P 8 YS I C A L R E V I E%' A VOLUME 13, NUMBER 6 Two-photon coherent states of the radiation field* Horace P. Yuen Research Laboratory of Etectronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received 26 June 1975; revised manuscript received 3 March 1976) The concept of a two-photon coherent state is introduced for applications in quantum optics. It is a simple generalization of the well-known minimum-uncertainty wave packets. The detailed properties of two-photon coherent states are developed and distinguished from ordinary coherent states. These two-photon coherent states are mathematically generated from coherent states through unitary operators associated with quadratic Hamiltonians. Physically they are the radiation states of ideal two-photon lasers operating far above threshold, according to the self-consistent-field approximation. The mean-square quantum noise behavior of these states, which is basically the same as those of minimum-uncertainty states, leads to applications not obtainable from coherent states or one-photon lasers. The essential behavior of two-photon coherent states is unchanged by small losses in the system. The counting rates or distributions these states generate in photocount experiments also reveal their difference from coherent states. I. INTRODUCTION In general, a two-photon coherent state differs from a coherent state in several ways: they are In a recent study' of quantum communication the- generated by different photon processes, they have ory it was found that if optical radiations can be different quantum statistical properties, and they generated in a minimum-uncertainty state"' xather have different coherence properties. Basically, than a coherent state, significant performance coherent states are generated from ideal one- improvement may result in an optical communica- photon stimulated processes, whereas TCS's are tion system. For a fixed radiation mode of fre- obtained from ideal stimulated two-photon process- quency v with photon annihilation operator u, the es for two photons of the same mode. The usual coherent states ~a) (a ~o.) =a~ a)) have relatively multiphoton parametric processes' are one-photon large quantum fluctuations when &u/2m ~ 10"Hz. processes in the present sense, as they involve a Writing a = a, + ia, for self-adjoint a„a„acoherent single photon from each separate mode. The state ~o) gives (AaP =(ha', ) = —,. A noise energy' quantum noise properties of TCS's are basically —,h~ is then obtained if either a, or a, is measured the same as those of minimum-uncertainty states. in optical homodyne detection. This quantum noise Minimum-uncertainty states and cohexent states is frequently dominant over other noise sources in can be easily confused. The above illustration in optical communications; for example, an equiva- terms of (M', ) and (M,') is a good indication of lent noise temperature of —3400 K is obtained for their difference, if we note that they describe the &Su at the YAlG laser frequency. However, states fluctuations of the field variables directly mea- ' with (ha', ) « —, and correspondingly larger (da', ) sured in a homodyne experiment. Further careful are permitted by the uncertainty principle discussions will be found in Sec. IIIE and else- ~ (M', ) (M2) —,'„such as the minimum uncertainty where in the paper. states with (4a,') (rhg22) =,—', . If one can generate Minimum-uncertainty states were studied at the such small (4a', ) states by an explicit physical very early days of quantum mechanics; the Ken- process, it should not be surprising that they may nard packets' are particularly familiar in the con- be profitably used in many applications. text of electrons. It is therefore no surprise that Minimum-uncertainty states cannot be obtained, much previous work has touched on various as- however, from available optical sources, all of pects relating to the present two-photon coherent which generate coherent states and their random states, including much work on quantum field the- or classical superpositions. An investigation of ory and interacting Bose fluids. It is impossible the possible ways to generate minimum-uncer- for the present author to give an accurate histori- tainty states leads to a broad class of radiation cal survey of the literature pertaining to minimum- states that we call "two-photon coherent states" uncertainty states and their generation by quadrat- (TCS's). They include the minimum-uncertainty ic Hamiltonians. The particularly relevant papers states of which the coherent state is a special now known to the author include Refs. 9-13 in case. A detailed study of these TCS's is x eported quantum optics and Refs. 14-17 in quantum field in this article. A few of our results have been theory. Nevertheless, it appears that TCS's have