Design and Characterization of Essential Components for a Magneto-Optical Trap (Project BS)

Design and Characterization of Essential Components for a Magneto-Optical Trap Rahul Das∗ Abstract This project is focused on the development and assembly of essential components for a Magneto-Optical Trap (MOT) used in ultracold atom experiments. Key tasks included constructing a mechanical shutter system to selectively block or allow the atomic beam into the trapping region, implementing a field-compensating Helmholtz coil to reduce background magnetic interference, and designing a radio-frequency antenna for evaporative cooling. These efforts contribute foundational tools for quantum control in ultracold atom research. 1 Introduction A Magneto-Optical Trap (MOT) is a fundamental and widely utilized apparatus in atomic physics designed to cool and confine neutral atoms by exploiting the interaction between laser light and a spatially varying magnetic field. The MOT combines Doppler cooling, which reduces the kinetic energy of atoms via laser photon momentum exchange, with a quadrupole magnetic field that provides a position-dependent restoring force. This configuration enables atoms to be slowed down from thermal velocities and trapped at ultralow temperatures typically in the micro-Kelvin range. MOTs serve as the foundational tool in the preparation of ultracold atomic samples, forming the initial stage for a variety of advanced experiments in quantum optics, atomic clocks, precision spectroscopy, and quantum simulation. The ability to precisely manipulate atomic motion and internal states using a MOT has revolutionized experimental atomic physics and has facilitated breakthroughs in fundamental science as well as in emerging quantum technologies. ∗Corresponding author: rahul [email protected]. 1 2 Contents •Pneumatic Cylinder for the beam shutter –How the pump works –Schematic Diagram •Earth’s Magnetic Field compensator –Calculating the Magnetic Field –Heat Generation, Power Dissipation, and Temperature Fluctuation in a Resistive Wire •Antenna for the Evaporative Cooling step –1st Trial –2nd Trial –Trying out Combination of Different Antennas •Conclusion •Acknowledgment •Mathematical codes using Python –For plotting Alpha versus Temperature –For Heat calculations of coil •References 2 3 Pneumatic cylinder Initially, the design incorporated a motor in place of a pneumatic cylinder. However, after extended operation, the motor exhibited inefficiency due to coil heating, which adversely affected the shutter’s performance. As a result, the system was revised to use a pneumatic cylinder, which offers a heavy-duty cycle and is capable of sustained operation over long periods without performance degradation. 3.1 How the pump works 3.1.1 Power Supply and Actuation Control The solenoid coil of the valve is powered by a DC power supply (e.g., 24V). The return action is achieved using a spring inside the valve, which resets the spool to its default position when the coil is de-energized. Actuation is controlled via a switch or digital controller. 3.1.2 Initial State In the initial stage, the solenoid is in the off position. The internal spring holds the spool in the default position. 3.1.3 Energized State (Solenoid ON) The solenoid is energized and shifts the internal spool. Compressed air flows from Port P to Port A, causing the cylinder to extend (piston moves upward). Port B is vented to the atmosphere through Port S. 3.1.4 Initial State (Solenoid OFF) While returning to the initial state, the solenoid is de-energized. The internal spring returns the spool to the initial position. The piston retracts due to air flowing into Port B again. Compressed air flows from Port P to Port B, causing the cylinder to retract (piston moves downward). Port A is vented to the atmosphere through Port R. 3.1.5 Mechanical Setup The pneumatic cylinder is mounted vertically on an optical table using a metal base bracket. The cylinder rod points upward, and the body is fixed securely to prevent displacement during actuation. The 5/2 solenoid valve is fixed nearby using a mechanical support bracket. 3 3.2 Schematic Diagram Figure 1: Schematic Diagram of the pneumatic cylinder 4 Earth’s magnetic Field Compensator The Earth’s magnetic field causes distortions in the trapping step when the atom is inside the Molecular trap. The Earth’s magnetic field is nullified using the square shaped Helmholtz coil. A square is preferred over circle to reduce the volume of the space occupied and that the coil gets fitted easily over the Optical table. 