Weak Measurement Combined with Quantum Delayed-Choice Ex- | I 2√2 −| I | I −| I | I Periment, We Can Amplify the Mirror’S Displacement

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arXiv:1509.00641v1 [quant-ph] 2 Sep 2015 xeietadipeetto notmcaia system Li, Gang optomechanical in delayed-choice implementation quantum and with experiment combined measurement Weak orsodneto Correspondence 3 2 1 eti 9 sotie yrtiigteKr hs nthe in phase Kerr the retaining by obtained [11]. is Ref. [9] ef- the in amplification in fect The proposed embedded [10]. is is interferometer amplified system March-Zehnder mir- optomechanical be the can the where photon of [9], single displacement by caused the ror that scheme mea- measurement mir- system amplification weak be the optomechanical the to measurement, of the weak spread hard with in combined the Recently, is than packet. smaller displacement photon wave much When the ror the is cavity by it mirror. since the caused the sured in mirror on photon the force the one of only radiation in is a light there give the to attached can where mirror cavity [7,8], tiny a a oscillator with to micromechanical cavity refers optical usually finesse system rarely high optomechanical was optomechanics The in reported. application its weak applications, of application [6]. in and reviewed basics been have the deflec- measurement Recently, beam and [4,5]. [3] de- as function tion directly such wave of experiment, not measurement in are direct small techniques amplify conventional which and by effects measure tected to or used quantities para- also is physical quantum it help counter-intuitive and is can [2] some It pointer doxes measured. understand be a to to where system us the by [1], to Vaidman coupled proposed weakly and first Albert was Aharonov, measurement weak Postselection Introduction 1 wl eisre yteeditor) the by inserted be (will No. manuscript EPJ colo hsc n polcrnc ehooy uinN Fujian Technology, Optoelectronics Ch and Physics 130012, Changchun of University, School Un Jilin Physics, Dalian Technology, of Optoelectronic College and Physics of School lhuhpsslcinwa esrmn aemany have measurement weak postselection Although 1 , h aeo eep n cetnesol eisre later inserted Abstract. be should acceptance and receipt of date the nqatmmcaisla st ceefripeetto o PACS. implementation for scheme a to us system. lead mechanics quantum in h ytmt eadcntb xlie yasmcasclwav semiclassical a amplific The by splitter. explained beam quantum caus be in can’t is ancilla the and effect and be system this surp to that a system show to We the rise give effect. splitter amplification beam the negative of instead splitter beam ∗ a Wang, Tao 25.k–4.5H 03.65.Ta – 42.65.Hw – 42.50.Wk : ∗ [email protected], ekmaueet[,9 obndwt unu delayed-cho quantum with combined [1,19] measurement Weak 2 igYn Ye, Ming-Yong † [email protected] 3 n eSa Song He-Shan and ina ra nvriy uhu300,China 350007, Fuzhou University, ormal vriyo ehooy ain162,China 116024, Dalian Technology, of iversity h e datg fteaclai unu emsplit- beam quantum splitter. in beam ancilla the the of of stead advantage key in use The splitter a that beam give experiment quantum first delayed-choice com- the will quantum [1,19] we with measurements paper, bined weak the about there In discussion splitter whether features. general beam measurement new weak the some of with are stead combined in in that splitter and experiment beam delayed-choice quantum quantum use to on want we based naturally and know (classical) splitter beam the from series a by verified the is at scheme [16–18]. appear this experiments can after, of photon wave Soon the the time. both of same that behaviours so particle open and can and being interferometer closed of the simultaneously scheme, superposition be this in In put absent. be and can present [12,13] controlled ancilla splitter the output beam by the quantum of the stead where in splitter, splitter beam beam quantum experiment on delayed-choice based quantum Ionicioiu [15] for recently scheme splitter However, a beam removed. propose output or the inserted classical whether is the i.e., on devices, depend duality detecting wave-particle that and show [14] it complementarity in- is Bohr’s Mach-Zehnder demonstrate [12,13] to the output experiment proposed delayed-choice on the the Based if the is absent. behaviour terferometer, splitter is particle beam splitter a be- output as beam photon the behaved the if or behaviour splitter, present wave beam a input as out- When the haved duality. the enters wave-particle and photon show splitter can a beam splitter, input beam the put including [10], eter 1 , † h aueo h unu emslte sdifferent is splitter beam quantum the of nature The interferom- Mach-Zehnder the that all to known is It iigapicto ffc,ie,counterintuitive i.e., effect, amplification rising to ehns bu aepril duality wave-particle about mechanism ation db h aeadpril eaiusof behaviours particle and wave the by ed hoy u oteetnlmn fthe of entanglement the to due theory, e ekmaueeti optomechanical in measurement weak f c xeietta s quantum use that experiment ice tal et 2 Gang Li et al.: Weak measurement implementation in optomechanical system

