Older projects

Quantum coherent control

Spin evolution governed by second-order self-refocusing pulse Q₁ [29].

We are working on physics and real-life applications of pulse-based control methods known as dynamical recoupling (also, "bang-bang", in the case of hard, δ-function-like pulses). In the simplest setup, the system is a collection of individually-controlled weakly-coupled parts (e.g., qubits). Individual qubits undergo a rapid forced precession, while the overall long-time evolution of the system is governed by the effective Hamiltonian averaged over the precession of the qubits. For example, the interaction Jσ₁^zσ₂^z between the two qubits is cancelled on average if one of them is rapidly precessing around the x-axis. Such cancellation of the quantum evolution is called "dynamical decoupling" or "refocusing" it is related to the spin-echo experiment.

Average refocusing error for a single spin in the presence of the classical magnetic field fluctuating along all three axes for the K=2 symmetric 8-pulse refocusing sequence "8p" using pulses Q₁. Symbols represent data averaged over 900 realizations of random field, dashes are the linear fits for data further than Δt=3τ₀ from the ends of the interval. The decoherence rate (proportional to the slopes B) is reduced dramatically for the correlation time τ₀ exceeding the duration of the refocusing sequence τ. The corresponding initial decoherence is proportional to the intercepts A. For this symmetric sequence initial decoherence scales down quadratically with increased noise correlation time τ₀. This results in an excellent refocusing accuracy already for τ₀=2τ. [31]

A systematic way to approach the design problem is to consider a cumulant (Magnus) expansion of the evolution operator, treating the strong control fields exactly. The cumulants give the expansion of the effective Hamiltonian in powers of the system Hamiltonian. Then, pulse sequences can be classified by their order K, defined as the number of terms in the effective Hamiltonian which give exactly the desired unitary evolution. For dynamical decoupling, or refocusing, K is the number of terms which exactly equal to zero. The locality of the cumulant expansion ensures that the classification by sequence order remains meaningful even for large systems. The corresponding calculation can be done efficiently by constructing a time-dependent perturbation theory expansion on small clusters [29]. The computational scheme was illustrated by designing 2nd-order self-refocusing π-pulses (please see illustration for Q₁ pulse on the left) and a 6th-order 8-pulse refocusing sequence for a chain of qubits with nearest-neighbor Ising couplings. The same sequence produces K=2 refocusing for random xxz spin chain with weak frequency mismathces at each qubit.

It is intuitively evident that refocusing should also remain effective in the presence of low-frequency environment, as long as the parameters of the system Hamiltonian are varying slowly compared to the refocusing period τ. We studied this effect systematically for sequences of order K≤2 by constructing the Floquet expansion of the non-Markovian master equation for an open multi-qubit system in the presence of continuous refocusing fields [31].

The main results can be summarized as follows. With order-one refocusing, K=1, the reduction of the dephasing rate Γ is due to effective shortening of the correlation time τ₀ of the thermal bath, which gives Γ⁽¹⁾ reduced by a factor of (τ/τ₀). With order-two refocusing, K=2, the second-order classical contribution to decoherence rate vanishes in the limit of static fluctuations. Then, the dephasing rate should be determined by either the commutators of the bath couplings, or by their time derivatives. We demonstrated that in the latter case all terms in the derivative expansion vanish, which results in a very rapid suppression of the second-order contribution to dephasing rate Γ⁽²⁾ with decreasing τ/τ₀. Additionally, the analysis of the corresponding prefactor reveals that even with very small dephasing rate the refocusing coherence may not be perfect because of the transient effects. The corresponding initial decoherence generally scales as the square of the refocusing period τ. However, additional cancellations characteristic of symmetric refocusing sequences cause a reduction of the initial decoherence by an additional factor of (τ/τ₀)².

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Plots of d-wave superconducting susceptibility as a function of temperature for t-J-V model in the physical regime (left) and the unphysical regime (right). Note the scale difference on the two plots [26].

High-temperature superconductivity

We are trying to understand the high-T_c superconductivity using the high-temperature series. In the recent paper [26], the high-temperature series for thermodynamical susteptibilities for a number of order parameters were computed, particularly for d-wave superconducting order. While with J>t, the susceptibility is large and increasing with decreasing temperature down to the smallest accessible temperature T=J/2, for J<t the susceptibility is much smaller and it is further decreasing below certain temperature scale. This implies that the canonical form of the t-J model does not contain high-temperature superconductivity.

