The harmonic oscillator: bosonic states

The quantum harmonic oscillator (HO) is the most natural description for field excitations. Basically, it consists in the endless possibility to create particles through a creation (or ladder) operator $\ud{a}$ . It is then the perfect match for bosons. Bosons are particles, quasi-particles or composite particles that have an integer total spin and can be many to occupy the same state. The electromagnetic field, composed of photons, is exactly modelled by HOs. Also, matter excitations, such as excitons in semiconductors, that are composite bosons in the regime^2.1 $a_\mathrm{B}^D d\ll 1$ , can be well represented in this basic picture, since when the density is very low, their energy levels are far from saturated and the Pauli effects arising from the fermionic components (electrons and holes) are negligible. We will see in Sec. 2.2 how to deal with matter excitations when fermionic effects are important.

Let us now go quickly through the basic properties of the HO and its possible realizations. To begin with, a state with one particle is simply defined as the application of a creation operator $\ud{a}$ on the vacuum, $\ket{1}=\ud{a}\ket{0}$ . The -particle state is obtained through recursive creations:

$\displaystyle \ket{n}=\frac{(\ud{a})^n}{\sqrt{n!}}\ket{0}\,,$

(2.1)

with a normalization prefactor $1/\sqrt{n!}$ that depends on the state of the field. The hermitian conjugate

of $\ud{a}$ , annihilates a particle. Therefore, these operators act on the number states $\ket{n}$ (with

particles,

an integer), as:

$\begin{subequations}\begin{align}a\ket{n}&=\sqrt{n}\ket{n-1}\,,\\ \ud{a}\ket{n}&=\sqrt{n+1}\ket{n+1}\,,\\ \ud{a}a\ket{n}&=n\ket{n}\,. \end{align}\end{subequations}$

With such properties, $\ud{a}a$ is the number operator. The Hamiltonian of a free, single-mode, field finds its most compact expression as:

$\displaystyle H_a=\omega_a\ud{a}a\,,$

(2.3)

where $\omega_a$ is the frequency of the monomodal field. In this section, we work in Schrödinger representation where states carry the temporal dynamics and operators are time-independent. Heisenberg picture, where operators--and not states--evolve with time, will be more suitable later on, when we deal with two-time correlations. In this picture, from the commutation rules of bosons, $[a,\ud{a}]=1$ , other relations follow, for example, those related to normal ordering of the operators (which requires to move all creation operators to the left):

$\begin{subequations}\begin{align}a\ud{a}^n&=\ud{a}^na+n\ud{a}^{n-1}\\ a^n\ud{a}&=\ud{a}a^n+na^{n-1} \end{align}\end{subequations}$

In order to further investigate interesting states of the HO, one can imagine an ideal detector that absorbs field particles of all frequencies one by one. A ``detection'' means removing one particle from the initial field state $\ket{i}$ to get the final state $a\ket{i}$ . As described by Glauber (1963b), the probability per unit time to detect a particle whatever the final state, $\ket{f}$ , is given by

$\displaystyle \mathrm{Probability}(1)=\sum_f\vert\bra{f}a\ket{i}\vert^2\,,$

(2.5)

which is equal to the mean number of particles,^2.2 $\langle n_a\rangle =\bra{i}\ud{a}a\ket{i}$ . Therefore, the probability of counting a particle per unit time is proportional to the intensity of the field. It is possible to generalize this idea to the probability of counting

photons at the same time

$\displaystyle \mathrm{Probability}(M)=\sum_f \vert\bra{f}a^M\ket{i}\vert^2=\bra{i}\ud{a}^Ma^M\ket{i}\,.$

(2.6)

Here, we see the importance of normal order and how it is linked to observable quantities in photon counting experiments. The most celebrated of those is the two-particle coincidence experiment developed by Hanbury Brown (1956) with photons. Taken at zero delay (we will see more general two-time expressions in Sec. 2.7), the probability of detecting two photons informs about the statistics of the particle number distribution, which is an important property of the quantum state of the field. A widely used quantity is the degree of second-order coherence (or, equivalently, second-order correlation function at zero delay)

$\displaystyle g^{(2)}=\frac{\langle\ud{a}\ud{a}aa\rangle }{\langle\ud{a}a\rangle ^2}\,.$

