Quadratic squeezing: An overview

N92-22059 QUADRATIC SQUEEZING: by M. Hillery, Dept. D. Yu, and J. Bergou of Physics Hunter and Astronomy College 695 Park New AN OVERVIEW* York, of CUNY Ave NY10021 I. Introduction The amplitude fixed quantity, having both mode of the electric there are always a magnitude annihilation parts of the electromagnetic mechanical fluctuations. is a complex a. It is also which of a mode quantum and a phase, operator real and imaginary field possible correspond number The field amplitude, and is described to characterize to the Hermitian is not a by the the amplitude by its and anti-Hermitian parts a, Xl=?a++a) respectively. relation These X2=_(a+-a) operators do not commute this relation complex state, plane we see whose are minimum squeezed state and, (1.1) as a result, obey the uncertainty (h=l) AX1AX2 From , applying area is at least uncertainty not be a minimum the squeeze (1.2) 1 that the amplitude in the X1 direction, need > -4 states 1/4. Coherent within states, an "error among with AX_ = AX2 = 1/2. has the property uncertainty that state, them A squeezed AX1 < I/2 (Refs.l-3). but those that box" in the the vacuum state, A squeezed are can be obtained operator a 2. s(c,) = e_" to a coherent fluctuates state(Ref.1). The phase a+2 _ , (1.3) of the complex parameter ( determines the *This work is supported by National Science Foundation under Grant No. PHY-900173 and by a grant from the City University of New York. PRECEDING PAGE BLANK 125 NOT FILMED by of direction of squeezing Squeezed described field states in terms that photons. even are examples state though cannot variables more involves variables mode one considers ab and a+b can mathematically can nonlinear optical processes We shall consideration now of these This a variable squeezing inspired. the kinds quadratic the simplest in the mode of a single we break commutator to the uncertainty number of classical behavior in generalization In the case of a single observable. products If such as more in these quadratic amplitude be measured by certain by standard squeezing and the properties example of quadratic amplitudes. It describes mode, this variable a 2 (Refs.5,6). to which they of these squeezing, possess. Following operators the example [Y1, Y2] = i(2N+l), in the square of standard (2.1) where N=a+a, and this leads relation is amplitude-squared squeezed (2.2) 2 _><N+I> in the Y1 direction 126 in parts Y2=_-(a+2-a 2) is i.e. squeezing the fluctuations into its real and imaginary AY]AY A state a large appears mode of higher-order leads be at fluctuations a and b, then can variables It should to a 2 , is one such fluctuations that a of photons(Ref.4). The simplest after disscuss means field. with by looking of a single Yl=_-(a+2+a 2) The number into fluctuations they number this procedure However, which have in the amplitude. operators At first glance This be Squeezing is perhaps of the amplitude linear, with annihilation quadratic I1. Amplitude-Squared a large amplitude. corresponds physically be converted than that is they cannot it can photon which be considered. than quantities techniques. of the amplitude, two modes have of large the mode rather of the squeezing. stochastic is nonclassical the idea of squeezing than quadratic, states, as a classical must association complicated the square state state to generalize the extent P representation(Ref.3). be modeled squeezed Thus we see that the usual is not correct. It is possible definite a squeezed In fact, a highly determines of nonclassical of a nonnegative in a squeezed noted and its magnitude if (AY1) 2 < <N+1/2>. States with this property can be written the double ordered term squared dots - <YI>) that commutation those the name the fact that (AY1) 2 K1, K2, and K2] SU(1,1). commutation relations to study onset For a classical of nonclassical was first disscussed squeezing state Y1, Y2, whose Kl= Y]/2, are satisfied. This higher-order The and by Wodkiewicz reason N relations [K2, K3] = iK] of amplitude- behavior. for this name are closely this Lie algebra commutation the normally in a paper (Ref.7). In particular = -iK3 the identification be used ordering. to the SU(1,1) K3, (2.3) 2 - <N+l> of the operators If one makes can from so one can see that the onset squeezing relations [Kb normal corresponds of the Lie algebra operators = (AY1) nonnegative Amplitude-squared under 2 "> indicate is always squeezing and Eberly This follows as < :(Y1 where are nonclassical. is described are given means squeezing to by three by (2.4) [K3, K1] = iK2 K2 = -Y2/2 and related is K3 = (N+1/2)/2, the above that the representations and this has been done of SU(1,1) by a number of authors(Refs.8-10). It is possible squeezing, to find i.e. states equality(Ref. minimum for which 11). This is done uncertainty the inequality by solving states for amplitude-squared in Eq.