Ultracold atoms in multiple-radiofrequency dressed adiabatic potentials

TL;DR

This paper demonstrates a novel method for creating and controlling configurable double well potentials for ultracold atoms using multiple radiofrequencies, enabling precise manipulation of atomic confinement geometries.

Contribution

It introduces the first experimental implementation of multiple-radiofrequency dressed potentials for ultracold atoms, allowing adjustable and reconfigurable magnetic trapping geometries.

Findings

Successful creation of a double well potential with adjustable barrier height

Good agreement between experimental spectroscopy and Floquet theory calculations

Potential for extending the method with additional dressing frequencies

Abstract

We present the first experimental demonstration of a multiple-radiofrequency dressed potential for the configurable magnetic confinement of ultracold atoms. We load cold $^{87}$Rb atoms into a double well potential with an adjustable barrier height, formed by three radiofrequencies applied to atoms in a static quadrupole magnetic field. Our multiple-radiofrequency approach gives precise control over the double well characteristics, including the depth of individual wells and the height of the barrier, and enables reliable transfer of atoms between the available trapping geometries. We have characterised the multiple-radiofrequency dressed system using radiofrequency spectroscopy, finding good agreement with the eigenvalues numerically calculated using Floquet theory. This method creates trapping potentials that can be reconfigured by changing the amplitudes, polarizations and frequencies of the applied dressing fields, and easily extended with additional dressing frequencies.

Equations

$${\mathbf{B}}_0(\mathbf{r})=B_q^{\prime }(x{\hat{\mathbf{e}}}_x+y{\hat{\mathbf{e}}}_y-2z{\hat{\mathbf{e}}}_z)$$

$${\mathbf{B}}_{\text{MRF}}(t)=\sum _j{B}_j\frac{\cos ({\omega }_{j}t+{\varphi }_j){\hat{\mathbf{e}}}_x-{\kappa }_j\sin ({\omega }_{j}t+{\varphi }_j){\hat{\mathbf{e}}}_y}{\sqrt{1+{\kappa }_j}}$$

$$V=\sum _j\hslash {\omega }_j{a}_j^{†}{a}_j+g_F{\mu }_B\mathbf{F}\cdot \left[{\mathbf{B}}_0(\mathbf{r})+{\mathbf{B}}_{\mathrm{MRF}}\right]$$

$$\mathbf{F}\cdot {\mathbf{B}}_{\mathrm{MRF}}=\sum _j{\mathcal{E}}_j\left(\frac{{\alpha }_j}{\sqrt{2}}{F}_{+}+\frac{{\beta }_j}{\sqrt{2}}{F}_{-}+{\zeta }_j{F}_z\right)a^{†}+\text{HC.}$$

$${\zeta }_j(\mathbf{r})=\frac{1}{\sqrt{1+{\kappa }_j}}(\sin \theta \cos \varphi +i{\kappa }_j\sin \varphi ).$$

$${\Omega }_1=\frac{g_F{\mu }_B{B}_1}{2\sqrt{2}\hslash }\left(1\pm \frac{2z}{\sqrt{x^2+y^2+4z^2}}\right)$$

$${\Omega }_1=\frac{g_F{\mu }_B{B}_1}{2\hslash }{\left(\frac{r_{\perp }^2+4z^2}{r_{\perp }^2+r_{\parallel }^2+4z^2}\right)}^{1/2}$$

Full text

††preprint: APS/123-QED

I Introduction

Our understanding of quantum systems has been shaped by the ability to study ultracold atoms in a variety of trapping geometries. These range from regular potentials such as lattices Bloch2012 , waveguides Navez2016 , rings Ramanathan2011 ; Jendrzejewski2014 and box traps Gaunt2013 ; Chomaz2015 to more arbitrary configurations such as tunnel junctions Husmann2015 or disordered potentials Choi2016 .

Such traps are often implemented using optical methods, exploiting their versatility in spite of drawbacks such as unwanted corrugations from fringes, sensitivity to alignment and off-resonant scattering processes that require large detunings and associated optical powers.

The application of a radiofrequency (RF) field to a static magnetic trap dramatically changes the character of the confinement Zobay2001 ; Colombe2004 , providing additional parameters to control the potential while retaining the advantages over optical dipole force traps. A single RF applied on an atom chip Reichel2011 has been used to coherently split a 1D quantum gas Schumm2005 , a technique since used to shed light on the nature of thermalisation in near-integrable 1D quantum systems Gring2012 . RF ‘dressed’ adiabatic potentials (APs) have also been employed to probe 2D gases Merloti2013 ; DeRossi2016 . Ring traps can be implemented by time averaging Lesanovsky2007 ; Gildemeister2010 or by adding an optical dipole potential Heathcote2008 , and are used to study superflow or for matter-wave Sagnac interferometry Navez2016 . The introduction of a multiple-radiofrequency (MRF) field provides an additional means by which to shape these potentials Courteille2006 , further increasing the versatility of magnetic traps.

