Asymptotic enumeration of sparse nonnegative integer matrices with specified row and column sums

Let svec = (s_1,...,s_m) and tvec = (t_1,...,t_n) be vectors of nonnegative integer-valued functions of m,n with equal sum S = sum_{i=1}^m s_i = sum_{j=1}^n t_j. Let M(svec,tvec) be the number of m*n matrices with nonnegative integer entries such that the i-th row has row sum s_i and the j-th column has column sum t_j for all i,j. Such matrices occur in many different settings, an important example being the contingency tables (also called frequency tables) important in statistics. Define s=max_i s_i and t=max_j t_j. Previous work has established the asymptotic value of M(svec,tvec) as m,ntoinfty with s and t bounded (various authors independently, 1971-1974), and when svec,tvec are constant vectors with m/n,n/m,s/n >= c/log n for sufficiently large (Canfield and McKay, 2007). In this paper we extend the sparse range to the case st=o(S^(2/3)). The proof in part follows a previous asymptotic enumeration of 0-1 matrices under the same conditions (Greenhill, McKay and Wang, 2006). We also generalise the enumeration to matrices over any subset of the nonnegative integers that includes 0 and 1.

Cooling in strongly correlated optical lattices: prospects and challenges

Optical lattices have emerged as ideal simulators for Hubbard models of strongly correlated materials, such as the high-temperature superconducting cuprates. In optical lattice experiments, microscopic parameters such as the interaction strength between particles are well known and easily tunable. Unfortunately, this benefit of using optical lattices to study Hubbard models come with one clear disadvantage: the energy scales in atomic systems are typically nanoKelvin compared with Kelvin in solids, with a correspondingly miniscule temperature scale required to observe exotic phases such as d-wave superconductivity. The ultra-low temperatures necessary to reach the regime in which optical lattice simulation can have an impact-the domain in which our theoretical understanding fails-have been a barrier to progress in this field. To move forward, a concerted effort to develop new techniques for cooling and, by extension, techniques to measure even lower temperatures. This article will be devoted to discussing the concepts of cooling and thermometry, fundamental sources of heat in optical lattice experiments, and a review of proposed and implemented thermometry and cooling techniques.

Thermometry with spin-dependent lattices

We propose a method for measuring the temperature of strongly correlated phases of ultracold atom gases confined in spin-dependent optical lattices. In this technique, a small number of impurity atoms--trapped in a state that does not experience the lattice potential--are in thermal contact with atoms bound to the lattice. The impurity serves as a thermometer for the system because its temperature can be straightforwardly measured using time-of-flight expansion velocity. This technique may be useful for resolving many open questions regarding thermalization in these isolated systems. We discuss the theory behind this method and demonstrate proof-of-principle experiments, including the first realization of a 3D spin-dependent lattice in the strongly correlated regime.

Disordered insulator in an optical lattice

Disorder can profoundly affect the transport properties of a wide range of quantum materials. Presently, there is significant disagreement regarding the effect of disorder on transport in the disordered Bose-Hubbard (DBH) model, which is the paradigm used to theoretically study disorder in strongly correlated bosonic systems. We experimentally realize the DBH model by using optical speckle to introduce precisely known, controllable, and fine-grained disorder to an optical lattice5. Here, by measuring the dissipation strength for transport, we discover a disorder-induced SF-to-insulator (IN) transition in this system, but we find no evidence for an IN-to-SF transition. Emergence of the IN at disorder strengths several hundred times the tunnelling energy agrees with a predicted SF--Bose glass (BG) transition from recent quantum Monte Carlo (QMC) work. Both the SF--IN transition and correlated changes in the atomic quasimomentum distribution--which verify a simple model for the interplay of disorder and interactions in this system--are phenomena new to the unit filling regime explored in this work, compared with the high filling limit probed previously. We find that increasing disorder strength generically leads to greater dissipation in the regime of mixed SF and Mott-insulator (MI) phases, excluding predictions of a disorder-induced, or re-entrant, SF (RSF). While the absence of an RSF may be explained by the effect of finite temperature, we strongly constrain theories by measuring bounds on the entropy per particle in the disordered lattice.

Lattice Thermodynamics for Ultra-Cold Atoms

We measure the temperature of ultra-cold Rb-87 gases transferred into an optical lattice and compare to non-interacting thermodynamics for a combined lattice--parabolic potential. Absolute temperature is determined at low temperature by fitting quasimomentum distributions obtained using bandmapping, i.e., turning off the lattice potential slowly compared with the bandgap. We show that distributions obtained at high temperature employing this technique are not quasimomentum distributions through numerical simulations. To overcome this limitation, we extract temperature using the in-trap size of the gas.

Strongly interacting bosons in a disordered optical lattice

Disorder, prevalent in nature, is intimately involved in such spectacular effects as the fractional quantum Hall effect and vortex pinning in type-II superconductors. Understanding the role of disorder is therefore of fundamental interest to materials research and condensed matter physics. Universal behavior, such as Anderson localization, in disordered non-interacting systems is well understood. But, the effects of disorder combined with strong interactions remains an outstanding challenge to theory. Here, we experimentally probe a paradigm for disordered, strongly-correlated bosonic systems-the disordered Bose-Hubbard (DBH) model-using a Bose-Einstein condensate (BEC) of ultra-cold atoms trapped in a completely characterized disordered optical lattice. We determine that disorder suppresses condensate fraction for superfluid (SF) or coexisting SF and Mott insulator (MI) phases by independently varying the disorder strength and the ratio of tunneling to interaction energy. In the future, these results can constrain theories of the DBH model and be extended to study disorder for strongly-correlated fermionic particles.

Phase-slip induced dissipation in an atomic Bose-Hubbard system

Phase slips play a primary role in dissipation across a wide spectrum of bosonic systems, from determining the critical velocity of superfluid helium to generating resistance in thin superconducting wires. This subject has also inspired much technological interest, largely motivated by applications involving nanoscale superconducting circuit elements, e.g., standards based on quantum phase-slip junctions. While phase slips caused by thermal fluctuations at high temperatures are well understood, controversy remains over the role of phase slips in small-scale superconductors. In solids, problems such as uncontrolled noise sources and disorder complicate the study and application of phase slips. Here we show that phase slips can lead to dissipation for a clean and well-characterized Bose-Hubbard (BH) system by experimentally studying transport using ultra-cold atoms trapped in an optical lattice. In contrast to previous work, we explore a low velocity regime described by the 3D BH model which is not affected by instabilities, and we measure the effect of temperature on the dissipation strength. We show that the damping rate of atomic motion-the analogue of electrical resistance in a solid-in the confining parabolic potential fits well to a model that includes finite damping at zero temperature. The low-temperature behaviour is consistent with the theory of quantum tunnelling of phase slips, while at higher temperatures a cross-over consistent with the transition to thermal activation of phase slips is evident. Motion-induced features reminiscent of vortices and vortex rings associated with phase slips are also observed in time-of-flight imaging.

The asymptotic volume of the Birkhoff polytope

Let m,n be positive integers. Define T(m,n) to be the transportation polytope consisting of the m x n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case B(n)=T(n,n) is the much-studied Birkhoff-von Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determine the asymptotic volume of T(m,n) as n goes to infinity, with m=m(n) such that m/n neither decreases nor increases too quickly. In particular, we give an asymptotic formula for the volume of B(n).