(2.2) the eigenvalue is replaced equation (Y1 + i_.Y2) IS°> = 13IhU> , where X is real and positive, equation have the and property 13 is complex. these equations 0 < X < 1, then imaginary parts it is clear Y1 is squeezed of A particularly The states IW> which (AY2)2 = -1- <qsl N+I that Z. plays Y2 is squeezed. to the mean values of Yl of these minimum subset I_> satisfy this 127 (2.6) . the role of a squeezing and if _. > 1, then 13are related simple (2.5) that (AY 1 )2 = _, <tI'q N-_-2 IqJ> From by an parameter. The If real and and Y2, respectively. uncertainty states occurs when 13 and _. are related. If _.> 1 and 13= (X2_ 1)1/2 (m+1/2), then the minimum uncertainty states where m is a nonnegative I°,°> = Cm(_.) S(_) Hm(i_(;_) a+) 10> Here Cm(X) is a normalization parameter _ depends The cases m=0 and m=l photon states, minimum constant, uncertainty (2.7) S(_) is a squeeze on _., Hm is the m th Hermite correspond respectively. to the squeezed for both operator polynomial Note that this implies state integer, are of the form squeezing uncertainty state the squeeze and "g_.)= [(X2- 1)a_/2z.]1_. vacuum and squeezed that the squeezed normal where one- vacuum state is a and for amplitude-squared squeezing. A second kind of minimum vacuum 10,_.>. These property that <Y1 >=<Y2 > = O. Such whose numbers satisfy are multiples We now come amplitude states mode, If the has frequency 20) then states b. This mode the with 13=0 which implies are superpositions this Hamiltonian, of fluctuations is accomplished described Hamiltonian transferred from mode using that they of photon number a to mode b. First harmonic of second 0) and that described corresponds theory, define B(t) = e2i_tb(t) = 1[ B+(t) + B(t) ] to this process Y1A(t) = _-[ A+(t) 2 + A(t) 2 ] is initially one can find the slowly varying operators (2.9) , Y2A(t) = _- [ A+(t) 2- A(t) 2 ] 128 state, is how fluctuations X2B(t) = _- [ B+(t) - B(t) ] in a coherent by b (2.8) (2.10) find, if the b mode states by means and We then the of the mode by a has frequency which perturbation A(t) = eimta(t) X1B(t) have in a2 into fluctuations H = 0)a+a + 20)b+b + k2(a+2b + a2b +) . From squeezed of 4. to the conversion of a second generation(Ref.5). Eq.(2.5) is the amplitude-squared that after a time t are (AX1B(t))2=1__4 + (k2t)2[(AY2A)2- <NA + 1>] (2.11) (AXzB(t))2=¼+ (ket)2[(AY1A)2- <NA + 1>] where quantities at t=0. What squeezed sense without these a time equations direction. the Y1 direction harmonic normal squeezing. Because normal results first the signal second Similarly, squeezing suggests harmonic. into converts If it is squeezed, doubler then squeezed in the and then the original the be detected. measures signal the into detection can in Therefore, squeezing via homodyne squeezing normal squeezed amplitude-squared amplitude-squared a frequency amplitude-squared in the X2 direction. can be measured how to be evaluated is amplitude-squared squeezed process is initially will become if the a mode will become generation preceding sends then the b mode the b mode second e.g. (AY1A) 2, are assumed tell us is that if the a mode in the Y2 direction in the XI argument, One the squeezing of the was amplitude-squared squeezed. Finally, let us see how The fact that the squeezed that a degenerate states. parametric 2). Well above the principle, be arbitrarily studied produce II1. state amplifier threshold of amplitude-squared threshold been vacuum amount large. in connection The with squeezing, effect(Ref.14). frequencies imaginary fourth can be produced. squeezed amplitude-squared parametric the cavity can shows squeezed oscillator reach can as a maximum