In this work we demonstrate MRF APs for the first time, creating a highly configurable double well potential with three radiofrequencies. Dynamic control over these potentials, which take the form of two parallel sheets, can be achieved by manipulating the RF polarisation and amplitude, or properties of the underlying static field Hofferberth2006 ; Gildemeister2012 ; Navez2016 . These traps are intrinsically state- and species-selective Extavour2006 ; Courteille2006 ; Bentine2017 , with demonstrably low heating rates when created using macroscopic coils located a few cm from the atoms Merloti2013 . Magnetic double well potentials have previously been demonstrated using a single RF on an atom chip Schumm2005 ; Hofferberth2006 , and by time-averaging either a bare magnetic trap Thomas2002 ; Tiecke2003 or AP Lesanovsky2007 ; Gildemeister2010 ; our MRF method builds upon these works to offer increased tuneability through independent control of the constituent dressing field components. This double well potential could be developed to investigate tunnelling dynamics or cold-atom interferometry Lesanovsky2006 ; Schumm2005 between pairs of 2D sheets. As a natural extension, additional frequency components can be applied to produce lattices Courteille2006 , continuous potentials, or wells connected to a reservoir Hunn2013 .

Our discussion begins with an introduction to the theory of MRF dressed potentials in Sec. II, focusing on the experimentally demonstrated three-frequency field. In Sec. III we present our experimental results, exploring the manipulation of atoms in our MRF double well potential. We describe the experimental apparatus and methods in Sec. III.1 and demonstrate precise control over the potential landscape in Sec. III.2.

After a discussion of RF spectroscopy methods in Sec. III.3, we use this technique to probe the MRF potential landscape and validate our theoretical model in Sec. III.4. We conclude in Sec. IV by outlining the new experimental possibilities arising with complex trapping geometries controlled by multiple RF fields.

II Atoms in a multi-component RF field

The dressed-atom picture of atom-radiation interaction Cohen-Tannoudji1969 ; Muskat1987 can be used to describe atoms trapped in optical, microwave Agosta1989 ; Spreeuw1994 , and RF fields Zobay2001 . An RF-dressed adiabatic potential (AP) provides a trapping mechanism for cold atoms subjected to uniform RF and inhomogeneous static magnetic fields Zobay2001 ; GarrawayPerrin2017 . We describe the theory of MRF dressed potentials in two parts: Sec. II.1 presents the calculation of the quasi-energy spectrum using Floquet theory, and Sec. II.2 describes the resulting potential surfaces and practical considerations of their implementation.

II.1 Atom-photon interactions

In this work we consider ${}^{87}\text{Rb}$ atoms in the $F=1$ hyperfine ground state, originally confined in the static magnetic quadrupole field

[TABLE]

with $B^{\prime}_{q}$ the radial quadrupole gradient and $\hat{\mathbf{e}}_{x},\hat{\mathbf{e}}_{y},\hat{\mathbf{e}}_{z}$ the Cartesian unit vectors. This inhomogeneous field introduces a spatial dependence to the Zeeman splitting between hyperfine sublevels. We apply the homogeneous MRF dressing field

[TABLE]

where $B_{j}$, $\omega_{j}$, and $\phi_{j}$ are the amplitude, angular frequency and relative phase of each frequency component respectively. In our experimental implementation we use three RF components $\omega_{j}=\omega_{1,2,3}=2\pi\times(5,6,7)\times$0.6\text{,}\mathrm{MHz}$$, producing circularly polarised dressing fields for $\kappa_{j}=1$, and linearly polarised fields for $\kappa_{j}=0$. The following discussion describes either linear or circularly polarised RF fields, for which the dressed-atom Hamiltonian of the system reads

[TABLE]

where

[TABLE]

In this expression, $\mathbf{B}_{\mathrm{MRF}}(t)$ now describes the second quantised operator for the MRF field with mode densities $\mathscr{E}_{j}$, and amplitudes $\alpha_{j},\ \beta_{j}$ and $\zeta_{j}$ as defined in Eqs. 5 and II.1. The Hermitian conjugate is indicated by HC, while $g_{F}$ denotes the Landé g-factor and $\mu_{B}$ the Bohr magneton.

The first term in Eq. 3 accounts for the energy of the RF field component $j$ with angular frequency $\omega_{j}$ and corresponding photon creation and annihilation operators $a_{j}^{\dagger}$ and $a_{j}$. The second term describes the interaction between the atomic spin $\mathbf{F}$, defined following the convention in Ref. Foot2005 , and the total magnetic field comprising static and RF components with operators $\mathbf{B}_{0}(\mathbf{r})$ and $\mathbf{B}_{\mathrm{MRF}}(t)$ respectively.

