Our laboratory is involved in a diverse set of research projects in the areas of quantum optics, control and synchronization of chaos in optical and electronic systems, and characterizing and controlling the dynamics of biological systems. A brief overview of each general area is given below.

### Current Projects

- Boolean chaos
- Collective nonlinear optics in cold atoms
- Hybrid quantum memories
- Quantum Information, Entanglement, and Free-Space Propagation

### Past Projects

- All-optical switching
- Chaos synchronization
- Characterizing and controlling cardiac dynamics
- Control and Synchronization of Chaos in Optical and Electronic Systems
- Information velocity research
- Nonlinear feedback in wave chaotic cavities
- Opto-electronic chaos
- Slow light with a swept-frequency source
- Spatially extended optical systems
- Stored Light
- Two-photon lasers

## Boolean Chaos

Deterministic Chaos is a dynamical state characterized by an exponentially fast divergence of solutions starting at nearby states and a broadband power spectrum. We are interested in deterministic chaos exhibited by systems that can be efficiently modeled by Boolean networks. We have found deterministic chaos in electronic devices comprised of high-speed digital gates. The voltage produced by the electronic device shows clearly defined transitions between high and low values and thus should be well described by Boolean delay equations. The electronic signal has a ultra-wideband frequency spectrum and therefore can be used as an inexpensive source in spread-spectrum communication systems

A Boolean network with three nodes. |

This research is performed at Duke University by Hugo L. D. de S. Cavalcante, Rui Zhang, Zheng Gao, Seth D. Cohen, and Daniel Gauthier, in collaboration with Joshua E. S. Socolar CNCS, Duke, and Daniel P. Lathrop IREAP and Department of Physics, University of Maryland. To find out more, please visit our Boolean chaos page.

This material is based upon work supported by the Office of Naval Research under Grant No. N00014-07-0-0734.

back to top## Collective nonlinear optics in cold atoms

Light-based technologies, like fiber optics, have revolutionized how we communicate and receive information. In order to push optical information networks to their fundamental quantum limit, we require the capacity to control the degree to which light and matter interact (even down to the most fundamental, quantum level). To this end, we are looking into new ways to prepare matter so that incident photons interact with the matter in novel ways.

In particular, we use cold atoms to study how to simultaneously control the internal and external atomic states so as to give rise to the largest nonlinear optical response. The experiment consists of shining a pair of laser beams on the atoms; some of this light reflects off the atoms, giving them a kick that alters their location and, possibly, their internal state. Both the applied laser beams and the scattered light affect the positions of the atoms, giving rise to a feedback mechanism that causes the atoms to self-assemble into a spatially-organized configuration. Because this configuration is influenced by both the incident and scattered light, it is just right to allow the atoms to cooperate to maximize nonlinear optical (light-matter) interactions.

The strong laser beams (red arrows) give rise to forces organize the atoms (red spheres) into a spatial structure that enhances further scattering. As a result of this positive-feedback process, new beams of light (yellow arrows) are generated along the direction of maximum gain. |

This interaction gives rise to new scaling relations for four wave mixing processes that lead to the occurrence of optical instabilities at reduced thresholds. One example is the spontaneous generation of new optical fields via a superradience-like process. In addition, because the light acts to effectively hold the atoms in place, the coherence time (time over which the atoms remain in a given quantum state) is greatly extended. This means that any information imparted to the atoms by the light will remain stored in the sample for a longer time, thus making this system a potential candidate for a quantum memory.

This research is performed at Duke University by Bonnie Schmittberger, Joel Greenberg and Daniel Gauthier. To find out more, please visit our Boolean chaos page.

This material is based upon work supported by the National Science Foundation under Grant No. PHY-0855399 and PHY-1206040.

## Hybrid Quantum Memories

Quantum information has made rapid progress and it is now common to think of quantum teleportation of qubits over tens of kilometers, and quantum computations that are conducted by controlling multiple single quanta. We are now at the frontier of seeing the development of the next generation quantum telecommunication and computing devices, which brings concepts such as quantum cryptography and fast quantum algorithms to everyday life, providing solutions to the growing demand for computing power and secure communication channels.

