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Harmonic oscillators, coupled harmonic oscillators, and Bosonic elds

Koji Usami

(Dated: November 16, 2015) We rst study why harmonic oscillators are so ubiquitous and see that not only a point mass in a harmonic potential but also an LC circuit behave like a harmonic oscillator. We then learn an important idea ofnormal modesto deal with coupled harmonic oscillators. Finally we see that taking thecontinuum limita (1+1)-dimensional Bosonic eld is emerged from coupled (0+1)-dimensional harmonic oscillators. As examples we deal with a longitudinal acoustic phonon mode in a 1D atomic chain and an itinerant microwave photon mode propagating in a 1D transmission line.

I. HARMONIC OSCILLATORS

A. Point mass in a harmonic potential

1. Lagrangian and Hamiltonian formalism [1]

Let us begin by considering a point mass with massmand coordinatexsituated in a potentialU(x). Suppose

that the mass is oscillating with small amplitude around the equilibrium positionx0. Then the potential energy of

the mass can be Taylor-expanded aroundx0:

U(x) =U(x0) +@U(x0)

@x x+1 2

2U(x0)

@x

2x2:(1)

Since the force,F=@U(x0)

@x , should be zero in the equilibrium position, neglecting the potential offsetU(x0) we have

U(x) =1

2

2U(x0)

@x 2| {z kx 2:(2)

This suggests that any potential can be considered as a harmonic potential when we are interested in the small motion

in the vicinity of the equilibrium position.

With the kinetic partK=1

2 m_x2we have the standard Lagrangian for the harmonic oscillator:

L(x;_x) =K(_x)U(x) =1

2 m_x21 2 kx2:(3) The motion of the mass from timet1tot2can be determined so as to minimize the action integral I=t

2∫

t

1L(x;_x)dt:(4)

The minimum ofIcan be obtained by avariational principle (Hamilton's principle), which leads to the Euler-Lagrange

equation of motion: d dt @L(x;_x) @_x) @L(x;_x) @x = 0;(5) that nothing but Newton's second law: mxkx= 0:(6)

Electronic address:usami@qc.rcast.u-tokyo.ac.jp

2 In the formal procedure the conjugate momentum,p, can then be obtained by p=@L(x;_x) @_x=m_x:(7) We have thus the HamiltonianH(x;p) from the Legendre transformation:

H(x;p) = _xpL(x;_x) =1

2mp2+1

2 kx2:(8)

2. Canonical quantization

We can promotexandpto the quantum-mechanical operators by imposing the commutation relation, [^x;^p] =ih:(9)

Let the annihilation and creation operators be

m! 2h( ^x+i m! ^p) (10) m! 2h( ^xi m! ^p) ;(11) k m . The Hamiltonian Eq. (8) can then be written in a diagonal form as

H(^x;^p) =1

2m^p2+1

2 m!2^x2 = h!( ^ay^a|{z} ^n+1 2 :(12) The mean value of energy in thermal equilibrium⟨^H(^x;^p)⟩= Tr[^H] becomes ^H(^x;^p)⟩= h!( ⟨^n⟩+1 2 ;(13) where ⟨n⟩=1 e h! k

BT1;(14)

which in high temperature limit becomes⟨n⟩ !kBT h!.

B. LC circuit [2]

1.Q-representation (loop variable representation)

Let us see that an LC circuit can also be viewed as a harmonic oscillator. Replacing the coordinatexby the

chargeQ, the massmby the inductanceL0, and the spring constantkby the inverse of capacitanceC0, we have the

Lagrangian for the LC circuit:

L(Q;_Q) =1

2

L0_Q21

2C0Q2:(15)

The rst term is the inductive energy and the second is the charging energy. The conjugate momentum is

@L(Q;_Q) _Q=L0_Q=φ;(16) 3 which is identied as the ux. The Hamiltonian is thus

H(Q;φ) =_QφL(Q;_Q) =1

2L0φ2+1

2C0Q2 1

2L0φ2+1

2

L0!2Q2;(17)

where!=1 p L 0C0.

The commutation relation:

[Q;φ] =ih(18)

Let the annihilation and creation operators be

L 0! 2h( ^Q+i L

0!^φ)

(19) L 0! 2h( ^Qi L

0!^φ)

:(20)

The Hamiltonian Eq. (17) can then be written as

H(^Q;^φ) =1

2L0^φ2+1

2

L0!2^Q2

= h!( ^by^b+1 2 :(21)

2.φ-representation (node variable representation)

We can equally use the

uxφas the coordinate of the LC circuit system, which is more relevant when we deal with a transmission line. Then, the Lagrangian is a function ofφand _φ, and is given by

L(φ;_φ) =1

2

C0_φ21

2L0φ2;(22)

where the roles ofC0andL10are the mass and the spring constant, respectively, and are switched from the rst

case. The rst term is then the charging energy and the second is the inductive energy. The conjugate momentum

becomes @L(φ;_φ) @_φ=C0_φ:(23) Since _φ=L0_I=V(Faraday's law of induction) the conjugate momentum of the uxφis indeed the charge: C

0_φ=C0V=Q:(24)

Consequently, the Hamiltonian is

H(φ;Q) = _φQL(φ;_φ) =1

2C0Q2+1

2L0φ2

1

2C0Q2+1

2

C0!2φ2:(25)

The commutation relation:

[φ;Q] =ih(26)

Let the annihilation and creation operators be

C 0! 2h( ^φ+i C 0!^Q) (27) C 0! 2h( ^φi C 0!^Q) :(28) 4

TABLE I: Point mass - LC circuit - EM cavity mode

Point mass

LC (Q-rep./ loop rep.)

