Research

“The behavior of things in the small scale is so fantastic! It is so wonderfully different! So marvelously different than anything that behaves on a large scale.” – Feynman

Here you will find descriptions of my current and past research, illustrated with images, slides, posters, and animations. To skip directly to publications, see below here or go to arXiv.

Position

I am a postdoctoral research associate in the Theory of Condensed Matter group at Cavendish Laboratory, University of Cambridge, working with Prof. Nigel Cooper and collaborators (ORCID). Previously, I worked as a graduate student on the theory of cold atoms with Prof. Erich Mueller at Cornell University, having explored other areas as an undergrad (see below).

Overview

Technical

My research is focused on understanding the physics of interacting quantum gases by theoretical modeling and numerical simulation. This is in part motivated by a remarkable growth of versatile experimental platforms for trapping and manipulating systems of atoms and photons (see this review). I have studied exotic phases of matter, collective excitations, and quantum dynamics of such systems. My recent work has explored novel out-of-equilibrium properties arising in the presence of dissipation and/or periodic driving, and how they can be probed in experiments. I am also actively involved in developing widely applicable numerical techniques, such as DMRG for continuum systems. The broad themes in my research are the following:

  • Exotic quantum phases
  • Collective excitations
  • Non-equilibrium dynamics
  • Open quantum systems
In plain English

I play with the strange mathematical rules of quantum mechanics to explore the wonderful small-scale behavior in Feynman’s quote above. Although everything is made up of these tiny objects or “atoms,” their quantum nature is often blurred by thermal fluctuations (random jiggling motion). As matter is cooled down, these fluctuations die out and one enters the quantum realm in full glory. Thanks to experimental advances, it is now possible to “cook” such scenarios in the lab and “look” at how individual particles behave. Understanding this quantum world is central to modern electronics and will play a key role in future technologies. I use a blend of analytical and numerical approaches to explore the rich features which emerge in these systems. For example, below are two articles written for a general audience: the first on ripples on a superfluid (a novel phase of matter), and the latter on “fractional” particles on a thin film of light!

Publications

  1. Shovan Dutta and Nigel R. Cooper, “Long-range coherence and multiple steady states in a lossy qubit array,” arXiv:2004.07981 (2020) [pdf].
Click here to read more!

Environmental coupling typically drives a quantum system to a unique steady state with very little coherence, which is a major obstable for quantum control and information processing. Here we identify a simple experimental setting of two-level systems with local pump and loss that can stabilize multiple highly coherent steady states. These states arise from a hidden symmetry that conserves entangled particle-hole pairs over arbitrarily long distances, giving rise to controllable long-range entanglement. We show how to selectively prepare and observe the steady states in existing setups. Our conclusions apply to a broad range of two-level quantum systems.

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  1. Shovan Dutta and Nigel R. Cooper, “Critical Response of a Quantum van der Pol Oscillator,” Phys. Rev. Lett. 123, 250401 (2019) [pdf] [supplement] [arXiv] [slides] [poster].
Click here to read more!

Self-sustained oscillations are ubiquitous in nature, from heartbeats to electrical circuits. Physical systems on the verge of such oscillations are incredibly sensitive to external drives and play a key role in biological sensors. For example, this is how we can hear faint whispers (see the power of hearing). In this work, we find this heightened sensitivity persists even in the microscopic world of atoms and photons, where the disparate rules of quantum mechanics dominate. An actively driven quantum oscillator not only shows enhanced sensitivity, but also exhibits surprising new features that are unique to the quantum realm. These findings could have important applications in quantum sensing and shed new insights into the out-of-equilibrium dynamics of quantum systems.

We model a prototypical scenario where a quantum oscillator is subject to energy dissipation. Ordinarily, the response of such oscillators is limited by damping. However, we find this limit can be negated via a strong “pump,” leading to enhanced sensitivity. Additionally, the study reveals novel features such as a regime where the response diminishes with increasing drive. These predictions can be tested in a variety of setups, e.g., atoms trapped in a well, light in a cavity, or a micromechanical membrane. Such platforms have already been used to build quantum sensors that are transforming the landscape of precision metrology and medical imaging (see https://www.bbc.co.uk/news/business-47294704). Active sensing could promote further advances, yielding a “quantum ear” that can detect ultraweak signals.