Two-pump driven nonlinear four-sideband interactions. The signal beam s (red solid arrow) is translated to the frequency-converted idler beam bs (red dashed arrow) via the nonlinear Bragg scattering process. |

The project “Quantum-Optical Circuits of Hybrid Quantum Memories” aims at realizing a scalable quantum information processing system that integrates different quantum memories using quantum information conveying circuits. It is a Multidisciplinary University Research Initiative (MURI) project, with Prof. Christopher Monroe at the University of Maryland as the principle investigator. As the Duke team of this project, Professor Gauthier’s group (Physics Department) and Professor Kim’s group (in ECE department) are specifically concerned with the challenge to share quantum information between hybrid quantum memory resources such as trapped ions and quantum dots. Typically in experiment, optical photons emitted by the quantum memories are used as the carrier of the quantum information (flying qubits). However, the frequency, spatial profile and temporal distribution of these photons are highly sensitive to the specific physical implementation. Quantum-state-preserving frequency conversion and reshaping of the temporal profile of the carrier photons is required to effectively communicate between different memories. Moreover, to disseminate quantum information over long distances, a two-step frequency translation is needed. The optical photons emitted from the original quantum memory are first frequency converted to the telecommunication near-infrared band for long-distance transmission. The flying qubits are then frequency translated again to match the receiving quantum memory after arriving.

This material is based upon work supported by the Army Research Office under Grant Nos. W911NF09-1-0496 and W911NF10-1-0395.

back to top## Quantum Information, Entanglement, and Free-Space Propagation

The goals of most information schemes are to encode more secure information and transfer it at a faster rate: quantum information schemes are no exception. However, quantum information schemes make use of the counter-intuitive ideas that quantum theory provides (such as entanglement). Entangling pairs of photons allows the sender to encode information that can be robust against eavesdropping because if the bi-photon wavefunction that they are sending is somehow intercepted and measured, the wave function collapses and the entanglement is lost. There have been schemes devised to be able to detect if anyone is "listening in" on the information being sent. In order to produce an entangled bi-photon state we use the nonlinear optical process of spontaneous parametric down-conversion, which transforms an incoming pump photon into two outgoing photons each with twice the wavelength. This process can only occur in a nonlinear medium: for our specific experiments we use a Bismuth tri-Borate crystal (BiBO). The outgoing photons can be emitted at any angle that satisfies certain conditions called phase-matching conditions and so when SPDC occurs in our nonlinear (BiBO) crystal we can see a bright ring of pairs of entangled photons on our CCD camera.

In order to have a high level of entanglement we put two BiBO crystals in our set-up and orient them such that their optic axes are 90 degrees from one another. This allows for photons in either state of polarization to interact with one of the two crystals and thus we get two down-conversion rings, which should in theory, completely overlap. If we can achieve this overlap, it will be impossible to distinguish which photons came from which crystal and so we would have polarization entanglement. Polarization entanglement has already been achieved using this scheme but for only two overlapping points on the rings. We are trying to produce a much higher level of entanglement, namely, complete entanglement around the entire down-conversion ring.

The second part of this project is to then propagate the beams of entangled photons in free-space. The reason we would want to do this is that free-space communication has some advantages over optical fiber based communication systems. The main advantage is that free-space communication does not have the dispersion and loss that is associated with optical fibers. However, the down-side to free-space communication is that any information sent is subject to atmospheric turbulence and so the beam is modified along its trip from sender to receiver. To this end we have built a turbulence cell to simulate what happens in the atmosphere and characterized it so that we can now observe what happens to our entangled photons.

This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) Defense Science Office (DSO) InPho Program.

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### Past Projects

## All-optical Switching

We are interested in understanding nonlinear interactions between light and matter. By exploiting such interactions, we demonstrated an all-optical switch where a weak ``switching'' beam of light controls a much stronger ``output'' beam. We can operate the switch with a switching beam containing as few as 300 photons. Our goal is to optimize the experiment and achieve few-photon switching.

This research is performed here at Duke University by Dan Gauthier, Andrew Dawes, Lucas Illing, and Susan Clark. To find out more, please visit the all-optical switching section of our webpage.

