We propose a scheme to engage phase shifting interferometry on cold atomic samples and present the simulation results under several experimentally achievable conditions nowadays. This method allows far-detuning, low power probing, and is intrinsically nondestructive. This novel detection means yields image quality superior to the conventional phase contrast imaging at certain conditions and could be experimentally realized. Furthermore, the longitudinal resolution of imaging by this manner is mainly set by optical interference and can be better than the diffraction limit. This scheme also provides special advantages to diagnose the surface-trapped clouds, with which phase imaging on the fabricated wires and atoms altogether is possible as well.
© 2011 Optical Society of America
Destructive imaging by means of near-resonance fluorescence and absorption are often utilized on cold atomic samples. This type of detection heavily counts on photon absorption and always accompanies with large photon scattering rate. It thus causes undesirable heating and state flip-ping on the atoms and inevitably damages the clouds to a large extent. Nondestructive imaging method mostly relies on photon refraction and is engaged under the conditions of far-detuning from the atomic resonances and low light level. Therefore, it is less harmful to the cold atoms and are more favorable for in-situ imaging on cold clouds [1–3], especially when the samples are delicate and with short lifetime such as Bose condensates [4, 5].
Besides the single-beam detection techniques, nondestructive spatial heterodyne imaging (SHI) done by two beam interference was also demonstrated previously . This method is implemented by sending a weak probe beam through the cold cloud along one arm. A lens subsequently images the probe beam onto a CCD camera. Another strong reference beam simultaneously propagating through the second arm is also imaged using a separate lens to interfere the probe beam at an angle θ on the CCD plane. The phase image of the atomic cloud is thus incorporated with a set of parallel fringes with a spatial frequency fs ∼ θ/λ, assuming the light wavelength is λ. The signal, namely the phase image, can be effectively demodulated and reconstructed from a single interferogram as long as fs is much higher than the desired characteristic spatial frequency of the cloud . Although further tilting the reference beam wavefront allows to increase fs, the CCD pixel size a sets the upper limit on fs to be 1/(2a) where two neighboring fringes are just resolvable . Furthermore, similar to the temporal heterodyne detection, the signal on the CCD camera is proportional to the square root of the product of probe and reference beam intensities, this allows to lower the probe power while raising the reference power in the same proportion to still keep the same signal magnitude. Using the spatial heterodyne detection method, Kadlecek et al. demonstrated much higher spatial contrast can be reached while with much fewer photons scattered.
In parallel to the tremendous progress on creating ultracold atomic gases in the past decade, developing on destructive optical imaging also advances significantly to achieve spatial resolution allowing for single neutral atom position measurements [9, 10], and even overcome the diffraction limit . On the other side, the phase-shifting interferometry (PSI) is widely used for nondestructive detection on living cells , and has long been demonstrated to reach depth resolution of few nm for microfabricated chip surface measurements [12–14]. However, to our knowledge, it has never been applied on cold atomic samples.
Motivated by the unique feature of the conventional phase shifting interferometry and the advantages offered by the spatial heterodyne imaging, in this letter, we propose and analyze another nondestructive detection way for cold atoms, with a combination of both methods, substituting the tilting of wavefronts in Ref.  by control of the relative phase between probe and reference beams. We will describe how PSI is implemented on cold samples and its excellent performance on high longitudinal resolution. Besides, we also present several potential applications using PSI.
2. Theory: phase shifting interferometry on cold atomic clouds
The transmission (Mach-Zehnder) type phase shifting interferometer for cold atomic cloud is illustrated in Fig. 1. First, a laser beam with Gaussian profile and linear polarization parallel/perpendicular to the incident plane, is collimated to a waist size w0. It is then divided into two after the 50%–50% beam splitter BS1. The reference beam of intensity IR(x, y) propagates through free space in one arm, whereas the probe beam with intensity IP(x, y) passes along the other arm where the atomic cloud locates. Here we assume the transverse coordinates of each beam are represented by (x, y) with respect to its longitudinal propagation direction along z axis.
