Imaging of microwave fields using ultracold atoms
We report a technique that uses clouds of ultracold atoms as sensitive, tunable, and non-invasive probes for microwave field imaging with micrometer spatial resolution. The microwave magnetic field components drive Rabi oscillations on atomic hyperfine transitions whose frequency can be tuned with a static magnetic field. Readout is accomplished using state-selective absorption imaging. Quantitative data extraction is simple and it is possible to reconstruct the distribution of microwave magnetic field amplitudes and phases. While we demonstrate 2d imaging, an extension to 3d imaging is straightforward. We use the method to determine the microwave near-field distribution around a coplanar waveguide integrated on an atom chip.
Today, Monolithic Microwave Integrated Circuits (MMICs) are of great importance in science and technology. In particular, they constitute key building blocks of today’s communication technology.Robertson01 () MMICs also serve as main components of superconducting quantum processors.DiCarlo09 () In our group, a simple MMIC structure has recently been used as a tool for quantum coherent manipulation of ultracold atoms on an atom chip.Boehi09a ()
Function and failure analysis is of crucial importance for the design of MMICs as well as for simulation verification.Boehm94 () External port measurements (e.g. using a network analyzer) offer only limited insight. The microwave (mw) near-field distribution on the device gives much more information, enabling specific improvement. Therefore, different methods have been developed to measure the spatial distribution of mw near-fields.imagingpapers () These methods use diverse physical effects to measure the mw electric or magnetic field. They have in common that they scan the field distribution point-by-point.
Here we propose and experimentally demonstrate a highly parallel method that allows for non-invasive and complete (amplitudes and phases) imaging of the mw magnetic field distribution using clouds of ultracold atoms.NatureInsight02 () In this method, the mw magnetic field drives resonant Rabi oscillationsGentile89 () between two atomic hyperfine levels that can be detected using state-selective absorption imaging.Matthews98 () The method offers spatial resolution and a mw magnetic field sensitivity in the range at frequencies of a few GHz. It is a frequency-domain, single-shot technique to measure a 2d field distribution. The method can be extended to measure 3d distributions slice by slice. Data extraction is simple, it offers a high dynamic range, and it is intrinsically calibrated since only well-known atomic properties enter in the analysis.
For the proof-of-principle experiment presented here, we use our atom chip setup,Boehi09a () see Fig. 1. The chip has integrated mw coplanar waveguide structures that were designed for quantum manipulation of ultracold atoms.Boehi09a () To demonstrate our method, we analyze the mw magnetic field near this structure. In general, the device to be tested does not have to be integrated on an atom chip.
Our method works as follows. On the room-temperature atom chip, we prepare clouds of atoms at a temperature of using laser and evaporative cooling techniques.Boehi09a (); NatureInsight02 () The magnetically trapped atoms are initially in the ground state hyperfine sublevel , see Fig. 1a. The trap is moved close to the mw structure to be characterized, where it is switched off and the atoms are released to free fall. During a hold-off time , the cloud drops due to gravity and expands due to its thermal velocity spread, filling the region to be imaged (Fig. 1b+c). We maintain a homogeneous static magnetic field of order T. It provides the quantization axis and splits the frequencies , () of the three hyperfine transitions , () by due to the Zeeman effect (cf. Fig. 1a). A mw signal on the MMIC is subsequently switched on for a duration (typically some tens of s). We select one of the transitions by setting the mw frequency . The mw magnetic field couples to the atomic magnetic moment and drives Rabi oscillations of frequency on the resonant transition. For the three transitions of interest,
Here, and are the real-valued amplitude and phase of the component of parallel to , and () are the corresponding quantities for the right (left) handed circular polarization component in the plane perpendicular to , see Supplementary Information. After the mw pulse, a spatial pattern of atomic populations in and results, see Fig. 1c. The probability to detect an atom in is
Here, () is the density of atoms in (), which can be measured using state-selective absorption imaging.Matthews98 () From we can reconstruct and thus the spatial distribution of the resonant mw polarization component , as shown below.
For a reconstruction of , we measure , , and with pointing along the , , and axis. In each of the nine measurements (cf. Fig. 2), we tune into resonance with the desired transition . Alternatively, to test a device at a given frequency , one can achieve resonance by Zeeman-tuning of via . The measurements allow a reconstruction of the amplitudes of all three cartesian components of as well as their relative phases. The measurements of for the three orientations of directly yield the amplitudes , , and . The relative phases can be reconstructed from the other measurements, see Supplementary Information. It is also possible to measure the spatial dependence of the global phase of using an interferometric method, see Supplementary Information.
Absorption imaging integrates over the direction of propagation of the imaging laser beam. Structures to be characterized should therefore have a characteristic length scale along the beam larger than the size of the atom cloud. The cloud size can be adjusted through the magnetic trap frequencies, , and .
