Cavity quantum electrodynamics with single atoms 

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At the heart of the experimental setup is a high-finesse, fiber based Fabry-Pérot (FFP) cavity[58, 69]. The primary aim of the setup is to provide an ensemble of trapped ultracold atoms well controlled spatially, and to couple them to the FFP cavity. To optimize coupling, the atoms need to be positioned at an antinode of the cavity mode. Figure 2.1 gives a schematic overview of how this aim is achieved. A MOT and an optical molasses are used to trap and pre-cool a large number of atoms. The atoms are then transferred into a magnetic trap created by the atom chip. The atoms are magnetically transported to the chip region where the cavity is positioned. Here, evaporative cooling in a strongly confining magnetic trap is employed to achieve Bose-Einstein condensation. After an optional phase of surface evaporation, the BEC is positioned at the exact center of the cavity mode, and an intracavity dipole trap is ramped up to trap the atoms inside the cavity. The atom-cavity system can now be investigated at will. The following sections provide more details on each of the experimental steps.
The design and construction of the core of the apparatus goes back a few years and is documented in [70]. A number of changes and improvements were implemented during the work for this thesis, and most of these changes are described in [66].

Cold atoms and BEC on an atom chip

Atomchip and magnetic micro traps

The energy of an atom with angular momentum F , magnetic quantum number mF and g-factor g in a magnetic field is given by E(mF ) = g BmF B, where B is the Bohr magneton. Particles in states for which gmF > 0 experience a force towards magnetic field minima. Atoms in such low-field seeking states can therefore be trapped in magnetic field distributions with a minimum (Maxwell’s equations do not allow maxima of the magnetic field in free space [71]). Traditionally, arrangements of coils have been (and are being) used to create magnetic traps [72]. This approach, while successfully employed in many experiments, allows a limited range of geometries and control over the magnetic traps. Atom chips present some advantages in this regard.
An atom chip [73–75] uses a combination of microfabricated wires and macroscopic coils to generate magnetic field distributions well suited for trapping atoms. The use of standard microfabrication methods allows the generation of relatively complex wire configurations and therefore enables the creation of a large variety of magnetic potentials. The small structure size of the current carrying structures results in the possibility of microstructuring the magnetic field [76] and gives the experimenter high-precision control of the position of the magnetic trap, with a resolution much better than optical wavelengths.

Our atom chip

Our atom chip is constituted of a basis chip and a science chip glued together with a vacuum compatible epoxy glue. Both layers are shown in figure 2.2. The basis chip contains wire structures with a typical width of 0.2 mm – 1 mm and a thickness of 7 m.
It is glued on top of the glass cell, thereby closing the experiment chamber. PCI-connectors are use to connect the 48 contacts of the basis chip. 12 of these contacts connect wires directly on the basis chip, whereas the remaining contacts are used to connect the science chip via bonding wires. The science layer is glued onto the basis layer. It is smaller and lies entirely within the vacuum chamber. This chip contains a large number of wires that can be connected individually using any of 34 contacts. The wire structures on the science layer typically have a width of at least 50 m and a thickness of 7 m. A dielectric mirror covers the surface of the science chip. It is separated from the microwires by a layer of epoxy glue of about 10 m thickness.
A macroscopic U-shaped copper wire is placed on top of the basis substrate, outside the vacuum chamber. A current of 60 A through the copper U generates the quadrupole magnetic field for the initial MOT. The copper-U is inside a block of copper which covers the backside of the basis chip. A Peltier element on top of this copper block actively stabilizes the atom chip temperature. The heat extracted by the Peltier is removed by a second copper block which is water-cooled.

Vacuum apparatus

The vacuum apparatus consists of the experiment chamber, a Rubidium dispenser and an ion getter pump. The geometry of the setup is shown in figure 2.3. The chamber itself is formed by a cubic glass cell of dimensions 30 mmx30 mmx30 mm, onto the top of which the atom chip is glued. The opposite side of the cube has a hole of 27 mm diameter. Epoxy glue was used to connect a glass-metal transition to the cube via this hole. The feedthroughs for both the optical fibers that form the FFP-cavities as well as the wires contacting the cavity piezo elements were realized by cutting grooves into two opposing walls of the cubic glass cell. The epoxy glue used to glue the atom chip to the cell also seals these feedtrough grooves. A 25 l ion getter pump maintains the vacuum in the chamber. A titanium sublimation pump can be used to improve the pressure. As source of Rubidium, commercially available dispensers are used and contacted via electrical feedthroughs. The resulting background pressure is around 10 9 mbar.