(2.7)

It is linked to the variance (or second cumulant) $\Delta n_a^2=\langle(n_a-\langle n_a\rangle )^2\rangle$ of the distribution of particles:

$\displaystyle g^{(2)}=1+\frac{\Delta n_a^2-\langle n_a\rangle }{\langle n_a\rangle }\,.$

(2.8)

More generally, the degree of Mth-order coherence reads:

$\displaystyle g^{(M)}=\frac{\langle\ud{a}^Ma^M\rangle }{\langle\ud{a}a\rangle ^... ...rac{\langle n_a(n_a-1)(n_a-2)\hdots(n_a-M+1)\rangle }{\langle n_a\rangle ^M}\,.$

(2.9)

The number state or Fock state that we already introduced, has zero variance around the mean number of particles , that is completely determined. This results in $g^{(2)}=1-1/n$ --which jumps from 0 at to at , as it corresponds to a two-photon observable. It is always below . This feature of $g^{(2)}<1$ is associated to some kind of quantum behavior. $\ket{n}$ is a very ``quantum'' state, in the sense that each quantum counts: the change in number has some strong impact. This is in contrast with a classical continuous field where a photon would be an infinitesimal contribution, which removal or addition has no effect whatsoever, as we shall see shortly.

If one photon is detected from an initial state $\ket{i}=\ket{1}$ , no second photon can be expected as it gets projected into vacuum $\ket{f}=\ket{0}$ when measuring the first photon. For the number states, the probability of emission decreases as photons get detected. At high numbers, one particle more or one particle less does not make much difference ( $n\approx n\pm1$ ). A classical description and understanding of the state starts to be valid at this point and $g^{(2)}$ tends to . Similar behavior is found for higher orders of coherence:

$\displaystyle g^{(M)}=\frac{n!}{(n-M)!\,n^M}\,.$

(2.10)

The probability of having

particles in the field can be written as a Kronecker delta $\mathcal{P}_p=\vert\bra{p}n\rangle\vert^2=\delta_{n,p}$ .

Another interesting state is the coherent state $\ket{\alpha}$ , derived by Schrödinger for the first time in 1926 but fully developed in its quantum optical context by Glauber (1963a). It is characterized by being the eigenstate of the destruction operator:

$\displaystyle a\ket{\alpha}=\alpha\ket{\alpha}$

(2.11)

with eigenvalue a complex number, $\alpha=\vert\alpha\vert e^{i\phi}$ . Eq. (2.11) shows that removing one particle does not change the coherent state. This is an essentially classical property where all detections are statistically independent, in stark contrast to the case of the number state. Therefore, the coherent state is a good quantum description for the classical monochromatic wave. Let us take as an illustration of this point, one mode of a transversal free electromagnetic field. The electric field operator

is composed of photons (bosons) and at some point of space can be written (skipping constants) as a sum of two contributions

$\displaystyle E=E^{(+)}+E^{(-)}=\frac{1}{2} (a e^{-i\omega_a t}+\ud{a} e^{i\omega_at})\,.$

(2.12)

This can be also considered the expression a general bosonic field. In a coherent state, the expectation value of the electric field, the intensity operator and the field variance, respectively, are given by

$\displaystyle \langle E\rangle$	$\displaystyle =$	$\displaystyle \bra{\alpha} E \ket{\alpha}=\vert\alpha\vert\cos{(\omega_at-\phi)}\,,$	(2.13)
$\displaystyle \langle E^2\rangle$	$\displaystyle =$	$\displaystyle \bra{\alpha}E^2 \ket{\alpha}=\frac{1}{4}+\vert\alpha\vert^2\cos^2{(\omega_at-\phi)}\,,$	(2.14)
$\displaystyle \Delta E^2$	$\displaystyle =$	$\displaystyle \langle E^2\rangle -\langle E\rangle ^2=\frac{1}{4}\,,$	(2.15)

which basically means that the quantum fluctuations of the field $\Delta E$ are independent of its intensity $\langle E\rangle$ and become negligible at large $\vert\alpha\vert$ . This is the regime where the coherent state can be considered a classical wave. On the other hand, for number states the situation is radically different

$\displaystyle \langle E\rangle$	$\displaystyle =$	$\displaystyle 0\,,$	(2.16)
$\displaystyle \langle E^2\rangle$	$\displaystyle =$	$\displaystyle \frac{1}{2}\big(\frac{1}{2}+n\big)\,,$	(2.17)
$\displaystyle \Delta E^2$	$\displaystyle =$	$\displaystyle \langle E^2\rangle \,,$	(2.18)

having no electric mean field but quantum fluctuations.