ter is to control the choice of the measured system basis. Suppose that the state 0 s is the initial state of the | i 1 When the quantum beam splitter possessing the entangle- system, where 0 s is an eigenstate ofσ ˆz. Set + = ( 0 s+ | i | i √2 | i ment of the system and the ancilla play a role of the post- 1 s) andσ ˆz + = , where + and is eigenstates of | i | i |−i | i |−i selection, we find that the wave and particle behaviours of σˆx. The ancilla is an arbitrary state cos α 0 anc+sin α 1 anc. the measured system give rise to a surprising amplification Hadamard gate usually plays a role of the| i beam splitters| i effect and this amplification effect can’t be explained by in quantum circuits [21]. The Hadamard transformation Maxwell’s equations [20], due to the entanglement of the is system and the ancilla in quantum beam splitter. We first show that the wave-particle duality can be used as the gen- ′ 1 0 s = ( 0 s + 1 s), eration mechanism for weak measurement amplification, | i √2 | i | i that has not previously been revealed. The scheme in this ′ 1 paper provide a new method for studying and exploiting 1 s = ( 0 s 1 s), (2) √ postselection and weak measurement [1,19] in three-qubit | i 2 | i − | i system. On the other hand, the perspective in this paper where 0 s and 1 s are two input modes entering the beam lead us to a scheme for implementation of weak measure- | i | i splitter, 0′ s and 1′ s are two output modes exiting the ment [1,19] in optomechanical system. The amplification | i | i mechanism achieved here is different from the one in [9] beam splitter. It is obvious that when Hadamard gate which is caused by the Kerr phase. Moreover, the displace- is present, the measured system show wave behaviours. ment of the mirror’s position in optomechanical system However, if Hadamard gate is absent, the transformation can appear within a large evolution time zone, which is a between input and output modes is counterintuitive negative amplification effect. 0′ s = 0 s, The rest of this paper is organized in the following way. | i | i In Sec. 2, we give a general discussion about weak mea- 1′ s = 1 s, (3) | i | i surements combined with quantum delayed-choice experiment. In Sec. 3, we present a scheme for implementation showing particle behaviours. However in Ref. [15], the of weak measurement in optomechanical system, and In Controlled-Hadamard C(H) gate is equivalent to the quan- Sec. 4, we give the conclusion about the work. tum beam splitter, where the quantum beam splitter controlled by ancilla [12] can be put in superposition of being present and absent. In other word, the state after quantum 2 Weak measurement amplification via beam splitter in the interferometer is given by quantum beam splitter ψQBS = cos α particle 0 anc + sin α wave 1 anc, (4) | i | i| i | i| i 1 To set the scene, we assume a general schematic diagram where particle = ( 0′ s + 1′ s) and wave = 0′ s | i √2 | i | i | i | i illustrated in Fig. 1. In the following the scheme involved describe the particle and wave behaviour states of the a simple three-qubit system. The pointer is a continuous measured system, respectively, and the ancilla 0 anc and system and initially prepared in the ground state 0 m. | i | i 1 anc controls if the quantum beam splitter is absent and In the standard scenario of weak measurement, the in- |not.i teraction Hamiltonian of the pointer and the system is 1 Defining an annihilation operatorc ˆ = qˆ+i σ pˆ, where expressed as 2σ h¯ qˆ is position operator (the canonical variable conjugates HÎ =¯hχ(t)ˆσz p,ˆ (1) ⊗ top ˆ and [q,p]= i¯h). According to the result of Ref. [19], the Hamiltonian above can be rewritten as whereh ¯ is Planck’s constant, χ is a small coupling constant, σ is a spin-like observable of the system to be mea- 2 z ¯h χ(t) sured andp ˆ is the momentum operator of the pointer. HÎ = i σˆz(ˆc cˆ†), (5) − 2σ −

