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Accueil du site > Atomes, cavités et photons > Principes de l’electrodynamique quantique en cavité > Le spin : un atome à deux niveaux

Le spin : un atome à deux niveaux

The two level atom is equivalent to a spin 1/2 system. We recall here its main properties.

A spin system and the Bloch sphere

A two-state system can be described without loss of generality, as a `pseudo-spin’ $S$ . This analogy will lead us to a geometrical representation of the system, widely used in NMR physics, which will prove very useful. The component of this spin along an arbitrary direction in three-dimensional space can take only one of the two values $\pm\hbar/2$ . The most general observable of this system can be expressed as a linear combination with real coefficients of the $2 \times 2$ unity operator $1$ and of the three Pauli operators $\sigma_i =2S_i/\hbar\ (i=X,Y,Z)$ , which in the basis of the $\sigma_Z$ eigenstates, are :

$\sigma_{X}=\left(\begin{array}{cc} 0&1\\1&0\end{array}\right);\quad \sigma_{Y}=\left(\begin{array}{cc} 0&-i\\i&0\end{array}\right); \quad \sigma_{Z}=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)\ .$

These operators satisfy the commutation rules :

$[\sigma_{i},\sigma_{j}]=2i\epsilon_{ijk}\sigma_{k}\ ,$

where $\epsilon_{ijk}$ is zero if two of the indices are equal and $+1$ or $-1$ , depending on the parity of the permutation of $ijk$ when they are all different.

The eigenvalues of $\sigma_{i}$ are $\pm 1$ . We will call $|{0}\rangle$ and $|{1}\rangle$ the eigenstates of $\sigma_Z$ with eigenvalues $+1$ and $-1$ respectively. In the spin language, they correspond to the `up’ and `down’ states along the $Z$ axis. The $0,1$ notation adopted here is more in the spirit of quantum information. The eigenstates of $\sigma_{X}$ are the symmetric and anti-symmetric linear combinations $|{0/1_X}\rangle=(|{0}\rangle\pm|{1}\rangle)/\sqrt{2}$ whereas the eigenstates of $\sigma_{Y}$ are $|{0/1_Y}\rangle=(|{0}\rangle\pm i|{1}\rangle)\sqrt{2}$ .

The most general traceless observable of the two-level system corresponds to a spin component along the direction defined by the unit vector $u$ with polar angles $\theta$ and $\phi$ (figure above). It is simply expressed in terms of the Pauli matrices as :

$\sigma_{u}=\cos \theta\, \sigma_{Z}+\sin \theta \cos \phi \, \sigma_{X} +\sin \theta \sin \phi\, \sigma_{Y}=\left(\begin{array}{cc} \cos \theta&\sin \theta e^{-i\phi}\\ \sin \theta e^{i\phi}&-\cos \theta\end{array}\right)\ .$

The observable $\sigma_{u}$ has also a $\pm 1$ spectrum, with the eigenstates :

$|{0_{u}}\rangle= \cos (\theta/2)\ket{0} + \sin (\theta/2) e^{i\phi}\ket{1}\ ;$

$|{1_{u}}\rangle= \sin (\theta/2)|{0}\rangle - \cos (\theta/2) e^{i\phi}|{1}\rangle\ .$

When the unit vector $u$ rotates in space, $|{0_u}\rangle$ explores the entire Hilbert space of the pseudo-spin. The most general spin state, $c_0|{0}\rangle+c_1|{1}\rangle$ , is indeed the eigenstate with eigenvalue $+1$ of the spin component along the direction $u$ of polar angles $\grt$ and $\phi$ defined by the relation $\tan(\theta /2) e^{i\phi}=c_1/c_0$ . The tip of this vector belongs to the sphere of radius unity, called the Bloch sphere in honour of the pioneer of NMR.

In other words, the Hilbert space of a two-level system is `mirrored’ onto the Bloch sphere, each of its points representing a possible superposition of $|{0}\rangle$ and $|{1}\rangle$ . These basis states are respectively imaged on the `north’ and `south’ poles. The $|{0_X}\rangle$ and $|{0_Y}\rangle$ states are associated to points on the $X$ and $Y$ axes, on the Bloch sphere’s equator. The state $|{1_u}\rangle$ , orthogonal to $|{0_u}\rangle$ in Hilbert space, corresponds to the point along the direction $-u$ . The identity $|{1_u}\rangle=|{0_{-u}}\rangle$ can easily be checked from their explicit expressions. Two orthogonal states in Hilbert space are thus associated to antipodes on the Bloch sphere.

The probability of finding $+1$ when measuring $\sigma_Z$ on $|{0_u}\rangle$ is $\cos^2(\theta/2)$ . The projection postulate implies that $\cos^2(\theta/2)$ is also the conditional probability of finding the same result ( $+1$ , $+1$ ) or ( $-1,-1$ ) when measuring successively two spin components along directions making the angle $\theta$ .

Two-level atom

We use two-level atoms whose upper level $|e\rangle$ is connected to level $|g\rangle$ by an electric dipole transition at angular frequency $\omega_{eg}$ . This system is equivalent to a spin-1/2 evolving in an abstract space, with a magnetic field oriented along the `vertical’ $Z$ axis accounting for the energy difference between $e$ and $g$ . These states correspond to the eigenstates of the spin along $Z$ , $|{0}\rangle$ and $|{1}\rangle$ . The assignment of $e$ and $g$ with qubit states is of course arbitrary. Here we will make the correspondence $|e\rangle\rightarrow|0\rangle$ and $|g\rangle\rightarrow|1\rangle$ , which, with the conventions of quantum information, make $e$ and $g$ eigenstates of $\sigma_Z$ with eigenvalue $+1$ and $-1$ respectively, the atomic Hamiltonian being :

$H_a=\frac{\hbar\omega_{eg}}{2}\sigma_Z\ ,$

where we have set the zero of energy half-way between the two levels. Note that the chosen qubit assignment means that the state $|1\rangle$ is less excited than the state $|0\rangle$ , which can be surprising. In some cases, it might be more convenient to use the opposite qubit choice and we will in the following shift freely from one qubit definition to the other.

Let us introduce also the atomic raising and lowering operators $\sigma_\pm$ :

$\sigma_\pm=(1/2)(\sigma_X\pm i\sigma_Y)\ .$

In terms of the spin eigenstates along the $Z$ axis, these operators are :

$\sigma_+=|{0}\rangle\langle{1}|\ ;\qquad \sigma_-=\sigma_+^\dagger= |{1}\rangle\langle{0}|\ .$

These atomic excitation creation/annihilation operators have a fermionic commutation relation :

$[{\sigma_-},{\sigma_+}]_+=1\ ,$

where $[,]_+$ denotes an anti-commutator. This relation results from the fact that the atom carries at most one excitation. There is a clear analogy between $\sigma_\pm$ and the photon creation and annihilation operators. The atomic Hamiltonian is in terms of these operators :

$H_a={\hbar\omega_{eg}}(\sigma_+\sigma_--\sigam_-\sigma_+)/2=\hbar\omega_{eg}(\sigma_+\sigma_--1/2)\ .$

The atomic dipole operator is :

$D=d(\epsilon_a\sigma_-+\epsilon_a^*\sigma_+)\ ,$

where $\epsilon_a$ is a complex unit vector describing the transition polarization. For the circular states with principal quantum numbers 51 and 50, the transition is $\sigma$ -polarized and $d=1776$ atomic units, a remarkably large value due to the huge size of the Rydberg states.

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