Deterministic preparation of non-classical states of light in cavity-optomechanics
DDeterministic preparation of non-classical states of light incavity-optomechanics
Yuxun Ling and Florian Mintert
Physics Department, Blackett Laboratory, Imperial College London,Prince Consort Road, SW7 2BW, United Kingdom (Dated: February 25, 2021)Cavity-optomechanics is an ideal platform for the generation non-Gaussian quantum states dueto the anharmonic interaction between the light field and the mechanical oscillator; but exactly thisinteraction also impedes the preparation in pure states of the light field. In this paper we derivea driving protocol that helps to exploit the anharmonic interaction for state preparation, and thatensures that the state of the light field remains close-to-pure. This shall enable the deterministicpreparation of photon Fock states or coherent superpositions thereof.
I. INTRODUCTION
Optomechanical experiments provide accuratecontrol over the quantum dynamics of mesoscopicmechanical oscillators and light fields on the singlephoton level [1]. In particular because the interac-tion between such oscillators and light fields is an-harmonic, there is great potential to generate non-classical, non-Gaussian states [2–7] with variousapplications including quantum metrology [8–10],quantum cryptography [11–13], and more [14–17].A widely pursued goal is the creation of single pho-ton Fock states [4, 18–22], but also multi-photonFock states and superposition of Fock states are ofuse in practical applications [23–27].While the anharmonic interaction is essential forthe realization of non-Gaussian states, its flip sideis that it results in the evolution towards quan-tum states with correlations between the opticaland mechanical degrees of freedom. The prepara-tion of a pure state of one of the subsystems canbe achieved in terms of a projective measurementon the other subsystem [2, 3, 28]. Such a scheme,however, is intrinsically probabilistic with a suc-cess probability limited by the multitude of possi-ble measurement outcomes. In this work, we pro-pose a driving scheme for optomechanical systemsfor the deterministic preparation of close-to-pure,non-classical states of light, and we exemplify thescheme with two-photon Fock states and the co-herent superposition of this state and the vacuumstate.The main body of the paper is divided into twosections. In Sec.II, a perturbative solution to thedriven evolution of the system is derived and theoptimal driving protocol is constructed. In Sec.III,numerical results for both the coherent dynamicsand the dissipative dynamics are presented andanalysed.
II. THEORYA. System Hamiltonian
We consider a generic model of an opto-mechanical system composed of a Fabry–P´erotcavity with one stationary and one movable mirror, i.e. a mechanical oscillator. The system Hamilto-nian H = H + H I consists of the non-interactingpart H = ω c a † a + ω m b † b with resonance frequen-cies ω c and ω m of the optical field and the mechani-cal oscillator, as well as corresponding creation andannihilation operators a , b , a † and b † . The inter-action H I = − g a † a ( b † + b ) is cubic in these op-erators, and thus can overcome the restriction toGaussian dynamics that is inherent to quadraticHamiltonians.Pumping of the cavity with an external classicallight field is described by the Hamiltonian H d = d ( t ) a † + d ∗ ( t ) a , (1)where the time-dependent function d ( t ) character-izes the frequency and any other time-dependenceof the light field, such as a temporal modulation.On the one hand, in the absence of the interac-tion H I , driving the cavity with classical light canonly result in the creation of a classical state. It isonly the interaction that can result in nonclassicalfeatures, as indicated for example by negative val-ues of the Wigner function [29]. On the other hand,in addition to nonclassical features, the interactionwill also result in correlations between the opticaland mechanical degrees of freedom. Each of theseparts of the system alone is then described by amixed states, and this mixing typically preventsthe observation of nonclassical effects [30].A way around this predicament is to performa measurement on the light field resulting in theprojection of the mechanical oscillator into a purequantum state. Such a probabilistic method is,however, not easily generalizable to the creation ofstates of light, because the mechanical degree offreedom is substantially less accessible for a mea-surement.For the creation of pure, non-classical states oflight it is thus highly desirable to ensure that the a r X i v : . [ qu a n t - ph ] F e b state of the full system is given as direct productof the states of each of the subsystems. It is possi-ble to deterministically create highly non-classicalstates of the mechanical oscillator in terms of suit-ably shaped pump profiles [6], but the extension ofsuch an approach for the creation of states of lightis rather challenging, since the control of the sys-tem is realized in terms of driving the light field,which needs to interact with the mechanical oscil-lator before excitations in the mechanical oscilla-tor can cause the light field to adopt non-classicalfeatures. At the same time, the driving tends tocreate Gaussian features of the state of light thattend to over-shadow non-classical effects. B. Driving profiles