After propagating an equal distance from BS1, owing to the density distribution of the atoms ρ(r⃗) the probe beam picks up an extra spatial dependent phase than its counterpart in the reference arm while passing a distance dz across the cloud, where λ is the laser wavelength and n(r⃗)is the refractive index of the atoms. Hence the overall additional phase accumulated by the probe is , by integrating through the whole atomic cloud in the longitudinal direction. Using two-level approximation n(r⃗) is shown to be 
The two beams, reference and probe, are subsequently recombined by another 50% – 50% beam splitter BS2. An imaging lens is placed to refocus the beams at the Fourier plane and simultaneously images the probe and reference beam wavefronts locating at positions (a) and (b), where both are equally displaced from BS2, onto a CCD camera. To be more specific, our scheme shown in this letter is in another limit of the spatial heterodyne imaging, in which the probe and reference beams are overlapped on the CCD camera with zero intersection angle, namely θ = 0. Under this condition the intensity pattern on the CCD is simply6] is no longer applicable. Instead, phase image can still be obtained by engaging phase-shifting interferometry with which several interferograms I(x,y,t) are sequentially recorded at different time t while Φ(t) is precisely controlled using an injected phase shifter.
To proceed PSI it usually requires taking at least 3 phase shifting steps [12, 15], though two-step PSI was recently invented . To illustrate the idea how PSI is applied on a cold atomic cloud, here we adopt the standard procedure using 4-step of equal phase shift by π/2 in our scheme. To tune the optical phase Φ a locking beam of wavelength λL, propagating in parallel to the PSI beam, is also injected [12, 17]. With this setup it allows to take four images IΦ(x,y) from CCD camera in sequence for Φ= 0, π/2, π, and 3π/2, respectively. From the four 2D intensity images the phase image of the cold cloud is obtained by12]. The advantages exhibited from PSI to retrieve the phase image by using Eq. (3) is now evident. First, the required calculation is rather straightforward and usually can be very fast. Second, the precision of the retrieved phase strongly depends on the setting phase shift and can be very well controlled using the presently existing techniques  and possibly reaches sub-wavelength scale. Moreover, as shown in Eq. (3), since each of the related image pair is processed by subtraction first, hence the offset effects are automatically canceled; and yet the gain-related effects can also be eliminated by taking the ratio of these image signals.
Furthermore, a 2D phase image ϕ (x,y) basically shows the additional optical phase accumulated along z direction at each position (x,y) in the probe arm than its counterpart in free space. It thus not only allows to map out the cloud density distribution in xy-plane using ϕ (x,y), but also to extract the spatial thickness by combining Eqs. (1) and (3).
Different from most other systems, conditions for employing PSI on cold atomic samples are more stringent. This is due to that the clouds are very dilute and delicate. Usually, the lifetime of the cold samples with spatial density less than 5×1014 atoms/cm3 is in order of few seconds or longer and is mainly set by the inelastic collisions with other atoms and photons. Therefore, to engage PSI, all the measurements have to be carried out before the samples die down or significantly degrade. However, since fast CCD using frame-transfer technique  and real-time phase tuning scheme  are both available nowadays thus the above issue can be completely resolved.
Moreover, in order to implement nondestructive measurement using the PSI scheme, the deposited heat from the probe beam must be kept small thus not to increase the cloud temperature or disturb its density distribution. To fulfill this requirement, photon absorption and reemission via dissipative process needs be much suppressed. The heating rate from the probe photons can be estimated by first knowing the spontaneous photon scattering rate of a two-level atom 
The condition to treat cold samples as transparently dispersive media is also evident from Eq. (4) and can be achieved by taking large probe detuning. Once the detuning is chosen, in normal nondestructive single-beam imaging method, the lowest probe intensity is then set by the desired signal-to-noise (S/N) ratio. Under the same experimental condition as in the single-beam detection, in our two-beam scheme an enhancement factor on S/N ratio by is automatically provided . This permits to lower the probe beam power and thus deposit less heat to the cold samples while still maintaining similar image quality.
