Photon number state

The energy levels of a molecule placed in an off-resonant microwave field can be calculafed by diagonalizing fhe mafrix of fhe Floquef Hamiltonian in the basis of direct products y) ), where y) represents in the eigenstates of the molecule in the absence of the field and ) - fhe Fourier componenfs in Eq. (8.21). The states k) are equivalent to photon number states in the alternative formalism using the quantum representation of the field [11, 15, 26], The eigensfales of the Floquet Hamiltonian are the coherent superpositions... 

The coupled channels expansion can be further simplified by introducing the (number state) rotating-wave approximation (RWA), valid only when the field is jjsfif moderate intensity and the system is near resonance. As pointed out above, igtyen an initial photon number state [JVf), the components of E, n, N — 1") of. . greatest interest for a one-photon transition are (JV, , n",JV, —1 ) and y (Nj dt l[Ji, n, Nj — 1 ). If [ , ) is the ground material state, then the (Nj+m,... 

This incoherent statistical average over the phases also gives exactly the same result as the coherent average (52) in a photon number state. 

Due to the relative character of the number operator — /S/S0, all the physical predictions of the Floquent model must be invariant with respect to a global translation of the relative photon numbers. We show that this is indeed the case for the properties discussed above. The propability P(L, t) is independent of the particular initial photon number state chosen that is, it is independent of k since... 

If one considers an initial coherent state for the photon field instead of a photon-number state, the superpositions of states have the additional optical phase, giving for (286)... 

The NS operation is successful when one and only one photon reaches the detector in mode 4. Provided the photons are indistinguishable, the two paths leading to exactly one photon in mode 4 will interfere. The two interfering processes are depicted in Fig. 6 b for n input photons. Either all n + 1 photons are reflected, or n — 1 photons in mode 1 are reflected and 1 photon in each mode 1 and 2 is transmitted. When a single photon ends up in mode 4, the photon number state undergoes the following transformation ... 

Since for the photon-number states the expectation values of the field operators vanish, all the information about the state of the system is contained in the intensities of the corresponding fields... 

Figure 1. Examples of discrete Wigner function on a torus in 19-dimensional Hilbert space i s = 18) (a) vacuum 0) (b) single-photon number state 1) (c) FD preferred phase state ( phase vacuum ) 0o)(5) (d) FD coherent state, a) ss a) (e) FD displaced number state, oc, 1)(j) oc, 1)(j) (0 FD phase coherent state, P, 0o)(j) P,0o)(5), with equal displacement parameters, 7. — 7. — P — P — 0.5 and 0o = 0. The darker is a region, the higher is the value of the Wigner function.

The one-atom maser can be used to investigate the statistical properties of non-classical light [1298, 1299]. If the cavity resonator is cooled down to T < 0.5 K, the number of thermal photons becomes very small and can be neglected. The number of photons induced by the atomic fluorescence can be measured via the fluctuations in the number of atoms leaving the cavity in the lower level n — 1). It turns out that the statistical distribution does not follow Poisson statistics, as in the output of a laser with many photons per mode, but shows a sub-Poisson distribution with photon number fluctuations 70 % below the vacuum-state limit [1300]. In cavities with low losses, pure photon number states of the radiation field (Fock states) can be observed (Fig. 9.77) [1301], with photon lifetimes as high as 0.2 s At very low... 

B.T. Varcoe, S. Brattke, M. Weidinger, H. Walther, Preparing pure photon number states of the radiation field. Nature 403,743 (2000)... 

A Fock state is a state containing a fixed number of photons, N. These states are very hard to produce experimentally for A > 2. Their photon number probability density distribution P (m) is zero everywhere except for m = N, their variance is equal to zero since the intensity is perfectly determined. Finally, the field autocorrelation function is constant... 

This obtainsonly if y in the sum on the rhs of Eq. (23) closely approachessuch that the number of photons in state 1, of lowest frequency a>, becomes much larger than any other number n, as energy pumping, S = ES, increases. This condensation of the photons into the lowest-frequency state is formally equivalent to the Bose condensation (138), and y, Eq. (18), plays a role analogous to the chemical potential in the Einstein condensation of a Bose-Einstein gas. 

The state of the radiation field is determined by a set of photon numbers nx-The vacuum state, designated 0), contains no photons. The state U) contains one photon of energy hcox, propagation vector kx and polarization ex- The state 12x) contains two such photons, while I lx, Ix ) contains two different photons, A and A. The most general state ofthe radiation fieldwould be designated nx, xs x" ) If the enclosure also contains an atom in quantum state lr , the composite state is designated n x, x, ) ... 