## Chaos Synchronization

In this project, we investigated how coupling can cause two chaotic systems to synchronize their behavior. Synchronized chaos is interesting from the perspective of fundamental nonlinear dynamics, and may be the cornerstone of communication systems based on masking information in a chaotic signal. In our work, we have been investigating a recently discovered dynamical behavior known as attractor bubbling. During attractor bubbling, small random perturbations occurring in the chaotic systems (due to the ever present `noise,' for example) cause brief desynchronization events for coupling strengths where previous theories would predict the occurrence of high-quality synchronization. We are investigating the transition from attractor bubbling to high quality synchronization in one-way coupled chaotic (and hyperchaotic) electronic circuits and comparing our experimental results to emerging theories on this behavior.

This research is in collaboration with Prof. Edward Ott of the University of Maryland and Dr. Lou Pecora of the Naval Research Laboratory.

Jon Blakely, Joshua Bienfang, and Seth Boyd worked on these projects. This material is based upon work supported by the US Army Research Office under Grant No. DAAD19-02-1-0223.

back to top## Characterizing and Controlling Cardiac Dynamics

Microelectrode study of a small piece of periodically-paced frog heart |

One intriguing application of the chaos control methods we have developed is in the biological area. We have initiated a program to characterize* in vitro* the dynamics of small pieces of rapidly paced cardiac muscle and to use feedback methods to suppress or control the observed bifurcations by applying small perturbations to the tissue. We find that there are only a small number of classes of bifurcations in the tissue, but that there is significant variation in the prevalence of these behaviors from animal to animal.

This research is collaboration with John Cain (Math), Hana Dobrovolny, Colleen Mitchell (Math), Marie Guerraty (Math), Ann Pitruzzello (BME), Soma Sau (BME), Dr. Monica Romeo (Math), Dr. Elena Tolkacheva, and Profs. Wanda Krassowska (Biomedical Engineering), David Schaeffer (Mathematics), Joshua Socolar (Physics), and Patrick Wolf (Biomedical Engineering). This material was based upon work supported by the National Science Foundation under Grant No. PHY-9982860.

An in vivo sheep heart with the plaque of electrodes |

In addition, we are using similar methods to control *in vivo* a fibrillating sheep atrium. The eventual long term goal of this project is to develop an implantable defibrillator that will maintain a healthy rhythm in humans prone to the onset of atrial fibrillation using only small electrical shocks. In our current experiments with sheep, we use a high-density mapping system to record the spatial-temporal complexity occurring on the surface of the heart during atrial fibrillation. Small control shocks are applied to a single electrode attached to the surface of the heart based on real-time measurements of the cardiac dynamics at a nearby spatial location.

This research is collaboration with Mr. Robert Oliver (graduate student in Biomedical Engineering), and Profs. Wanda Krassowska (Biomedical Engineering), David Schaeffer (Mathematics), Joshua Socolar (Physics), and Patrick Wolf (Biomedical Engineering). This project was supported by the National Institute of Health.

## Control and Synchronization of Chaos in Optical and Electronic Systems

Set up for studying high speed optical chaos |

Chaos and instabilities often limit the performance of devices and hence it is important to understand their control or suppression. Also, recent research suggests that controlled chaotic systems may find useful applications in the areas of communication and computation. The basic idea underlying the control methods is to take advantage of the large number of unstable periodic orbits and unstable steady-states that are embedded in a chaotic attractor. We are investigating methods for controlling chaos that occurs in optical and electronic systems on a fast (nanosecond) time scale and in spatially extended optical systems.

Due to technological constraints, a practical limit exists on the response time of an electronic or optical controller. Any system whose dynamics evolve faster than this limit presents a serious challenge to our ability to achieve control. Potential control methods must be as computationally simple as possible to cope with the rapid evolution of the system. We have developed time-delay feedback methods that can be implemented on very fast time scales and are successful in stabilizing the desired behavior using only small perturbations to the dynamical system. Our current experimental effort involves applying these methods to an dynamical system consisting of a diode laser, a Mach-Zehnder interferometer, and a time-delayed electrical feedback from the interferometer output to the injection current of the laser. The time delay in the feedback loop increases the complexity of the dynamics while the use of RF electronics in the loop allows for fast oscillations on the time scale of 1 ns. No system this fast has been successfully controlled... yet.

