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Home , Interaction picture, Quantum fluctuation

ЖЭТФ

FEYNMAN DIAGRAM APPROACH TO DYNAMICAL

Yusong Cao a,b*, Yanxia Liu a

a Beijing National Laboratory for , 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 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 transferring of the process, which encourages us to use the method in weak coupling regime. We show that the observed frequency, mirror oscillation frequency and 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 of the mirror with quantum , 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 , one of the most anti- quantum field to the boundary, one can only get the intuitive thing is the Heisenberg’s , result at an average level [11–15]. The infinities in the as a result of which, the quantum harmonic oscillator backaction force from the 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 . 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 is properly modulated, it chanics and 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], from anomaly magnetic moment, 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 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 of Law [21]. The Hamiltonian (~ = c = 1) is L given as H = H0 + V , with the free term † † H0 = ωca a + ωmb b, (1)

Fig. 1. The sketch map for our system: A optical cavity where a is the cavity field and the frequency with one side fixed, while another side attached to a spring is ωc = nπ/L, with n for the mode number of the given which forms a simple harmonic oscillator. The radiation frequency ωc. 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 mωm (x + ip ), with x and p are the position state energy from quantum fluctuation of the vacuum 2 mωm 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 ωm = 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 † † 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 . 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 + (a†)2](b + b†) (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 ωc, ωmg, 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 . 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 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.

3. FREQUENCY SHIFT OF HARMONIC OSCILLATOR of our model is shown in Fig.(2). The time arrow in In this section we study the backaction of the vac- Fig.(2) and the Feynman diagrams used in the follow- uum field exerted on the mirror. There are some previ- ing sections is from left to right. There are two dif- ous works on the fluctuation of the force with mirror’s ferent things between our Feynman rules and those in motion quantized [17, 18]. Here we develop a different standard quantum field theory (QFT) textbooks [3,23], type of treatment. Since we have already admitted the one, there are two kinds of vertices exist, and two, the quantum nature of the motion of the harmonic oscilla- left and right direction of legs attached to the vertices tor, then we can obtain the information of its dynamical matters. Speaking in detail, Fig.(2c) demonstrates two properties directly from Hamiltonian, which allows us processes, emission and absorbtion of a phonon from to avoid the computation of the backaction force. As a photon; Fig.(2d) shows four processes: (1) a phonon it is in the QFT, the information of the motion of a turns into a photon pair, (2) a photon pair turns into a subject is encoded in the in and out relation while the phonon, (3) a photon pair and a phonon emerges from quantities needed to be focused on is the four momen- vacuum and (4) a photon pair and a phonon annihilate tums of incoming and outgoing particles. into vacuum. Note that the cross symbols in Fig.(2a) and Fig.(2b) denote the vertices connect to the particle Like the emission and absorbtion of virtual photons lines, and their type of origin is irrelevant to the value and virtual electrons of a electron gives an additional of the lines. electromagnetic mass term to the electron bare mass in quantum electrodynamics, the phonon in our sys- tem can also experience an additional frequency shift There are three more things of Fig.(2): One, the en- term due to the emission and absorbtion of the virtual ergy conserves at each vertices. Two, take the integral photons and virtual phonons. To the order g2, there are of all the undermined energy appear in loop diagrams two such processes of shown in the diagrams in Fig.(3). R dE as 2π . And three, energy conservation of the whole Since the system is weakly coupled, we can obtain the diagram. frequency shift δωm by chain approximation [23]. In