bright port where σ is the zero-point ﬂuctuation and √2σ is the width

of the Gaussian distribution. Then the time evolution of darkport 0' s darkport

0 m the system and the pointer is given by H I 0 s 1' s H H i ˆ R Hdt e− h¯ + 0 = exp[ ησˆz(ˆc cˆ†)] + s 0 m cosα 0+ sin α 1 1 anc anc ( 0− 1 ) | i| i − − | i | i 2 anc anc 1 . = ( 0 sD(η) 0 m √ | i | i Fig. 1. Weak measurement combined with quantum delayed- 2 + 1 sD( η) 0 m), (6) choice experiment. HI is the weak interaction between the | i − | i pointer and the system. Hadamard gate is equivalent to the beam splitter. Controlled-Hadamard gate is equivalent to hχ¯ where D(η) = exp[(ηcˆ† η∗cˆ)] with η = 2σ is a displace- quantum beam splitter and play a role of the postselection. The ment operator and η −1. After the quantum beam split- ancilla is prepared in an arbitrary state cos α 0 anc +sin α 1 anc ≪ | i | i ter and using the transformation of Eq. (2) and Eq. (3), Gang Li et al.: Weak measurement implementation in optomechanical system 3 then the state of the total system becomes dark port Da′′ cos α ( 0 sD(η) 0 m + 1 sD( η) 0 m) 0 anc (a) a′ PBS1

D portbright √2 | i | i | i − | i | i PBS C 2 sin α D + [( + (D(η) 0 m + D( η) 0 m) PDBS b′′

TinyMirror D 2 | i | i − | i Oscillator Db′ + (D(η) 0 m D( η) 0 m)] 1 anc. (7) Cav. A |−i | i − − | i | i DH When the state 1′ is detected, in the language of weak measurement we| simultaneouslyi measure the system in DV EOM BS PBS 3 1 Cav. B the orthogonal postselection state of wave behaviours ( 1,H 1,H + 1,V 1,V ) |−i ancs anc s [11,9] and in the state of the particle behaviour 1 s before 2 | i Ancilla photon apparatus System photon apparatus the ancilla, i.e., before choosing if quantum beam splitter (b) is present or absent, then the state of the pointer and the ancilla becomes PBS

cos α BS ψ m = D( η) 0 m) 0 anc | i √ − | i | i PBS PBS 2 PDBS sin α PBS + (D(η) 0 m D( η) 0 m)] 1 anc. (8) . 2 | i − − | i | i Fig. 2. (a) Quantum delayed-choice experiment in optome- Results indicate that even though the system is measured, chanical system. The one of the entangled photons enters the we can vary the system from wave to particle behaviour first beam splitter of March-Zehnder interferometer with an op- through adjusting the angle α in ancilla. It is interesting tomechanical cavity A and a conventional cavity B on the right to consider the fact that when the ancilla is measured in side. The quantum beam splitter consist of the polarization- 1 a superposition state, such as ϕ = ( 0 anc 1 anc), dependent beam splitter (PDBS) and the erasure device, i.e., | i √2 | i − | i then the final state of the mirror is a superposition state, the polarizing beam splitter (PBS1 and PBS2). The ancilla photon of the entangled state is sent to ancilla photon apparatus on the left side where EOM denote electro-optic phase cos α ψ m = D( η) 0 m modulator and PBS3 is the polarizing beam splitter oriented | i √2 − | i at the H,V basis (b) Polarization-dependent beam splitter sin α (PDBS){ yield} the 100/0 reflection/transimission ratio for H (D(η) 0 m D( η) 0 m)]. (9) | i − 2 | i − − | i and 50/50 reflection/transimission ratio for V . In PDBS, the four polarizing beam-splitter (PBS) orient at| thei H,V basis We note that cos α is caused by the particle behaviour of the and the BS is an ordinary 50/50 BS. { } system, while sin α is due to wave behaviour of the system. From the derivation of Eq. (9), it will be shown that the key advantage of the ancilla measurement is to control the choice of the measured system basis. This indicate that the Controlled- where the coefficient (π/2 α)/√2 is due to cos α generated Hadamard gate or quantum beam splitter play a role of the − postselection. Moreover, the state of Eq. (9) can also be ob- by the particle behaviour of the system, while the coefficient η tained when we simultaneously measure the system and the is due to η sin α caused by the wave behaviour of the system. ancilla in the entangled states, i.e., the postselection state The average displacement of the pointer positionq ˆ is

