Squeezed Light

Squeezed Light Thomas Rotter University of New Mexico, Department of Physics and Astronomy, Albuquerque, NM 87131, USA Dated: January 28, 2000 Abstract This pap er gives an overview on squeezed light with an emphasis on its generation. Squeezing means, that the uctuation in one of two conjugate comp onents e.g. amplitude and phase is suppressed while enhanced in the other. Consequently squeezing can b e generated by phase sensitive pro cesses. Such pro cesses are known from nonlinear optics, e.g. second harmonic generation, parametric down{ and up conversion and four wave mixing, and they have b een successfully used to generate squeezed light. Application of squeezed light is still rare b ecause it is dicult to maintain the squeezed character. PACS numb er: 42.50 1 I. INTRODUCTION A classical electromagnetic eld consists of waves with well de ned amplitude and phase. However, in a quantum treatment, uctuations are asso ciated with b oth conjugate variables. Equivalently the eld can be describ ed in two conjugate quadrature comp onents and the uncertainties in the two conjugate variables satisfy the Heisenb erg uncertainty principle. A coherent state, that most nearly describ es a classical electromagnetic eld, has an equal amount of uncertainty in the two variables, and the pro duct is the minimum uncertainty. This case is usually called the shot noise limit. While the pro duct of the uncertainties in the two conjugate quadrature comp onents cannot be minimized further, it is p ossible to reduce the uctuations in one of them well b elow the quantum limit. The canonically conjugate quadrature comp onent must then display enhanced uctuations in order to ful ll the Heisenb erg uncertainty principle. Those states are called squeezed states. Squeezed light with uctuations b elow the standard quantum limit in one of the quadrature comp onents has many attractive applications, e.g. in optical communication, in pre- cision and sensitive measurements such as gravitational wave detection, or in noise free ampli cation. Squeezed light is however extraordinarily fragile and may be degraded even by a b eam splitter or a mirror, as those admit the vacuum uctuations from outside to enter, which exceed the squeezed uctuations. In this pap er I will brie y describ e the theory of squeezed states and their prop erties. The main fo cus will b e on generation of squeezed light by a variety of pro cesses, including 2 3 - and - nonlinearities. I will present exp eriments that have b een successfully carried out, as well as brie y describ e theoretical approaches to investigate the generation pro cesses. Comparisons b etween theoretical and exp erimental results will b e given. II. SQUEEZED STATES A Theoretical considerations Consider a quantized single-mo de electric eld of frequency [1]: i t y i t ~ E t=E~âe +â e 1 y where a^ and a^ are the non-Hermitian annihilation and creation op erators, resp ectively, y ob eying the commutation relation [â; a^ ] = 1. The eld can also be written in terms of ^ ^ Hermitian quadratures X and Y , ~ ^ ^ E t=2E~X cos t + Y sin t: 2 1 1 y y ^ ^ Here X = â +â and Y = â a^ are dimensionless versions of the p osition and 2 2i momentum op erators x^ and p^. The uncertainty relation 1 ^ ^ X Y 3 4 i ^ ^ . If the equality in the uncertainty relation follows from the commutation relation [X; Y ]= 2 3 holds, then the minimum amount of uctuation, the shot noise limit, is reached. Further reduction of uctuation, squeezing, is p ossible only in one of the two canonically conjugate variables 1 2 ^ X < i = 1 or 2 4 i 4 at the exp ense of increasing uctuation, antisqueezing [11], in the other, so that 3 is still satis ed. An ideal squeezed state is obtained if in addition to 4 the equality in the uncertainty relation 3 also holds. Figure I I A illustrates a coherent state in the phasor plane. The exp ectation value of the ^ ^ annihilation op eratora ^ = X + iY has amplitude and phase . The hatched areas indicate the probability distribution for an event yielding a phasor terminating in one of the p oints in the phasor plane, the circle indicating the ro ot-mean-square deviation of the distribution. This isaschematic illustration of the Wigner distribution. FIG. 1: Representation of a coherent state in the complex phasor plane with X along the real axis and Y along the imaginary, and b elow Y versus phase = !t. [2 ] Figure I I A shows a squeezed state. The Wigner