Coherent state density matrix

As an example of a reconstructed number state density matrix, we show in Fig. S our result for a coho-oit superpoation of n=0) and n=2) number states. This state is ideally suited to demonstrate the sensitivity of the reconstruction to coherences. Our result indicates that the prepared motional states in our system are very close to pure states. 

In molecules, as noted by de Vries and Wiersma, the application of free induction decay to study optical dephasing may be frustrated by the presence of an intermediate triplet state. The level scheme, which is representative for most molecules with an even number of electrons, is shown in Fig. 26. For an applied laser field E =EQCOs t-k ), that is resonant with the (2 <- 1) transition, we may write, in the RWA approximation, the following steady-state density matrix equations, which describe the coherent decay after laser frequency switching ... 

This calculation makes it clear that light beats are associated with the time evolution of the off-diagonal elements of the excited-state density matrix. Consequently they can only be observed in pulsed experiments if the light which excites the atoms of the sample is also polarization coherent. Only then is the necessary Hertzian coherence created in the excited-state density matrix. The theory predicts that modulation at the angular frequencies and will be detectable depending on the geometry and polarization used in the experiment. 

Quantum mechanically the modulation of the fluorescent light is associated with the radio frequency coherence of the excited state density matrix. In a standard Brossel-Bitter double-resonance experiment Tt-polarized light excites the atoms initially to the m=0 state of the excited level. Fig. 16.13(b), and then interaction with the r.f. magnetic field transforms each atom into a coherent superposition of the m=0, l states. The relative phases of the probability amplitudes of these states are fixed by the phase of the r.f. field and are the same for every atom of the sample. Thus the r.f. field is able to generate substantial hertzian coherence in the excited state density matrix. The fluorescent light emitted in the direction of B is then a coherent... 

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

A quantum state loses quantum coherence (decoheres) when Sqd wave functions are peaked along classical trajectories. And it decoheres when each trajectory loses quantum coherence with its neighbors. Quantum decoherence is realized when the diagonal term density matrix dominates over the off-diagonal term (fts-... 

In Ref. [4] we have studied an intense chirped pulse excitation of a molecule coupled with a dissipative environment taking into account electronic coherence effects. We considered a two state electronic system with relaxation treated as diffusion on electronic potential energy surfaces with respect to the generalized coordinate a. We solved numerically equations for the density matrix of a molecular system under the action of chirped pulses of carrier frequency a> with temporal variation of phase [Pg.131]

At T = 0 the hindered rotation is a coherent tunneling process like that studied in Section 2.3 for the double well. If, for instance, the system is initially prepared in pure state localized in one of the wells, then the density matrix in the coordinate representation is given by... 

The next two chapters are devoted to ultrafast radiationless transitions. In Chapter 5, the generalized linear response theory is used to treat the non-equilibrium dynamics of molecular systems. This method, based on the density matrix method, can also be used to calculate the transient spectroscopic signals that are often monitored experimentally. As an application of the method, the authors present the study of the interfadal photo-induced electron transfer in dye-sensitized solar cell as observed by transient absorption spectroscopy. Chapter 6 uses the density matrix method to discuss important processes that occur in the bacterial photosynthetic reaction center, which has congested electronic structure within 200-1500cm 1 and weak interactions between these electronic states. Therefore, this biological system is an ideal system to examine theoretical models (memory effect, coherence effect, vibrational relaxation, etc.) and techniques (generalized linear response theory, Forster-Dexter theory, Marcus theory, internal conversion theory, etc.) for treating ultrafast radiationless transition phenomena. 

In the coherent (Hamiltonian) approach to the four-state spin system, the state populations are just the diagonal elements of the corresponding density matrix p(r, f), which obeys the Bloch or Redfield equation [211] ... 

Together, these 16 product operators describe the 16 matrix elements in the 4 x 4 density matrix representation of a two-spin system (Chapter 10). In the matrix, each element represents coherence between (or superposition of) two spin states. As there are four spin states for a two-spin system (aid s i/3s, P as, and PiPs), there are 16 possible pairs of states, which can be superimposed or share coherence. The product operators are closer to the visually and geometrically concrete vector model representations, so in most cases they are preferable to writing down the 16 elements of the density matrix, especially as only a few of the elements are nonzero in most of the examples we discuss. 

Note that each of these coherences corresponds to a transition between two energy levels (e.g., c ci corresponds to the /3h c to q hQ c transition) in the two-spin energy diagram. The four SQCs correspond to the four peaks in the 13 C and XH spectra (two doublets), and the MQCs are not observable. Later, we will see that all of these numbers (the four populations, the four single-quantum coherences, and the ZQC and DQC) can be fit into a 4 x 4 matrix that provides a succinct summary of everything we could ever want to know about the spin state of this ensemble of N spin pairs. This matrix is called the density matrix. 

If the average is taken over all of the spin pairs in the sample for each term in the matrix, we have the density matrix a that describes the state of the whole system. The superpositions (or coherences) between states can be described as follows ... 

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