ψf = 1 s 0 anc 1 anc. (10) | i | i | i − |−i| i Therefore, the superposition of the pointer in Eq. (9) caused by ψ mqˆ ψ m the particle and wave behaviours of the system is no classical qˆ = h | | i 0 qˆ 0 . (13) h i ψ ψ m −h | | i analog. h | i Based on the Eq. (9), we can then perform a small quantity expansion about η till the second order, where η 1, then ≪ For Eq. (12), it can be seen that qˆ is none-zero (amplification) cos α h i ψ m ( 0 m + η 1 m) η sin α 1 m. (11) since the superposition between the states 0 m and 1 m. Such | i ≈ √2 | i | i − | i a result stems from the wave and particle behaviours| i | ofi the sys- For α = π/2, the beam splitter is present, i.e., only wave be- tem controlled by quantum beam splitter that play a role of haviour occur, and we find the state ψ m proportional to 1 m, the postselection. It will be shown that wave-particle duality which correspond to the case that the| post-selectedi state of| ithe in quantum mechanics can be used as generation mechanism system is orthogonal to the initial state of the system [1,19], of this amplification effect, that has never before been revealed the displacement of pointer’s position is zero. in weak measurement. Moreover, the weak measurement pro- For π/2 α 1, the particle and wave behaviour simulta- vided here relies on the entanglement of the system and the neously occur,− here≪ we use the approximation cos α π/2 α ancilla in quantum beam splitter, mean that the results can’t and sin α 1 to get the superposition state ≈ − be explained by Maxwell’s equations alone [20]. This scheme ≈ provide a new method for studying and exploiting postselec- ψ m (π/2 α)/√2 0 m η 1 m, (12) tion and weak measurement in three-qubit system. | i ≈ − | i − | i 4 Gang Li et al.: Weak measurement implementation in optomechanical system

3 Implementation in optomechanical system 1.0

0.5 In the following we will consider an weak measurement model in optomechanical system combined with quantum delayed- Σ L\

t 0.0 choice experiment. H q X -0.5

3.1 The optomechanical model -1.0 0 Π 2 Π 3 Π 4 Π 5 Π 6 Π Ω Based on a Mach-Zehnder interferometer, a schematic dia- mt . gram of quantum delayed-choice experiment to be considered Fig. 3. is shown in Fig. 2. The optomechanical cavity A is embed- q(t) /σ as a function of ωmt with k = 0.005 for different α, i.e,h α i= 0.996 π/2 (red-solid), 0.9995 π/2 (orange- ded in one arm of the March-Zehnder interferometer and a × × stationary Fabry-P érot cavity B is placed in another arm. dashed) and π/2 (blue-dotted-dashed). And the first beam splitter is symmetric, but the second one is quantum beam splitter [18] which is made up of two com- is input into the interferometer, where H (V ) represent hor- ponents. The first is a polarization-dependent beam splitter izontal (vertical) polarized photonic modes, then the state of (PDBS) which shows 100% reflection for horizontal polarized the photons after the first beam splitter becomes photons and 50/50 relection/transmission for vertical polarized 1 photons. The second is erasure device consisting of polarizing ψi = [ 1,H anc( 1,H A + 1,H B) beam splitters (PBS1 and PBS2). In Fig. 2 on the left side, | i 2 | i | i | i + 1,V anc( 1,V A + 1,V B)]. (15) the electro-optic phase modulator is to rotate the polarization | i | i | i state of the the ancilla’s photon by an angle α. After interacting weakly with the mirror prepared at ground The Hamiltonian of optomechanical system in the interfer- state 0 m that can be achieved using sideband-resolved cooling ometer is expressed as followed: technique| i [24,25], the state of the photons and the mirror is

