First Observations of an S-wave Contact on the Repulsive Side of a Feshbach Resonance. Scott Smale University of Toronto

Transcription

1 First Observations of an S-wave Contact on the Repulsive Side of a Feshbach Resonance Scott Smale University of Toronto

2 Contents 1 Introduction 3 2 Theory Feshbach Resonances Important Feshbach Resonances Tan s Contact Tan s relations Large k Momentum Scaling of n σ ( k) Density-density Correlator at Short Distances Adiabatic Relation Qualitative Meaning of the Contact Measuring the contact with RF Spectroscopy Experiment RF Spectroscopy and Imaging Contact Signature in rf Spectroscopy State Selective Imaging RF Spectra Rescaling Molecules in RF spectra Observations Contact for Repulsive Iterations Spectroscopy Near a P-wave Feshbach Resonance Conclusions and Future Work 29 2

3 Chapter 1 Introduction Understanding the physics of strongly interacting many-body systems is one of the current challenges of physics. Ultracold atomic systems provide unique experimental controls to probe properties of many-body systems that in other areas of physics are more difficult. One such experimental knob is the control of the strength of the interactions by use of Feshbach resonances [1]. Many-body systems with strong interactions result in strong correlations between particles which are difficult to treat theoretically. However for systems with a short range interactions there exist universal relations that relate properties of the system such as thermodynamic quantities and the tails of distributions. These universal relations are commonly called Tan s relations after Shina Tan who derived the first set of relations [2, 3, 4]. All of the universal relations derived so far involve a property of the system called the contact. Measuring the contact gives information, through the universal relations, about more conventional quantities such as the kinetic, potential, and interaction energies of the system without needing to worry about the details of the microscopic interactions. The contact has been measured before for a strongly interacting Fermi gas with attractive interactions [5, 6]. We present the first measurements of the contact for a strongly repulsive Fermi gas. We also present spectroscopy measurements around a p-wave Feshbach resonance that show evidence of an analogous quantity to the s-wave contact. This work comes at the end of a fruitful year and a bit for me. I have done many things in the past year. My first real project consisted of gluing a cell phone to a microcontroller and teaching it to send texts when the power goes out. Earlier in the year my cohort Chris Luciuk and I spent many months returning the system to working order after an upgrade caused us to lose our atoms. Around that time I also spent a period improving the software that runs our experiment which is described in the main text. Lately, in addition to the 3

4 work described in this thesis, we made measurements to increase our confidence in findings that are described in a paper that will be published at the end of this month in PRL [7]. 4

5 Chapter 2 Theory The theory presented herein is aimed at providing a qualitative understanding of the necessary physics to understand both Feshbach resonances and Tan s contact. 2.1 Feshbach Resonances A Feshbach resonance occurs when the energy of two states scattering into one another becomes equal to the energy of a bound state composed of those two states [13]. To gain qualitative understanding consider a system in which there are two spin species of free atoms. These two spin species are capable of elastic two particle scattering and of inelastic scattering into a bound state of the two spin species. We assume that the magnetic moment of a pair of atoms and the bound state are different so that their relative energies can be controlled by changing the magnetic field. The uncoupled energies of the pair of free atoms and the bound state are shown in Fig. 2.1a. The states consisting of a pair of free atoms and a bound pair are coupled since there is a finite overlap in the scattering wave function between elastic two particle scattering and inelastic bound pair formation. The result is an avoided crossing which is shown in Fig. 2.1b. The resulting energy levels are called the upper and lower branch. When the magnetic field is greater than the Feshbach resonance we say we are above the resonance and when the field is less we are below the resonance. Above the Feshbach resonance on the lower branch the scattering between the pair of free atoms is attractive since the avoided crossing energy level is less than the uncoupled energy. Similarly below the resonance on the upper branch the scattering between a pair of free atoms is repulsive since the energy has been increased. Below the resonance on the lower branch the pair of atoms form bound pairs or dimers. Note that there are other Feshbach resonances for which the correlation between 5

6 Figure 2.1: Coupling between a pair of free atoms and a bound state produces a generic avoided crossing. a) Bound state energy, blue, and the energy of a pair of free atoms, green, as a function of magnetic field. b) Avoided crossing resulting from coupling between the two states. the terms above and below and the attractive and repulsive side of resonance is reversed. For our resonance above the resonance is also called the BCS (Bardeen, Cooper, Schrieffer) side of the resonance because the physics of these weakly attractive pairs is similar to the BCS theory of superconductors [8, 1]. Below the resonance is also called the BEC side of the resonance since the bound pairs are bosons [8, 1]. We will discuss further the upper branch including its instabilities later. So far we have not specified the form of the interatomic scattering potential V ( R). The specifics of the potential in the binding region are determined by the specific states and orbitals involved. At long distances the dominant interaction is the van der Waals interaction, C 6 /R 6. The van der Waals interaction has an effective range, r e = (m r C 6 / 2 ) 1/4 where m r is 6