For a coherent state, the variance of the particle number distribution is the same as the mean number

$\displaystyle \langle n_a\rangle =\Delta n_a^2=\vert\alpha\vert^2\,.$

(2.19)

In fact, all cumulants of the distribution converge to this value and the state is coherent at all orders (in Glauber's sense): $g^{(M)}=1$ for all

. One can double check this by obtaining the explicit expression of the coherent state in terms of number states

$\displaystyle \ket{\alpha}=e^{-\vert\alpha\vert^2/2}\sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}\ket{n}$

(2.20)

and analyzing the particle number distribution

$\displaystyle \mathcal{P}_p=\vert\bra{p}\alpha\rangle\vert^2=e^{-\langle n_a\rangle }\frac{\langle n_a\rangle ^p}{p!}\,.$

(2.21)

This is a Poissonian distribution.^2.3 A distribution with $g^{(2)}<1$ , such as that of the number state, is called subpoissonian, and a distribution with $g^{(2)}>1$ , is called superpoissonian.

States like $\ket{n}$ and $\ket{\alpha}$ that are completely described with a wavefunction (one ket) are known as pure states. They can be a good description for a field in some limiting cases where it is very well isolated from the environment, and it experiences only coherent dynamics given by a Hamiltonian. For example, the evolution of $\ket{\alpha}$ through the free Hamiltonian (2.3) (a phase rotation in its complex parameter) remains always perfectly determined by a wavefunction, $\ket{e^{-i\omega_a t}\alpha}$ . However, in general, one should consider the contamination of this dynamics due to the field being irremediably in contact with the exterior world. In principle one could model all possible interactions with the environment with a more comprehensive Hamiltonian that includes all the processes that affect the field . This is, of course, an impossible task if one takes it seriously (having to model the whole universe!), and quite a difficult one even with drastic approximations. One cannot and does not want to keep track of all the degrees of freedom affecting the field. This lack of interest on the external world results in decoherence for our system.

In the previous example of the evolution of a coherent state, one can imagine that the field is affected by an incoherent process that interrupts its coherent free evolution (like a measurement that randomizes its phase). We are not interested in this process by itself and therefore only retain its effect on our field: the rate at which the perturbation happens. After some time , when the probability that a first event has happened is $\mathcal{P}_e$ , we cannot say anymore that the state of the system is defined by $\ket{e^{-i\omega_a t_e}\alpha}$ . We only know that this is so with a probability $1-\mathcal{P}_e$ and that the state of the system is $\ket{\alpha}$ with a probability $\mathcal{P}_e$ . Therefore we need a mixture of two wavefunctions, rather than only one like for the pure state. Following this idea, the dynamics of the system can be understood as a succession of coherent periods and incoherent random (from our ignorant point of view) events that project the wavefunction into a given state. Those are the so-called quantum jumps. One can guess that after some time and a complicated mixture of quantum trajectories, we loose track completely of the phase of the state. This means that the steady state (SS) of this system is expected to be a mixture of coherent states where all possible phases have the same probability, $\mathcal{P}(\phi)d\phi=1/(2\pi)$ .

A consistent way to express this situation, and the most general description of the state of the system, is using the density matrix operator $\rho$ . In general, the density matrix can always be put in its diagonal form, as a linear superposition of projectors,^2.4

$\displaystyle \rho=\sum_i \mathcal{P}_i \ket{\Psi_i}\bra{\Psi_i}\,.$

(2.22)

