Towards Magnetic Resonance Imaging of Bose-Einstein Condensates

Transcription

1 Towards Magnetic Resonance Imaging of Bose-Einstein Condensates Author: Prasanna Pakkiam Supervisors: Dr Lincoln Turner Dr Russell Anderson A thesis submitted for the degree of Bachelor of Science (Honours) in the School of Physics of Monash University 14 November 2014

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3 Abstract Conventional techniques used to image Bose-Einstein condensates (BEC) are diffraction limited in resolution and require complete destruction of the BEC to surpass this limit This project worked towards a minimally destructive imaging method that beats the diffraction limit by combining techniques in magnetic resonance imaging (MRI) and Faraday polarimetry The project included the design and implementation of gradient coils where analytics showed the possibility of compact topologies to generate magnetic field gradients required for MRI A differential autobalancing photodetector required to perform Faraday polarimetry was designed and characterised This photodetector was experimentally proven to be quantum shot-noise limited and showed 40 db common mode noise suppression over its operational bandwidth of approximately 5 MHz A custom theoretical framework was created to perform image extraction from MRI signals generated from sinusoidally modulated gradients The Faraday polarimetry technique was verified on an actual BEC where the detector able to resolve a Faraday signal showing atomic spin precession iii

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5 Acknowledgements I would first like to thank my supervisors Dr Lincoln Turner and Dr Russell Anderson for agreeing to supervise this project They were extremely patient and never ceased helped me when I faced theoretical or experimental brick walls during the course of the honours project I would also like to thank the entire Monash BEC lab group; specifically Martijn Jasperse and Alexander Wood for their help in the laboratory when I performed bench-testing I really valued their insight into practical optics and BEC experimentation v

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7 Contents List of Figures ix 1 Introduction 1 2 MRI using Faraday rotation 5 21 Resonant and off-resonant light-atom interactions 5 22 Conventional imaging techniques 7 23 Magnetic resonance imaging 9 24 Difficulties in detecting the spins in a BEC using MRI Using Faraday rotation spectroscopy to measure spins in a BEC Imaging a BEC using the Faraday effect MRI on a BEC using the Faraday effect Possible limitations to MRI of BEC Resolving issues due to dephasing and Stern-Gerlach forces Optimal MRI resolution on a BEC Making a minimally destructive and non-invasive measurement 20 3 Coils for off-diagonal magnetic field gradients Magnetostatics Gradient coil design Construction and bench-testing 30 4 Autobalancing photodetector for Faraday polarimetry Detector overview Design Differential and transimpedance stages Bandwidth tuning of transimpedance stage Autobalancing Practical considerations and construction Detector testing and characterisation Noise characterisation Testing detector autobalancing 47 vii

8 viii Contents 443 Photodetector bandwidth Common mode rejection ratio Faraday polarimetry of a spinor BEC 54 5 MRI Reconstruction with sinusoidal gradients Overview of the problem Square wave gradient modulation Overview of restrictions in the square wave gradient modulation Forward problem for a sinusoidal gradient Solution to inverse problem of the sine wave case (B 0 = 0) Solution to inverse problem of the sine wave case (B 0 0) Misalignment of condensate and magnetic field origin Comparing MRI with a square-wave and sinusoidal gradient modulation Previous work on sinusoidally modulated gradients 75 6 Conclusion and future work 77 Bibliography 79 A Integral transforms 83 A1 Fourier transform properties 83 A2 Modified integral transform properties 84 A3 Jacobi-Anger expansion 85 B Considerations for the photodetector design 87 B1 Input capacitance of the transimpedance stage 87 B2 Operation of the autobalancing stage 89 C Magnetic field and gradient of the 4 current-bar configuration 91

9 List of Figures 21 Schematic of apparatus to perform MRI on a BEC using the Faraday effect Dephasing of Larmor precessing spins over time Sequence of forces to enable minimally destructive imaging of the BEC Schematic showing how four current bars produce a B y / x magnetic field gradient Four current-bar configuration used to produce off-diagonal magnetic field gradients Loci of regions where four current bars produce zero field gradient Schematic of the final geometric gradient coil configuration Number of coil windings to produce a 39 G/cm magnetic field gradient over different gradient coil dimensions Coil inductance for gradient coils producing a 39 G/cm magnetic field gradient Final coil design as seen in Radia Simulated magnetic field profiles generated by the designed gradient coils Measured magnetic field of constructed gradient coils Simulations of winding imperfections in the constructed gradient coil Decomposition of polarisation axis into vertical and horizontal components Interface between the Faraday beam and the photodetector (the subtraction module) Photodetector circuit Small signal equivalent of the differential stage in Figure Electronic noise simulation of photodetector Schematic of the final optics in the Faraday polarimetry used to interface the photodetector with the BEC 43 ix

10 x List of Figures 47 Test Bench to perform noise characterisation Note that the COMP photodiode was blocked in this experiment Plot showing that the photodetector is shot-noise limited Plot showing the region in which the photodetector can be considered as shot-noise limited Test bench to perform noise characterisation Experimental verification of the autobalancing stage Test Bench to find the operational bandwidth of the photodetector Measured photodetector transfer function showing its bandwidth Simulation of photodetector transfer function for differing values of c f Test Bench to find the CMRR of the photodetector Measured CMRR values over different V LOG outputs for a 50kHz common mode tone Measured CMRR values over different frequencies and powers Measured noise spectrum of laser Spectrogram from a Faraday beam passing through a BEC to measure the Larmor precessing spins Flow chart of reconstruction for the constant-magnetic field case Test density distribution used in simulations Simulated reconstruction using a square wave modulated gradient Simulated reconstruction using a square wave modulated gradient under the presence of noise Simulated square wave modulation used to produce multiple timesequenced images of the BEC Exponential rise and fall of currents seen when using square wave gradient modulation Inadequacy of Fourier transform alone when performing image extraction using sinusoidally modulated gradients Flow chart of Reconstruction for the constant-magnetic field case Simulated reconstruction of the density distribution for a sinusoidally varying magnetic field gradient Simulated reconstruction of the density distribution for a sinusoidally varying magnetic field gradient under the presence of noise Simulations when using shorter periods or smaller magnetic field gradients in the square wave modulated gradient case Simulations when using longer periods or larger magnetic field gradients in the sinusoidally modulated gradient case 74 B1 Small signal equivalent of the differential stage in Figure B2 A simplification of the circuit shown in Figure B1 to find the output impedance 88 C1 A single Finite Current Bar 91

11 Chapter1 Introduction A BEC (Bose-Einstein condensate) is a macroscopic wavefunction That is, the condensate displays an order parameter that extends over a large region of space (up to hundreds of microns) This wavefunction is highly controllable and mostly immune to thermal perturbations This makes it an ideal platform for quantum simulation To measure the result of such a simulation, one simply takes an image of the BEC (effectively measuring the magnitude of the wavefunction) Conventional methods to take an image involve the act of shining light through the BEC and observing the resulting intensity patterns (as discussed in Section 22) These methods are diffraction limited in resolution Any attempt to beat this limit involves dropping the BEC and letting it undergo ballistic expansion However, this makes the technique inherently destructive Thus, it is difficult to observe chaotic or stochastic dynamics or other emergent phenomena (such as vortices) that are too small to be resolved in-situ via absorptive imaging methods In this project I investigated a method of using well-known magnetic resonance imaging (MRI) techniques to work towards an imaging method that is not only minimally destructive, but also of resolution better than the diffraction limit Although the technique uses one dimensional MRI, it should inherit the tomographic abilities of medical MRI to enable three dimensional imaging The ultimate aim is to gain the ability to measure or observe stochastic systems with features that are smaller than that resolvable by the diffraction limited imaging techniques The initial phases of this project involved the feasibility study to ensure that the experimental parameters for an MRI (that is, to surpass the diffraction limit in resolution while remaining minimally destructive) were physically achievable in the given BEC apparatus This involved the trade-off between measurement time and 1

12 2 Chapter 1 Introduction the tuning of the magnetic field gradient (required to perform MRI as described in Section 22) The trade-off balances the need for higher resolution while accounting for Stern-Gerlach effects; an effect that is only prevalent when performing MRI on a BEC (see Sections 28 to 210) After I ensured that the magnetic parameters were feasible, I designed, constructed and tested the magnetic field gradient coils These coils had to produce a magnetic field that pointed upwards (the y-axis) with strength increasing along the BEC (the orthogonal x-axis) as a linear gradient (that is, to produce a B y /dx gradient) Finding analytic forms of the magnetic fields brought insights into optimal coil construction for this particular gradient term The constraints placed on the design were geometric constraints (due to pre-existing apparatuses) where the goal was to minimal inductance whilst obtaining the required magnetic field gradients The coil inductance had to be minimised as the imaging sequence requires fast switching of the magnetic field gradient direction This process is described in Chapter 3 One result from the feasibility study was that the BEC produces an MRI signal (the signal upon which one may perform image reconstruction) that is too weak to be measured via the conventional MRI techniques (that is, using a pick-up coil) However, an alternative method to measure this same MRI signal is to use the Faraday effect in which the BEC spins couple to the off-resonant probe beam To utilise this effect, optoelectronics is required to measure the variation of the polarisation axis of linearly polarised light passed through the BEC (that is, to perform Faraday polarimetry ) I created the optoelectronics in the form of a differential photodetector I used a design by PCD Hobbs and modified it for the parameters of the MRI apparatus I simulated the circuit and created the fully assembled PCB I ran tests on the final photodetector on an optical test bench to ensure that it fulfilled the requirements as a detector to measure the MRI signal Details of the photodetector can be found in Chapter 4 Although the inductance of the coils was minimised, the geometric constraints in the experiment meant that the inductance was still too high to switch the coils fast enough to perform multiple-image MRI However, sufficiently fast oscillations were predicted to be possible if the coils were driven by a sinusoidally modulated current waveform This required modifications to the conventional image reconstruction techniques used in medical MRI After finding that the Fourier transform alone was inadequate for the sinusoidally time-modulated magnetic field gradient condition, I created a modified integral transform that solved the inverse problem I simulated this new algorithm on a computer to ensure the robustness of the algorithm in

13 3 the presence of noise I also showed that switching the magnetic field gradients sinusoidally does not have any negative effects when compared to the square wave gradient modulation This eases the power supply requirements for the experiment and may readily allow the extension to resonant coils for stronger gradients The development and simulation of this new algorithm is shown in Chapter 5

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15 Chapter2 MRI using Faraday rotation In this project we propose a BEC (Bose-Einstein condensate) imaging technique using MRI (magnetic resonance imaging) In the following sections I review the conventional approaches to BEC imaging and progressively build up a technique to incorporate well-known processes in medical MRI and Faraday spectroscopy This chapter ends with a feasibility study to highlight the practical considerations involved in this new BEC imaging technique 21 Resonant and off-resonant light-atom interactions Imaging techniques on ultracold atomic clouds (like a BEC) involve passing a monochromatic beam of laser light through the sample The effect of the photon-atom interactions is important when one gauges the destructivity and eventual image-quality given by the imaging technique When an atom absorbs a photon via some resonant atomic-transition, it will recoil in the direction of the photon momentum It spontaneously emits a photon and subsequently starts recoiling in random directions Such random recoil momentum experienced by atoms in a BEC contributes to a temperature rise If the temperature exceeds the critical temperature, the atoms cease to be a BEC and may leave the trap (the trap potential depth is of similar magnitude to the condensate energy) For example, atoms at the BEC critical temperature of 400 nk (as with 87 Rb) yield speeds of order 6 mm/s (assuming a random distribution of velocities) Absorbing a 780 nm photon yields velocities of approximately 60 mm/s, which corresponds to a temperature of approximately 40 µk Thus, the recoiled atoms are no longer a part of the BEC Therefore, for every photon the BEC absorbs resonantly, one atom is 5

