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Accueil du site > Atomes, cavités et photons > Principes de l’electrodynamique quantique en cavité > Un oscillateur : mode de la cavité

Un oscillateur : mode de la cavité

The single cavity mode is a quantum harmonic oscillator. We recall here its main properties and those of quantum states of interest, Fock and coherent states.

In these sections, we provide only a short presentation. More information can be found in [1]

Fock states

A single field mode is equivalent to a one-dimensional harmonic oscillator. The non-degenerate energy eigenstates are the Fock or `photon number states’ $\{|n\rangle}\ n>0$ , whose energy is $\hbar\omega_c (n+1/2)$ , where $\omeaga_c$ is the cavity mode angular frequency. The ground state is the vacuum $|0\rangle$ . The Fock states are an orthogonal set :

$\langle n|p\rangle=\delta_{np}\ .$

The photon annihilation and creation operators $a$ and $a^\dagger$ connect the Fock states :

$a|n\rangle=\sqrt{n}|n-1\rangle\ ;\qquad a^\dagger |n\rangle=\sqrt{n+1}|{n+1}\rangle\ .$

The action of $a$ on $|0\rangle$ gives a null vector (it is not possible to annihilate a photon in vacuum). All Fock states can be generated from the vacuum by repeated applications of the photon creation operator : $|n\rangle=a^{\dagger n}|0\rangle\sqrt{n!}$ . These operators obey a bosonic commutation rule : $[a,a^\dagger]=1$ .

The cavity hamiltonian is $H_c=\hbar\omega_c(a^\dagger a+1/2)$ . Since we are dealing with a single-mode situation, we can redefine the energy origin to get rid of the non-vanishing vacuum state energy. We can thus also use the simpler hamiltonian :

$H'_c=\hbar\omega_c a^\dagger a\ .$

The cavity mode electric field operator at position $r$ writes :

$E_c=i E_0\left[ f(r)a-f^*(r)a^\dagger \right]\ ,$

where $E_0$ is a normalization factor. The dimensionless vector function $f(r)$ describes the spatial structure of the field mode. At the point where the field mode amplitude is maximum, which we also take as the origin, $|f|=1$ .

The normalization factor is obtained by equating the Fock states energies with the integral over space of the expectation value of the electromagnetic energy density $\varepsilon_0|E_c|^2$ :

$E_0=\sqrt{\frac{\hbar\omega_c}{2\varepsilon_0V}}\ .$

where we define the cavity effective volume $V$ by :

$V=\int\,|f(r)|^2\,dV\ .$

Our Fabry-Perot cavity is made of two spherical mirrors facing each other (figure above). The mode is then a standing wave with a gaussian transverse profile and a sinusoidal field variation in the longitudinal direction normal to the mirrors, separated by the distance $L$ . The waist $w$ characterizes the minimum width of the gaussian. The mode volume is then $V={\pi Lw^2/ 4}$ . For the specific parameters of the experiment ( $L=2.7$ cm ; $w= 6$ mm) we have $V = 0.7\$ cm $^3$ . The field per photon is then :

$E_0=1.5\,10^{-3}\ \mbox{V/cm}\ ,$

a rather large value in S.I. units for a quantum field.

Field quadratures

The field quadrature operators correspond to a mechanical oscillator’s position and momentum :

$X = \frac{{a + a^ \dagger }}{2}\quad ;\quad P = \frac{{a - a^ \dagger }}{{2i}} = \frac{{e^{ - i\pi /2} a + e^{i\pi /2} a^ \dagger }}{2}\ .$

More generally, phase quadratures are linear combinations of $a$ and $a^{\dagger}$ :

$X_\varphi \; = \frac{{e^{ - i\varphi } a + e^{i\varphi } a^ \dagger }}{2}\;;X_{\varphi + \pi /2} \; = \frac{{e^{ - i\varphi } a - e^{i\varphi } a^ \dagger }}{{2i}}\ .$

They satisfy the commutation rules $\left[ {X_\varphi ,X_{\varphi + \pi /2} } \right] = i/2$ , which correspond to the uncertainty relations $\Delta X_\varphi \Delta X_{\varphi + \pi /2} \ge 1/4$ , where $\Delta X_{\varphi}$ and $\Delta X_{\varphi + \pi /2}$ are conjugate phase quadrature fluctuations.

