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Register a new userBack and forth from Fock space to Hilbert space: a guide for commuters
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Quantum states of systems made of many identical particles, e.g. those described by Fermi-Hubbard and Bose-Hubbard models, are conveniently depicted in the Fock space. However, in order to evaluate some specific observables or to study the system dynamics, it is often more effective to employ the Hilbert space description. Moving effectively from one description to the other is thus a desirable feature, especially when a numerical approach is needed. Here we recall the construction of the Fock space for systems of indistinguishable particles, and then present a set of recipes and advices for those students and researchers in the need to commute back and forth from one description to the other. The two-particle case is discussed in some details and few guidelines for numerical implementations are given.
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We analyze the quantum evolution of a weakly nonlinear resonator due to a classical near-resonant drive and damping. The resonator nonlinearity leads to squeezing and heating of the resonator state. Using a hybrid phase-space--Fock-space representation for the resonator state within the Gaussian approximation, we derive evolution equations for the four parameters characterizing the Gaussian state. Numerical solution of these four ordinary differential equations is much simpler and faster than simulation of the full density matrix evolution, while providing good accuracy for the system analysis during transients and in the steady state. We show that steady-state squeezing of the resonator state is limited by 3 dB; however, this limit can be exceeded during transients.
In this letter we present a protocol to engineer interactions confined to subspaces of the Fock space in trapped ions: we show how to engineer upper-, lower-bounded and sliced Jaynes-Cummings (JC) and anti-Jaynes-Cummings (AJC) Hamiltonians. The upper-bounded (lower-bounded) interaction acting upon Fock subspaces ranging from $leftvert 0rightrangle $ to $leftvert Mrightrangle $ ($leftvert Nrightrangle $ to$ infty$), and the sliced one confined to Fock subspace ranging from $leftvert Mrightrangle $ to $leftvert Nrightrangle $, whatever $M<N$. Whereas the upper-bounded JC or AJC interactions is shown to drive any initial state to a steady Fock state $leftvert Nrightrangle $, the sliced one is shown to produce steady superpositions of Fock states confined to the sliced subspace $left{ leftvert Nrightrangle text{,}leftvert N+1rightrangle right} $.
I defend the extremist position that the fundamental ontology of the world consists of a vector in Hilbert space evolving according to the Schrodinger equation. The laws of physics are determined solely by the energy eigenspectrum of the Hamiltonian. The structure of our observed world, including space and fields living within it, should arise as a higher-level emergent description. I sketch how this might come about, although much work remains to be done.
In quantum mechanics, physical states are represented by rays in Hilbert space $mathscr H$, which is a vector space imbued by an inner product $langle,|,rangle$, whose physical meaning arises as the overlap $langlephi|psirangle$ for $|psirangle$ a pure state (description of preparation) and $langlephi|$ a projective measurement. However, current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement. We establish a theoretical framework for such a changing inner product, which we show is consistent with standard quantum mechanics. Furthermore, we show that this change is described by a quantum channel, which is tomographically observable, and we elucidate how our result is strongly related to the exploding topic of PT-symmetric quantum mechanics. We explain how to realize experimentally a changing inner product for a qubit in terms of a qutrit protocol with a unitary channel.
Given a set of correlations originating from measurements on a quantum state of unknown Hilbert space dimension, what is the minimal dimension d necessary to describes such correlations? We introduce the concept of dimension witness to put lower bounds on d. This work represents a first step in a broader research program aiming to characterize Hilbert space dimension in various contexts related to fundamental questions and Quantum Information applications.
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