Cooling an Optically Trapped Ultracold Fermi Gas by Periodical Driving

Summary

We present a parametric driving method to cool an ultracold Fermi gas in a crossed-beam optical dipole trap. This method selectively removes high-energy atoms from the trap by periodically modulating the trap depth with frequencies that are resonant with the anharmonic components of the trapping potential.

Abstract

We present a cooling method for a cold Fermi gas by parametrically driving atomic motions in a crossed-beam optical dipole trap (ODT). Our method employs the anharmonicity of the ODT, in which the hotter atoms at the edge of the trap feel the anharmonic components of the trapping potential, while the colder atoms in the center of the trap feel the harmonic one. By modulating the trap depth with frequencies that are resonant with the anharmonic components, we selectively excite the hotter atoms out of the trap while keeping the colder atoms in the trap, generating parametric cooling. This experimental protocol starts with a magneto-optical trap (MOT) that is loaded by a Zeeman slower. The precooled atoms in the MOT are then transferred to an ODT, and a bias magnetic field is applied to create an interacting Fermi gas. We then lower the trapping potential to prepare a cold Fermi gas near the degenerate temperature. After that, we sweep the magnetic field to the noninteracting regime of the Fermi gas, in which the parametric cooling can be manifested by modulating the intensity of the optical trapping beams. We find that the parametric cooling effect strongly depends on the modulation frequencies and amplitudes. With the optimized frequency and amplitude, we measure the dependence of the cloud energy on the modulation time. We observe that the cloud energy is changed in an anisotropic way, where the energy of the axial direction is significantly reduced by parametric driving. The cooling effect is limited to the axial direction because the dominant anharmonicity of the crossed-beam ODT is along the axial direction. Finally, we propose to extend this protocol for the trapping potentials of large anharmonicity in all directions, which provides a promising scheme for cooling quantum gases using external driving.

Introduction

In the past two decades, various cooling techniques have been developed for generating Bose-Einstein condensates (BEC) and degenerate Fermi gases (DFG) from hot atomic vapors^1,2,3,4,5. BEC and DFG are novel phases of matter that exist in extremely low temperatures, usually one millionth of a degree above absolute zero temperature, far below those normally found on Earth or in space. To obtain such low temperatures, most cooling methods rely on lowering the trapping potential to evaporatively cool the atoms. However, the lowering scheme also decreases the collision rate of the atoms, which limits the cooling efficiency when the gas reaches the quantum regime⁶. In this article, we present an "expelling" method to evaporatively cool an ultracold Fermi gas in an ODT without lowering the trap depth. This method is based on our recent study of parametric cooling⁷, showing several advantages compared to the lowering schemes^7,8,9.

The key idea of the parametric scheme is to employ the anharmonicity of the crossed-beam ODT, which makes the hotter atoms near the edge of the trapping potential feel the lower trapping frequencies than the colder atoms in the center. This anharmonicity allows the hotter atoms to be selectively expelled from the trap when modulating the trapping potential at frequencies resonant with the high-energy atoms.

The experimental protocol of parametric cooling requires a pre-cooled noninteracting Fermi gas near the degenerate temperature. To implement this protocol, an acousto-optic modulator (AOM) is used to modulate the intensity of the trapping beams by controlling the modulation frequency, depth and time. To verify the cooling effect, the atomic cloud is probed by absorption imaging of time-of-flight (TOF), where a resonant laser beam illuminates the atomic cloud and the absorption shadow is captured by a charge coupled device (CCD) camera. The cloud properties, such as the atom number, energy, and temperature, are determined by the column density. To characterize the cooling effect, we measure the dependence of the cloud energies on the various modulation times.

Protocol

NOTE: This protocol requires a home-built ultracold atom apparatus including the following equipment: two external cavity diode lasers (ECDL), a locking setup for the ECDL offset frequency locking¹⁰, a fiber laser for the ODT, an AOM for laser intensity modulation, an radio frequency (rf) antenna system with a source generator and a power amplifier, an absorption imaging system with a CCD camera, a computer program for timing sequence and data acquisition (DAQ), a computer program for imaging processing and data analysis, a pair of electromagnets for the MOT and bias magnetic fields, and an ultrahigh vacuum chamber including a ⁶Li vapor oven and a Zeeman slower (shown in Figure 1).

Caution: Three lasers of different powers and wavelengths are used. Please consult the relevant laser safety data sheets and choose the proper laser safety goggles.

