It's really amazing to think of the incredible group of people who went through Jeff's group. Therefore, considerable effort has been devoted to understanding the utility of microspheres for cavity QED with strong coupling.
Background
This trap represents an improvement of a factor of 102 over the first realization of trapping in cavity QED [2], and of approximately 104 over previous results for atomic trapping [5] and localization [6] by cavity field QED. himself. Therefore, considerable effort has been devoted to understanding the utility of microspheres for strong-coupling cavity QED.
A History of My Involvement in the Kimble GroupGroup
My work focused on understanding the boundaries of microspheres for achieving cavity QED in the strong coupling regime. After this, I moved to Lab 1 to work with Theresa Lynn and Kevin Birnbaum on a QED experiment with holes.
Electronics Projects
The circuit has a buffered output, so that the photon counting module can be connected to the tachometer box as well as a counting card in the computer. While the data stored by the computer provides a complete record of the pulse arrival time, the tachometer output provides a real-time signal that can be used for control of the experiment.
Organization of the Thesis
It turns out that our system is sensitive to the thermally excited motion of the cavity mirrors. Many situations including most aspects of the photoelectric effect can be understood using the semiclassical theory.
Hamiltonian
A two-state description of an atom associated with a single mode of the electromagnetic field is valid if the two atomic states are resonant or nearly resonant with the driving field and all other fields are strongly out of tune. In the semiclassical treatment, the atom is treated as a quantum two-state system and the field is treated classically.
Dipole Approximation and Radiation Gauge
Rabi Oscillations
Definition of the Rabi frequency. we can now express the interaction Hamiltonian in terms of the Rabi frequency Hint =−~ΩR eiφ|gihe|+e−iφ|eihg|. 2.24). This example illustrates the usefulness of the interaction picture in solving the time evolution of a system.
Hamiltonian
Now the interaction term is found by noting that e− →r =X. where σij is the atomic transition operator |iihj| is, and ehi|−→r|ji is the electric dipole transition matrix element. If we place the atom at the origin, the electric field operator, −→. εn is the unit polarization vector, 0 is the permittivity of free space and Vn is the electromagnetic mode volume. the Hamiltonian for the system can now be expressed as H = X.
The Jaynes-Cummings Ladder
This scheme is for the case of zero cavity detonation, ωc =ωa, where ωc is the cavity resonance and ωa is the atomic resonance. Therefore, for the case of mode non-penetration in cavity mirrors, the mode volume.
Saturation Photon Number
Critical Atom Number
This plot assumes no cavity mode penetration into the mirrors. These plots assume no cavity mode penetration into the mirrors.
Cavity Transmission and Losses
The mirrors are glued to aluminum v-blocks, which are mounted on shear-mode piezoelectric transducers. These are then glued to a solid copper base which is mounted on a vibration isolation array inside the chamber.
Modes of a Fabry-Perot Cavity
Here we have assumed that the cavity has hard edges at x= (0, Leff) and the mode is a pure standing sine wave. If we neglect the penetration of the mode into the mirror substrates, L = Lef f and the mode volume is given by.
Cavity Length Stabilization
- Gas Flow in a Vacuum System
- Analysis of Gas Flow in Our Vacuum System
- Relation of Pressure to Kinetic Properties of Particles
- Mean-Free Path and Collision Times
- Collisional Lifetime for Background in Our System
Product is defined as the product of pressure and pumping speed at a given cross section of the system. If the pressure is monitored at the pump, the pressure at the other end of the connecting pipe may be very different.
Magneto-Optical Traps
Optical Molasses
The atom's velocity leads to a Doppler shift, so the atom sees a laser frequency of ωL 1−vc. The scattering probability is related to the detuning, so maximum scattering occurs when the Doppler-shifted frequency matches the atom's resonance frequency.
Magneto-Optical Trap
These forces are velocity dependent due to the Doppler shifts due to the atomic motion. As the atom moves to the right of the origin, the magnetic field it sees increases, causing the state mF = +1(−1) to go up (down) in energy.
Sub-Doppler Cooling in a MOT
As the atom moves away from the origin, the magnetic field causes Zeeman splitting, which affects the scattering rates. In addition, we would like to create a trapping potential that is independent of the internal state of the atom.
Classical Dipole Force
Furthermore, we will see that multiple levels of a real atom can be exploited to create a trap that is insensitive to the state of the atom. Fortunately, as we will see in the next section, the scattering velocity can be arbitrarily small.
Two-Level Atom
In large dislocations, heating due to absorption corresponds to an increase in thermal energy in the direction of light propagation equal to the return energy, Erec, per scattering event. Spontaneous emission also results in an increase in energy equal to the return energy per scattering event, however, this occurs in a random direction.
Multi-Level Alkali Atoms
State Insensitive Trap
The AC Stark shift for the state|eid to this field has the opposite sign to that of the ground state,-Uge. In this situation, the AC Stark shift of the ground and excited states will be equal.
Trap Vibrational Frequencies
To check the timescales for the off-resonant scattering rates, we first empty the F = 4 population and monitor the reequilibration of the population between F = (3,4).
Heating Due to FORT Intensity Fluctuations
The red trace is for the case of no repump light and an initial depletion of the F=4 state population. This suggests new possibilities for sensing and controlling the quantum dynamics of an individual system.
The Experimental Setup
Delivering Atoms to the Cavity Mode Volume
Similarly, the cloud of atoms from the upper MOT must be able to fall through the differential pump hole to be efficiently transferred to the lower MOT. We are able to control the density of atoms delivered to the cavity state volume.
