Optimum quantum states for interferometers with fixed and moving mirrors Alfredo Luis* Departamento de Óptica, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain (Received 9 November 2003; published 22 April 2004) We address a systematic approach to the study of the optimum states reaching maximum resolution for interferometers with moving mirrors. We find a correspondence between the optimum states for interferometers with fixed and moving mirrors. DOI: 10.1103/PhysRevA.69.045801 PACS number(s): 42.50.St, 03.65.2w Interferometric measurements are among the most sensi- tive measuring strategies currently available. This has prompted a significant amount of effort devoted to the inves- tigation of the limits that quantum fluctuations may impose on the precision of interferometric measurements. This means determining two items: the maximum resolution and the optimum input states required to achieve it. A well studied example is the case of the interferometers with moving mirrors, devised to detect minute forces acting on them, such as the pass of a gravitational wave, for ex- ample, Ref.[1]. While the issue of maximum resolution has been well studied[2–8], much less effort has been devoted to the topic of the optimum states. Leaving aside a very recent work [9], only quadrature coherent and squeezed states have been considered[2–8]. This situation is in sharp contrast to the case of standard interferometry with fixed mirrors, where many different families of states have been proposed in order to reach the quantum limits[10–13]. In this work we address a systematic approach to the study of the states reaching maximum resolution for interfer- ometers with moving mirrors. We solve the problem by find- ing a one-to-one correspondence between the optimum states for interferometers with fixed and moving mirrors. The key point to this result is a fully quantum description of the in- terferometer as a nonlinear input-output transformation em- bodying the mobility of the mirrors[5–8]. This approach has been already useful to demonstrate that the precision can be arbitrarily increased by increasing the power of the input light, as demonstrated in Refs.[4,6,8], in sharp contrast to the previous idea of an upper bound independent of the in- tensity (the standard quantum limit) [2,3,5,7]. In this work we show that the limits to the resolution of interferometers with moving and fixed mirrors are the same. We also show that the input-output approach allows us to easily solve the question of the optimum states. For definiteness we focus on the Michelson interferometer schematized in Fig. 1 showing the inputa1,2, internal b1,2, b̃1,2, and outputã1,2 modes whose complex amplitude opera- tors are coupled at the beam splitter: b1 = 1 Î2 sa1 + a2d, ã1 = 1 Î2 sb̃1 + b̃2d, b2 = 1 Î2 sa1 − a2d, ã2 = 1 Î2 sb̃1 − b̃2d. s1d Taking into account the mobility of the mirrors the input- output transformation can be expressed as ãj = UM † ajUM, b̃j = UM † bjUM, s2d where UM = UFUP = UPUF, s3d and UF = ei2ksL1b1 †b1+L2b2 †b2d, UP = ei2gfsb1 †b1d2+sb2 †b2d2g, s4d L1, L2 being the lengths of the arms in darkness,k the wave number, andg a constant depending on the mass and restor- ing forces of the mirrors, assumed identical[3–8]. The signal to be detected would produce a change of the phase differ- encef=2ksL2−L1d which in turn is observed as a variation of the output intensities. In the above expressionsUF represents the input-output transformation for the same interferometer but with fixed mirrors, while the nonlinear termUP accounts for the effect of the radiation pressure. It is worth noting that both factors commutefUF,UPg=0. This is the key point that allows us to establish the equivalence between interferometers with fixed *Electronic address: alluis@fis.ucm.es FIG. 1. Scheme of a Michelson interferometer showing the in- put, output, and internal modes. For the sake of illustration the incidence has been represented as non-normal. PHYSICAL REVIEW A 69, 045801(2004) 1050-2947/2004/69(4)/045801(4)/$22.50 ©2004 The American Physical Society69 045801-1 and moving mirrors. To this end let us express the input state uclM for interferometer with moving mirrors as uclM = UP †uclF, s5d so that the output state is uc̃lM = UMuclM = UFuclF. s6d This is the same output state for an interferometer with fixed mirrors illuminated by the