7 the reduced mass. For a spherically symmetric interaction potential the interatomic potential is given by V l (R) = V (R) + l(l + 1) 2m r R 2 (2.1) where l is the relative angular momenta between the two scattering atoms, m r is the reduced mass, and R is the interatomic distance [9]. A generic scattering potential is shown in Fig 2.2 for the l = 0, s-wave, scattering case, and the l = 1, p-wave, scattering case. The energy barrier in the p-wave scattering case is on the order of 100 µk [10, 11, 12]. Once cooled, the temperature of our cloud of 40 K atoms is between 300 nk and 1200 nk [10]. Thus p-wave scattering is typically energetically prevented from happening. However, as discussed later, near a p-wave resonance the strong interactions can overcome this barrier. Figure 2.2: A generic scattering potential for s-wave, blue, and p-wave, red, scattering. The height of the p-wave energy barrier energetically forbids p-wave scattering at ultracold temperatures unless near a resonance. Feshbach dimers are the least bound state that such a potential can support. Feshbach dimers can decay into deeply bound molecular states with energies typical of chemical reactions. Ultracold atoms are commonly in the regime where the de Broglie wavelength corresponding to the relative momentum k of scattering atoms is much greater than the effective range, r e, of the scattering potential. In this low energy regime the s-wave scattering amplitude 7

8 can be expanded in terms of k as f(k) = r a e k2 ik (2.2) 2 where a is the s-wave scattering length.[13] The scattering length is said to parameterize the strength of the scattering interaction. For small k the phase of the scattered wavefunction at long distances is given by ka. If a is negative then the wavefunction has been shifted into the scattering potential and the effective interaction is attractive. If a is positive then the phase of the wavefunction has been shifted away from the scattering center at the origin and the effective interaction is repulsive. Consider the avoided crossing shown in Fig. 2.1b. Above the resonance scattering between free atoms in the lower branch is attractive since the energy level is lower than for the uncoupled case and so the s-wave scattering length is negative. Below the resonance on the upper branch the interaction between free atoms is repulsive because the energy level has increased due to the avoided crossing and so the s-wave scattering length is positive. At the resonance the scattering length diverges because of the nature of the coupling with the bound state.[13] The scattering length as a function of field can be parameterized as[13] ( a(b) = a bg 1 B ) B B 0 (2.3) where a bg is the background scattering length, B is a measure of the width of the resonance, and B 0 is the magnetic field at the resonance. A plot of the s-wave scattering length for the resonance between F = 9/2, m F = 9/2 and F = 9/2, m F = 7/2 atoms is shown in Fig When the scattering length diverges the scattering cross-section between atoms is maximized. Instead of working with a divergent quantity is it convenient to work with 1/ka. For a degenerate Fermi gas the characteristic k to use is the Fermi wavevector k F and so we often parameterize the distance to the Feshbach resonance by (k F a) 1. Two expressions for the binding energy of the Feshbach dimers on the lower branch are[8] E b = 2 ma 2 and E b = 2 m(a r e ) 2 (2.4) The first is most accurate near the resonance. The bound pairs formed from scattering at the Feshbach resonance are the least bound state possible. There exist other bound states with deeper binding energy since the typical depth of the interatomic potential is on the 8

9 1000 Scattering Length a Magnetic Field G Figure 2.3: S-wave scattering length as a function of magnetic field for the resonance between F = 9/2, m F = 9/2 and F = 9/2, m F = 7/2 atoms at (7) G. Note the zero crossing at 209 G and the non-zero background scattering length. order of 1 ev or 100 THz whereas the binding energy of the dimers is on the order of 1 MHz. The dimers can decay into these deeply bound molecules [15, 16]. 2.2 Important Feshbach Resonances The following is a summary of the properties of the Feshbach resonances that are near the 9/2 and 7/2 resonance. All values are taken from [11]. The s-wave resonance between 9/2 and 7/2 atoms is centered at B 0 = (7) G. There are two separate measurements of B listed in [11], B = 7.0(2) G and B = 7.5(1) G. There is an s-wave resonance between 9/2 and 5/2 atoms centered at B 0 = (5) G. Again there are two measurements of B, B = 9.7(6) G and B = 7.6(1) G. There is a p-wave resonance between 7/2 and 7/2 atoms around G. A p-wave resonance is strictly composed of a m l = 0 resonance and a m l = ±1 resonance. The m l = ±1 p-wave resonance is centered at B 0 = (2) G and the m l = 0 p-wave resonance is centered at B 0 = (5) G. 2.3 Tan s Contact We now give an introduction to the quantity known as the contact. The following discussion follows primarily from Braaten [18]. Note that the following only considers s-wave interactions. 9