The range of achievable states will strongly de-pend on the chosen driving profile d ( t ). In par-ticular strong driving can modify the system dy-namics with great potential for state preparation,but practical constraints demand sufficiently weakdriving with sufficiently simple spectra.Suitable choices for driving profiles can beidentified from the basic properties of the opto-mechanical interaction a † a ( b † + b ) that couples theabsorption and emission of a phonon to the photonnumber-operator n c = a † a . Since the number op-erator is quadratic, and thus does not contributeto the creation of non-classicality, it will not behelpful to use driving that supports the absorp-tion and emission of single phonons. An effectiveprocess, comprised of two successive interactionevents, on the other hand would involve the op-erator n c , which is no longer quadratic. We willthus employ a driving profile that favours the ab-sorption and emission of pairs of phonons.A reasonably elementary driving profile satisfy-ing this requirement is given by d ( t ) = Ee − i ( ω c t − ψ ) cos(2 ω m t ) , (2)with a detuning from the optical resonance fre-quency ω c that amounts to twice the mechanicalfrequency ω m . The real-valued amplitude E andphase ψ are not determined yet, and the freedomto choose these parameters will be utilized for thedesign of suitable driving patterns. C. System dynamics
Given the cubic character of the interaction, itis not possible to solve the dynamics of the driven-interacting system exactly. Since, however, thenon-interacting system is described by a quadraticHamiltonian, the dynamics of the driven, but non-interacting system can be found analytically. It isthus natural to solve the system dynamics pertur-batively in the interaction strength. The propagator of the non-interacting system isgiven by [31] U ( t ) = e iξ ( t ) e − iω c n c t e − iω m n m t e f ( t ) a † − f ∗ ( t ) a , (3)with f ( t ) = (cid:90) t dτ e iω c τ d ( τ ) (4)and a real, scalar, time-dependent global phase ξ ( t ).The interaction Hamiltonian ˜ H I = U † H I U inthe frame defined by U reads˜ H I = − g (cid:16) a † + f ∗ ( t ) (cid:17)(cid:16) a + f ( t ) (cid:17)(cid:16) b ( t ) + b † ( t ) (cid:17) (5)with the time-dependent annihilation operator b ( t ) = be − iω m t and creation operator b † ( t ) = b † e iω m t of a phonon.Since the cavity frequency ω c is much larger thanthe frequency ω m of the mechanical oscillator, onecan take ω c to be an integer multiple of ω m . In thiscase, the propagator of non-interacting system U is periodic with period T = 2 π/ω m . Furthermore,the Hamiltonian ˜ H I also becomes periodic, and thepropagator induced by ˜ H I thus admits a decom-position into a periodic part that reduces to theidentity after full periods and a part exp( − iH e t )that is induced by the effective Hamiltonian H e which fully captures the dynamics of the systemafter full periods.The effective Hamiltonian for the evolution afterone mechanical period can be constructed pertur-batively [32] in powers of k = g /ω m . It will be in-sightful to distinguish between the parts of the ef-fective Hamiltonian that capture processes of onlythe cavity field, processes of only the mechanicaloscillator and interaction processes each. That is,the explicit expansion of the effective Hamiltonianis given by H e = ω m (cid:88) j k j (cid:16) M Cj + M Mj + M Ij (cid:17) , (6)with the symbols C , M and I referring to cavity,mechanical and interaction.With the explicit driving profile given in Eq. (2)the lowest order effective Hamiltonian (linear in k )vanishes exactly, and the second order contributionis the dominant term.Also the second order term M M of the mechan-ical oscillator vanishes, but the term M C is gener-ally finite and reads M C = − n c − η n c + η a e − iψ + h.c.) , (7)with the scaled driving amplitude η = E/ω m . Thesecond order interaction term reads M I = −√ ηP ψ (cid:16) b + ( b † ) (cid:17) , (8)with the phase-shifted momentum P ψ = i √ ae − iψ − a † e iψ ) (9)of the cavity field. D. Effective Hamiltonian