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  1. Shovan Dutta and Erich J. Mueller, “Coherent generation of photonic fractional quantum Hall states in a cavity and the search for anyonic quasiparticles,” Phys. Rev. A 97, 033825 (2018) [pdf] [supplement] [arXiv] [news story] [slides1] [slides2] [poster] [animation].
Click here to read more!

Wave-particle duality is a key principle of quantum mechanics. Central to modern electronics, it explains that matter has dual nature: “particles” like electrons can act as waves and “waves” like light can act as particles. Following new lines of research which build on this concept, we model an experimental setting where the particles of light (photons) can mimic a very special behavior of electrons in semiconductors. In particular, we show how one can use light in a cavity to create and manipulate exotic quantum excitations, known as anyons, which could form the hardware for future quantum computers. This protocol might enable the first direct probe of these exotic entities which have remained elusive.

In the protocol, the anyons are formed in the waist of an optical cavity built by carefully aligning a set of high-quality mirrors. Such a setup already exists in Jon Simon’s lab in Chicago. We show how one can drive the cavity with lasers to sequentially inject photons, building up a quantum state which has vortex-like excitations with unusual properties. These excitations are the desired anyons: they act like particles, but due to their collective nature behave unlike any known elementary particle. In particular, when two of them are exchanged, the quantum-mechanical wavefunction gains a fractional phase. We explain how one use laser beams to create and move these anyons, and measure the fractional phase. The proposed experiment uses existing technology and can be considered the simplest “braiding” protocol, which forms the basis of topological quantum computing schemes.

anyons-in-cavity-summary

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  1. Shovan Dutta and Erich J. Mueller, “Protocol to engineer Fulde-Ferrell-Larkin-Ovchinnikov states in a cold Fermi gas,” Phys. Rev. A 96, 023612 (2017) [pdf] [arXiv].
Click here to read more!

Following up on our result in the previous paper (see below), here we propose a two-step experimental protocol to directly engineer FFLO states in a cold Fermi gas loaded into a quasi-1D trap. First, one uses phase imprinting to generate a series of domain walls in a superfluid with equal number of ↑- and ↓-spins. Second, one applies a controlled radio-frequency sweep which selectively breaks Cooper pairs near the domain walls and transfers the ↑-spins to a third noninteracting spin state, leaving behind a stable FFLO state with one unpaired ↓-spin in each domain wall. We show how the protocol can be implemented with high fidelity for a wide range of parameters available in experiments.

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  1. Shovan Dutta and Erich J. Mueller, “Collective Modes of a Soliton Train in a Fermi Superfluid,” Phys. Rev. Lett. 118, 260402 (2017) [pdf] [arXiv] [news story] [slides] [poster].
Click here to read more!

We study the collective motion of a train of domain walls or solitons in a quasi-1D Fermi superfluid by analyzing the Bogoliubov-de Gennes equations. We uncover a variety of unexpected modes, including long-lived gapped modes describing oscillations of the soliton cores and an instability where pairs of solitons collide and annihilate. The instability rate is sensitive to the separation of solitons and the interaction between atoms, both of which can be tuned in experiments. In addition, the instability is prevented by magnetizing the gas – forming an exotic FFLO state which has eluded direct experimental detection despite much effort over decades. We discuss how such stable FFLO states can be directly engineered in cold Fermi gases.

soliton-train-dynamics-summary

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  1. Shovan Dutta and Erich J. Mueller, “Dimensional crossover in a spin-imbalanced Fermi gas,” Phys. Rev. A 94, 063627 (2016) [pdf] [arXiv] [slides].
Click here to read more!

We study the relative stability of exotic superfluid states, such as the FFLO and breached-pair states, in a spin-imbalanced Fermi gas confined in a cylindrical harmonic trap. We calculate the mean-field phase diagram in the density–imbalance plane as a function of the confinement, strength of interactions between atoms, and temperature. The phase diagram changes from 1D-like to 3D-like as one increases the interactions or reduces the confinement. We map the system to an effective 1D model, finding significant density dependence of the 1D scattering length. We discuss the prospects of observing the superfluid states in similar ongoing experiments.

dimensional-crossover-summary

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  1. Shovan Dutta and Erich J. Mueller, “Kinetics of Bose-Einstein condensation in a dimple potential,” Phys. Rev. A 91, 013601 (2015) [pdf] [arXiv] [slides].
Click here to read more!

We model the dynamics of condensate formation in a bimodal optical trap, consisting of a large reservoir region and a tight, tunable “dimple” potential at the center. We simulate the quantum Boltzmann rate equations with two-body scattering and three-body loss processes to provide detailed quantitative estimates for condensate yields, lifetimes, thermalization timescales, and temperature variations. We study the dependence of these quantities with the trap parameters, explaining the principal trends in physical terms and extracting optimal parameters for future experiments.

condensation-kinetics-summary

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  1. Shovan Dutta and Erich J. Mueller, “Variational study of polarons and bipolarons in a one-dimensional Bose lattice gas in both the superfluid and the Mott-insulator regimes,” Phys. Rev. A 88, 053601 (2013) [pdf] [arXiv] [slides].
Click here to read more!

We model the spread of an impurity in a 1D Bose gas across the Superfluid-Mott transition through a computationally tractable variational ansatz. When the interactions between atoms are strong, we find the impurity binds with a hole, forming a quasiparticle known as a polaron. We characterize the dynamics and stability of this polaron and explain the features observed in a recent experiment. We find that two polarons can bind together to form a bipolaron. At weaker interactions, the polaron becomes unstable over a growing range of momentum and decays into particle-hole excitations. We predict how this instability can be observed in experiments by measuring the impurity-hole correlation.

impurity-dynamics-summary

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E-prints

  1. Shovan Dutta, “Collective Phenomena in Quantum Gases,” PhD thesis (2018).
Click here to read more!

Motivated by rapid experimental progress in the fields of ultracold atoms and quantum optics, I present a series of theoretical studies which explore collective phenomena in quantum gases of atoms and photons. In Chapter 1, I highlight the major developments in the research field and identify the overarching themes and motivations. I also provide a roadmap for the rest of the thesis and summarize the main results. The remaining eight chapters contain original studies, organized along three broad motifs. In Chapters 2 through 5, I investigate how the nature of collective excitations and quasiparticles can be explored in modern experiments. More specifically, I model the dynamics of a spin impurity in a Bose lattice gas, develop a protocol for observing fractionalized excitations or anyons in an optical cavity, and characterize the collective dynamics of Bogoliubov quasiparticles and domain walls in a Fermi superfluid. In Chapters 6 and 7, I examine unconventional superfluid phases in spin-imbalanced Fermi gases. In particular, I propose a novel technique for engineering the long-sought-after Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase and study the relative stability of exotic phases across a dimensional crossover. Finally, Chapters 8 and 9 are devoted to studies of kinetics in out-of-equilibrium systems. I model the formation of a Bose-Einstein condensate in a dimple trap and characterize the approach to thermal equilibrium in quasi-one-dimensional geometries.

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  1. Shovan Dutta and Subhankar Ray, “Damped bead on a rotating circular hoop – a bifurcation zoo,” arXiv:1201.1218 (2012) [slides].
Click here to read more!

We investigate the evergreen problem of bead on a rotating hoop, but with damping. The introduction of damping alters the nature of the fixed points, giving rise to a multitude of new bifurcations. We show phase portraits and trajectories corresponding to different motions of the bead, characterizing its dynamics over the full parameter space. For certain critical values of the damping coefficient and rotation speed, linear stability analysis is insufficient to classify the nature of the fixed points. We present a rigorous technique involving transformation of coordinates and order of magnitude arguments to resolve such cases, which might provide a general framework to treat such borderline cases in more complex nonlinear systems.

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  1. Shovan Dutta and Subhankar Ray, “Bead on a rotating circular hoop: a simple yet feature-rich dynamical system,” arXiv:1112.4697 (2011).
Click here to read more!

We perform an in-depth analysis of the nonlinear dynamics of an undamped bead on a rotating hoop using elementary calculus and symmetry arguments. We characterize the different types of motion the bead can undergo and show simulations of its beautiful trajectories. At a critical rotation speed, the system undergoes a pitchfork bifurcation where two new equilibrium points emerge on either side of the hoop. We find a dramatic change in the relation between time period and amplitude of bead oscillations as the rotation speed is varied. The study would be particularly useful to students as it illustrates such concepts as phase portraits, bifurcations, symmetry breaking, critical slowing down, and the use of Lagrange multipliers to determine constraint forces.

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  1. Shovan Dutta, Subhankar Ray, and J. Shamanna, “Continuous Time Random Walk with time-dependent jump probability: a direct probabilistic approach,” arXiv:1112.3253 (2011).
Click here to read more!

We tackle the problem of a continuous-time random walk in 3D under time-varying external fields: the random walker, on arriving at position ρ at time τ, stays there for a time t´, after which it jumps to a new position at time t = τ + t´. The waiting time t´ is distributed with a probability density function ψ(t´). The probability that a jump occurring at time t results in a displacement between and + d is φ(|t) d3r´. The goal is to find the probability density p(r,t) that the random walker is at position r at time t. Continuous-time random walks are good models of anomalous diffusion.

We use direct probability arguments to derive recurrence relations for all moments of p(r,t) for arbitrary choices of ψ(t´) and φ(|t). For a memoryless walk, where ψ(t´) is exponential, we simplify these equations further to find a closed form expression for p(r,t). We also consider the special case of a 1D lattice with nearest-neighbor jumps, which was modeled in prior work by a Fractional Fokker-Planck Equation (FFPE). Our equations reproduce the mean and standard deviation in the FFPE formulation but has additional terms for the higher moments which can markedly alter the asymmetry (skewness) and peakedness (kurtosis) of p(r,t). We show that the missing terms are an artifact of the approximation in taking the continuum limit to derive the FFPE.

  1. Shovan Dutta, “A simple circuit model showing feature-rich Bogdanov-Takens bifurcation,” IEEE link [best paper in National Students Paper and Circuit Design Contest (NSPCDC) 2011].
Click here to read more!

I propose an easy-to-implement circuit model for the Bogdanov-Takens bifurcation, exhibiting three local (spiral-to-node, saddle-node, Andronov-Hopf) and one global (Homoclinic) bifurcations. The bifurcations have a profound effect on the system stability. For example, in the Homoclinic bifurcation, a stable limit cycle collides with a saddle and disappears. Thus the physical variables such as currents and voltages executing sustained oscillations suddenly increase in an unbounded manner. Such dramatic changes are useful in describing voltage collapse in power systems, excitability of neurons, and several other phenomena. With the proposed circuit, one can experimentally measure the drastic changes in the dynamics simply by altering the values of some linear circuit elements.

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Work in progress

  1. Hard-core bosons with local dissipation [with Nigel Cooper].
Click here to read more!

With the recent experimental creation of a Mott insulator of photons (Nature 2019), locally engineered dissipation allow for the exciting prospect of novel out-of-equilibrium physics arising from an interplay between interactions and dissipation. We’re exploring this physics, as an extension of our recent work above.

  1. Dynamically induced phase transitions by resonant shaking [with Ulrich Schneider and Nigel Cooper].
Click here to read more!

We have all seen wine glasses being shattered by a high pitch! The resonant sound excites increasingly strong vibrations in the glass until the glass order is broken. Here we study a scenario where one can resonantly shake an insulator and turn it into a superfluid.

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  1. Matrix product states (DMRG) for continuous 1D systems [with Erich Mueller (Cornell), Anton Buyskikh (Riverlane), and Andrew Daley (Strathclyde)].
Click here to read more!

What prohibits the exact simulation of large quantum systems is that the space of all possible states grows exponentially with the system size. For discrete 1D systems, the “Density Matrix Renormalization Group” (DMRG) technique provides a viable solution by letting one systematically truncate this space based on the degree of entanglement between subsystems. Indeed, DMRG cast in terms of Matrix Product States is the state of the art for simulating such systems. There has been attempts to generalize this technique to continuous systems, but they have severe limitations for non-homogeneous or time-dependent settings.

We’re developing a new algorithm which allows one to study continuum systems via the standard DMRG/TEBD techniques, without distorting the physics with an artificial lattice.

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Talks

  • 2019
    • “Critical Response of a Quantum van der Pol Oscillator,” Centre for Condensed Matter Theory (CCMT) Seminar, Indian Institute of Science (IISc), December 11, 2019.
    • “Critical Response of a Quantum van der Pol Oscillator,” Joint DesOEQ-QSUM meeting, University of Oxford, September 16, 2019.
    • “Critical Response of a Quantum van der Pol Oscillator,” Collective Phenomena Group Meeting (CPGM) talk, University of Cambridge, July 16, 2019.
    • “Critical Response of a Quantum van der Pol Oscillator,” Condensed Matter and Quantum Materials (CMQM) Conference, University of St Andrews, July 03, 2019.
    • “Paving an enlightened path to anyons and quantum computation,” Science Lunchtime Seminar, Darwin College, Cambridge, April 25, 2019.
    • “Creating and Braiding Anyons in an Optical Cavity,” Department of Physics Seminar, University of Strathclyde, January 2019.
  • 2018
    • “Creating and Braiding Anyons in an Optical Cavity,” Collective Phenomena Group Meeting (CPGM) talk, University of Cambridge, October 10, 2018.
    • “Creating and Braiding Anyons in an Optical Cavity,” Public talk (thesis defense), Department of Physics, Cornell University, April 27, 2018.
  • 2017
    • “Collective Dynamics of Solitons in Superfluids,” Research highlight for prospective graduate students, Department of Physics, Cornell University, March 27, 2017.

Posters

  • 2019
    • “Critical Response of a Quantum van der Pol Oscillator,” Bose-Einstein Condensation (BEC) 2019, Sant Feliu de Guixols, Spain, September 09, 2019.
    • “Creating and Braiding Anyons in an Optical Cavity,” MPIPKS International Workshop on Synthetic Topological Matter, Dresden, Germany, May 21, 2019.
    • “Critical Response of a Quantum van der Pol Oscillator,” DesOEQ (Designing Out-of-Equilibrium Quantum systems) Review Meeting, Glasgow, Scotland, March 14, 2019.
  • 2018
    • “Creating and Braiding Anyons in an Optical Cavity,” DesOEQ Annual Meeting & DOQS Workshop, Glasgow, Scotland, October 15, 2018.
    • “Creating and Braiding Anyons in an Optical Cavity,” 49th Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics (DAMOP), Fort Lauderdale, Florida, May 30, 2018.
    • “Creating and Braiding Anyons in an Optical Cavity,” Cornell Center for Materials Research (CCMR) Symposium on Advances in Photonics and Quantum Optics, Ithaca, New York, May 23, 2018.
  • 2017
    • “Creating and Braiding Anyons in an Optical Cavity,” ARO/AFOSR Quantum Matter MURI Program Review, Gaithersburg, Maryland, October 12, 2017.
    • “Creating and Braiding Anyons in an Optical Cavity,” ITAMP workshop on Many-Body Cavity QED, Boston, Massachusetts, October 10, 2017.
    • “Collective Modes of a Soliton Train in a Fermi Superfluid,” 48th Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics (DAMOP), Sacramento, California, June 6, 2017.
  • 2016
    • “Collective Modes of a Soliton Train in a Fermi Superfluid,” ARO/AFOSR Quantum Matter MURI Program Review, Chicago, Illinois, September 27, 2016.

Unpublished work

  1. Understanding a long-standing puzzle in liquid Helium-3: How does the superfluid B phase nucleate in the supercooled liquid? (2018) [with Erich Mueller and Jeevak Parpia].
Click here to read more!

Cut a long story (see below) short: the puzzle is still open as ever! We tried sophisticated approaches for finding the nucleation energy, such as the nudged elastic band method, but that did not explain the observations.

Helium is one of the few elements which refuses to solidify at normal pressure no matter how much it is cooled. This is a consequence of the Heisenberg uncertainty principle of quantum mechanics which prevents everything from coming to a standstill even at absolute zero temperature. The residual energy, called the zero-point energy, is larger for lighter elements like Helium. Moreover, being a noble gas, the interatomic forces in helium are weak — too weak, in fact, to lock the atoms into a regular grid and form a solid. So it remains a liquid even at absolute zero. But that is only the start – the story gets much more interesting!

There are two stable isotopes of Helium: He-3 and He-4. As early as 1937, He-4 was cooled below a few K, where it suddenly became a superfluid, displaying some bizarre properties! On the other hand, He-3 remained a rather “boring” liquid even at much lower temperatures. But in November 1971, in a cold Ithaca winter (~250 K), He-3 was cooled much further (~2 mK), and lo! It turned superfluid. Not only that, two different types of superfluid — called rather unimaginatively “A” and “B” 🙂 . This vast difference between the two isotopes has its origin in quantum statistics. He-4 has an even number of fermions (protons, neutrons, and electrons), so it obeys Bose statistics. Conversely, He-3 has an odd number of fermions and obeys Fermi statistics. And that makes all the difference. Whereas bosons can condense into the same state at low temperatures, fermions have to pair up before they can condense. This Cooper pairing occurs at much lower temperatures, and depending on the internal structure of the pairs, He-3 can form different superfluid states which are much more exotic than the structureless superfluid of He-4! Read this entertaining review.

Now comes the puzzle: the superfluid A phase is stable only above a critical temperature, below which the superfluid B phase is stable (see phase diagram). However, He-3 can be “supercooled” below this critical temperature and still remain in the A phase. Now, the puzzle comes in how the liquid eventually turns into the B phase as it is cooled further. Experiments have seen such nucleation of the B phase, but we still don’t understand the mechanism! Despite some rather crazy proposals, there is yet no definitive theoretical understanding which explains this nucleation phenomena. However, new experiments underway in Jeevak Parpia’s lab at Cornell.

  1. 1D-to-3D crossover in the superfluidity of a spin-imbalanced Fermi gas in an array of coupled tubes (2016) [with Erich Mueller].
Click here to read more!

This is an extension of our work on dimensional crossover in a cylindrically confined spin-imbalanced Fermi gas (see above). Here we consider a 3D gas broken up into tubes by a 2D lattice, as in experiments at Rice, and calculate the (BdG) phase diagram taking into account the higher energy bands. The 1D-to-3D crossover occurs in two different manner depending on whether the lattice depth is decreased or the interactions are increased. For weak interactions, when the (average) chemical potential lies within an energy band, we find 3D-like behavior, whereas if the chemical potential lies between the 1st and 2nd band, we find 1D like behavior. As the lattice depth is decreased, these features are qualitatively unchanged, however the energy bands get wider and eventually the spectrum becomes gapless, making the entire phase diagram 3D like. On the other hand, stronger interactions cause mixing between the energy bands and the 1D-like behavior is suppressed.

    Note: In the figures, gray regions indicate locations of the energy bands. Some of the figures only show the critical imbalance for the BCS state. Generically the region just outside is FFLO. The curve has a positive slope in 3D and a negative slope in 1D.

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  1. Thermalization in a quasi-one-dimensional quantum gas (2015) [with Erich Mueller and Mukund Vengalattore] [manuscript] [slides].
Click here to read more!

We model thermalization in a quantum gas via binary elastic collisions after the gas is loaded into an array of weakly-coupled 1D tubes by turning on an optical lattice in the transverse plane. When the lattice is turned on adiabatically (slow compared to the collision rate), the quasimomentum distribution evolves smoothly into a thermal profile. For small intertube coupling J, the rate of thermalization grows as J2 log J. We show that the equilibration times in two recent experiments [Nature 467, 567 (2010)] and [Nature 440, 900 (2006)] differ hugely from one another, which explains why one of them saw a thermal cloud whereas the other didn’t. When the lattice is turned on suddenly (fast compared to the collision rate), the momentum distribution develops multiple isolated peaks which eventually merge into a thermal distribution. These nonequilibrium peaks originate from the exchange of particles between different energy bands and can be resolved for about 50 collision times.

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Undergraduate research

  • Bifurcation in classical dynamical systems [with Subhankar Ray]: Exploring simple nonlinear systems with feature-rich bifurcation diagrams, coming up with techniques to characterize borderline cases where linear stability analysis fails (see eprints above).
  • Random walks modeling anomalous diffusion [with Subhankar Ray and Jaya Shamanna]: Using probability arguments and integro-differential equations to model a continuous-time random walk under an external field, leading to subdiffusive transport (see eprint above).
  • Photoemission from thin films [with Manas Bose and Chayanika Bose]: Using Boltzmann rate equations to model photocurrent from a semiconductor film as a function of the film thickness and the frequency and polarization of incident light (senior-year project).
  • PT-symmetric quantum mechanics [with Subhankar Ray]: Exploring the properties of a class of non-Hermitian Hamiltonians that are symmetric under spacetime reflection, and their equivalence to Hermitian Hamiltonians used in ordinary quantum mechanics.
  • Liquid-gas phase transition of nuclear matter [with Subhankar Ray and Jaya Shamanna]: Using the Bethe-Peierls approximation to model the liquid-gas phase transition in a cubic lattice gas model of cold nuclear matter.

Other activities