10 A system with a large scattering length is strongly interacting and as such has strong correlations between particles. Many theoretical methods are inadequate for dealing with such strongly correlated systems. Fortunately it is this large scattering length that sets a set of universal relations that constrain the behaviour of the system. These relations hold whether the system be a few-body or many-body system and the hold whether the system is in the ground state or has a non-zero temperature. They hold whether the density of the system is homogeneous or inhomogeneous such as our cloud of trapped atoms. They also hold whether the system is a superfluid or not and regardless of the balance of spin populations in the system. There exist many universal relations that have been derived so far that connect various properties of the system which range from thermodynamic quantities to large momentum and high frequency tails of distributions. We now discuss the universal relations for a system composed of two fermionic spin species which we denote in this section by σ = {1, 2}. In our experiment these fermionic spin species are represented by the lowest two Zeeman sublevels of 40 K. All of the relations derived thus far involve a property of the system known as the contact. The first universal relationships were derived by Shina Tan and have since been known as the Tan Relations. [2, 3, 4] The Tan relations apply in the regime where the scattering length, a, is large compared to the effective range, r e, of any interactions. For a many-body system the length scales associated with the density, n σ ( r), and temperature, T, must also be large compared to the effective range. If the atoms are in a trap then the length scale associated with the trap should also be much greater than the effective range. For a spherically symmetric trap V ( r) = 1 2 mω2 r 2 we require (/mω) 1/2 r e. Thus for the Tan s relations to be valid we require ( ) 2π a r e, n 1/3 2 1/2 r e, λ T = r e, mk B T ( ) 1/2 r e (2.5) mω where λ T is the thermal de Broglie wavelength. These requirements that the effective interaction scale be much less than any other length scale is at the root of the name contact. The contact, C, is an extensive quantity that is the integral over real space of the contact density C ( r) C = d 3 r C( r) (2.6) 2.4 Tan s relations We now present a selection of Tan s relations. 10

11 2.4.1 Large k Momentum Scaling of n σ ( k) The tail of the momentum distribution for each spin state, n σ ( k) in the limit of large k scales according to n σ ( k) C k 4 (2.7) This scaling is independent of spin state and the coefficient C is the contact. This asymptotic behaviour only holds for k such that a 1 ( ) 1/2 k r e, k n 1/3, k λ 1 T, k (2.8) mω The momentum distributions have been normalized according to N σ = d 3 k (2π) 3 n σ( k) (2.9) where N σ is the total number of atoms in spin state σ. This universal relation implies that the contact is positive definite and has dimensions of (length) 1. Accordingly the contact density has dimensions of (length) Density-density Correlator at Short Distances At short separations, R, the correlation between the densities of opposite spin separated by R diverges as 1/R 2 and is proportional to the contact density: n 1 ( r + R 2 ) n 2 ( r R 2 ) 1 ( 1 16π 2 r 2 ) C( r) (2.10) 2 ar This Tan s relation will be used later to give a qualitative meaning to the contact Adiabatic Relation The previous sections showcased universal relations that are concerned with the tails of distributions. The adiabatic relation shows that the contact is a central property of the system. The rate of change of the total energy of the system with respect to changing the inverse scattering length very slowly is ( d E d a 1 ) S 11 = 2 4πm C (2.11)

12 at contact entropy S. Implicitly the total particle numbers, N σ are also held fixed. This relation holds whether the system is in an energy eigenvalue of energy E or in a statistical mixture of eigenstates so long as the occupation numbers are held fixed. The contact is not just a quantity that governs the tails of distributions it is one of the fundamental properties of the system. 2.5 Qualitative Meaning of the Contact To understand the meaning of the contact, consider the density-density correlator, Eqn 2.10 described earlier and its Tan relation. An alternative expression for that Tan relation is n 1 ( r + R ) 1 n 2 ( r R ) π 2 R 1 R C( r) (2.12) 2 2 Integrating R 1 and R 2 over a ball of radius s gives N pair ( r, s) s4 C( r) (2.13) 4 The left hand side is the number of pairs of opposite spins in a sphere of radius s. Consider a gas of atoms with no interactions and a sphere of volume V with radius s f embedded in the gas. Counting pairs of opposite spins such that each atom of a given spin in a volume V is counted as pairing with all other opposite spins in that volume V, as previously, then the number of pairs scales as V 2 s 6 f. This change in scaling dimension due to interactions is generic. The difference is dimension, -2, is known as the anomalous dimension. Thus a non-zero contact implies that there are interactions and that consequently the number of pairs of opposite spins in a small volume is greater than the non-interacting case. This abnormal scaling behaviour only applies in the regime a s r e, s n 1/3, s λ T, s 1/2 mω (2.14) where the scaling behaviour extends to small s on the order of r e. 12