16 6 Chapter 2 MRI using Faraday rotation lost from the condensate However, one should note that as they leave the BEC, they can still contribute to the image by scattering multiple photons during their journey out of the trap To minimise destruction of the BEC one may use either less-intense light or offresonant light The use of less-intense light simply lowers the number of photons passing through the BEC per second Thus, the BEC lasts a longer period However, in the context of detectors, this implies that fewer photons reach the detector Thus, the signal-to-noise ratio (SNR) of the resulting image is lower Hence, one exploits of off-resonant light driven at the same light-intensity Although every scattered photon has the same destructive effect, a large detuning lowers the photon-to-atom interaction rate This manifests in a complex refractive index [1]: n ref = ( 1 3λ3 ρ 8π ) ( δ 3λ 3 ρ 1 + δ 2 + i 8π ) δ 2, (21) in which, the real part contributes to a phase-shift and the imaginary-part contributes to attenuation in the amplitude Here λ is the vacuum wavelength of the incoming light, ρ is the atomic density and δ ω ω 0 Γ/2 is the detuning per atomic half-line width (where ω is the vacuum angular frequency of incident light and the ω 0 is the resonant angular frequency of the given atomic transition) transmission coefficient (ratio of output amplitude to incident amplitude) is: Thus, the ( ) ( T = exp 3λ3 n 8π δ 2 exp D ) (22) 2 Here D is the optical density The corresponding phase shift is: φ = 3λ3 n 8π δ 1 + δ 2 (23) The transmission coefficient is inversely proportional to the square of detuning for large detuning However, the phase is inversely proportional to the detuning for large detuning To better gauge the degree of destruction, one should consider a metric that describes whether the photon is likely to collide with an atom in the BEC One such metric is the absorption cross-section: σ = 7λ3 10π δ 2 (24) Thus, the degree of destruction reduces at an inverse proportion to the square of the detuning, for a fixed flux of photons (that is, a fixed intensity)

17 22 Conventional imaging techniques 7 22 Conventional imaging techniques This section highlights the current optical imaging methods along with their advantages and disadvantages This helps gauge the true novelty and advantages of the technique proposed in this project The metrics highlighted in this survey are the resolution and degree of destruction (that is, the number of atoms lost in the BEC due to a single imaging process) The common techniques involve some variant of absorption-based imaging In these techniques, one sends a wide beam of light through the BEC The imaging process involves the interpreting of the resulting intensity pattern The basic absorption imaging technique involves a beam of light passing through the BEC and focused onto a CCD camera via a lens The density distribution can be inferred from changes in the intensity of this light beam (captured in the CCD) as compared to the light when no BEC is present (see Equations 22, 23 and 24) The optical setup in such an imaging apparatus is simple The goal is to tune the light such that the atoms optimally absorb the incoming photons (for example, one may utilise the D2-Line which is a S 1/2 F = 2 S 3/2 F = 3 transition in 87 Rb corresponding to 780 nm light [2]) During this process, the atoms that absorb an incoming photon are knocked out of the trap Thus, this is a destructive measurement To make a less destructive measurement, one may increase the detuning away from the resonant transition [3] This technique is known as off-resonant absorption imaging (as opposed to resonant absorption imaging) Equation 24 shows that if one increases the detuning, the degree of destruction lowers by the square of the detuning for the absorption cross-section decreases Now one measures the resulting image quality via the signal-to-noise-ratio (SNR) The SNR for an absorption-based technique is proportional to the square root of the cross-section [4, 5] Therefore, while the degree of destruction lowers by an inverse-square relationship with respect to the detuning, the SNR only lowers by an approximate inverse-relationship Thus, the trade-off in making the measurement less destructive is the lower SNR when comparing to the resonant-imaging case where the SNR is at its peak It is not ideal to pass resonant light through the BEC in the trap even though one attains the maximal SNR This is because of the high optical density (see Equation 22) of a BEC under resonant light (for example, it is approximately 300 for sodium to give a transmission coefficient of e 150 [3]) Thus, one obtains a black shadow of the BEC without any contrast in the density of the BEC cloud One may have the effect of lowering the density by performing a destructive measurement where one drops

18 8 Chapter 2 MRI using Faraday rotation the BEC to allow it to undergo ballistic expansion Alternatively, one can obtain contrast in the densities using off-resonant light However, this is still an absorptive technique that has limited contrast (that is, it only utilises the amplitude-lowering characteristics of the BEC as given in Equation 22) Thus, other methods consider the use of the phase-shifting characteristics of the BEC Techniques that utilise the phase-shifting properties of the BEC are known dispersive techniques (see Equation 23) as opposed to absorptive techniques One may utilise standard microscopy techniques such as defocus or phase-contrast imaging to recover the phase-shifting characteristics of the BEC [6] Similar to phase-contrast imaging, one may utilise techniques utilised in holographic microscopy [7, 8] The SNR of these methods are follow the same characteristic (with respect to detuning) as absorption images [5] Another method that utilises off-resonant light is fluorescence imaging [1] Here one detects the photons that were re-emitted in a different direction due to fluorescence One images the few photons detected from an otherwise dark background as opposed to the previously outlined imaging techniques that compare the loss in intensity of the incident beam due to the presence of the BEC Unlike the scattering effects in off-resonant absorptive imaging (where the BEC acts as a lens for the incident light), interference fringes do not form in fluorescence imaging Thus, the image need not require processing to remove background-fringing effects This method is not optimal on thicker clouds for it cannot distinguish higher densities within BEC samples This is because the fluorescent light in the middle of the BEC will undergo reabsorption within the BEC cloud itself Thus, only the outer atoms in the BEC will contribute fluorescent light to the image leaving no contrast with respect to the central density distribution The major shortcoming of all the outlined optical imaging methods is that the image formation is coupled to the distribution of the absorbed, dispersed or reemitted photons Thus, the image formation is diffraction limited (the Abbe limit) Henceforth, when using the common infrared-to-red laser light, the resolution is bounded by approximately 400 nm Note that this bound is the best-case scenario where one requires a lens with a large numerical-aperture However, it is usually impractical in the experimental setup to place such a lens due to limited opticalaccess to the BEC itself Decreasing the laser wavelength would be possible by moving to a blue transition such as the 421 nm energy transition in 87 Rb and using a blue laser However, this method is not used due to cost and the inexistence of high frequency, high-intensity and coherent light sources Another issue is that this light must be compatible with the new transition (for example, 421 nm)

19 23 Magnetic resonance imaging 9 A method to overcome the diffraction limit is to drop the BEC Dropping the BEC and letting it undergo expansion increases the size of the BEC This expansion is primarily due to mean-field interactions Although this achieves a larger resolution, it is a clearly destructive measurement and prevents one from observing dynamics in the single BEC The partial-transfer technique achieves a further minimally destructive measurement by partially transferring a small uniform proportion of the atoms the BEC into an auxiliary state [9] This fraction of atoms drops out of the trap and undergoes ballistic expansion before being imaged at the bottom of BEC chamber One repeats this process to continue imaging the BEC The fundamental disadvantage is that one does not utilise the entire BEC in the imaging process and thus, one sacrifices the SNR when considering smaller fractions 23 Magnetic resonance imaging The issue with the conventional BEC imaging techniques is that they do not surpass the diffraction limit in resolution without destroying the BEC itself Thus, this project sought to investigate the potential of utilising magnetic resonance imaging techniques on the BEC MRI is well known in the field of medicine where physicians use it for non-invasive diagnostic probes on living tissue This project will employ this technique to image the density distribution of a BEC This section shall highlight the aspects of MRI relevant to this project including the differences to conventional MRI [10, 11] The procedure begins by passing a uniform magnetic field directed along the x- axis through the material sample In conventional MRI, this aligns a fraction of the nuclear proton spins along the x-axis Upon alignment, a radio frequency π/2 pulse tips the spins such that they are on the y-z plane This induces Larmor precession about the x-axis at the angular frequency: ω L = g F µ B B γb (25) where g F is the Landé g-factor, µ B is the Bohr magneton, γ is the gyromagnetic ratio and B is the applied magnetic field strength After the spins start precessing, the applied magnetic field is switched to a linear gradient such that the magnitude increases along the z-axis Thus, each segment of the sample experiences a field of a different magnitude Henceforth, each segment undergoes a different frequency of Larmor precession, as the Larmor frequency is proportional to the applied magnetic field strength Detecting the total time-varying

20 10 Chapter 2 MRI using Faraday rotation oscillations of the spins (due to Larmor precession) and inferring the number of atoms precessing at a given Larmor frequency yields a function proportional to the density distribution of the atoms along the z-axis In conventional MRI, a pick-up coil detects these Larmor precessing spins (the magnetic spins induce a voltage signal in the coil) 24 Difficulties in detecting the spins in a BEC using MRI Conventional MRI uses strong magnetic fields to precess the spins of 1 H nuclei (that is, protons) in the sample, while this project will precess the total spins of the actual rubidium atoms in the BEC (electrons hyperfine-coupled to their atomic nuclei) This difference is significant when one considers the gyromagnetic ratio in conventional MRI and that in this project (see Table 21) Thus, a high frequency signal (to surpass low frequency noise) can be achieved with a relatively lower magnetic field strength However, the major difference is that the total number of precessing spins is large in conventional MRI Even though the BEC has a larger percentage of aligned spins under precession, the BEC does not have enough precessing spins to induce an experimentally measurable voltage Thus, even if the pick-up coil could be placed within microns of the BEC (experimentally unfeasible), the induced voltage is too low to overcome electrical thermal noise for any coil or amplifier Thus, a different method is required to measure the precessing spins MRI (H-nuclei) BEC ( 87 Rb) Gyromagnetic Ratio (f L /B) 4258 MHz/T 6991 MHz/T Percent of aligned spins 0001% 100% Total number of spins Signal strength 1 µv 01 pv Table 21: Comparison of conventional MRI (on 1 cm 3 of water at room temperature) to an 87 Rb BEC for a 1T field and a 1 Gauss per Ampere pick-up coil

