ЖЭТФ FEYNMAN DIAGRAM APPROACH TO DYNAMICAL CASIMIR EFFECT Yusong Cao a;b*, Yanxia Liu a a Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China b School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China In this paper we study a optomechnical system enclosed by a optical cavity with one mirror attached to a spring as a closed quantum system. We provide a different sight of view of studying the backaction effect from the quantum field to the mirror. Instead of concentrating on the force exerted on the mirror, we focus on the energy transferring of the process, which encourages us to use the Feynman diagram method in weak coupling regime. We show that the observed photon frequency, mirror oscillation frequency and ground state energy are shifted as a result of emission and absorbtion of virtual particles. We emphasis that the integrals in the loop diagrams are convergent, which means by treating the motion of the mirror with quantum mechanics, we get finite result in energy transferring process. At last we use Feynman diagrams to give the transition amplitudes of general dynamical Casimir process. 1. INTRODUCTION (1) the motion of boundary must be introduced prior [10–13] and (2) when examining the backaction from In quantum mechanics, one of the most anti- quantum field to the boundary, one can only get the intuitive thing is the Heisenberg’s uncertainty principle, result at an average level [11–15]. The infinities in the as a result of which, the quantum harmonic oscillator backaction force from the vacuum to the boundary has is still moving even at the ground state. When it comes plagued the construction of a consistent dynamical the- to the systems that allow the creation and annihilation ory of the boundary for some time. To solve this prob- of particles, say, in quantum field theory, it tells us two lem, several regularization methods have been devel- things (1) there are virtual particles emerges from vac- oped [14,16]. In this paper, we tackle the problem from uum and quickly annihilates again [1, 2] and (2) the a different angle: we quantize the motion of boundary absolute energy of the vacuum state is infinity without and treat the whole system as a closed system [17–19]. normal ordering technique [3]. One of the astonishinng To get a hold of the picture, we noticed that one of the results is if the background field or boundary condi- most significance difference between the quantum me- tion of the quantum vacuum is properly modulated, it chanics and classical mechanics is the existence of quan- is possible to convert virtual particles of vacuum into tum fluctuation. New effects of purely quantum nature real ones. In practice, such phenomenons are called can emerge if we quantize the classically moving sub- dynamical Casimir effect [4, 5] due to the modulation jects. For example, when we treat the charged current of boundary condition of the cavity field, cosmologi- in electromagnetic vacuum with quantum mechanics, cal particle creation as a result of varying size of the such effects as Lamb shifts of electronic energy levels, background flat spacetime [6], Hawking radiation from anomaly magnetic moment, renormalization of charge, curved spacetime [7, 8], Unruh effect for a accelerated effective photon-photon interaction will take place due observer in vacuum [9] and spontaneous radiation for arXiv:2103.10657v1 [quant-ph] 19 Mar 2021 to the fluctuation of electromagnetic vacuum [3]. a charged current is placed in electromagnetic vacuum [3]. In this paper, we study a optomechanical system en- Most of the treatments for such systems are semi- closed by a cavity with one mirror attached to a spring classical: treat field with quantum mechanics and the as a closed quantum system. In weak coupling regime, background field/moving boundary with classical me- we use Feynman diagram method to study three things, chanics [4-14]. Such treatments has two drawbacks, (1) the backaction effect of photons on the mirror, (2) the effect of quantum fluctuation of the quantized mir- * E-mail: [email protected] ror on the photons and (3) the shift of the ground 1 Yusong Cao, Yanxia Liu ЖЭТФ ity when the spring is at its relaxion. Together with the spring, the mirror behaves like a simple harmonic oscillator. The radiation pressure on the mirror can be k viewed as driving force to the harmonic oscillator. And the motion of the mirror will squeeze the cavity field. The radiation pressure is small compared to the elastic c force from the spring, so the mirror’s sphere of activity is much smaller compared to