4.1 Calculating the Magnetic Field 4.1.1 Finding equations for the Square Helmholtz coil In our case we have z=L/2 Using Biot Savart Law, vectorcially we can show, Bx=µ0iL2 2π 1 x2+L2 4!  1 qx2+L2 2 (1) 4 Figure 2: Schematic figure of the square loop Figure 3: Magnetic Field Comparison between 3 loops We compared our formula with the known formulas for a circular coil of radius aand √2a. We got the result as expected leading to our understanding that the equation should work properly. The parameters we took here as, N= 100, d = 0, a = 0.24m, I= 2Amps, just for rough estimation. 5 [1] We now refer to this figure and do our calculations with respect to this, Figure 4: Image of the Helmholtz square loops Here we consider that the total length of one side of the coil is 2aand the total length of the distance between the two coils is 2d Bz=Nµ0i2a2 π"1 (z+d)2+a2 1 p(z+d)2+ 2a2! +1 (z−d)2+a2 1 p(z−d)2+ 2a2!# (2) We aim to cancel the Earth’s magnetic field using a coil system. We begin by analyzing the magnetic field along one dimension (the z-axis), which can later be generalized to three dimensions. Let Btotal z(z) = Bcoil z(z) + Bearth z,(3) where Bearth z≈const near the origin. To nullify the Earth’s field, we want Btotal z(z) = 0 in a region around z= 0.(4) Since perfect cancellation at all points is not possible, we flatten the total field near the origin by requiring its derivatives to vanish: dnBtotal z dznz=0 = 0 for n= 1,2,3, . . . (5) 6 Given Bearth zis approximately constant, dnBearth z dzn= 0 for all n≥1,(6) so we require: dnBcoil z dznz=0 = 0 for n= 1,2,3, . . . (7) Now, expand Bcoil z(z) in a Taylor series about z= 0: Bcoil z(z) = Bcoil z(0) + dBcoil z dz z=0 z+1 2 d2Bcoil z dz2z=0 z2 +1 4! d4Bcoil z dz4z=0 z4+··· (8) Due to symmetry of the coil configuration (two identical coils symmetrically placed about z= 0), the magnetic field is an even function of z, i.e., Bcoil z(z) = Bcoil z(−z),(9) which implies that all odd derivatives vanish at z= 0: d2n+1Bcoil z dz2n+1 z=0 = 0.(10) We are thus left with the condition that all even-order derivatives vanish: d2Bcoil z dz2z=0 = 0,d4Bcoil z dz4z=0 = 0, . . . (11) Imposing the condition d2Bcoil z dz2z=0 = 0 (12) and solving for d/a, we obtain: d a= 0.5445.(13) This defines the Helmholtz condition, which ensures the second derivative of the magnetic field vanishes at the center, giving a flat field region around the origin. 4.1.2 The Helmholtz Configuration d= 0.5445a [2] 7 Figure 5: Image of the Helmholtz Configuration square loop We now plot the graph between the Magnetic field vs the spatial coordinate fixing some parameters and taking d= 0.5445a[1] Figure 6: Magnetic field vs spatial coordinates The parameters we took here as, N= 24, z = 0, a = 0.24m, I= 2.05Amps just for rough estimation. 8 In this equation, Bz=Nµ0i2a2 π"1 (z+d)2+a2 1 p(z+d)2+ 2a2! +1 (z−d)2+a2 1 p(z−d)2+ 2a2!# (14) We put z= 0, as we are interested to find the field in the center of the two coils. We start the evaluation with taking a= 0.2m and a Magnetic field requirement for each coil to be 2 Gauss. We end up in the equation as, N×i= 49.1175 turns-Amps. 4.1.3 The Cubical Configuration d=a Figure 7: Image of the Cubical Configuration square loop The parameters we took here as, N= 43, z = 0, a = 0.24m, I= 2.01Amps just for rough estimation. In the same way, we proceed with the calculations here and end up in the equation, N×i= 86.6 turns-Amps. 9 5 Antenna for the evaporative cooling step 5.1 1st Trial We tried to build an antenna for the evaporative cooling step, like an LC Oscillator circuit using a bend metal strip. [3], Figure 14: 1st trial of the Antenna We formed the antenna by bending a copper strip of length land width w. The inductance is approximated by the solenoid formula: L=µ0N2A l,(24) where the strip is modeled as a solenoid, Nis the number of turns, and Ais the cross-sectional area. The capacitance is given by C=ϵ0A d,(25) where Ais the area of the capacitor plate and dis the separation between the plates. The resonance frequency is then f=1 2π√LC .