“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 and postdoc
I study the quantummechanical properties of ultracold gases with Prof. Nigel Cooper in the Theory of Condensed Matter group at the University of Cambridge. Previously I worked as a graduate student on the theory of cold atoms with Prof. Erich Mueller 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 BoseEinstein 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 quantum gases in this review article [pdf] or Chapter 1 of my thesis. To skip directly to my publications, click here. To skip to my undergrad research, click here.
The projects I have worked 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 excitations such as domain walls and polarons, experimental signatures of FFLO states.
 Quantum phase transitions and crossovers: SuperfluidMott transition in Bose gases, BECBCS crossover in Fermi gases, dimensional crossovers, dynamically induced phase transitions.
 Nonequilibrium dynamics: Spread of impurities, condensate formation, thermalization, soliton dynamics, collective oscillations and waves, driven optical cavities.
 Exotic quantum states: Preparing and detecting often fragile manybody quantum states with exotic features, such as FFLO and breachedpair superfluidity in spinimbalanced Fermi gases, and fractional quantum Hall states of light in 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: Physics emerging from the competition of drive and dissipation, engineering lightmatter coupling to observe novel states and excitations (such as anyons), nonlinear oscillators and limit cycles in the quantum regime.
 Algorithm development: Developing new theoretical tools and numerical techniques, such as a DMRG framework for continuous 1D systems.
Journal Publications
 Shovan Dutta and Erich J. Mueller, “Variational study of polarons and bipolarons in a onedimensional Bose lattice gas in both the superfluid and the Mottinsulator 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 SuperfluidMott 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 particlehole excitations. We predict how this instability can be observed in experiments by measuring the impurityhole correlation.

Also see these slides [pdf].
 Shovan Dutta and Erich J. Mueller, “Kinetics of BoseEinstein 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 twobody scattering and threebody 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.

Also see these slides [pdf].
 Shovan Dutta and Erich J. Mueller, “Dimensional crossover in a spinimbalanced 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 breachedpair states, in a spinimbalanced Fermi gas confined in a cylindrical harmonic trap. We calculate the meanfield 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 1Dlike to 3Dlike 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.

Also see these slides [pdf].
 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 quasi1D Fermi superfluid by analyzing the Bogoliubovde Gennes equations. We uncover a variety of unexpected modes, including longlived 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.

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.
 Shovan Dutta and Erich J. Mueller, “Protocol to engineer FuldeFerrellLarkinOvchinnikov 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 twostep experimental protocol to directly engineer FFLO states in a cold Fermi gas loaded into a quasi1D 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 radiofrequency 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.
 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] [animation] [slides] [poster].
Click here to read more!
Waveparticle 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 highquality 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 vortexlike 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 quantummechanical 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.

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.
Conference Posters
 Shovan Dutta and Erich J. Mueller, “Collective modes of a soliton train in a Fermi superfluid”
 ARO AFOSR MURI Program Review, September 2628, 2016, Chicago, Illinois [poster]
 48th Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics (DAMOP), Vol. 62, No. 8, June 59, 2017, Sacramento, California [website] [poster]
 Shovan Dutta and Erich J. Mueller, “Creating Laughlin states and braiding anyons in an optical cavity”
 ITAMP workshop on ManyBody Cavity QED, October 911, 2017, Boston, Massachusetts [website] [poster]
 ARO AFOSR Quantum Matter MURI review, October 1213, 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 28June 1, 2018, Ft. Lauderdale, Florida [website] [poster]
 DesOEQ Annual Meeting & DOQS Workshop on designing outofequilibrium manybody quantum systems, October 1519, 2018, Glasgow, Scotland [website]
 Shovan Dutta and Nigel R. Cooper, “Critical response of a quantum van der Pol oscillator”
Work in progress
 Critical response of a quantum van der Pol oscillator [with Nigel Cooper] [poster]
Click here to read more!
Classical nonequilibrium systems close to a dynamical critical point (or bifurcation) are known to act as efficient sensors. We study the response of such systems in the quantum regime and find new, characteristic features which can be realized in experiments.

In particular, a large class of physical systems can be modeled as nonlinear oscillators, which exhibit spontaneous “limit cycle” oscillations beyond a certain critical point. It can be shown, quite generically, that the response of such “van der Pol” oscillators to a resonant drive F is purely nonlinear at the critical point, growing as F^{1/3}. Thus, the susceptibility (slope) diverges at weak drives. This feature is at the heart of important biological sensors, as in hearing sound over a large dynamic range [PRL 84, 5232 (2000)].