by (AY1) 2 / <N + 1/2> = 1/2, but just below squeezing in the output subharmonic generalized generation field can, process, squeezed states(Ref.13), to amplitude-squared squeezing, in which can has also squeezing(Ref.6). Squeezing Sum squeezing. inside of amplitude-squared amplitude-squared Sum produce given states amplitude-squared a degenerate the field squeezing squeezed is also can As one of us (D. Yu) has shown, well(Ref.1 level amplitude-squared as opposed In fact, amplitude-squared Let us consider two modes (oa and O)b. The variables parts of the product squeezing is the degenerate with annihilation involved in sum is a two operators squeezing limit V2 = _- (a+b+- ab) 129 of sum a and b and are the ab, i.e. V1 = { (a+b + +ab) mode (3.1) real and The commutator of these NB = b÷b, which yields operators the uncertainty AV 1 AV 2 A state a state closely 2 < V2 ] = _- ( NA + NB + 1 ), where NA = a+a and relation NA + NB + l > 1< 4 (3.2) in the V1 direction l<NA if + NB +1> (3.3) is nonclassical. The commutation and > is said to be sum squeezed (AV1) Such is [ Vl, related to those relations of the operators of the SU(1,1) K3 = 9]---(N,_+ NB +1) one obtains V1, V2 amd Lie algebra. the SU(1,1 In fact NA + NB +l if one sets ) commutation are also K1 = Vl, K2 = - V2 relations given in Eq.(2.4). The name, sum squeezing, comes converted into normal squeezing frequency generation is a three-mode from the fact that this kind of squeezing by the process process of sum which frequency is described by the H = O)aa+a + o.,'bb+b + (OcC+C+ ks ( ca+b + + c+ab ) , where oc=%+%. similarly As before we define for B(t) and C(t). We also 1 V](t)=_-(A If the c mode is initially as before, in a coherent this equation quantities to Eq.(3.3) the a and b modes state then without +NB+ a time argument A(t)=ei_ta(t), and order 1>] in ks we find (3.6) , are evaluated will be squeezed at t=0. Comparing in the Xc2 direction if in the V1 direction. are uncorrelated, modes operator (3.5) to second [ (AV1)2-1<NA are sum squeezed in the individual Hamiltonian (3.4) i XC2=_-(C+-C) we see that the c mode If the a and b modes squeezing varying is Sum define + + B +AB) (AXc2(t))2=¼+(kst)2 where, the slowly generation. then there and sum squeezing. 130 is a connection In particular, between if neither mode is squeezed, then and the other modes the state is in a coherent are squeezed, This then connection considering by the squeezed. state, then produced If one of the two the state the resulting disappears the state is described is not sum state is sum squeezed. are correlated. the vacuum This by a parametric when again, that if both is in a highly the a and b modes possible two the amount source modes squeezing IV. amplifier. light. this Hamiltonian state, increases A further then (3.8) with time calculation Therefore, and this device shows that for correlated is a neither modes squeezing is also a two-mode to difference-frequency operators effect(Ref.14). generation. a and b, and we assume We Its name again describe that COb> O3a.The commutator is given the uncertainty (4.1) W2 = _-(ab + - a+b) by - NB) , (4.2) relation AW] AW2-> 1]<NA - NB >] . 131 comes the observables it are yields we find of the normal for sum squeezing. [W], W21 = ½(NA which Using in the vacuum sense. Wl = 21-(ab + + a+b) Their state. Squeezing connection describe system to that in Eq.(3.4) + NB(t) + 1> =- 2Lsinh 2 (gt) in the normal is not a prerequisite annihilation coherent of sum squeezing is squeezed Difference This by (3.7) is an approximation are originally of sum squeezed Difference close Hamiltonian excited (AVI(t)) 2 " ¼<NA(t) Therefore, be seen Hamiltonian e0c = o_a+ eb. This the c mode if both squeezed. can H = (Oaa+a + CObb+b + g (e-i_ta+b + + elm°tab) where, is squeezed Finally, may or may not be sum if the modes from modes (4.3) from modes which its by A state is said to be difference squeezed (AWl)2 Note that for a state Difference regard latter, same. them condition For difference squeezing (AW which is not the same squeezed states instead used 4]__ <NA + NB > (Ref.15). The SU(2) Lie algebra commutation relations are [Jk, Jd = ieldmJm where all indices run from states. in this For both nonclassical is nonclassical Eq.