Doctoral Thesis
Dissipative and non-dissipative
many-body physics with cold Rydberg
atoms
Author:
Cristiano Simonelli
Supervisor: Dr. Oliver Morsch
Thesis submitted for the degree of Ph.D.
but that’s not why we do it.” R. P. Feynman
Penso che i ”Ringraziamenti” siano una parte complessa e doverosa di questa Tesi, sia perch´e ultimo atto dopo la molta fatica accumulata in questi mesi di scrittura, sia perch´e, durante il percorso di dottorato, sono molte le persone a cui devo la mia gratitudine. Un ringraziamento speciale va ovviamente al mio relatore Dr. Oliver Morsch, che in questi anni mi ha guidato con infinita pazienza e comprensione nell’affascinante campo della Fisica multi-corpo. Vorrei estendere i miei ringraziamenti anche al resto del Gruppo. Professori, colleghi, tecnici e personale universitario che hanno saputo rendere il posto di lavoro un luogo produttivo dove stare volentieri fino a tarda sera e tornare col sorriso il mattino seguente. Un ultimo ringraziamento va alla mia famiglia e ai miei amici che mi hanno sostenuto e supportato nelle varie difficolt`a incontrate durante questo lungo percorso.
Acknowledgements ii
Contents iii
List of Figures vi
Abbreviations xxi
Physical Constants xxiii
Symbols xxiv
1 Introduction 1
1.1 The many-body problem . . . 2
1.2 The cold atoms opportunity . . . 4
1.3 A general introduction to Rydberg atoms . . . 6
1.3.1 The radiative lifetime . . . 11
1.3.2 The interaction term . . . 13
1.4 Conclusions . . . 17
2 Experimental Setup 18 2.1 Introduction: General overview of the apparatus . . . 18
2.1.1 Laser sources . . . 18
2.1.2 Magneto Optical trap and vacuum cell . . . 22
2.1.3 Excitation Scheme . . . 24
2.2 Experimental calibrations . . . 25
2.2.1 Calibration of the 420 nm beam power . . . 25
2.2.2 Calibration of the 1013 nm beam power . . . 26
2.2.3 Moving the atoms cloud . . . 27
2.2.4 Estimate of the MOT Temperature . . . 30
2.2.5 Ionization and detection . . . 32
3 Non-Dissipative Regime 36 3.1 Introduction . . . 36
3.2 Two level System . . . 37
3.2.1 Towards the incoherent regime . . . 40
3.2.2 Incoherent Regime . . . 42
3.3 Many-body effect on the dynamics: kinetic constraints . . . 43
3.3.1 On-Resonant excitation: the Blockade constraint . . . 44
3.3.2 Resonant experimental results . . . 45
3.4 Off-Resonant excitation: the facilitation constraint . . . 46
3.4.1 Off-Resonant experimental results . . . 48
3.4.2 Geometrical model of the dynamics . . . 49
3.5 Controlled avalanche process: seed technique and bimodal model . . . 52
3.5.1 Experimental results . . . 53
3.6 Conclusions . . . 58
4 Deexcitation 60 4.0.1 Introduction . . . 60
4.0.2 Implementation of the technique . . . 61
4.0.3 Experimental results for resonant excitation in the weakly inter-acting regime . . . 62
4.0.4 Experimental results on resonance from weakly to strongly inter-acting regime . . . 64
4.0.5 Experimental results off-resonance . . . 65
4.0.6 Numerical simulations on resonance . . . 68
4.0.7 Conclusions and Outlook . . . 69
5 Dissipative Regime 71 5.1 Out of equilibrium system . . . 71
5.1.1 Introduction and definition . . . 71
5.1.2 Motivation of studying Out of equilibrium systems . . . 72
5.1.3 Experimental implementation using Rydberg atoms . . . 73
5.2 External degrees of freedom . . . 74
5.2.1 Introduction: spatial motion of the excitations . . . 74
5.2.2 van der Waals interactions . . . 75
5.2.3 The calibration: space-time . . . 75
5.2.4 Rydberg excitations time of flight measurement . . . 80
5.2.5 Spatial measurement . . . 80
5.2.6 Conclusions . . . 83
5.3 Internal degrees of freedom . . . 84
5.3.1 Introduction . . . 84
5.3.2 Rydberg states migration . . . 85
5.3.3 State selective detection . . . 88
5.3.4 The analytical model . . . 88