As shown in Fig. 16.11, two pairs of entangled photons are produced by sources A and B, which are actually realized by passing a UV pulse twice through the same BBO crystal. Source A emits photons numbers 1 and 2, while source B emits photons numbers 3 and 4. The state of the four-photon system at this point can be represented as... 

The initial photon state can be a number state (with a not well-defined phase) or a linear combination of number states, for instance a coherent state. We formulate the construction of coherent states in the Floquet theory and show that choosing one as the initial photon state allows us to recover the usual semiclassical time dependent Schrodinger equation, with a classical held of a well-defined phase (see Section II.C). 

The formal hypothesis (31) must be interpreted in relation to the functions on which — /S/00 acts. The statement is that if all the states elk()) that are relevant in the dynamics are such that k exchanged between light and matter compared to the average photon number n contained in the laser field — then the coupled Hamiltonian //[ () can be identified with the Floquet Hamiltonian K. 

For a general initial condition of the photon field i (0) C jSf, we first remark that the evolution of the initial condition (that we take here at t — to —0) 4>(jc) / c( 0j can be obtained from the one of the initial condition 4>(v) 0 1 (where the constant function 1 = e relative number state of zero photons) ... 

This property is quite remarkable In the large photon number regime the coherent quantum average on a number state gives the same result as the incoherent statistical average over coherent states. 

In adiabatic passage processes with pulsed lasers, as we will discuss in the forthcoming sections, one often encounters the following particular situation If the initial condition of the photon field were a number state, that is,... 

In the following, we will consider for simplicity the case A] = — A2 so that 8 = — 2Ai. For frozen values of the two fields fij and 02, we calculate dressed states and dressed energies by diagonalizing Keff. The eigenelements can be labeled with two indices One, denoted n, refers to the levels of the atom and another one, denoted ft, refers to the relative photon numbers. The index k stands for the number of the coi photons absorbed and the number of 0)2 photons emitted. The eigenvalues and eigenvectors have the following property of periodicity ... 

This is confirmed by the numerical solution of the dressed Schrodinger equation (308) with a number state as the initial condition for the photon field 11 0,0) It shows that the solution dressed state vector v /(t) (the transfer state, which in the bare basis is given by / /(0,0) mainly projects on the transfer eigenvector during the process. Additional data of the dressed solution during time are shown in Fig. 21a and 21c. Figure 21a displays the probabilities of being in the bare states 1, 2, and 3 ... 

To that effect, we choose the parameters 8 = 2flo and Qmax = 4.4 f>o, corresponding to the path (c) on the surfaces in Fig. 19. As shown in Fig. 22, the solution of the semiclassical Schrodinger equation (321) leads to nearly complete population transfer from state 11) to state 3). The analysis of the surfaces shows that the state 1 0,0) connects 3 —3,3). Thus the complete population transfer from the bare state 1) to the bare state 3) must be accompanied with absorption of three pump photons and emission of three Stokes photons at the end of the process. This is confirmed by the numerical solution of the dressed Schrodinger equation (308) with the initial state as a number state for the photon field 11 0,0), shown in Fig. 23a the dressed state vector /(f) approximately projects on the transfer eigenvectors during the process. It shows... 

The natural line width of the spectral line is a significant result of the dissipative quantum process which accompanies the spontaneous emission of an atom. We will treat this emission process in a dissipative two-state model. We consider the two states of the atom as the zeroth and the first occupation number state of a linearly damped oscillator. In this case, the spontaneous emission of a photon is the consequence of the transition from the first occupations number state to the equilibrium state of the damped oscillator. In this model, the spectrum density of the emitted photon follows from Equation (92)... 

The central feature of QED is that there is a closed system of particles and fields that act on each other, the particles associated with the radiation field being photons. The states of the entire system can be dealt with, and its properties, including rates of transitions in which the number of particles changes, calculated as accurately as desired. Typically in applications to molecules (MQED) the Hamiltonian is written in the form (5.2) with the interaction between molecule and fields separated out as a small term to be treated as a perturbation,... 

For our first measurement we act on the photon-number entangled states in mode 3. Since the NS operation is an interference effect, it only proceeds when the entangled photons and the ancilla photon arrive at BS2 within their coherence time rcoh- In this case, the operation performs the following ... 

While parametric down-conversion techniques have recently been used to generate multi-photon states [Waks 2004], it remains experimentally challenging to implement schemes that allow for simultaneous control over both photon number and spatio-temporal properties of the pulse. 

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