The cavity operator M C in Eq. (7) contains sev-eral terms that are very useful for the generationof non-Gaussian states. The terms ( a † ) and a describe the creation and annihilation of pairs ofphotons. Those processes alone, however, are stillwithin the set of Gaussian dynamics, but the quar-tic operator n c breaks this restriction. It resultsin a deviation from the evenly spaced level struc-ture of the quantum harmonic oscillator. The ef-fectively larger spacing between higher lying levelsmakes it possible to populate the two-photon Fockstate starting from the vacuum state, while mak-ing sure that population of the four-photon Fockstate and higher-lying states are sufficiently far off-resonant to be negligible.On the other hand, in second order there arealso processes that can impede the generation ofnon-Gaussian states. The term linear in n c thatis contained in the cavity operator M C increasesthe constant spacing between energy levels of thequantum harmonic oscillator and thus reduces theeffective anharmonicity resulting from the n c term.Furthermore, the interaction term M I in Eq. (8)results in correlations between the cavity and themechanical oscillator to build up.With the driving profile specified in Eq. (2), itis not possible to have these undesired terms van-ish without also having terms describing the cre-ation and annihilation of pairs of photons vanish.Goal of the following analysis will therefore be theconstruction of a sequence of driven and undrivenintervals such that the effective Hamiltonian forthe dynamics over all these intervals contains thedesired terms, but in which terms describing un-desired processes are no longer present.To this end, we will first consider the effectiveHamiltonian for the dynamics of several periodsof driven dynamics as basis for the constructionof the effective Hamiltonian H f of the full dy-namics. Because of non-commutativity, the effec-tive Hamiltonian H f would need to be constructedwith the Baker-Campbell-Hausdorff series. How-ever, because the leading contribution to the indi-vidual effective Hamiltonian is of order k , the firstnon-trivial contribution to the Baker-Campbell-Hausdorff series is of order k , which is smallerthan the highest order ( i.e. k ) that is includedin the individual effective Hamiltonians so far, andsmaller than the highest order ( k ) that will beincluded later-on. Within the given level of ac-curacy, the complete effective Hamiltonian H f is thus given by the sum of the individual effectiveHamiltonians.The interaction term M I (Eq. (8)) depends onthe phase ψ of the driving field (Eq. (2)) via themomentum operator P ψ given in Eq. (9). After N periods of driven dynamics, with driving strength η j and phase ψ j in period j , the interaction termin the full effective Hamiltonian thus contains thefactor (cid:80) Nj =1 η j P ψ j . Any series of driving periodssatisfying (cid:80) Nj =1 η j exp { iψ j } = 0 can thus ensurethat there are no interaction effects in leading orderat the end of the dynamics.The creation and annihilation of pairs of pho-tons in Eq. (7), on the other hand depend onthe driving field via η exp {± iψ } , and one caneasily find choices for the driving fields suchthat (cid:80) Nj η j exp { iψ j } is finite while the condition (cid:80) Nj η j exp { iψ j } = 0 is satisfied. To leading or-der, this prescription would result in an effectiveHamiltonian H f = ω m N k (cid:32)(cid:18) ζ a + h.c. (cid:19) − n c − χn c (cid:33) , (10)with ζ = 1 N N (cid:88) j =1 η j e − iψ j , and χ = 1 N N (cid:88) j =1 η j . (11)This is a viable effective Hamiltonian for the cre-ation of non-classical states, with a non-linearitythat breaks the restriction to Gaussian dynam-ics. Yet, in practice, it is desirable to have a non-linearity n c that is strong as compared to the linearterm ∝ n c in order to obtain a spectrum of the di-agonal part of H f in Eq. (10) with strongly un-evenspacing between neighbouring energy levels. Sincethe linear part ∝ n c in Eq. (10) is getting strongas compared to the non-linear part ∝ n c in theregime of strong driving, it is necessary to find amechanism that effectively reduces this linear part.The central idea that allows to achieve this, isthat the free evolution induced by n c results in thetype of phase shift described by the ψ j . That is, aramp in the phases ψ j has approximately the sameeffect as a true free phase evolution.In order to formalize this, it is helpful tonote that the effective Hamiltonians M C ( ψ ) and M I ( ψ ) in Eqs.