1. Timing Control

NOTE: All timing sequences are controlled by a 128 channel PCI DAQ card through a timing control program. The resolution of the timing sequence is 100 µs. Several instrumentation control programs are used to control the settings of the instruments, such as fiber laser arbitrary function generator (AFG), ODT AFG, arbitrary pulse generator (APG), parametric modulation AFG, MOT multiplexer, rf generator, etc.

1. Open the timing control program and the control programs for the instruments.
   NOTE: The timing control program sends TTL (Transistor-transistor logic) signals to the control terminals for running the timing control files. Some instruments are connected to the computer by GPIB (IEEE 488) for real-time control.
2. Write the experiment timing file and set the timing parameters as listed in Table 1.
   NOTE: The after MOT timing sequence is also illustrated by Figure 2.

2. CCD Camera Preparation

NOTE: CCD camera is used to record the absorption imaging of the cold atoms, which is the main diagnostic tool of cold atoms.

1. Turn on the CCD camera driver and its control program. Set the CCD camera to Particle image velocimetry (PIV) mode¹¹. Set the CCD exposure time to 5 ms.
   NOTE: PIV mode reduces the time gap between the signal and reference frame, which increases the signal-to-noise ratio of the absorption imaging.
2. Use an external trigger to control the CCD exposure
   NOTE: The CCD trigger time is listed in Table 1.

3. 671 nm Laser Preparation

NOTE: A 671 nm single frequency ECDL with 500 mW output power is used to generate the MOT cooling and trapping beams. Another 671 nm ECDL of 35 mW is used for absorption imaging. A digital laser current modulation method (DLCM) is applied for laser frequency stabilization¹⁰. The related ⁶Li energy levels are shown in Figure 3a. Room temperature stability of 20 ± 1 °C is required for the optimal stability of laser frequency locking.

1. MOT Laser Preparation
   NOTE: The optical setup and relevant results of the DLCM method is presented in Reference 10.
   1. Turn on the ⁶Li atomic vapor cell heater and warm it up to 340 °C.
   2. Warm up the laser locking AOM for 1 h.
   3. Turn on the laser frequency lock controller and open its software. Turn on the laser grating and current modulation of the ECDL in the software.
      NOTE: The modulation frequency and amplitude of the grating modulation are set to 5 Hz and 1.0 V respectively. The modulation frequency and amplitude of the current modulation are set to 100 kHz and 0.0015 V_pp respectively to reduce the laser linewidth¹⁰.
   4. Turn on the ECDL emission.
      NOTE: The laser light passes through the MOT optical setup and reaches the experiment vacuum chamber.
   5. Slightly adjust the current of the ECDL laser manually to tune the laser frequency until the lock-in error signal of the ⁶Li D₂ line is observed, as shown in Figure 3b.
   6. Set the lock point in the control software to the 2²S_1/2 (F = 3/2) → 2²P_3/2 transition (see Figures 3a, 3b). Then lock the laser frequency to this transition, and adjust the lock point to the center of the transition¹⁰.
      NOTE: Once the laser frequency is locked, the lock-in error signal shows a small fluctuation at the lock point corresponding to the frequency fluctuation around the lock point.
2. Imaging Laser Preparation
   NOTE: The optical setup and relevant results of the offset locking method are presented in Reference 10.
   1. Turn on the offset locking rf signal generator.
   2. Turn on the modulation of the grating, and increase the modulation amplitude to 2 V.
   3. Repeat the frequency tuning process in 3.1.4.-3.1.5. to get the laser frequency beating error signal in the oscilloscope and the rf spectrum analyzer.
   4. Lock the laser frequency to the beating signal of the offset locking through two PID feedback modules.
      NOTE: Once the laser frequency is locked, the spectrum of the beating signal in the rf spectrum will stop at the locking point.

4. Absorption Imaging Preparation

NOTE: The atoms are probed with absorption imaging, which needs two image frames. The first one with the atoms is the signal frame, and the second one without atoms is the reference frame.

1. Turn on an APG and the imaging beam AOM.
2. Set the imaging pulse duration to 10 µs, and set the separation time between the two imaging frames to 5.5 ms.
3. Set the imaging beam intensity to about 0.3I_sat, where I_sat = 2.54 mW/cm² is the saturated absorption intensity of the ⁶Li D₂ line.