21 25 Using Faraday rotation spectroscopy to measure spins in a BEC Using Faraday rotation spectroscopy to measure spins in a BEC It is well known that a linearly polarised electromagnetic wave undergoes a rotation in its polarisation axis when travelling through a magnetised medium can explain this effect by considering the fact that a linearly polarised wave is a superposition of both left and right circularly polarised light When the light passes through the magnetised material, the material exhibits a different refractive index on these two circularly polarised components This imparts a different phase shift to the circularly polarised components when the wave leaves the material to yield a net rotation in the polarisation axis of the linearly polarised wave One The different refractive indices n for different circular polarisations of the incident waveform inside an alkali vapour are [12] ( n ± = 1 λ3 Γρ 16π 2 2 ± ˆF ) x, (26) F where λ is the wavelength of the incident light, Γ is the atomic line width of an electric dipole transition of angular frequency ω 0, ρ is the atom density of the vapour (or BEC) and is the detuning (ω ω 0 ) away from resonance (ω is the angular frequency of the incident light) F denotes the magnitude of the total spin vector (sum of nuclear and electron spins) The key aspect of this difference in refractive indices is that it is proportional to ˆF x, the spin-projection of F onto the x-axis (the optical axis) The acquired phase difference between the two circularly polarised components can be found by considering the optical path-lengths (product of refractive index n and length of material L) of the two components when travelling through the material Since the phase difference causes the net rotation in the polarisation axis of the linearly polarised wave, the Faraday rotation angle is proportional to the ratio of the optical path-length and the wavelength: θ 2πl λ L 0 (n + n ) dx = Γλ2 4π L 0 ˆF x ρ (x) dx (27) F Since the Faraday rotation angle is proportional to the spin-projection ˆF x, the sign changes depending on whether the spin F is parallel or anti-parallel to the direction of the incident light Thus, Larmor precession of the spins induces an oscillating rotation of the polarisation angle This angle can be inferred by

22 12 Chapter 2 MRI using Faraday rotation splitting the polarisation components (via a polarisation beam splitter) and precisely measuring each component individually with a photodetector Although imaging via Faraday rotation spectroscopy is common in materials engineering, the usage of Faraday rotation in the realms of imaging a BEC is uncommon when compared to the techniques outlined in Section 22 However, several groups have utilised the Faraday effect to measure the total spins in a BEC In 2001 a group in Glasgow performed Faraday-rotation spectroscopy in cold lithium atoms [13] Although they did not utilise the technique to image the cloud of cold lithium atoms, they successfully observed the Faraday effect The details of their experimental setup and analysis provide insight into the experimental setup for this project This experiment only aligned the spins via a magnetic field directed parallel to the beam of light passing through the cold atoms Thus, the group did not observe Larmor precession In 2003 the Optical Sciences Centre at the University of Arizona performed a similar experiment with cold caesium atoms [12] However, this experiment utilised a magnetic field directed along the y-axis (that is, perpendicular to the optical beam) The magnitude of the magnetic field was constant along the imaging beam axis Thus, all atoms experienced precession at the same Larmor frequency yielding no chance of spatial density encoding In 2005, a group in University of California Berkeley utilised atomic birefringence to image the Larmor precession of spins in an 87 Rb BEC [14] Atomic birefringence is similar to the Faraday effect in which the atoms display a different refractive index to different polarisations of light The Clebsch-Gordan coefficients for the F = 1 to F = 2 transition for 87 Rb, are such that when passing σ + polarisation light, the probability of photon absorption for the m F = 1 state is lower than that for the states m F = 0 and m F = +1 The experiment utilises polarisation-contrast-imaging to obtain time-resolved minimally destructive images of the Larmor precessing spin states of the BEC However, the images are still diffraction limited and only a small number of images can be taken with a single BEC 26 Imaging a BEC using the Faraday effect Although there is no documented evidence of a group using the Faraday effect and magnetic resonance to image a BEC, several groups have utilised the Faraday effect to measure the spins in a BEC while one group wrote a theoretical paper detailing MRI on a BEC In 2002, a group in Kyoto University published a theoretical paper that outlined

23 26 Imaging a BEC using the Faraday effect 13 an imaging technique specific to Bose Einstein condensates that utilised Faraday rotation and techniques in MRI [15] This paper provides some analysis of the imaging technique to be utilised in this project In addition, it included analysis on the achievable resolution and SNR However, this paper utilises the time-of-flight method where the BEC undergoes ballistic expansion Thus, the imaging technique is destructive as opposed to the non-invasive in-situ method proposed in this project (see Section Section 211) It should be noted that if one were to take a destructive (dropped BEC) measurement, a normal off-resonant absorption imaging technique provides a simple setup with a large SNR The paper cites issues with the Stern- Gerlach effect that will cause the BEC to separate spatially as discussed in detail in Section 28 However, this paper only states that the measurement should be quick to avoid errors in measurement due to this spatial separation without any notion of a solution to this problem This could be because the proposed technique in this paper is a completely destructive single-measurement method and thus it does not concern itself with the reconstruction of the BEC after the Stern-Gerlach separation In June 2013, Gajdacz replicated a previous experiment [14], but used a darkfield technique [16] This setup yielded the same results along with the advantages and disadvantages of the previous experiment However, as a dark-field technique, the measured photons correspond only to the perturbed polarisation axis That is, if the beam were vertically polarised on incidence, the small Faraday rotation would yield a small change to the amplitude on the horizontal polarisation axis from what was initially at zero amplitude The polarisation beam splitter discards the vertical axis and only considers the small perturbations on the horizontal axis to infer the Faraday rotation angle The paper claims an improvement in the signal-to-noise ratio (SNR) One should finally note that a CCD-based technique has an advantage over a technique utilising photodiodes with respect to photons that miss the BEC (and subsequently undergo no Faraday rotation) If the photon misses the BEC, one may simply discard the corresponding pixel in the CCD (as it will show no Faraday rotation) However, with a photodiode, such light will constitute extra shot-noise and raise the noise floor Thus, one will have a lower SNR if such light remains unfiltered before reaching the photodiode Such spatial filtering can be achieved via simple optics such as an aperture or slit to block unnecessary light from reaching the photodiode

24 14 Chapter 2 MRI using Faraday rotation 27 MRI on a BEC using the Faraday effect This project exploits the concepts from MRI to precess the spins at a rate proportional to their positions along the BEC When passing linearly polarised light though the BEC, the polarisation axis oscillates to mirror the time-resolved MRI signal (via the Faraday effect) Optoelectronics detect this oscillation to produce a measurable voltage signal from which one may perform image reconstruction to obtain the density distribution of the BEC Before precession can begin, the BEC must have its spins aligned along the y- axis Unlike conventional MRI, the nature of the BEC trap enables one to start the experiment with all the atomic spins completely aligned along the positive y-axis Thus, one does not require strong magnetic fields to perform initial spin-alignment After state preparation, a constant magnetic field (pointing along the y-axis) is initialised Upon initialisation of the constant magnetic field, the spins are then tipped onto the x-z-plane via a rotation operation induced by a radio frequency π/2 pulse Since the applied magnetic field is along the y-direction, the atoms precess at a uniform Larmor frequency about the y-axis Now the magnetic field gradient is switched on (pointing along the y-axis and increasing linearly in magnitude along the x-axis) This causes the Larmor precession frequency to increase along the x-axis due to the increasing magnitude of the magnetic field strength of the gradient-field along the x-axis (see Figure 21) To begin the imaging process, one passes a beam of linearly polarised light through the BEC along the x-axis The light-beam undergoes a rotation in its polarisation axis due to the Faraday effect The magnitude of this rotation is dependent on the density of the cloud and the azimuthal phase of the spins at a given position (see Equation 27) Since the spins undergo Larmor precession, the polarisation axis oscillates at the Larmor frequency This rotation angle has the greatest magnitude when the spins are aligned or anti-aligned with the direction of the lightbeam (along the x-axis) and zero-magnitude when the spins are orthogonal to the x-axis The frequency of this Faraday rotation increases along the x-direction due to the presence of the magnetic-field gradient, while the amplitude of the sinusoidal signals varies with the density of the cloud at the given x-plane Optoelectronics detect these polarisation oscillations to yield a voltage output that corresponds to the time-domain MRI signal s(t) The final image is a time-domain Fourier-transformation that unravels the final time-varying polarisation (axis oscillation) signal into a spatial density distribution

25 28 Possible limitations to MRI of BEC 15 y x Linearly Polarised Light Optoelectronics Σ = Figure 21: Schematic of apparatus to perform MRI on a BEC using the Faraday effect The BEC shown in blue is immersed in magnetic field gradient while linearly polarised light probes the MRI signal via the Faraday effect φ(t) is the phase angle of the Larmor precessing spins This is because the amplitude of a given frequency of the final signal encodes the density at a particular plane in the x-axis Note that if the magnetic field-gradient were replaced with a uniform magnetic field (along the x-axis), the Fourier-transform only shows a single tone corresponding to the Larmor-frequency in which all atoms precess and thus, reveals no spatial distributions[17] 28 Possible limitations to MRI of BEC Several issues inherent in MRI based techniques need to be addressed Such issue include the relaxation of spins, Stern-Gerlach force and the choice of spin-echo sequences during dephasing When the spins (nominally precessing about the y-axis) lose energy to the surrounding environment, they start tiling such that they are no longer perfectly orthogonal to the y-axis (known as spin-relaxation time in medical MRI and described by the time constant T 1 ) This causes the projection of the spin along the x-axis to reduce and one receives a weaker Faraday signal (see Equation 27) This is prevalent in conventional MRI for the constituents in the material can lose energy to

26 16 Chapter 2 MRI using Faraday rotation neighbouring matter during precession In the case of a BEC there is no significant mechanism to cause a tilt in the spins However, there is still a decay in the signal due to number loss [14] That is, stray atoms (for example, nitrogen or oxygen) inside the cell containing the BEC can scatter off atoms in the BEC and thus, forcing them to leave the trap Since the decay is uniform over the entire condensate, if the measurement time is significantly less than the decay time, the image will not lost integrity in describing the spatial density distribution Similarly, if one assumes that there is minimal condensate loss over multiple images, one may normalise the density distributions to allow valid comparisons of adjacent time-sequenced density distributions This assumption is valid as the measurement time is in the order of several hundred microseconds, while the time it takes to diminish the measured signal (proportional to BEC atom number) by 1/e is is longer than one second Consequently, this issue shall be discarded The loss of coherent phase in the precessing spins causes issues when taking multiple images of the BEC That is, the relative phase between different atomic spins along the x-axis will diverge away from zero due to the spins precessing at different Larmor frequencies (see Figure 22) The overall superposition causes a decay in signal amplitude over time One requires the system to be in phase before taking another image Therefore, if one does not bring the spins back into phase, the measurement does not provide information encoded during the initial section of the time domain signal One possible resolution is to utilise the conventional MRI spin-echo technique [10, 11, 18] B y (x) x V (t) t Figure 22: Dephasing of a lattice of seven spins (viewed from above) along the x-axis due to a magnetic field gradient inducing different Larmor precession frequencies (see Equation 25) The final issue is that the presence of the spatially varying magnetic field will cause the Stern-Gerlach effect to manifest Given the magnetic moment of the spins µ and the magnetic field B, the Stern-Gerlach force is[19]: F SG = (µ B) (28)