(7) and (8) satisfy the relation M C/I ( ψ ) = V ψ M C/I (0) V † ψ , (12)with V ψ = e iψn c . The propagator induced by theeffective Hamiltonian can thus be written as U ψ = e − iH e ( ψ ) T = V ϕ e − iH e ( ψ − ϕ ) T V † ϕ , (13)with a phase ϕ that can be chosen at will.Because of the identity V † ϕ j +1 V ϕ j = V † φ j , with φ j = ϕ j +1 − ϕ j , the product of two propagators ofconsecutive periods simplifies to U ( j +1) U ( j ) = V ϕ j +1 e − iH e ( ψ j +1 − ϕ j +1 ) T V † φ j ×× e − iH e ( ψ j − ϕ j ) T V † ϕ j . (14)The full propagator over N periods can thus beexpressed as U = V ϕ N +1 N (cid:89) j =1 V † φ j e − iH e ( ψ j − ϕ j ) T . (15)The first factor V ϕ N +1 describes a free phase evo-lution after all the time-intervals of driven dynam-ics. Since this is merely a rotation in phase space,it has no bearing on the classical or quantum me-chanical character of the final quantum states.Each factor V † φ j e − iH e ( ψ j − ϕ j ) T in Eq. (15) is aproduct of a term e − iH e ( ψ j − ϕ j ) T induced by theeffective Hamiltonian and a term of free phase evo-lution induced by n c . In the limit of infinitesimallyshort intervals T →
0, this is equivalent to an evo-lution induced by an effective Hamiltonian with amodified term n c . In practice, the duration T willalways be finite, but the approximation V † φ j e − iH e ( ψ j − ϕ j ) T (cid:39) e − i ( H e ( ψ j − ϕ j ) T + iφ j n c (16)is sufficiently good for the purpose of state prepa-ration for realistic values of T .Now the effective Hamiltonian for each periodcontains an extra term ∝ n c in addition to H e ,and the freedom to choose values for the phaseangles ψ j and φ j can be used to ensure that unde-sired terms cancel. For any choice of the drivingparameters η j and ψ j satisfying N (cid:88) j =1 η j e − i ( ψ j − ϕ j ) = 0 , N (cid:88) j =1 η j e − i ( ψ j − ϕ j ) (cid:54) = 0 , (17)the effective Hamiltonian after N periods will re-duce to ω m k N (cid:32)(cid:18) ζ (cid:48) a + h.c. (cid:19) − n c (cid:33) −− ω m N (cid:88) j =1 (cid:32)(cid:18) k η j − φ j π (cid:19) n c (cid:33) + O ( k φ ) , (18)with ζ (cid:48) = 1 N N (cid:88) j =1 η j e − i ( ψ j − ϕ j ) (19)With the specific choice of φ j = 4 / πk η j , termslinear in n c will vanish in the leading order during each period. Since φ j is the accumulated phaseshift of the driving pattern, this amounts to theindividual phases ϕ j = 43 πk j − (cid:88) l =1 η l , (20)that increase by an amount determined by thedriving amplitude η j . E. Driving pattern
While, in principle, it is only required that theeffective Hamiltonian for the dynamics over the en-tire interval of interest matches the desired Hamil-tonian, it is preferable that such a condition issatisfied at in-between points in time. We willtherefore consider driving protocol in which thefull time-window of duration
N T is divided into N blocks of duration 2 T , and require that the dy-namics over each block is induced by the desiredeffective Hamiltonian within the perturbative ap-proximation.In each such block, we will consider a constantdriving amplitude, so that the phases ψ j = π + 83 πk j − (cid:88) l =1 η l ,ψ j +1 = 43 πk η j + 2 j − (cid:88) l =1 η l , (21)of the driving profiles are suitable solutions ofEq. (20). With this choice the resulting effectiveHamiltonian over N periods of driven dynamicsreads H g = ω m k N (cid:88) j =1 (cid:18) η j (cid:16) a + ( a † ) (cid:17) − n c (cid:19) . (22)This can be taken as a starting point for statepreparation, but it is worth exploring higher or-der perturbative corrections, since this will help tosubstantially increase the accuracy of state prepa-ration with only slightly more involved driving pat-terns. F. Third Order Corrections
The third order term M C of the cavity van-ishes, and the term M M of the mechanical oscilla-tor reads M M = yη (cid:18) b † cos 5 π
12 + b sin 5 π (cid:19) + h.c. , (23)with y = 16 √ / (cid:39) . M I is of theform M I = A ψ X m + B ψ P m + X ψ G m , (24)with A ψ = √ (cid:16) η P ψ + 35 η ( n c P ψ + P ψ n c ) (cid:17) ,B ψ = 2 √ iη (cid:16) ( a † ) e iψ − a e − iψ (cid:17) ,G m = 3 yi η (cid:18) b † cos 5 π − b sin 5 π (cid:19) + h.c. , (25)and X ψ = 1 √ ae − iψ + a † e iψ ) ,X m = 1 √ b + b † ) ,P m = i √ b − b † ) . (26)With the driving pattern derived above inSec.II E, the effective Hamiltonian for the dynam-ics over two periods of driving with constant driv-ing amplitude is given by H p = H g + ω m k ( M M + B P m ) , (27)in which H g is the second order effective Hamilto-nian in Eq. (22) and B is the operator B ψ eval-uated at ψ = 0. The contributions from A ψ and X ψ in M I (Eq. (24)) average out in the effectiveHamiltonian H p . However, there is still a finitecorrection due to the term B P m in the interaction M I ; and a contribution to the effective Hamilto-nian from the term M M of the mechanical oscilla-tor.In addition to the interaction term B P m , it isalso desirable to remove