(4.4). of the are the if Therefore, difference regime. which consists for being to a Lie algebra representation <NB>. (4.5) condition related > is a difference , In fact, the operators in the Schwinger <NA> squeezed A state the nonclassical is also have but there and the condition this is not true. within squeezing of SU(1,1). are nonclassical as the squeezing are well Difference those 1 )2 < we must or amplitude-squared for squeezing if (4.4) squeezed states and sum direction A_ NB > to be difference squeezed between the < I<N in the Wl describe but this time difference of the angular of three squeezing momentum operators it is SU(2) are operators J1, J2 and J3 whose (4.6) , ! to 3 and eke= is the completely antisymmetric tensor of rank 3. If the modes difference coherent squeezing state A necessary, correlated, describing I(x>, the state then it is no longer then at least one of them If the b mode wil be difference is squeezed squeezed must be squeezed and the a mode but only if Io_12 is large condition is that <NB > < _/-I(zl 2. If the modes true that squeezing in the individual modes for is in a enough. are is required squeezing. Finally, normal to be present. but not sufficient for difference into are uncorrelated, as might squeezing this process be suspected by difference from the name, frequency is 132 difference generation. The squeezing Hamiltonian is turned H = maa+a + o_ob+b + COcC+C+ kd(a+bc + + ab +c) where COc= O)b- (0,. We define section and then Wl(t) Using slowly varying , (4.7) A(t), B(t), and C(t) as in the previous set = 1(A(t)B+(t) perturbation theory + A+(t)B(t) ) Xc2 = _- (C+(t) - C(t) ) we find that if the c mode (4.8) is originally in a coherent state, then (AXc2(t)) This equation Therefore, 2 = _- + (kdt)2[(AWl)2 shows - ¼ <NA us that Xc2 becomes difference frequency squeezed generation can (4.9) - NB>] if W1 is difference be used to detect squeezed. difference squeezed light. V. Amplification of Higher-Order An amplifier ground states We shall This Poissonian these gain, Friberg, photon effects and limit Recently (Ref.18). particular, Mandel statistics and two. We found of us looked that thoroughly where at time normal the effect at the situation 1 An input signal than signal N1. is put amplitudes at G is the amplitude of amplification amplification gain. 133 on sub- found state in. This that both of is, if the intensity as the photon goes cloning limit gain has stood as behavior. for amplitude-squared for gains will be present ' They what the input one that of nonclassical I G 12< 2 + <No> + 1/2 N2 is greater t. The squeezing(Ref.17). no matter squeezing are in their by Carusotto(Ref.16). examined it can survive amplitude-squared states of this system. out for every for the amplification two excited The gain IGt2= 2 is known photons N1 of which by <a(t)> = G <a(0)> where at the output, than one gets two an upper rather atoms regime at the output are related disappear IGI 2, is greater because are in their at t=0 and emerges was analyzed Hong, of two-level that we are in the linear and the output system of a collection and N2 of which assume into the amplifier the input consists Squeezing slightly at the output (5.1) squeezing greater if than two. In where <No > is the photon greater than squared two this suggests squeezed shows that there at the output the photon cloning nonclassical behavior. nonclassical states which substantially larger than squeezed limit does which greater side two. Further is investigation 10,;L>, with _,<<1 is such not, at least in principle, interest nonclassical the right-hand will still be amplitudethan vacuum, be of considerable remain Because are states if [GI 2 is slightly It would can state. when represent to know they a state. a barrier if there to are are amplified at gains two. Conclusion Quadratic this property which of the input that the amplitude-squared Therefore, Vl. number squeezing have fluctuations is quadratic quadratic represents in mode squeezing can a new class smaller creation than of nonclassical is possible and annihilation be converted into normal for classical operators. squeezing effects. light States with in a variable As we have by Z (2) type seen, nonlinear interactions. 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