(26) 16 Figure 15: Schematic circuit of the antenna system. Figure 16: Amplitude versus frequency (747–947 MHz). A 50,Ω resistor is incorporated into the circuit to ensure proper impedance matching with the pickup antenna. Impedance matching is crucial in RF systems to maximize power transfer and minimize signal reflections that can lead to losses or distortions in the received signal. By matching the antenna’s impedance to the characteristic impedance of the transmission line, the resistor helps maintain signal integrity across the system. This analysis serves as a preliminary approximation of the antenna’s expected performance across a range of radio frequency (RF) fields. The resonance behavior observed, specifically the peak in power emission, aligns with the design intent, as the resonance frequency of the antenna’s LCR (inductor-capacitor-resistor) circuit was deliberately tuned within the band of 870 MHz to 900 MHz. The presence of this resonance peak indicates the antenna’s ability to efficiently convert electromagnetic energy at the target frequency, which is vital for effective evaporative cooling procedures. Such cooling relies on precise RF field control to manipulate atomic states, and any mismatch or inefficiency in the antenna response could compromise the cooling dynamics. 5.2 Second Trial Building upon insights from the initial trial, the second trial adopted a more analytical approach to antenna design by focusing on the calculation of the Rabi frequency [3]. The Rabi frequency quantifies the rate of coherent oscillations between atomic energy levels induced by the applied RF field and is a critical parameter in optimizing evaporative cooling and other quantum control techniques. 17 This approach allows for a more targeted antenna configuration, where the amplitude and spatial profile of the RF magnetic field can be engineered to produce desired transition rates. By directly relating antenna parameters to quantum mechanical interaction strengths, the design process becomes more predictive and efficient, paving the way for improved experimental outcomes. Figure 17: Schematic of RF evaporation. Radiation with frequency vRF drives the |2,2⟩ → |1,1⟩transition for atoms with sufficiently high energy. The RF frequency vRF is decreased over time, ejecting atoms and leaving only a small fraction of very cold atoms. We have a Magnetic field Gradient of dB dx = 400 Gauss/cm and we assume the trap nearly harmonic for the ease of calculations, so, we have, Ukinetic =Upotential (27) KBT=µBBtrap ⇒Btrap =KBT µB≈1.384 (28) 18 now, secondly we have, 1 2mv2=KBT⇒v=r2KBT mLithium ≈0.55 m/s (29) Now, Btrap ≈1.54 = xdB dx ≈400 Gauss/cm ×x⇒x≈0.4×10−4m (30) ⇒t=x v≈10−4secs (31) and thus we have, frequency as, Freq = 1 T≈10 KHz.(32) Now we go for calculating the Rabi frequency. We know that ΩRabi ≫Freq, so we assume ΩRabi ≈1GHz. Now from the graph, we see, |2,2⟩→|1,1⟩ ⇒ ∆E= ∆(mFgF)µBB≈(2gF=2 −1gF=1)µB(33) ⇒∆E= (1 + 1)µBB(34) ⇒2µBB⇒∆E= 2µBB(35) So, now we have, ∆E= ¯hΩRabi,and thus, 2µBBcenter = ¯hΩRabi ⇒Bcenter =¯hΩRabi 2µB (36) ⇒Bcenter ≈10 mG (37) So, we need an antenna which will have a magnetic field of 10mG at the center of the MOT and with a high freq of ≈1 GHz. [4] We have, Hr≈a2I0e−jkr 2r3(38) and µ0Hr=Bfor near field loop antennas (kr < 1), we have λ=c f≈30 cm and k=2π λ. Taking Real parts, we see, µ0I0a2cos(kr) 2r3= 10 mG (39) Where ais the radius of the loop antenna and ris the distance of the atoms from the loop antenna and kis 20.94 rad/m. We are given the expression for the magnetic field due to a loop antenna in the near field: 19 µ0I0a2cos(kr) 2r3=B(40) In our case we have a distance r= 16 mm = 0.016 m, from the atomic cloud to the antenna. And a loop of radius a= 42 mm = 0.042 m. Rewriting the expression and solving for I0: I0=2r3B µ0a2cos(kr)(41) First, compute kr: kr = 20.94 ×0.016 = 0.33504 rad (42) cos(kr) = cos(0.33504) ≈0.944 (43) After evaluating the whole problem, we get, I0≈0.003914 Amps or 3.91 mA (44) A current of approximately 3.91 mA is required in the frequency range of 800 MHz–1000 MHz to effectively drive the antenna. For antennas with a radius smaller than 42 mm, the required current for efficient radiation is expected to be even lower, which is advantageous. Although explicit calculations were not performed for each possible antenna radius, it can be reasonably estimated that the current remains on the order of milliamperes and does not deviate significantly from this reference value. While tuning across the wide frequency range of interest, minor fluctuations in current are acceptable and do not critically impact system performance, provided they remain close to the target value. Thus, achieving a plateau in the emitted RF power within the 800 MHz–1000 MHz band is crucial for ensuring reliable Majorana spin-flip transitions. 