The combined system of magnetically-confined atom, RF radiation, and the interaction between them can be intuitively described in the dressed-atom picture, as illustrated in Fig. 1 for a single- and triple-frequency field. In the absence of interactions with the RF field, the dressed eigenstates $\ket{n_{1},n_{2},...,m_{F}}$ are the tensor products of the Fock states of each RF field $\ket{n_{j}}$ and the atomic Zeeman substates $\ket{m_{F}}$. These form a ladder of eigenenergies $g_{F}\mu_{B}m_{F}\absolutevalue{\mathbf{B}_{0}}+\sum_{j}n_{j}\hbar\omega_{j}$ in which the three Zeeman substates are repeated with a spacing of $\omega_{f}$, the highest common factor of RF photon frequencies $\omega_{j}$. The interaction described by Eq. 4 drives transitions between dressed states, turning energy level crossings into avoided crossings.

While the dressed-atom picture provides an intuitive visualisation of the RF dressing process, the large mean photon number of the RF field allows it to be represented classically by replacing $\mathscr{E}_{j}a_{j}^{\dagger}$ and $\mathscr{E}_{j}a_{j}$ by their mean field value $\frac{1}{2}B_{j}$. This is performed within the context of the interaction picture, in which $V\rightarrow U_{\mathrm{RF{}}}^{\dagger}VU_{\mathrm{RF{}}}$ and $\ket{\psi}\rightarrow U_{\mathrm{RF{}}}^{\dagger}\ket{\psi}$ with $U_{\mathrm{RF{}}}=\exp[i\sum_{j}a_{j}^{\dagger}a_{j}\omega_{j}t]$.

The RF field is decomposed into components parallel and perpendicular to a local axial vector $\hat{\mathbf{z}}^{\prime}$ where $\mathbf{F\cdot B}_{0}=B_{0}F_{z^{\prime}}$. The parallel component is given by $\frac{1}{2}B_{j}\zeta_{j}\exp[i(\omega_{j}t+\phi_{j})]+\text{CC}$, where CC indicates the complex conjugate, with

[TABLE]

From the definition of the static quadrupole field, $\cos\theta=-2z(x^{2}+4z^{2})^{-1/2}$ and $\cos\phi=[(x^{2}+4z^{2})/(x^{2}+y^{2}+4z^{2})]^{1/2}$. The anticlockwise and clockwise rotating components of the perpendicular field are $\frac{1}{2}B_{j}\alpha_{j}\exp[i(\omega_{j}t+\phi_{j})]+\mathrm{CC}$ and $\frac{1}{2}B_{j}\beta_{j}\exp[i(\omega_{j}t+\phi_{j})]+\mathrm{CC}$ respectively, with {IEEEeqnarray*}rCl α_j (r) &= 12+2κj(cosθ- i sinθsinϕ- κ_j cosϕ),

β_j (r) = 12+2κj(cosθ+ i sinθsinϕ+ κ_j cosϕ). \IEEEyesnumber

In this basis the semiclassical version of the Hamiltonian presented as Eq. 3 becomes {IEEEeqnarray*}rl V(t) =& g_{F}$$\mu_{B}B_0 F_z

• g_{F}$$\mu_{B}2 ∑j [ ( αj2 F- + βj2 F_+ + ζ_j F_z ) B_j e^i (ω_j t+ϕ_j)

+( αj2 F_+ + βj2 F_- + ζ_j^* F_z ) B_j e^-i (ω_j t+ϕ_j) ], \IEEEyesnumber

which is periodic in time with period $T=2\pi/\omega_{f}$. The coefficients $\alpha_{j},\ \beta_{j}$ and $\zeta_{j}$ give the projection of the field operator in the local circular basis, with $\absolutevalue{\alpha_{j}}^{2}+\absolutevalue{\beta_{j}}^{2}+\absolutevalue{\zeta_{j}}^{2}=1$.

Using Floquet’s theorem, the eigenstates of this time-periodic Hamiltonian, with period $T$, can be expressed in the form $\ket{\psi(t)}=\exp[iE^{\prime}t/\hbar]\ket{\Psi(t)}$, a product of a phase term and the time-periodic state vector $\ket{\Psi(t)}$, where $\ket{\Psi(0)}=\ket{\Psi(T)}$. Alternatively, one can write $\ket{\psi(t)}=U(t)\ket{\psi(0)}$, where $U(t)$ is the time evolution operator. We calculate $U$ through numerical integration of the Schrödinger equation with the interaction Hamiltonian of Eq. II.1. By comparing these two equations for $\ket{\psi(t)}$, we find $U(T)\ket{\psi(0)}=\ket{\psi(T)}=\exp[iE^{\prime}T/\hbar]\ket{\psi(0)}$. The phases $E^{\prime}T/\hbar$ can be associated with the energy of the dressed eigenstates of Eq. 3 at time $T$ Shirley1965 ; Yuen2017 such that the dressed state eigenenergies modulo $\hbar\omega_{f}$ are given by the $2F+1$ eigenvalues of $(-i\hbar/T)\log U(T)$. These eigenenergies are illustrated in Fig. 1 for the three-RF example that we investigate experimentally.

