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N.Christodoulides, Phys. Rev. Lett. , 093902 (2009).[10] C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N.Christodoulides, M. Segev, and D. Kip, Nat Phys ,192 (2010).[11] A. Regensburger, C. Bersch, M.-A. Miri, G. On-ishchukov, D. N. Christodoulides, and U. Peschel, Na-ture , 167 (2012).[12] B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda,G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat Phys , 394 (2014).[13] L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang,G. Li, G. Wang, and M. Xiao, Nat Photon , 524 (2014).[14] Z.-P. Liu, J. Zhang, i. m. c. K. ¨Ozdemir, B. Peng, H. Jing,X.-Y. L¨u, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu,Phys. Rev. Lett. , 110802 (2016).[15] B. He, L. Yang, and M. Xiao, Phys. Rev. A , 031802(2016).[16] X.-W. Xu, Y.-x. Liu, C.-P. Sun, and Y. Li, Phys. Rev.A , 013852 (2015).[17] M. Znojil, Phys. Rev. D , 085003 (2008).[18] J. Gong and Q. hai Wang, Journal of Physics A: Mathe-matical and Theoretical , 485302 (2013).[19] A. Mostafazadeh, International Journal of GeometricMethods in Modern Physics , 1191 (2010).[20] S. Schmidt and S. P. Klevansky, Philosophical Transac-tions of the Royal Society of London A: Mathematical,Physical and Engineering Sciences (2013).[21] S. Deffner and A. Saxena, Phys. Rev. Lett. , 150601(2015).[22] D. C. Brody, Journal of Physics A: Mathematical andTheoretical , 10LT03 (2016).[23] L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang,Science , 972 (2014).[24] H. Hodaei, M.-A. Miri, M. Heinrich, D. N.Christodoulides, and M. Khajavikhan, Science , 975 (2014).[25] R. Fleury, D. L. Sounas, and A. Al`u, Phys. Rev. Lett. , 023903 (2014).[26] F. Monticone, C. A. Valagiannopoulos, and A. Al`u,Phys. Rev. X , 041018 (2016).[27] M. O. Scully and M. S. Zubairy, Quantum Optics (Cam-bridge University Press, Cambridge, UK, 1997).[28] H. Schomerus, Phys. Rev. Lett. , 233601 (2010). [29] G. Yoo, H.-S. Sim, and H. Schomerus, Phys. Rev. A ,063833 (2011).[30] G. S. Agarwal and K. Qu, Phys. Rev. A85