A great deal of this thesis concerns the design of quantum devices which, as the name suggests, take advantage of the uniquely quantum properties of light fields. We have already introduced technologies such as the single-photon source, quantum memory and quantum computer in Chap. 1, albeit without the essential theory of their operation. In the remainder of this chapter we will explore key properties of light which cannot be explained in terms of classical fields: for instance the degree of anti-bunching, quadrature squeezing and entanglement. To do so we will need to introduce the most elementary single-mode, pure states of the light field and their properties.
2.2.1 Fock states
The Fock states|niare eigenstates of the photon number operator with photon number n
|ni = (ˆa†) n p n! |0i (2.13) ˆ n|ni = n|ni (2.14)
and form an orthonormal Hilbert space basis for single mode states of the quantum field. Although a foundational concept for quantum optics and an elementary component of proposed quantum technologies, pure Fock states are difficult to generate in the lab (we’ll explore this problem in some detail when we discuss single photon sources in Chap. 7). Fock states of a fixed frequency mode, such as plane waves, are also energy eigenstates with energy
ˆ
H |nki=~!k(ˆn+ 1/2)|nki (2.15)
where the factor of 1/2 is the zero-point energy of the zero-photon Fock state, or ‘vacuum field’, |0i. In this thesis we will usually consider energy exchange between atomic and
§2.2 Quantum states of light 17
optical systems, and in this context the constant factor can be neglected from the Hamil- tonian, but it shouldn’t be forgotten that the vacuum is a meaningful optical state of a mode. For example, the uncertainty in the electric field strength of the nth Fock state is [77]
E =
q
hEi2 hE2i (2.16)
= p2u(r, t)pn+ 1/2. (2.17) Even the field strength of then= 0 ‘vacuum’ state fluctuates. These vacuum fluctuations are responsible for both the spontaneous decay and energy level shifts of excited atomic states. Importantly, even the vacuum fluctuations depend on the allowed mode amplitudes
u(r, t) such that the vacuum fluctuations may be shaped by surfaces with appropriate boundary conditions. We will take advantage of this fact for the design of experiments with precision fabricated mirrors in Chap. 10.
2.2.2 Coherent states
According to Eqn. 2.7, the amplitude and phase quadratures of the optical field do not commute, so we can see that the product of phase and amplitude uncertainty has a lower limit fixed by Heisenberg’s uncertainty relation. Coherent states are the optical states that minimize this uncertainty product, they have the smallest possible quadrature fluctuations. In this sense, they are the closest possible quantum state to the noiseless classical field. They are eigenstates of the annihilation operator
ˆ
a|↵i=↵|↵i (2.18)
and can be expanded in the Fock state basis
|↵i=e 12|↵| 2X1 n=0 ↵n p n!|ni (2.19)
which is to say that the coherent state |↵i is a Poissonian superposition of photons with mean intensity ↵2 as shown in Fig 2.3(b). Coherent states have the same quadrature un- certainty as the vacuum, and can be described as a displaced vacuum state with amplitude
|↵|and phase arg(↵)
ˆ
D = exph↵ˆa† ↵⇤ˆai (2.20)
|↵i = Dˆ(↵)|0i . (2.21)
Coherent states are not mutually orthogonal, because they are not the eigenstates of a Hermitian operator, but coherent states with ↵ > 2 are approximately orthogonal. Because coherent states are not eigenstates of the creation operator, any physical opera- tion (Hamiltonian or unitary operator) containing ˆa necessarily changes the energy in a coherent state. An ideal laser produces a coherent optical field in a single mode, usually a Gaussian mode with a narrow bandwidth.
2.2.3 Squeezed states
Squeezed states, like coherent states, have only the minimum quadrature uncertainty necessary to satisfy Heisenberg’s relation. Unlike coherent states, the squeezed states distribute this uncertainty asymmetrically between the two quadratures, reducing uncer- tainty in one quadrature at the expense of a corresponding increase in the complementary quadrature. We can write the set of single-mode quadrature-squeezed states in terms of the operator ˆS(⇣) [79]
|⇣,↵i = Dˆ(↵) ˆS(⇣)|0i (2.22) ˆ
S(⇣) = exph(⇣ˆa2 ⇣⇤ˆa†2)/2i, (2.23) where ⇣ = rei is the squeezing parameter with magnitude r at angle . The squeezing operator contains only even powers of creation and annihilation operators, and may be written as the action of a Hamiltonian
ˆ
H =i~r(ˆa2 ˆa†2)/2 (2.24) that creates or annihilates photons in pairs. Applied to the vacuum state, the squeezing operator produces pairs of photons with a Poissonian distribution as shown in Fig. 2.3(a). For this reason, an early name for the squeezed-vacuum state was a ‘two photon coherent state’.
In practice, any unitary Hamiltonian that is at least quadratic in the ladder operators can generate some degree of squeezing. The most common means of generating squeezed light is spontaneous parametric down-conversion (SPDC), in which a bright pump field propagates through a medium with a second-order nonlinear optical response. Each pump photon has some probability to be split into two photons of lower energy, with modes that satisfy a phase-matching condition.
The squeezing operation ˆS is best pictured as a Bogoliubov transformation of the creation and annihilation operators, corresponding to compression and elongation of the two quadratures by a factor R = er. The quadrature variances of a squeezed coherent
state are
x2 = 1/(2R2), (2.25)
p2 = R2/2, (2.26)
although the squeezing operator also changes the field amplitude ↵ for all |↵|>0. This operation may be visualized by stretching the plane of the Wigner quasi-probability dis- tributionW(x, p) which we shall now introduce. We will make further use of these Wigner functions by deriving the photon-number distribution of a general squeezed state.