II.2 Adiabatic potentials

The interaction $g_{F}\mu_{B}\mathbf{F\cdot B}_{\mathrm{MRF}}$ couples the states to form avoided crossings at values of the static field for which the energy splitting $g_{F}\mu_{B}B_{0}$ is resonant with an integer multiple of $\hbar\omega_{f}$. When this interaction is sufficiently strong and the static field orientation varies sufficiently slowly with position, an atom traversing an avoided crossing can adiabatically follow this new eigenstate, labelled by the quantum number $\widetilde{m}_{F}$ Lesanovsky2006a .

In the case of a single applied RF with angular frequency $\omega_{1}$ shown in Fig. 1 (a), atoms trapped in $\widetilde{m}_{F}=1$ experience a trapping potential $U_{AP}(\mathbf{r})=\widetilde{m}_{F}\hbar\sqrt{\delta^{2}(\mathbf{r})+\Omega^{2}_{1}(\mathbf{r})}$ where $\delta(\mathbf{r})=|g_{F}\mu_{B}B_{0}(\mathbf{r})/\hbar|-\omega_{1}$ gives the angular frequency detuning of the RF from resonance and the Rabi frequency is determined by the applied RF amplitude and polarisation.

The spatial variation of the static field amplitude $B_{0}(r)$ translates the detuning-dependence of the potential to a spatial dependence, such that for the static quadrupole of Eq. 1 the resultant trapping potential forms an oblate spheroidal ‘shell trap’. Atoms are trapped on the surface of this resonant spheroid, over which the spatial variation of the coupling strength is dictated by the RF polarisation.

The Rabi frequency for a circularly polarised RF field is given by

[TABLE]

with $B_{1}$ the magnetic field amplitude of the $\omega_{1}$ RF field and $x,y,z$ Cartesian coordinates with an origin at the centre of the quadrupole field. The sign of the second term depends on the handedness of the RF field polarisation; in this work the handedness is chosen such that the coupling is maximised at the south pole of the resonant spheroid. For the case of an RF field linearly polarised in the $xy$ plane the Rabi frequency instead takes the form

[TABLE]

where $r_{\parallel}$ and $r_{\perp}$ describe the coordinates parallel and perpendicular to the polarisation direction of the linear RF field. The resonant spheroid therefore has maximum coupling at points for which the parallel component is zero, and zero coupling at the points on the equator for which the perpendicular component is zero.

As illustrated in Fig. 1 (b), this principle can be easily extended to the MRF case, in which the three first-order avoided crossings form two trapping wells separated by an anti-trapping barrier for an atom in $\widetilde{m}_{F}=1$. This results in trapping on two concentric spheroids forming a spatially-extended double well in which the relative heights of the barrier and both wells are controlled by the three separate input RFs. Multi-photon interactions lead to cross-talk between these features, and the impact of the amplitude $\Omega_{j}$ of each avoided crossing on the properties of its neighbours is investigated experimentally in Sec. III.2 and III.4. Also studied in Sec. III.4 is the effect of the relative phase $\phi_{j}$ between RF components; this alters the overall shape of the MRF waveform and thus influences the strength of nonlinear multi-photon processes that occur.

Adiabaticity constraints motivate the choice of parameters including the frequency separation, RF amplitudes and static field gradient. An atom with constant velocity $v$ moving through this spatially-varying potential will remain trapped with a probability approximately given by the Landau-Zener model: this states that $P_{\text{LZ}}=(1-\exp[-h\Omega^{2}/4g_{F}\mu_{B}\partial_{t}B_{0}(vt)])^{2}$ where the time derivative of the static field $B_{0}$ indicates the field gradient as experienced by the moving atom Courteille2006 ; Burrows2017 . Minimising the well spacing requires a dressing RF frequency separation comparable to the Rabi frequency of each RF component.

As the piecewise approach presented in Ref. Courteille2006 is invalid in this limit Morgan2014 Chakraborty2017ARf-fields , Floquet theory is employed to calculate the MRF dressed state eigenenergies. Numerical artefacts are removed by appropriate meshing over the range of magnetic field values considered, while an intuitive depiction of MRF dressing that uses the resolvent formalism to discard these artefacts is explored in Ref. Yuen2017 .