13 Chapter 3 Measuring the contact with RF Spectroscopy Most methods of measuring the contact make use of the high momentum tail of the single particle momentum distribution in the cloud. For example the Jin group measured the contact with three methods: measuring the single particle momentum distribution directly, performing rf spectroscopy, and using atom photoemission spectroscopy [5] which are each different facets of measuring the high momentum tail. They found that using each method produced consistent results for the contact. For the measurements in this thesis only rf spectroscopy was used. 3.1 Experiment Detailed descriptions of the entire experimental cycle can be found in previous PhD theses from our group [10, 19, 20]. Most of the experimental cycle concerns preparing the ultracold cloud of atoms before we can actually use the cloud to perform experiments. We first trap 40 K and 87 Rb in a Magneto-optical trap. The atoms are then transferred to a pure magnetic trap and moved to a position 200 µm from a microfabricated chip. This chip has wires etched onto its surface that we use to create a magnetic trap and deliver the rf pulses to the atoms. The 40 K atoms are cooled by sympathetic evaporation of the 87 Rb and then transferred to a crossed optical dipole trap (ODT). After spin manipulation to put the atoms in the ground state a final sympathetic evaporation step is performed to reduce the temperature of the cloud to the requested value. The entire experimental cycle is controlled by an ADwin controller which in turn is instructed by a computer running custom lab written software. The ADwin must receive 13

14 instructions in between every cycle but the software takes time to convert its inputs into a format that the ADwin understands. I rewrote parts of the software so that the overall cycle time was decreased by 4 seconds. A typical experimental cycle with my improvements is currently 30 seconds. This 10% decrease in cycle time allows us to collect more statistics in the same amount of time. After the preparation we are left with a cloud of spin polarized 40 K atoms trapped in our ODT cooled typically to an initial temperature of around 0.2 T/T F where T F is the Fermi temperature. Since the atoms are trapped in an optical trap the magnetic field is a free experimental parameter. The magnetic field at the atoms points in the direction we call the z-axis and this defines our quantization axis. Initially all of the atoms are in the ground state, 9/2, which is adiabatically connected to the F = 9/2, m F = 9/2 state at zero field. We apply a rf pulse resonant with the state 7/2 which induces Rabi flopping. The power and length of this pulse are calibrated to produce an equal coherent superposition of 9/2 and 7/2 atoms. Such a pulse is commonly called a π/2-pulse since on the Bloch sphere representing these two spin states this pulse corresponds to a rotation about an axis by π/2. In addition to the total magnetic field that controls the distance from the s-wave Feshbach resonance we apply a small magnetic field gradient in the direction of the magnetic field along the z-axis. When the atoms are in the 9/2 state they are aligned with the field and as such the gradient has no effect. With the application of a π/2-pulse all of the spins point transversely to the magnetic field. The spatial gradient in magnetic field then results in a spatial gradient of Larmour frequencies as each spin precesses around the local magnetic field. Viewed in a rotating frame, each spin rotating at a slightly different frequency results in a spin spiral along the z-axis which increasingly twists as each end of the cloud become increasingly out of phase. If the spin spiral were not present then Pauli exclusion would prevent the atoms from interacting via an s-wave collision and so the gas would remain as if it were an ideal gas composed of two spin species. With the spin spiral induced by the gradient, neighbouring spins are no longer identical and so they can interact via an s-wave interaction. As the twist of the spin spiral increases the atoms can interact more and more. Thus the trap averaged s-wave contact of the cloud starts at zero when the π/2-pulse is applied and grows as time progresses. As the spin spiral twists, spin currents are induced which drive demagnetization by diffusive processes. When the cloud is unpolarized the trap averaged contact reaches its maximal value. Only the timescale for the growth of contact is governed by the demagnetization dynamics. The final value of the contact for an unpolarized gas is independent of 14