3 Yusong Cao, Yanxia Liu ЖЭТФ this way, the frequency shift is given as 4

(2) (4) δωm = Σ2 (ωm) + Σ2 (ωm), (4) 3 (2) (4) where −iΣ2 (ωm) and −iΣ2 (ωm) are the values of Fig.(3a) and Fig.(3b). The superscripts are for the or- m 2 der of diagrams while the subscripts are for the number of intermediate particles. Fig.(3a) represents the pro- cess that a phonon transforming into a virtual photon 1 pair and back again with two dynamical Casmir effect vertices. Fig.(3b), however, is different from the usual way of constructing a Feynman diagram from Feyn- 0 1.5 2 2.5 man rules in the textbooks of QFT [3,22,23]. In our k case, the vertices in Fig.(3b) can be viewed as an ex- ternal phonon line glued to a vacuum-vacuum diagram, Fig. 4. The frequency of the physical phonon (Ω ) with in which a virtual photon pair and one virtual phonon m respect to the ratio of the bare phonon frequency to the emerging from vacuum and annihilating into vacuum bare phonon frequency k = ωm . In this picture we set with dynamical Casmir effect vertices. Calculate the ωc ωc = 1, and blue solid line for g = 0.1 with red dashed value of Fig.(3a) with Feynman rules in Fig.(2) gives line for g = 0.05. Both lines goes to infinity at k = 2. This is a result of the unstableness of phonon in resonance 2 Z +∞ (2) g dE1dE2 i case ωm = 2ωc. −iΣ2 (ωm) = − 4 −∞ 2π E1 − ωc + i i (5) × δ(E1 + E2 − ωm) is also convergent. The derivation of this formula can E2 − ωc + i g2 1 be tricky and we leave the details in Appendix. Taking = −i , the results in Eqs.(5) and (6) into Eq.(4) gives the value 4 2ωc − ωm of frequency shift of harmonic oscillator from quantum where the integrated elements E1 and E2 are the en- fluctuation of cavity vacuum ergy of the virtual particles. Note that the integral in g2 1 1 Eq.(5) is convergent. This suggests that due to the δωm = − ( + ), (7) quantum fluctuation of the position of the mirror, the 4 2ωc − ωm 2ωc + ωm from photon to mirror does not takes places only which is identical to the result from at one space point at a time. So the backaction from perturbation theory around state |0, 1i, where the first the vacuum field on the mirror is not determined by the number in the ket denotes photon number and the sec- value of the energy-momentum tensor of the field at a ond denotes phonon number. We can see that the ob- specific point, which gives divergent result in calcula- served frequency of the harmonic oscillator is shifted tion of the force exerted on the mirror [12–15]. This due to the virtual processes presented in Fig.(3). As a result also suggests, unlike the case in QED [24], there result, the observed frequency of the harmonic oscilla- is no internal structure of the phonon below a typical tor is no longer only determined by the strength of the time scale. The value of Fig.(3b) is spring and the mass of mirror, but also by the geomet- rical parameters of the cavity and coupling strength. 2 Z +∞ (2) g dE1dE2dE3dE4 From Eq.(7) we can see the frequency shift goes diver- −iΣ4 (ωm) = − 3 4 −∞ (2π) gent at resonance ωm = 2ωc, which is demonstrated in i i Fig.(4). × E1 − ωc + i E2 − ωc + i However, this does not introduce any inconsistency i i (6) of our discussion. As we can see, in this case, the dy- × E3 − ωm + i E4 − ωb + i namical Casimir effect term VDCE will introduce such

× δ(E1 + E2 + E3 + E4 − ωm) process b→2a, which is forbidden by energy conserva- tion in the detuning cases. This means the phonon is no g2 1 = −i , longer a stable particle in resonance case. As a result, 4 2ω + ω c m the energy of the phonon itself will not be determined where the integrated elements E1, E2 and E3 are the but experiences a distribution width due to the time- energy of the virtual particles. The integral in Eq.(6) energy uncertainty [25]. The detailed computation of

4 ЖЭТФ Feynman diagram approach to Dynamical Casimir effect the decay width and the life time of the phonon will be E given in the next section. 1