III Experimental implementation of the MRF potentials

III.1 Trapping atoms in an adiabatic potential

In standard operation, we routinely produce BECs of $3.5\times 10^{5}$ 87Rb atoms in the $\ket{F=1,m_{F}=-1}$ hyperfine state using a time-orbiting potential (TOP) trap Petrich1995 , via an experimental sequence that we can truncate to load thermal atoms into an AP prior to a final stage of evaporation. The TOP is formed by applying a bias field, rotating at $7\text{\,}\mathrm{kHz}$, to the static quadrupole field of Eq. 1. This bias field sweeps the quadrupole field in a horizontal circular orbit with a rotation radius given by $B_{T}/B_{q}^{{}^{\prime}}$, with $B_{T}$ the amplitude of the TOP field.

The TOP and dressing RF fields are generated by a coil array that surrounds the atoms, with an extent of a few cm. This array is illustrated in Fig. 2. The RF signals for each coil and frequency component are independently generated by direct digital synthesis (DDS) 111Analog Devices AD9854. This digital control over the amplitude and polarisation of each dressing field component enables us to precisely sculpt the waveform and resultant potential as a function of time. The signals for each coil are combined using splitters 222Mini-Circuits ZSC-2-2, and amplified by 25 W amplifiers 333Mini-Circuits LZY-22+. The RF coil array has a self-resonance of approximately $7\text{\,}\mathrm{MHz}$ such that, with a custom wideband impedance match, we can confine atoms in APs with dressing frequencies in the range $2\pi\times 2.7$ to $2\pi\times 4.4$ MHz without additional amplification. Mixing processes in the amplifiers constrain us to use only combinations of dressing frequencies with a common fundamental $\omega_{f}$, ensuring that the resulting intermodulation products are far detuned from transitions between dressed states such that we avoid losses.

We load a single-RF shell with thermal atoms as described in Gildemeister2010 ; Sherlock2011 ; Gildemeister2012 , combining the TOP field with dressing RF to produce a time-averaged adiabatic potential (TAAP) as illustrated in Fig. 2(b-e). The dressing RF is switched on while the TOP field satisfies $2\hbar\omega_{1}/g_{F}\mu_{B}>B_{T}>\hbar\omega_{1}/g_{F}\mu_{B}$ such that the TOP field sweeps the resonant spheroid in an orbit outside the location of the atom cloud. With an RF amplitude on the order of $\Omega_{j}=g_{F}\mu_{B}B_{j}/(\sqrt{2}\hbar)=2\pi\times 400$ kHz at the south pole of the spheroid, decreasing $B_{T}$ allows us to load the atoms into the TAAP formed at the lower of the two intersections of the spheroid with the rotation axis under the influence of gravity. The RF field is circularly polarised in the laboratory frame, with a handedness that maximises the interaction strength at the bottom of the resonant spheroid. Using an additional weak field we then optionally perform forced RF evaporation to BEC in $2\text{\,}\mathrm{s}$, exploiting the enhanced radial trap frequencies inherent to the TAAP. Reducing $B_{T}$ to zero subsequently loads atoms from the TAAP onto the lower surface of the shell. This reliably loads condensates of greater than $3\times 10^{5}$ atoms into the shell trap with negligible heating.

III.2 Potential shaping and the double shell

This single-RF configuration forms the starting point for the MRF double well potential, with atoms initially confined in the shell corresponding to either $\omega_{1}$ or $\omega_{3}$ and ultimately transferred into the combined $\omega_{1,2,3}=2\pi\times(5,6,7)\times$0.6\text{,}\mathrm{MHz} potential. In our apparatus the $2\pi\times 0.6$ MHz frequency difference between RF components maps to a spatial well separation of $\sim 140\mathrm{\SIUnitSymbolMicro m}$at a quadrupole gradient$B_{q}^{{}^{\prime}}=62.45G/cm, allowing the trapping wells to be clearly resolved with our low-resolution imaging system. The double shell loading procedure is shown in Fig. [3](#S3.F3) for the case of loading from a single shell at\omega_{3}$. We first ramp up$\Omega_{1}$, which has a minimally perturbative effect on the potential near the atoms but establishes this resonance in preparation for the subsequent application of the field at$\omega_{2}. As shown in Fig. [1](#S2.F1), the avoided crossing formed by \omega_{2}$takes the form of an anti-trapping barrier. As$\Omega_{2}$increases, the barrier is lowered and the MRF potential is flattened, rounded out, or tilted slightly according to the desired loading scheme and relative values of$\Omega_{1}$,$\Omega_{2}$and$\Omega_{3}$. To minimise any sudden changes in the width of the potential experienced by the atoms as the barrier is lowered,$\Omega_{1}is held at an artificially high value, and lowered to the value at which atoms can be transferred only once the barrier has been ramped down fully. Once atoms equilibrate within this new potential, we raise the barrier to separate the wells and complete the loading process. This method is illustrated in Fig. [3](#S3.F3) for the RF ramps used to split a BEC between the two shells, and variants on this loading scheme were used in the remaining figures. The second-order resonances apparent in Fig. [3](#S3.F3) place an upper limit to the well depth of\hbar\omega_{f}$; the combination of RF amplitudes and frequency separation are therefore chosen to complement the temperature of atoms loaded into the potential.