the length L. The Hamiltonian of the model was first proposed in the work of Law [21]. The Hamiltonian (~ = c = 1) is L given as H = H0 + V , with the free term y y H0 = !ca a + !mb b; (1) Fig. 1. The sketch map for our system: A optical cavity where a is the cavity field operator and the frequency with one side fixed, while another side attached to a spring is !c = nπ=L, with n for the mode number of the given which forms a simple harmonic oscillator. The radiation frequency !c. Note that the polarization of the cavity pressure of cavity field acts on the mirror as drive force field is not important in the processes we will study of the oscillator while the motion of the mirror squeezes later, so we choose only one polarization of the cavity the cavity field. field in our Hamiltonian [19]. And b is the annihila- tion operator of the simple harmonic oscillator of form b = p m!m (x + ip ), with x and p are the position state energy from quantum fluctuation of the vacuum 2 m!m and momentum of the oscillator. The frequency of har- field and the mirror. With Feynman diagram method, q k we compute the transition rates of general dynamical monic oscillator is given by !m = m . The exciton Casimir effect processes (i.e. the transition of phonons number of harmonic oscillator is also refered as phonon into photon pairs). This paper is organized as follows. number [1]. The interaction term is split into two parts: In Sec.2, we give a brief introduction of our model V = Vom + VDCE, the first part Hamiltonian and demonstrate the Feynman rules of y y Vom = ga a(b + b ) (2) our model. In Sec.3, we calculate the frequency shifts of the mirror’s oscillation due to the backaction from is called the standard optomechanical coupling term. field vacuum and show that the integrals in the formu- This term conserves the number of photon [20]. The las are convergent, and why the infinity of frequency parameter g is called the coupling constant. And the shift of phonon in resonance brings no inconsistency. second term In Sec.4, we compute the energy shift of the system’s g V = [a2 + (ay)2](b + by) (3) ground state from vacuum-vacuum process. In Sec.5, DCE 2 we calculate the frequency shift of cavity photon due is called dynamical Casimir effect term being respon- to the quantum fluctuation of the mirror. In Sec.6, we sible for dynamical Casimir effect process, which in- compute the decay width of the mirror’s oscillation en- troduces the transformations between the phonons and ergy quanta at resonance and transition amplitudes of photons. In the weak coupling regime !c;!mg, the some dynamical Casimir processes and compare them optomechanical coupling term Vom plays the dominant with the results qualitatively in previous work [20]. In role and the effects of dynamical Casimir effect term is Sec.7, we give a conclusion of our results and give some hard to observe [20]. In this paper, we will neglect the discussion. dispassion and treat the system as a closed quantum system. Like it is in the case of Rabi model [22], the optome- 2. MODEL HAMILTONIAN AND FEYNAMN RULES chanical system under consideration can be viewed as an anology to the cavity QED system, which is consid- The system we will study in this paper is a cavity ered a simplified version of quantum electrodynamics. with one mirror fixed and another one attached to a In this sense, we are encouraged to apply the Feyn- spring, the schematic of which will be demonstrated in man diagram method. In weak coupling regime, it is Fig.(1). The mass of the movable mirror is m and the of high accuracy computing the processes to the low- strength of the spring is k. L is the length of the cav- est order of g. The momentum space Feynman rules 2 ЖЭТФ Feynman diagram approach to Dynamical Casimir effect (a) (b) i = (a) E − ωm + iϵ E1 = 1 i ωm ωm = E − ωc + iϵ (c) g = − i E2 2 d ( ) (b) E1 ig = − E2 ωm ωm E3 Fig. 2. Basic Feynman diagrams in our system. Note that there are two kinds of vertices of different coupling E strength: (c) square for the optomechanical interaction 4 and (d) dot for the Dynamical Casimir effect interaction. The crosses in (a) and (b) denote the vertices attached to the particle line, whose type of origin are irrelevant Fig. 3. The irreducible diagrams of phonon propagator. to the value of the lines. The legs of the lines attached (a) Diagram of two virtual photons in virtual process. to the vertices looks short thus we name them "Corgi (b) Diagram of a propagating phonon glued to a vacuum- diagrams". vacuum diagram with one virtual phonon and two virtual photons.