The eigenstate of the quadrature $X_{\varphi}$ corresponding to the real and continuous eigenvalue $x$ is a non-normalizable state (infinite energy), which obeys the orthogonality and closure relationships :

${}_\varphi \left\langle {x} \mathrel{\left | {\vphantom {x {x'}}} \right. \kern-\nulldelimiterspace} {{x'}} \right\rangle _\varphi = \delta (x - x');\quad \int {\left| x \right\rangle } _{\varphi \;\varphi } \left\langle x \right|dx = \unity ,$

the transformation from the $| x \rangle_{\varphi}$ basis to the conjugate basis $| x \rangle_{\varphi + \pi/2}$ being a Fourier transform :

$\left| x \right\rangle _{\varphi + \pi /2} \; = \quad \int {dy\left| y \right\rangle } _{\varphi \varphi } \left\langle {y} \mathrel{\left | {\vphantom {y x}} \right. \kern-\nulldelimiterspace} {x} \right\rangle _{\varphi + \pi /2} = \frac{1}{{\sqrt \pi }}\int {dy\,e^{2i\,x\,y} \left| y \right\rangle } _\varphi \ .$

Note that two conjugate field quadratures provide coordinates for the quantum field phase space, equivalent to the Fresnel plane for classical fields.

The expectation value $\left\langle n \right|X_\varphi \left| n \right\rangle$ of any phase quadrature in a Fock state is zero. There is thus no preferred phase neither in the vacuum nor in any Fock state, a feature which shows that these quantum states are quite different from classical fields. For the vacuum, the quadrature fluctuations are isotropic and correspond to the minimum value compatible with Heisenberg uncertainty relations : $\Delta X_\varphi ^{(0)} = \sqrt {\left\langle 0 \right|X_\varphi ^2 \left| 0 \right\rangle } = 1/2$ . The probability distribution $P^{(0)}(x)$ of the field quadrature is then a gaussian :

$P^{(0)} (x) = \left| {{}_\varphi \left\langle {x} \mathrel{\left | {\vphantom {x 0}} \right. \kern-\nulldelimiterspace} {0} \right\rangle } \right|^2 = \left( {\frac{2}{\pi }} \right)^{1/2} e^{ - 2x^2 } \ .$

In summary, the vacuum field in each mode has isotropic gaussian fluctuations around zero field.

Coherent states

To describe situations in which the phase of the field is relevant, it is convenient to expand the field on the basis of the coherent states, which are more physical than Fock or quadrature states and are experimentally more accessible.

A coherent state of a single field mode is defined as resulting from the translation of the vacuum field in phase space. This translation is represented, in its most general form, by the unitary displacement operator :

$D(\alpha)=e^{\alpha a^{\dagger}-\alpha^{*} a}\ ,$

where $\alpha=|\alpha|\exp(i\phi)$ is a $C$ -number whose real and imaginary parts are the projections along the $X_0$ and $X_{\pi/2}$ directions respectively of the translation vector. The translated vacuum state is the coherent state $| \alpha \rangle$ :

$|\alpha\rangle=D(\alpha)|0\rangle\ .$

The translated `packet’, whose evolution is determined in the Schrödinger picture by the free field hamiltonian, [subsequently rotates at frequency $\omega_c$ in phase space, without deformation : $|\alpha(t)\rangle=|\alpha e^{-i\omega_c t}\rangle$ . This corresponds to the best possible approximation of a classical free oscillator motion. The two parts of the figure above show how the vacuum state is transformed by translation into a coherent state and how this state freely evolves in phase space. In (b), the coherent states are pictorially shown as uncertainty circles of radius unity at the tip of the classical amplitude, a representation that we shall repeatedly use in the following.

In order to make the translation operation more explicit, it is convenient to split the exponential in the displacement operator in two, separating the contributions of the real and imaginary parts of $\alpha$ . We make use of the Glauber relation $e^{A+B} = e^{A} e^{B} e^{-[A,B]/2}$ (valid if $[A,B]$ commutes with $A$ and $B$ ) and we get :

$D(\alpha ) = e^{ - i\alpha _1 \alpha _2 }\,e^{2i\alpha _2 X_0}\,e^{- 2i\alpha _1 X_{\pi /2}}\ ;$

Using a mechanical oscillator analogy, the displacement $D(\alpha)$ can thus be viewed as a translation along space by an amount $\alpha_1 =\Re (\alpha)$ , followed by a `momentum kick’ of magnitude $\alpha_2 =\Im (\alpha)$ .

Let us compute the probability amplitude for finding the value $x$ when measuring the quadrature operator $X_0$ on a field in state $|\alpha \rangle$ :

$\left\langle {x} \mathrel{\left | {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} {\alpha } \right\rangle = \left( {\frac{2}{\pi }} \right)^{1/4}\,e^{- i\alpha _1 \alpha _2 }\,e^{2i\alpha _2 x}\,e^{ - (x - \alpha _1 )^2}\ ,$

a gaussian wave packet centered in $\alpha_1$ , with a phase modulation at frequency $\alpha_2$ describing the momentum kick. The probability for finding the value $x$ for the quadrature is thus :

$P(x)=\left( {\frac{2}{\pi }} \right)^{1/2} \exp \left[ { - 2\left( {x - \alpha _1 } \right)^2 } \right]\ ,$

a translated ground state distribution.