4. VACUUM EFFECT E3 In this section we explore further in backaction phe- nomenon. In semiclassical treatments, we have vac- uum radiation if the mirror’s acceleration varies in time E [11, 12]. In our case, the mirror is not still even in its 2 ground state, which means there may be virtual vac- uum radiation process in our system. From the Feyn- man rules, we have the lowest order process of vacuum Fig. 5. The vacuum-vacuum process, which will modify emission and absorbtion of a phonon and a photon pair the ground state energy of the system. shown in Fig.(5). If we give the vacuum state a "propa- gator" like those of the phonons and the photons, then we can use the chain approximation in Sec.3. Fig.(5) photons. The physical cavity frequency can be derived demonstrates the existence of virtual particles ground numerical methods in previous work [20]. However, state of the whole system. As a result of which, the with help of the diagrammatic method we can get a ground state energy is shifted by detailed observation of the mechanism how the virtual phonon can affect the frequency of the observed cavity δEg = Λ, (8) mode. To the order of g2, the photon experience two where the value Λ is that of Fig.(5) and is given by virtual processes demonstrated in Fig.(6), which gives the frequency shift as 2 Z +∞ g dE1dE2dE3 i −iΛ = − 2 4 −∞ (2π) E1 − ωc + i (2) (2) δωc = Πom(ωc) + ΠDCE(ωc), (11) i i × δ(E + E + E ) E − ω + i E − ω + i 1 2 3 (2) (2) 2 c 3 m where −iΠom(ωc) and −iΠDCE(ωc) are the values of g2 1 Fig.(6a) and Fig.(6b). The superscript (2) means sec- = i , 4 2ωc + ωm ond ordered while the subscripts om and DCE de- (9) note the type of vertices. Fig.(6a) shows the emission and absorbtion of a virtual phonon of a photon while where the integrated elements are the energy of the vir- Fig.(6b) can be viewed as a vacuum-vacuum diagram tual particles. Note that the ground state energy of the glued to a photon line like in Fig.(3b). Compute the free Hamiltonian H is zero, so the ground state energy 0 diagrams with Feynman rules gives their values of the full Hamiltonian H is g2 1 Eg = − , (10) Z ∞ dE dE i 4 2ωc + ωm −iΠ(2) (ω ) = −g2 1 2 om c 2π E − ω + i which is also an exact match to that from stationary −∞ 1 c i perturbation theory around |0, 0i. Note that the inte- × δ(E1 + E2 − ωc) gral is also convergent. This means, although there will E2 − ωm + i 1 be no photon radiated from the vacuum if the cavity = ig2 , field and the harmonic oscillator are at their ground ωm 2 Z +∞ state, the zero-point fluctuation of the mirror will still (2) g dE1dE2dE3dE4 −iΠDCE(ωc) = − 3 generate "virtual radiation" and will have 4 −∞ (2π) (12) effect. This is a result of pure quantum nature of the i i × mirror’s motion. E1 − ωc + i E2 − ωc + i i i × E − ω + i E − ω + i 5. FREQUENCY SHIFT OF CAVITY 3 c 4 m × δ(E1 + E2 + E3 + E4 − ωc) In this section we study how zero-point fluctuation g2 1 of harmonic oscillator affects the frequency of cavity = i , 4 2ωc + ωm

5 Yusong Cao, Yanxia Liu ЖЭТФ

(a) 1 E1 0.998 ωc ωc 0.996

E c 2 0.994

0.992 (b) E1 g=0.1 0.99 g=0.05 E2 ωc ωc 0.988 1 2 3 4 E 3 k

Fig. 7. The frequency of the physical photon (Ω ) with E c 4 respect to the ratio of the bare phonon frequency to the bare phonon frequency k = ωm . In this picture we set ωc ωc = 1, and blue solid line for g = 0.1 with red dashed Fig. 6. The irreducible diagrams of photon propagator. line for g = 0.05. (a) Diagram of emission and absorbtion of virtual phonon. (b) Diagram of a propagating photon glued to a vacuum- vacuum diagram with one virtual phonon and two virtual oscillator to the photons, which in scattering language, photons. corresponds to process kb→2a when parameters are properly tuned (kωm = 2ωa). With the Feynman dia- gram method we can easily show the mechanism behind where the integrated elements are the energy of the vir- those processes. First we will deal with the mostly fa- tual particles. Note that the integrals are convergent. mous case: a phonon decays into photon pairs, whose Combining the two results we obtain the frequency shift Feynman diagram of leading order is given in Fig.(8a). of photons From Fig.(8a) we can see directly that the decay width 2 2 1 1 2 1 g 1 is Γ = g , so the life time of phonon is τ = Γ = g2 . δωc = −g − . (13) ωm 4 2ωc + ωm The transition amplitude of 2b→2a in Fig.(8b) is of order g2, and note that both dynamical Casimir effect This result is identical to that from stationary state vertices and optomechanical vertices take place in this perturbation theory around state |1, 0i. The frequency process. The transition amplitude of 3b→2a in Fig.(8c) shift of photon as a function of ratio ωm was demon- ωc is of order g3. In this case we have two optomechanical strated in Fig.(7). And we can see that due to the vertices and one dynamical Casimir effect vertice. The virtual processes in Fig.(6), the photon frequency is transition amplitude of 3b→2a in Fig.(8d) is of order shifted even we put the spring at its ground state. The g4 with three optpmechanical vertices and one dynam- existence of this phenomenon is resulted from the quan- ical Casimir effect vertices. We can deduce that for tum fluctuation of the mirror’s position. And this phe- general case kb→2a, there will be k − 1 optomechani- nomenon will not take place in semiclassical treatments cal vertices and one dynamical Casimir effect vertices [10-13,15]. in the diagram of the lowest order, so the leading order of transition amplitude is 6. DECAY WIDTH AND DYNAMICAL 1 CASIMIR EFFECT PROCESS A ∼ gk. (14) k→2 2k−1 Here we apply our Feynman diagram method to more some dynamical Casimir effect processes to com- Since the coupling constant g is assumed small, the pute the corresponding transition amplitudes. In the transition amplitude decrease rapidly with k growing, study of dynamical Casimir effect, one of the most fasci- which is qualitatively in agreement with the previous nating effects is the energy transferring from harmonic work [20].