other term M M from thefull effective Hamiltonian because it induces exci-tations in the mechanical oscillator, and the im-pact of interactions on the optical field remainingin higher order corrections tends to be increasingwith growing excitations of the mechanical oscilla-tor. Therefore, we devote the following paragraphsto modifying the driving pattern such that theseterms are reduced as much as possible.The basic idea is that half a period of free evo-lution of the mechanical oscillator corresponds toa phase-shift of π in the creation and annihilationoperators b † and b . Since M M + B P m is an oddfunction in b and b † , a sequence of two periodsof dynamics induces by M M + B P m with an in-between half-period of free evolution results in aneffective cancellation. Any driving protocol withalternating driven dynamics and intervals of freeevolution, with the phase of the driving fields sat-isfying Eq. (21), and the duration of free evolutionbeing half a period of free evolution thus realisesthe dynamics described in Sec.II E, and it ensures cancellation of the dominant terms deviating fromthe desired effective Hamiltonian.In practice, however, it is not possible to real-ize an exact free evolution of the mechanical oscil-lator because of the intrinsic interaction betweenthe mechanical oscillator and the light field. As wewill see in the following, an interval of un-drivendynamics can be used to achieve a similar effect.Following Eq. (3) the propagator for the un-driven dynamics over half a mechanical periodreads U T = e πik n c e − √ ikn c P m e − πin m , (28)given that the optical resonance frequency is aneven multiple of mechanical resonance frequency.The adjoint of this propagator satisfies the relation U † T = U T e − πik n c . (29)Any sequence including two intervals of drivendynamics and an interval of un-driven dynamicsbefore each interval of undriven dynamics thus re-sults in the propagator U d U T U d U T = U d U † T e πik n c U d U T (cid:39) e − iH p T U † T e πik n c − iH p T U T , (30)where terms of order k in the Baker-Campbell-Hausdorff relation are neglected. Within the sameapproximation, the propagator over the four inter-vals reads (cid:39) exp (cid:32) − iT (cid:18) H p + U † T H p U T (cid:19) + 2 πik n c (cid:33) . (31)To leading orders ( i.e. k ), the term U † T ( M M + B P m ) U T reduces to − ( M M + B P m ). The term −M M thus cancels the corresponding term in H p so that undesired processes of the mechan-ical degree of freedom disappear in leading or-ders. For the interaction terms, this cancellationis not perfect, but there is a residual interaction ω m k B P m / U T commutes with the operator V φ j in Eq. (14), the dynamics of any time-window oftwo periods of driven dynamics is still described
With the driving profiles devised so far, it isensured that undesired processes are largely sup-pressed, and the amplitude of the driving field canbe chosen in accordance with the state that is tobe prepared. In order to realise high fidelity statepreparations, it is helpful to consider a series ofseveral intervals T of driven dynamics, and to op-timize over the driving amplitudes of each of thoseintervals. Since such optimizations require the analysis ofsystem dynamics with several different patternsof driving amplitudes, efficiency in the numericalpropagation is essential. We therefore use the ef-fective Hamiltonian H (2) in Eq. (33). Since to thislevel of approximation, there is no interaction be-tween the cavity field and the mechanical oscilla-tor, this permits to restrict the numerical propa-gation to the dynamics of the cavity field only.All the pulses discussed in Sec.III are optimizedbased on simulations with a Hilbert space for thecavity field that is truncated to the lowest 60Fock states, and constraints on maximally admit-ted driving amplitude η . These constrains ensurethat the obtained solutions are compatible withpractical constraints, and they help to avoid trun-cation errors. III. OPTIMIZED STATE PREPARATION
