We propose to use quantum tomography to characterize the state of a perturbed Bose-Einstein condensate. We assume knowledge of the number of particles in the zero-wave-number mode and of density distributions in space at different times, and we treat the condensate in the Bogoliubov approximation. For states that can be treated with the Gross-Pitaevskii equation, we find that the reconstructed density operator gives excellent predictions of the second moments of the atomic creation and annihilation operators, including the one-body density matrix. Additional inclusion of the momentum distribution at one point of time enables somewhat reliable predictions to be made for the second moments for mixed states, making it possible to distinguish between coherent and thermal perturbations of the condensate. Finally, we find that with the observation of the zero-wave-number mode anomalous second moment $⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{0}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{0}+{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{0}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{0}^{\ifmmode\dagger\else\textdagger\fi{}}⟩$, the reconstructed density operator gives reliable predictions of the second moments of locally amplitude squeezed states.
We propose to use quantum tomography to characterize the state of a perturbed Bose-Einstein condensate. We assume knowledge of the number of particles in the zero-wave-number mode and of density distributions in space at different times, and we treat the condensate in the Bogoliubov approximation. For states that can be treated with the Gross-Pitaevskii equation, we find that the reconstructed density operator gives excellent predictions of the second moments of the atomic creation and annihilation operators, including the one-body density matrix. Additional inclusion of the momentum distribution at one point of time enables somewhat reliable predictions to be made for the second moments for mixed states, making it possible to distinguish between coherent and thermal perturbations of the condensate. Finally, we find that with the observation of the zero-wave-number mode anomalous second moment ⟨a\ifmmode^\elseˆ\fi0a\ifmmode^\elseˆ\fi0+a\ifmmode^\elseˆ\fi0\ifmmode†\else\textdagger\fia\ifmmode^\elseˆ\fi0\ifmmode†\else\textdagger\fi⟩, the reconstructed density operator gives reliable predictions of the second moments of locally amplitude squeezed states.
Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as N−1/2 in the limit of large N.
The impact of interparticle correlations on the behavior of Bose-Einstein Condensates (BECs) is discussed using two approaches. In the first approach, the wavefunction of a BEC is encoded in the N-particle sector of an extended "catalytic state". Going to a time-dependent interaction picture, we can organize the effective Hamiltonian by powers of N−1/2. Requiring the terms of order N1/2 to vanish, we get the Gross-Pitaevskii Equation. Going to the next order, N0, we obtain the number-conserving Bogoliubov approximation. Our approach allows one to stay in the Schrödinger picture and to apply many techniques from quantum optics. Moreover, it is easier to track different orders in the Hamiltonian and to generalize to the multi-component case. In the second approach, I consider a state of N=l×n bosons that is derived by symmetrizing the n-fold tensor product of an arbitrary l-boson state. Particularly, we are interested in the pure state case for l=2, which we call the Pair-Correlated State (PCS). I show that PCS reproduces the number-conserving Bogoliubov approximation; moreover, it also works in the strong interaction regime where the Bogoliubov approximation fails. For the two-site Bose-Hubbard model, I find numerically that the error (measured by the trace distance of the two-particle reduced density matrices) of PCS is less than two percent over the entire parameter space, thus making PCS a bridge between the superfluid and Mott insulating phases. Amazingly, the error of PCS does not increase, in the time-dependent case, as the system evolves for longer times. I derive both time-dependent and -independent equations for the ground state and the time evolution of the PCS ansatz. The time complexity of simulating PCS does not depend on N and is linear in the number of orbitals in use.
We study higher moments of quantum fluctuations and microscopic correlation energy in the three-dimensional Gross--Pitaevskii regime. For a smooth, compactly supported, nonnegative radial interaction and an N-independent normalized initial condensate wave function in H6, we propagate the third spectral moment of a positive correlated energy along the exact many-body evolution. A uniform initial bound controls the third moment of the depletion number and sixth moments of collective one-particle fluctuations on every compact time interval, independently of N. For a nonzero interaction and a smooth, compactly supported initial condensate, we also identify the excitation-energy measure at scale N2, weighted by the first energy. This limit holds for norm-convergent pure-state families with a uniform bound on the initial third correlated-energy moment. Its positive-energy part is a spectral measure of the relative scattering operator −2Δ+v, multiplied by a time-dependent dressed contact. We compute the radial density and prove strict positivity of the contact for nonzero references with finite kinetic energy and finite mean excitation number. The same evolving states have bounded physical fluctuation moments and excitation-energy moments of order p>1 bounded below by a positive multiple of N2p−2, in the extended spectral sense. Inverse powers of the positive initial energy construct admissible pure and mixed families. For the norm-convergent pure-state class, we determine the scaled interaction-energy moments for 1≤p<2 and bound the second moment by O(N2). A finite cubic approximation has norm error O(N−3/16) on the energy form domain and O(N−5/4) on a stronger initial domain of its generator.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number N of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator A for the total number of particles and the sum of a ``condensate wave function'' $\ensuremath{\xi}(\mathbf{x})$ and a phonon field operator $\ensuremath{\chi}(\mathbf{x})$ in the form $\ensuremath{\psi}(\mathbf{x})\ensuremath{\approx}A{\ensuremath{\xi}(\mathbf{x})+\ensuremath{\chi}(\mathbf{x})/\sqrt{N}}$ when the field operator acts on the N particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of N, and thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different N. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if $\ensuremath{\xi}(\mathbf{x})$ is a solution of the time-independent Gross-Pitaevskii equation for N particles and any chemical potential $\ensuremath{\mu}$, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed N is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of N for the equations of motion. In this expansion the condensate wave function has the time-dependent form $\ensuremath{\xi}(\mathbf{x},t)$, and the condition for the validity of the expansion is that $\ensuremath{\xi}(\mathbf{x},t)$ satisfies the time-dependent Gross-Pitaevskii equation $\ensuremath{\partial}\ensuremath{\xi}/\ensuremath{\partial}t=\ensuremath{-}({\ensuremath{\Elzxh}}^{2}/2m){\ensuremath{\nabla}}^{2}\ensuremath{\xi}+V\ensuremath{\xi}+Nu|\ensuremath{\xi}{|}^{2}\ensuremath{\xi}$. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as $\ensuremath{\chi}(\mathbf{x},t)={\ensuremath{\sum}}_{k}{\ensuremath{\xi}}_{k}(\mathbf{x},t){\ensuremath{\alpha}}_{k},$ where the operators ${\ensuremath{\alpha}}_{k}$ are time-independent annihilation operators, and the state vector has a time evolution described by a Schr\"odinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function $\ensuremath{\xi}(\mathbf{x},t)$.