Laser system for cold atoms

The laser system used can be conveniently grouped into two mostly independent subsys-tems. Three laser diodes are used to laser-cool the atoms, three more are required for the locking and probing of the FFP cavity. The two systems are described independently, the system for laser cooling here, the cavity-related system in chapter 2.3.2.
The level scheme of the 87Rb D2-line is shown in figure 2.4, along with the different frequencies needed for laser cooling. The main cooling laser operates red detuned to the jF = 2i ! jF 0 = 3i transition. A repump laser resonant to the jF = 1i ! jF 0 = 2i transition prevents pumping into jF = 1i. A circularly polarized pump beam resonant to the jF = 2i ! jF 0 = 2i transition is used to optically pump the atoms into jF = 2; mF = 2i.
The light with the required frequencies is generated by three diode lasers: the master, the slave and the repumper. The optical setup is shown in figure 2.5. Tunable external-cavity diode lasers in the Littrow configuration [77] are used for all but one of the laser sources. Frequency modulation Doppler-free saturation spectroscopy setups [78–80] are used to lock these lasers to well-determined detunings with respect to 87Rb transitions. The slave laser is frequency locked by light injected from the master. In the following section, we describe the complete setup used for the generation the required light fields.
The cooling light is provided by two laser sources in a master-slave setup. The master is locked on the jF = 2i ! jF 0 = 2i jF = 2i ! jF 0 = 3i cross-over resonance of an saturated absorption spectroscopy. Part of the light from the master diode injects the slave laser diode. The injection light is frequency shifted by an acousto-optical modulator (AOM) in double pass configuration running at a frequency from 60 MHz to 115 MHz. This allows to bring the frequency of the slave laser to a range of +2 to -14 with respect to the jF = 2i ! jF 0 = 3i transition. Most of the slave output is used as MOT cooling light after further frequency shifting by an AOM running at 80 MHz. It is therefore divided into four parts using polarization optics and coupled into the MOT fibers. Since we use a mirror MOT (see section 2.2.4), only four beams are required for the MOT.
The repump laser is locked on the jF = 1i ! jF 0 = 1i jF = 1i ! jF 0 = 2i cross-over resonance of a saturated absorption spectroscopy. It is frequency shifted by a single-pass AOM operating at 83 MHz, leading to a frequency resonant to the jF = 1i ! jF 0 = 2i transition. It is superposed with the 45 MOT beam before fiber coupling.
The light required for the pump beam is obtained from the master laser output. It is frequency shifted in a double-pass AOM configuration running at -67 MHz to make it resonant to the jF = 2i ! jF 0 = 2i transition.
Absorption imaging requires light resonant to the jF = 2i ! jF 0 = 3i transition. A small portion of the slave beam is separated from the MOT light for imaging. A single-pass AOM at -55 MHz shifts the frequency to the right value. Since there are two spatially separated regions of interest for absorption imaging on the chip (the MOT region and the cavity region), the imaging beam is split into two before being coupled into optical fibers.

The optical setup around the vacuum chamber

There are three groups of optical components around the vacuum chamber (more pre-cisely, around the glass cell), pertaining to three different functions: the MOT beams, two independent imaging systems, and cavity side-excitation beams aligned vertically to the cavity axis. Figure 2.6 shows a picture of the optics around the chamber.
The MOT uses the mirror MOT configuration [73, 81] that only uses four laser beams, two of which are reflected from a dielectric mirror on the chip surface. For absorption imaging, two independent systems are installed. One allows imaging in the region where the MOT is created, the other images the cavity region, where the BEC is created. The two side-excitation beams allow the pumping of atoms coupled to the cavity. One side excitation beam is aligned parallel to the chip surface, the other perpendicular to it (see [66, 70] for details).