briefly stated in Ref. 6. not been systematically studied before, at least in 2226 TWO- P HO TON COHE RENT STATES 0 F T HE RADIATION FIE LD 222'7 the context of quantum optics. These and other ad- frequency v, we have the usual canonical variables ditional papers will be mentioned in relation to (q, p), some of our results whenever appropriate. —= (ff/2&d)'/'(a +a), —= i( 'fi—m)'/'(a —a) . (2.4) This work is not at all a study of possible math- q P , ematical generalizations of coherent states. In In general, we can introduce two dimensionless fact, the TCS's can be considered as "coherent self-adjoint operators (a„a,), states" in a broad sense, as discussed later. How- a -=a, +ia, =a, . (2.5) ever, they should be distinguished from the coher- ; a, =a„a, ent states of Glauber, ' because in quantum optics We also write the term coherent state is usually understood in Q=-Q, +iQ„. (2.8) the Glauber sense which carries the implication Q„Q, real, of equal fluctuations in the quadratures a, and a,. and refer to (a„a,) or (n„n,) as the quadrature In Sec. II we establish notations and briefly re- components (quantum or classical) of the radiation capitulate the basic results in quantum optics used mode. These quadrature components are the nat- in this paper. For a single degree of freedom, the ural variables in describing a TCS, even when the definition and properties of two-photon coherent mode has a single frequency. Defining the eigen- states are developed in Sec. III, together with a states of o. of l n, ), a„ l ,), a„ discussion of the contrast between TCS's and ordi- nary coherent states. The generation of TCS's 9 Q =Q Q, Q~ Q ~=Q~ Q~ (2.7) from quadratic Hamiltonians is presented in Sec. have IV, with illustrative examples. The general struc- we ture and properties of the corresponding unitary o.') = (2/v)'/' —n')'+ & n'(n'- n'*)] . , (o&, l exp[-(n, transformation are fully determined. Physical interpretation of quadratic Hamiltonians is devel- (2.8) oped in Sec. V in terms of stimulated two-photon From (2.6) it follows easily that processes, which include two-photon lasers and degenerate parametric amplifiers. We argue that (2.9) TCS's describe ideal two-photon lasing states in (2.10) analogy to the coherent-state description of one- where =— — photon lasers. The approximation which leads to (ttM') ((M (M))') for an operator M. this conclusion is formalized in the Appendix. We The angular bracket (M) represents the expected also show that the mean-square noise behavior of value trpM of M with respect to a quantum state which TCS's remains essentially the same when the ef- p may be either pure or mixed. The uncer- fect of small damping and additive noise are in- tainty principle on (a„a,) is cluded. In Sec. VI we briefly discuss the coher- (~', )(~', & - —,', . (2.11) ence properties and counting statistics of two-pho- ton coherent states. The usefulness of TCS's in Thus ln) is a minimum-uncertainty state. From certain applications is brought out in Sec. VII. The the arithmetic-geometric mean inequality, (2.11) detailed theory of multimode TCS's will be given implies in a separate treatment. (Aa', ) +(4a', ) - —,', (2.i2) so that can be characterized as the with l n) states II. COHERENT STATES AND OPERATOR both minimum-uncertainty product and minimum- REPRESENTATION uncer tainty sum. For a single mode of the field with photon anni- The differential operator representation of hilation operator a coherent state'@ &9 is an in the coherent-sta. te representation is" " a, l n) M(a, a) eigenstate of a, aQ =QQ (2.i) It can be obtained from the vacuum state l0) via from which one can show — the unitary displacement operator D(o&) = e ' (n a) =(ct p)M&"'(ct*, lM(a, l p) l p), (2.S4) =D(o) . (2.2) le) l 0) where From (2.2), M&"'(n*, a) = (n lMl n&- (2.i5) )-&/ ae- I n& / 2 an(s ) (2.3) is the normal-order form' of M. Note that it is If the mode has a harmonic time variation with important to keep the separate dependence of M~"' 2228 HORACE P. YUEN "' on o(* and o( in (2.14)—M is a function of two ability density function" describing the outcome complex variables ((x*, a). Equation (2.14) explic- statistics of a simultaneous quantum measurement itly shows the fact" that M'"'(o. *, n) un;quely de- of two noncommuting observables"'" (or a mea- termines M. In terms of U("'(n*, a;t, t,), the surement of the non-self-adjoint operator a, opti- Schrodinger equation ik SU/St =H(a, a)U becomes" cal heterodyning). " We will use p("'(a*, a) or the characteristic functions to represent a density (2.16) operator, particularly because P(c(*, n) for a TCS is too singular. In terms of p'"', we have These results are at least valid for bounded oper- ators or unbounded operators which are polynomi- p= e"~ ~" p'"'(n*, n)e "' e" 'd'ed'g m', 2.25) als in a and a .