This research is in collaboration with Prof. Joshua Socolar of the Duke Physics Department.

back to top## Information Velocity Research

We are generally interested in the velocity of information on optical pulses. The best-known question in this area is, of course, can information go faster than the speed of light in vacuum, c? We address this question and the more general question of how fast does information go? by creating pulses of light that travel very fast (much much faster than c) or very slow (much much slower than c) and measuring information encoded on them.

For more about our information velocity research, please visit the Information Velocity Research section of our web site.

back to top## Nonlinear feedback in wave chaotic cavities

Conventional optical imaging has a resolution limited by diffraction. This minimum resolution scales as λ/4 (Rayleigh criterion), where λ is the minimum wavelength of the system. New techniques for better resolution are a current research topic of great interest. We propose a novel scheme that combines two fields of research: nonlinear delayed-feedback systems and wave chaotic cavities. This method allows us to detect and track changes in the location of a sub-wavelength water scatterer with extreme sub-wavelength resolution.

In a wave chaotic cavity, electromagnetic (EM) waves fill the entire cavity through many multipath reflections [1]. Broadband chaos has been observed previously in a simple transistor-based nonlinear circuit with time-delayed feedback through a single coaxial cable [2]. In such a nonlinear delayed feedback system, the dynamical state depends heavily on the delay time and the gain of the feedback loop.

We replace the single feedback loop with a multipath delay system. The system consists of stadiumshaped wave-chaotic cavity containing broadband antennas and a sub-wavelength water scatterer. This creates a new nonlinear feedback system where the multipath reflections of the radio waves inside the cavity become the delayed feedback loops of the dynamical system. By translating the location of the water scatterer, the path lengths and coupling strengths of these feedback delays change.

From scatterer movements as little as 10 micrometers, we observe bifurcations in the system’s output voltage between periodic, quasi-periodic, and chaotic attractors. The primary purpose of this work is to explore and describe the bifurcations observed in this multipath, time-delay dynamical system. The minimal detectable change in scatterer position is 15,000 times smaller than the minimum wavelength of radiation inside the cavity (15 cm). By exploiting this novel technique for sub-wavelength sensitivity, we hope to improve traditional methods of intrusion detection systems and tracking devices with through-wall capabilities.

[1] B. Taddese et al, J. Appl. Phys. 108, 114911 (2010).

[2] L. Illing and D. J. Gauthier, Chaos 16, 033119 (2006).

This research is performed at Duke University by Seth D. Cohen, Hugo L. D. de S. Cavalcante, and Daniel Gauthier.

This material is based upon work supported by the Office of Naval Research under Grant No. N00014-07-0-0734.

back to top## Opto-electronic Chaos

The study of chaos has been an active area of interdisciplinary research since the 1970s. Today, researchers are interested in practical applications of chaos, such as chaos communications and ranging, which require simple devices that produce complex and high-speed dynamics. Furthermore, since all physical signals travel at finite speeds, it is important to understand how inherent time delays in these devices interact with nonlinearities to affect their behavior. The continuing development of optical, electronic and optoelectronic systems with time-delayed feedback has emerged as a top contender for these applications and for fundamental investigations of time-delay systems. We study an optoelectronic device with bandpass filtered feedback that displays a rich variety of dynamics, including ‘featureless’ broadband chaos, multi-timescale dynamics (‘breathers’), and a number of different pulsing behaviors.

This material is based upon work supported by the Office of Naval Research under Grant No. N00014-07-0-0734.

back to top## Slow light with a swept-frequency source

We introduced a new concept to realize SBS slow light applicable to broadly-swept sources. This method allows slow light to be achieved, in principle, over the entire transparency window of the optical fiber (many 100’s of nm at telecommunication wavelengths). The key idea is to pump the SBS process with a beam that is derived from the linearly swept source, but shifted to a higher frequency equal to the Brillouin frequency shift of the fiber - using a Mach-Zehnder modulator. In this way, the pump beam frequency automatically tracks the swept-source signal frequency as they enter the fiber and hence are always near the SBS resonance frequency where the slow-light effect is largest. The fact that the pump and signal beams counterpropagate through the fiber causes a small detuning - between the beams, which decreases the slow light effect. This detuning increases with increasing fiber length L and the source sweep rate R and must be accounted for to optimize the slow light delay.