We are currently looking at related models with spatial modulation of parameters in order to find a model in the same class where the high-temperature superconductivity would actually be present, and also to establish a bridge to observed stripe phases.

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Phase diagram of the extended Bose Hubbard model with on-site energy U=20 and hopping t=1 on the square lattice. V is the nearest-neighbor repulsion energy, and ρ is the average particle density. SF is superfluid, SS is supersolid, PS is phase-separated, MI is Mott insulator (a boson at each site), CDW I and CDW II respectively are checkerboard crystals with 1 and 2 particles at every other site [28].

Cold atoms in optical lattices

Experimentalists now have the ability to load different species of atoms (bosonic, fermionic, or both) in optical lattices and indepentently control interactions and the hopping matrix elements for different species. [See, e.g., the paper M. Greiner et al., Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature 415, 39 (2002).] We are studying different novel phases which may occur in such systems.

In the paper [28], a combination of a numerical quantum Monte Carlo method and analytical strong-coupling expansion technique were used to prove the existence of a "supersolid" phase, a combination of a crystal-like ordering and a superfluid Bose condensate in a model which describes strongly-interacting bosons on a lattice. The results of a number of previous studies were invalid because the phase is thermodynamically unstable (system phase separates) if the on-site repulsion between particles becomes too strong.

Phase diagram of the extended Bose-Fermi Hubbard model for two values of the boson on-site energy U boson-fermion repulsion V and equal hoppings t_B=t_F=1 in one dimension. μ_B and μ_F are the chemical potentials for bosons and fermions respectively. F-F, B-B, and B-F mark pure-fermion, pure-boson, and boson-fermion mixtures. Boxes connected by dotted lines mark numerical results for chains of length L≤128, while solid lines are the analytical results for the phase boundaries. Dashed lines separate the regions of commensurate density where the system has charge gap. [32].

In the paper [32] we combined a quantum Monte Carlo method with model mapping and exact spectral bounds to study the ground state phases of a mixture of ultracold bosons and spin-polarized fermions in a one-dimensional optical lattice. We rigorously proved the existence of a quantum boson-fermion mixture along a certain symmetry plane in the parameter space of the Hamiltonian. The stability boundaries of this mixture are also established analytically, in a spectacular agreement with numerics. The main feature of such a mixture is an absence of a gap towards boson to fermion conversion, even in the phase with the charge gap, where the total density is n_F+n_B=1. We rigorously proved the existence of this phase and studied the equal-time density correlations. Other properties of this phase were studied in detail numerically by L. Pollet, M. Troyer, K. Van Houcke, and S. M. A. Rombouts in a paper "Phase diagram of Bose-Fermi mixtures in one-dimensional optical lattices".

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Transport in disordered correlated systems

Physics of many-body systems with interacting particles becomes even more complicated in the presence of disorder. We are trying to understand how disorder affects transport properties (e.g., the flow of an electrical current) in such systems.

In the paper [25], the low-frequency conductivity was calculated for a very low-density strongly-correlated liquid, a temperature-melted Wigner crystal of electrons in the lowest Landau level. Surprisingly, in the regime where Coulomb interaction is much larger than the temperature, which in turn is much larger than the amplitude of short-range disorder, the low-frequency conductivity of this liquid can be computed very precisely.

The large-drive part of the effective mobility μ as a function of the driving force F for different values of the power n in the disorder correlation law at large distances, <V(x) V(y)> = V₀² ξ₁ⁿ | x−y|⁻ⁿ. V₀ is the rms disorder amplitude, T is the temperature in energy units, and ξ₁ is a distance scale [27].

In the paper [27] we considered a classical system of diffusive weakly-coupled particles in one dimension in the presence of very strong disorder. If the driving force is very large, the particles are constantly moving downhill, and their mobility μ=μ₀ is the same as in the absence of disorder. With very small driving force, particles have to move between the minima and the maxima of the disorder potential, and the effective mobility μ is exponentially suppressed. We identified a crossover between these two regimes. Depending on the nature of disorder correlations, the logarithm of the effective mobility is either linear in the applied driving force F, or is proportional to a power of F.

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For other projects please see Leonid's annotated publication list or the full publication list. New publications will appear at the arXiv.