5. Cooling Atoms with MOT

NOTE: MOT is a widely-used cooling method in ultracold atoms experiments. This section generates a MOT of around one billion ⁶Li atoms at about 300 µK.

1. Slow Atom Source
   1. Turn on the oven heaters.
   2. After the oven temperatures reach the operational region (refer to Table 2), turn on the cooling fans for the Zeeman slower. Then slowly increase the current of the slower to 9.2 A. Turn on the current of the two crossover coils to 7 A and 1 A respectively.
      NOTE: The temperature distribution of the oven listed in Table 2 is optimized for collimation and lifetime of the atomic source¹². The location of the heaters on the oven is shown in Figure 4.
   3. Unblock the Zeeman slower laser beam manually by opening the atomic shutter. Set the frequency of the laser beam to 192 MHz red-detuned with the 2²S_1/2 (F = 3/2) → 2²P_3/2 transition.
      NOTE: With this setup, the speed of the atoms is slowed down from 1,400 m/s to 100 m/s. The Zeeman slower is shown in Figure 5.
2. Magnetic Field Gradient
   NOTE: This apparatus uses a pair of coils controlled by an H-bridge switch circuit to produce either an anti-Helmholtz or Helmholtz magnetic field. The coils are water cooled to prevent overheating.
   1. Slowly turn on the water flow to 6 gal/min.
   2. Set the H-bridge for anti-Helmholtz magnetic field configuration by running the timing control program with the MOT loading timing file.
   3. Turn on the magnets' power supplies, and set the current of each coil to about 18 A via its control program, which creates a magnetic field gradient of about 22 G/cm for the MOT.
      NOTE: A static MOT is observed in the experiment chamber after the magnetic field gradient is turned on.
3. Dynamic MOT
   NOTE: The optical setup of the ⁶Li MOT contains three pairs of counter propagating MOT beams with all pairs orthogonal to each other. Each MOT beam includes a cooling beam and a repumping beam. The intensities and frequency detunings of the beams, which are controlled by AOMs, are varied for the three phases. The control voltages of the AOMs are set via multiplexer circuits commanded by a timing control system. The parameters for three phases are listed in Table 3. The optical layout of the MOT beams is shown in Figure 6.
   1. Load, compile and run the experiment timing file in the timing control program on a loop with the software control. The experiment timing starts with the MOT loading phase. Monitor the MOT fluorescence signal in the photodetector to reach 2 V, which indicates around 10⁹ atoms in the MOT.
      NOTE: The fluorescence of the MOT is collected by a lens with spatial angle of about 10^-4 rad. The loading phase atom number can be calculated by the method in Reference 13.
   2. Use the optical shutter to block the slowing beam before the loading phase ends.
      NOTE: The timing of the slowing beam shutter is also under control of the experiment timing, which is listed in Table 1.
   3. Set intensities and frequency detunings of the MOT laser beams according to Table 3 for the cooling phase.
      NOTE: After the cooling phase, the temperature of the MOT is reduced to about 300 µK.
   4. For the pumping phase, program the experiment timing file to turn off the repumping beams with the AOM.
      NOTE: The pumping phase pumps all the atoms into the lowest hyperfine states 2²S_1/2 (F = 1/2).
   5. Turn off the MOT beams and shift the laser frequency 30 MHz below the atomic transition resonance by AOM, and block the leaking light from the AOMs with optical shutters.
      NOTE: After the MOT stage, any leakage of the resonant light to the atomic cloud will result in atom loss. The timing of the AOM control and MOT beam shutter are all listed in Table 1.
   6. After the dynamic MOT, acquire the imaging frames from the camera. Get the absorption imaging of the MOT.
      NOTE: The atomic number of the MOT is about 10⁷ after the pumping phase. A typical absorption image of the MOT is shown in Figure 7a.

