The zero point energy in the quantised electromagnetic field, encapsulated in the Hamiltonian given in equation3.4, was mentioned at the end of section3.2. The Fock state with zero photons is found to have nonzero energy, hence the vacuum state has some energy associated, equivalent to half the energy of a single photon. In this section we will further explore some of the possible states of light relevant to this thesis, including the vacuum state.
For the purpose of this background section we only consider minimum uncertainty states of light where the uncertainty principle is treated as an equality
∆Xˆ1∆Xˆ2=1. (3.16)
A more complete mathematical treatment of minimum uncertainty states may be found in Walls and Milburn [172]. Examples of states with uncertainty greater than the limit of the uncer- tainty principle include thermal states or states with additional classical noise above the quantum noise limit.
Representing states of light
We will use two pictures describing these states of light: the ball-and-stick picture and the sideband picture. The ball-and-stick picture is conceptually intuitive, light is represented as a phasor, and uncertainty in phase and amplitude as a fuzzy ball at the end of the phasor. The sideband picture represents the real and imaginary parts of the carrier field amplitude from a reference frame co- rotating at the carrier frequency.
Modulation of the carrier field – that is phase, frequency or amplitude modulation – at Fourier frequencies of ±∆ may be represented simply in the sideband picture. The amplitude of the sidebands indicates the modulation depth, and the phase of the sidebands indicates the type of modulation. To introduce the sideband picture, amplitude and phase modulation are illustrated in figure3.1aand3.1brespectively. In the case of amplitude modulation the rotation of the sideband phasors causes the amplitude of the carrier to be modulated at the sideband frequency. The phase modulation sidebands result in the phase of the carrier being modulated at the sideband frequency. This is shown as a function of time by Chua [39].
Quantum noise may also be represented in the sideband picture as a spectrum of sidebands in phase and amplitude. For the minimum uncertainty state with equal noise in phase and amplitude, these sidebands are randomly oriented, and for squeezed states these sidebands are correlated, increasing the noise in one quadrature and reducing it in the other. The quantum noise sidebands in the sideband picture are shown in figure 3.2. These states of lights will be explored more completely in the following sections.
(a)Amplitude modulation (b)Phase modulation
Figure 3.1: Amplitude (left) and phase (right) modulation at frequency∆in the sideband picture. As the
phasors evolve in time the sidebands add in and out of phase. In the case of amplitude modulation this
causes the amplitude of the carrier to be modulated at frequency∆, while phase modulation modulates the
phase of the carrier.
The vacuum state
The ground state of the harmonic oscillator corresponds to the vacuum state of the field, defined as
ˆ
ak|0i=0, (3.17)
where ˆak is the dimensionless annihilation operator, and|0iis the vacuum state. While there are
no photons in the vacuum state, there is energy. From the Hamiltonian of the quantised electro- magnetic field (equation3.2) the energy of the vacuum state is given by
h0|H|0i=1/2
∑
k
¯
hωk, (3.18)
whereωk is the angular frequency of thekthmode.
The variance (defined in equation3.13) of the vacuum state in each quadrature may be calcu- lated using the quadrature operators defined in equations3.7 and3.8. This calculation is shown completely in Wade [169], and the variances are found to be unity,
V(Xˆ1|0i) =1 V(Xˆ2|0i) =1. (3.19)
Hence the vacuum state is a minimum uncertainty state with equal variance in each quadrature, and no coherent amplitude. The vacuum state is represented in the ball-and-stick picture in figure 3.3c.
The coherent state
A field from a well-stabilised laser can be represented by the coherent state,|αi. Coherent states are eigenstates of the annihilation operator,
a|αi=α|αi. (3.20)
The coherent state is generated mathematically from the vacuum state using the displacement operator,
3.2 Quantisation of the electromagnetic field
(a)A coherent state (b)A coherent squeezed state
Figure 3.2:Coherent and squeezed states in the sideband picture. The carrier is shown in red, along the real axis, the quantum noise sidebands are uncorrelated in the case of the coherent state, and correlated when the state is squeezed.
ˆ D(α) =eαaˆ †− α∗aˆ (3.21) =e−|α|2/2eαaˆ† e−α∗aˆ,
whereα is the complex amplitude of the field. The coherent state is generated by applying the displacement operator to the vacuum state,
|αi=Dˆ(α)|0i. (3.22)
Expanding the coherent state in the number state basis reveals that the state has a Poisson distribution of photons, with the mean number of photons given by|α|2[172].
The coherent state is a minimum uncertainty state of light with coherent amplitude and equal variance in the phase and amplitude quadratures,
V(Xˆ1|αi) =V(Xˆ2|αi) =1. (3.23) The variance of a coherent state hence is unity. This state may be represented by an uncertainty ball in the ball-and-stick phasor type picture as shown in figure3.3a, or as uncorrelated sidebands about a carrier field represented in figure3.2a.
The squeezed state
The squeezed state is a minimum uncertainty state with unequal variance in the two quadratures. One quadrature has less noise than a coherent state and, to satisfy the uncertainty principle, the noise in the other quadrature must be greater than a coherent state.
The squeezed state is generated mathematically with the unitary squeeze operator, defined as
ˆ
S(ε) =e1/2ε
∗aˆ2
e−1/2εaˆ†2
, (3.24)
where ε =re2iφ quantifies the level of squeezing, r =|ε| is the “squeeze factor”, andφ is the phase angle relative to the quadrature axis. The general squeezed state is generated by applying the squeeze operator to the vacuum state, and then applying the displacement operator [31]
|α,εi=Dˆ(α)Sˆ(ε)|0i. (3.25) The displacement operator does not modify the quadrature variances (which is why the co- herent state has the same variance as the vacuum state), the squeeze operator does modify the
(a)The coherent state (b)The squeezed coherent state
(c)The vacuum state (d)The squeezed vacuum state
Figure 3.3:Ball-and-stick picture of four minimum uncertainty states, whereαis the amplitude of the state,
φis the phase measured with respect to the basis defined by the orthogonal∆Xˆθ and∆Xˆθ+π/2quadratures.
∆Xˆ+is the uncertainty in the amplitude quadrature, and∆Xˆ−is the uncertainty in the phase quadrature.
variances. Applying the squeeze operator transformation to the creation and annihilation operat- ors,
ˆ
S†(ε)aˆSˆ(ε) =aˆcosh(r)−aˆ†e2iφsinh(r) (3.26)
ˆ
S†(ε)aˆ†Sˆ(ε) =aˆ†cosh(r)−aˆ−2iφsinh(r).
Applying the squeeze operator transformation to the quadrature operators yields
ˆ
S†(ε)Xˆ1,2Sˆ(ε) =Xˆ1,2 cosh(r)∓e−2iφsinh(r)
, (3.27)
where the minus sign corresponds to the ˆX1 quadrature. From these quadratures the quadrature
variances can be calculated
V(Xˆ1|α,εi) =e−2r, V(Xˆ2|α,εi) =e2r. (3.28)
Hence the variance in one quadrature is attenuated by the squeeze factor, while the variance in the opposite quadrature is amplified by the same factor. The squeezed state is a minimum uncertainty state as the product of the two quadrature variances is unity.
Also note that the mean number of photons in the squeezed state, given by