6 ЖЭТФ Feynman diagram approach to Dynamical Casimir effect

directly into such cases. But our strategy of avoiding (a) (b) computation of force may of use in studying the quan- tum features in gravity systems (such as black holes) [26] at a semiclassical level, which may provide some insights in the tasks of modeling the black hole evapo- ration and construction of theory. c d ( ) ( ) Acknowledgements We thank J. P. Cao for comments on early drafts of this paper. Y. S. Cao thanks K. F. Lv for discussion about the strategy in loop diagram computation.

A. ON FEYNMAN DIAGRAM WITH FOUR Fig. 8. Diagrams of the lowest order dynamical Casimir INTERMEDIATE PARTICLES effect process: (a) Diagram of decay of a phonon, (b) Di- agram of 2b→2a process, (c) Diagram of 3b→2a process, In this section we demonstrate the detailed compu- (d) Diagram of 4b→2a process. Note that we dropped tation steps in deriving Fig.(3b) and its mathematical the markings for because they can be easily be formula Eq.(6). The processes in Fig.(3b) is part of the deduced by the energy conservation rules at each ver- S matrix element h0, 1| S |1, 0i to the second order of g. tices and of the whole process (which gives the relation In coordinate space ( in ), it is between ωm and ωc.) Z +∞ 2 † 2 † dt1dt2 h0, 1| T a (t1)[a (t2)] b(t1)b (t2) |0, 1i −∞ 7. CONCLUSION AND DISCUSSION + t1↔t2 Z +∞ We have shown in this paper that how the vacuum 2 + † = dt1dt2Ga(t1 − t2) h1| b (t1)b(t1)b(t1)b (t2) fluctuation can affect the observed quantities of our sys- −∞ † tem. Unlike the authors in [17,18] focusing on the force ×b (t2)b(t2) |1i exerted on the boundary in a fully quantized model, we Z +∞ 2 † focus on the energy transferring in the scattering pro- = dt1dt2Ga(t1 − t2) h1| b (t1) |0i h0| b(t1)b(t1) cesses. This allows us to obtain the dynamical informa- −∞ † † tion of the processes without computing the backaction ×b (t2)b (t2) |0i h0| b(t2) |1i Z +∞ force. We demonstrated that the integrals in loop di- 2 2 † = dt1dt2Ga(t1 − t2)Gb (t1 − t2) h1| b (t1) |0i agrams are convergent, which gives us a description of −∞ the motion of the mirror under backaction without the × h0| b(t2) |1i , requirement of renormalization procedure. Using Feyn- (15) man diagram method, we computed the transition am- plitudes for some dynamical Casimir effect processes where Ga(t) and Gb(t) are the free propagators of pho- and gave the universal results for general dynamical ton and phonon in coordinate space. What is notable is Csimir effect processes. that we used the trick nb(t) |1i = |1i in the calculation The idea of treating both quantum field and back- the last step of Eq.(15). Applying a Fourier transfor- ground field/boundary with quantum mechanics can mation to Eq.(15) gives Eq.(6). be extended to more relative systems. And the idea of concentrating on the four momentum, which is energy in our case, in developing dynamical theory of the back- REFERENCES action on background field/boundary instead of force may be of help in avoiding the infinities. We also hope 1. P. D. Nation, J. R. Johansson J.R., M. P. Blencowe our results can be help in studying the backaction of the and F. Nori, Rev. of Mod. Phys. 1, 84 (2012). quantum fields to their background dynamcial curved 2. P. W. Milonni, The Quantum Vacuum: An Introduc- spacetime. Since the gravity itself can not be directly tion to Quantum Electrodynamics, Academic Press quantized canonically, our strategy can not be applied (New York) (1994).

7 Yusong Cao, Yanxia Liu ЖЭТФ

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