Rydberg atoms in static electric fields: Stark effect
Rydberg atoms are notoriously sensitive to static electric fields. This has pros and cons. On the positive side, this allows us to tune nearly at will the atomic transition by making use of moderate fields, in the V/cm range. In the same vein, the high sensitivity to electric field leads to the field ionization method, which offers an excellent efficiency combined with a nearly perfect state selectivity.
On the negative side, Rydberg atoms are very sensitive to stray electric fields, which must be controlled as carefully as possible in all experiments. It is not unfair to say that most of the experimental work presented in this PhD has been devoted to the control of the stray fields. In order to quantify properly the stray fields encountered in this work, we need a detailed understanding of the position of the Rydberg levels in an electric and/or magnetic field. When a Rydberg atom is placed in an external electric field F, a Stark effect term must be added to the Hamiltonian H = − 1 2r2 + V (r) + HF.
Rydberg atoms under static magnetic fields
The Zeeman effect plays a major role in these experiments, since the atoms are confined in a Ioffe-Pritchard trap, whose minimum magnetic field is different from zero, varying from 4 to 9 Gauss approximately. For such fields, we deal with an intermediate case for the Zeeman effect, in which neither the fine structure nor the Hamiltonian describing the magnetic field contribution dominate. The Zeeman shift in the LS-coupling picture is given by HZ = μB ~ (Lx + 2Sx)Bx = gJμB ~ mJBx (I.28).
The factor gj is the Landé g-factor, μB the Bohr magneton. As we have mentioned before, the hyperfine structure can be neglected. For this intermediate case, two basis are equally appropriate, either the |J,mJ i one or that of the uncoupled states |Li, |mSi. Nevertheless, the second makes it more difficult to express the matrix elements of the Hamiltonian of the system without Zeeman or Stark effect. The Landé gJ factor for Rb87 can be obtained from the total quantum angular momentum, orbital angular momentum and spin momentum J, L and S, respectively, as: gJ = 1+ J(J + 1) + S(S + 1) − L(L + 1) 2J(J + 1) .
Theoretical description of the dipole-dipole interactions
For a first theoretical insight into the dipole-dipole interaction, we examine here the simple case of two atoms separated by a distance R = |~R | (scheme show in figure I.7) excited to the Rydberg level |e1i, |e2i = |e1i ⌦ |e2i. Precise numerical calculations will be done for the evolution of the pair state |60S, 60Si.
We assume that the distance between the two atoms, R, is much larger than the distance between each core and its electron. Hence higher multipole terms can be neglected. The interaction Hamiltonian is then: H = H1 + H2 + Vdd(r).
Simulations of dipole blockade regime in a small BEC
In order to guide the design of these experiments, we have performed numerical simulations of the dipole blockade mechanism. We used the context of a small Bose Einstein Condensate on the chip, since it provides the highest confinement and density, favorable conditions for the excitation of a single Rydberg state.
We simulate excitation of a small BEC, containing only 300 atoms, with a Thomas- Fermi profile and rx, ry, rz = (2.34, 1.6, 1.08) μm (radius at e−1). The trap frequencies are (!x,!y,!z) = 2⇡(37(1), 107(1), 121(1)) Hz corresponding, in the experimental section of this work, to a Ioffe Pritchard trap at 455 μm from the chip surface. The critical BEC temperature TC in this trap is given by  kbTC ⇡ 0.94~¯!N1/3 (I.54) where ¯! = (!x!y!z)1/3 and kB is again the Boltzmann constant. For our trap TC = 23.6 nK. Note that the diameter along the z direction of the BEC is below 1.6 μm which is the distance of the first anticrossing. We are there in a situation of complete blockade.
At zero temperature, all atoms are in the condensed phase. At a more realistic finite temperature, T = 14 nK, only 80% of the atoms are in the condensed phase. The BEC is thus surrounded by a thermal cloud with dimensions (radius at e−1) given by rx, ry, rz = (5.37, 1.83, 1.64) μm. We will consider in the following simulations both the pure BEC case and the more realistic finite temperature one.
D2 transition line of 87Rb.
The Rubidium 87 transitions involved in the cooling and trapping are shown in figure II.7 . We work with the hyperfine transition |F = 2i ! |F0 = 3i of the D2 line for all cooling, trapping, imaging and optical pumping process.
