This paper considers the problem of finding the quantum states that minimize the products of the (even) Nth-order fluctuation of two canonically conjugate operators. The problem is first attacked in an abstract form and an equation derived for the desired states. A consideration of the N=2 case then leads to a connection with the new concept of ‘‘squeezed states’’ of the electromagnetic field, and the usual exact solution involving the coherent state. The concept of ‘‘higher-order squeezing’’ is touched on to motivate the further discussion. The cases N=4, 6, and 8 are then taken up, and approximate solutions to them found via a new numerical technique. During these calculations, it is noted that only the first few terms in the expansion in number states figure in the solution; this observation is then exploited to find an approximate closed solution to the problem valid for all N.
This paper considers the problem of whether mixed states can arise in the Nth order minimum uncertainty expression UN=<Ψ‖‖x‖N‖Ψ≳<Ψ‖‖p‖N‖Ψ≳. Using a method due to R. Jackiw [J. Math. Phys. 9, 339 (1968)], and finding the extremum of the appropriate functional, a set of equations is derived which must be satisfied by such a mixed state. It is easily shown that a sufficient condition that the equations have a solution is that the mixed state in fact reduces to a pure state. It does not appear possible to rigorously prove that such a reduction is necessary, although the authors present approximate calculations which make such a conclusion plausible. In the course of the approximate calculations generalized Nth order ‘‘squeezed state’’ values for UN are also obtained which are of some interest in and of themselves.
Starting from a convenient measure for the phase uncertainty delta phi , a rigorous solution to the variational problem in which ( delta phi )2 is minimized under the constraint that the mean photon number N is held fixed, was found. In an approximation valid for large values of N, closed-form expressions for both ( delta phi )2 and the variance of the photon number ( Delta n)2 were obtained. In the final result, the approximation is very similar to an approximation scheme recently discussed by Summy and Pegg Opt. Commun., vol.77, p.75, 1990. The phase uncertainty for the phase optimized states is found to decrease as (N+1)-1 with growing N.
We study d-variate approximation for a weighted unanchored Sobolev space having smoothness m ≥ 1. Folk wisdom would lead us to believe that this problem should become easier as its smoothness increases. This is true if we are only concerned with asymptotic analysis: the nth minimal error is of order n^-(m-δ) for any δ > 0. However, it is unclear how long we need to wait before this asymptotic behavior kicks in. How does this waiting period depend on d and m? We prove that no matter how the weights are chosen, the waiting period is at least m^d, even if the error demand ε is arbitrarily close to 1. Hence, for m ≥ 2, this waiting period is exponential in d, so that the problem suffers from the curse of dimensionality and is intractable. In other words, the fact that the asymptotic behavior improves with m is irrelevant when d is large. So, we will be unable to vanquish the curse of dimensionality unless m = 1 , i.e., unless the smoothness is minimal. We then show that our problem can be tractable if m = 1. That is, we can find an ε-approximation using polynomially-many (in d and ε^-1) information operations, even if only function values are permitted. When m = 1, it is even possible for the problem to be strongly tractable, i.e., we can find an ε-approximation using polynomially-many (in ε^-1) information operations, independent of d. These positive results hold when the weights of the Sobolev space decay sufficiently quickly or are bounded finite-order weights, i.e., the d-variate functions we wish to approximate can be decomposed as sums of functions depending on at most ω variables, where ω is independent of d.
We study theoretically a weak measurement of an observable
A and its impact on a subsequent strong measurement of another observable
B. We assume that both measurements are described by the Gaussian point detectors. It is shown that a decrease of the system-detector interaction during the measurement of
A results in an increase of the fluctuations of
A, while the mean value of
A is kept unaltered. The ratio ${(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{weak}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}/{(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{strong}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}$ turns out to be universal, i.e., independent of the physical system, its quantum state, and the quantity measured. Most of our attention is focused on analyzing outcomes of the subsequent strong measurement of the quantity
B, namely, on the mean value
B and the variance ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$. We derive explicit analytic corrections for
B and ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$ arising due to the prior weak measurement of
A. Importantly, the leading-order impact of the weak measurement of
A on
B and ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$ is encoded in the terms ${\phantom{\rule{0.16em}{0ex}}}_{s}\ensuremath{\langle}\ensuremath{\psi}|[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A},[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A},{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}}^{k}]]{|\ensuremath{\psi}\ensuremath{\rangle}}_{s}$, where
k=1,2 and ${|\ensuremath{\psi}\ensuremath{\rangle}}_{s}$ is the quantum state vector of our system before the measurements. Our additional contribution consists of deriving Heisenberg-like uncertainty relations for the product ${(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{weak}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}\phantom{\rule{0.28em}{0ex}}{(\mathrm{\ensuremath{\Delta}}B)}_{\mathrm{strong}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}$ and analyzing the case of minimum uncertainty. Finally, our theoretical developments are illustrated numerically via studying the
s states of the hydrogen atom, assuming $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}$ (atomic radius) and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ (energy).