5.3 Trying out Combination of Different Antennas Our objective is to design an antenna that emits power consistently across a plateau region approximately spanning 800 MHz to 1000 MHz. We explored various antenna configurations and evaluated their performance within this frequency range. 20 Power emitted = Reflected to Signal/ Function Generator (SG/FG) + To Spectrum Analyzer (SA) + Wasted as heat (45) Figure 18: Circuit for observing how much power went through the antenna after emission and reflection back due to impedance mismatch. Figure 19: Circuit used to analyze the shape of the power emitted by the antenna. We had several options for positioning the antenna: •Inside the Bitter magnet (radius approximately 1.5 cm), •Outside the Bitter magnet but inside the MOT chamber (radius approximately 3.5 cm-4 cm), •Outside the MOT, near an optical window. However, in this configuration, the distance between the antenna and the atomic cloud would be significantly larger, requiring higher emitted power to effectively induce Majorana transitions. The antenna was not placed inside the vacuum chamber. [3] To identify the most suitable antenna configuration for RF cooling, various antenna designs were tested and compared. To streamline this process, I developed a custom Python-based interface that allowed simultaneous control of the signal generator and RF spectrum analyzer. This tool enabled real-time data acquisition and plotting, significantly enhancing the speed and efficiency of graphical analysis and comparison. 21 5.3.1 Small Loop vs. Large Loop Antenna (Parallel and Perpendicular) We constructed a small single-loop antenna using copper wire and a coaxial cable. In both setups, the small loop acts as the radiating source and the large loop as the receiving antenna.(Refer Figure:19) Two configurations were tested: •Parallel Configuration: Both loops are positioned parallel to each other. •Perpendicular Configuration: The large loop is oriented perpendicular to the small loop. (a) Parallel configuration. (b) Perpendicular configuration. Figure 20: Comparison between parallel and perpendicular configurations of small loop (radiating) and large loop (receiving) antennas. 5.3.2 Strip Antenna and Small Loop Detector A copper strip (10 cm ×1 cm) was curved to form a radiating antenna. A small loop antenna (diameter 3 cm) was used as the detector.(Refer Figure:19) 22 Figure 21: Receiver placed 3 cm from strip antenna. Figure 22: Receiver placed 1.5 cm from strip antenna. Inside the Bitter Magnet Figure 23: Strip antenna inside bitter magnet; receiver at 1.5 cm. Figure 24: Strip antenna inside bitter magnet; receiver at 3 cm. 5.3.3 Strip Antenna with Capacitor and Loop Detector A copper strip was curved to form a loop of the sane dimensions, described in the previous case, and a capacitor was added using another copper strip placed nearby. (Refer Figure:19) ( Same as described in the section of 1st Trial). 23 Figure 25: Strip antenna with capacitor; receiver at 1 cm. Inside the Bitter Magnet Figure 26: Strip antenna with a capacitor inside bitter magnet with loop detector at 1.5 cm. 24 5.3.4 Spiral Antenna with Loop Detector A spiral antenna (7 loops, outer radius 3 cm, inner radius 1.5 cm) with loop detector of diameter 3 cm at varying distances.(Refer Figure:19) (a) The Spiral Antenna (b) Loop detector at 1.5 cm (c) Loop detector at 2 cm (d) Loop detector at 3 cm Figure 27: Spiral antenna and loop detector measurements at 1.5 cm, 2 cm, and 3 cm distances. 25 # Use fsolve safely within this bracket T_solution = fsolve (equation , x0 =( T_low + T_high ) / 2) [0] # Plotting plt . figure ( figsize =(10 , 6) ) plt . plot (T_vals , alpha_vals , label =r’$\ alpha (T)$’, color =’blue ’) plt . axhline (y=1 , color = ’red ’, linestyle =’--’, label =r’$\ alpha = 1$’) plt . scatter ( T_solution , 1, color = ’black ’, zorder =5) plt . text ( T_solution , 1.05 , f’T = { T_solution :.2f} K’, ha=’center ’) plt . xlabel (’T (K)’) plt . ylabel (r’$\ alpha (T)$’) plt . title (r’Solution of $\ alpha (T) = 1$’) plt . grid ( True) plt . legend () plt.tight_layout() plt .show () # Print result print (f" Temperature T where alpha (T) = 1: T = { T_solution :.6 f} K ") 8.2 For