input stateuclF. Therefore, ifuclF is optimum for fixed mirrors, thenuclM is optimum for mov- ing mirrors, irrespective of the measurement performed at the output fields. This implies two conclusions:sid the limits to the resolution of interferometers with moving and fixed mirrors are the same andsii d the corresponding optimum states are related by Eq.s5d. Concerning the first conclusion, it is known that for inter- ferometers with fixed mirrors the minimum phase uncer- tainty scales as the inverse of the mean number of photons employed in the measurement(this is referred to as the Heisenberg limit) [10,13]. From the first result listed above we get that this is also the case of interferometry with mov- ing mirrors. This is in sharp contrast with the idea of a lower limit independent of the number of photons(the standard quantum limit) [2,3,5,7]. Previous works on the subject have already demonstrated that the standard quantum limit may be surpassed by a proper choice of the input states[4,6,8,14]. In this work we have demonstrated in very general terms that any improvement of the precision beyond the standard quan- tum limit is actually bounded by the Heisenberg limit, as being the ultimate quantum limit for interferometry with moving mirrors. The second conclusion above greatly in- creases the possibilities to reach the quantum limits by using very diverse input states and measuring strategies that have already been studied in the context of interferometry with fixed mirrors, involving input states such as SU(2) squeezed states or number states[10–12]. As an application illustrating the results of this work this last example will be examined below in some detail. We can regard this last result from two different perspec- tives. On the one hand we can consider that the interferom- eter is actually illuminated by the state(5). On the other hand, Eq.(6) suggests that the optimization consists essen- tially of the removal of the radiation pressure while the in- terferometer is illuminated directly by the input stateuclF. This idea of compensation of radiation pressure has been studied in Refs.[3,4]. In any case, the transformationUP † can be accomplished in practice by propagation in nonlinear Kerr medium. The two conclusions above are direct consequences of a fully quantum description of the interferometer as a nonlin- ear input-output transformation embodying the mobility of the mirrors as introduced and developed in Refs.[5–8]. The uniqueness of the present paper with respect to the previous works are the three following results. We have used the input-output approach to derive a complete equivalence be- tween interferometers with fixed and moving mirrors. We have found that the ultimate quantum limit to the sensitivity of interferometry with moving mirrors is the Heisenberg limit. We have found the quantum states reaching this limit for different measuring strategies previously introduced within the context of interferometry with fixed mirrors. Next we illustrate these points with a suitable particular example. We consider as input stateuclF= unl1unl2 a product of an equal number of photons in the input modesa1,2 [12]. For the sake of completeness we include the possibility that the transformationUP in Eq. (5) is slightly different fromUP in Eq. (3), so that the removal of the radiation pressure is not complete. This can be taken into account by considering that unl1unl2 is the input state for the transformationUM after replacing the parameterg by a new parameterx which would be zero in the case of perfect compensation of the radiation pressure. Since the input state is eigenvector of the total number operator we find it helpful to rearrangeUP in the form sb1 †b1d2 + sb2 †b2d2 = 1 2sS0 2 + Sx 2d, s7d whereSj, j =0,x,y,z denote the Stokes operators S0 = b1 †b1 + b2 †b2 = a1 †a1 + a2 †a2, Sx = b1 †b1 − b2 †b2 = a1 †a2 + a2 †a1, Sy = isb1 †b2 − b2 †b1d = isa2 †a1 − a1 †a2d, Sz = b1 †b2 + b2 †b1 = a1 †a1 − a2 †a2. s8d Performing a similar rearrangement for the linear partUF we get that for this input state the total input-output transforma- tion can be expressed as UM = eisf/2dSxeixSx 2 . s9d Since for the output statekSl=0 the simple measurement of the output intensities does not provide suitable phase- dependent observables and other strategies must be adopted f12g. For example, we can considerM =Sz 2 as the measured observable at the output. A suitable estimator for the phase uncertainty is Df = kDMl udkMl/dfu . s10d For x=0 (perfect removal of the radiation pressure) we get kMl . 