The high-finesse cavity

Introduction

The core of the experiment is the FFP cavity. Just as conventional high-finesse cavities [16, 53, 82], it is formed by two highly reflective, multilayer dielectric mirrors. The FFP cavity however uses the end facets of optical fibers as substrate for the reflective coatings, rather than finely polished blocks of glass. Replacing relatively cumbersome mirror substrates with fibers results in a number of advantages. In particular, cavity probe light is easily accessible since the cavity output is contained in a single mode fiber. Additionally, the fiber design enables relatively easy integrability on an atom chip. The magnetic traps generated by atom chips typically are at a distance of a few tens to hundreds of micrometers from the chip surface, and the trap frequency scales inversely to the chip-trap distance. If one wishes to couple atoms in these traps to a cavity in a well controlled manner, the cavity mode axis should not be more than around 100 m away from the chip surface . Using bulk substrates with diameters on the order of many millimeters, this is only possible when placing the mirrors around the chip rather than on top of it. This however leads to a large cavity length and large cavity mode volume, strongly limiting the atom-cavity coupling strength. The fibers used here have a diameter of 125 m, allowing to mount the cavity directly onto the atom chip. Therefore, a FFP system is not subject to these constraints and allows to achieve a very small cavity length. The small cavity length also means that small radii of curvature can be used for the mirror surfaces1, reducing the mode waist and enabling even smaller mode volumes.
The fiber based construction of the cavity also makes the system scalable in the sense that many cavities can be mounted next to each other on one and the same single chip. In our setup, there are two cavities parallel to each other.
The production process of the cavities is presented in detail in reference [69], and we only give an overview here. The end facets of two fibers are shaped into a concave structure by CO2-laser machining. The CO2-laser is focused on the fiber end facet and a short pulse abruptly heats the fiber endfacet. A combination of evaporation from and material flow on the surface leads to the formation of a concave structure with very low surface roughness ( 0.2 nm). The structure shape can be made close to circular. The radius of curvature at the bottom of the structure and the structure size can be fine-tuned by adjusting pulse length, pulse energy, CO2 laser beam waist and thermal contact between the fiber and its holder. After the CO2-laser machining, the fiber end facets are coated in a commercial coating facility.
To mount the cavities on the atom chip, V-groove holders are used. Each fiber is glued into one groove of a holder itself glued onto a shear piezo. The piezos allow to adjust the cavity length by approximately 1 m over a voltage range of 400 V. No other degree of freedom is adjustable once the fibers are glued to the V-groove. The shear piezos are mounted on a ceramic bridge which is glued to the atom chip at one end. Figure 2.7 shows a picture of the two mounted cavities. The fibers are positioned such that the cavity mode center lies above a crossing of microwires on the chip. A magnetic dimple trap can therefore be generated inside the cavity mode, allowing optimal control over the position and extension of the atomic cloud. A hole in the ceramic bridge enables optical access transversally to the cavity mode along the vertical z-axis.

Cavity parameters

Both cavities in the experiment consist of a non-polarization maintaining single mode fiber used as input port and a multi mode fiber used as output port. The multi mode fiber on the output side was chosen to make alignment of the two fibers forming the cavity less critical. For the experiments described in this thesis, only one cavity (called science cavity) was used for measurements, with the second cavity supporting the cavity lock (see chapter 2.3.2).
Table 2.1 lists the most important cavity parameters for both cavites. The science cav-ity has mirrors with radii of curvature of 450 m on the single mode fiber and 150 m on the multimode fiber. Note that these radii are estimates based on profilometer measure-ments and are prone to errors. They are of importance especially in their role of defining the cavity mode volume, and therefore g0. The coupling constant g0=2 = 240 MHz given in the table however is the measured g0 (see chapter 3.5).

FFP laser system and locking scheme

The FFP probe laser

The laser used to probe the atom-cavity system needs to fulfill a few specific require-ments. A variety of experimental situations call for a flexibility of the locking scheme. The atom-probe detuning j apj may vary from zero to up to 60 GHz depending on the experimental regime. Also, the atomic reference frequency might be any transition of the D2 multiplet starting in either jF = 1i or jF = 2i, separated by 6.8 GHz. Little power is required, typically on the order of picowatts. A laser diode with antireflection coating in a setup identical to the master and repump laser sources is used. Frequency locking is achieved with a beat lock, in which a beam picked up from either the repump or master beam is used as reference. A chain of voltage controlled oscillators (VCO) and mixers allow detunings from either repump or master frequency of up to 4 GHz. After the double pass AOM, the beam is coupled into an optical fiber. The beam is coupled out of the fiber on the second optical table and frequency shifted in a single pass AOM. It then passes an optional neutral density filter before being coupled to the optical fiber connected to the main experimental table.
For experiments in which a large detuning larger than the 4 GHz enabled by the offset lock is required, the fiber after the double pass AOM can easily be replaced by a fiber integrated electro-optical modulator (EOM). This allows phase-modulation of up to 20 GHz, with a significant amount of power in the second and third sidebands at 40 GHz and 60 GHz respectively. Using a sideband instead of the carrier as cavity probe allows for experiments with a total detuning of 60 GHz.
On the main experimental table, the probe laser is superposed with the dipole trap laser (see figure 2.5) before being coupled into the FFP cavity input fiber. The polar-ization of the probe is adjusted so as to excite only of the two cavity polarization eigen modes.