The framework developed in Sec.II permits toidentify driving patterns that result in the de-sired evolution towards nonclassical pure states oflight in the regime of strong optomechanical cou-pling. In subsection III A we will discuss a rangeof achievable states and assess the validity of theperturbative approximation. In subsection III B,we will discuss the impact of dissipative effects onthe achievable states. All simulation results arecomputed using the Python toolbox Qutip [33, 34].The accuracy of the state preparation will beassessed in terms of the fidelities F ( (cid:37), ρ ) = (cid:18) Tr (cid:113) √ ρ(cid:37) √ ρ (cid:19) , (34)and F ( (cid:37), | Ψ (cid:105) ) = (cid:104) Ψ | (cid:37) | Ψ (cid:105) (35)that specify the similarity between the state (cid:37) anda mixed or pure state ρ and | Ψ (cid:105) .In order to discriminate between the limited ac-curacy of the perturbative expansion and the qual-ity of the optimized driving profiles, it will be help-ful to define three different fidelities in terms of thenumerically exact propagator Λ n of the coherentsystem dynamics, the propagator Λ l of the dissi-pative system dynamics, and the second order per-turbative propagator Λ p of the system dynamics.With the cavity and mechanical oscillator ini-tialized in their ground state and when the systemis lossless, the numerically exact final state of thecavity is given by (cid:37) n = Tr M Λ n ( | (cid:105) (cid:104) | ⊗ | (cid:105) (cid:104) | ) , (36)where the symbol Tr M denotes the trace over themechanical degree of freedom. Similarly, the finalstate in perturbative approximation reads (cid:37) p = Tr M Λ p ( | (cid:105) (cid:104) | ⊗ | (cid:105) (cid:104) | ) . (37)Finally, when any system imperfection is involved,the numerically exact final state reads (cid:37) l = Tr M Λ l ( | (cid:105) (cid:104) | ⊗ | (cid:105) (cid:104) | ) . (38)For any given target state | Ψ t (cid:105) of the cavity, wecan thus define the fidelity F n = F ( (cid:37) n , | Ψ t (cid:105) ) , (39)that specifies how well the goal of optimization isachieved in a lossless system, and the fidelity F l = F ( (cid:37) l , | Ψ t (cid:105) ) , (40)that specifies how well the goal of optimizationis achieved when relevant experimental noises areconsidered. Lastly, the fidelity F i = F ( (cid:37) n , (cid:37) l ) (41)characterizes the impact of the relevant experimen-tal noises with other imperfections isolated. A. Coherent dynamics
1. Fock state
Given the suitability of the present controlscheme for the creation of photon pairs, the cre-ation of the Fock state | (cid:105) of the light field, start-ing from the cavity field and the oscillator in theirground state is a natural task.Fig. 5 depicts the dynamics of the cavity fieldunder optimised driving for an evolution time of16 T with a maximum admissible driving strength η max = 4 and the coupling strength k = 1 / b ) and ( c ) depict the time-dependent oc-cupation of the lowest 6 Fock states. Due to thesuppression of the creation of single photons, dis-cussed in Sec.II D and Sec.II E, the populationsof odd Fock states remain orders of magnitudessmaller than the populations of even Fock states.There is a sizeable population of the state | (cid:105) and | (cid:105) during the dynamics; that is, despite the sup-pression of excitations to higher-lying states, thesestates do become occupied. The numerically op-timized driving pattern, however, induces a dy-namics in which these undesired states becomeun-occupied in the final state, and a final fidelity F n = 0 .
997 is obtained.In the idealized situation of lossless dynamics,one would expect to obtain best results in the limitof long evolution times with weak interactions ( i.e. k (cid:28) (a) Populations of even Fock states evolution time ( ) (b) Populations of odd Fock states FIG. 2. Plots of population of Fock states in the cavity.Inset (a) plots population of even Fock states and inset(b) plots population of odd Fock states. The relativecoupling strength is: k = 1 /
26, the relative drivingstrengths η are optimized with a maximally admissiblevalue η max = 4, and the length of evolution is t f =16 T . A data point is plotted after each five mechanicalperiods T . The final state is almost a Fock state | (cid:105) . For any given evolution time, one would thusexpect to find an optimal value of the interactionconstant k . Fig. 3 depicts the fidelity F n (solid)obtained with optimized driving and the interac-tion strength that was found to be optimal for anygiven evolution time. Dashed lines indicate thefidelity F n obtained with optimized driving pro-files with fixed interaction strengths k = 1 /
13 and k = 1 /
26 that are optimal for the evolution times2 T to 16 T . Comparison of dashed and solid lineshighlights the substantial gain in fidelity of statepreparation that can be obtained by selecting asuitable combination of interaction strength andevolution time. Fig. 4 shows the coupling strengththat is optimal for a given evolution time rangingfrom 2 T to 16 T . Even though the range of opti-mal coupling strength varies only by a factor of 2,the optimal choice of the interaction constant hasa strong impact on the achievable state fidelities.For example, a fidelity of 0 .
990 can be achievedwith an evolution time as short as 10 T with theoptimal interaction strengths, while an interactionstrength of k = 1 /
26 would only result in a fidelityof 0 .
836 within the same time.