27 29 Resolving issues due to dephasing and Stern-Gerlach forces 17 Thus, the direction of the force is only dependent on the spatially varying magnetic field strength That is, the Stern-Gerlach force manifests along the x-direction Now if the atoms move along the x-direction, the resulting image will have blur for it would have a superposition of the original image and the shifted image Another problem is that the spatial translation is not uniform across the BEC This is because the BEC (in the state after tipping onto the x-z plane) has three components enumerated by three magnetic quantum numbers (assuming that the atoms are in an F = 1 state) +1, 0 and 1 The proportions in the +1 and 1 states will move along the x-axis in the opposite directions such that the BEC eventually separates into three parts As Larmor precession requires the overlap of the m F = 1 and m F = 1 states, the three spatially separated parts will no longer undergo Larmor precession Thus, there is will be no detected signal Note that in conventional MRI techniques, the Stern-Gerlach effect is irrelevant as the material is in a rigid condensed phase (solid or liquid) Thus, the Stern- Gerlach force is small compared to the interatomic interactions (for example, bonding or thermal excitations) Therefore, this force is not an issue as it will not cause significant deviation in the sample to cause blurring during the imaging process However, in a BEC, this is a major constraint in the imaging resolution as discussed in the next section 29 Resolving issues due to dephasing and Stern-Gerlach forces The Stern-Gerlach shift is a problem in both single-image experiments and multiimage experiments This is because one requires a certain period of time to have enough information to perform a Fourier transform However, if the Stern-Gerlach shift too large during this measurement period, the retrieved image will be blurred Several proposed resolutions shall be explored in this section The first proposed solution is to add a reverse magnetic field gradient The idea is that any velocity imparted on the BEC due to the initial Stern-Gerlach force will be reversed to leave the system in its initial rest-state Thus, one has a nearly non-invasive imaging technique Note that this places demands on the apparatus to have the ability to switch the direction of a large current (in a short time interval) through a predominantly inductive load Another proposed solution (not implemented in the project) is to keep the magnetic field switched on over the duration of the imaging process (this includes the

28 18 Chapter 2 MRI using Faraday rotation duration over multiple images) while applying radio frequency pulsed sequences The idea is that one could send an radio frequency pulse coupled to all Larmorprecessing spins and perform a 180 spin-flip Since the magnetic moment of spins is now in the opposite direction, the direction of the force reverses (see Equation 28) Thus, any velocity imparted on acceleration in one direction will be decelerated back to rest The radio frequency pulse must be capable of imparting a 180 flip to the atomic spins across a continuous range of Larmor frequencies present in the BEC Dephasing causes a loss in the signal strength One can recover the original signal strength by either restarting the experiment (that, is realign the spins to the x-direction and tip the spins again) or using the spin-echo technique used in MRI The spin-echo technique requires the spins to travel in reverse This effectively reverses dephasing The techniques described to solve the Stern-Gerlach issues in fact, cause the spins to reverse magnetic moments via either a gradient-field reversal or the application of a radio frequency π-pulse Thus, one may use either of the proposed techniques to resolve both the Stern-Gerlach shift and the dephasing issue without restarting the experiment 210 Optimal MRI resolution on a BEC Unlike conventional optical imaging techniques, the magnetic field gradient encodes the longitudinal density distribution along the x-axis rather than absorptive encoding of the two dimensional density distribution along the transverse y-z plane The light may freely diffract for the density encoding is embedded in the polarisation axis oscillations in any one of the diffracted light rays (diffraction does not affect the polarisation axis orientation) Thus, the proposed experimental technique is not diffraction limited The key parameters determining the optimal resolution x are the measurement time τ m and the applied magnetic field strength along the y-direction B y = B G x (where B G is the field gradient) Now combining this with Equation 25 and differentiating with respect to x yields the Fourier-constrained resolution (noting that the frequency step is simply ω = 2π τ m ): x F = ( ) 2π 1 (29) γ B G τ m The loss in resolution due to the blurring caused by the Stern-Gerlach separation must also be considered To find the Stern-Gerlach separation along z in the BEC

29 210 Optimal MRI resolution on a BEC 19 during the imaging process, a simple lower-bound approximation in the resolution can be found using basic linear mechanics 1 Thus, using the Stern-Gerlach force given in Equation 28, the resolution-limit due to the Stern-Gerlach shift is: x SG = µb ( ) G γ 2m τ f 2 B G τ 2 2m m (210) The physical interpretation is that the Fourier theory infers that a longer measurement time yields greater resolution However, measuring for too long a period yields a large Stern-Gerlach blur The resolution limit due to the signal-to-noise ratio of the photodiodes is smaller than both the Fourier and Stern-Gerlach constraints [20] Therefore, one only needs to optimise the two constraints given in Equations 29 and 210 to find the optimal resolution Thus, equating the field-gradients from both equations yields an equation for the measurement time in terms of the optimal resolution x opt : τ m = The associated magnetic field gradient is: ( ) 2m x 2 h opt (211) B G = 4mπ γ 2 τ 3 2 m (212) Now one selects a desired resolution to obtain the associated measurement time and magnetic field gradient (see Table 22) A resolution of 500 nm only requires a 260 G/cm field gradient with a measurement time of 110 µs These two parameters are technically feasible in the lab To provide a proof of concept, this project aimed for a 1 µm resolution That is, a 33 G/cm gradient and 434 µs measurement time 1 Note that d = 1 2 at2 given the distance travelled d, acceleration a and time taken to perform the translation t The acceleration is the quotient of the force applied and the mass of the atoms in the BEC

30 20 Chapter 2 MRI using Faraday rotation Resolution (µm) Magnetic Field Gradient (G/cm) Measurement Time (µs) Table 22: The required magnetic field gradients and measurement times for different desired resolutions 211 Making a minimally destructive and non-invasive measurement When imaging the BEC, the process should be minimally invasive so that it does not affect the time-evolution of the BEC In the MRI process, in order to bring the experiment back into the initial state, one must reverse the dephasing and Stern- Gerlach effects The proposed methods to do this (see Section 29) have the net effect of accelerating the spins in the direction opposite to that of the Stern-Gerlach separation and the reversing the direction of precession to fix the dephasing issue If the time one waits during this reversing process is the same as the measurement time, the system would have dephased and the atoms would be at rest However, the net effect is that the spins now have a net displacement given by: x disp = γb Gτ 2 m m (213) The net effect yields a 1 µm separation of spins in either direction if tuning for a resolution of 500 nm To reverse this displacement, the sequence of forces already applied to the BEC (via magnetic field gradients) are applied once more but in reverse (see Figure 23) This yields no net displacement of the atoms and places the atoms at rest with all spins in-phase This process yields two pairs of signals where one of the signals in each pair is a time-reversed image of the other (assuming that the spins have not displaced during this sequence of applied forces) Thus, concatenating these two time domain signals in each pair yields symmetric timedomain signals and thus, improves the quality of the final images in the frequency domain Alternatively, one may consider the two signals in each pair to be of separate images to yield four separate images in total

31 211 Making a minimally destructive and non-invasive measurement 21 Upon applying this sequence of forces to obtain non-invasively captured images, one may wait a certain period (to allow the BEC to evolve through time) before taking another image During this period, the Faraday beam can be switched off to ensure that photons do not scatter atoms away from the trap To keep the acceleration due to the Stern-Gerlach force zero, one simply deactivates the magnetic field gradient Note that one still requires a constant magnetic field directed towards the positive y-axis to ensure that the spins stay aligned During this period, the spins stay in phase for they all precess at the same Larmor frequency To make the Faraday probe minimally destructive, one may use of a particular wavelength of light near 790 nm wavelength such that the BEC has no net force imparted upon it due to off-resonant scattering This so-called magic wavelength ensures the imaging technique is overall non-invasive Note that although the BEC is brought to rest at the same initial position, the technique still assumes that the BEC undergoes minimal time-evolution during the imaging phase In addition, the BEC must be stable under temporary spatial displacements caused by the Stern-Gerlach forces

32 22 Chapter 2 MRI using Faraday rotation a/(10-12 ms -2 ) t(μs) v/(10-9 ms -2 ) d/(10-9 ms -2 ) s(t) t(μs) t(μs) t(μs) Figure 23: Sequence of forces to enable the optimal non-invasive measurement of the BEC at 500nm resolution The graphs show the acceleration, velocity and displacement of atoms in the BEC The red and blue curves are the opposite magnetic spin components (m F = 1 and m F = 1) that displace due to the Stern-Gerlach force The positive and negative acceleration components can be due to a reversal of the magnetic field gradient or a spin-flip due to a radio frequency π/2 pulse The measurements are made when the acceleration is positive for that corresponds to minimal displacement due to the Stern-Gerlach force The bottom plot shows the dephasing when the magnetic field gradient is switched on; when the magnetic field gradient is switched off, the spins precess about a constant magnetic field directed along the y-direction

33 Chapter3 Coils for off-diagonal magnetic field gradients To perform Faraday MRI on the spinor BEC apparatus, we require coils that produce an off-diagonal B y / x magnetic field gradient to encode the positions and densities of the BEC with the frequency of its Larmor precessing spins (see Figure 21) The coils on the current BEC apparatus are inadequate for creating this type of magnetic field gradient This necessitated the design and construction of custom coils that produce this off-diagonal magnetic field gradient Due to power supply and geometric restrictions in the laboratory, the target gradient was taken to be 39 G/cm Performing MRI with this gradient will yield images with 1 µm resolution (see Section 210) even if the BEC is misaligned from the geometric centre of the coils The four custom magnetic field coils were simulated and optimised numerically in Mathematica using the magnetostatics package Radia The coils were verified to produce the correct magnetic field gradient via a magnetometer probe on a test bench 31 Magnetostatics The coil design is a variant of the four current-bar configuration This configuration produces a uniform off-diagonal magnetic field gradient in the required direction as seen by applying the Ampere s right hand screw rule in Figure 31 Now consider the four bars to be of length l each carrying a current I as shown in Figure 32 The magnetic field gradient at the origin of the four current-bar configuration (where each bar is of a distance w x and w y from the origin along the 23

34 24 Chapter 3 Coils for off-diagonal magnetic field gradients y y x x Figure 31: A schematic to show that the four current bar configuration produces the required B y / x magnetic field gradient (red = current bars, blue = magnetic field lines) w y I I y I I x y I I z w x Figure 32: Four current bar configuration to produce off-diagonal B y / x magnetic field gradients x and y respectively) is: B y x (4-bar) (x,y,z)=0 = µ 0I π l ( w 2 x + wy) 2 2 w 2 y w 2 x w 2 x + w 2 y + ( l 2 ) 2 ( wx 2 ( w 2 x + wy 2 ) w 2 x + w 2 y + ( l 2 ) 2 ) 3 2, (31) as shown in Appendix C It is useful to define the aspect ratio d w x /w y to simplify the equation to: B y x (4-bar) (x,y,z)=0 = µ 0I πwy 3 l (1 + d 2 ) 2 1 d d 2 + ( l 2w y ) 2 d 2 ( 1 + d 2) ( ( ) ) d l 2w y (32) This form clearly shows that the magnitude of the gradient scales with the inverse cube of w y This analytic formula has limited utility as real wires are not line elements of infinitesimal diameters Thus, the implemented design will be optimised via numerical simulations However, the utility of this expression is seen by the