Cavity locking chain

The science cavity needs to have a well defined length in order to remain resonant at the same frequency throughout the experimental cycle. An active stabilization of the cavity length is therefore required. The stabilization needs to accommodate some key requirements. It has to be fast enough to correct for vibrational as well as thermal
disturbances. A large range of possible set points is required: the detuning between cavity and atomic resonance can vary from 0 to many tens of gigahertz. The lock point should be independent of the probe laser frequency. Finally, the cavity needs to remain locked throughout the experimental sequence. Figure 2.8 conceptually shows the chain of lock systems implemented to stabilize the science cavity length. The science cavity is stabilized using a Pound-Drever-Hall (PDH) locking scheme, stabilizing the cavity on a resonance of a TEM00 mode of the 830 nm locking laser. The laser itself is frequency stabilized on a macroscopic cavity, called transfer cavity, using another PDH setup. Using a third PDH setup, we stabilize the transfer cavity length to be resonant with a 780 nm laser, here called auxiliary laser. This auxiliary laser itself is frequency stabilized to a given offset frequency to the master laser using a beat lock setup. As seen above (page 34), the master is locked relative to a 87Rb transition, thereby giving a fixed frequency anchor to the whole locking chain.
In the following, a more detailed description of this lock chain is given.
Figure 2.9 shows a schematic of the science cavity lock. The main contribution to the lock comes from the 830 nm PDH setup. This scheme is used to lock the cavity length to d=Lc/2, where Lc = n8302c =!830: (2.1)
Here, n830 is the longitudinal mode number of the dipole trap laser, and Lc the effective cavity length. For the implementation of the PDH lock, the locking light at 830 nm is phase modulated at 1.3 GHz by a fiber-integrated EOM, before being coupled to the FFP input fiber. The reflection of the locking light from the cavity is separated from the probe reflection using an interference filter and detected by a fast photodiode2. After amplification and demodulation, a PDH error signal is obtained and used as input for a PI-lock with feedback on the science cavity shear piezo.
A second contribution to the science cavity lock comes from the second fiber cavity. Since the two cavities are separated by just 500 m and mounted on the same ceramic bridge, they are subject to the same thermally induced drifts. This effect is used to enhance the science cavity lock by locking the second fiber cavity and adding the resulting correction signal to the shear piezo of the science cavity.
The lock of the second fiber cavity relies on a mechanism similar to the tilt-locking-scheme [84], where the reflection of a non-TEM00 mode provides a dispersion-like profile. This effect is used to lock the cavity without modulating either the probe frequency or cavity length. To obtain the error signal, it suffices to couple a part of the 830 nm laser to the second cavity input fiber and detect the reflection signal on a photodiode. The resulting error signal is processed in a PI-circuit and the correction signal is fed to both the shear piezo of the second cavity and the science cavity. While the science cavity could be locked by the PDH-lock alone, this additional lock provides some advantages. It leads to an enhanced stability of the science cavity lock; it gives a correction signal even when the locking light in the science cavity has to be switched off (see chapter 2.4.1); and it enables locking when a very shallow intracavity dipole trap (and therefore low locking light power for the science cavity) is required. This last point is important since the lock laser also serves as intracavity dipole trap.
The third contribution to the science cavity lock is used during times in which locking light switched off completely, i.e. before the loading of the intracavity dipole trap (see chapter 2.4.1). Due to the slightly different thermal drifts between the two cavities, the correction signal from the second cavity is not sufficient to keep the science cavity within from its lock point for more than a few milliseconds. However, the necessary additional correction to the science cavity shear piezo is identical in each run. A computer generated signal (called feedforward signal) is therefore used to compensate for this drift. With a correctly adjusted feedforward correction, the science cavity does not noticeably drift from its lock point during up to 100 ms even with its PDH-lock completely disabled.

