Industry

Spectral properties of the single beams generated by an optical parametric oscillator

Description
Spectral properties of the single beams generated by an optical parametric oscillator
Categories
Published
of 5
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  I NSTITUTE OF  P HYSICS  P UBLISHING  J OURNAL OF  O PTICS  B: Q UANTUM AND  S EMICLASSICAL  O PTICS J. Opt. B: Quantum Semiclass. Opt.  4  (2002) S313–S317 PII: S1464-4266(02)37864-9 Spectral properties of the single beamsgenerated by an optical parametricoscillator A Porzio 1 , 2 , C Altucci 2 , 4 , P Aniello 1 , 3 , A Chiummo 1 , C de Lisio 1 , 2 and S Solimeno 1 , 2 1 Dipartimento Scienze Fisiche, Universit´a di Napoli ‘Federico II’, Complesso Universitariodi Monte Sant’Angelo, Via Cintia 80126, Napoli, Italy 2 Istituto Nazionale per la Fisica della Materia, Napoli, Italy 3 Istituto Nazionale de Fisica Nucleare, Sezionale di Napoli, Italy 4 Dipartimento di Chimica, Universit´a della Basilicata, Potenza, Italy Received 31 December 2001Published 29 July 2002Online at stacks.iop.org/JOptB/4/S313 Abstract The quantum nature of the parametric interaction affects the spectralproperties of the single output beams in a way which depends critically onthe damping coefficients of the signal, idler and pump modes. When they areof the same order relaxation oscillations appear in the single-beam spectra.The single outputs are described in the frequency domain by analytic transferfunctions, representing the system response to white noise sources. Wereport numerical simulations of the single-beam spectra referring to differentoptical parametric oscillator working conditions. In particular we study howthese spectra depend on detuning, pumping level and cavity dampings tobetter clarify the conditions for which relaxation oscillations appear. Keywords:  Quantum optics, quantum noise, OPO 1. Introduction For its peculiar quantum properties the optical parametricoscillator (OPO) has been considered since the earliest daysof quantum optics as a prominent subject for theoreticalanalysis [1] and experimental investigation [2, 3]. Particularattention has been paid to the spectral correlations of thetwo mode fluctuations. While the spectrum difference of the two modes is shaped like an inverted Lorentzian with arather good suppression at zero frequency, the single-beamone depends critically on pump level and cavity dampings atsignal, idler and pump frequencies. In the absence of pumpexcess, squeezing should appear in the single-beam spectrumof a perfectly doubly resonant OPO for a pump intensity wellabove four times the threshold value [4]. To reach such aregime it is necessary to lower the threshold by reducing thepump decay rate. In so doing the OPO becomes a triplyresonant device (TRO) with single output spectra markedlydifferentfromthedoublyresonantcase,asitfeaturesrelaxationoscillations [3, 5, 6]. In these conditions the squeezing isconfined to a low frequency region blurred by pump phaseand amplitude excess noise.The aim of this paper is to discuss the properties of theintensity spectrum for a single beam generated by a TRO bydwelling on the role of cavity dampings, pump noise, crystalloss, detunings and cavity imbalance.With regard to the fluctuations, an OPO behaves as alinear system forced by different noise sources. The Fouriertransforms  ˜ α k  (ω)  of the combination  δ a k   =  µ k   −  i φ k   of the relative mode amplitude  µ k   and phase  φ k   fluctuations ( a