The final population imbalance between the wells is influenced by the relative amplitudes of each RF component during the ramp. The effect of barrier height is illustrated in Fig. 4, where we vary the maximum value of $\Omega_{2}$ to load a controllable proportion of atoms between the lower and upper wells, formed by $\omega_{3}$ and $\omega_{1}$ respectively. Starting from a cloud of thermal atoms in the lowest shell, the RF components $\omega_{1}$ and $\omega_{2}$ are turned on adiabatically following a similar procedure to that described in Fig. 3 in which $\Omega_{1}$ is ramped directly to its final value. Initially, few atoms possess sufficient energy to cross the high barrier that results from a small $\Omega_{2}$, and minimal population redistribution between the wells occurs. Increasing $\Omega_{2}$ to lower the barrier allows more atoms to populate the second well. At around $\Omega_{2}=2\pi\times 400$ kHz the barrier vanishes and the atoms distribute themselves across the broad single well formed by the three RF dressing frequencies as shown in Fig. 1(c). Finally, $\Omega_{2}$ is decreased to raise the barrier and split the population distribution into two distinct wells, with the proportion reflecting any imbalance between the lowest energy of each well. Figure 4(a) illustrates a loading process that transfers 52% of the atoms into the well defined by $\omega_{1}$. This could be corrected or exacerbated by adjusting either $\Omega_{1}$ or $\Omega_{3}$ to raise or lower the potential energy minimum of each well.

Figure 5 illustrates the atom density arising from two possible transport sequences. Keeping the lowest energies of each well approximately equal allows us to load the balanced double shell with approximately $75\text{\,}\mathrm{\char 37\relax}$ efficiency in atom number, while deliberately mismatching these energies allows a full population transfer between the wells. Crucially, Fig. 5 also demonstrates the effect of the barrier amplitude on the positions of the two trapping wells that is shown in the calculated energy levels in Fig. 1: the $\omega_{1}$ and $\omega_{3}$ potential minima are drawn closer together as the barrier is lowered to form the broad single well.

The simple potential shaping schemes demonstrated here for three frequencies comprise single wells, a double well, and a flattened three-frequency well. We have also demonstrated a method of dynamic control that provides the intermediate stages for loading. This approach can be extended in a straightforward manner by applying additional dressing RFs.

III.3 RF spectroscopy

RF spectroscopy is an experimental technique commonly used to precisely characterise bare magnetic traps and adiabatic potentials Hofferberth2007 ; Easwaran2010 . A weak probe RF is applied to atoms held within the trap, causing expulsion of atoms when the probe RF is resonant with a transition between trapped and untrapped states. With these resonances appearing as dips in the measured atom number, the probe frequency is varied to map out the spectrum of transitions. For a BEC, this resonance has a width on the order of the chemical potential (typically $\mathrm{kHz}$) while for a thermal cloud the resonance is broadened due to the thermal distribution of atoms in the trap Easwaran2010 .

RF spectroscopy is employed here to characterise the key components of our trapping fields: the TOP field magnitude $B_{T}$, amplitudes of applied dressing RF components, and ultimately the MRF eigenenergies. $B_{T}$ is measured by RF spectroscopy of a condensate confined in the TOP, and $B_{q}^{\prime}$ calibrated by measuring the trap frequency of the centre of mass mode of a condensate oscillating in this approximately harmonic potential for a known current through the quadrupole coils.

To calibrate the RF amplitudes, transition frequencies are measured for single-RF shells at $\omega_{1,2,3}=2\pi\times(3,3.6,4.2)$ $\mathrm{MHz}$. We use linearly polarised RF to measure the RF fields in x and y directions independently. The Rabi frequencies are calculated from these measured resonances through Floquet theory as described in Sec. II. This calculation incorporates the Bloch-Siegert shifts Bloch1940 ; AdvAtPhys . We also include the effect of gravity by adding the potential energy term $H_{\text{grav}}=mgz$ to the Hamiltonian of Eq. 3, which typically shifts the transition by a few kHz. The amplitude of each RF component used in the MRF APs is deduced using a co-wound pickup coil; we convert the measured voltage amplitudes into a magnetic field amplitude using the single-RF Rabi frequency calibration measurements. The linearity of the pickup coil response was verified by repeating the single-RF spectroscopy measurements for a variation in RF amplitude of up to $50\text{\,}\mathrm{\char 37\relax}$. We note that the combined MRF input approaches a value close to the saturation of the amplifier, resulting in an up to $4\text{\,}\mathrm{\char 37\relax}$ compression of the amplitudes of each RF component for the highest dressing RF powers applied; this saturation is accounted for by the RF pickup measurement.