distribution is represented by an ellipse. There are two phases to be considered, that of the exp ectation of the phasor and that of the orientation of the ellipse of the Wigner distribution or in Fig. I I A. The area of the ellipse is the same as the area of the circle of the phase-indep endent zero-p oint uctuations, since the uncertainty principle must b e ob eyed. As shown in Fig. I I A, the ma jor and minor 1 1 +s s e and e , where s is a squeezing parameter. axes of the ellipse dep end on each other as 2 2 s = 0 corresp onds to a coherent state with no squeezing, as depicted in Fig. I I A. For the general case of Fig. I I A and Fig. I I A the unitary squeeze op erator [7] is given by 1 1 2 y2 a^ a^ ^ 2 2 S = e ; 5 where is the complex squeeze parameter, which dep ends on s and from Fig. I I A, i = se ; 0 s<1; 0 2: 6 The vacuum coherent state j0i corresp onds to Fig. II A with = 0, likewise squeezed ^ vacuum j0 i = S j0i corresp onds to Fig. II A with = 0, or to Fig. II A. In contrast to s FIG. 2: Representation of a squeezed state in the complex phasor plane. X along the real axis and Y along the imaginary, and b elow Y versus phase = !t. [2 ] squeezed vacuum, the situation 6=0 is called bright squeezed light. Every squeezed state can then be describ ed as ^ ^ j ; i = D S j0i 7 where y a^ a^ ^ D =e 8 is a coherent displacement op erator [7]. The squeeze op erator 5 transforms the annihilation op erator according to the Bogoliub ov-transformation 1 y i ^ ^ a^ ! S âS =a ^cosh s a^ e sinh s: 9 ^ ^ The mean of the quadrature op erators hX i, hY i corresp ond then to the real and imaginary part of , resp ectively, as shown in Fig. I I A. The variances are 1 1 1 2s 2 2s 2 2 e cos +e sin ; 10 hX i = 4 2 2 1 1 1 2s 2 2s 2 2 e sin +e cos ; 11 hY i = 4 2 2 FIG. 3: Uncertainty ellipse of the ideal squeezed vacuum state. [7] corresp onding to Fig. I I A. Detection of squeezed states requires a phase sensitivescheme that measures the variance of a quadrature 4 of the eld. In homodyne detectors the squeezed input state is sup erp osed by a b eamsplitter on the mo de from a lo cal oscillator with the same frequency, and this sup erp osition is measured by a photo detector. The principle is somewhat similar to that of a lock in ampli er, which is familiar to every exp erimentalist. Squeezing in the input mo de is then revealed by sub-Poissonian photo count statistics. In a balanced homo dyne detector b oth output p orts of the b eamsplitter are measured and subtracted from each other. This technique removes the noise contributions that are made by the input signal and the lo cal oscillator alone. A thorough discussion of these and other detection schemes can be found e.g. in [7] or [1]. III. GENERATION OF SQUEEZED STATES To generate squeezed light from coherent light is to reduce uctuation in one quadrature comp onent, while enhancing it in the canonically conjugate comp onent. As can be seen from 2, this is a phase sensitive pro cess. Such phase sensitive pro cesses are known from the nonlinear interaction of light with matter, e.g. second harmonic generation, parametric down- and upconversion and four wave mixing. All of these pro cesses have in fact b een successfully employed to generate squeezed light. A Second harmonic generation Although most of the earlier exp eriments on squeezing have concentrated on the frequency downconversion pro cess, the reverse pro cess, namely second harmonic generation SHG, 2 has also attracted considerable attention. It is one of the simplest - or second order nonlinear optical pro cesses and it can create nonclassical light for b oth the fundamental and the harmonic elds [3, 4, 5]. Substantial squeezing arises if the nonlinear crystal is suitably long or if it is placed in an optical cavity to resonate the fundamental or the second-harmonic mo de or b oth. Although noise reduction as high as 52 was observed with an actively stabilized doubly resonant cavity [10], it is quite dicult to maintain doubly resonant condition for a suciently long time. Therefore singly resonant cavities, in which only the fundamental mo de is con ned, have b een prop osed [9] to generate squeezed light. A theoretical analysis by Paschotta et al. [9] predicted that such a system can pro duce 9.5 dB of squeezing in the second harmonic output and demonstrated this in an exp eriment featuring a monolithic nonlinear device.

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