$\{\mathcal{P}_i\}$ are the probabilities for the field to be in the states of a given basis $\{\ket{\Psi_i}\}$ , in the Hilbert space. The pure state is a particular case where $\rho=\ket{\Psi_1}\bra{\Psi_1}$ , all eigenvalues of $\rho$ are zero but one ( $\mathcal{P}_1=1$ and $\mathcal{P}_{i\neq1}=0$ ). In this case, it is straightforward to see that $\rho^2=\rho$ . On the other hand, a mixture is characterized by $\rho^2\neq\rho$ , which yields $\Tr{(\rho^2)}<\Tr{(\rho)}=1$ . These properties are independent of the choice of basis and so are others such as $\Tr{(\rho)}=1$ (normalization) or $\rho=\ud{\rho}$ (hermiticity). In any other basis than that of the eigenstates, $\rho$ has off-diagonal elements that give an account of the interplay or coherence between each pair of pure states. For example, the density matrix of a coherent state,

$\displaystyle \rho^a_{\alpha}=\ket{\alpha}\bra{\alpha}=e^{-\vert\alpha\vert^2}\sum_{m,n}\frac{\alpha^{m}\alpha^{*n}}{\sqrt{m!\,n!}}\ket{m}\bra{n}\,,$

(2.23)

has all off-diagonal terms in the number state basis. On the other hand, in our previous example of a mixture of coherent states with a random phase, the SS density matrix can be constructed as

$\displaystyle \rho^a_{\vert\alpha\vert}=\int d\phi \frac{1}{2\pi} \ket{\vert\alpha\vert e^{i\phi}}\bra{\vert\alpha\vert e^{i\phi}}\,,$

(2.24)

from which follows that

$\displaystyle \rho^a_{\vert\alpha\vert}=e^{-\vert\alpha\vert^2}\sum_{n}\frac{\vert\alpha\vert^{2n}}{n!}\ket{n}\bra{n}\, .$

(2.25)

In each basis we can see two aspects of the decoherence that the coherent state of Eq. (2.23) has suffered. In the first one, the most direct consequence of the phase randomization manifests in the lack of off-diagonal elements between states with different phases. The second basis of number states evidences that the particle number distribution is still Poissonian but also that the off-diagonal elements between number states have been washed out. As it is the case for any mixture diagonal in the number state basis, the average of the field is zero, $\langle E\rangle =0$ , and its intensity is time independent,

$\displaystyle \langle E^2\rangle =\frac{1}{2}\big(\frac{1}{2}+\langle n_a\rangle \big)\,.$

(2.26)

These results are closer to those of a number state (2.16) than of a coherent state (2.13). However, the state is still coherent at all orders.

The next important state to discuss is the thermal mixture. It is the state whose bosonic excitations, the particles of the field, are thermally spread among the energy levels. We will see in Sec. 2.4 that this is the result of the interaction with a reservoir of particles at a given temperature . The density matrix for a given mode $\omega_a$ can be derived from the Bose-Einstein statistics as

$\displaystyle \rho^a_\mathrm{th}=\frac{e^{-\frac{H}{K_BT}}}{\Tr{(e^{-\frac{H}{k... ...^{-\frac{\omega_a\ud{a}a}{k_BT}}}{1\big/\big(1-e^{-\frac{\omega_a}{k_BT}}\big)}$

(2.27)

where

is the Boltzmann constant and the denominator is the partition function. The thermal density matrix is also diagonal in the number basis

$\displaystyle \rho^a_\mathrm{th}=\sum_{n}\frac{\langle n_a\rangle ^n}{(1+\langle n_a\rangle )^{1+n}}\ket{n}\bra{n}\,,$

(2.28)

with the average occupation being the Bose-Einstein distribution,

$\displaystyle \langle n_a\rangle =\frac{1}{e^{\frac{\omega_a}{k_BT}}-1}\,.$

(2.29)

This formula was guessed by M. Planck in 1900 to fit the experiments on blackbody radiation and later derived by Bose from a statistical arguments for photons: Bose merely required that the particles be indistinguishable. As the system is in thermal equilibrium with a bosonic bath, their average occupation at the frequency $\omega_a$ are the same:

$\displaystyle \langle n_a\rangle =\bar n_T\,.$

(2.30)

Let us analyze the process that leads to the thermal equilibrium. We suppose that the reservoir population is not influenced by the interaction with our system (approximation discussed in Sec. 2.4). The system does evolve from vacuum into the SS of thermal equilibrium and the mean value depends on time $\langle n_a(t)\rangle$ . The total rate of incoming particles from the reservoir to the system is given by $\kappa_a \bar n_T [1+\langle n_a(t)\rangle ]$ . The effective rate of excitation to the system (analogue for bosons of the Einstein B-coefficient), is

