Quadrature variances

Therefore, whenever the normal form of the quadrature variance is negative, this component of the field is squeezed or, in other words, the quantum noise in this component is reduced below the vacuum level. For classical fields, there is no unity coming from the boson commutation relation, and the normal form of the quadrature component represents true variance of the classical stochastic variable, which must be positive. 

Figure 8. The quadrature variances (a) squeezed variances ([AQa 2) (solid line) and [APj,]2) (dashed line) (b) nonsqueezed variances [APJ2) (solid line) and ([Agt]2) (dashed line) for Na = 10. The dotted lines are the linearized solutions.

Similarly, we obtain corresponding expressions for the quadrature variances... 

Figure 17. Quadrature variances (a) [AQ (t)]2) and (b) [APa(x)]2) for the signal mode with t = -n/2 and Nb = 10 (solid line), Nb = 40 (dashed line), and Nb = 100 (dashed-dotted line). Dotted lines represent parametric approximation.

Since the covariance is different from zero if y / 0, the initial vacuum state of the field is transformed to the correlated quantum state [223,271,272], One should remember, however, that the values of Up, Vy., and yield the (co)variances of the field quadratures only at the moment t T (when the wall stopped oscillating). At the subsequent moments of time the quadrature variances exhibit fast oscillations with twice the frequency of the mode. For example (omitting the mode index), one obtains... 

Therefore the physical meanings do not have the values U, V, and Y themselves, but rather the minimal cimin = and maximal CTmax = values of the quadrature variances during the period of fast oscillations [273,274]... 

This distribution possesses the photon-number variance ct = sinh2(2ooo f). Similar formulas for the amount of photons created in a cavity filled with a medium with a time-dependent dielectric permeability (and stationary boundaries) have been found [222], The quadrature variances change in time as (now... 

Squeezing occurs if one of the quadrature variances ((Ap)2), ((Aqj2) is below the coherent state level. A more general definition deals with single and compound mode principal squeeze variances Xj and V [140]... 

For example, (A) and (B) can be computed using Eqs. (49) and (50), respectively. Note that instead of Eq. (55), we could use the simpler expression for given by Eq. (33), which avoids the need for numerical quadrature. In both cases, the mean and variance of the mixture fraction are identical (and thus both models account for finite-rate mixing effects.) In practical applications, the differences in the predicted values of () can often be small (Wang and Fox, 2004). 

The phase-dependent directionality of photocurrents produced by such a detector entails advantageous properties of the photocurrents cross correlations in nonoverlapping time intervals or spatial regions (considered in Section 4.2.2). These directional time-dependent correlations are measured with one detector only. They involve solely terms dependent on LO phases, in contrast to similar correlations measured by conventional photocounters, which inevitably contain terms depending on photon fluxes such as the LO excess noise. Owing to these properties, the mean autocorrelation function of the SL quadrature is shown in the schemes considered here to be measurable without terms related to the LO noise. LO shot noise, which affects the degree of accuracy to which this autocorrelation is measured (i.e., its variance) is easily obtainable from zero time delay correlations because the LO excess noise is suppressed. The combined measurements of cross correlations and zero time delay correlations yield complete information on the SL in these schemes. 

Let us now switch our attention to the quantum statistical effects and entanglement production for the case of perfect symmetry between the modes (7i = 72 = 7> Ai = A2 = A). In order to obtain general expressions for the variances, we first write them in terms of the boson operators corresponding to the Hamiltonian (lb). We perform the transformations at —> at exp (i (I>j), which restore the previous phase structure of the intracavity interaction and find, quite generally, the variances at some arbitrary quadrature phase angles 0i, 02 as... 

Now, assuming that the two modes are not correlated at time x = 0, it is straightforward to calculate the variances of the quadrature field operators and check, according to the definition (12), whether the field is in a squeezed state. If the initial state of the field is a coherent state of the fundamental mode and a vacuum for the second-harmonic mode, /0) = wa(0)) 0), for which we have... 

We can thus expect from the short-time approximation that quantum noise does not significantly affect the classical solutions when the initial pump field is strong. We will return to this point later on, but now let us try to find the short-time solutions for the evolution of the quantum noise itself—let us take a look at the quadrature noise variances and the photon statistics. Using the operator solutions (94) and (95), one can find the solutions for the quadrature operators Q and P as well as for Q2 and P2. It is, however, more convenient to use the computer program to calculate the evolution of these quantities directly. Let us consider the purely SHG process, we drop the terms containing b and b+ after performing the normal ordering and take the expectation value in the coherent... 

For the case of two-dimensional CS, there have been computed quantities such as the mean values and variances of the various operators, including N and 4> quadratures and their commutators [16,17]. Most of these quantities can easily be displayed on the Poincare sphere and expressed by means of the Stokes parameters. We find that the following mean values and variances are given respectively by... 

The minimal variance u monotonously decreases from the value at t = 0 to the constant asymptotic value 2/tx2 at x 1, confirming qualitatively the earlier evaluations [107,110] and giving almost 50% squeezing in the initial vacuum state. The variance of the conjugate quadrature monotonously increases, and for x 1 it becomes practically linear function of time ui(x 1) 16x/ti2. The asymptotic minimal value u (x = oo) does not depend on y provided y < 1 (only the rate of reaching this asymptotic value decreases with y as J 1 — y2). In the strongly detuned case, y > 1, the minimal variance oscillates as a function of x (it is always greater than 2/7X2), since in this case the function k(x) oscillates between y 1 and y 1. 

The time dependence of the energy and the second-order statistical moments (variances) of the field mode quadrature components, Oab = (ab + ba) — (ia)(b), is governed by the equations following from Eq. (220) ... 

Our discussion is focused on photon-number statistics rather than squeezing or other phase-related properties. Nevertheless, by analyzing the g-function evolution presented in Fig. 4, we can draw the conclusion that squeezing cannot be observed for initial coherent fields at interaction times exceeding the relaxation time. In fact, the quadrature squeezing variances (k = 1,2)... 

By analogy with our quantum analysis, we calculate the semiclassical Fano factor, defined by Eq. (1.4), and quadrature squeezing variance... 

To Calculate Noise from a Set o/H.Hf Values The square of the noise in the narrow spectral band around each frequency is the PSD times the spectral bandwidth. Noise adds in quadrature, so the square of the total noise (also known as the variance) is the sum over all frequencies of the PSD times the frequency intervals between the PSD values ... 

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