Although the cold clouds have much shorter lifetime compared with the biological and semiconductor samples, however they are highly reproducible in many research laboratories worldwide. This allows to engage PSI, instead of taking many sequential images for different phase shifts on the same cloud during a period much shorter than the cloud lifetime, we could take only one image on one cloud under a fixed phase shift and take a new one on a fresh cloud under another phase shift. This would make PSI operation much easier since it only requires ordinary CCD camera instead the frame-transfer one and simple phase shifter setup such as that shown in Ref. .
Furthermore, it is straightforward to switch our scheme to a reflection type (Michelson type) and yield another powerful way to image the surface-trapped cold atomic samples . Since the optical path is doubled in this scheme the phase shift is also increased by twice. More importantly, using PSI measurement it is possible to retrieve the cloud depth profile better than the diffraction limit for the 2D atomic clouds confined in atom chip surface . Additionally, all wires fabricated on the surface of an atom chip are essentially phase objects and can be directly mapped out using the reflection-type PSI too. This thus allows to diagnose the wires as well as precisely monitor the cloud location altogether, which can never be achieved by other means.
3. Theoretical simulation and results
The theoretical simulation to illustrate the transmission-type PSI scheme is described below. For simplicity, we assume a Gaussian laser beam is collimated to the beam size w0 all along through the propagation distance L. This condition is fulfilled as long as the Rayleigh length is much larger than L. Due also to this, we can ignore the dipole force exerted on the cloud. In order to set the power ratio IR(x, y)/IP(x, y) to the desired value β, a power attenuator such as neutral density filter (not plotted) is placed in the probe arm after BS1. As shown in Fig. 1, each individual beam walks for an equal distance from BS1 and arrives position (d) along the reference arm, and to position (c) where the probe beam just reaches the atomic cloud. Assume the electric fields at positions (d) and (c) are ER0(x, y) and EP0 (x, y)eiΦ, respectively, with . Note a controllable phase factor Φ is added in probe arm for the reason to implement PSI which is just described above.
Since only the probe beam passes the cloud, the electric field gains an additional phase ϕ(x,y) while moving from position (c) to (a), relative to its counterpart in the reference arm from position (d) to (b). Therefore, the electric fields at positions (a) and (b) are and , respectively. If the absorption is negligible while passing the cloud, the probe beam is only altered in its phase. To measure the wavefronts at positions (a) and (b) a thin imaging lens, with focal length f and aperture diameter D, is placed after BS2 and sets the object distance p for both arms. The wavefronts are thus imaged on the CCD at a distance q from the lens and is magnified by M = −q/p. We are especially interested in the experimentally feasible and favorable condition: and σx,y ≪ D, where σ is the transverse radius of the cold cloud. Under these assumptions we simulate the wavefront imaging by applying Fresnel diffraction integral and the thin lens approximation  to obtain the electric fields at CCD position after propagating through the probe and reference arms asEq. (2), where c is the speed of light and ɛ0 the vacuum permittivity.
For the incident cylindrically symmetric Gaussian beam of total power P we rewrite the electric fields at positions (c) and (d) as and , where , and r2 = x2 + y2; PR and are the reference and probe beam powers and satisfy the weak absorption condition P ≈ PR + PP. Again, for simplicity, the atomic cloud containing N atoms is also assumed to have a Gaussian density distribution as , where ρ0 is the peak density, and σx, σy, σz are the root-mean-square (rms) radii in the corresponding directions. By defining the effective volume Ve ≡ (2π)3/2σx ·σy · σz, the total atom number is simply related to the peak density by N = ρ0Ve.
Knowing the cloud density distribution allows to calculate the phase factors on EP1 (x, y), and on ER1 (x, y), respectively. We also notice the phase integral on the probe electric field already converges to the asymptotic value as long as the total integration limit along z reaches 14σz. This means the additional phase picked up on the probe electric field is mostly accumulated in the ±7σz range around the cloud center. Accordingly, the object plane (a) in the probe arm, for the imaging lens, sits at a distance 7σz after the cloud center. The electric fields at the CCD camera from both beams can thus be calculated once the object and image distances p and q are determined. Additionally, to guarantee sufficient dispersive light signal due to phase lag across the cloud is collected, the beam waist w0 must be much larger than σx,y. Whereas, the highest achievable transverse resolution for the phase image is still limited to due to diffraction .
