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Nanoscale Device PhysicsScience and Engineering Fundamentals$

Sandip Tiwari

Print publication date: 2017

Print ISBN-13: 9780198759874

Published to British Academy Scholarship Online: August 2017

DOI: 10.1093/oso/9780198759874.001.0001

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(p.655) P Quantum oscillators

(p.655) P Quantum oscillators

Source:
Nanoscale Device Physics
Author(s):

Sandip Tiwari

Publisher:
Oxford University Press

FIELD QUANTIZATION is essential for understanding quantum optical interactions, and ladder operators, also called creation and annihilation operators, provide a very compact and convenient mechanism to tackle the elementary excitation changes that are pervasive with such interactions. Here, we summarize the preliminary elements of this approach, with the goal of describing the coupling between mechanical oscillations and the field oscillations of interest to us.

The potential V=mω2x2/2 describes a classical simple harmonic oscillator. Using just one spatial dimension x, the position xˆ and momentum operators pˆ satisfy the commutation relation [xˆ,pˆ]=i, and the Hamiltonian of the simple harmonic oscillator is

(P.1)
Hˆ=pˆ22m+mω2xˆ22.

To write energy in terms of quanta, we use a normalization that allows the positional and momentum operators to be suitably mapped to energy forms. These normalizations for position and momentum are

(P.2)
x0=2mωxˆ0=xˆ2x0,andp0=mω2pˆ0=pˆ2p0,

which reduces the Hamiltonian to

(P.3)
Hˆ=ω(xˆ02+pˆ02),

with the position and momentum energy contributions on identical energy-normalized numerical footing and a form that is more symmetric than that of Equation P.1. We now define two non-Hermitian operators,

(P.4)
aˆ=xˆ0+ipˆ0=122mωxˆ+i122mωpˆ,and its conjugateaˆ=xˆ0ipˆ0=122mωxˆi122mωpˆ,

(p.656) whose operation results in the normalized amplitude and its complex conjugate for the classical oscillator. These ladder operators have a number of interesting properties. The following relations hold:

(P.5)
[aˆ,aˆ]=I,xˆ0=122mωxˆ=12(aˆ+aˆ),pˆ0=122mωpˆ=i2(aˆaˆ),andHˆ=ω(aˆaˆ+12).

Looking at the Hamiltonian in Equation P.5, one sees aˆaˆ=nˆ—a number operator—and a fractional term of ω/2—the vacuum fluctuation energy. aˆ is the annihilation operator, and the aˆ is the creation operator. If the fluctuation energy, in an analysis of the problem, appears as a constant shift and does not have a direct implication, one could just work with a modified Hamiltonian, of shifted origin, of

(P.6)
Hˆs=ωaˆaˆnˆ.

The shifted Hamiltonian Hˆs or the number and ladder operators settle for us nearly all of the spectrum-related properties of the quantum oscillator and is thus useful in coupled-energy problems. They have a number of interesting properties:

(P.7)
nˆn=nn,[aˆ,nˆ]=aˆ,[aˆ,nˆ]=aˆ,aˆn=nn1,aˆn=n+1n+1,andn=aˆnn!0.

n is a Fock state that forms the ladder in n, as described by the first of the operations in Equation P.7. These constitute an orthonormal basis set in the Hilbert space. The shifted Hamiltonian eigenenergy is nω—it is the energy associated with n of ω quanta, for example, the quanta of phonons. The number operator and the ladder operators do not commute, as stated by the second and third relationship. The annihilation operator removes an elementary excitation, and the creation ladder operator adds an elementary excitation. And the final relationship states that Fock states may be generated by repeated use of the creation operator from the ground state 0.

(p.657) In the Schrödinger view, Fock states are stationary states—probability amplitudes evolve with time-dependent, eigenenergy-specified phases. In the Heisenberg view, oscillator states are stationary, and the operators evolve as

(P.8)
daˆdt=i[aˆ,H]=iω[aˆ,nˆ]=iωaˆ

so that

(P.9)
aˆ(t)=aˆexp(iωt).

This description applies to the field mode of a cavity such as a Fabry-Pérot cavity, where the quanta are that of the photon, and the frequency ω=ωcav, the cavity mode’s angular frequency; so, in our notation, Hˆsn=nωcav. The electric field is expressible as

(P.10)
ε(r)=iε0[u(r)ϵaˆu(r)ϵaˆ],

with the following descriptions: u(r) is a function describing the normalized amplitude spatial dependence of the field mode, ϵ is the unit polarization vector, ε0 is a normalization factor, and the form of the equation is consistent with that of the Helmholtz equation. Since field is related to vector potential as ε=ψA/t, including the operator’s time evolution,

(P.11)
Aˆ(r,t)=ε0ωcav[u(r)ϵaˆexp(iωcavt)+u(r)ϵaˆexp(iωcavt)]

in the Heisenberg representation, and

(P.12)
Aˆ(r)=ε0ωcav[u(r)ϵaˆ+u(r)ϵaˆ]

in the Schrödinger representation. The Fock state 0, the ground state, is the vacuum state. The expectation values of the field and the vector potential vanish in this state. But, field fluctuations do not. The energy is the square of the field, so terms resulting from aˆaˆaˆaˆ are non-zero. The field is zero, but a finite energy density also exists showing up as fluctuations, uncertainty and exchange between kinetic and potential energies in this uncertainty. The normalization can be thought of in terms of mode volume. Since the energy of state n is ωcav(n+1/2),

(P.13)
n|ϵ0|ε|2d3r|n=n|ϵ0ε02V(2n+1)|n=ωcav(n+12),

with V=|u(r)|2d3r being the mode volume. Our field normalization is then related to the energy and mode volume through

(P.14)
ε0=ωcav2ϵ0V,

(p.658) which depends only on the frequency and cavity geometry. This captures the essence of the quantization of field in the finite-sized cavity. If mode is propagating and is a single mode, then one may imagine it as a mode of a single-mode circulating cavity, such as a ring laser, of a size larger than the size of the system of interest. Here, the vacuum field amplitude is very small. Spontaneous emission conditions can now be taken to be an expansion over a continuum of modes in a volume that goes to infinity.