By an explicite calculation of Michelson interferometric output intensities
in the optomechanical scheme proposed by Marshall et al. (2003), an oscillatory
factor is obtained that may go down to zero just at the time a visibility
revival ought to be observed. Including a properly tuned phase shifter offers a
simple amendment to the situation. By using a Pockels phase shifter with fast
time-dependent modulation in one arm, one may obtain further possibilities to
enrich the quantum state preparation and reconstruction abilities of the
original scheme, thereby improving the chances to reliably detect genuine
quantum behaviour of a nanomechanical oscillator.Comment: For Proc. DICE-2010 (Castiglioncello), to be published in J. Phys.
  Conf. Ser., 201

Adler S L

Saleh B E A

Tamás Geszti

Vanner MRef al.

Journal of Physics : Conference Series

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To make a nanomechanical Schrödinger-cat mew

Geszti, T

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arXiv:1101.4308v1  [quant-ph]  22 Jan 2011To make a nanomechanical Schrödinger-cat mewTamás GesztiDepartment of Physics of Complex Systems, Eötvös University, Budapest, HungaryE-mail: geszti@elte.huAbstract. By an explicite calculation of Michelson interferometric output intensities in theoptomechanical scheme proposed byMarshall et al. [1], an oscillatory factor is obtained that maygo down to zero just at the time a visibility revival ought to be observed. Including a properlytuned phase shifter offers a simple amendment to the situation. By using a Pockels phaseshifter with fast time-dependent modulation in one arm, one may obtain further possibilities toenrich the quantum state preparation and reconstruction abilities of the original scheme, therebyimproving the chances to reliably detect genuine quantum behaviour of a nanomechanicaloscillator.Detecting genuine quantum effects in nanomechanical systems is a highly desirable scope,expected to be reached in the near future. The major challenges are: efficient cooling closeto the ground state, sufficiently strong coupling to a well-identified quantum system, reliablepreparation and identification of non-classical states of the nanomechanical component. Most ofthe experimental studies are done on mechanical oscillators, considered harmonic for the smalldisplacements characteristic for the quantum domain. For those systems ground-state coolingmeans kBT < ~ω which is relatively easy to fulfil for hard (high-frequency) oscillators [2]. Forthem, however, preparing and analyzing quantum states is overly demanding [3]. The efforts toreconcile these conflicting requirements have seen rapid and most competitive advance with theparticipation of a number of experimental groups. A still more ambitious, so far elusive prospectof related studies would be to detect deviations from standard quantum mechanics, predictedby several theoretical studies [4].An original and promising project, using a soft (low-frequency) nanomechanical oscillator,coupled by a mirror to one of the paths of a Michelson interferometer, and using Fabry-Perotcavities (on both paths, to preserve interference) to strengthen the optomechanical coupling, hasbeen started by Marshall et al. [1], and following extensive development of an optical feedbackcooling technique [5], is now approaching the level of seeing quantum effects.According to the argument of Ref. [1], for a single-photon source of frequency ωc, and avibrating mirror of effective mass M and vibration frequency ωm, acting as one of the mirrorsdelimiting a Fabry-Perot resonator cavity of length L, the dimensionless coupling constant isκ = (ωc/ωm)(√(~/2Mωm)/L). With the vibrating-mirror cavity on interferometer arm A, andthe cavity with two rigid mirrors on arm B, starting with a mirror in its quantum mechanicalground state |0〉m, coupling during time t results in an entangled photon-mirror state|Ψ(t)〉 = e−iωct√2(eiϕ(t)|A〉|α(t)〉m + |B〉|0〉m)(1)where |A〉 = |1〉A|0〉B and |B〉 = |0〉A|1〉B denote one-photon states with the photon in arm Aand in arm B resp., whereas, |α(t)〉m and |0〉m are two coherent states of respective complexamplitudes α(t) and 0 of the vibrating mirror, appearing as a pair of Schrödinger cat statesentangled to the respective orthogonal photon states |A〉 and |B〉. Finally,ϕ(t) = κ2(ωmt− sinωmt); α(t) = κ(1− e−iωmt). (2)The density matrix R̂(t) = |Ψ(t)〉〈Ψ(t)| corresponding to the pure state |Ψ〉 can be expandedon the photon state basis |A〉 =(10), |B〉 =(01)in the form of a two-by-two matrixR̂(t) =(ρ̂AA(t) ρ̂AB(t)ρ̂BA(t) ρ̂BB(t)), (3)where the matrix elements are operators acting on variables of the vibrating mirror. It hasbeen pointed out [1, 6] that the crucial quantity characterizing the interference pattern to beobserved is the nondiagonal element on that basis, ρ̂AB(t) = 〈A|Ψ(t)〉〈Ψ(t)|B〉, which carriesextra information about the motion of the vibrating mirror, missing from the more usual reduceddensity matrix ρ̂m(t) = ρ̂AA(t) + ρ̂BB(t), obtained by tracing out R̂(t) over photon variables.Interference visibility, as expressed by ρ̂AB(t), carries a signature of the motion of thevibrating mirror, but the usefulness of that signature is seriously limited by interactions of themirror with the environment. Neglecting direct environmental effects on the photon subsystem,the above expansion remains valid, allowing decoherence and friction of the vibrating mirror tobe included in the dynamics of ρ̂AB(t) and analyzed by standard tools [6, 7].Returning for a moment to the decoherence-free case covered by Eq. (1), let us quote theresult of Ref. [1]:Trm ρ̂AB(t) =eiϕ(t)2Trm [|α(t)〉m m〈0|] =12eiϕ(t)−|α(t)|2/2, (4)where Trm means trace over mirror variables. From here one concludes that the crucial quantityto keep under control is |α(t)|, and - using Eq. (2) - the observation should be carried outat multiples of time t = 2π/ωm, where |α(t)| returns to 0 and the interference signal can belarge enough to be observed, under the extremely stringent condition that decoherence could besuppressed by sufficient cooling and isolation from mechanical supports.That treatment catches the essential point of the phenomenon but fails to give a correctdescription of the underlying oscillatory behaviour, which is relevant for a reliable identificationof superposition states of the mirror. To show that, we proceed by noticing that Eq. (1) is notthe final state observed by the photon detectors, since the photons pass once more through thebeam splitter, before reaching output ports C (facing A) and D (facing B). On the time scaleof whatever happens to the vibrating mirror, this is a fast and coherence-preserving unitaryoperation on the photon subsystem, transforming the intermediate state (1) into the final one12e−iωct[|C〉(|0〉m + ieiϕ(t)|α(t)〉m)+ i |D〉(|0〉m − ieiϕ(t)|α(t)〉m)](5)with |C〉 = |1〉C |0〉D = (|A〉 + i|B〉)/√2 and |D〉 = |0〉C |1〉D = (i|A〉 + |B〉)/√2. Carrying outthe above unitary transformation on the full density matrix, now we can trace over mirror (andeventually, for a full analysis, environmental) variables to obtain directly the density matrix ofthe photons to be detected, in the formRphot =(Trm ρ̂CC Trm ρ̂CTrm ρ̂DC Trm ρ̂DD,)(6)with the diagonal elementsICID}=Trm ρ̂CCTrm ρ̂DD}=12∓ Im (Trm ρ̂AB) (7)describing the output intensities on the two respective ports, and the off-diagonal elementsTrm ρ̂CD = [Trm ρ̂DC ]∗ =12Re (Trm ρ̂AB) + i(12− Trm ρ̂AA)(8)describing cross-correlations between the two detectors.Of particular interest are the output intensities, and we see from Eq. (7) that, instead of themodulus of Trm ρ̂AB(t), they display the imaginary part of the same quantity. Evaluating, likebefore, the decoherence-free case, we obtainICID}=12(1 ∓ e−|α(t)|2/2 sin [ϕ(t) + χ(t)])(9)=12(1 ∓ e−κ2(1−cos ωmt) sin[κ2(ωmt− sinωmt) + χ(t)]), (10)where we have used Equation (2), and for the discussion to follow, included an additional phaseshift χ(t) created by a phase shifter