15 the demagnetization dynamics. By tuning the magnetic field near the Feshbach resonance between 9/2 and 7/2 atoms at G we can control the strength of the interactions which results in a changing value of the trap averaged contact of the unpolarized cloud. To measure the trap averaged contact we perform rf spectroscopy after holding the cloud in the trap for a given hold time after the initial π/2-pulse. In the next sections we describe rf spectroscopy and how we extract the contact from rf spectra. 3.2 RF Spectroscopy and Imaging Radio frequency spectroscopy consists of applying a rf pulse to the cloud which excites a portion of atoms from one of the spin states, 9/2 or 7/2, into a higher probe state. In this thesis we consider the probe state to be non-interacting with either of the two initial states. The number of atoms in the probe state and the initial state are then measured using state selective imaging. In our setup we use the state, 5/2, which is the state adiabatically connected to the F = 9/2, m F = 5/2 state at zero field, as our probe state. We choose to transfer atoms from the 7/2 state to the probe state. The 9/2 to 5/2 transition is not excited by our probe pulse. When the rf probe pulse is resonant with the 7/2 to 5/2 transition the number of atoms transferred is maximal. This peak in 5/2 number as a function of frequency is called the single particle peak since this feature is produced by exciting a single free 7/2 particle to the probe state. By changing the frequency of the rf probe pulse we probe the number of atoms that can be excited into the 5/2 state as a function of frequency. In rf spectroscopy the signature that indicates a non-zero contact is an excess of transferred probe atoms in the high frequency tail of the single particle peak. This is due to the high frequency tail having a different scaling with frequency than the low frequency tail for non-zero contact. In the next section we give a qualitative picture for this excess. Since we are interested in the high frequency tail of the single particle peak it is important to use an rf pulse that is as close to a delta function in frequency space as possible so that we do not contaminate the high frequency tail with weight from the single particle peak. A square pulse in time is a sinc 2 function in frequency space which has appreciable spectral weight in the sidebands [10]. We use a Blackman pulse for our rf spectroscopy because Blackman pulses have negligible weight in its sidebands [10]. 15

16 3.2.1 Contact Signature in rf Spectroscopy For a qualitative picture of how a non-zero contact results in an excess consider an unpolarized cloud with a mixture of 9/2 and 7/2 atoms in equal proportion. Consider also that the rf pulse consists of many photons all of the same frequency and that only one photon can interact with a single 7/2 atom. In the absence of a contact, or alternatively interactions, as the rf frequency is scanned the natural linewidth of the 7/2 to 5/2 transition results in a Lorentzian shaped peak in the number of 5/2 atoms transferred as a function of frequency. A photon with a higher frequency than the 7/2 to 5/2 transition by many linewidths would have negligible probability of causing an excitation. Since contact is a measure of the number of opposite spin pairs in a small volume a nonzero contact implies the presence of 9/2 atoms very close to the 7/2 atoms. Therefore if a rf photon at a higher frequency than the single particle peak by many linewidths interacts with a 7/2 atom the 9/2 atom is close enough to carry away the excess energy. A photon at a lower frequency than the single particle peak is unaffected. Thus the high frequency tail of the single particle peak has an excess and extends further that the low frequency tail. 3.3 State Selective Imaging After the rf spectroscopy pulse we first jump the field to 209 G where the scattering length, a, has a zero crossing and then perform state selective imaging. State selective imaging is a variant of standard time of flight(tof) imaging. Time of flight imaging involves dropping the atoms from the trap, waiting as the atoms expand during TOF, and then imaging the atoms. State selective imaging in general consists of an initial Stern-Gerlach pulse after turning off the trap which gives each m F state a different momentum impulse. The atoms then fall in a comparatively weak magnetic field gradient. During the TOF we use resonant rf pulses to swap spin states such that the initial states that we want to count are mapped to the spin state that is imaged. Our absorption imaging system only uses light on resonant with the F = 9/2, m F = 9/2 to F = 11/2, m F = 11/2 transition and so after TOF only the atoms that are in the 9/2 state will be imaged and the others will be invisible. Previous experiments[10, 14] used a previous state selective imaging protocol. This protocol counted the number of atoms initially in the 9/2 and 5/2 states. To count the atoms in the 9/2 and 7/2 states simultaneously the atoms in the 7/2 and 5/2 states would be swapped via adiabatic rapid passage. For some of the data taken for this thesis a new protocol was used. This protocol allows the number of atoms in all 16

17 three states, 9/2, 7/2, and 5/2 to be counted simultaneously at the cost of some potential cross-counting between high momenta atoms. We now detail the new state selective imaging protocol, see Fig The optical dipole trap is turned off and a pulse of current through one of the wires on the microfabricated chip generates a magnetic field with a strong gradient in the vertical direction. The gradient from this Stern-Gerlach pulse gives each spin state a different acceleration. The clouds then fall in a comparatively weak magnetic field gradient. When the clouds have begun to separate a first rf pulse is applied that is calibrated to be a π-pulse at the spatial position of the 5/2 atoms which puts the 5/2 atoms into the 7/2 state. Due to the gradient not all 7/2 atoms are transferred to the 5/2 state. After the first rf pulse the atoms continue to fall and separate. A second rf pulse to swap atoms in the 9/2 and 7/2 states is then applied. This pulse is calibrated so that maximal number of 7/2 atoms, from the initial 5/2 atoms cloud, are transferred. Some of the atoms in the other two clouds are also swapped between the 7/2 and 9/2 states. This results in each initial spin state being mapped to three spatially separated clouds of 9/2 atoms. As the clouds expand they separate due to the differential kicks they receive from the Stern-Gerlach pulse. However they also expand due to their kinetic energy which reduces the effective rate that the clouds separate. To allow the clouds to be clearly distinguished the total TOF has been increased compared to the previous protocol. Even after a longer TOF the clouds are closer together which necessitates a more careful analysis to prevent cross-counting between spin states. Work to improve the separation and analyze images with slightly overlapping clouds is ongoing. 17

