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.

Research during PhD

I do theoretical research on the quantum-mechanical properties of ultracold gases with Prof. Erich Mueller in the Laboratory of Atomic and Solid State Physics at Cornell University. As matter is cooled to near absolute zero, thermal fluctuations die out and the strange rules of quantum mechanics dictate the behavior of atoms, often giving rise to exotic and beautiful forms of matter such as Bose-Einstein condensates, superfluidity, and fractional quantum Hall states. Quantum gases provide a remarkably versatile and controllable testbed to investigate such quantum phenomena. My research consists of analyzing mathematical models of such gases of atoms and photons to understand the rich features which emerge out of their interactions, and make predictions which could be tested in experiments. To achieve this goal, I use a blend of analytical and numerical techniques, and work in close contact with experimental groups.

    You can read more about the field of ultracold quantum gases in our group website, in this review article [pdf], or Chapter 1 of my thesis. To skip directly to my publications as a graduate student, click here. To skip to my undergraduate research, click here.

The projects I work on fall under the following broad research areas:

  • Superfluidity in ultracold gases of fermions and bosons: kinetics of Bose condensation, spread of impurities, stability and dynamics of collective excitations such as domain walls and polarons, experimental signatures of different superfluid states.
  • Quantum phase transitions and crossovers: Characterizing quantum phase transitions and crossovers by studying the variation of different physical properties such as density, superfluid order, and correlation functions. Examples include Superfluid-Mott transition in Bose gases, BEC-BCS crossover in Fermi gases, and dimensional crossovers.
  • Nonequilibrium dynamics: Spread of impurities, condensate formation, thermalization, soliton dynamics, collective oscillations and waves, driven optical cavities. Different theoretical tools – rate equations, Fermi’s golden rule, Heisenberg/Schrodinger equations, Bogoliubov-de Gennes equations, Lindblad type master equations, and variational ansatz.
  • Exotic quantum states: Proposing experimental protocols to prepare, detect, and manipulate often fragile many-body quantum states with exotic features, such as FFLO (Fulde-Ferrell-Larkin-Ovchinnikov) and breached-pair superfluidity in spin-imbalanced Fermi gases, and fractional quantum Hall states of light in specially designed optical cavities.
  • Emergent macroscopic structures: Stability and dynamics of persistent nonlinear waves or solitons and quasiparticle excitations such as polarons and bipolarons in superfluids.
  • Open quantum systems: Modeling driven dissipative quantum systems by master equations to study the formation of many-body states. Of particular interest are novel many-body phenomena which become accessible through strong light-matter coupling in optical cavities and how one can harness them by engineering drive, dissipation, and feedback. Examples include the formation of photonic Laughlin states in a driven optical cavity with carefully aligned mirrors.

Journal Publications

  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].
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

    Also see these slides [pdf].

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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].
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

    Also see these slides [pdf].

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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].
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

    Also see these slides [pdf].

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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].
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

    See a movie (yes!) of the instability and more on our group website. Also see this poster presented at DAMOP 2017, these slides (key, 14 MB) [pdf] presented to prospective grads at Cornell, and this presentation [key, 24 MB] by Erich.

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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, 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.

engineering-C-FFLO-summary

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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].
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. Our protocol might enable the first direct probe of these exotic entities which have remained elusive in previous experiments.

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 using interferometry. 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

    See our protocol in action in this video! More simulations can be found in the Supplement. Also see these slides (key) [pdf] and this poster presented at an ITAMP workshop.

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Conference Posters

  1. Shovan Dutta and Erich J. Mueller, “Collective modes of a soliton train in a Fermi superfluid”
    • ARO AFOSR MURI Program Review, September 26-28, 2016, Chicago, Illinois [poster]
    • 48th Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics (DAMOP), Vol. 62, No. 8, June 5-9, 2017, Sacramento, California [website] [poster]
  2. Shovan Dutta and Erich J. Mueller, “Creating Laughlin states and braiding anyons in an optical cavity”
    • ITAMP workshop on Many-Body Cavity QED, October 9-11, 2017, Boston, Massachusetts [website] [poster]
    • ARO AFOSR Quantum Matter MURI review, October 12-13, 2017, Gaithersburg, Maryland
    • CCMR Symposium on Advances in Photonics and Quantum Optics, May 23, 2018, Ithaca, New York [website] [poster]
    • 49th Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics (DAMOP), May 28-June 1, 2018, Ft. Lauderdale, Florida [website] [poster]

Work in progress

We are trying to understand a long-standing puzzle in liquid Helium-3: How does the superfluid B phase nucleate after the liquid is supercooled below the critical temperature?

Click here to read more!

Helium is one of the few (only?) elements which refuses to solidify at normal pressure no matter how much it is cooled down. It turns out 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, liquid He-4 was cooled below a few Kelvin, where it suddenly became a superfluid, displaying some really 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 to much colder temperatures (~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), hence it obeys Bose statistics. Conversely, He-3 has an odd number of fermions and thus 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 Cooper 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. That is not the puzzle! 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. With new experiments underway in Jeevak Parpia’s lab, we are having a crack at the problem!

Noteworthy unpublished work

  1. Thermalization in a quasi-one-dimensional quantum gas [with Prof. Erich Mueller and Prof. Mukund Vengalattore]
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.

    Also see this manuscript and these slides.

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  1. 1D-to-3D crossover in a spin-imbalanced Fermi gas in an array of coupled tubes [with Prof. 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 of the lattice. 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 causes 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 +ve slope in 3D and a -ve slope in 1D.

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Research as an undergrad

Before starting my PhD, I did some theoretical research at Jadavpur University under Prof. Subhankar Ray, Prof. Jaya Shamanna, Prof. Chayanika Bose, and Dr. Manas Bose. To skip to my publications as an undergrad, click here. I worked on the following topics:

  • Bifurcation in dynamical systems: Analyzing simple nonlinear systems with feature-rich bifurcation diagrams, coming up with theoretical techniques to characterize borderline cases where linear stability analysis fails, and proposing tunable electronic circuits which will possess a given set of bifurcations.
  • Random walks modeling anomalous diffusion: Using probability arguments to construct and solve integro-differential equations describing the dynamics of a particle executing continuous-time random walk under an arbitrary time-varying external field, leading to subdiffusive transport seen in disordered media.
  • Photoemission from thin films: Analyzing quantum Boltzmann rate equations to study the photocurrent from a semiconductor film as a function of the film thickness and the frequency and polarization of the incident light. This was my final-year project supervised by Dr. Manas Bose and Prof. Chayanika Bose.
  • PT-symmetric quantum mechanics: Exploring the mathematical properties of a class of non-Hermitian Hamiltonians that are symmetric under spacetime reflection, and showing how they are physically equivalent to the Hermitian Hamiltonians used in ordinary quantum mechanics.
  • Liquid-gas phase transition of nuclear matter: Using the Bethe-Peierls approximation of quantum statistical mechanics to model the liquid-gas phase transition in a cubic lattice gas model of cold nuclear matter.

Undergrad papers

  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 and Subhankar Ray, “Damped bead on a rotating circular hoop – a bifurcation zoo,” arXiv:1201.1218 (2012).
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.

    Also see these slides.

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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.” Selected as the best paper in the National Students Paper and Circuit Design Contest (NSPCDC) 2011, organized by IEEE Students Branch, Jadavpur University, in collaboration with IEEE Calcutta Section. Available here.
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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