5.3.5 The experimental implementation . . . 90
5.3.6 Conclusions and Outlook . . . 94
6 Absorbing state phase transition 96 6.1 Introduction . . . 96
6.2 Directed Percolation . . . 97
6.3 Directed percolation models . . . 98
6.4 Implementation using Rydberg atoms . . . 100
6.4.2 The motion of the atoms . . . 103
6.4.3 The perturbation . . . 104
6.5 Experimental results . . . 105
6.5.1 High number of seed excitations . . . 106
6.5.2 Low number of seed excitations . . . 108
6.5.3 Detuning dependence . . . 110
6.5.4 Conclusions . . . 112
7 Conclusions 113
1.1 Graphic representation of the system and the environment. The system (light blue circle) is coupled to the environment (green blob). If the coupling is weak and the environment relaxes quickly to equilibrium (so that the Born approximation and Markov approximation can be ap-plied) the evolution of the system is described by the master equation in Lindblad form [30]. If the system is made up of a large number of interacting components (blue circles) its evolution is a many body problem. 2 1.2 Cold atoms system. a) At high temperature T each atom of the
ensem-ble moves isotropically with a finite thermal velocity vth(T ). b) As the temperature decreases, the motion of the atoms is also reduced and the thermal de Broglie wavelength λthincreases. c) When the wavelength λth is higher than the relative distances between two atoms of the ensemble, their wavefunctions overlap and form a single macroscopic quantum object. 5 1.3 Rydberg atoms of a) H and b) Rb. The electron e orbits around
the nuclear charge. In the H atom the electron orbits around the point charge of the proton. In the Rb atom, the electron orbits around the nuclear charge of Z = 37 protons and the Z − 1 inner electrons. For high l states, the electron orbits away from the nucleus and the Rb atom behaves identically to the H. For low l states, the electron penetrates and polarizes the distribution of the inner electron charge. . . 7 1.4 Electric dipole and dipole-dipole interaction. a) A displacement of
the center of mass of the electron and nuclear charge distributions gen-erates a dipole moment ~p. The interaction between two Rydberg atoms can be described as a classical interaction between electric dipoles. . . 13 1.5 Graphic representation of the two interaction regimes for an
ensemble of atoms. a) In the resonant dipole-dipole regime the con-tribution of all the surrounding atoms to the interaction energy is not negligible and the system has to be treated as a many-body system. b) In the van der Waals regime the interaction is dominated by the pair in-teractions with the nearest neighbour and the system can be treated as an ensemble of interacting pairs. . . 16
2.1 Schematic representation of the experimental setup in Pisa. The apparatus is divided into two optical tables: table a) and table b). All the laser sources, in master-slave configuration, required to cool and trap the atoms, as well as the Fabry-Perot cavity, lie on the table a). The radiation is transferred via monomode optical fibers to table b) where the actual experiment takes place. The excitation laser sources, in master-slave configuration, as well as the vacuum cell where the atoms are cooled and confined, are placed on table b). The temperature of the environment is maintained constant at 20 degrees. . . 19 2.2 Possible geometries in our experimental setup. The overlap of the