We experimentally and theoretically investigate the slow light effect and its dependence on R and other experimental parameters. We demonstrate that there is an optimum value of L to obtain the largest delay for a given R. We observe a delay of 10 ns using a 10-m-long photonic crystal fiber (PCF) with R=400 MHz/μs and a pump power Pin=200 mW. Larger delays can be obtained by increasing Pin until spontaneous Brillouin scattering dominates the process. We find that the maximum obtainable delay for an optimum-length fiber of the same type used in our experiment is ~38 ns independent of R. A pump power of 760 mW is required to obtain the maximum delay for R=400 MHz/μs for our fiber.

This research provides a path toward practical applications of the SBS slow light technique that uses broadband swept sources. The total frequency sweeping range using this method is limited up to 1 nm due to the varying Brillouin resonance as a function of pump frequency. However, by adjusting the RF frequency to simultaneously track the Brillouin resonance of the pump frequency, the slow light can be achieved over the entire transparency window of the optical fiber. It has a potential to work with the high-speed commercial frequency-swept sources, which makes it applicable to optical coherence tomography and Fourier transform spectroscopy.

This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) Defense Science Office (DSO) Slow Light Program.

back to top## Spatially extended optical systems

Transverse intensity profile of a laser beam in a light valve based dynamical system. Roll or hexagon patterns form when diffraction length is sufficient. |

We are also investigating control in spatially extended optical systems. One such system uses a 'Liquid Crystal Light Valve' as a nonlinear element in a diffractive optical system with feedback. A pattern of light applied to the 'Write' side of the light valve is imprinted onto light reflected from the 'Read' side by phase modulation. This pattern diffracts through a fixed difference and is fed back to the Write side. The resulting dynamics include spatial pattern formation and spatio-temporal chaos. We are currently interested in exploring the limits of control of spatio-temporal chaos in this complex dynamical system. For example, must the control signal be spatially extended to control spatial dynamics? Or perhaps a few control signals applied to certain spatial points will suffice. If so, do these point controllers require global knowledge of the dynamics?

Jonathan Blakely and Lucas Illing are currently working on these projects. This material is based upon work supported by the US Army Research Office under Grant No. DAAD19-02-1-0223.

back to top## Stored Light

We are interested in understanding nonlinear interactions between light and matter. By exploiting such interactions, we demonstrated a method for storing the information encoded on a beam of light in the form of an acoustic wave and then converting the information back to the optical domain after a controllable storage time. The method uses a standard optical fiber at room temperature and hence is potentially compatible with existing telecommunication infrastructure. Our overal research goal is to use other materials (in a fiber format) that will allow for longer storage time and reduce the peak power of the laser beam needed for storing and reading out the information.

This research is performed here at Duke University by Zhaoming Zhu and Daniel Gauthier, in collaboration with Robert Boyd at the University of Rochester. To find out more, please visit the stored light section of our webpage.

## Two Photon Laser

An illustration of the two-photon stimulated emission process |

In the area of quantum optics, we worked to develop a new type of quantum oscillator known as a two-photon laser. It differs from all other known laser sources in that it is based on the two-photon stimulated emission process whereby two photons induce an atom to make a transition to a lower energy state and four photons are scattered. We create a two-photon gain medium by strongly driving potassium atoms with a (normal) laser beam. The interaction between the potassium atoms and the laser beam is so strong that it creates a `composite' atom+field system that has very interesting quantum optical properties. These composite atoms are placed in an very high finesse optical resonator that selectively enhances the two-photon stimulated emission process while simultaneously suppressing other competing nonlinear optical processes. We have recently achieved two-photon lasing in our laboratory using this technique. To the best of our knowledge, this is the only two-photon optical laser operating in the world.

A close up image of the two-photon laser cavity |

It is expected that the photons emitted from the two-photon laser cavity will posses unique quantum correlations that might be useful for precision measurement, quantum computing, and quantum communication, for example.

Andy Dawes, Michael Stenner, Jean-Phillipe Smits, and Heejeong Jeong worked on this project. This material was based upon work supported by the National Science Foundation under Grant No. PHY-0139991.

back to topAny opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF), the US Army Research Office (US ARO), or the National Institute of Health (NIH).