An alternative and useful expression of the displacement operator is obtained by using again the Glauber relation, separating this time the $a$ and $a^{\dagger}$ terms :

$D(\alpha ) = e^{ {\alpha a^ \dagger - \alpha ^* a} } = \quad \exp \left( - \frac{{\left| \alpha \right|}}{2}^2 \right)\,e^{\alpha a^ \dagger}\,e^{- \alpha ^* a}\ .$

This form corresponds to the `normal ordering’ in quantum optics. If we expand the exponential of operators in series, all the $a^n$ terms are on the right and the $a^{\dagger n}$ terms on the left. The action of the $\exp(-\alpha^{*}a)$ operator on the right leaves the vacuum unchanged, since only the zero-order term in the expansion yields a non-zero result. We get :

$\left| \alpha \right\rangle = \sum\nolimits_n {c_n (\alpha )\left| n \right\rangle } \quad\mbox{with}\quad c_n (\alpha ) = \exp \left( - \frac{{\left| \alpha \right|}}{2}^2 \right)\frac{{\alpha ^n }}{{\sqrt {n!} }}\ .$

The distribution of photon numbers in a coherent state obeys a Poisson statistics. The figure above this paragraph shows this distribution for $\alpha = 1$ and $\alpha= \sqrt{20}$ . The average photon number $\overline{n}$ and photon number variances $\Delta n$ are :

$\overline n = \left| \alpha \right|^2 \quad ;\quad \frac{{\Delta n}}{\overline n} = \frac{1}{{\left| \alpha \right|}} = \frac{1}{{\sqrt {\overline n } }}\ .$

The relative fluctuation of the photon number is thus inversely proportional to the square root of its average. For large fields, this fluctuation becomes negligible (classical limit).

Coherent states are eigenstates of the photon annihilation operator $a$ :

$a\left| \alpha \right\rangle = \alpha \left| \alpha \right\rangle \quad\mbox{and}\quad \left\langle \alpha \right|\;a^ \dagger \; = \;\left\langle \alpha \right|\;\alpha ^* \ .$

It is also useful to recall the expression of the scalar product of two coherent states :

$\left\langle {\alpha } \mathrel{\left | {\vphantom {\alpha \beta }} \right. \kern-\nulldelimiterspace} {\beta } \right\rangle = e^{ { - \left| \alpha \right|^2 /2\; - \left| \beta \right|^2 /2\; + \;\alpha ^* \beta } } \quad;\quad \left| {\left\langle {\alpha } \mathrel{\left | {\vphantom {\alpha \beta }} \right. \kern-\nulldelimiterspace} {\beta } \right\rangle } \right|^2 = e^{ - \left| {\alpha - \beta } \right|^2} \ ,$

which shows that the overlap of two such states decreases exponentially with their `distance’ in phase space. Although they are never strictly orthogonal, they become practically so when the distance of their centers is much larger than $1$ , the radius of the uncertainty circle.

The coherent states constitute a complete set of states in the mode’s Hilbert space :

$\frac{1}{\pi }\int {d\alpha _1 d\alpha _2 \left| \alpha \right\rangle } \left\langle \alpha \right| = \unity \ . \label{EQ_CLOSURECOHERENT}$

Note however that the non-orthogonal coherent state basis is over-complete. The expansion of a state over it is not unique.

Coherent states are easily produced experimentally, by the classical microwave source $S$ weakly coupled to the cavity mode. The evolution operator for the mode state under the coupling with a classical current is the displacement operator, transforming the initial vacuum in a coherent state whose amplitude and phase are under experimenter’s control.

As a classical field, a coherent state is defined by a complex amplitude, evolving in the Fresnel plane. The non-vanishing quantum fluctuations of the field quadratures is pictoriallly represented by an uncertainty circle. For very small fields (about one photon on the average) the amplitude is comparable to the uncertainties and quantum fluctuations play an important role. For very large amplitudes, quantum fluctuations are negligible and the coherent state can be viewed as a classical object, with well defined phase and amplitude. Coherent states stored in a cavity thus span the quantum to classical transition, with the mere adjustment of the source controls.

[1] S. Haroche et J.M. Raimond, Oxford University Press, 2006 : “Exploring the quantum : atoms, cavities and photons »

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