2. superposition states
Similarly to the creation of Fock states, thepresent framework can also be used to find drivingpatterns for the creation of coherent superpositionsof Fock states. This will be exemplified in the fol-lowing with the target state | Ψ ϑ (cid:105) = 1 √ (cid:16) | (cid:105) + e iϑ | (cid:105) (cid:17) . (42) evolution time ( ) f i d e li t y (a)(b)(c) FIG. 3. Plot of the fidelity F n against the evolutiontime in T when η max = 4. (a) For each time point, thecoupling strength k is optimized to achieve the highestpossible fidelity. (b) The coupling strength is fixed tobe k = 1 /
26. (c) The coupling strength is fixed to be k = 1 /
13. Whether or not a suitable combination ofinteraction strength and evolution time is select has agreat impact on the final fidelity F n . evolution time ( ) c o u p li n g s t r e n g t h k FIG. 4. Plot of the coupling strength that is optimalfor a given evolution time ranging from 2 T to 16 T when η max = 4. Optimal coupling strengths doubleswhen the evolution time is restricted from 16 T to 2 T . In addition to the optimization of the driving pro-file, the following optimization includes an opti-mization over the relative phase ϑ , i.e. it identifiesthe target state that is best suited among all bal-anced superpositions of the Fock state | (cid:105) and | (cid:105) .Similar as in the case of Fock states, the drivingstrengths are optimized for a maximally admissibledriving strength η max = 4 and coupling strength k = 1 /
26. However, because the expected numberof photons in the cavity is less for an equal super-position state | Ψ ϑ (cid:105) than for a Fock state | (cid:105) , theevolution time can be reduced from 16 T to 10 T while keeping the fidelity as high as F n = 0 . (a) Populations of even Fock states evolution time ( ) (b) Populations of odd Fock states FIG. 5. Plots of population of Fock states in the cav-ity. Inset (a) plots population of even Fock states andinset (b) plots population of odd Fock states. Therelative coupling strength k = 1 /
26, the relative driv-ing strengths η are optimized with a maximally admis-sible value η max = 4, and the length of evolution is t f = 10 T . A data point is plotted after each five me-chanical periods T . The final state is almost an equalsuperposition state 1 / √ (cid:16) | (cid:105) + e − . i | (cid:105) (cid:17) . Fock states that do not contribute to the desiredsuperposition state.
B. Dissipative dynamics
The final question to be discussed is the im-pact of experimental noise on the final state [35].The two most significant experimental imperfec-tions are leakage of light from the cavity and ther-malization in the mechanical oscillator. The lat-ter can result in thermal excitations in the initialstate of the mechanical oscillator and it can resultin dissipative dynamics during the process of statepreparation.In order to analyse the impact of experimentalnoise, this section addresses the accuracy of statepreparation under various imperfections. This willbe exemplified with the control pulses identifiedas optimal for the noiseless system and with thethree optimal pairs (1 / , T ), (1 / , T ) and(1 / , T ) of parameter values of coupling strengthand evolution time. For all three parameter sets,the maximally admissible driving strength is fixedto be η max = 4.
1. Thermal Initial States
In this subsection, we consider noiseless, unitarydynamics, but thermal initial state of the mechan-ical oscillator. Fig. 6 depicts the fidelities F l and F i as functions of the mean thermal phonon num-ber of the initial mechanical state. Fidelity F l as- (a) Fidelity with |2 ( F l ) ( )( )( )0 1 2 3 4 5 mean thermal phonon numer of the initial state f ii d e li t y (b) Fidelity with n ( F i ) ( )( )( ) FIG. 6. Plot of the fidelities (a) F l and (b) F i against the mean thermal phonon number ¯ n th of ini-tial states. Curve (i) , (ii) , and (iii) represent caseswhen parameters ( k, η max , t f ) equal to (1 / , , T ),(1 / , , T ), and (1 / , , T ) respectively. All fi-delities remain high when ¯ n th < / , , T ). The exception is caused by the higherorder entanglement that is amplified by excitations inthe mechanical oscillator. sesses the overall accuracy of the state preparation,whereas F i isolates the impact the thermal excita-tions. The difference between these two fidelitiesbecomes best apparent in the case (1 / , , T ), inwhich F n does not reach the ideal value of unityfor vanishing thermal excitations.In addition to the fact that imperfections reducethe state fidelities most strongly for strong interac-tions and fast protocols, also the impact of thermalexcitations is most pronounced in these cases.This can be explained by the fact thatshorter dynamics require either stronger couplingstrengths or stronger driving strengths, and theincrease of either of the two parameters will leadto larger coefficients before undesired terms suchas k H (3) in the perturbative solution in Eq. (32).These undesired terms including entanglement be-tween the cavity and the oscillator are further am-plified by the non-vanishing phononic occupation,and thus will lead to lower final fidelities.