H =hω ¯ 0(a†a + b†b) + ¯hωmc†c ¯hga†a(c† + c), (14) 1 ψi = [ 1,H anc( 1,H A ξ(t) m + 1,H B − | i 2 | i | i | i | i whereh ¯ is Planck’s constant, ω0 is frequency of the optical 0 m)+ 1,V anc( 1,V A ξ(t) m cavity A, B and the corresponding annihilation operators are | i | i | i | i + 1,V B 0 m)]. (16) aˆ and ˆb, ωm is frequency of mechanical system and the cor- | i | i responding annihilation operator isc ˆ, and the optomechanical After the photon passes through the polarization-dependent ω0 beam splitter (PDBS), the state of the overall system becomes coupling strength g = L σ, where L is the length of the cavity 1/2 A or B, σ = (¯h/2mωm) which is the zero-point fluctuation 1 ψPDBS = 1,H anc( 1,H C ξ(t) m + 1,H D and m is the mass of mechanical system. Here it is a weak | i 2 | i | i | i | i measurement model where the mirror is used as the pointer to 1 0 m)+ 1,V anc[ 1,V C ( ξ(t) m measure the number of photon in cavity A. Noted that because | i 2√2 | i | i −| i a†a of the Eq. (14) corresponds toσ ˆz of the Hamiltonian in the + 0 m)+ 1,V D( ξ(t) m + 0 m)]. (17) standard scenario of weak measurement and c+c† corresponds | i | i | i | i top ˆ. In order to erasure of all polarization information about In the optomechanical cavity A (see Fig. 2), if the ini- PDBS output, we make the state of Eq. (17) go through the tial state of the mirror is prepared at the ground state 0 polarizing beam splitter (PBS1 and PBS2) oriented at 45◦ to and when one photon interacts with the mirror, the mirror| i H, V basis. Then Eq. (17) becomes { } will become a coherent displacement state [23,22], ξ(t) = 1 iφ(t) iφ(t) | i ψ = ( 1,H anc particle sm + 1,V anc wave sm) (18) e ϕ(t) m, where e is the Kerr phase of one photon | i √ | i | i | i | i | i 2 iωmt 2 with φ(t) = k (ωmt sin ωmt), ϕ(t) = k(1 e− ) and − − k = g/ωm. According to the expression of the position displace- with ment ξ(t) qˆ ξ(t) 0 qˆ 0 , then the displacement of the mirror 1 h | | i−h | | i particle sm = ( 1 ′ + 1 ′′ ) ξ(t) m +( 1 ′ is not more than 4kσ for any time t and can not be bigger than | i 2 | ia | ia | i | ib ′′ the zero-point fluctuation of the mirror since k = g/ωm can not + 1 b ) 0 m (19) be bigger than 0.25 in weak coupling condition [26]. Therefore, | i | i and the displacement of the mirror caused by one photon can not be detected. However, if considering optomechanical cavity with 1 wave sm = ( 1 a′ + 1 a′′ )( ξ(t) m + 0 m) weak measurement combined with quantum delayed-choice ex- | i 2√2 −| i | i −| i | i periment, we can amplify the mirror’s displacement. +( 1 ′ + 1 ′′ )( ξ(t) m + 0 m). (20) −| ib | ib | i | i Using the electro-optic phase modulator (EOM) rotate the polarization state of ancilla by an angle α, we can obtain 3.2 Amplification with postselected weak 1 measurement in optomechanics ψQBS = [ 1,H anc(cos α particle sm sin α | i √2 | i | i − Suppose the one of the photons from the maximally polarization- wave sm)+ 1,V anc(cos α wave sm + 1 | i | i | i entangled Bell state Φ = ( 1,H anc 1,H s+ 1,V anc 1,V s) + sin α particle sm)] (21) | i √2 | i | i | i | i | i Gang Li et al.: Weak measurement implementation in optomechanical system 5 1 - 1.5 L 0.00