k   =  r  k  e − i φ k  ( 1 +  µ k  ))  ( k   =  0 (pump), 1 (signal), 2 (idler)),and the adjoint  ˜ α ‡ k  (ω)  ≡ ˜ α † k  ( − ω ∗ )  form an algebraic system.AcompleteanalysisoftheOPOfeatureshasbeenreportedin[7] and wewill make use ofthe formalism therein todiscussdetailsofthesingle-beam properties. Inparticular, wewillusethe frequency responses  K  0 , π 2 kk  ′  (ω)  of the  k  th complex modequadrature to the  k  ′ th noise source. These responses areanalytic functions of   ω  with, in general, six complex poles(systemresonances)inthehalf-planeIm (ω)  0,theirlocationdepending on cavity parameters and pumping level.An important role is played by the detuning of the cavitymodes, which transforms the decay constant  γ  k   of the  k  thmode into a complex quantity  κ k   =  γ  k   + i ϕ k  . Consequently, 1464-4266/02/040313+05$30.00 © 2002 IOP Publishing Ltd Printed in the UK  S313  A Porzio  et al the detuning has the effect of breaking the symmetry of thefrequency responses, that is  K  0 , π 2 kk  ′  (ω)  =  K  0 , π 2  ∗ kk  ′  ( − ω) .The paper is organized as follows. Section 2 reviewsbriefly the equations of motion for the system and theirlinearization in terms of small fluctuations. Section 3introduces the formalism of the transfer functions  K  0 ,π/ 2 kk  ′  (ω) and discusses their behaviour in some noticeable cases.Section 4 gives the expressions for the photocurrent spectrumand analyses the role of cavity dampings. 2. Equations of motion The evolution of the mode amplitudes  a 0  (pump),  a 1  and  a 2 (signal+idler)ofanOPOisdescribedbythefollowingsystemof Langevin equations: ∂∂ t   a 1 a 2 a 0  =  − κ ′ 1 a 1  + 2 χ a 0 a †2  +  R ′ 1 − κ ′ 2 a 2  + 2 χ a 0 a †1  +  R ′ 2 − κ ′ 0 a 0  − 2 χ ∗ a 1 a 2  +  ǫ  +  R ′ 0   (1)with  R ′ k  ( t  )  independent fluctuating delta-correlated Langevinforces and  ǫ  = | ǫ | e − i φ  p . The coefficients  κ ′ k   ( k   =  0 , 1 , 2 ) stand for κ ′ k   =  γ  k   +  ̹ k   + i ϕ k   =  γ  ′ k   + i ϕ k   =  κ k   +  ̹ k   (2)with  γ  k   and  ̹ k   the cavity mode damping rates, associatedrespectively with the output mirror and the other lossmechanisms (mainly crystal absorption),  ϕ  j  the detunings of thesignalandidlermodesand2 χ  thestrengthoftheparametricinteraction.In particular, the detuning angles are proportional tothe damping coefficients through the tangent of an angle  ψ (i.e.  ϕ  j /γ  ′  j  =  tan ψ) , which will be used for measuring thedeviation from the resonance.The steady-state solutions of (1) are represented by theproduct  a k   =  r  k  e − i φ k  of the amplitude  r  k   and phase factore − i φ k  . In particular the amplitudes and phases are interrelated: | κ ′ 1 | r  21  = | κ ′ 2 | r  22  =  ( E   − 1 ) | κ ′ 0 | r  20  ≡  C  2 φ 1  +  φ 2  − φ 0  =  ψφ  p  − φ 0  =  ψ  − ψ  p where E   standsfortheeffectivepumpparameterinthepresenceof detuning: E   =    E  2 − sin 2 ψ  + 1 − cos ψ with  E  :  E   =  ǫ 2 χ | κ ′ 0 |   | κ ′ 1 κ ′ 2 |≡  ǫǫ th . Accordingly, the relation between the frozen phases of thethree beams depends on the detuning through the phases  ψ and  ψ  p  =  arcsin ( sin ψ/  E  ) .The system (1) can be linearized with respect to thefluctuating phases  φ k   and amplitudes  µ k   (see [8]). Next,changing the phases  φ k   and  φ  p  into  φ ′  j  =  φ  j  −  12 ψ  p ,φ ′ 0  = φ 0  +  ψ  −  ψ  p  and  φ ′  p  =  φ  p  +  ψ  p , and keeping for simplicitythe old symbols for the new ones, yields δ ˙ a 1 κ ′ 1 +  δ a 1  − δ a 2  − δ a 0  =  Z  ′ 1 δ ˙ a 2 κ ′ 2 +  δ a 2  − δ a 0  − δ a 1  =  Z  ′ 2 e i ψ 0 E   − 1  δ ˙ a 0 κ ′ 0 +  δ a 0  +  δ a 1  +  δ a 2  =  Z  ′ 0  +  Z  ǫ  +  Z  φ (3)with  E   an effective pump parameter that also includes thedetuning effect (see [7]), and  ψ ′  the detuning angle while  Z  ′  j  = e i φ  j  R ′  j κ ′  j r   j (  j  =  1 , 2 )  (4 a )  Z  ′ 0  = 1 E   − 1e i ψ 0 e i φ 0  R ′ 0 κ ′ 0 r  0 (4 b )  Z  ǫ  =  E  E   − 1e i ψ  p µ ǫ  (4 c )  Z  φ  = − i  E  E   − 1e i ψ  p φ  p  (4 d  )having denoted by  µ ǫ  =  δǫ/ǫ  and  φ  p  the relative laser pumpexcess noise and phase fluctuation, respectively. The outputfield, having intensity  ¯  I   j  =  2 γ   j r  2  j  =  ( 2 γ   j /γ  ′  j ) cos ψ C  2 , isgiven by b out   j  =   2 γ   j r   j e − i φ  j  1 +  µ  j  − κ ′  j 2 γ   j  Z   j  .  (5) 3. Frequency analysis and transfer functions For analysing the spectral characteristics of the single outputbeam of the OPO it is worth moving to the frequency domainby applying the Fourier transforms δ a k  ( t  )  =  1 √  2 π    ∞−∞ e i ω t  ˜ a k  (ω) d ω,δ a † k  ( t  )  = 1 √  2 π    ∞−∞ e i ω t  ˜ a ‡ k  (ω) d ω (6)with  ˜ a ‡ k  (ω)  the adjoint of the Fourier component relative to thefrequency  − ω , i.e.  ˜ a ‡ k  (ω)  = ˜ a † k  ( − ω ∗ ) .Thistransformwillbeextendedtothequadrature  X  θ  c ( t  )  = 12 e i θ  c  +  12 e − i θ  c † of an operator  c ( t  )  X  θ  c ( t  )  =  1 √  2 π    ∞−∞ e i ω t   12 ( e i θ  ˜ c (ω)  + e − i θ  ˜ c ‡ (ω)) d ω ≡  1 √  2 π    ∞−∞ e i ω t  ˜  X  θ  c (ω) d ω.  (7)Using the transformed mode amplitudes the OPOdynamics (see equations (3)) is described by the algebraicsystem  1 ˜ a 1  − ˜ a 0  − ˜ a ‡2  = ˜  Z  ′ 1  2 ˜ a 2  − ˜ a 0  − ˜ a ‡1  = ˜  Z  ′ 2  0 ˜ a 0  +  ˜ a 1  +  ˜ a 2  = ˜  Z  ′ 0  +  ˜  Z  ǫ  +  ˜  Z  φ (8)S314  Spectral properties of the single beams generated by an optical parametric oscillator Figure 1.  Density plot of  | K  0  jj | 2 (amplification of the whitequantum noise introduced into the  j th mode) versus  E   andnormalized frequency ( ω/γ  ). It is peaked around the threshold andat low frequency, with a peak height reducing to 1 / 3 of itsmaximum. while moving to higher frequency. with   j  =  1 + i  ωκ ′  j ,  0  =  1 + i  ωκ ′ 0   e i ψ ′ 0 E   − 1 . Accordingtoequations(4 c )and(4 d  )  ˜  Z  ǫ  =  [  E  /( E   − 1 ) ]e i ψ ′  p ˜ µ ǫ and  ˜  Z  φ  = − i[  E  /( E   − 1 ) ]e i ψ ′  p  ˜ φ  p  depend, respectively, on thelaser amplitude excess noise  ˜ µ ǫ  and phase  ˜ φ  p .In order to discuss the OPO output properties, thesystem (8) has to be completed by adding the ‡-transformedequations, thus obtaining for  ˜ a k  (ω) , and  ˜ a ‡ k  (ω)  a sixth-order algebraic system depending on the forcing terms ˜  Z  ′ k  ,  ˜  Z  ǫ ,  ˜  Z  φ ,  ˜  Z  ′ ‡ k   ,  ˜  Z  ‡ ǫ ,  ˜  Z  ‡ φ . Once this system is solved itis possible to calculate the Fourier-transformed amplitudequadrature  ˜ µ β k   =  12 ( ˜ β k   +  ˜ β ‡ k  ) of the fluctuating parts  δ  ˜ β k   of theoutput modes: ˜ µ β  j  =  K  0  jj  − κ ′  j 2 γ   j  ⊗ ˜  X  0  j  +  K  0  jj  ⊗ ˜  X  0 ̹  j +  K  0  jj ′  ⊗ ˜  X  ′ 0  j ′ +  K  0  j 0  ⊗  ˜  X  ′ ψ ′ 0 0  +  E  E   − 1 ˜  X  ψ ′  p ǫ  −  E  E   − 1 ˜  X  π 2  + ψ ′  p φ  p   (9)with ⊗ defined by  P  ⊗  X  Q  =  12 ( PQ  +  P † Q † ) .The functions  K  0 kk  ′  represent the frequency responses of the OPO to the quantum noise sources of the three modesand to the classical fluctuations of the pump laser. Detailedexpressions for each of them for different cavity