18 Figure 3.1: Schematic of state selective imaging procedure. See text for details. Note that the fraction of atoms shown in each pie slice is only for illustrative purposes. The initial proportions of atoms will change due to the rf spectroscopy. This figure is adapted from [10]. 18

19 3.3.1 RF Spectra Rescaling To formally state the scaling of the high frequency tail of the single particle peak of an rf spectrum consider the two level system of 7/2 and the probe state 5/2. In the so called linear regime the transfer rate to the probe state is Γ(δ) = N 5/2(δ) t p (3.1) where Γ(δ) is the transfer rate from 7/2 to 5/2, N 5/2 (δ) is the number of atoms transferred into the 5/2 state, t p is the length of the rf pulse used to transfer the atoms, and δ = ω ω 0 is the detuning from the single particle peak. We are sure to remain in the linear regime by changing the power of the rf pulse such that less than 20% of the 7/2 atoms are transferred to the probe state. Higher transferred fraction would result in non-linear effects. At and around the s-wave Feshbach resonance, such that the interactions are strong, the transfer rate, Γ(δ), for large positive detunings, δ scales as lim Γ(δ) = δ F δ δ er Ω 2 R 8π m/δ 3/2 C (3.2) where Ω R is the on resonance Rabi frequency, and m is the mass of 40 K. The lower limit is the detuning corresponding to the Fermi energy δ F = E F. The upper limit is given by the effective range such that δ er = a 2 /(2mr 2 e). For the 9/2 and 7/2 resonance δ er is on the order of 20 MHz[10] and so is well outside the region we can probe. Note that we assume no final state effects due to interactions between the probe state and the initial states. Instead of detuning in units of radians per second, δ it is convenient to define the dimensionless detuning = δ/e F. We define the normalized transfer rate, Γ( ), in terms of as Γ( ) = and putting in the form for Γ(δ) gives Γ( ) = E F Ω 2 R πn Γ(δ) (3.3) 7/2 E F Ω 2 R π N 5/2 N 7/2. (3.4) 19

20 The behaviour of the normalized transfer rate for large detunings, 1, is Γ( ) C 2 3/2 π 2 3/2 k F N 7/2 (3.5) Inverting this relationship we find that at each detuning the contact per particle is given by C = k F N Γ( )2 3/2 π 2 3/2 = E F 2 3/2 π 3/2 N 5/2 ( ) 7/2 Ω 2 R t (3.6) p N 7/2 This relationship is only valid in the limit 1. A plot of contact as a function of detuning will display a plateau when the normalized transfer rate scales as 3/2. The definitions for the global Fermi energy, E F, and the Fermi wavevector, k F, are those of an unpolarized gas. In terms of the total local density, n, the local Fermi wavevector and local Fermi energy are k F = (3π 2 n) 1/3 ɛ F = 2 (3π 2 ) 2/3 2m n2/3 (3.7) and in terms of the total number, N, the global Fermi energy and global Fermi wavevector are ( ) 1/3 ( ) 1/6 3 2mω 3 E F = ω 2 N k F = 2 N (3.8) where ω = (ω x ω y ω z ) 1/3 is the geometric average of the trap frequencies Molecules in RF spectra The presence of Feshbach dimers results in a molecular feature in the spectra. The spectra shown in Fig. 4.1 has a small molecular feature. As stated in Eqn. 2.4 the binding energy of the dimers is given by E b = 2 m(a r e ) 2 (3.9) We see these dimers in our rf spectra when the rf photons disassociate the dimers and then excite the 7/2 atom to the probe state. For this to happen an rf photon must be of energy greater than the sum of the single particle transition energy and the binding energy. As a consequence for large binding energies the molecular feature in the rf spectra produced by the dimers is distinct from the single particle peak. The separation between the single particle peak and the molecular feature is governed by the binding energy of the dimers. Far away from resonance the molecular feature is clearly separated from the single particle peak. 20