Furthermore, to reduce the cloud heating during the probe period, lower light power is always desirable. Nevertheless, the photon detector sets a lower bound on it. We consider a CCD camera having the quantum efficiency η (λ) for wavelength λ, and also accompanying with a dark current Nd. Thus the total photon count Npe is Np · η corresponding to the incident photon number Np. Both Npe and Nd are in unit of electron count. Assume a minimum photon count to dark current ratio α = Npe/Nd is required for the CCD to clearly identify the Gaussian probe beam up to the size s · σx,y, with s ≤ 3 ∼ 5, the minimum beam power under a probe time tp is estimated to be , where rp is the pixel size of the CCD. For the conventional single-beam phase contrast imaging, the lowest power for the probe beam is just Pm. However, using the interference scheme shown in this letter we can set the probe beam power down to Pm/(β + 1) and meanwhile raise the reference power to βPm/(β + 1) to still maintain the same signal magnitude as the single-beam one. Judging from the lower heating rate offered by the two-beam-based detection method the PSI scheme clearly shows its advantage for nondestructive measurement.
To verify the PSI scheme provides a practical means for nondestructive imaging on cold atoms, we examine several cold 87Rb clouds of densities from 1011 to 1014 atoms/cm3 and temperatures between 200 μK and 2 μK; i.e., the condition for atoms in magneto-optical trap (MOT) to Bose-Einstein condensates. The simulations are carried out based on the above discussions. Since 87Rb atoms are used the probe beam wavelength λ is now ∼780 nm, and for most of the cases its size w0 is set to be at least ten times larger than the rms radius of each simulated cloud. For convenience, through all simulations, we set the probe beam intensity Ip to be 1/100 of the saturation intensity Is ∼ 2.5 mW/cm2 of 87Rb D2 transition . The probe beam power is also determined by Pp = Ipπw02/2. Additionally, since the probe beam is collimated with long Rayleigh length and has the extremely low intensity, any effect due to dipole and scattering force can be ignored.
The heating rate and peak phase shift ϕ0 ≡ ϕ (x = 0, y = 0) for each simulated cloud under different probe beam detunings are calculated and plotted in Fig. 2. To set the optimum probe detuning one must take cloud parameters such as its temperature, density and size, heating rate and desired phase shift into consideration and get a good compromise among them. For all simulated clouds, the optimum probe detuning is around 100 MHz at where the phase shift is the largest while heating is still not yet a problem.
Figure 3 shows the four simulated PSI interferograms and retrieved phase image for a cigar-shape cloud containing 8 × 106 atoms with peak density ρ0 = 1.0 × 1012 atoms/cm3, the rms radii in x and z directions are both 55μm and with the aspect ratio σy/σx = 3 in y and x directions. This is a typical cloud in the period of evaporative cooling, usually having temperature in tens of μK. The simulation is done by assuming β = 1 and the following imaging parameters: f = 5 cm, D = 2.5 cm, p = 8 cm, q = 13.3 cm, hence M = 1.67. When applying a probe beam with detuning 100 MHz, the total heating on the atoms is about 63 nK during a probe time of 1 ms and is very small compared to the cloud temperature.
The final retrieved phase image is extracted directly using Eq. (3) by taking the simulated images, I0(x, y), Iπ/2 (x, y), Iπ(x,y), I3π/2(x, y) corresponding to the four sequential phase shifts of π/2, respectively. The probe beam size used in the simulation is about 13σx and therefore the total probe power is about 200 nW, which is larger than the minimum required power Pm = 10 nW under the following practical detection parameters: α = 10, η = 0.3, Nd = 20 and rp = 7 μm. As what we discuss above the heating rate can be further lowered by a factor of while still keeps the same S/N ratio, by increasing the reference beam power a factor of .