parametersare given in [7]. Density plots of   | K  0  jj | 2 ,  | K  0  jj ′ | 2 and  | K  0  j 0 | 2 versus frequency/pump strength are reported in figures 1–3.Figure 1 refers to  | K  0  jj | 2 , the self-coupled noise contribution,withapeaklocatednearthreshold(  E   →  1)andlowfrequency.As the pump grows the peak moves toward higher frequenciesand, most important, reduces its height to less than 1 / 3. Infigure 2 we have plotted the response  | K  0  jj ′ | 2 of the  j th modeto the  j ′ th quantum noise. Also in this case the peak responseis located in proximity to the oscillation threshold and at low Figure 2.  Density plot of  | K  0  jj ′ | 2 (amplification of the whitequantum noise introduced into the  j ′ th mode) versus  E   andnormalized frequency ( ω/γ  ). It is peaked around the threshold andat low frequency. In contrast to | K  0  jj | 2 the peak does not move sofast to higher frequenciesbut reduces even more critically. Figure 3.  Density plot | K  0  j 0 | 2 (amplification of the white quantumnoise + classical pump amplitude-phasenoise) versus  E   andnormalized frequency ( ω/γ  ). The peak value increases with  E  . frequency. Moving toward higher pump rates the peak movesto higher frequencies but remains more localized than in thecase of   | K  0  jj | 2 . The third transfer function  | K  0  j 0 | 2 presentsa different behaviour, shown in figure 3. This contributionbecomes important only for high pumping rates. In fact, thefunction is peaked in the upper-right part of the plot whereit reaches a value twice the peak one of   | K  0  jj | 2 and  | K  0  jj ′ | 2 .This last transfer function controls the coupling of the lasernoise withthe single-beam spectrum (see equation (9)). Beingnegligibleatlowpumpratesthepumpexcessnoiseisirrelevantin devices working close to the oscillation threshold, in whichcase the output is dominated by the quantum contributions of both twin modes.S315  A Porzio  et al ( a ) ( b )( c ) ( d  ) Figure 4.  Intensity spectra  S   j  of a single output beam for a balanced, resonant OPO. ( a ) Damping ratio  γ  0 /γ   varying between 1 and 10(1, 2, 3, 5, 10) with  E   =  3. Reducing  γ  0  results in an increasing peak height and lower frequencylocation. The very low frequency regionappears to be below the SNL. In fact ( b ) the same curve, plotted for a narrower frequency range, shows a spectrum starting from below theSNL and crossing it at a frequency depending on the damping ratio value. ( c ) Damping ratio as in ( a ),  E   =  5. The peaks move to greaterfrequenciesby becoming more pronounced. ( d  ) Spectra for increasing  E   (1 . 5 , 1 . 75 , 2 . 0 ,..., 3 . 25) with  γ  0 /γ   =  5. The spectrum modifiesfrom a Lorentzian-likeshape to one dominated by the relaxation oscillationspeak. The transition to the below SNL regime, as usualconfined in the low frequencyregion, occurs for 2 . 75  <  E   <  3. 4. Intensity spectrum Thesignalandidlerinputmodesareinthevacuumstatesothatonly antinormally ordered terms contribute to the variances.Consequently, the noise correlations for the right-hand sideterms of equations (8) have the following expressions in thefrequency domain : ˜  Z  ′  j (ω) ˜  Z  ′ ‡  j  (ω ′ )  ′  = 2 γ  ′  j | κ ′  j |=  ς  ′  j ˜  Z  ′ 0 (ω) ˜  Z  ′ ‡0  (ω ′ )  ′  = 2 γ  ′ 0 | κ ′ 0 | 1 E   − 1  =  ς  ′ 0 ˜  Z  ǫ (ω) ˜  Z  ‡ ǫ (ω ′ )  ′  = ˜  Z  ‡ ǫ (ω) ˜  Z  ǫ (ω)  ′  =   E  E   − 1  2 S  ǫ (ω) ˜  Z  φ (ω) ˜  Z  ‡ φ (ω ′ )  ′  = ˜  Z  ‡ φ (ω) ˜  Z  φ (ω)  ′  =   E  E   − 1  2 S  φ (ω) (10)all the other terms vanishing identically.  