Andrew C. Millard

• foundations of quantum mechanics
• non-linear optical microscopy

We address the issue of when generalised quantum dynamics, which is a classical symplectic dynamics for noncommuting operator phase space variables based on a graded total trace Hamiltonian H, reduces to Heisenberg picture complex quantum mechanics. We begin by showing that when H = Tr H, with H a Weyl-ordered operator Hamiltonian, then the generalised quantum dynamics operator equations of motion agree with those obtained from H in the Heisenberg picture by using canonical commutation relations. The remainder of the paper is devoted to a study of how an effective canonical algebra can arise, without this condition simply being imposed by fiat on the operator initial values. We first show that for any total trace Hamiltonian which involves no non-commutative constants, there is a conserved anti-self-adjoint operator C with a structure which is closely related to the canonical commutator algebra. We study the canonical transformations of generalised quantum dynamics, and show that C is a canonical invariant, as is the operator phase space volume element. The latter result is a generalisation of Liouville’s theorem, and permits the application of statistical mechanical methods to determine the canonical ensemble governing the equilibrium distribution of operator initial values. We give arguments based on a Ward identity analogous to the equipartition theorem of classical statistical mechanics, suggesting that statistical ensemble averages of Weyl-ordered polynomials in the operator phase space variables correspond to the Wightman functions of a unitary complex quantum mechanics, with a conserved operator Hamiltonian and with the standard canonical commutation relations obeyed by Weyl-ordered operator strings. Thus there is a well-defined sense in which complex quantum field theory can emerge as a statistical approximation to an underlying generalised quantum dynamics.

• Coauthored with Stephen L. Adler.
• Appeared in Nuclear Physics B, 473 199 (1996).
• Available here from LANL E-print Archive.

A structure theorem concerning projective quaternionic representations is stated and proved. A corollary to the structure theorem is then proved, answering a question recently posed by Adler.

• Coauthored with Terry Tao.
• Appeared in Journal of Mathematical Physics, 37 5848 (1996).
• Available here as PDF file (122k).
• Also available here as gzipped DVI file (19k).

We develop Perelomov’s coherent states formalism to include the case of a quaternionic Hilbert space. We find that, because of the closure requirement, any attempted quaternionic generalisation of the special nilpotent or Weyl group reduces to the normal complex case. For the case of the compact group SU(2), however, coherent states can be formulated using the quaternionic half-integer spin matrices of Finkelstein, Jauch and Speiser, giving a non-trivial quaternionic analogue of coherent states.

• Coauthored with Stephen L. Adler.
• Appeared in Journal of Mathematical Physics, 38 2117 (1997).
• Available here as PDF file (145k).
• Also available here as gzipped DVI file (16k).

Quantum mechanics over an associative ring with a conjugation operation can be recast in a form familiar as a classical dynamical system. The generators of transformations on the classical phase space are the expectation values of anti-self-adjoint operators and are closed under a Poisson bracket that is in direct correspondence with the quantum mechanical commutator. A prescription also exists for determining when a classical flow is equivalent to a quantum mechanical evolution.

• Appeared in Journal of Mathematical Physics, 38 6230 (1997).
• Available here as PDF file (238k).
• Also available here as gzipped DVI file (34k).

Non-commutativity appears in physics almost hand in hand with quantum mechanics. Non-commuting operators corresponding to observables lead to Heisenberg’s Uncertainty Principle, which is often used as a prime example of how quantum mechanics transcends “common sense”, while the operators that generate a symmetry group are usually given in terms of their commutation relations. This thesis discusses a number of new developments which go beyond the usual stopping point of non-commuting quantities as matrices with complex elements. Chapter 2 shows how certain generalisations of quantum mechanics, from using complex numbers to using other (often non-commutative) algebras, can still be written as linear systems with symplectic phase flows. Chapter 3 deals with Adler’s trace dynamics, a non-linear graded generalisation of Hamiltonian dynamics with supersymmetry applications, where the phase space coordinates are (generally non-commuting) operators, and reports on aspects of a demonstration that the statistical averages of the dynamical variables obey the rules of complex quantum field theory. The last two chapters discuss specific aspects of quaternionic quantum mechanics. Chapter 4 reports a generalised projective representation theory and presents a structure theorem that categorises quaternionic projective representations. Chapter 5 deals with a generalisation of the coherent states formalism and examines how it may be applied to two commonly used groups.