Based on the polarizing beam splitter (PBS3) and detecting Σ -0.02 2

H -0.04 the ancilla’s photon in DH , then we ﬁnd that Ñ -0.06

L\ -0.08 t 1 1.0 H -0.10 p - X 0 Π 2 Π 3 Π 4 Π 5 Π 6 Π

1 L Ω ψQBS = (cos α particle sm sin α wave sm) (22) Σ mt

| i √2 | i − | i 2 0.5 H Ñ Here we only consider detector D ′ . When a photon is detected

a L\

t 0.0 at the dark port D ′ , in the language of weak measurement we H

a p X simultaneously measure the single-photon in the orthogonal -0.5 1 postselection state of wave behaviours ( 1 A 1 B ) [11,9] √2 | i − | i and in the state of the particle behaviour 1 A. Then the state -1.0 0 Π 2 Π 3 Π 4 Π 5 Π 6 Π of the mirror after postselection is | i Ωmt cos α sin α . ψm = ξ(t) m ( ξ(t) m 0 m). (23) 1 | i 2√2 | i − 4 | i − | i Fig. 4. p(t) /¯h(2σ)− as a function of ωmt with k = 0.005 for differenth αi, i.e., α = 0.996 π/2 (red), 0.999 π/2 (green) We note that cos α is due to the particle behaviour of the pho- and π/2 (blue). × × ton, while sin α is due to wave behaviour of the photon. The average displacement of the mirror’s position operator qˆ = (ˆc +c ˆ†)σ is given by 0, 1, 2, ). In the same way as the Eq. (12), for Eq. (23) we · · · Tr( ψm ψm q) can perform a small quantity expansion about time T till the q = | ih | Tr( 0 m 0 mq). (24) second order. Suppose that ωmt T 1, and k 1, then h i Tr( ψm ψm ) − | i h | | |ih | | − |≪ ≪ cos α Substituting Eq. (23) into Eq. (24), then ψm ( 0 m + 2k 1 m) k sin α 1 m. (26) | i≈ √2 | i | i − | i 2 q(t) = σ[(2 sin α √2 sin 2α)(ϕ(t)+ ϕ∗(t)) h i − − When π/2 α 1, here we use the approximation cos α ϕ t 2 2 | ( )| iφ(t) − ≪ ≈ + (sin 2α/√2 sin α)e− 2 (e ϕ(t) π/2 α and sin α 1 to get the superposition state − − ≈ iφ∗(t) + e ϕ∗(t))]/[2 √2 sin 2α + (sin 2α/√2 − ψm (π/2 α)/√2 0 m k 1 m. (27) 2 iφ(t) iφ∗(t) | i≈ − | i − | i + sin α)(e + e )] (25) Noted that the coefficient (π/2 α)/√2 is due to cos α by To see this clearly, in Fig. 3, we plot the average dis- generated the particle behaviour of− the photon, while the coef- placement q(t) /σ of the mirror versus the time ωmt with ficient k is due to k sin α caused by the wave behaviour of the h i k = 0.005 for different angles α [α = 0.996 π/2 (red-solid), photon. Substituting Eq. (27) into Eq. (24), the average value × 0.9995 π/2 (orange-dashed) and π/2 (blue-dotted-dashed)]. of the displacement operatorq ˆ is given by It can be× seen clearly from Fig. 2 that when α = 0.996 π/2 × √ 2 the amplification (red-solid) is a time function of period 2π and q(t) ωmt T 1 = σ 2k(π/2 α)/[k h i| − |≪ − − at time ωmt = (2n + 1)π (n = 0, 1, 2, ) the amplifications + (π/2 α)/2], (28) can reach strong coupling limits (the level· · · of the ground state − fluctuation) q = σ [26]. However, when α π/2, such which can get its minimal value σ when k = (π/2 α)/√2. h i − → − − as α = 0.9995 π/2, the amplification around time ωmt = Hence the state of the mirror achieving the maximal negative × 1 2nπ (n = 1, 2, ) appears, but the rest of the amplifica- amplification is ( 0 m 1 m). The results show that the √2 | i − | i tions (orange-dashed)· · · is decreasing. Until α = π/2 which indi- key to the amplification is the superposition of the vacuum cates that the quantum beam splitter corresponds to a beam state and one phonon state of the mirror, which is due to the splitter, the amplification (blue-dotted-dashed) around time superposition of wave and particle behaviours of the photon. ωmt = 2nπ (n = 1, 2, ) is very prominent and reaches the Such result has not previously been revealed. For Eq. (27), the level of the ground state· · · fluctuation q = σ. The result is amplification mechanism is accord with standard weak mea- caused by the kerr phase which can’t beh i explained± by the stan- surement [1,19] and it is caused by proper near-orthogonal dard weak measurement [1,19] and has been discussed in de- postselection but different from the amplification mechanism tail in Ref. [9]. Note that in the optomechanical cavity A (see for Eq. (16) in Ref. [9] which is caused by the Kerr phase in Fig. 2) the maximal displacement of the mirror caused by one this case of orthogonal postselection. photon is 4kσ and the displacement σ can be obtained using Next we would like to discuss the amplification of momen- quantum beam splitter in weak measurement,− hence the ampli- tum variable p of the mirror. Substituting Eq. (23) into Eq. h¯ fication factor can be Q = 1/4k which is 50 when k = 0.005. (24) which uses p = i(c c†) 2σ instead of q, then we have Moreover, the negative amplification is a− counterintuitive re- − − ¯h 2 sult since intuitively there cannot exist negative displacement p(t) = i [(2 sin α √2 sin 2α)(ϕ(t) ϕ∗(t)) relative to the direction of the photon propagation. h i − 2σ − − − 2 ϕ(t) 2 iφ(t) + (sin 2α/√2 sin α)e−| | (e ϕ(t) − iφ∗(t) 3.3 Small quantity expansion about time for e ϕ∗(t))]/[2 √2 sin 2α + (sin 2α/√2 − −∗ amplification sin2 α)(eiφ(t) + eiφ (t))] (29) − To explain the amplification phenomenon shown in Fig. 3, we The average displacement p(t) / h¯ of the mirror is shown in h i 2σ only give an analysis about the special case that the maxi- Fig. 4 as a function of ωmt with k = 0.005 for different angle mal amplification appearing at the time T = (2n + 1)π (n = α [α = 0.996 π/2 (red-solid), 0.999 π/2 (green-solid) and × × 6 Gang Li et al.: Weak measurement implementation in optomechanical system

is considered in [11] is small, i.e., ωm = 4κ (red line), the photon arrival rate is almost distributed from 0 to 2π. Combined 0.6 with the ampliﬁcations from Fig. 3 within time ωmt = 2π, the implementation of our scheme is feasible in experiment.

0.4

0.2 4 Conclusion Photon arrival rate density In conclusion, we use the quantum beam splitter in stead of 0.0 0 Π Π 3 Π 2 Π 5 Π 3 Π the beam splitter in the delayed-choice experiment. When com- 2 2 2 bined with the weak measurement and the quantum beam Ω mt Hmecahanical periodsL . splitter play a role of the postselection, it will be shown that Fig. 5. For k = 0.005 and α = 0.996 π/2, photon arrival the wave and particle behaviours of the measured system give probability density vs arrival time for a successful× postselection rise to a surprising amplification effect. We find that the wave- with ωm = κ (blue), ωm = 2κ (green) and ωm = 4κ (red). particle duality of the measured system can be used as generation mechanism for weak measurement amplification, that has not previously been revealed. This amplification effect can’t π/2 (blue-solid)]. We can see clearly from Fig. 4 that the am- be explained by Maxwell’s equations [20], due to the entangle- plification effect (green-solid) around the vibration periods of ment of the system and the ancilla in quantum beam splitter. the mirror ωmt = 2nπ (n = 0, 1, 2, ) is very prominent and This amplification mechanism about the wave-particle duality reaches strong coupling limits (the· · level · of the ground state of the measured system lead us to a scheme for implemen- h¯ tation in optomechanical system. Moreover, the displacement fluctuation) p = 2σ [26] if the angle α is very close to π/2. However, whenh i α =± π/2 the amplification of the mirror’s mo- of the mirror’s position in optomechanical system can appear mentum p (blue-solid) is suppressed around time ωmt = 2nπ within a large evolution time zone, which is a counterintuitive (n = 0, 1, 2, ) (see the inset in Fig. 4). negative amplification effect. The results provide us an alter- In order· to · · understand the amplification phenomenon around native method to discuss weak measurement and also deepen time ωmt = 2nπ (n = 0, 1, 2, ), for Eq. (23) we can also our understanding of the weak measurement. make a small quantity expansion· · · about time T = 2nπ (n = 0, 1, 2, ) till the second order. Suppose that π/2 α 1, · · · − ≪ ωmt T 1 and k 1, then | − |≪ ≪ Acknowledgments