Transfer cavity

The frequency of the locking laser !830 determines the science cavity length as given by equation 2.1. The value for !830 is chosen such that the cavity resonance frequency is at a detuning cp with respect to the TEM00 mode of the probe laser at frequency !780: Lc = n7802 c=(!780cp); (2.2)
In the special case cp = 0, the cavity is therefore doubly resonant to the locking laser and the probe laser. Equations 2.1 and 2.2 constrain the locking laser frequency to a single possible value !830. The locking laser needs to be stabilized at this frequency. This is achieved by using the transfer cavity as shown in figure 2.10. A part of the 830 nm laser is phase modulated at 17 MHz by an EOM and coupled to the transfer cavity. The reflection from the cavity is recorded on a fast photodiode. The photodiode signal is demodulated, resulting in a PDH error signal which is used as input in a PI circuit. The resulting correction signal is fed back to the piezo regulating the external cavity length of the 830 nm diode laser.
To stabilize the transfer cavity length, the auxiliary laser is used. Part of its beam is coupled to the transfer cavity, and its frequency !aux is chosen such that it is resonant to the cavity. A PDH setup analogous to the one for the 830 nm laser is employed, see figure 2.10. However, the correction signal is fed back not to the auxilary laser but to the piezo regulating the cavity length.
The auxiliary laser’s frequency is stabilized at !aux by means of a beat lock. Part of the auxiliary laser beam is superposed with a part of the master laser beam. The resulting beat frequency is converted into a direct current signal by digital electronics3 and used to stabilize the external cavity length of the auxiliary laser via a PI loop.

Table of contents :

Introduction 
1. Cavity quantum electrodynamics with single atoms 
1.1. Ideal case: Two atomic levels, one light mode
1.1.1. The Jaynes-Cummings model of the closed system
1.1.2. Master equation for an open system
1.1.3. Steady state solution to the master equation
1.1.4. Analysis of the steady state solution
1.2. Real-world atoms and cavities
1.2.1. Master equation for many levels and two light modes
1.2.2. Solution to the full master equation
1.3. Conclusion
2. Setup 
2.1. Overview
2.2. Cold atoms and BEC on an atom chip
2.2.1. Atomchip and magnetic micro traps
2.2.2. Vacuum apparatus
2.2.3. Laser system for cold atoms
2.2.4. The optical setup around the vacuum chamber
2.3. The high-finesse cavity
2.3.1. Introduction
2.3.2. FFP laser system and locking scheme
2.3.3. Transfer cavity
2.4. Experimental sequence
2.4.1. From the magneto-optical trap to a BEC in the cavity
3. Coupling single atoms to the cavity 
3.1. Introduction
3.2. Coupling of single atoms in a waveguide to the cavity
3.3. Microwave based single atom extraction
3.3.1. Heralded preparation of single atoms
3.3.2. The reservoir and single atom extraction
3.3.3. Reservoir removal
3.3.4. Zeeman-state preparation
3.3.5. Two atom probability
3.4. Characterisation of the single atom-cavity system
3.4.1. Dipole trap lifetime
3.4.2. Single atom internal state dynamics
3.5. Vacuum Rabi splitting
3.5.1. Spectroscopy of the strongly coupled system
3.5.2. Fluorescence single atom vacuum Rabi spectrum
3.6. Conclusion
4. Detection of a single atom hyperfine state 
4.1. Introduction
4.1.1. Single atom state detection methods
4.1.2. Detection error and fidelity
4.2. Accessing the quantum state from photon counts
4.2.1. Counts thresholding in two dimensions
4.2.2. Maximum likelihood method
4.3. Experimental results
4.3.1. Measurement of the detection efficiency
4.3.2. Time dependency of the detection efficiency
4.4. Conclusion
5. Measurement backaction 
5.1. Information gain and scattering in cavities and in free space
5.1.1. The minimum detection error
5.1.2. Backaction and scattering
5.2. Non-ideal cavities and measurements
5.2.1. Helstrom bound for our cavity
5.2.2. Photon counting and the Chernoff information
5.2.3. Scattering in non-ideal cavities
5.3. Experimental results: Scattering and accessible information
5.3.1. Experimental parameters
5.3.2. Measurement of Zeeman state diffusion and scattering rate
5.3.3. Detection error vs. scattering
5.4. Measurement of the Helstrom bound of the atom-cavity system
5.4.1. The quantum Zeno effect
5.4.2. Zeno effect on a single atom coupled to the cavity
5.4.3. Zeno effect for a single atom
5.4.4. The Helstrom bound
5.5. Conclusion
6. Conclusion and Outlook 
A. Appendix
A.1. Rubidium data
A.1.1. Physical properties
A.1.2. Hyperfine structure
A.2. APD correction factor
A.3. Publications
Bibliography

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