The probe RF field must be sufficiently weak that it does not itself shift the transition. For the APs used here the Rabi frequencies of the dressing RFs are $100$s of $\mathrm{kHz}$ while that of the probe is below $100\text{\,}\mathrm{Hz}$. Selected RF spectroscopy measurements were repeated with probe amplitudes spanning one third to three times its standard value, with no measurable shift of the resonance observed.

III.4 RF spectroscopy in the MRF potential

The closely spaced ladder of dressed-atom energy levels resulting from the application of multiple dressing RFs leads to a large number of transitions between different Floquet manifolds that can be driven by an appropriate probe RF field GarrawayPerrin2016 ; GarrawayPerrin2017 . However, many of these correspond to higher-order multiple-photon processes with low transition rates. Determining the theoretical transition frequencies begins with a calculation of the AP eigenenergies using the Floquet method of Sec. II, followed by selecting a single energy level corresponding to the double well from the infinite ladder of periodicity $\hbar\omega_{f}$. The condensate is localised at the position of minimum energy within the well near resonance with $\omega_{1}$. Energy separations from this position in the trapped eigenstate to all untrapped eigenstates of the ladder are calculated, yielding a spectrum of possible transitions but with no information as to the strength of each individual transition.

The calculated eigenenergies are experimentally verified using a BEC confined in the $\omega_{1}$ shell, using a linearly polarised MRF field to minimise experimental variables and eliminate any experimental uncertainty arising from the phase between x and y field components. The spectroscopy method, calculated values, and measured results are illustrated in Fig. 6. We measure the dressed state transition as illustrated in Fig. 6(a). By separately varying $\Omega_{2}$ and $\phi_{2}$, the amplitude and phase of the barrier RF, we experimentally probe the effects of these two parameters. These results are plotted in Fig. 6(c) and (d) respectively. The theoretical transitions were calculated for each set of measured RF field amplitudes $\Omega_{j}$ and phases $\phi_{j}$, and plotted with a finite width corresponding to the uncertainty arising from quadrupole gradient and RF amplitude calibrations.

The RF amplitude ramps for these measurements follow a similar method to that discussed in Sec. III.1 but starting with a BEC in the shell formed by the linearly polarised $\omega_{1}$ field component, ramped from circular polarisation over $500\text{\,}\mathrm{ms}$. $\Omega_{2}$ and $\Omega_{3}$ are then ramped up to their final values with a set relative phase, to form the MRF potential in which RF spectroscopy is performed. For the barrier amplitude spectroscopy measurement plotted in Fig. 6(c), $\Omega_{1}=2\pi\times$187\text{,}\mathrm{kHz} and $\Omega_{3}=2\pi\times$248\text{\,}\mathrm{kHz}, while $\Omega_{2}$ takes values between [math] and $2\pi\times$332\text{,}\mathrm{kHz} with a quadrupole gradient $B_{q}^{\prime}=$82.5\text{\,}\mathrm{G}\text{\,}{\mathrm{cm}}^{-1}. Over the course of the $\Omega_{2}$ amplitude ramp, we measure a fall in $\Omega_{1}$ by 5% and rise in $\Omega_{3}$ by 1% due to amplifier saturation and nonlinearities. This amplitude sweep is performed with a fixed phase relationship between the RF components, with relative phase components $\phi_{(1,2,3)}=(0,0.302\pm 0.001,0.132\pm 0.002)\pi$ radians where the quoted uncertainty is given by the standard deviation of the measured relative phase of each RF component. The measured field amplitudes and relative phase values are accounted for in the calculated transition frequencies plotted as the theoretical grey line in Fig. 6. The phase variation measurement shown in Fig. 6(d) sees barrier amplitudes fixed at $\Omega_{1,2,3}=2\pi\times(177,310,245)$\mathrm{kHz} with $B_{q}^{\prime}=$82.8\text{\,}\mathrm{G}\text{\,}{\mathrm{cm}}^{-1} and $\phi_{2}$, the relative phase of the barrier component, varied over a $\pi$ range. The amplitudes $\Omega_{1}$ and $\Omega_{3}$ are set such that the condensate remains confined to the initial well for the spectroscopy measurements, during which the weak RF probe is applied for a duration of $40\text{\,}\mathrm{ms}$. The potential is deformed slowly to avoid sloshing of the condensate; ramps occur over an $800\text{\,}\mathrm{ms}$ duration that is slow compared to the inverse of the 200 to $400\text{\,}\mathrm{Hz}$ axial trap frequencies. The probe duration is sufficiently long that any residual sloshing in the wells would only manifest as a broadening of the measured RF spectroscopy resonances.