We experimentally demonstrate our method by measuring the mw magnetic field distribution near the coplanar waveguide (CPW) on our atom chip. The measurement is performed at a position where the CPW is translationally invariant along the imaging beam. An overview of the data is shown in Fig. 2. What appears in the images are essentially isopotential lines of the mw magnetic field components. Qualitative conclusions can be drawn directly by looking at the images – e.g. the left/right asymmetry visible in Fig. 2 reveals that there is an asymmetry in the mw currents on the CPW wires. This confirms independent findings in our previous paper.Boehi09a ()
It is possible to automatically extract from a single image. From we can calculate using Eq. (4) up to an offset of , where is an integer. The offset for each point can be calculated by a ray-tracing method, where rays are sent from the image periphery (where ) through the desired point to the center of the mw structure (where is maximal). is given by the sum of the number of minima and maxima of encountered on the ray.
Alternatively, we take series of image frames, scanning either the mw power or (the movie method). depends on the desired dynamic range, but can be as low as 10. The time to record one frame is s, but could be reduced to s.Farkas09 () For each image pixel, we thus obtain a sequence of datapoints showing Rabi oscillations, see Fig. 3a. We fit a function to the data, where and is the fit parameter. From the fit, we determine and thus, via Eqs. (1) - (3), at this pixel for a given . As an example, Fig. 3b shows an image of the cartesian mw field component near our CPW reconstructed in this way. Fig. 3c shows a corresponding simulation. Comparing data and simulation, we obtain information about the current distribution on the CPW.
The single-shot mw field sensitivity of our method is mainly determined by the interaction time . There is a trade-off between sensitivity and effective spatial resolution . Longer yields higher sensitivity, but at the same time the image blurs due to the movement of the atoms. For our parameters (optical resolution , , ) and , we obtain and a mw magnetic field sensitivity of T (Supplementary Information).
In a variant of the presented method, trapped atoms could be used as a scanning mw field sensor. For trapped atoms, can be much longer without increasing , thereby improving the field sensitivity. To record an image, the trap position has to be scanned from shot to shot of the experiment (Supplementary Information).
Our method is frequency selective. Individual components of a multi-tone signal could be resolved. The transition frequencies can be adjusted via . For up to T, which is a realistic effort e.g. for prototype testing, transition frequencies of - GHz are accessible with (Supplementary Information). Note that for T we start entering the Paschen-Back regime, where the matrix elements of Eqs. (1) - (3) change and the theory has to be modified. Using other atomic species, different frequency ranges are accessible. Alternatively, a two-photon transition could be used for imaging, where two mw fields or a mw and a radio frequency field are applied to the atoms. The first field (frequency ) is applied externally with known spatial distribution, while the other field (frequency ) is to be imaged. Resonant Rabi oscillations occur for . It is also possible to image off-resonant mw fields and other differential potentials using a Ramsey interferometry technique, see Ramsey56 () and Supplementary Information.
Our technique can be extended to measure 3d distributions of slice by slice, either by using a gradient of such that only a slice of atoms is resonant with , or by using a light sheet detection technique,Buecker09 () where slices perpendicular to the camera line of sight are imaged. Using similar techniques, it is also possible to shape the atomic cloud to prepare a thin sheet of atoms.
The method presented here seems promising for applications like prototype characterization. It allows for highly sensitive, parallel, high-resolution, and non-invasive imaging of the complete mw magnetic field distribution around an MMIC. Compact and portable systems for the preparation of ultracold atoms have been built,Farkas09 () and key components of such systems are commercially available.
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Imaging of microwave fields using ultracold atoms: Supplementary information
.1 Rabi Frequencies
We derive the Rabi frequencies Gentile89 () for the resonant coupling of ground state hyperfine levels of Rb with a microwave (mw) field. In the following, we consider an atom in a weak static magnetic field , so that the Zeeman splitting is small compared to the zero-field splitting of GHz between the two hyperfine states and of the electronic ground state of Rb. The atom is initially prepared in the hyperfine sublevel , and the microwave frequency is resonant with one of the transitions , () connecting to this level (see Fig. 1 of the main paper).
The real-valued mw magnetic field at position in the fixed cartesian laboratory coordinate system is with the complex phasor
Here, we have chosen , (). In the following, we consider a fixed position in space and suppress the dependence of and on to simplify notation. In our experiment, we apply a homogeneous static magnetic field along several directions in order to be able to reconstruct all components of the mw magnetic field. For a given , we choose a new cartesian coordinate system with the -axis pointing along , which defines the quantization axis for the atomic states . In this new coordinate system, the mw magnetic field phasor is given by
The mw magnetic field couples to the magnetic moment of the electron spin of the atom. The coupling to the nuclear magnetic moment is neglected, because it is three orders of magnitude smaller than the electron magnetic moment. The Rabi frequency on the hyperfine transition is given by
with the electron spin operator. Using we can write
Evaluating the matrix elements for the three transitions connecting to ,TreutleinThesis08 () we obtain the Rabi frequencies:
where we have used the definitions
with , (). We note that is proportional to the projection of onto , while is proportional to the right (left) handed circular polarization component in the plane perpendicular to .
