2 Quantization of the electromagnetic field Fock states and Fock spaceCoherent statesSqueezed statesCoherent representation of Thermal states
3 Quantization of the free electromagnetic field Electromagnetic field contained in a very large cube of side LPeriodic boundary conditionsElectric field linearly polarized in the x- directionClassical electromagnetic theory cannot describe optical phenomena involving small photon numbers and so we come to treat the field quantum mechanically.We will allow L to tend to infinity, any physical meaningful results should not depend on L.
4 Canonical momentum of the jth mode Classical hamiltonian for the fieldHilbert spaceHermitian operators satisfying the commutation relations.
5 Annihilation (absorption) operator Creation operatorHamiltonian is the same as Harmonic oscilator
6 Fock states Single mode of frequency Eigenstate Eigenvalue Energy eigenstate corresponding to eigenvalueWell defined energy- no time defenitionCan we lower and raise the eigen values as low and high as we like?N has only non negative values – the only way to stop the lowering process is to demand tht there must be a vacuum state.Ground state/ vacuum stateIt is useful to interpret the energy eigenvlaues as corresponding to the presence of n quanta or photons of energy hv.These eigenstates are called fock states or photon number statesThe operators a and a+ annihilate and create photons, respectively- they change a state with n photons into one with n-1 or n+1. They are not hermitian and do not represent observable quantities.EigenstateEigenvalue
8 Coherent states Eigen states of the annihilation operator Poisson distribution of Fock statesStates of minimum uncertainty productA product of the displacement operator on the vacuum state.The coherent state is an eigenstate of the annihilation operator a – we annihilate photons without changing the state!It is possible to absorb photons from an electromagnetic field in a coherent state repeatedly without changing the state in anyway.Coherent states turn out to be particularly appropriate for the description of electromagnetic fields generated by coherent sources, like lasers and parametric oscillators.The coherent states of the field come as close as possible to being classical states of definite complex amplitude.
9 Fock representation of the coherent state Considering only one mode- monochromatic light (sub space of Hilbert space)Since a is not hermitian – eigenvalue will be complexC_0 is determined from the requirement that v be normalizedNotice that when v becomes 0, |v> becomes the vacuum stateIt is partly because the coherent states are eigenstates of the absorption operator , that these states prove to be particularly convenient for the description of many properties of the field encountered in photoelectric measurements (photoelectric conductor, photomultiplier, photographic plate and the eye)Minimum uncertainty
10 The photon distribution p(n) for a coherent state Probability that n photons will be foundin the coherent stateMean number of photonsVarianceProbability that n photons will be found in the coherent statePoisson distribution in nMean number of photons present when the state is a coherent state |v>It is large or small according to |v|. No matter how small |v| may be (v=/o) there is always a probability that any number of photon is present in the field.The photon number is as random as possible in the coherent state.For |v|^2<1 p(n) is maximum at n=0For |v|^2>1 the peak is at n=|v|^2Variance- calculated by normal mode.
11 Complete setThe coherent states form a basis for the representation of arbitrary quantum states.There are no values of v1 v2 for which this term vanishes no two coherent states are orthogonal.For large v1 v2 the function acts like delta function.Despite the fact that they are not orthogonal, they span the whole Hilbert space of state vectors and form a convenient basis for the representation of other states.Actually, they are an over complete set- each state can be written in more than one representation.A resolution of the identity operator 1 in terms of coherent state projectors:Over- complete set
13 Squeezed states Squeezing a single mode field One part of the field fluctuates more and another part fluctuates less than a coherent state.Dimensionless variables representing the real an imaginary parts of the complex amplitudeDimensionless canonical conjugates , obey uncertainty relationshipsExpress the field vector for a single mode linearly polarized fieldThe canonical variables are the amplitudes of the quadraturesThe uncertainty product has it’s minimum valueIf there exists a state for which either Q or P has dispersion below unity, below the vacuum level, at the cost of a corresponding increase in the dispersion of the other variable then the corresponding space distribution takes on an elliptic shape- this is what we call a squeezed state.
14 Squeezed state with reduced phase uncertainty Vacuum stateCoherent stateSqueezed state with reduced phase uncertaintyMore general variables for any angle betaSame commutation and uncertainty relations, dispersion is unity in the vacuum state.Fock states are squeezed states taken to infinity.Squeezed state with reduced amplitude uncertainty
15 The unitary squeeze operator It is possible to generate a squeezed mode form an unsqueezed mode by the action of the unitary squeeze operatorThe unitary transformation of the operator a by S(z)A and A+ are pseudo –annihilation and creation operators
16 Two photon coherent state Squeeze operator act on the coherent state has been studied by Yuen (1976) who labeled it two photon coherent stateA and A+ bare the same relation and have the same eigenvalues with respect to |z,v> as do a and a+ to the coherent state |v>.We will show that |z,v> is indeed squeezed.The fluctuations of P exceed those in the vacuum state, while those of Q are below the vacuum level.R – the squeeze parameter.With the same choice of beta, the squeeze operator acting on any quantum state reduces the dispersion of Q by the same factor exp^-2r
17 Coherent representation of Thermal states Density operatorFock state representation- exponentCoherent state representation- Gaussian distribution (Distribution function)A thermal state is defined by the density operatorP can be used to evaluate the expectation values of any noramal ordered function of a and a+
18 BibliographyLeonard Mandel and Emil Wolf, Optical coherence and quantum optics, chap 10, 11 and 21 (Cambridge University press, Cambridge 1995)Marlan O. Scully and M. Suhail Zubairy, Quantum Optics, chap 1 and 2 (Cambridge University press, Cambridge 1997)
19 Minimum uncertaintyThe real field behaves as nearly as a classical field as possible not because ninfiThe real and the Imaginary parts of the complex amplitude a do not have well defined values in a coherent state, even though the complex amplitude itself is well defined.However, the Hermitian canonical variables q and p, which correspond to the vector potential and to the electric field in our single mode problem, are well defined as quantum mechanics allows, because the product of the uncertainties is h/2.