35 31 Magnetostatics 25 formulation of a four current-bar configuration where the magnetic field gradient is zero A zero-gradient configuration is useful when placing the return bars These bars are required as the current-bar cannot exist by itself to create current along wires shown in Figure 32 1 The return bars create a closed loop of wire (an element through which one can easily drive a given current) Closed loops also enable multiple turns on the coil to give a greater field gradient than that of a single loop of wire However, the return bars will produce a magnetic field gradient in the opposite direction to that of the primary current bars and subsequently cause partial cancellation of the initial gradient Thus, a configuration in which all four return bars produce a null gradient is useful when aiming to maximise the magnitude of the magnetic field gradient The net gradient from the return bars can also be determined by using Equation 32 That is, by setting the gradient to zero; solving for the aspect ratio d to determine the configuration of the return bars that produce no gradient yields: ( ) ) ( 2 ( ) ) 2 2d 4 + d (1 2 + l 2w y 1 + l 2w y = 0 (33) This is a simply quadratic equation in d 2 aspect ratio required to yield a null-gradient at the origin is: Thus, by the quadratic formula, the ( d null = 1 ( ) ) 2 2 ( ( ) ) ( 2 ( ) ) l 2w y l 2w y 1 + l 2w y 1 2, (34) where the negative solution has been ignored for it will give a non-physical complexvalued aspect ratio This aspect ratio is not fixed; it can be tuned by varying the bar length l and the vertical bar separation 2w y Another subtlety to note is that when crossing the locus configurations of null gradients, the gradient reverses in direction That is, if the aspect ratio of the primary current bars satisfies d > d null and the return bars satisfy d < d null, the return bars (with current moving in the opposite direction) actually reinforce the gradient created by the main four bars This insight can potentially allow for coils where the primary current-bars and the return current-bars act in concert rather than against each other to produce a larger gradient than the primary current-bars alone This can allow for smaller coils (as opposed to larger coils where the effect of the return bars are neutralised by 1 It can be achieved if one has a heated element on one end and uses a small loop to accelerate electrons and to catch them on the other end in some electron reservoir

36 26 Chapter 3 Coils for off-diagonal magnetic field gradients placing them a large distance away from the origin), and smaller coil inductances (important for fast switching times as discussed in Section 53) 32 Gradient coil design The design was numerically simulated and optimised in Mathematica using the Radia package [21] The geometric constraints are mainly set by the pre-existing apparatus Since Equation 32 shows that the magnitude of the gradient scales with the inverse cube of the vertical separation 2w y Thus, it is in our interest to place the bars as close as possible to the origin, where the BEC resides The smallest allowed w y is 15 mm as lasers entering the BEC chamber will be blocked if the coils were to be spaced closer The largest gradients can be achieved by placing the four primary current bars above the BEC cell, but other fixtures force the point of maximal gradient to be such that w x is 35 mm The placement of the return bars should be such that they are on the opposite side of the locus of null gradients (with respect to the primary current bars) However, the surrounding fixtures prevented placement of these return bars on locations of nulling gradients regardless of the current-bar length l (see Figure 33) In fact, the geometry constrained the return bars to be in the same vertical planes as the four primary current-bars (see Figure 34) y (mm) x (mm) l=10 mm l=200 mm l=1000 mm -50 Figure 33: The loci of regions in which placing current bars along those regions yields a zero gradient The blue box is the BEC cell, while the orange boxes are surrounding fixtures The red circles represent of the placement of the main current bars Thus, the only free variable now is the current-bar length l and the x-coordinate

37 32 Gradient coil design 27 y I z I w y x x w x w Figure 34: Schematic of the final geometric gradient coil configuration (red = primary current-bars, blue = return current-bars, green = wires linking each coil) placement of the return bars (or equivalently the width w of each coil as shown in Figure 34) The required current for a 39 G/cm field from any coil configuration where each coil is of a single turn of wire exceeds 50 A limit in the power supply Thus, each coil requires multiple turns of the wire Ideally the return loops should be placed at an infinite distance away and the coils should have as many loops as required to create the given field gradient However, making larger coils with more turns increases the resistance and inductance The total resistance must not exceed 2 Ω due to 100 V limitation in the power supply In addition, it is of interest to keep both the inductance and resistance as low as possible to ensure that the coils can be switched fast enough to take multiple images in the MRI process at optimal resolution (see Sections 210 and 211) The number of turns N required for a given coil length l and width w is shown in Figure 35 This function calculated by simply dividing 39 G/cm by the magnetic field due to a single turn coil (given by Equation 32) Note that this calculation includes the influence due to the return bars and ignores the side linking bars (the green bars in Figure 34) as they have no influence on the said gradient due to symmetry The plot clearly shows that as the length is increased, the magnetic field gradient is stronger due to the contribution of more current elements, hence less turns are required Similarly, making the width larger implies that the return bars are placed further away, again decreasing the required number of turns While the magnitude of the magnetic field gradient increases with the number of turns N, the inductance increases with N 2 For a given coil dimension (length l and width w), the required number of turns is shown in Figure 35 Each coil size (along with the number of required turns) will yield a value for inductance This inductance is calculated by noting that the inductance of a single rectangular loop is [22]:

38 28 Chapter 3 Coils for off-diagonal magnetic field gradients 400 number of turns w (mm) l (mm) 40 Figure 35: This plot shows the required number of turns per coil to produce a 39 G/cm magnetic field gradient over different coil dimensions The maximum current through the wires is taken to be the power supply limitation of 50 A L rect N 2 µ 0 π ( 2(w + l) + 2 l 2 + w 2 l ln ( l+ l 2 +w 2 w ) w ln ( w+ l 2 +w 2 l )) (35) Assuming that the mutual inductance between the four separate coils is negligible, the total inductance is estimated to be L tot = 4L rect ; that is, that of four coils in series where each coil is of N turns The total inductance of the coils is plotted in Figure 36; from which it can be seen that there exists a global minimum This minimum is at l = 141 mm and w = 745 mm The required number of turns is N = 42, while the estimated total inductance of all four coils is L tot = 4 mh The implemented design utilises a race-track geometry for each coil that spirals inwards from the optimal dimensions (that is, the outer coils are of the optimal dimensions) To form 42 turn coils, the coils were made from 1 mm diameter wire and stacked into 5 layers as shown in Figure 37 The simulated estimate of the coil inductance was L tot = 19 mh, while the predicted coil resistance was 16 Ω The switching time to get from a current of 0 A to 50 A was 21 ms for the predicted inductance, resistance, and power supply limitations (which has a maximum output voltage of 98V) The simulated magnetic field variation is shown in Figure 38 Note that the coils are predicted to produce the required B y / x gradient at the origin They

39 32 Gradient coil design 29 Ltot (mh) w (mm) ℓ (mm) Figure 36: This plot shows inductance for coils of a given size and number of turns (given in Figure 35) producing a 39 G/cm magnetic field gradient Note that there is a global minimum in the coil size where the inductance is minimised also produce another off-diagonal gradient Bx / y This is physically unavoidable as Maxwell s equations require that B = 0 This field is however, not a problem as the net magnetic field along the y-axis is much larger than that along the x-axis once a uniform magnetic bias field is applied (via Helmholtz coils) The simulation predicted a magnetic field gradient By / x = Bx / y = 39 G/cm at the origin Figure 37: Final coil design as seen in Radia The coils (shown in red) are placed near the BEC glass cell (shown in blue)

40 30 Chapter 3 Coils for off-diagonal magnetic field gradients Bx(G) By(G) Bz(G) x (mm) y (mm) z (mm) Figure 38: Simulated magnetic field profiles generated by the designed gradient coils Each Cartesian component of the magnetic field is plotted against every spatial direction intersecting the origin Note that the only significant gradients generated by the designed coils are B y / x and B x / y All other magnetic field gradients (illustrated in light blue) are small numerical fluctuations inherent to the simulation 33 Construction and bench-testing The constructed coils were tested using a magnetometer on a travelling stage to ensure that they produced the required magnetic field gradient along the x-axis A 1 A stimulus was driven through all four coils Before each measurement, the coils were switched off and the magnetometer was zeroed, to circumvent the spurious measurement of ambient magnetic field fluctuations The measured magnetic field profile B y (x) is shown in Figure 39 From the measured field profile, the magnetic field gradient for 1 A current was 065 ± 002 G/cm Thus, the gradient for a 50 A current will be 33 ± 1 G/cm This is clearly lower than the the simulated gradient of 39 G/cm Using an impedance analyser, the measured total inductance and resistance of the entire set of gradient coils totalled to 160 mh and 155 Ω respectively This resistance indicates that the coils have the correct number of turns The discrepancy between the predicted and measured magnetic field gradient was found to be due to an imperfection in the construction In the designed and simulated coils, the wire windings were such that their edges were straight, and parallel to the z-axis However, due to the nature of winding the coils on a lathe, the coils have an ovoid shape as opposed to a strict rectangular geometry Based on

41 33 Construction and bench-testing 31 B y (G) x (cm) Figure 39: Measured magnetic field B y along the x-axis for a 1 A current drive The linear fit yielded a gradient of 065 ± 002 G/cm When scaling to 50 A, this yields 33 ± 1 G/cm the measured dimensions of the outer winding, a simulation was run to determine the effect of the winding imperfection (see Figure 310), yielding an approximately 15% decrease in the magnitude of the B y / x magnetic field gradient Thus, the simulated gradient in the presence of the winding imperfection is approximately 332 G/cm, which falls within the uncertainty bound of the measured value The lower than specified magnetic field gradient achieved by the constructed coils is however not a problem, for it is shown in Section 53 that the coils are to be driven with a sinusoidal current waveform as opposed to a square waveform The sinusoidal waveform can be driven by the power supply with a higher maximum current than with the square waveform This allows for the 50 A restriction to be surpassed and subsequently, one may achieve larger magnetic field gradients using the modified waveform

42 32 Chapter 3 Coils for off-diagonal magnetic field gradients Y Y -20 X 0 50 X Z Z Figure 310: A Radia simulation used to investigate the effect of having a winding imperfections in the coils Red: conducting wires; Blue: glass vacuum cell in which the BEC is prepared Left: single-turn outer winding of the designed coils using strictly rectangular corners; Right: single-turn outer winding of the constructed coils with rounded corners due to imperfections in construction This difference in the profile of the outer windings accounts for the 15% discrepancy between the predicted gradient of the designed coils and the measured gradient of the constructed coils

43 Chapter4 Autobalancing photodetector for Faraday polarimetry To perform MRI using the Faraday effect, the optoelectronics must be able to measure very small and very fast oscillations in the polarisation axis (see Figure 21) A standard polarimetry setup involves the use of a polarisation beam splitter to split the orthogonal components of the linearly polarised beam to deduce the axial rotation A half wave plate is used rotate the axis of polarisation to ensure that the components are split equally The photodetector subsequently detects the small changes in the differential intensity of the two components to deduce the polarisation rotation Keeping the split ratio equal is important (for it leads to a better common mode noise rejection as explained later) in the detector operation and can be done by constantly rotating the waveplate to ensure that the split ratio is sustained It is desirable to have the detector handle the splitting electronically (autobalancing) The detector must operate at a high bandwidth up to several megahertz to ensure that the MRI signal can be sufficiently sampled past the Nyquist limit The detector must also display a strong common mode rejection ratio (CMRR) to reject intensity noise inherent in the incoming laser To achieve the best possible signal to noise ratio, bright beams with approximately 7 mw in power will be used in the experiment The detector must be shot-noise limited in these bright beams; that is, the highest physically attainable signal to noise ratio for a given beam power The design must trade-off the fact that, with brighter beams and smaller photodiodes, the photodiode PIN junction saturates, while larger diodes display a large capacitance and stifle the operational 33

44 34 Chapter 4 Autobalancing photodetector for Faraday polarimetry bandwidth To solve this design problem, I used a design inspired by Dr Phillip CD Hobbs [23, 24] This design was analytically derived, simulated in PSPICE, constructed and fully bench-tested before being tested on the BEC 41 Detector overview The photodetector must be able to measure the rotation of the polarisation axis of an incoming Faraday signal (see Figure 21) In this project a differential photodetector that measures the difference between the intensities of two orthogonal components of polarisation axis of the incoming light suffices to measure this rotation This can be seen by considering Figure 41 Since the rotations in this experiment will be small (and measured with respect to the diagonal to ensure an equal split ratio), clearly p v p h = 1 sin(θ + π 2p 2 4 ) 1 cos(θ + π 2 4 ) = sin θ θ (41) Thus, the differential signal (signal found by subtracting the vertical and horizontal components of polarisation) is proportional to the rotation angle y p p h θ p v z Figure 41: Decomposing the polarisation axis P into the vertical and horizontal components Note that the small oscillations occur with respect to the diagonal as this ensures an equal split ratio of the horizontal and vertical polarisation components p h and p v To interface the Faraday signal (see Figure 21) with the differential photodetector, the polarisation axis must be split into two orthogonal components This can be achieved via a polarisation beam splitter (PBS) such as Wollaston prism Lasers exhibit intensity noise across a wide range of time scales Since the photodetector is a differential detector, it naturally has noise cancellation properties with regards to common mode noise Thus, it should demonstrate a high commonmode-rejection-ratio (CMRR) where any common component of the signal sent to the differential inputs is rejected via subtraction of the photocurrents However, this fundamentally requires the inputs to be balanced That is, the polarisation

45 41 Detector overview 35 λ/2 Linearly Polarised Light after leaving BEC PBS Photodiodes s(t) Figure 42: Interface between the Faraday beam and the photodetector (the subtraction module) beam splitter (PBS) splitting the two orthogonal components must ensure an equal splitting of light intensities to achieve perfect subtraction This also causes the DC level of the output signal (and common intensity) to be zero This can be achieved manually by turning a half-wave plate before the polarisation beam splitter (see Figure 42) The half-wave plate rotates the polarisation axis such that the orthogonal components with respect to the PBS are of equal magnitude However, wave-plates drift the polarisation axis rotation slightly over time This will imbalance the photodetector and in turn yield a lower CMRR Thus, the user must frequently adjust the wave-plate to continue achieving an equal split ratio (seen when the DC level of the output signal is zero) An optomechanical photodetector would have a motor that turns the wave plate to sustain the equal splitting It is desirable however to have a purely optoelectronic implementation for simplicity of design The detector should employ an electronic auto-balancing technique to enable a large CMRR across a wide band of frequencies Finally, the detector must be able to resolve small differential signals in bright beams that may range up to 7 mw This is because the detuning is large (approximately 47 THz 1 ) and requires bright beams to compensate for the smaller photon interaction with the BEC (to produce a measurable signal) To best recover the time-domain MRI signal, the detector must be shot-noise limited This implies that the detector must not introduce electronic noise above the physical limitation set by the photon-shot noise If the electronic noise floor is well below the shot-noise level (that is, more than 10 db), then any further reduction in the electronic noise floor yields very marginal improvements to the signal to noise ratio 1 As explained in Figure 211, the laser used in the Faraday beam is of 790 nm wavelength 47 THz is the frequency detuning when interact with the 780 nm energy level in 87 Rb

46 36 Chapter 4 Autobalancing photodetector for Faraday polarimetry 42 Design The photodetector design was adapted from one done by Hobbs [23] Hobbs design is a differential detector that features automated electronic polarimetry that mimics an optomechanical servo loop where a motor turns the half wave plate to ensure an equal split ratio The circuit has three distinct stages being the differential stage, transimpedance stage and the auto-balance stage (see Figure 43) D A V + Transimpedance Stage C f R f Q 3 I IN I SIG - V + U 1 + V V LIN D B Q 4 V Differential Stage I C(Q1) I C(Q2) Q 1 Q 2 I D COMP COMP V 1k 26 V + C - U 2 + V V LOG 1k Autobalance Stage Figure 43: Photodetector circuit 421 Differential and transimpedance stages The differential stage yields a current signal I IN that is the difference in the diode currents I A and I B This can be seen by first considering Kirchoff s current law 2 : I IN (I DA I DB ) (42) The purpose of the transistors (in a cascode topology) is to simply shield the transimpedance stage from the large diode capacitance This largely solves the issue of the diode capacitances hindering the operational bandwidth (on appropriate selection of C f shown in Section 422) 2 This is seen by noting that the I IN I C3 I C4 Since transistors operate as I C = βi B and I E = I C + I B (C, E and B refer to collector, emitter and base respectively), it can be shown that I IN β β+1 (ID A ID B ) It is assumed that the transistors have the same β To make the β prefactor near unity, the Darlington transistors (MPSA64 and MPSA14) with high β factors β+1 were chosen

47 42 Design 37 The transimpedance stage converts the input current I SIG into an output voltage V LIN Noting that the op-amp fundamentally outputs V out A v (V + V ) (where A v is the open-loop gain of the op-amp and V + and V are the voltages on the noninverting and inverting inputs respectively), the output voltage is (ignoring C f ): A v V LIN = A v + 1 R f (43) That is, as A v 1, this behaves as an ideal current to voltage converter The choice of R f depends on the noise considerations In order to be shot-noise limited, the shot-noise must be of a greater magnitude than the electronic noise of the system The shot-noise per root frequency bandwidth for a given photocurrent is: I shot f = 2eI d (44) where I d is the current through a given photodiode and e is the elementary charge Since the shot-noise power is white (within the given frequency band), the noise power in the difference signal is the additive sum of the shot-noise powers due to light falling on D A and D B The corresponding noise photocurrents add in quadrature Given Equation 43 for a large A v, the voltage noise (per root frequency bandwidth) on V LIN, due to shot-noise from photocurrents flowing through D A and D B is V shot f = R f 2e (I DA + I DB ) R f 2eη (P DA + P DB ) (45) where P DA and P DB are the respective powers of the laser light falling upon the photodiodes D A and D B, while η is responsivity of the photodiode (typically 052 A/W) Sources contributing to electronic noise include the thermal (or Johnson) noise from the resistor, the characteristics of the op-amp U 1 and semiconductor shot noise from the transistors Noise characterisation shall be referenced to the output voltage V LIN in this analysis The dominant noise source comes from the resistor R f in the form of Johnson noise: V elec(j) f = 4k B T R f, (46) where k B is the Boltzmann constant and T is the temperature of the resistor As a transimpedance amplifier, the main source of noise from the op-amp U 1 is due to the current noise at the inputs of op-amp Thus, it is of interest to choose an op-amp with lower current noise The op-amp must also have a supply rail that can span to ±15 V to both ensure a large dynamic range as well as the minimisation of

48 38 Chapter 4 Autobalancing photodetector for Faraday polarimetry the capacitance on the photodiodes when reverse biased (discussed later in Section 422) The chosen op-amp was the AD This op-amp has an input noise current density of 24 pa/ Hz Thus, in this case, the electronic voltage noise at V LIN due to the op-amp is effectively V elec(o) f = ( ) R f (47) Note that this Johnson noise dominates the op-amp input voltage noise of the opamp (09 nv/ Hz) at low frequencies when R f > 1 Ω Thus, adding the noise powers (that is, the voltages in quadrature) yields the total electronic noise: V elec 4k B T R f + ( ) 2 R 2 f f (48) Note that the electronic noise is independent of the incoming laser power, while the shot-noise is proportional to the square root of the laser power It is desired that the system is shot-noise limited Therefore, the shot-noise must be substantially greater than the electronic noise The ratio of the shot-noise and electronic noise is V shot V elec = 2eη (PDA + P DB ) (49) 4kB T R f + ( ) 2 Taking the voltage shot-noise to be nominally ten times greater than the electronic voltage noise, η = 05 and 7 mw total power to each diode yields a resistance of: R f 1000 Ω (410) If this resistance were increased from 1000 Ω, the noise in the output signal is decreased (as seen by Equation 48) However, increasing R f comes at the expense of the operational bandwidth as discussed in Section Bandwidth tuning of transimpedance stage The transimpedance stage should convert the differential current signal I SIG over as large a bandwidth as possible While the selection of R f was based on noise considerations, the eventual selection of C f is due to operational bandwidth considerations Now as shown in Appendix B1, the input capacitance of the transimpedance stage is

49 42 Design 39 C IN = 2C CB + C op-amp, (411) where C CB is the collector-base capacitance of the transistors in the differential stage and C op-amp if the the intrinsic differential input capacitance of the op-amp U 1 Note that for this condition to be met (and to ensure the stability of the upcoming pole placements), the photodiode capacitances C d must be minimised This can be done by placing as large a reverse bias voltage on the photodiodes as possible To ensure compatibility of a single dual rail supply to all electronics in the circuit, a nominal ±15V rail was selected This yields approximately 10pF for the Hamamatsu S1223 photodiodes used in the circuit With an input capacitance on U 1, the equivalent circuit is now shown in Figure 44 By Equation 43 (except including the effect of the parallel capacitor C f ), V LIN = A v A v + 1 R f (412) 1 + jωr f C f The optimisation of C f to maximise bandwidth requires one to consider the open loop gain of the op-amp to be band-limited That is, A v A 0 A 0 + jω ω 0, (413) where A 0 is the open loop gain at DC and ω 0 is the cutoff frequency of the opamp The remaining analysis requires pole-zero cancellation via cumbersome algebra The reader may either consult the author or refer to Hobbs [23] The resulting capacitance from the optimisation is C f = C IN For the components chosen, this capacitance is approximately 22 pf C f I SIG C IN - R f V + U 1 + V V LIN Figure 44: Small signal equivalent of the differential stage in Figure 43

50 40 Chapter 4 Autobalancing photodetector for Faraday polarimetry 423 Autobalancing If the polarimeter formed by the half wave-plate and PBS (see Section 42) splits the orthogonal polarisation components of the Faraday beam at equal intensities, the differential stage of the photodetector ensures that common mode noise (that is, laser intensity noise) is cancelled with a high CMRR However, the optical polarimeter exhibits slow drift in the polarisation angle This will cause the differential inputs to be imbalanced and the CMRR degrades The autobalancing stage cancels this drift in the optical polarimeter to ensure that the effective split ratio of the photocurrent subtraction equal Thus, the differential stage may continue laser intensity noise cancellation at a high CMRR Now consider intensity noise in the laser light and a polarimeter that has drifted in its rotation angle (see Figure 42) The noise n(t) is intrinsically present on all axes of polarisation in the laser (and subsequently acts as a common mode noise source on the differential inputs of the photodetector) The polarimeter splits the polarisation components via the ratio 1 : k (k > 0), where k = 1 implies equal splitting of the polarised beams If k 1, then the differential noise component of the signal I IN is: I IN(noise) = 1 k ηn(t), (414) 1 + k where η is responsivity of the photodiode Thus, imperfect balancing yields poor rejection of common mode noise (that is, noise present in both split beams) For the autobalancing stage to operate, several constraints must be satisfied (see Appendix B2) The first constraint is that: P DA > P DB (415) That is, the split ratio of intensities is set by the polarimeter such that photodiode D A receives slightly more laser power This implies that the differential photocurrent I IN has a positive bias and subsequently, the differential stage alone does not yield perfect subtraction of common mode noise (that is, laser intensity noise) The transistor Q 2 provides a path for this excess current (including uncancelled common mode noise) This now effectively yields the required perfect subtraction such that the laser intensity noise in I SIG has been cancelled with a high CMRR The autobalancing stage controls the portion of sunk current by sampling a portion of the laser beam (before it enters the BEC) via the photodiode D COMP This leads to the second constraint where:

51 43 Practical considerations and construction 41 P COMP > P DA P DB (416) This equivalently implies that I COMP > I SIG With these conditions set, the autobalancing stage can automatically balance the signal I IN The resulting voltage V LOG (see Appendix B2) is: ( ) ICOMP V LOG = ln 1 (417) I IN Since V LOG is the output to an op-amp stage, it must be ensured to be within the ±15 V supply rails to ensure proper autobalancing operation Thus, the V LOG output can be used as a diagnostic tool to check if the detector meets the conditions to perform automatic balancing The autobalancing stage only cancels low-frequency drift in the polarimeter as given by the value of capacitor C (as seen by the low pass filter formed around U 2 ) The low frequency cutoff is given by ω = (RC) 1 Given the 1 kω resistor placed at the input of the op-amp U 2, for a nominal ω = 500 rad/s, the value of C is: C = 2 µf (418) Now note that although the autobalancing stage only cancels the effects of lowfrequency drift in the polarimeter s polarisation rotation angle, the key fact is that this suffices to sustain the an equal split ratio That is, k 1 in Equation 414 Since the CMRR of differential stage is only is only dependent on effective split ratio, the photodetector produces wide band cancellation of the laser intensity noise as a result of the autobalancing stage maintaining the equal effective split ratio [24] 43 Practical considerations and construction In order to meet the final technical requirement of yielding a signal that can be measured by an analogue to digital converter without introducing further noise, I had to make several modifications to the circuit The two main modules were the capacitance multiplier to reduce power supply ripples and the use of line drivers to amplify the output signal Power supply ripples can cause time-varying scaling of the op-amp outputs These ripples will also cause time-varying scaling of the diode capacitances and emitter voltages in the transistors Q 1 and Q 2 This will subsequently introduce noise on the output signal V LIN To help mitigate this power supply noise, an active

52 42 Chapter 4 Autobalancing photodetector for Faraday polarimetry low-pass filter in the form of a BJT (also known as Capacitance Multiplier ) was placed on the two power supply rails To make the signal large enough to be read by an analogue to digital converter, several line drivers (voltage to voltage amplifiers) were implemented using AD829 op-amps These drivers provided unity buffering on the LOG and LIN outputs, while providing an additional 100 amplification of the LIN output The 100 output was AC coupled to ensure that op-amp voltage offsets did not cause output saturation The printed circuit board (PCB) design was done using the computer aided design package Altium Designer 2010 This software features schematic capture, PCB layout, physical constraints via a 3D view and active design rule checks for robust PCB design The package also provided opportunities to simulate the circuit via SPICE to ensure the integrity of the design For example, a simulation of the electronic voltage noise as measured at the output V LIN in response to a current stimulus through the photodiode D A is shown in Figure 45 The noise is seen to be frequency dependent (due to the op-amp) with a value of 47 nv/ Hz over the sub-megahertz domain in which the MRI signal exists Note that at 7 mw, the voltage noise due to photon shot noise at V LIN is approximately 47 nv/ Hz Thus, the design satisfies the nominal requirement that shot-noise to be ten times the electronic noise is satisfied in the sub-megahertz band where MRI is to be performed 50 VLIN ( 100)/ f (pv/ Hz) Electronic noise Shot noise at 7mW f (Hz) Figure 45: An electronic noise simulation of the photodetector done in Altium Designer 10 as measured at the output V LIN in response to a noise stimulus at the photodiode D A Note that shot-noise plotted in blue dominates the electronic noise The final practical consideration is the optical bench itself Figure 46 shows

53 44 Detector testing and characterisation 43 the final apparatus to include the use of the photodiode D COMP to sample the initial laser noise by sampling a portion of the beam before it enters the BEC The polarimeter formed by wave plate W 1 and PBS S 1 is used to split a portion of the initial laser beam to D COMP to satisfy the condition given in Equation 416 Similarly the polarimeter formed by W 2 and S 2 is used to control the final split ratio between D A and D B to satisfy the condition given in Equation 416 The test benches used in Section 44 to characterise the photodetector use variants of these optics to closely simulate the final experimental conditions (when performing MRI) 790nm Laser λ/2 PBS COMP W 1 S 1 BEC λ/2 W 2 PBS S 2 A B LOG LIN (x100) PHOTODETECTOR Oscilloscope Spectrum Analyser Figure 46: Schematic of the final optics in the Faraday polarimetry used to interface the photodetector with the BEC 44 Detector testing and characterisation 441 Noise characterisation The detector is required to be shot-noise limited The general test bench is to use two constant laser light-sources on the differential diode inputs A and B, while letting no light into the reference beam input COMP (see Figure 47) By Equations 45 and 46, the total voltage noise (noting that noise voltages add in quadratures) is V 2 noise f = 2eηR 2 f P T + 4k B T R f, (419) where P T P D1 + P D2 The first term is due to shot-noise and the second term is due to electronic noise If the detector were shot-noise limited, then for a given range of laser powers, the squared-noise-voltage per unit frequency should be linear with laser power; with a gradient of 2eηR 2 f

54 44 Chapter 4 Autobalancing photodetector for Faraday polarimetry If the detector was balanced (that is, the same laser beam power on input A and B), then the subtraction should yield zero output as both signals are equal That is, the differential signal and noise powers are zero respectively However, when each photodiode detects the incoming photons, the discrete nature of the photons leads to photon shot noise that is not equal on each photodiode This is because the photon shot noise is a random Poisson process in which the probability of receiving a given photon is independent of the other photon arrivals Thus, photon shot noise will simply add in quadratures as shown by Equations 45 The experimental test bench (see Figure 43), used to verify that the detector was shot-noise limited, used this principle by ensuring that both photodiodes received equal portions of the same laser beam 790nm Laser λ/2 PBS W 1 S 1 λ/2 W 2 PBS S 2 A B LIN PHOTODETECTOR Spectrum Analyser Figure 47: Test Bench to perform noise characterisation Note that the COMP photodiode was blocked in this experiment To ensure that any common-mode noise (for example, laser drift) induced on the two laser beams was propagated equally to both photodiodes, the two laser beams were split components of the initial linearly polarised beam The half wave plate W 1 and beam splitter S 1 (acting as a conventional polarimeter) control the total power on both photodiodes A and B, while the splitting ratio is controlled by the half wave plate W 2 and the beam-splitter S 2 The photodiode COMP was blocked as this test was to simply ensure that the photodetector was shot-noise limited (that is, the photodetector was not autobalancing) Thus, this test was only concerned the noise characteristics of the photodiode and the electronics involved in converting the differential current signal, from photodiodes A and B, into an output signal voltage Now by Equation 419, plotting squared voltage noise per frequency bandwidth Vnoise 2 /f against the total input beam power P T yields a linear plot with gradient 2eηRf 2 and y-intercept of 4k BT R f as shown in Figure 48 Note that the intercept

55 44 Detector testing and characterisation 45 can be larger if the spectrum analyser itself introduces a non-zero noise floor that adds to the electronic noise from the detector The measured Johnson noise (square root of the y-intercept as given by Equation 46) was (110 ± 003)nV/ Hz The expected gradient was (16 ± 01) V 2 /(W Hz) (where the uncertainty stems from the 5% tolerance on the 1000 Ω feedback resistor R f ) The gradient from the measured data was (16 ± 002) V 2 /(W Hz) Note that the points after 10 mw tend to stray above the line of best fit This is due to classical laser intensity noise This noise is proportional to the set laser power Therefore plotting the square of voltage noise per frequency bandwidth against the laser power would yield a quadratic relation with laser power Note that this effect is only prevalent when moving to higher laser powers (limited here by the power rating of the photodiode) The reason that the photodetector fails to fully reject laser intensity noise at higher powers is due to the PIN junction saturation in the photodiodes Subsequently, the CMRR is degraded at these powers and can only be fixed by either spreading the beam (to cover a larger area of the photodiode) or lowering the beam power V noise / f ((nv) 2 /Hz) P T (mw) Figure 48: Plot of square of voltage noise (per frequency bandwidth) against total input beam power The red line is the fitted line of best fit The fact that the data shows a linear gradient implies that the photodetector is shot-noise limited The area in which shot-noise dominates the Johnson noise is better seen in a logarithmic plot (see Figure 49) It was appropriate to use a hyperbolic fit on the

56 46 Chapter 4 Autobalancing photodetector for Faraday polarimetry data to show the two asymptotes On a logarithmic scale, when electronic noise dominates, the function has a zero gradient as seen by the asymptote in the region of lower powers When photon shot-noise dominates, Equation 419 clearly shows that the function scales linearly with log 10 P T as seen by the second asymptote in the region of higher powers The turning point is at approximately 1 mw Thus, at powers above 1 mw, we refer to the photodetector as shot-noise limited Vnoise/ f (nv/ Hz) P T (mw) Figure 49: Plot of square of voltage noise (per frequency bandwidth) against total input beam power The red line is the fitted hyperbola and the dashed lines are the hyperbolic asymptotes

57 44 Detector testing and characterisation Testing detector autobalancing When the photodetector is successfully autobalancing, the effective split ratio of the two photocurrents is unity If the intensity of the lase beams were suddenly increased on one of the differential photodiodes (D A or D B ), then the effective split ratio is not equal The autobalancing stage should compensate and bring the effective split ratio back to unity One may monitor the state of the effective split ratio by monitoring the DC voltage level of the signal V LIN Under perfect subtraction (that is, a unity split ratio), the DC level of the V LIN output should be at 0 V and non-zero when the split ratio is non-unity The experimental test bench involved the use of a Faraday crystal in a solenoid This crystal rotates the axis of polarisation (of light passing through it) at an amount proportional to the applied magnetic field in the solenoid (similar to the Faraday effect seen with a BEC) As crystal rotates the polarisation axis by an amount proportional to the current fed into the coil, injecting a square wave current into the coil the results in alternating polarisation axis rotations Thus, this optoelectronic device enables one to bias the beam intensities arbitrarily to input A and input B to force the effective split ratio to stray away from unity The action of the photodetector autobalancing is observed by viewing the DC level as it is brought back to zero via an exponential decay (see Figure 410) 790nm Laser λ/2 PBS COMP W 1 S 1 Faraday Crystal λ/2 PBS A LIN Oscilloscope S 2 B ±10V/10Hz Square Wave W 2 PHOTODETECTOR Figure 410: Test bench to perform noise characterisation The measured response in the LIN output (see Figure 411) clearly shows that the photodetector responds to the stimulus and exponentially decays towards the balanced state The balanced state is offset by approximately 24 mv due to the op-amp DC voltage offset in U 1 The exponential decay 1/e time constant was

58 48 Chapter 4 Autobalancing photodetector for Faraday polarimetry 77 ± 02 ms This is fast enough to compensate for thermal drift in the polarisation optics which is in the order of seconds This time constant by the capacitor C and the 1 kω resistor on the inverting input of the op-amp U 2 A 2 µf capacitor yields a time constant of 2 ms This discrepancy was attributed to the insertion of a capacitor of too large a value (10 µf as this was the only capacitor in stock while using smaller values may start cancelling the desired differential signal) V (V) t (ms) DC Stimilus LIN (x1000) Figure 411: DC step response of the autobalancing stage The DC stimulus is the coil voltage that biases the beam intensity via the Faraday crystal (see Figure 410) The LIN output is the response The exponential decays show the autobalancing stage at work Note that the offset is due to the intrinsic voltage offset on the op-amp U Photodetector bandwidth The photodetector converts differential light intensities into an output voltage The operational bandwidth refers to the range of frequencies (of same amplitude) in which the differential signal may undertake such that the output amplitude does not vary to within a certain amplitude margin A nominal 3 db band was taken as the acceptable region To measure the photodetector bandwidth, the experimental test bench involved the use of the Faraday crystal (see Figure 412) Injecting a 15 dbm sinusoidal frequency sweep (generated by the vector network analyser) into the coil around Faraday crystal induces polarisation rotations The combination of the half wave plate W 2 and polarisation beam splitter S 2 converts this into a sinusoidal differential intensity input for the photodetector The amplitude of this input stimulus is proportional to the magnetic field produced by the solenoid Since, magnetic field in the solenoid is proportional to the injected current, the voltage V sense (voltage across a sense resistor in series with the coil) was taken to be the input stimulus voltage (as the voltage across this resistor is proportional to the current in the coil) The VNA took the ratio of the output voltage V out and the sense resistor voltage V sense

59 44 Detector testing and characterisation 49 to obtain the relative transfer function That is, the transfer function is: T (f) V out V sense (420) This is not the absolute transfer function (however, it is still correct to a constant factor) as the absolute induced rotation caused by the Faraday crystal is unknown 790nm Laser λ/2 PBS COMP W 1 S 1 Faraday Crystal λ/2 PBS A LIN (x100) Vector Network Analyser 15dBm 50 V sense W 2 S 2 B PHOTODETECTOR Figure 412: Test Bench to find the operational bandwidth of the photodetector The measured transfer function is shown in Figure 413 The passband region (region where the transfer function is within a 3 db band) is from 20 khz to 4 MHz f (Hz) Vout/Vsense (db) Figure 413: Measured transfer function of the photodetector using the apparatus shown in Figure 412 The transfer function highlights the passband and bandwidth of the amplifier Note that it is only correct to a constant factor as the absolute induced rotation caused by the Faraday crystal (driving the input stimulus) is unknown The 2 db peak at approximately 5 MHz is due to the error in the soldered ca-

60 50 Chapter 4 Autobalancing photodetector for Faraday polarimetry pacitance due to the unavailability of low tolerance capacitors of the exact required value This peak was already noted during simulations where perturbations of just the capacitance value for C f already displayed peaking in the transfer function (see Figure 414) It should be noted that this transfer function displayed a similar sensitivity to the capacitance values of several feedback capacitors used in the line drivers The 3 db cutoff before the large drop in the transfer function is at approximately 8 MHz This large drop is expected as this is the circuit was designed at this cutoff frequency using a multi-stage amplifier that constitutes the 100 line driver Simulations showed an approximately 70 db per decade roll-of as observed by the measured transfer function 120 VLIN ( 100)/IIN (db) f (Hz) Figure 414: Simulation of the transfer function in Altium Designer 10 for differing values of C f The roll-off before 10 khz is because the 100 LIN output is AC coupled while the rapid roll-off after 10 MHz is a consequence of using multi-stage amplifier at that cut-off frequency Red, Blue and Green represent C f being 5 pf, 22 pf and 35 pf respectively This operational characteristic ensures that the detector should faithfully measure the MRI signal uniformly over all required frequencies (from approximately 100 khz to 1 MHz) It is of note that in order to realise the full available bandwidth, an ADC (analogue to digital converter) with at least 18 MHz sampling frequency is required (to satisfy Nyquist s theorem) 444 Common mode rejection ratio A noise signal is common mode if it is present equally in both differential inputs The common mode rejection ratio (CMRR) defined as

61 44 Detector testing and characterisation 51 CMRR P OUT P IN(CM) (421) where P IN(CM) is noise power present in the laser before it has entered the BEC and P OUT is the output power after this noise power has propagated through the photodetector Thus, the CMRR measures the ability of the photodetector to reject noise in the laser If the photodetector is perfectly balanced, it should reject all common mode noise An experimental test bench tested the autobalancing stage by measuring the CMRR for input noise tones injected into the initial laser The test bench consisted of an acousto-optic-modulator (AOM) which was driven by a double sideband, low modulation index amplitude modulation signal to produce small 50kHz intensity fluctuations (see Figure 415) The combination of the half wave plate W 1 and polarisation beam splitter S 1 controls the total power sent into inputs A and B From Equation 417, it is clear that the combination of wave plate W 2 and PBS S 2 will allow for the selection of different V LOG outputs It is ideal to keep the V LOG near zero as this not only implies that the voltage is away from the voltage rails (that is, greater dynamic range) but also allows optimal operation of the transistors in the differential pair That is, V LOG = 0 implies that I COMP = 2I IN Thus, the excess current in I IN is sunk through the transistors Q 1 and Q 2 equally If the transistors sink heavily unequal currents, then transistor non-linearities will cause degradation of the autobalancing stage As the photodetector operates, this voltage V LOG will fluctuate to counter the drift in the value of I IN (due to drift in the polarimeter formed by W 2 and S 2 ) It is undesirable for the CMRR to change with different V LOG values as this occurs during normal operation of the photodetector Thus, the CMRR for was measured for different total laser powers (on the photodiodes A and B) and V LOG voltage values The different total laser powers were controlled via the polarimeter formed by W 1 and S 1, while the subsequent V LOG values were generated by changing I IN (via the polarimeter formed by W 2 and S 2 ) The actual CMRR for a given V LOG value was measured via a two step process In the first step, the common mode noise tone was first measured by splitting the entire beam to input A, using W 2, and blocking the COMP diode The amplitude of the noise tone seen in output voltage signal is proportional to P IN(CM) In the second step, the COMP diode was unblocked and W 2 was turned to give a specific V LOG value The P OUT value is proportional to amplitude of the noise tone seen in output voltage signal converted to decibels The CMRR is then calculated via Equation 421 and

62 52 Chapter 4 Autobalancing photodetector for Faraday polarimetry λ/2 790nm Laser AOM PBS COMP W 1 S 1 λ/2 W 2 PBS S 2 A B LOG LIN PHOTODETECTOR Oscilloscope Spectrum Analyser Figure 415: Test Bench to find the CMRR of the photodetector The plot in Figure 416 shows a high CMRR ranging from 40 db to 60 db for a 50 khz noise tone Hobbs showed that this design can reach down to 80 db The limitation is due to the optical apparatuses used in the test bench which requires the entire beam (carrying the common mode noise) to fill the photodiodes A and B equally without being too small such that there is saturation in the PIN diodes Hobbs noted that this CMRR of 80 db came from careful accounting of all reflections and resulting diffraction fringes while incorporating mild touches to the waveplates and mirrors to ensure peak CMRR In addition, the beam coming from the AOM should have been passed through a fibre to ensure that the noise tone was spatially homogeneous across the entire beam waist This is in contrast to the test bench which was designed to mimic the final optics in the MRI experiment 0 V LOG (V) CMRR (db) x x o Δ x o Δ x x x x x x Δ Δ Δo o Δ o o o o o o o x o Δ 02mW 2mW 15mW -70 Figure 416: Measured CMRR values over different V LOG outputs for a 50kHz common mode tone (lower is better) The CMRR should be sufficiently large to cancel common mode laser intensity

63 44 Detector testing and characterisation 53 noise over the photodetector bandwidth The spectral response was measured over different total laser powers (to inputs A and B) while keeping the V LOG value constant at 1 V (not a coarse approximation as the CMRR is seen to be approximately constant over a range of V LOG value in Figure 416) As seen by the plot in Figure 417, the CMRR is at least 40 db over the useful operating range of 10 khz to approximately 5 MHz The overall CMRR is acceptable for there are no spikes in the laser intensity noise spectrum (that would manifest as peaks in the output signal spectrum V LIN ) such that they require over 40 db common mode suppression (see Figure 418) That is, the laser intensity noise is white without any peaks that cannot be rejected by the photodetector f (Hz) CMRR (db) mW 62mW 145mW -80 Figure 417: Measured CMRR values over different frequencies and powers for a constant V LOG = 1 V (lower is better) Thus, we have a photodetector that is shot-noise limited at power approximately above 1 mw The photodetector has a functioning autobalancing stage that can counter the effects of drifting optical polarimeters (which can degrade the signal to noise ratio if the split ratio was left imbalanced) The photodetector functions to a wide band ranging from 10 khz to approximately 9 MHz, while the CMRR is below 40 db from 10 khz to approximately 5 MHz (enough to cancel any classical laser intensity noise) The photodetector is thus, ready to measure Larmor precessing spins in the BEC

64 54 Chapter 4 Autobalancing photodetector for Faraday polarimetry f (Hz) Vnoise/ f (dbv/ Hz) Figure 418: Measured noise response when the laser is allowed to bypass the AOM, the COMP diode is covered The diodes A and B are fed 73 mw in total 45 Faraday polarimetry of a spinor BEC Due to other experiments being done on the BEC apparatus, the gradient coils could not be mounted on the BEC apparatus during the course of the project However, using pre-existing coils, a constant magnetic field was placed on the BEC to cause all the spins in the BEC to precess at a single Larmor frequency of 697 MHz The goal was to at least test the photodetector by using it to successfully measure this frequency tone Due to issues with the current optic fibre, the laser could only be operated in 1 mw laser power (as opposed to the 7 mw) However, as shown in the measured spectrogram Figure 419 the detector yields successful isolation of the tone (from background shot-noise) using this laser power The spectrogram shows a deviation in the 697 MHz tone at a rate of 50 MHz due to background magnetic fields create by surrounding AC power lines modulating the Larmor frequency The spectrogram also shows oscillation in the signal amplitude as well as an exponential decay This dephasing effect is not expected to occur theoretically for there is only one tone present However, on further investigation during the submission of this thesis, it was found that this effect was due to the quadratic Zeeman effect causing a different energy splitting from the m = 1 and m = +1 states with respect to the m = 0 energy level This causes two Larmor precessing tones to be present (that is, the spins in the m = +1 states and m = 1 states precess at slightly different Larmor frequencies) This effect was found to be nulled via an appropriate choice of radio

65 45 Faraday polarimetry of a spinor BEC 55 frequency pulse sequencing Similarly, the 50 Hz oscillations can be nulled using a magnetic field servo where a large coil counters the magnetic field due to AC power lines f (khz) t (ms) 10-4 Figure 419: Spectrogram from a Faraday beam passing through a BEC residing within a constant magnetic field as measured by the photodetector described in this Chapter The signal was sampled at 2 MSPS, while the short time Fourier transform was binned to samples