2. Optical Loss
Leakage of photons from the cavity at a rate κ can be modelled with the Lindbladian L (cid:37) = κ D [ a ] ρ ≡ κ (cid:18) a(cid:37)a † − { a † a, (cid:37) } (cid:19) , (43)which, together with the system Hamiltonian de-fines a Master equation.Fig. 7 depicts the fidelities F l and F i as functionof the optical decay rate κ . Just like in Fig. 6 thefidelities F l remain smaller than 1 for κ →
0; but, (a) Fidelity with |2 ( F l ) ( )( )( )0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 optical decay rate
1e 30.600.700.800.900.951.00 (b) Fidelity with n ( F i ) ( )( )( ) FIG. 7. Plots of (a) F n and (b) F i against relative op-tical decay rate κ/ω m of the lossy cavity. Curve (i), (ii) , and (iii) represent cases when parameters( k, η max , t f ) equal to (1 / , , T ), (1 / , , T ),and (1 / , , T ) respectively. The x-axis is in logscale while the y-axis is in linear scale. The plot sug-gests that the fidelity drops dramatically with the in-crease in optical decay rate when the decay rate ex-ceeds 10 − ω m although shorter evolution time leads toa more resilient state. in contrast to the case of initial thermal excita-tions, the faster protocols in systems with strongerinteractions become favourable with stronger opti-cal decay. Fig. 6 a) thus indicates at what valuesof κ the coherent imperfections outweigh the in-coherent imperfections, and helps to identify thecoupling strength and corresponding duration thatis best adopted for a given level of optical loss.In order to reach fidelities over 0 .
9, the opticaldecay rate may at most be on the order of 10 − ω m .The best value achieved by the current state-of-the-art [36] in the strong coupling regime is stillapproximately two orders of magnitude away fromthe required value, but given the steady pace overthe past decades, the regime may eventually bereached in the future.
3. Mechanical Thermalization
Finally, systems with vanishing optical decayrates but with finite mechanical decay rates γ areconsidered. The thermalization of the mechani-cal oscillator can be modelled with the Lindbla-dian [37] L (cid:37) = γ (¯ n b + 1) D [ b − ka † a ] ρ + γ ¯ n b D [ b † − ka † a ] ρ + 4 γk log (1 + n b ) D [ a † a ] ρ , (44)with ¯ n b being the mean thermal phonon number ofthe environment. The unusual shift of the mechan-ical annihilation and creation operators depending0 (a) Fidelity with |2 ( F l ) ( )( )( ) mechanical decay rate
1e 20.600.700.800.900.951.00 (b) Fidelity with n ( F i ) ( )( )( ) FIG. 8. Plots of (a) F n and (b) F i against relative op-tical decay rate κ/ω m of the lossy cavity. Curve (i), (ii) , and (iii) represent cases when parameters( k, η max , t f ) equal to (1 / , , T ), (1 / , , T ),and (1 / , , T ) respectively. Solid, dashed, and dot-ted curves represent cases when the thermal bath con-tains on average ¯ n b = 1, ¯ n b = 10, and ¯ n b = 100 re-spectively. The x-axis is in log scale while the y-axis isin linear scale. The plot suggests that the infidelity re-sulting from mechanical imperfection is negligible com-pared to that from optical imperfection when the ther-mal phonon number n bath (cid:28) on the photon number operator a † a and the de-phasing term depending on the temperature of theoscillator result from the non-vanishing value ofthe coupling strength k .The fidelities F n and F l are plotted in Fig. 8as functions of the mechanical decay rate γ . Theplot indicates that faster protocols which requirestronger optomechanical interactions are slightlyfavourable for higher F i but the advantage is notsignificant when taking into the account the factthat faster protocols have lower F n .When the thermal bath contains 10 or lessphonons, effect of mechanical thermalization is completely negligible as compared to effect of opti-cal decay when, as observed experimentally [1], themechanical decay rate γ is smaller than the opti-cal decay rate κ . Especially when ¯ n b ≤
1, a fidelity F i > .
95 can be achieved with γ < − ω m whichcan be realized under several existing experimentalsetups [1, 36]. However, for systems that cannotmaintain a low thermal phonon number ¯ n b (cid:28) γ is required to be at leastsmaller than 10 − ω m to reach a fidelity of 0 . IV. CONCLUSIONS
The tools for state-preparation developed here,help to overcome the low success rates of proba-bilistic protocols. Even though the non-linear in-teraction between light-field and mechanical oscil-lator tends to result in growing entanglement be-tween the two degrees of freedom, the present driv-ing patterns achieve close-to-perfect unitary dy-namics of the light field while ensuring that thebenefits of the non-linear interactions are preservedfor the realization of non-Gaussian states.The extra freedom of pulse shaping gives accessto a variety of quantum states of light, beyondthe two-photon Fock state and superposition statesdiscussed here in more detail. The high fidelitiesthat can be obtained in the presence of optical lossand mechanical heating highlight the experimentalfeasibility of deterministic state preparation in up-coming generations of optomechanical experimentswith increasing coupling strength between opticaland mechanical components.
V. ACKNOWLEDGEMENT
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