In the experiment, we choose a sufficiently strong static field so that . Furthermore, we choose the microwave frequency resonant with one of the transitions, . In this way, Rabi oscillations are induced only on the resonant transition in a given run of the experiment, which allows us to selectively image the individual microwave magnetic field components .
.2 Reconstruction of the microwave magnetic field
The amplitudes , , and of the cartesian components of in laboratory coordinates can be easily determined by measuring with the quantization axis pointing along , , and , respectively. The extraction of the field components from absorption images is described in the main text. In the following, the upper index indicates the direction of the quantization axis in laboratory coordinates, e.g. () means () for pointing along the y-axis.
To reconstruct the relative phases and between the cartesian components of , we also measure the amplitudes of the circularly polarized components , , , , , and . Having measured these components, we can reconstruct the relative phases according to the following recipe.
We choose a new coordinate system with along , resulting from the following coordinate transformation:
In this rotated coordinate system, the mw magnetic field phasor reads
From this we obtain:
A similar calculation as before yields
In this case, we obtain
.3 Reconstruction of the absolute microwave phase
While Sec. .2 describes the reconstruction of the relative phases and between the cartesian components of , it is also possible to reconstruct the spatial dependence of the global phase of . The procedure uses two mw pulses. During the whole sequence, stays the same. In the following, we assume is pointing along the x-axis, so that is measured. The other two phases, and , can then be determined from the already known relative phases.
After releasing the atoms from the trap, they are prepared in an equal superposition of states and by application of a -pulse at frequency . This mw pulse is applied from a well-characterized source, so that it has negligible (or at least known) intensity gradients and negligible (or known) phase gradients across the atomic cloud. This can be achieved by using an external mw horn.Boehi09a () The duration of the pulse is , where is the Rabi frequency of the pulse. The state after this preparation pulse is (in the rotating wave approximation) Scully97 ()
Immediately after the end of this preparation pulse, the mw in the MMIC to be characterized is pulsed on at frequeny for a duration . The Rabi frequency and phase of this second mw pulse are denoted by and , respectively. After the second pulse, the state of an atom at position is
The probability of finding an atom at position in state is given by
To calculate , the quantities and have to be known. can be measured as described in the main text. If is pointing along the -axis, then .
The calculation above assumes that there is zero delay between the end of the first preparation pulse and the second pulse in the MMIC. A similar calculation is also possible for non-zero delay between the two pulses.
.4 Sensitivity & spatial resolution
In this section we estimate the maximum sensitivity of our technique for our set of experimental parameters. As described in the main text, the mw magnetic field sensitivity is mainly determined by the interaction time of the atoms with the mw pulse on the MMIC. A longer interaction time leads to a higher mw magnetic field sensitivity, because then a weaker mw field can already drive a substantial fraction of a Rabi cycle. However, at the same time the effective spatial resolution decreases as the image blurs due to the movement of the atoms during . In the following, the symbol always refers to an r.m.s. width. is determined by the average moving distance of the atoms during the mw pulse, by the optical resolution of the imaging system , and by the movement of the atoms during the imaging laser pulse, where thermal motion () and diffusive motion due to photon scattering () contribute. In the following, we will calculate , , , and .
.4.1 Movement of atoms during the microwave pulse -
A free-falling atom in a cloud at temperature has a mean thermal velocity perpendicular to the line of sight of .Ketterle99 () After releasing the atom from the trap and waiting for , the atom has furthermore acquired a velocity along the direction of gravity. During the interaction with the mw for a time , the atom moves ballistically by an average distance
For , , and we obtain . For short times, the last term in the equation above dominates (as it is the case for our parameters). The displacement of the atoms during the mw pulse can then be approximated by a Gaussian function of r.m.s. width .
.4.2 Movement of atoms during the imaging pulse -
During the imaging pulse of duration , the atoms move ballistically by an average distance given by
due to gravity and thermal motion. For we get . Again, for short times the atomic density distribution after the imaging pulse can be approximated by a Gaussian with r.m.s. width .
.4.3 Diffusive movement of atoms due to photon scattering -
During the imaging laser pulse of duration , the atoms randomly scatter photons. The associated momentum recoils lead to a diffusive motion of the atoms, which leads at the end of the pulse to an average displacement perpendicular to the line of sight of Ketterle99 ()
where is the atomic recoil velocity for , and the number of scattered photons, with and the natural linewidth MHz. For our experimental parameters (saturation parameter , ) we get .