6. Preparing an Ultracold Fermi Gas with ODT

1. Optical Dipole Trap
   NOTE: ODT is the main tool to generate ultracold Fermi gases. In order to generate a deep ODT, a fiber laser with 100 W emission power at 1064 nm wavelength is used. The setup of ODT is shown in Figure 8.
   1. Turn on the water flow for cooling the laser beam dumps.
   2. Set the ODT AOM control voltage to 1 V manually. Turn on the fiber laser with 13 W emission power.
   3. Check the ODT optics with an infrared light viewer, and remove any dust with argon gas flow.
      NOTE: Dust on the optics can change the spatial profile of the ODT, and cause instability of the ODT.
   4. Command the fiber laser AFG to generate a laser pulse via the AFG control program.
      NOTE: The output of the laser pulse is triggered by the experiment timing, and the starting time of this pulse is set to 14 ms before the end of the MOT loading phase. The pulse sequence control is shown in Figure 1, and the timing is listed in Table 1.
   5. Manually set the ODT AOM control voltage to 8 V (80% of the saturated rf power).
      NOTE: The maximum rf power of the AOM driver should be limited to 80% of the saturated power to reduce the thermal lensing effect.
   6. Acquire the absorption images of the MOT and ODT from the camera.
      NOTE: Check the overlap of the MOT and ODT through their absorption imaging. Figure 7b shows typical absorption images of the MOT and ODT, respectively.
2. Bias Magnetic Field and Spin Mixing rf Field
   NOTE: In order to generate an interacting Fermi gas, a bias magnetic field in the vertical direction is applied to tune the s-wave scattering length.
   1. Set the H-bridge in the experiment timing program so that the magnetic field configuration changes from anti-Helmholtz to Helmholtz.
      NOTE: The Helmholtz coils generate the bias magnetic field for tuning interatomic interaction.
   2. Set the bias magnetic field to 330 G in channel 2 and 527.3 G in channel 3 of the magnets control program.
   3. Program the experiment timing sequence to sweep the magnetic field from 0 G to 330 G after the MOT is turned off.
      NOTE: This magnetic field sweep prepares a weakly interacting ⁶Li Fermi gas for standard evaporative cooling.
   4. Program a magnetic field sweep from 330 G to 527 G for a noninteracting Fermi gas¹⁴.
      NOTE: The magnetic field sequence from 6.2.1-6.2.4. is shown in Figure 1, and the timing is listed in Table 1.
   5. Apply a noisy rf pulse to create a 50:50 mixture of the two lowest hyperfine states 2²S_1/2(F = 1/2, m_F= ±1/2) of ⁶Li.
   6. Tune the locked laser frequency resonant with the atoms at 527.3 G (corresponding to the transition 2²S_1/2 (F = 1/2, m_F = -1/2) → 2²P_3/2 at the low magnetic field) by changing the output frequency of the rf signal generator.
      NOTE: The resonant frequency maximizes the atom number of the absorption imaging, which is used to guide the frequency adjustment. Only the spin-down atoms are imaged to present the atomic cloud because the 50:50 spin mixtures are used for the experiment.
3. Evaporative Cooling by Trap Lowering
   NOTE: A standard evaporative cooling is used to cool the fermionic atoms of ⁶Li near the degenerate regime. The first stage of evaporative cooling is controlled by the pulse of the fiber laser and the second is controlled by the ODT AOM. The near-degenerate Fermi gas will be used as the sample for parametric cooling.
   1. Start the first stage of evaporative cooling with the control software by pulsing the fiber laser power, which increases the trap depth of the ODT to U_0, then back to 0.1U₀ (U₀ is the full trap depth with the laser power of 100 W). The total time of this stage is 0.5 s.
      NOTE: The pulse duration corresponding to U₀ should be limited to 0.5 s to avoid the thermal lensing effect.
   2. Program the ODT AOM with an exponential curve as shown in Figure 1. After the first stage of evaporative cooling is finished, wait 30 ms, and then start the second stage of evaporative cooling by lowering the trap depth from 0.1U₀ to 0.01U₀ through the ODT AOM. The total time of this stage is 1.5 s.
   3. Acquire the absorption imaging of the cold atoms after the evaporative cooling.
      NOTE: About 10⁵ atoms are left in the ODT after evaporative cooling, which can be calculated from the absorption image.