The resonance point is extracted from the asymmetric spectroscopy profile Easwaran2010 by fitting a function of the form $a(x-b)+c/\sqrt{x-d}$. This function provides a good approximation to the asymmetric lineshape of the resonance profile from which the resonant probe frequency that minimises the atom number can be extracted. Only the data points lying within the range of the resonance were included in the fit, such that the asymmetric parabola captures the centre of the resonance with minimal free parameters.

The actual lineshape can be simulated numerically Easwaran2010 , and is influenced by the amplitudes of both dressing and probe RF fields, and the chemical potential of the trapped condensate. With these factors, a separate fit for each spectroscopy data set is impractical and at risk of overfitting. Qualitative comparisons between the simulated lineshape and chosen fit function suggest that the systematic uncertainty arising from a discrepancy between these models would be smaller than a kHz. The uncertainty in the fitted resonance location for both single-RF calibration and MRF potentials is estimated from the $99\text{\,}\mathrm{\char 37\relax}$ confidence interval of the fitted minimum, and is of order 1 to $3\text{\,}\mathrm{kHz}$, although with a statistical accuracy limited by the sample size. This forms the dominant source of uncertainty in the measured transition frequencies, with a smaller influence from uncertainty in measuring dressing RF amplitudes with the pickup coils. Agreement is found with calculated values for the transition frequencies for both amplitude and phase measurements.

The total width of each MRF spectroscopy resonance is of order 10 kHz, with the peak itself identifiable to within 3 kHz. The 40 kHz shift of the resonance peak over the full range of the parameter sweep is thus clearly resolved. The widths of each resonance are comparable to Ref. Hofferberth2007 although broader than those presented in Ref. Merloti2013 . This arises from the relatively weak vertical trap frequencies of 290 Hz in this work, as compared to 2 kHz in Ref. Merloti2013 , and the consequent increase in the broadening effect of the gravitational sag.

As shown in Fig. 6, increasing $\Omega_{2}$ to lower the barrier reduces the energy separation between trapped and untrapped states for the measured transition. A shift in the measured RF spectroscopy resonance on the order of tens of kHz is observed as $\Omega_{2}$ is varied, in agreement with the theory. The variation in transition energy with phase $\phi_{2}$ relative to $\phi_{1,3}$, resulting from the dependence of nonlinear processes on the overall shape of the waveform, demonstrates a periodicity in $\pi$ expected from the numerical calculations; the same calculations suggest that a $2\pi$ periodicity would arise from varying $\phi_{3}$.

IV Conclusions and outlook

We have performed the first experimental implementation of a multiple-RF adiabatic potential, using three separate dressing RFs to produce a double well configuration with independent control over each trapping well and the barrier between them. We have demonstrated potential shaping through manipulation of the individual RF amplitudes, achieving transport from one well to another, a reliable loading sequence for this double well, and dynamic control over the barrier height. Experimental characterisation of the MRF potential by RF spectroscopy of a trapped BEC validates the theoretical calculation of MRF eigenenergies by Floquet theory.

The separation of the wells in our scheme is determined by the quadrupole gradient and frequency spacing of the MRF components. In this work, we have demonstrated a large spacing of order $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

This choice was motivated by the desire to image the double well in situ with a low NA imaging system. Far smaller separations are possible using smaller frequency intervals and higher quadrupole gradients, limited only by the constraint that atoms follow the potential adiabatically Burrows2017 . For example, we have confined a BEC in a double well with a separation of $7.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$, using a frequency interval of 200 kHz, which is sufficient for matter-wave interference experiments. Exploiting the anisotropic character of RF dressed potentials Merloti2013 , our technique could be used to probe the behaviour of 2D systems Mathey2010 . Further reduction to a separation suitable for the observation of tunnelling or Josephson oscillations is possible within the constraints imposed by adiabaticity.

Dressing with multiple independently generated radiofrequencies opens a range of new opportunities beyond the existing single-RF adiabatic potential experiments while retaining their characteristic smoothness and low heating rates. As an extension of this work, additional frequency components enable the implementation of more complex geometries such as lattices Courteille2006 , box traps, or wells coupled to larger reservoirs. Independent control over both the polarisation and amplitude of each RF component permits further manipulations, for example to connect our two trapping potentials at different locations through the spatial variation of the coupling strength. The MRF technique can also be combined with existing proposals to produce AP lattices using micro-structured arrays of conductors Sinuco-Leon2015 ; Sinuco-Leon2016 , or provide a means of independent species-selective confinement for mixtures of atomic species with different $g_{F}$ values Bentine2017 .

Acknowledgements

The authors would like to thank Rian Hughes for comments on the manuscript. This work was supported by the EU H2020 Collaborative project QuProCS (Grant Agreement 641277). TLH, EB, KL and AJB thank the EPSRC for doctoral training funding.