S  ǫ (ω)  = |˜ µ ǫ (ω) | 2  ′ and  S  φ (ω)  =  ν 2  L /ω 2 stand for the spectral densities of thepump amplitude and phase, respectively. The apex  (  ′ ) indicates the omission on the right-hand side of the factor δ(ω ∗  +  ω ′ )/ C  2 .Inside the cavity the intensity variances of the singlebeamsarerepresentedbythefourth-ordertime-normalorderedcorrelation for the mode amplitude  a  j :  :  a †  j (τ) a  j (τ), a †  j ( 0 ) a  j ( 0 )  :  =  r  4  j ( 2  µ  j (τ)µ  j ( 0 ) +  µ  j ( 0 )µ  j (τ)  + i  [ φ  j ( 0 ),µ  j (τ) ]  ).  (11)TheFouriertransform  S   j  ofthefirsttermontheright-handsideof (11) (see also [9]), normalized with respect to the intensityfactor  r  4  j / 2 C  2 , separates into the three contributions S   j  =  S  µ  j  +  S  ǫ S  ǫ  j  +  S  φ S  φ  j  (12)of the quantum sources S  µ  j (ω)  =  σ  2 µ  j (ω)  +  σ  2 µ  j ( − ω)  (13)and pump amplitude and phase fluctuations S  ǫ  j (ω)  =   E  E   − 1  2 [ σ  2 ǫ  j (ω)  +  σ  2 ǫ  j ( − ω) ] S  φ  j (ω)  =   E  E   − 1  2 [ σ  2 φ  j (ω)  +  σ  2 φ  j ( − ω) ] . In turn,  σ  2 µ  j ,  σ  2 ǫ  j and  σ  2 φ  j stand for σ  2 µ  j =  k  | K  0  jk  | 2 ς  ′ k  σ  2 ǫ  j =  e i ψ  p K  0  j 0  + e − i ψ  p K  0‡  j 0 2  2 ,σ  2 φ  j =  e i ψ  p K  0  j 0  − e − i ψ  p K  0‡  j 0 2  2 . (14)Outside the OPO cavity the above expressions aremodified, with  r  2  j  replaced by  ¯  I   j  =  2 γ   j r  2  j  and  σ  2 µ  j by σ  2 µ  j  =  k  =  j | K  0  jk  | 2 ς  ′ k  +  K  0  jj − κ ′  j 2 γ   j  2 γ   j γ  ′  j + | K  0  jj | 2 ̹  j γ  ′  j  ς  ′  j .  (15)S316  Spectral properties of the single beams generated by an optical parametric oscillator In figures 4( a )–( d  ), we have plotted  S   j  (equation (13)),the normalized spectrum for different values of the OPOparameters, by omitting the laser excess noise and phasefluctuations. Figure 4( a ) represents the spectra for  E   =  3 andpump normalized damping coefficient  γ  0 /γ   (2 γ   =  γ  1  +  γ  2 )ranging from 1 to 10 (1, 2, 3, 5, 10). The value of this ratiois critical:  S   j  starts at low values of   γ  0 /γ   with a pronouncedpeak which reduces and moves to higher frequencies as theratio increases. For all the investigated values the intensityspectrum starts from below the shot-noise-level (SNL), thenit reaches a peak and finally decays monotonically towardthe SNL. The peaks coincide with the relaxation oscillationswhose frequencies move to the left as  E   increases (seefigures 4( c ), ( d  )). Figure 4( b ) shows the same spectra in areduced frequency range. In this plot is evident the reductionof the squeezing bandwidth due to the change of the dampingratio  γ  0 /γ  , and its location in a region dominated by the pumpamplitude and phase noise [7].Figure 4( c ) represents the intensity spectra calculated forthe same damping ratios as figure 4( a ) but different pumpingparameter  E   =  5. For the same ratio  γ  0 /γ   the relaxationoscillation peaks are more pronounced and located at higherfrequencies, the noise reduction is more evident in the lowfrequency range and its bandwidth is wider.The last spectra (figure 4( d  )) are calculated for fixeddamping ratio ( γ  0 /γ   =  5) and increasing pumping parameter(1 . 5  <  E   <  3 . 25). The spectrum slowly changes from aLorentzian profile toward a profile characterized by relaxationoscillations. In this case the sub-shot-noise character appearsfor pump parameter  E   >  2 . 8. 