We find the response is dramatically altered in the quantum regime. It is linear for weak drives, although the susceptibility still diverges as one approaches the critical point. Further, the competition between drive and dissipation yields a region of negative susceptibility. The classical limit emerges for strong drives. The response can be probed in current experimental setups, including trapped ions, optomechanical cavities, and superconducting circuits.
 Dynamically induced insulatorsuperfluid transitions by resonant shaking [with Nigel Cooper and Ulrich Schneider]
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.

Consider bosonic atoms loaded into an optical lattice. One can control the tunneling between lattice sites by changing the lattice depth. When the atoms are in the lowest band, it is well known that they form an insulator for weak tunneling, and a superfluid for strong tunneling. However, one can also increase the effective tunneling by coupling the atoms to a higher band, e.g. by resonantly shaking the lattice. Thus, it is possible to start with an insulator in the lowest band and then cause it to melt simply by turning on this shaking. We’re studying the various phase transitions which are possible under this controllable dynamical setting.
 Matrix Product States (DMRG) for continuous systems [with Erich Mueller, Anton Buyskikh, and Andrew Daley]
Click here to read more!
What prohibits the exact simulation of large quantum systems is that the space of all possible states grow 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 the 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 a few attempts to generalize this technique to continuous systems, but they have severe limitations for nonhomogeneous or timedependent 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. In brief, one divides the system up into multiple segments, describing the local physics with Fock wave functions while imposing continuity across neighboring segments. We wish to apply this framework to studying a number of topical problems.
Unpublished work
 Thermalization in a quasionedimensional quantum gas [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 weaklycoupled 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 J^{2} 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.
 1Dto3D crossover in a spinimbalanced Fermi gas in an array of coupled tubes [with Erich Mueller]
Click here to read more!
This is an extension of our work on dimensional crossover in a cylindrically confined spinimbalanced 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 1Dto3D 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 3Dlike 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 1Dlike 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.
 Understanding a longstanding puzzle in liquid Helium3: How does the superfluid B phase nucleate after the liquid is supercooled below critical temperature? [with Erich Mueller]
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Cut a long story (see below) short: we did not resolve the puzzle.
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: He3 and He4. As early as 1937, liquid He4 was cooled below a few Kelvin, where it suddenly became a superfluid, displaying some really bizarre properties! On the other hand, He3 remained a rather “boring” liquid even at much lower temperatures. But in November 1971, in a cold Ithaca winter (~250 K), He3 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. He4 has an even number of fermions (protons, neutrons, and electrons), hence it obeys Bose statistics. Conversely, He3 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, He3 can form different superfluid states which are much more exotic than the structureless superfluid of He4! 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, He3 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. However, new experiments underway in Jeevak Parpia’s lab at Cornell.
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 featurerich 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 integrodifferential equations describing the dynamics of a particle executing continuoustime random walk under an arbitrary timevarying 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 finalyear project supervised by Dr. Manas Bose and Prof. Chayanika Bose.
 PTsymmetric quantum mechanics: Exploring the mathematical properties of a class of nonHermitian Hamiltonians that are symmetric under spacetime reflection, and showing how they are physically equivalent to the Hermitian Hamiltonians used in ordinary quantum mechanics.
 Liquidgas phase transition of nuclear matter: Using the BethePeierls approximation of quantum statistical mechanics to model the liquidgas phase transition in a cubic lattice gas model of cold nuclear matter.
Undergrad papers
 Shovan Dutta and Subhankar Ray, “Bead on a rotating circular hoop: a simple yet featurerich dynamical system,” arXiv:1112.4697 (2011).
Click here to read more!
We perform an indepth 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.
 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.
 Shovan Dutta, Subhankar Ray, and J. Shamanna, “Continuous Time Random Walk with timedependent jump probability: a direct probabilistic approach,” arXiv:1112.3253 (2011).
Click here to read more!
We tackle the problem of a continuoustime random walk in 3D under timevarying 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 r´ and r´ + dr´ is φ(r´t) d^{3}r´. The goal is to find the probability density p(r,t) that the random walker is at position r at time t. Continuoustime 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(r´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 nearestneighbor jumps, which was modeled in prior work by a Fractional FokkerPlanck 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.
 Shovan Dutta, “A simple circuit model showing featurerich BogdanovTakens 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 easytoimplement circuit model for the BogdanovTakens bifurcation, exhibiting three local (spiraltonode, saddlenode, AndronovHopf) 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.