LC (φ-rep./ node rep.)

EM cavity mode

Mass m L 0 C 0 0

Spring const.

k 1 C 0 1 L 0 1 0

Ang. freq.

k m !=1 p C 0L0 !=1 p C 0L0 k=ck

Position

x Q A k

Velocity

_x=v _ Q=I _φ=V _ Ak=Ek

Momentum

p=m_x=mv

φ=L0_Q=L0I

Q=C0_φ=C0V

k=ϵ0_Ak=ϵ0Ek Force

F= _p=kx

V= _φ=Q

C 0

I=_Q=φ

L 0 _ k=Ak 0

The Hamiltonian Eq. (25) can then be written as

H(^φ;^Q) =1

2C0^Q2+1

2

C0!2^φ2

= h!( ^cy^c+1 2 :(29)

Compared with the point mass case, theQ-representation choosesQas a position variable whileφ-representation

choosesφfor that. In view of standard circuit theory, these choices are related to Kirchhoff's laws; the former chooses

loop currents as a set of independent degrees of freedom of the circuit while the latter chooses node voltages for

that [2]. We will mainly use the latterφ-representation, which has more direct relevance to 1D transmission line when

taking continuum limit of coupled LC circuits.

Table I shows the correspondances of the physical quantities of point mass, LC circuit, as well as electromagnetic

cavity mode, all of which can be viewed as a harmonic oscillator.

II. COUPLED HARMONIC OSCILLATORS

A. Coupled masses

B. Coupled LC circuit

C. Electro-mechanics

Let us consider rst the situation in which a metallic membrane oscillator with the angular frequency of!mis

capacitively coupled to a LC circuit with the angular frequency of!LC. The coupled system's potential can then

be given byH(x;Q), wherexis the membrane displacement andQis the charge in the capacitor of the LC circuit.

Suppose that with certain external voltage the equilibrium position isx=X0, and the equilibrium charge isQ=Q0.

Then, around the equilibrium point, the potential can be written as

H(x;Q) =H(X0;Q0) + (@H

@x ^x+@H @Q ^Q) + (1 2 2H @x

2^x2+1

2 2H @Q

2^Q2+@2H

@x@Q ^x^Q);(30)

withx=X0+^x;Q=Q0+^Q. The linear terms in Eq. (30), however, vanish becasue of the denition of the equilibrium

condition @H @x jx=X0= 0 and@H @Q jQ=Q0= 0. Neglecting the equilibrium potential energyH(X0;Q0), we have

H(^x;^Q) =1

2 m!m^x2+1

2C^Q2+G^x^Q;(31)

where @2H @x

2jx=X0=k=m!2m,@2H

@Q

2jQ=Q0=1

C , and@2H @x@Q jx=X0;Q=Q0=G.

By adding the kinetic energy parts,

1

2mp2=1

2 m(dx dt )2for mechanics and1

2Lϕ2=1

2 L(dQ dt )2for LC-circuit, we have the Hamiltonian, H=1

2m^p2+1

2 m!m^x2 {z mechanics+ 1

2L^ϕ2+1

2C^Q2 {z

LC+G^x^Q|

{z coupling:(32) 5

The Hamiltonian Eq. (32) can be rewritten as

H=1

2m^p2+1

2 m!m^x2+1

2L^ϕ2+1

2

L!LC^Q2+G^x^Q

= h!m( ^ay^a+1 2 + h!LC( ^by^b+1 2 +G( h h

2L!LC(

^by+^b)) = h!m( ^ay^a+1 2 + h!LC( ^by^b+1 2 +h 2 G p m! mp L! LC| {z g( ^ay+ ^a)(^by+^b) :(33)

Using therotating-wave approximationwhich neglect rapidly oscillating terms ^a^band ^ay^byin the last term in Eq. (33)

we have thecanonicalHamiltonian for the coupled oscillator system:

H= h!m^ay^a+ h!LC^by^b+h

2 g( ^ay^b+^by^a) :(34) Here the vacuum energy terms are omitted since the energy can be offset arbitrary.

Let us analyze the energy level structure for the coupled system. First, suppose that the two oscillators are resonant,

that is,!m=!LC=!. We then easily guess that thenormal modes, which diagonalize the Hamiltonian Eq. (34),

are ^c=1 p 2 ^a^b) (35) d=1 p 2 ^a+^b) :(36) With these normal mode operators the Hamiltonian can be indeed rewritten in a diagonal form as H=( h!hg 2 ^cy^c+( h!+hg 2 ^dy^d:(37)

The eigen-energies are shifted from the originally degenerate h!by hg, which callednormal mode splitting.

Next, let us consider the situation where the mechanical and LC oscillators have different resonance angular fre-

quencies,!mand!LC=!m+ ∆, respectively. The normal modes in this case become ^c= cos^asin^b(38) d= sin^a+ cos^b;(39) where themixing angleis dened by cot2=∆ :(40)

The resultant diagonalized Hamiltonian is

H=0 B

BB@h!m+h∆

2 {z 1 2 (!m+!LC) hg 2 1 sin21 C

CCA^cy^c+0

B

BB@h!m+h∆

2 {z 1 2 (!m+!LC)+ hg 2 1 sin21 C

CCA^dy^d:(41)

D. 1D atomic chain -phonon modes [3]

Let us consider one-dimensional monatomic atomic chain with the periodic (Born-von Karman) boundary condi-

tion [4],q(Naa) =q(0), whereais the inter-atomic distance. The potential energy of an atom in the chain is now

dependent on the congurations of the nearest-neighbor atoms and the total potential energy is V=1 2quotesdbs_dbs17.pdfusesText_23