2n2sin2f, ksDMd2l . 2n4sin2f + 2n2sin2s2fd, s11d so that sDfd2 . tan2f 8 + 1 2n2 s12d and the maximum resolutionsHeisenberg limitd is obtained for f=0,p. BRIEF REPORTS PHYSICAL REVIEW A69, 045801(2004) 045801-2 On the other hand, forxÞ0 (partial removal of the radia- tion pressure) and for high enoughn we can use in Eq.(9) the approximationSx.2n cosu, whereu is defined by the ladder operators(for n.1) e±iuunl1unl2 = un ± 1l1un 7 1l2. s13d For n@1 these operators can be treated as effectively unitary e±iue7iu.e7iue±iu.1. Since the radiation pressure is a small effect, and for preparations of the interferometer around the above optimum working pointf=0, we can expand the re- sult in powers off, x retaining up to second order kMl . 2f2n2 + 32x2n4 + ¯ , ksDMdl2 . 8f2n2 + 512x2n4 + ¯ , s14d and sDfd2 . 1 2n2 + 32x2 f2 . s15d In this case the optimum resolution requires thatx!f. Otherwise very large phase fluctuations would arise. In any case, whenn→`, the phase uncertainty attains a lower boundDf=4Î2x /f that no longer depends on the number of photons employed. Therefore, the input number state is extremely sensitive to errors and imperfections in the com- pensation of the radiation pressure. It is worth extracting some further conclusions from the above calculations. For interferometers with moving mirrors there are two basic sources of quantum uncertainty: the shot noise and the fluctuations of the radiation pressure. Two ma- jor approaches have been followed to compute the joint ef- fect of these sources. On the one hand, there is a separate calculation that considers the noise sources as if they were statistically independent, leading to the standard quantum limit [2,9]. On the other hand, in a unified calculation(the approach followed in this work) the mobility of the mirrors is embodied in the input-output transformation, so that all the uncertainties manifest as shot noise[3–8]. In comparison with the separate calculation, the results of this work provide two points of discrepancy. In a separate calculation the shot noise is computed as if there were no radiation pressure. Since this last one does not depend onf, the separate analysis of this particular strategy followed in Ref. [9] concludes thatf=0 is the optimum working point with and without radiation pressure. On the other hand, we have shown above that according to the uni- fied calculation the optimum value forf is no longerf=0. As a matter of fact,f=0 would provide very bad results whenxÞ0. We think that the unified approach provides the correct result. This is because aroundf=0 we get dkMl /df.0 and the small uncertainty inM caused by the fluctuations of the radiation pressure must imply a very large phase uncertaintyDf. We can also appreciate that the dependence of the contri- bution of radiation pressure with the input intensity[the last terms in Eqs.(14)] differs from the result of a separate cal- culation [9]. This is because in the unified calculation the radiation pressure effects are seen through the fluctuations of the square ofSz. This modifies the dependence with the in- tensity in comparison with a direct calculation, where the effect of radiation pressure is computed irrespective of the detection mechanism. It is worth stressing that this refers to the comparison between separate and unified calculations just for a particular measuring arrangement(input number states and detection of the squared difference of output pho- ton numbers) so that the conclusion would not apply to other different strategies such as the one considered in Ref.[2], for example. Summarizing, we have presented and completed a sys- tematic approach to the optimization of interferometers with moving mirrors by finding an equivalence with the same problem for fixed mirrors. This allows us to translate to the former all the results already proved for the latter, including the determination of the optimal input states. For the demon- stration of the above results it is crucial to describe the in- terferometer with moving mirrors by a fully quantum nonlin- ear input-output transformation embodying the mobility of the mirrors. 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