• Princeton University Physics Department Doctoral Thesis (June 1997).
• Available here as PDF file (360k).
• Also available here as gzipped DVI file (70k).

• Text of presentation based on above thesis.
• Available here as PDF file (122k).
• Also available here as gzipped DVI file (19k).

We investigate using collinear Type II SHG FROG to measure the pulse intensity and phase at the focus of high NA oil objectives. Because of the strong focusing for such objectives, it is not clear theoretically that such a measurement should work. Such objectives can produce severe distortions of the pulse as a function of radius in the objective. In addition, the standard SHG FROG algorithms are based on the assumption that the fundamental and second harmonic fields are plane waves that can be described by the paraxial approximation, and for high NA objectives, such assumptions are suspect. We show that such measurements work remarkably well. The tight focus, while a theoretical difficulty, eliminates many of the problems traditionally associated with SHG FROG including the difficulty of phase matching and walkoff of different polarizations in the crystal. Specifically, we use collinear Type II SHG FROG to measure the intensity and phase at the focus of a Zeiss CP-Achromat 100x, 1.25 NA, infinity-corrected oil objective, and accurately retrieve 20 fs pulses.

• Invited paper for thirty-fifth anniversary feature issue.
• Coauthored with David N. Fittinghoff, Jeffrey A. Squier and Michiel Müller.
• Appeared in IEEE’s Journal of Quantum Electronics 35 479 (1999).
• Also available here as PDF file (994k).

We demonstrate for the first time that the two-photon response of GaAsP photodiodes can be used to conveniently characterise the spatial and temporal profiles of ultrashort pulses at the focus of a high numerical aperture system.

• Coauthored with David N. Fittinghoff, Jeffrey A. Squier, Michiel Müller and Alexander L. Gaeta.
• Appeared in Journal of Microscopy 193 179 (1999).
• Available here as PDF file (124k).

As ultrafast multiphoton microscopes become more useful for biological imaging, a major challenge for researchers is to determine the exposure conditions that provide the best combination of image resolution, contrast and specimen viability. To do this requires an accurate understanding of the spatial and temporal evolution of ultrashort pulses at the focus produced by a microscope objective. The objective itself, however, can significantly alter the pulses. Some effects, such as the broadening of pulses due to group delay dispersion in materials along the path, are understood and partial compensation for them can be made. Other effects, such as radial variations in the propagation time and variations in the pulse width, are less well understood. In this work, we investigate the radially dependent propagation and focusing of ultrashort pulses through a Zeiss CP-Achromat 100x, 1.25 NA, infinity-corrected, oil immersion microscope objective. We also extend to this high numerical aperture case the technique of collinear type II second harmonic generation frequency-resolved optical gating which has previously been used to measure the temporal intensity and phase of ultrashort pulses at the focus of air objectives with lower numerical aperture.

• Coauthored with David N. Fittinghoff, Jeffrey A. Squier and Michiel Müller.
• Proceedings manuscript for presentation given at SPIE’s Photonics West ’99.
• Appeared in Proceedings of the International Society for Optical Engineering 3616 76 (1999).
• Available here as PDF file (9.7M).

We demonstrate the first use, to our knowledge, of a compact diode-pumped, femtosecond fiber laser for third harmonic generation (THG) microscopy. We discuss the utility of this technique, as well as the technical issues involved in using this compact source, and demonstrate the first use, to our knowledge, of imaging by THG backlighting.

• Coauthored with Paul W. Wiseman, David N. Fittinghoff, Kent R. Wilson, Jeffrey A. Squier and Michiel Müller.
• Appeared in Applied Optics 38 7393 (1999).
• Available here (with subscription to Optics InfoBase) as PDF file.
• Reprinted in Selected Papers on Multiphoton Excitation Microscopy, SPIE Milestone Series MS175, edited by Barry R. Masters (2003).

We find that several key endogenous protein structures give rise to intense second harmonic generation (SHG) — non-absorptive frequency doubling of an excitation laser line. Second harmonic imaging microscopy (SHIM) on a laser-scanning system proves, therefore, to be a powerful and unique tool for high resolution, high contrast, three dimensional studies of live cell and tissue architecture. Unlike fluorescence, SHG suffers no inherent photobleaching or toxicity and does not require exogenous labels. Unlike polarization microscopy, SHIM provides intrinsic confocality and deep sectioning in complex tissues. In this study, we demonstrate the clarity of SHIM optical sectioning within unfixed, unstained thick specimens. SHIM and two-photon excited fluorescence (TPEF) were combined in a dual-mode non-linear microscopy to elucidate the molecular sources of SHG in live cells and tissues. SHG arose not only from coiled-coil complexes within connective tissues and muscle thick filaments, but also from microtubule arrays within interphase and mitotic cells. Both polarization dependence and a local symmetry cancellation effect of SHG allowed the signal from species generating the second harmonic to be decoded, by ratiometric correlation with TPEF, to yield information on local structure below optical resolution. The physical origin of SHG within these tissues is addressed and is attributed to the laser interaction with dipolar protein structures that is enhanced by the intrinsic chirality of the protein helices.

• Coauthored with Paul J. Campagnola, Mark Terasaki, Pamela E. Hoppe, Christian J. Malone and William A. Mohler.
• Appeared in Biophysical Journal 82 493 (2002).
• Available here as PDF file (1.7M).

• Coauthored with Paul J. Campagnola, William A. Mohler, Aaron Lewis and Leslie M. Loew.
• Appeared in Methods in Enzymology 361 “Biophotonics Part B” 47 (2003) edited by Gerard Marriott and Ian Parker.

We show that structural protein arrays consisting largely of collagen, myosin, tubulin and their associated proteins can be imaged in three dimensions with high contrast and resolution by laser-scanning second harmonic generation (SHG) microscopy. SHG is a non-linear optical scheme and this form of microscopy shares several common advantages with multiphoton excited fluorescence, namely, intrinsic three-dimensionality and reduced out-of-plane photobleaching and phototoxicity. SHG does not arise from absorption and in-plane photodamage considerations are therefore also greatly reduced. In particular, structural protein arrays that are highly ordered and birefringent produce large SHG signals without the need for any exogenous labels. We demonstrate that thick tissues including muscle and bone can be imaged and sectioned through several hundred micrometers of depth. Combining SHG with two-photon excited green fluorescent protein (GFP) imaging allows inference of the molecular origin of the SHG contrast in Caenorhabditis elegans sarcomeres. Symmetry and organization of microtubule structures in dividing C. elegans embryos are similarly studied by comparing the endogenous tubulin contrast with that of GFP::tubulin fluorescence. It is found that SHG provides molecular level data on radial and lateral symmetries that GFP constructs cannot. The physical basis of SHG is discussed and compared with that of two-photon excitation as well as that of polarization microscopy. Due to the intrinsic sectioning, lack of photobleaching, and availability of molecular level data, SHG is a powerful tool for in vivo imaging.

• Coauthored with William A. Mohler and Paul J. Campagnola.
• Appeared in Methods 29 97–109 (2003).
• Available here as PDF file (884k).

• Coauthored with Paul J. Campagnola, William A. Mohler and Sergey V. Plotnikov.
• Proceedings manuscript for presentation given at SPIE’s Photonics West ’03.
• Appeared in Proceedings of the International Society for Optical Engineering 4963 73 (2003).

We report what is to our knowledge the first optical imaging of voltage-clamped cells by second-harmonic generation. For the membrane-staining styryl dye di-4-ANEPPS, we determined the sensitivity of second- harmonic generation to be 18% / 100mV at an excitation wavelength of 850nm. This sensitivity is significantly better than the optimal 10% / 100mV under fluorescence and further establishes the importance of second harmonic generation for the functional imaging of membrane potential in living cells.

• Coauthored with Lei Jin, Aaron Lewis and Leslie M. Loew.
• Appeared in Optics Letters 28 1221–1223 (2003).

The macromolecular structure of purified cellulose samples is studied by second harmonic generation (SHG) imaging microscopy. We show that the SHG contrast in both Valonia and Acetobacter cellulose strongly resembles that of collagen from animal tissues, both in terms of morphology and polarization anisotropy. Polarization analysis shows that microfibrils in each lamella are highly aligned and ordered and change directions by 90° in adjacent lamellae. The angular dependence of the SHG intensity fits well to a cos² theta distribution, which is characteristic of the electric dipole interaction. Enzymatic degradation of Valonia fibers by cellulase is followed in real time by SHG imaging and results in exponential decay kinetics, showing that SHG imaging microscopy is ideal for monitoring dynamics in biological systems.

• Coauthored with R. Malcom Brown, Jr. and Paul J. Campagnola.
• Appeared in Optics Letters 28 2207–2209 (2003).

In this paper we present results from the simultaneous non-linear (second harmonic generation and two-photon excitation fluorescence) imaging and voltage-clamping of living cells. Specifically, we determine the sensitivity to trans-membrane potential of second harmonic generation by ANEP-chromophore styryl dyes as a function of excitation wavelength and dye structure. We have measured second harmonic sensitivities of up to 43% per 100 mV, more than a factor of four better than the nominal voltage-sensitivity of the dyes under “one-photon” fluorescence. We find a dependence of voltage-sensitivity on excitation wavelength that is consistent with a two-photon resonance, and there is a significant dependence of voltage-sensitivity on the structure of the non-chromophore portion of the dyes.

• Coauthored with Lei Jin, Mei-de Wei, Joseph P. Wuskell, Aaron Lewis and Leslie M. Loew.
• Appeared in Biophysical Journal 86 1169–1176 (2004).
• Available here as PDF file (180k).

• Coauthored with Aaron Lewis and Leslie M. Loew.
• Appeared in Imaging in Neuroscience and Development, a new edition of Imaging Neurons: a Laboratory Manual, edited by Rafael Yuste and Arthur Konnerth (Cold Spring Harbor Laboratory Press, 2004).

Second harmonic generation offers an important alternative and complement to fluorescence for the imaging of cellular structure and function. Staining the eggs of the sea urchin, Lytechinus pictus, with the styryl dye di-8-ANEPPS, we have observed large changes in both second harmonic generation and two-photon fluorescence after fertilization, consistent with the dynamics of exocytosis of cortical granules. With non-linear imaging on a scanning microscope, we are able to visualize the wave of exocytosis in real time.

• Coauthored with Mark Terasaki and Leslie M. Loew.
• Appeared in Biophysical Journal 88 L46–L48 (2005).

We present a membrane-staining dye, di-4-ANEPPDHQ, which differentiates liquid-ordered phases from liquid-disordered phases coexisting in model membranes under both linear and non-linear microscopies. The dye's fluorescence emission spectrum is blue-shifted 60 nm in liquid-ordered phases compared with liquid-disordered phases, and shows strong second harmonic generation in the liquid-disordered phase compared with the liquid-ordered phase. The ease of staining and the ability of this single dye to detect both phases, should lead to broad applications in biophysical studies of lipid domains in model membranes and cells.

• Coauthored with Lei Jin, Joseph P. Wuskell, Heather A. Clark and Leslie M. Loew.
• Appeared in Biophysical Journal 89 L04–L06 (2005).

Second harmonic generation (SHG) imaging microscopy is an important emerging technique for biological research, complementing existing one- and two-photon fluorescence (2PF) methods. A non-linear phenomenon employing light from mode-locked Ti:sapphire or fiber-based lasers, SHG results in intrinsic optical sectioning without the need for a confocal aperture. Furthermore, as a second-order process SHG is confined to loci lacking a center of symmetry, a constraint that is readily satisfied by lipid membranes with only one leaflet stained by a dye. Of particular interest is "resonance-enhanced" SHG from styryl dyes in cellular membranes and the possibility that SHG is sensitive to transmembrane potential. We have previously confirmed this, using simultaneous voltage-clamping and non-linear imaging of cells to find that SHG is up to four times more sensitive to potential than fluorescence. In this work, we have extended these results in two directions. First, with a range of wavelengths available from a mode-locked Ti:sapphire laser and a fiber-based laser, we have more fully investigated SHG and 2PF voltage-sensitivity from ANEP and ASTAP chromophores, obtaining SHG sensitivity spectra that are consistent with resonance enhancements. Second, we have modified our system to coordinate the application of voltage-clamp steps with non-linear image acquisition to more precisely characterize the time dependence of SHG and 2PF voltage sensitivity, finding that, at least for some dyes, SHG responds more slowly than fluorescence to changes in transmembrane potential.

• Coauthored with Lei Jin, Joseph P. Wuskell, David M. Boudreau, Aaron Lewis and Leslie M. Loew.
• Appeared in Journal of Membrane Biology 208 103–111 (2005).

Several biologically important protein structures give rise to strong second-harmonic generation (SHG) in their native context. In addition to high-contrast optical sections of cells and tissues, SHG imaging can provide detailed structural information based on the physical constraints of the optical effect. In this study we characterize, by biochemical and optical analysis, the critical structures underlying SHG from the complex muscle sarcomere. SHG emission arises from domains of the sarcomere containing thick filaments, even within nascent sarcomeres of differentiating myocytes. SHG from isolated myofibrils is abolished by extraction of myosin, but is unaffected by removal or addition of actin filaments. Furthermore, the polarization dependence of sarcomeric SHG is not affected by either the proportion of myosin head domains or the orientation of myosin heads. By fitting SHG polarization anisotropy readings to theoretical response curves, we find an orientation for the elemental harmonophore that corresponds well to the pitch of the myosin rod α-helix along the thick filament axis. Taken together, these data indicate that myosin rod domains are the key structures giving SHG from striated muscle. This study should guide the interpretation of SHG contrast in images of cardiac and skeletal muscle tissue for a variety of biomedical applications.

• Coauthored with Sergey V. Plotnikov, Paul J. Campagnola and William A. Mohler.
• Appeared in Biophysical Journal 90 693–703 (2006).

In this article, we characterize the fluorescence of an environmentally sensitive probe for lipid membranes, di-4-ANEPPDHQ. In large unilamellar lipid vesicles (LUVs), its emission spectrum shifts up to 30nm to the blue with increasing cholesterol concentration. Independently, it displays a comparable blue shift in liquid-ordered relative to liquid-disordered phases. The cumulative effect is a 60nm difference in emission spectra for cholesterol containing LUVs in the liquid-ordered state versus cholesterol-free LUVs in the liquid-disordered phase. Given these optical properties, we use di-4-ANEPPDHQ to image the phase separation in giant unilamellar vesicles with both linear and nonlinear optical microscopy. The dye shows green and red fluorescence in liquid-ordered and -disordered domains, respectively. We propose that this reflects the relative rigidity of the molecular packing around the dye molecules in the two phases. We also observe a sevenfold stronger second harmonic generation signal in the liquid-disordered domains, consistent with a higher concentration of the dye resulting from preferential partitioning into the disordered phase. The efficacy of the dye for reporting lipid domains in cell membranes is demonstrated in polarized migrating neutrophils.

• Coauthored with Lei Jin, Joseph P. Wuskell, Xuemei Dong, Dianqing Wu, Heather A. Clark and Leslie M. Loew.
• Appeared in Biophysical Journal 90 2563–2575 (2006).

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