The retrieved phase image is shown in Fig. 3(b) and with the peak phase shift close to 23°. We can see the cloud can be clearly identified to the tail up to 2.2σy where the S/N ratio in phase is about 2 if we assume the phase noise on the PSI setup is ∼ 1° . The peak phase shift can be larger by decreasing probe detuning as long as the heating is not a problem. It is also possible to resolve the cloud tail even farther by injecting more stable phase shifter which will produce much lower phase fluctuations.
We would like to address an issue which might be encountered in a practical experiment situation while the above 4-step of equal phase shift is applied on cold atoms. As shown by the simulated interferogram for Φ = π in Fig. 3(a), since the overall phase shift between the probe and reference is π, hence the phase image is totally dark except around the cloud region. Under this condition the background noise might set in and contaminate the image. However, this can be technically avoided by raising the reference beam power to set a nonzero DC light background even when Φ = π, or still employing 4-step of equal phase shift of π/2 however starting from a nonzero phase shift to get away from the region near Φ = π.
Similar simulations are also applied on the two general cases for atoms in dark magneto-optical trap (dark MOT) and Bose-Einstein condensates. In Fig. 4(a), the isotropic cloud has N = 1 × 109 atoms and rms radius of 680 μm, with a peak density ρ0 = 2.0 × 1011 atoms/cm3. The cloud temperature is about 200 μK under the typical dark MOT condition. Using a probe beam of 200 MHz detuning for 1 ms PSI measurement, while p = 7 cm and with an imaging lens of 3-cm focal length and f-number of 1.2, a high contrast phase image, corresponding a peak phase shift of 29°, is obtained. Again, the total heating and possible induced trapping force from the probe beam can be ignored. In Fig. 4(b) the cloud is in a condition of Bose condensation and consists of 6×106 atoms while with temperature of ∼ 2μK. The near pancake cloud has the rms radii σy ∼ 38.5 μm, σx ∼ 30.8 μm, and σz ∼ 3.2 μm, corresponding to a peak density ρ0 = 1 × 1014 atoms/cm3. The simulation is carried out by taking the probe beam detuning as 500 MHz which is far enough and only heats up the cloud by about 3 nK for a probe time of 1 ms. The phase image shows a peak phase shift of about 27°.
One more concern that we would like to point out here is the lensing effect from the atoms which might degrade the image quality [4, 26]. This effect arises when the probe beam is shed into a small and dense cloud. If the cloud radius is d the effective focal length is fe = d/2(n – 1), where n is the refractive index of the cloud under the probe detuning . Assume a probe beam of 500 MHz detuning is shined into a cloud with an average radius of 30 μm as that in Fig. 4(b). The effective focal lens is estimated to be 1000 μm and is much longer than the field depth of 70 μm for an object distance of 6.7 cm and the given imaging lens with focal length of 5 cm and an f-number of 2 . Hence the lensing effect is insignificant in this situation although the cloud is already very small and dense. In addition, the field depth of the imaging system can be set up to as shallow as few micrometers nowadays . Therefore, under the circumstances lensing effect can be completely ignored. The simulated data convince that PSI implemented under the above conditions is experimentally feasible.
One interesting case occurs when the longitudinal thickness of the cloud is smaller than the optical diffraction limit. In this situation it is impossible to resolve the cloud size better than this limit if looking from transverse direction, either by traditional destructive methods or nondestructive phase contrast means. To demonstrate our PSI scheme allows to break this limit in principle we simulate an almost 2D cloud with its short axis along the z direction. We also assume the atom cloud is formed, under some special trapping condition and suitable cooling procedure, to have temperature close to 4 μK, total atom number 1 × 106 atoms and peak density ρ0 = 5 × 1014 atoms/cm3, while with σy = σx ∼ 35 μm and σz ∼ 0.25 μm. Figure 5(a) shows the 3D phase image using a probe beam detuning of 100 MHz. The peak phase shift is around 21° and the cloud heating during a 1 ms probe time is still small compared to the cloud temperature. Indeed, the heating rate can be further lowered by reducing the probe beam power if necessary.