7. Parametric Cooling

1. Trap Depth Modulation
   1. Wait 100 ms after the magnetic sweep to 527.3 G. Modulate the trap depth with the ODT AOM by U(t_m) = 0.01U₀(1+δ_cos(ω_m t_m)), where δ is the modulation depth and ω_m is the modulation frequency. Set the modulation time t_m in the parametric modulation AFG control program. The time sequence of the modulation is shown in Figure 1.
      NOTE: This is the key step of implementing parametric cooling.
   2. Program the APG to release the atoms from the ODT by abruptly turning off the trapping beams. Let the gas ballistically expand for 300 µs before applying absorption imaging.
      NOTE: The ballistic expansion is used with TOF absorption imaging to get the temperature of the cold atoms.
   3. Acquire the absorption image of the cold atoms after parametric cooling.
2. Time Dependence Measurement
   NOTE: In our previous work⁷, we found the optimized frequency of the parametric cooling to be 1.45ω_x, where ω_x is the radial trapping frequency of ODT at 0.01U₀. Using this frequency, we can selectively remove high-energy atoms along the axial direction.
   1. Set the modulation depth to δ = 0.5 via the parametric modulation AFG control program.
   2. Use the external trigger control function of the parametric modulation AFG to change the modulation time from 0 to 600 ms by varying the modulation cycle numbers.
      NOTE: With the increasing of modulation time, the size of the atomic cloud will be reduced, especially the axial direction. The relevant results are shown in Figure 9.
   3. Acquire the imaging frames from the camera. Save and analyze the images through the CCD control program.

Results

Using this protocol, we study the dependence of the parametric cooling on the modulation time with the optimized modulation frequency and amplitude, both of which have been determined in our previous publication⁷. We first prepare a noninteracting Fermi gas of ⁶Li atoms in the two lowest hyperfine states with a temperature of T/T_F ≈ 1.2. Here, T_F = (6N)^1/3ħω/

Discussion

We present an experimental protocol for parametric cooling of a noninteracting Fermi gas in a crossed-beam optical trap. The critical steps of this protocol include: First, the optically-trapped Fermi gas needs to be cooled close to the degenerate temperature by lowering the trap depth. Second, a modulation frequency is chosen that is resonant with the anharmonic component of the trapping potential. Third, the intensity of the trapping beam is modulated to cool the atomic cloud and measure the dependence of the cloud ene...

Materials

Name	Company	Catalog Number	Comments
500 mW 671 nm ECDL	Toptica	TA Pro	Quantity: 1
35 mW 671 nm ECDL	Toptica	DL-100	Quantity: 1
671 nm AOM	Isomet	1206C	Quantity: 3
671 nm AOM Driver	Isomet	630C-110	Quantity: 3
100 W 1,064 nm CW laser	IPG photonics	YLR-100-1064-LP	Quantity: 1
1,064 nm AOM	IntraAction	ATM-804DA6B	Quantity: 1
1,064 nm AOM Driver	IntraAction	ME-805EH	Quantity: 1
Arbitrary Function Generator	Agilent	33120A	Quantity: 3
Digital I/O Board	United Electronic Industries	PD2-DIO-128	Quantity: 1
System Design Platform	National Instruments	LabVIEW	Quantity: 1
Analog Voltage Output Device	Measurement Computing	USB-3104	Quantity: 1
CCD Camera	Hamamatsu	Orca R2	Quantity: 1
Arbitrary Pulse Generator	Quantum Composer	9618+	Quantity: 1
Analog Voltage Output Device	Measurement Computing	USB-3104	Quantity: 1
20 A power supply			Quantity: 1
10 A power supply			Quantity: 1
120 A power supply			Quantity: 2
Cooling Fans			Quantity: depends on apparatus design
671 nm Mirrors			Quantity: depends on apparatus design
671 nm Half-wave Plate			Quantity: depends on apparatus design
671 nm Quarter-wave Plate			Quantity: depends on apparatus design
500 mW Beam Shutter			Quantity: depends on apparatus design
671 nm Lenses			Quantity: depends on apparatus design
Faraday Isolator			Quantity: 2, one for each ECDL
671 nm Polarizing Beam Splitter			Quantity: depends on apparatus design
Photodetector	Thorlabs	SM05PD1A	Quantity: 1
Multiplexer	Analog Devices	ADG409	Quantity: 1
Multiplexer	Analog Devices	ADG408	Quantity: 2
1,064 nm plano-concave lens			Quantity: 1 for beam reducer
1,064 nm plano-convex lens			Quantity: 1 for beam reducer
1,064 nm Mirrors			Quantity: depends on apparatus design
1,064 nm Half-wave Plates			Quantity: depends on apparatus design
1,064 nm Lenses			Quantity: depends on apparatus design
1,064 nm Thin Film Polarizer			Quantity: 1
100 W, 1,064 nm Beam Dump			Quantity: 1
100 W, 1,064 nm Power Meter			Quantity: 1
RF Function Generator	Rigol	DG4162	Quantity: 1
RF Power Amplifier	Mini-Circuits	ZHL-100W-GAN+	Quantity: 1