21 At the resonance the molecular feature overlaps the single particle peak. The long high frequency tail of the molecular feature is due to the two-body nature of the disassociation process. Any excess energy above the binding energy is converted to kinetic energy of the final 9/2 and 5/2 particles. Performing rf spectroscopy at a much higher frequency than the combined single particle peak and binding energy can cause the energetic particles produced from rf spectroscopy to escape the trap. We investigated the effect of trap depth on the molecular feature by reducing the power in the optical dipole trap. Figure 3.2 shows the rf spectra for two different trap depths. It is evident that for reduced trap powers the high frequency tail of the molecular feature is dominated by loss from the trap. Unfortunately we have no reason to believe that at our highest trap depth that the high frequency tail is not dominated by loss. This is supported by comparisons with theory[17] which suggests the high frequency tail should be much flatter. We will not discuss it further but, just as this thesis shows it is possible to measure a repulsive contact for the upper branch, it should be possible to measure a repulsive contact for the dimers in the lower branch. To do this would require measuring the high frequency tail of the molecular feature accurately enough to distinguish the scaling of the contact from the disassociation physics. 21

22 N N khz Figure 3.2: The molecular feature is shown as a function of detuning from the single particle peak for a 20µK deep trap, upper, and a 30µK deep trap, lower. The 20µK deep trap shows molecular loss due to the disassociated atoms leaving the trap. Error bars are not shown. 22

23 Chapter 4 Observations We have made the first measurements of an s-wave contact on the repulsive side of a Feshbach resonance. The signature that indicates contact is an asymmetry of the single particle peak such that the high frequency tail is larger and scales differently than the low frequency tail. On the attractive side of the Feshbach resonance the scaling of this high frequency tail at large positive detuning from the single particle peak is related to the contact. On the repulsive side of the resonance we observe an asymmetric high frequency tail and analyze it using the same theory as for the attractive side. We first show that we observe the formation of the contact on millisecond timescale. We then show our results for measuring the contact as a function of (k F a) 1. Finally we present spectroscopy measurements near a p-wave resonance that show evidence of a quantity analogous to the s-wave contact. Note that our measurements of the contact are an average over all the densities of the trap. 4.1 Contact for Repulsive Iterations One of the first rf spectra taken on the repulsive side of the Feshbach resonance, at G, is shown in the left plot of Fig There is a clear high frequency tail on the single particle peak after 2.4 ms of hold time. The molecular feature is separated from the single particle peak by 120 khz. As discussed in Sec. 3.1 we observe that the contact increases on the millisecond timescale. This is consistent with previous measurements of the rise time of contact at the resonance [14]. An example of this behaviour is shown in Fig On the attractive side of the 9/2 to 7/2 Feshbach resonance measurements of the contact have been made previously as a function of (k F a) 1 [5, 6]. In previous work we 23

24 Fraction Fraction Frequency MHz Frequency MHz Figure 4.1: The left plot shows one of the first rf spectra taken to probe the contact on the repulsive side at G. The vertical axis is the fraction of atoms excited into the probe state. The circles are rf spectroscopy after a hold time of 0.1 ms. The dots are rf spectroscopy after a hold time of 2.4 ms. There is a clear high frequency tail on the single particle peak for the 2.4 ms hold time measurements and a small molecular feature. The right plot shows rf spectra performed with different powers with Rabi frequencies of 190/2π khz, black, and 18/2π khz, red at G. Note that the beginning of the molecular feature is closer to the single particle peak since the field is closer to the resonance at G. The different magnetic fields mean the single particle peak is also shifted due the Zeeman effect. have measured contact at the resonance [14]. We now present measurements of the contact on the repulsive side of the 9/2 to 7/2 resonance. We have also made measurements of the contact on the attractive side of the resonance. The analysis of the repulsive high frequency tail of the single particle peak is identical to the analysis at resonance and on the attractive side of the resonance but is complicated by the presence of the molecular peak. Since the binding energy of the dimers increases moving away from the resonance the separation between the single particle peak and the molecular feature is greater the further from resonance. The rf spectrum is only directly relatable to the contact for large positive detunings, 1, from the single particle peak. Therefore there is some minimum (k F a) 1 at which the molecular peak contaminates the high frequency tail and any value extracted for the contact cannot be trusted. In addition to this the contact decreases away from the resonance and we do not have sufficient signal to noise to resolve the high frequency tail. Therefore there is a range of (k F a) 1, or magnetic fields, for which we can measure the contact. This is also complicated by the presence of the two p-wave resonances between 7/2 and 7/2 atoms at G and G. During the analysis raw spectra are converted, using the equations given in Sec , into the scaled detuning,, and the normalized spectra, Γ. The contact is then computed and plotted as a function of detuning. Figure 4.3 shows Γδ 3/2 as a function of the detuning 24