5. Conclusions ThebeamsoutlininganOPOpresentspectralfeaturesrelatedtothequantumdynamicsofthesystem. Anamplitudecorrelationbetween the twin beams, well above the quantum limit, is onlyone of the properties depending on the OPO parameters.The observation of single-beam intensity spectra offersa chance to look into the role played by the different(quantum + classical)noise sources affecting the system. Thispaperhasfocusedonthespectralresponsetothequantumnoisepresent in the three interacting modes.The system has been analysed in terms of the transferfunctions  K  0 , π 2 kk  ′  (ω)  which determine the frequency responsesto the different noise sources. These functions result from theresolution of the equation of motion for the mode amplitudeoperators  a  j (ω)  in the frequency domain. These Fouriertransforms satisfy an sixth-order algebraic system, solvedin [7], characterized by six complex poles in the half-planeIm (ω)    0. Then, relying on the Gaussian statistics of theamplitude and phase operators, fourth-order correlations canbe expressed in terms of the residues of these functions [7].For triply resonant configurations relaxation oscillationsappearforexcitationparameter  E   exceedingavaluedependingon the cavity parameters. In these cases, the single beamspresent a sub-shot-noise character in a low frequency rangewhere the OPO spectrum is blurred by the contribution of thepump amplitude and phase noise. Neglecting pump excessnoise the photocurrent spectral density of a single beam starts,for  ω  =  0, from below the SNL, rapidly increases by reachinga peak in correspondence to the relaxation frequency, and thendecays monotonically toward the SNL. The height of the peak increases notably as the pump decay constant decreases withrespect to those of the signal/idler. References [1] Reynaud S 1987  Europhys. Lett.  4  427Drummond D, McNeil K J and Walls D F 1981  Opt. Acta 28  211Milburn G J and Walls D F 1981  Opt. Commun.  39  401Collett M J and Gardiner C W 1984  Phys. Rev.  A  30  1386Yurke B 1985  Phys. Rev.  A  32  300Collett M J and Walls D F 1985  Phys. Rev.  A  32  2887Savage C M and Walls D F 1987  J. Opt. Soc. Am.  B  4  1514Collett M J and Loudon R 1987  J. Opt. Soc. Am.  B  4  1525[2] Heidmann A, Horowicz R, Reynaud S, Giacobino E andFabre C 1987  Phys. Rev. Lett.  59  2555Nabors C D and Shelby R M 1990  Phys. Rev.  A  42  556Leong K W, Wong N C and ShapiroJ H 1990 Opt. Lett.  15  1058Teja J and Wong N C 1998  Opt. Express  2  65Peng K, Pan Q, Wang H, Zhang Y, Su H and Xie C D 1998  Appl. Phys.  B  66  755Porzio A, Altucci C, Autiero M, Chiummo A, de Lisio C andSolimeno S 2001  Appl. Phys.  B  73  763[3] Porzio A, Sciarrino F, Chiummo A, Fiorentino M andSolimeno S 2001  Opt. Commun.  194  373[4] Fabre C, Giacobino E, Heidmann A and Reynaud S 1989  J. Physique  50  1209[5] Bj¨ork G and Yamamoto Y 1988  Phys. Rev.  A  37  125Bj¨ork G and Yamamoto Y 1988  Phys. Rev.  A  37  1991[6] Lee D-H, Klein M E and Boller K-J 1998  Appl. Phys.  B  66  747[7] Porzio A, Altucci C, Aniello P, de Lisio C and Solimeno S 2002Resonances and spectral properties of a detuned OPOpumped by fluctuating sources  Appl. Phys.  B, at press( Preprint   quant-ph/0112180)[8] Graham R and Haken H 1968  Z. Phys.  210  276Graham R 1968  Z. Phys.  210  319Graham R 1968  Z. Phys.  211  469[9] Yamamoto Y, Imoto N and Machida S 1986  Phys. Rev.  A  33 3243 S317
Search
Similar documents
View more...
Tags
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks