EPJ manuscript No. (will be inserted by the editor) Weak measurement combined with quantum delayed-choice experiment and implementation in optomechanical system 1, 2 3 1, Gang Li, ∗ Tao Wang, Ming-Yong Ye, and He-Shan Song † 1School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China 2College of Physics, Jilin University, Changchun 130012, China 3School of Physics and Optoelectronics Technology, Fujian Normal University, Fuzhou 350007, China the date of receipt and acceptance should be inserted later Abstract. Weak measurement [1,19] combined with quantum delayed-choice experiment that use quantum beam splitter instead of the beam splitter give rise to a surprising amplification effect, i.e., counterintuitive negative amplification effect. We show that this effect is caused by the wave and particle behaviours of the system to be and can’t be explained by a semiclassical wave theory, due to the entanglement of the system and the ancilla in quantum beam splitter. The amplification mechanism about wave-particle duality in quantum mechanics lead us to a scheme for implementation of weak measurement in optomechanical system. PACS. 42.50.Wk – 42.65.Hw – 03.65.Ta 1 Introduction It is known to all that the Mach-Zehnder interferom- eter [10], including the input beam splitter and the out- put beam splitter, can show wave-particle duality. When Postselection weak measurement was first proposed by a photon enters the input beam splitter, the photon be- Aharonov, Albert and Vaidman [1], where a pointer is haved as a wave behaviour if the output beam splitter is weakly coupled to the system to be measured. It can help present or behaved as a particle behaviour if the output us to understand some counter-intuitive quantum para- beam splitter is absent. Based on the Mach-Zehnder in- doxes [2] and it is also used to measure and amplify small terferometer, the the delayed-choice experiment [12,13] is physical quantities or effects which are not directly de- proposed to demonstrate Bohr’s complementarity [14] and tected by conventional techniques in experiment, such as it show that wave-particle duality depend on the classical direct measurement of wave function [3] and beam deflec- detecting devices, i.e., whether the output beam splitter tion [4,5]. Recently, the basics and application of weak is inserted or removed. However, recently Ionicioiu et al measurement have been reviewed in [6]. propose a scheme for quantum delayed-choice experiment Although postselection weak measurement have many [15] based on quantum beam splitter in stead of the output applications, its application in optomechanics was rarely beam splitter, where the quantum beam splitter controlled reported. The optomechanical system usually refers to a by the ancilla [12,13] can be put in superposition of being high finesse optical cavity with a tiny mirror attached to present and absent. In this scheme, the interferometer can arXiv:1509.00641v1 [quant-ph] 2 Sep 2015 micromechanical oscillator [7,8], where the light in the be simultaneously closed and open so that both the wave cavity can give a radiation force on the mirror. When and particle behaviours of the photon can appear at the there is only one photon in the cavity the displacement same time. Soon after, this scheme is verified by a series of the mirror caused by the photon is hard to be mea- of experiments [16–18]. sured since it is much smaller than the spread of the mir- ror wave packet. Recently, in the optomechanical system The nature of the quantum beam splitter is different combined with weak measurement, the weak measurement from the beam splitter (classical) and naturally we want to amplification scheme that the displacement of the mir- know based on quantum delayed-choice experiment that ror caused by single photon can be amplified is proposed use quantum beam splitter in stead of the beam splitter [9], where the optomechanical system is embedded in the and in combined with weak measurement whether there March-Zehnder interferometer [10]. The amplification ef- are some new features. In the paper, we will first give a fect in [9] is obtained by retaining the Kerr phase in the general discussion about weak measurements [1,19] com- Ref. [11]. bined with quantum delayed-choice experiment that use the quantum beam splitter in stead of the beam splitter. Correspondence to: ∗[email protected], †[email protected] The key advantage of the ancilla in quantum beam split- 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 exper- iment. 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 con- trolled 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 HˆI =¯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 where ¯h is Planck’s constant, χ is a small coupling con- stant, 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. HˆI = 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 dark port dark 0' s port dark 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.