25 Fraction Frequency MHz Figure 4.2: RF spectra for different hold times of 0.1 ms black circles, 0.6 ms green dots, 1.2 ms blue dots, and 2.4 ms black dots. The smaller high frequency tail at for 0.6 ms shows the growth of contact. δ/2π. The set of contact values as a function of detuning is then analyzed to determine the value of the plateau. Note in Fig 4.3 that the plateau due to contact is interrupted by the molecular feature. The behaviour around δ = 0 is due to the scaling of the single particle feature. The contact as a function of (k F a) 1 is shown in Fig On the attractive side our data agrees qualitatively with previous measurements [5, 6]. For large negative (k F a) 1 our results agree with the BCS limit. On the repulsive side a naive theory would predict that the contact would increase without bound towards the resonance. However we suspect that non-linear effects will cause the contact to roll over towards resonance to some constant value as a function of (k F a) 1. The value of the contact for free particles on the repulsive side of the resonance is much smaller than on the attractive side. A theoretical prediction by H. Lianyi is shown in Fig. 4.4 as the black dashed line [21]. The data presented in Fig.4.4 was analyzed by Chris Luciuk using a definition of the Fermi energy E F = ω(3n) 1/3 in contrast to equation Eqn My work to reanalyze the data independently with the definition of the Fermi energy given in this thesis is in progress. Note that these definitions of the Fermi energy only differ by a factor of 2 1/3 which only affects the numerical values of the contact by the same factor. 25

26 Detuning 2 MHz Figure 4.3: Plot of Γδ 3/2 which is proportional to the contact as a function of detuning δ for 0.1 ms, circles, and 2.4 ms, dots. Note how the plateau is interrupted at a detuning of 0.12 MHz by the molecular feature. Errors are not shown. 4.2 Spectroscopy Near a P-wave Feshbach Resonance In the process of taking data to measure the repulsive contact we performed spectroscopy at G. There are two p-wave resonances at G and G. The first such rf spectra is shown in Fig We observed that the single particle peak had a high frequency tail for a hold time of 0.1 ms after the creation of the mixture of 9/2 and 7/2. After 3.0 ms the high frequency tail is no longer present. We believe the initial asymmetry to be due to an analogous mechanism as the s-wave contact but with p-wave interactions. Since p-wave interactions are antisymmetric, interactions between 7/2 atoms are not suppressed by Pauli exclusion due to our creating a coherent superposition. Thus the 7/2 atoms can interact immediately after the π/2-pulse. We know that in the absence of Pauli exclusion the contact will form rapidly [10]. This is why we see an asymmetry immediately after creating our superposition. The disappearance of the asymmetry for longer hold times could be due to the rapid loss of 7/2 atoms near the p-wave resonance [15]. For a p-wave interaction the momentum distribution for large k scales according to 1/k 2 [22]. For this momentum distribution the scaled spectra will scale as 1/2 for 1. To remove any effects due to residual 9/2 atoms we performed a π-pulse instead of the initial π/2-pulse such that only 7/2 atoms are present. With no 9/2 atoms around we still see an asymmetry at short hold times. Figure 4.6 shows a plot of Γ as a function of for an initial π-pulse. The inset shows a fit of 1/2 to the high frequency tail. Figure

27 1.5 Ck F N k F a Figure 4.4: The contact as a function of (k F a) 1. The red dashed line corresponds to the BCS limit, 4(k F a) 2 /3. The black dashed line is a theory plot by H. Lianyi [21]. is taken from analysis performed by Chris Luciuk Fraction Frequency MHz Figure 4.5: Spectra at G showing an asymmetry at short hold time, 0.1 ms, and none at long hold time, 3.0 ms. 27

28 Figure 4.6: Scaled spectra, Γ, as a function of detuning,. Inset: Fit of 1/2 to the high frequency tail. 28

29 Chapter 5 Conclusions and Future Work We have measured the contact for a repulsive Fermi gas for the first time. We see the contact grow as a function of time. Compared to the contact for an attractive gas the contact for a repulsive gas is smaller. We have also observed a signal analogous to the contact for a p-wave resonance. Immediate future work includes investigating the p-wave resonance as a function of magnetic field and as a function of temperature. Other future work involves increasing the size of our optical dipole trap with the addition of another beam. A larger trap will allow lower densities which will change the Fermi energy which will allow more before the molecular feature at the same field. This should allow us to probe closer to the repulsive side of the resonance without having the molecular feature contaminate the 1 limit of the high frequency tail. However since the density also affects k F there will still be a limit to the values of (k F a) 1 that we can probe. Another benefit of lowering the density is that the rate of dimer formation will be reduced. Adding this new beam also opens up the possibility of adding a lattice and probing the contact of a 2D gas. 29

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