Since the cold cloud has a non-uniform density distribution, to specify its average length along the longitudinal direction we thus define the effective thickness Ze(x, y) which satisfies the following equationEq. (7). For the atomic cloud with a Gaussian density distribution while trapped in a harmonic potential the corresponding effective thickness image is thus obtained as Figure 5(b) presents the effective thickness image and clearly shows the peak thickness is about 0.63 μm, smaller than the probe beam wavelength of 0.78 μm. We also estimate the effective focal length fe of the 2D cloud to be 270 μm under the light detuning of 100 MHz. Compared to the depth of field of about 60 μm, using the similar imaging parameters as in Fig. 4(b) except p = 6.5 cm here, the effective focal length is still long enough to avoid significant lensing. The simulation results also promise our PSI scheme allowing for in-situ imaging and resolving the cloud thickness better than the diffraction limit of 1.9 μm under the above imaging condition.
With this special feature, we would emphasize the following potential applications on the atom chip using the presented PSI scheme. 1. The in-situ imaging provides high longitudinal resolution for studying the surface effects . Additionally, the reflection-type PSI scheme applied on atom chip also enhances the phase signal due to double path. 2. The large magnetic field gradients on the atom chip surface produces significant spatial dependent detunings for the probe beam due to Zeeman shifts. The phase images taken from the clouds near the surface should also map out the magnetic fields. Therefore, PSI also provides a means to retrieve the local magnetic field distribution in the scale of the clouds, typically in few micrometers. 3. The signal to noise ratio in this PSI scheme may be significantly improved by coherently enhancing the atom refractive index using the method presented in . This also renders a possibility to detect extremely thin and dilute 2D atom clouds using phase-shifting interferometry in the future. Moreover, since each wire, including its shape and depth, on the atom chip can also be measured by the same PSI setup for the atoms, therefore exact measurements on the relative positions of the cloud to the wires are possible. This might be useful to help diagnosing the atom chip defects or delivering the cloud to the desired locations to very high precision.
To compare with the spatial heterodyne imaging presented in Ref. , the scheme shown in this letter requires a real-time phase-shifter to tune the relative phase between reference and probe beams while several interferograms are taken. However, due also to this, our scheme presents the following advantages. 1. The phase image reconstruction is rather straightforward simply using Eq. (3) and does not require other elaborative calculations such as FFT or digital filtering which are usually time consuming and case sensitive. 2. The spatial phase shift retrieved using PSI is determined by the injected phase shifter and essentially can be very precise. If the injected phase shifter (locking beam) also works as an active phase stabilizer of the interferometer such as that shown in Ref. , phase imaging implemented using this scheme is thus more robust against phase noise, for example due to vibration, and should also provide better longitudinal resolution. 3. The transverse resolution of phase image in our scheme is mainly set by the diffraction limit of the final imaging lens. In SHI, the phase image reconstruction fully counts on the spatial fringes. However, the optimum fringe space is a tradeoff between its resolution and contrast due to the finite pixel size of the CCD. This eventually leads to significant image resolution degradation [6, 7]. Therefore, our scheme is superior in its transverse resolution and more suitable for imaging small atomic cloud like Bose-Einstein condensate.
In conclusion, we theoretically analyze a scheme by applying phase-shifting interferometry on cold atomic clouds for nondestructive imaging. We carry out theoretical simulations on cold 87Rb clouds, in temperatures of 2 μK to 200 μK and density 2 × 1011 to 5 × 1014 atoms/cm3, under different probe beam detunings. The simulation results promise the PSI scheme can be experimentally realized and is suited for other atomic species as well. We also demonstrate the longitudinal resolution offered by the PSI scheme is better than the optical diffraction limit set by the optical imaging system and should provide a powerful tool for examining the 2D atom clouds in the future. Finally, we indicate the possible future applications using this scheme on the experiments related to atom chip, such as mapping out the magnetic field distribution with micrometer resolution, and imaging the cold cloud and the fabricated wires altogether for atom chip diagnosis and surface effect monitoring.
We acknowledge the support from the National Science Council of Taiwan (R.O.C.) under NSC grant No. 98-2112-M-194-001-MY3.
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