The trapping and cooling in the 2D-MOT and 3D-MOT stages are performed with a laser detuned by −20 MHz with respect to the transition |F = 2i ! |F0 = 3i. It is called the cooling beam in figure II.7. Later in the sequence, in one of the mirror-MOT stages, this detuning is changed in order to increase the density of the atomic cloud in phase space. A probe bean is set in resonance with the transition mentioned above for observing the atoms. For the optical pumping, which pumps the atoms in the level |F = 2,mF = 2i of the ground state just after the optical molasses stage and before turning on the magnetic trap, a laser beam is needed at resonance with the |F = 2i ! |F0 = 2i transition. It is referred to as the Zeeman pumper in the figure.
In the cooling and trapping process, some atoms can decay in the F = 1 state, impervious to the previously mentioned laser beams. Hence, a repumper beam in needed, tuned to the |F = 1i ! |F0 = 2i transition as can be seen in figureII.7.
Table of contents :
Introduction in English
I Experiment: Towards deterministic preparation of single Rydberg atoms
I Rydbergatomsanddipoleblockade: theoryandsimulations
I.1 Rydberg atoms
I.1.2 Rydberg atoms in static electric fields: Stark effect
I.1.3 Rydberg atoms under static magnetic fields
I.2 Dipole Blockade
I.2.1 Principle of the blockade effect
I.2.2 Theoretical description of the dipole-dipole interactions
I.2.3 Specific case of the target state: 60S − 60S
I.2.4 Dipole blockade effect
I.3 Simulations of dipole blockade regime in a small BEC
II Experimental setup
II.1 Cryogenic environment
II.2 A superconducting atom chip
II.3 Laser and imaging system
II.3.1 D2 transition line of 87Rb
II.3.2 Imaging the atoms
II.3.2.a Determination of the atom number
II.3.2.b Temperature measurement
II.4 From the 2D-MOT to the BEC: sequence for the optical cooling and trapping.
II.4.1 The source of slow atoms: 2D-MOT
II.4.2 The mirror MOT
II.4.3 The U-MOT
II.4.4 Optical molasses and optical pumping
II.4.5 Transfer into the magnetic trap
II.4.6 Getting a BEC: evaporative cooling
II.4.7 Decompressing and moving the magnetic cloud
II.5 Conclusion of the chapter
III First electric field studies
III.1 Laser stabilization system
III.2 Detection setup
III.3 First atomic spectrum
III.4 Fresh chip and deposit of Rubidium via MOTs
III.5 Macroscopic Rubidium deposit
III.6 Conclusion of the chapter
IV Long coherence time measurements for Rydberg atoms on an atom-chip
IV.1 Experimental conditions of microwave spectroscopy
IV.2 Characterization of residual electric field by microwave spectroscopy
IV.2.1 Electric field perpendicular to the chip surface
IV.2.2 Field parallel to the chip surface
IV.2.3 Electric field gradients
IV.3 Probing coherence times with Ramsey spectroscopy and spin echo sequences.
IV.3.1 Spectra and Rabi oscillations
IV.3.2 Ramsey spectroscopy
IV.3.3 Spin echo experiment
Conclusions and perspectives: Part I
II Theory: Applications to quantum information processing
V Atoms and photons. Theoretical description of the interaction
V.1 Quantum description of the electromagnetic field
V.1.1 Quantization of the electromagnetic field
V.1.1.a Coherent states and displacement operator
V.1.2 Quantum states and density operator
V.1.2.a Pure states
V.1.2.b Mixed states
V.1.2.c Quantum state of compound systems and degree of entanglement
V.1.3 Phase space representation
V.1.3.a Characteristic functions
V.1.3.b Wigner function
V.1.3.c Examples of the Wigner function
V.2 Two-level atoms
V.2.1 Atomic spin and Bloch sphere
V.2.2 Manipulation of atomic states
V.3 Light-matter interaction: quantum theory
V.3.1 Jaynes & Cummings Model
V.3.1.a Resonant quantum Rabi Oscillation
V.3.1.b Resonant MFSS generation
V.3.1.c Dispersive MFSS generation
V.4 Decoherence process
V.4.1 Master equation
VI Fast generation of mesoscopic field states superpositions in CQED
VI.1 Dicke model
VI.1.1 Factorization approximation
VI.2 Two atoms interacting with a cavity field without dissipation. Exact calculation.
VI.2.1 MFSS size and fidelity with respect to an ideal cat
VI.2.2 Case of an Initial atomic state |1i
VI.3 Numerical simulation: two atoms-field interaction including field dissipation.
VI.4 Case of more than two atoms
Conclusions and perspectives: Part II
A Broadening sources
B Calibration of perpendicular electric field
C Measurement of Electric field gradients