Atom Transits
The red trace on the left (A) is a typical transition for atoms dropped from the upper MOT at a height of ~25 cm. The blue trace on the right is a typical transition for atoms dropped from the lower MOT at a height of ~5 mm.
However, it was easier to implement a FORT wavelength of 906 nm as an intermediate step to check our lifetime improvement ideas. The next step was to move to a capture wavelength of 935.6 nm as described in the next section.
An example of the resulting probe transmission is shown in Figure 8.7, which shows the continuous observation of a single captured atom. Shown is the strength of the intracavitary field ¯m =|haˆi|2 derived from the heterodyne current as a function of time t, with an RF detection bandwidth of 1 kHz and ∆0C = 0 = ∆0p.
Modes of a Microsphere
Semi-log plot of the mode volume as a function of cutoff parameter (rQ−a) for the optimal sphere size discussed in Section 10.2. Therefore, the field would vanish everywhere outside the sphere and just outside the surface of the resonator.
Losses in Dielectric Spheres
Intrinsic Radiative Losses
The contribution to the quality factor for purely radiative effects, Qrad, can be derived following the arguments presented in Ref. However, for a sphere of 7 µm radius, Qrad ≈ 4×108 and radiative losses can play a decisive role in the characteristics of spheres that are optimal for use in cavity QED.
Material Loss Mechanisms
Because fused silica has a minimum absorption at 1550 nm, there is a maximum for the quality factor due to the large absorption of Qbulk. The graph contains the quality factor due to purely radiative losses (Qrad), the three loss mechanisms that include Qmat: (Qbulk, Qs.s., Qw) and the predicted Q for all four loss mechanisms.
The Strong Coupling Regime
In the strong coupling regime, important parameters for characterizing the atom-cavity system are the two dimensionless parameters: the saturation photon number, n0, and the critical atomic number, N0. Ideally, one hopes to minimize both the critical atomic number, N0, and the saturation photon number, n0, which corresponds to simultaneous maxima for both κγg2.
Strong Coupling with Cesium
Figures 10.1 and 10.2 are graphs of this dimensionless parameter β and of √1β as functions of the dimensionless size parameter ˜x = 2πnaλ. However, we have seen that Q is not the only relevant factor in determining the suitability of the WGMs for QED in cavities in a strong coupling regime.
Comparing Microspheres and Fabry-Perot CavitiesCavities
Encouragingly, the currently achievable results for the production of small spheres would already allow WGMs to compete favorably with the current state of the art in Fabry-Perot cavity QED. H is the current state of the art in 10 µm microspheres based on the results presented in Section 10.2.
Effect of Displacement on Cavity Output
Fabry-Perot Finesse
The measured cavity output power can be approximated as a Lorentzian for frequencies close enough to resonance and for modes that are sufficiently separated.
Important Assumptions
If we are interested in a frequency greater than the inverse cavity accumulation time, the effects will be attenuated at that frequency.
Simple Harmonic Oscillator in Thermodynamic Equilibrium
Quantum Harmonic Oscillator
Effective Mass Coefficients
We can then use the equalization theorem to calculate the root mean square motion of a thermally excited mirror mode. As a first approximation, most of the motion energy occurs within a bandwidth given by the quality factor of the acoustic mode.
Spectral Density Function for Displacement
Velocity Damping
Each mode will have energy kBT, where kB is the Boltzman constant and T is the temperature. This is consistent with the uniformity theorem, as the mode in thermodynamic equilibrium should have energy kBT, and the harmonic oscillator mode energy would be given by mω02∆x2.
Structural Damping
In the case of velocity damping, the spectral density function for displacement is the same. The root mean square shift is then found by integrating the spectral density function and then taking the square root. The spectral density function in the case of structural damping can be found from the velocity damping function by assigning the frequency dependence βn= ϕnmωω 2n to the friction coefficient for each mode.
Measuring Brownian Motion in Fabry-Perot CavitiesCavities
- Shot Noise in Photodetectors
- Measuring Displacements Due to Brownian Motion
- Sensitivity of the Mechanical System
- Longitudinal Modes of a Bar
- Corrections to Cylinder Modes
- Finite Element Analysis for a Cylinder
The conversion factor is the ratio between the optical power at the input of the detector and the voltage generated at the input impedance of the spectrum analyzer. We define the sensitivity of the system as the shift corresponding to the optical power representing the lightning noise level.
Normal Modes of Our Mirrors
From Figure 12.10 we see that we can qualitatively model the density of states and its translation into thermal noise. A comparison is made with the results calculated in Section 12.2 using finite element analysis (shown in red).
Quantum Shannon Noisy Coding Theorem
Furthermore, if E is fixed and only the maximization over p(x) is performed in equation (14.3), then the resulting expression will define the capacitance that can be achieved with the given measurement.
Limiting Cases for C n
E to denote the generalized quantum measure or positive operator-valued measure (POVM) [113] on the message Hilbert spaceHn, i.e. E = (Ek) is an infinite sequence of operators on Hn with only finite number Ek 6= 0 such that hψ|Ek|ψi ≥ 0 for all k and |ψi, and the Ek's form the decomposition of the identity operator on Hn. Furthermore, if E is fixed and only the maximization over p(x) is performed in equation (14.3), the resulting expression will define the performance that can be achieved with a given measurement. 14.7) For all intermediate cases, there is nothing better than an explicit search over all probabilities p(x) and all measurements of E.
R 2 : Rate for Two-Shot Collective Measure- mentsments
14.7) For all cases in between, there is nothing better to do than an explicit search over all probabilities p(x) and all measurements E.