Heat calculations of coil import numpy as np N = 24 # float ( input (" Say the number of turns :::::") ) I = 2.05 # float ( input (" Say the current we provide it in Amperes :::::")) half_side = 0.20 # float ( input (" Say the dimension of one sid ein metres :::::") ) Radius = 0.000725 # float ( input (" Say the radius of the copper wire we are using :::::") ) Time = 36000 # float ( input (" Time in seconds in which the device will remain on :::::") ) Area = np.pi * ( Radius )**2 rho_nominal = 1.68 e -8 error_percent = 0.10 # float ( input (" Enter the error in decimal point 32 for the resistivity of copper ( Mostly 0.10 ) :::::") ) error_percent_C_v = 0.02 # float ( input (" Enter the error in decimal point for the C_v of copper ( Mostly 0.02 ) :::::") ) error_percent_Density_of_copper = 0.01 # float ( input (" Enter the error in decimal point for the Density of copper ( Mostly 0.01 ) :::::")) error_percent_in_emissivity = 0.55 # float ( input (" Enter the error in decimal point for the Emissivity of copper ( Mostly 0.55 ) :::::")) error_percent_in_h = 0.5 # float ( input (" Enter the error in decimal point for the Convection coefficient of copper ( Mostly 0.5 ) :::::")) rho_lower = rho_nominal * (1 - error_percent ) rho_upper = rho_nominal * (1 + error_percent ) length_of_one_loop = 4 * half_side * 2 total_length = N * length_of_one_loop Volume = 2 * np.pi * Radius * total_length Density_of_copper = 8960 mass_of_copper = Volume * Density_of_copper C_v = 385 emissivity = 0.03 stefan_boltzmann_constant = 5.67e-8 T_ambient = 295 surface_area_exposed_to_radiation = np.pi * Radius * np.sqrt(N) * total_length h=5 # convective heat transfer coefficient (W/ m$^2$\cdot$K )$ C_v_upper = C_v * ( 1 - error_percent_C_v ) C_v_lower = C_v * ( 1 + error_percent_C_v ) Density_of_copper_upper = Density_of_copper * ( 1 - error_percent_Density_of_copper ) Density_of_copper_lower = Density_of_copper * ( 1 + error_percent_Density_of_copper ) mass_of_copper_upper = Volume * Density_of_copper_upper mass_of_copper_lower = Volume * Density_of_copper_lower emissivity_upper = emissivity * ( 1 + error_percent_in_emissivity ) emissivity_lower = emissivity * ( 1 - error_percent_in_emissivity ) h_upper = h * ( 1 + error_percent_in_h ) h_lower = h * ( 1 - error_percent_in_h ) Resistance_upper = ( rho_upper * total_length )/ Area 33 Resistance = ( rho_nominal * total_length / Area ) Resistance_lower = ( rho_lower * total_length )/Area Power_radiated = (I**2) * Resistance Power_radiated_upper = (I **2) * Resistance_upper Power_radiated_lower = (I **2) * Resistance_lower Heat_dissipated = Power_radiated * Time Heat_dissipated_upper = Power_radiated_upper * Time Heat_dissipated_lower = Power_radiated_lower * Time print (f"The Power radiated is :::::{ Power_radiated :.6f} Watt") print (f" With an uncertainty of ::::: { Power_radiated_lower :.6f} Watt to {Power_radiated_upper:.6f} Watt") print (f"The heat dissipated is ::::: { Heat_dissipated :.6f} Joule ") print (f" With an uncertainty of ::::: { Heat_dissipated_lower :.6 f} Joule to { Heat_dissipated_upper :.6 f} Joule ") Temperature = (( Power_radiated ) / (h * surface_area_exposed_to_radiation)) + T_ambient T_in_particular_upper = ((Power_radiated_upper) / ( h_upper * surface_area_exposed_to_radiation )) + T_ambient T_in_particular_lower = ((Power_radiated_lower) / ( h_lower * surface_area_exposed_to_radiation )) + T_ambient print (" Temperature after time" , Time ,"seconds is ", Temperature ,"K" ) print ("We have considered the ambient temperature as", T_ambient ," and after a time of", Time ," seconds we get a temperature difference of", Temperature - T_ambient ,"K") print ("If the Coils are turned on for ",Time ,"seconds , then the temperature will vary from",T_in_particular_upper ,"K to", T_in_particular_lower ,"K") References [1] X. Zhu, M. Xing, C. Liu, J. Ye, H. Cheng, and Y. Miao, ”Optimization of composite Helmholtz coils towards high magnetic uniformity”. [2] D. S. Batista, F. Granziera Jr., M. Tosin, ”Three–Axial Helmholtz Coil Design and Validation for Aerospace Applications,” IEEE Transactions on Aerospace and Electronic Systems, Oct. 2017, doi: 10.1109/TAES.2017.2760560.” [3] Z. A. Geiger, “An Apparatus for Dynamical Quantum Emulation Using Ultracold Lithium,” Ph.D. dissertation, Dept. of Physics, Univ. of California, Santa Barbara, CA, USA, Jan. 2018. [4] C. A. Balanis, *Antenna Theory: Analysis and Design*. 34