$\displaystyle P_a=\kappa_a \bar n_T\,.$

(2.31)

It vanishes at

. Similarly, the total transfer rate in the opposite sense is given by $\kappa_a(1+\bar n_T)\langle n_a(t)\rangle$ . The system is loosing excitations at an effective rate of:

$\displaystyle \gamma_a=\kappa_a(1+\bar n_T)\,.$

(2.32)

The new parameter $\kappa_a$ is the spontaneous emission (SE) rate at

, analogous to the Einstein A-coefficient. In terms of the effective parameters, $\gamma_a$ ,

, the rate equation for the system dynamics reads:

$\displaystyle \frac{d \langle n_a(t)\rangle }{dt}=-\gamma_a\langle n_a(t)\rangle +P_a[1+\langle n_a(t)\rangle ]$

(2.33)

and leads in the SS to Eq. (2.30), now

$\displaystyle n_a^{\mathrm{SS}}=\frac{P_a}{\gamma_a-P_a}\,.$

(2.34)

At very high temperatures, as the effective income of particles approaches the outcome, $P_a\approx\gamma_a$ , the number of particles grows as $n_a^{\mathrm{SS}}\approx k_B T/\omega_a$ but it never diverges because $P_a<\gamma_a$ . As long as $\gamma_a\neq0$ , any combination of parameters $\gamma_a,P_a$ corresponds to a physical thermal bath (with $\kappa_a=\gamma_a-P_a$ and

Logically, given its origin, the thermal state does not exhibit any coherence properties at any order (other than the first for a single mode^2.5),

$\displaystyle g^{(M)}=M!\,,$

(2.35)

and in particular $g^{(2)}=2$ . This means that the particle distribution in Eq. (2.28) is superpoissonian with fluctuations

$\displaystyle \Delta n_a^2=\langle n_a\rangle ^2+\langle n_a\rangle$

(2.36)

that exceed those of the Poissonian distribution by $\langle n_a\rangle ^2$ . A thermal state is a very fluctuating field indeed. We shall see that this has some important consequences for the spectral shapes.

In the following chapters and sections, we will study different configurations and processes which generate the states that we just described. Very rarely, the state of the system is completely thermal, or coherent, or has a purely Poissonian statistics. In most of the cases, the bosonic field (of light or matter) is a convolution of different states. For example, a cothermal state, the superposition of a coherent and a thermal state, first explored by Lachs (1965), has a distribution of particles

$\displaystyle \bra{n}\rho^a_\mathrm{coth}\ket{n}=\mathcal{P}_\mathrm{coth}(n)=e... ...}{(1+n_\mathrm{t})^{n+1}}L_n[-\frac{n_\mathrm{c}/n_\mathrm{t}}{1+n_\mathrm{t}}]$

(2.37)

where

are the Laguerre polynomials. Together with the total mean value $\langle n_a\rangle =n_\mathrm{c}+n_\mathrm{t}$ , this state is defined by the fraction of coherent particles $\chi=n_\mathrm{c}/\langle n_a\rangle$ . With this definition, the thermal fraction is $n_\mathrm{t}=\langle n_a\rangle (1-\chi)$ . This interpolates between a thermal (at $\chi=0$ ) and a Poissonian (at $\chi=1$ ) distribution. The variance is the sum of the variance of both components, Eq. (2.19) and Eq. (2.36), plus an ``interference term''

$\displaystyle \Delta n^2=n_\mathrm{c}+n_\mathrm{t}^2+n_\mathrm{t}+2n_\mathrm{c}n_\mathrm{t}\,.$

(2.38)

The density matrix of the optical field assumes a transparent expression in the

representation,^2.6 namely, the convolution of the coherent and thermal

functions (a $\delta$ function and a Gaussian, respectively). This results in a Gaussian centered at the mean thermal population. However, I shall not use this representation in what follows.

Cothermal states represent a simple but precise description of a system where coherent and incoherent processes compete. For instance, this is the case of Bose-Einstein condensation in the presence of decoherence, as it was shown in Ref. 16 of the list of my publications .