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Quantum Many-Body Physics of Ultracold Molecules in Optical Lattices

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

According to the postulates of quantum mechanics, a quantum system is completely specified by the state \(\vert \psi \left (t\right )\rangle\) whose evolution is provided by the Schrödinger equation

$$\begin{array}{rlrlrl} i\hslash \frac{\partial } {\partial t}\vert \psi \left (t\right )\rangle & =\hat{ H}\vert \psi \left (t\right )\rangle. &\end{array}$$
(1.1)

Here, \(\hat{H}\) is the Hamiltonian describing the interactions of all of the microscopic degrees of freedom in the system under study. Unfortunately, the dimension of the Hilbert space in which the state \(\vert \psi \left (t\right )\rangle\) lives grows exponentially with the number of constituents in a many-body system, rendering Eq. (1.1) essentially useless for extracting physically relevant information from systems with more than a few particles. Practical concerns aside, there is a more fundamental reason why Eq. (1.1) does not enable us to answer all relevant questions in many-body physics. This reason is put succinctly by P.W. Anderson in his now famous article “More is different” [1] when he says that “The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe.” That is to say, many-body systems can display very different, emergent, behavior from their microscopic constituents. In particular, the ground state of a many-body system need not have the same symmetry as its governing Hamiltonian due to the phenomenon of spontaneous symmetry breaking.

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Notes

  1. 1.

    The precise sense in which we mean the system is weakly interacting is that it is not strongly correlated, the latter of which will be defined below. In particular, by weakly interacting we do not mean that the interactions lie within the radius of convergence of a perturbation series.

  2. 2.

    In the absence of exponential decay, the correlation length may be taken to be a length scale on which correlations qualitatively shift to a long-distance behavior.

  3. 3.

    By ultracold, we mean temperatures less than 1 μK.

  4. 4.

    This is what differentiates a Feshbach resonance from a shape resonance. In the latter no bound state exists in the absence of the coupling.

  5. 5.

    Note that the effective range given in the context of scattering of slow particles, r ⋆, is related to the effective range defined here as \(r_{\star } = -2r_{B}\).

  6. 6.

    The spin states of real atoms are never purely singlet or triplet, but rather singlet-dominated or triplet-dominated.

  7. 7.

    An overview of molecular degrees of freedom is provided in Sect. 1.3.

  8. 8.

    The dynamic polarizability of LiCs has not yet been calculated, to our knowledge. We estimate that the ratio of the perpendicular and parallel polarizabilities will be similar to that of KRb and RbCs.

  9. 9.

    Explicitly, it would imply that a Quantum Merlin–Arthur (QMA)-complete problem lies in P, where P is the class of problems which can be solved in polynomial time by a deterministic Turing machine. QMA is the quantum analog to the classical complexity class NP.

  10. 10.

    Series expansions often employ a diagrammatic notation for the terms in the series, hence the name diagrammatic Monte Carlo.

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  1. Colorado School of Mines, Golden, CO, USA

    Michael L. Wall

  2. JILA, NIST, and University of Colorado, Boulder, CO, USA

    Michael L. Wall

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Wall, M.L. (2015). General Introduction. In: Quantum Many-Body Physics of Ultracold Molecules in Optical Lattices. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-14252-4_1

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