.4.4 Effective spatial resolution
The effective resolution can approximately be calculated by the convolution , where approximates the point spread function of the imaging system by a Gaussian with . As a result, we get . We take as the effective resolution for our parameters .
.4.5 Imaging noise
The noise on the absorption images is important for the sensitivity of our technique. It determines the minimum number of atoms that has to be transferred by the mw into the manifold during in order for the mw field to be detectable. We currently use an Andor iKon-M camera with a quantum efficiency of 90% for absorption imaging Note1 (). The optical resolution of our imaging system is , the imaging pulse duration , and we are imaging at saturation intensity (). With these parameters, we calculate an uncertainty in the number of atoms detected in an area of 1.4 atoms r.m.s. We measure a value of atoms. The difference can be explained by interference fringes on the image. This additional noise could certainly be decreased further.
Quantum projection noise due to the probabilistic nature of the measurement process is an additional contribution of noise on the images. The measurement process projects the atomic superposition state onto the and states, resulting in a number of atoms of and in the two states, respectively. Even if the total number of atoms is the same in each shot, and will show (anticorrelated) fluctuations. This projection noise has an r.m.s. amplitude of , where denotes noise in and . The total noise on is thus . We find that in order to obtain a signal-to-noise-ratio , we have to have .
.4.6 Microwave field sensitivity
In the experiment described in the main paper, we trap about atoms in the magnetic trap. The trapping frequencies are and . We calculate the average atomic density in the trap to .Ketterle99 () The trapped cloud has a -radius of along the -axis, which is the direction of the imaging beam. If we image the atoms with , we have about atoms inside a cylinder of radius and height . The mw magnetic field which transfers on average atoms to is obtained by requiring that
For the three transitions , , and , this is equivalent to
Solving the above equations with , we get , , and .
Note that the projection noise slowly increases with increasing . Therefore, the absolute mw field resolution of our method decreases with increasing values of .
The consideration above relies on the assumption that we can achieve perfect resonance . Solving Eq. (22) for , we obtain for . A change in of 0.53 kHz corresponds to a magnetic field instability of T for , T for and T for . Inside the magnetic shielding surrounding our experiment, we achieve a stability of of T r.m.s., which could certainly be improved such that the effect can be neglected.
The sensitivity of our field imaging technique can be increased by using colder or denser clouds. Suitable techniques to reduce the temperature further are adiabatic relaxation of the trap or further forced evaporative cooling.
.5 Measurement of the microwave field with trapped atoms
There are further operation modes of our field imaging method. Instead of releasing the atoms from the trap, it is also possible to prepare a Bose-Einstein condensate (BEC) Ketterle99 () in a trap (either magnetic or optical) or in an array of traps (e.g. an optical lattice) and scan its position spatially from shot to shot. The position can be scanned on a length scale in all three dimensions. A typical stepsize is given by twice the Thomas-Fermi radius of the BEC.Ketterle99 () If the BEC is held in a magnetic trap, transitions between and are detected via loss of atoms, because with are magnetically non-trappable states. If the BEC(s) are held in an optical trap or optical lattice, then transitions to the manifold can be detected via state-selective imaging.
.6 Tunability of transition frequencies , and
.6.1 Rubidium atoms
The transition frequencies , , and between the initial state and the target states , , and , respectively, can be adjusted by changing the static magnetic field . For small magnetic fields (i.e. ) the transition frequencies are approximately given by , and , where and .
For larger values of a more accurate calculation is required. The ground state hyperfine structure of is described by the Hamiltonian TreutleinThesis08 ()
with the nuclear spin operator and the operator for the total electronic spin, , , and .Steck09 () Even though , the coupling of to the magnetic field becomes important for large values of . An analytical formula exists for the ground state manifold of a transition (as it is the case for our states of interest), the Breit-Rabi formula Steck09 ():
.6.2 Other atomic species
By using atomic species other than , different frequency ranges become accessible, e.g. for Cs or for Na.
.7 Ramsey interferometry and off-resonant probing
Instead of having resonant with a hyperfine transition frequency, it is also possible to probe an off-resonant mw or light field (or anything else that causes a differential energy shift between the involved hyperfine levels) using a scheme based on Ramsey interferometry.Ramsey56 () In Ramsey interferometry, the interaction with the mw field of duration is enclosed by two -pulses. The first pulse prepares the atoms in an equal superposition of two hyperfine states such as
During time both states accumulate a differential phase shift . is the differential potential between the two hyperfine levels involved and is in general state-dependent. The state after this interaction is
After applying the second -pulse, the state is
where is the phase of the resonant mw in a frame rotating at the atomic transition frequency. The probabilities to detect an atom in state and are then given by
By measuring the relative populations and after the second pulse, it is possible to determine the value of and thereby . Details on the effect of an off-resonant mw field are discussed in Treutlein06b ().
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