device on arm A of the Michelson interferometer, not presentin the original setup [1].Having to measure the imaginary part of Trm ρ̂AB(t) makes a huge difference. Without thephase shifter, that factor oscillates as sinκ2(ωmt − sinωmt), which - depending on the actualvalue of the optomechanical coupling constant κ - may go down close to zero just about time(2π/ωm)n, when the nth visibility revival is expected. There is a simple amendment though:one should include a constant phase shifter and tune it toχn = 2π(2n+ 14− κ2), (11)to match the nth maximum of the oscillatory factor with the expected revival time. With that,revivals become observable in the interferometric intensity patterns, as soon as decoherenceeffects are suppressed to the desired level [8]; of course, to see at least n = 1 would be theminimum requirement. Alternatively, if noise level of data allows, scanning with χ to locate themaximum signal can be a way to measure the coupling constant κ.The above reasoning connects the Marshall et al. scheme to a simple token introduced byYurke and Stoler [9], used in various experiments [10, 11] to overcome some of the difficultiesassociated with the entanglement present e.g. in Eq. (1), and create a visible interferencepattern of the two Schrödinger cat components, as apparent in Eq. (5), observable throughstraightforward intensity measurement of a chosen component of the associated two-statequantum system. In the Marshall et al. scheme it is the last passage through the Michelsonbeamsplitter that - acting as a π/2 quasi-spin rotation of the two-state photon basis - offers akind of gratis implementation of the Yurke and Stoler procedure; this was the original motivationfor the present work.Alas, just seing the visibility revival would not prove quantumness [7]. Therefore to reachthat final scope it may be necessary to add a time-dependent phase shifter χ(t) to the photonsubsystem. That can strongly enhance the possibilities of preparing and analysing quantumstates of an oscillator, as demonstrated by the successful application of phase controllingprotocols in ion-trap Schrödinger cat experiments [10]. Optical phase shifters based on thePockels effect are commercially available, and admit electric modulation at frequencies up to afew GHz [12], with the potentiality to make the Marshall et al. nanomechanical Schrödinger catmew more clearly about his/her quantum behaviour. Some of the possibilities offered by thattoken will be analyzed in a forthcoming paper.AcknowledgmentsSupport by the Hungarian Scientific Research Fund OTKA under Grant No. T75129 isacknowledged. I thank Lajos Diósi for helpful discussions.References[1] Marshall W, Simon C, Penrose R and Bouwmeester D 2003 Phys. Rev. Lett. 91 130401[2] O’Connell A D et al. 2010 Nature 464 697[3] For a recent discussion, see Vanner M R et al. 2010 arXiv:1011.0879v1[4] Ghirardi G C, Rimini A and Weber T 1986 Phys. Rev. D 34 470; Ghirardi G C, Pearle P and Rimini A 1990Phys. Rev. A 42 78; Diósi L 1987 Phys. Lett. A 120 377; Diósi L 1989 Phys. Rev. A 40 1165; Penrose R1996 Gen. Rel. Grav. 28 581; see also Geszti T 2004 Phys. Rev. A 69 032110[5] Kleckner D and Bouwmeester D 2006 Nature 444 75[6] Adler S L, Bassi A and Ippoliti E 2005 J. Phys. A 38 2715; Bassi A, Ippoliti E and Adler S L 2005 Phys.Rev. Lett. 94 030401[7] Bernád J Z, Diósi L and Geszti T 2007 Phys. Rev. Lett. 97 250404[8] Another possibility is to measure cross-correlations between the two output ports, the oscillations of whichare shifted in phase by π/4 with respect to the one-port intensities.[9] Yurke B and Stoler D 1986 Phys. Rev. Lett. 57 13[10] Monroe C, Meekhof D M, King B E and Wineland D J 1996 Science 272 1131[11] Brune M, Hagley E, Dreyer J, Mâıtre X, Maali A, Wunderlich C, Raimond J M and Haroche S 1996 Phys.Rev. Lett. 77 4887[12] Saleh B E A and Teich M C 2007 Fundamentals of Photonics 2nd ed (Hoboken, NJ: Wiley) Chapter 20

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