ψm (π/2 α)/√2 0 m ik(ωmt T ) 1 m. (30) | i≈ − | i − − | i This work was supported by the National Natural Science Foundation of China under grants No. 11175033. Y. M. Y. ac- It is obvious from the Eq. (30) that the key to the amplification knowledges support from the National Natural Science Foun- is also the superposition of the vacuum state and one phonon state of the mirror, which is due to the superposition of wave dation of China (Grant Nos. 61275215) and the National Fun- and particle behaviours of the photon. It is not hard to see that damental Research Program of China (Grant No. 2011CBA00203). h¯ its maximal value is when (π/2 α)/√2 = k(ωmt T ), 2σ − − − h¯ √ and its minimal one is 2σ when (π/2 α)/ 2 = k(ωmt T ). Therefore the mirror− state achieving− the maximal positiv−e References 1 amplification is ( 0 m + 1 m) and the state achieving the √2 | i | i 1 1. Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. maximal negative amplification is ( 0 m 1 m). √2 | i − | i Lett. 60, 1351 (1988). 2. Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Weinheim: Wiley-VCH, 2005). 3.4 Discussion 3. J. SLundeen, S. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature (London) 474, 188 (2011). We shall show that how the photon arrival rates vary with time 4. O. Hosten and P. Kwait, Science 319, 787 (2008). after the single photon is detected at dart port. The probability 5. P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, density of a photon being released from optomechanical cavity Phys. Rev. Lett. 102, 173601 (2009). after time ωmt is 6. J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. κ exp( κt), (31) W. Boyd, Rev. Mod. Phys. 86, 307 (2014). − where κ is the decay rate of the cavity. The probability of a 7. F. Marquardt and S. M. Girvin, Physics 2, 40 (2009). successful post-selection being released after ωmt is 8. M. Aspelmeyer, T. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014). 1 2 iφ(t) 90 [2 √2 sin 2α + (sin 2α/√2 sin α)(e 9. G. Li, T. Wang, H. S. Song, Phys. Rev. A , 013827 4 − − (2014). ∗ +eiφ (t))]. (32) 10. M. O. Scully and M. S. Zubairy, Quantum Optics (Cam- bridge University Press, Cambridge, England, 1997), pp. Multiplying Eq. (31) and Eq. (32), then we obtain the photon 128-134. arrival rate in optomecanical cavity, which is shown in Fig. 5. 11. B. Pepper, R. Ghobadi, E. Jeffrey, C. Simon, and D. It can be seen clearly that if the decay rate of the cavity which Bouwmeester, Phys. Rev. Lett. 109, 023601 (2012). Gang Li et al.: Weak measurement implementation in optomechanical system 7

12. J. A. Wheeler, in Mathematical Foundations of Quantum Mechanics, edited by A. R. Marlow (Academic, New York, 1978), pp. 9-48. 13. J. A. Wheeler, in Quantum Theory and Measurement, edited by J. A. Wheeler and W. H. Zuurek (Princeton Univeraity Press, Princeton, NJ, 1984), pp. 182-213. 14. N. Bohr, in Quantum Theory and Measurement, edited by J. A. Wheeler and W. H. Zuurek (Princeton Univeraity Press, Princeton, NJ, 1984), pp. 9-49. 15. R. Ionicioiu, and D. R. Terno, Phys. Rev. Lett. 107, 230406 (2011). 16. J. S. Tang, Y. L. Li, X. Y. Xu, G. Y. Xiang, C. F. Li, and G. C. Guo, Nat. Photonics. 6, 602 (2012). 17. A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’Brien, Secience 338, 634 (2012). 18. F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and S. Tanzilli, Secience 338, 637 (2012). 19. C. Simon and E. S. Polzik, Phys. Rev. A 83, 040101(R) (2011). 20. D. Suter, Phys. Rev. A 51, 45 (1995). 21. M. A. Niesien and I. L. Chuang, Quantum computation and Quantum information (Cambridge university press, Cambridge, 2000). 22. S. Bose, K. Jacobs, and P. L. Knight, Phys. Rev. A 56, 4175 (1997). 23. S. Mancini, V. I. Man’ko, and P. Tombesi, Phys. Rev. A 55, 3042 (1997). 24. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippen- berg, Phys. Rev. Lett. 99, 093901 (2007). 25. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys. Rev. Lett. 99, 093902 (2007). 26. W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, Phys. Rev. Lett. 91, 130401-1 (2003).