excitation beams, 420 and 1013 nm, and the atom cloud determine the interaction volume. In our experiments we can choose between two dif-ferent optical path branches of the 420 nm laser beam: one co-propagates with the 1013 nm in free-space, the other is injected into a monomode optical fibre whose output is focused on the atom cloud forming an an-gle θ ' 10◦ with the 1013 nm beam. The waist of the 1013 nm beam is around 170 µm at the atom position, whereas for the 420 nm beam in the two cases it is around 40 µm or 7 µm, respectively. Two characteristic length scales are defined, rb ' 12 µm and rf ac= 5 − 7 µm, thanks to the interaction between Rydberg atoms, so that the geometry realized is 3D for the co-propagating beam and quasi-1D for the focused beam. . . 20 2.3 Excitation frequency scan.a) Frequency scan of the 6P3/2intermediate
state hyperfine structure (F0 = 3, F0 = 2 and F0 = 1) through two 420 nm photons. The radiation sent in the FP cavity and analyzed here is the master 840 nm. Triple-gaussian Fit results: gaussians centred at x1 = 0.10319 ± 4.59 × 10−5GHz, x2 = 0.16768 ± 7.3 × 10−5GHz and x3 = 0.24326 ± 8.27 × 10−5GHz. The corresponding widths are σ1 = 12.056 ± 5.41 × 10−2MHz, σ2 = 11.442 ± 8.77 × 10−2MHz and σ3 = 7.691 ± 8.42 × 10−2MHz. b) Frequency scan of the 70S1/2 Rydberg state through two-photons excitation. Excitation performed with a low number of excitations, i.e., in the non-interacting regime. The gaussian width is 1.70 ± 0.697 MHz. . . 22 2.4 Level scheme of the magneto-optical trap. The cooling (|5S1/2, F =
2i−|5P3/2, F0= 3i) and repumping transitions (|5S1/2, F = 1i−|5P3/2, F0= 2i) are evidenced in red while the spontaneous decay in green. . . 23
2.5 Level scheme for the excitation to the Rydberg state. The
atoms in the ground state |5S1/2, F = 2i are excited to the Rydberg state |70S1/2i via a two-photon coupling. Blue and red arrows represent the single transitions and Ω420 and Ω1013 are the corresponding Rabi frequencies. The population of the intermediate state |6P3/2i is reduced by applying a detuning ∆6P > Ω420. . . 24
2.6 Timing of one typical experimental run. The MOT beams are switched off during the excitation to avoid ionization from the |5P3/2, F0= 3i state. The pulses are labeled with letters A-H to simplify the compre-hension of the plot. The pulses are defined as follows: the excitation time tex = AB, the dark time tdark = BC, the deexcitation time (if present in the experiment) tdeex = CD and the (simplified) ionization time tion = EF . The detection tdet occurs typically after around 10 µs of ions time of flight. The experimental sequence is over and the MOT beams are switched on again. This experimental sequence is repeated hundreds of times at a rate of 4 times per second. . . 25 2.7 Calibration of the 420 nm beam. The power of the beam is measured
with a photodiode, which gives a reference voltage signal to our oscillo-scope, as a function of the control parameter of our Labview program. The voltage signal is converted to mW measuring the maximum power (at 100%) with a power meter. The typical error bar shown in the plot reflects the uncertainty of the power measurement and the voltage signal (±0.5 mW). . . 26 2.8 Calibration of the 1013 nm beam power. The power of the beam
is measured with a photodiode which gives a reference voltage signal to our oscilloscope. The voltage signal is converted to mW measuring the maximum power with a power meter. a) The power of the beam as a function of the RF input in the AOM. b) The RF amplitude value as a function of the RF set in our waveform generator (Agilent 33250A) to maintain a constant power: 30 ± 2 mW red circles and 13 ± 2 mW green circles. . . 27 2.9 Magnetic fields in the MOT. a) Square big coils around the vacuum
cell generate the bias magnetic field Bbs (light blue arrows). The atomic cloud (red spot) experiences both the quadrupole magnetic field (dark blue arrows) and Bbs. b) The zero point of the total magnetic field, i.e. the atom cloud, is shifted by adding the bias magnetic field. . . 28 2.10 Example of coil current calibration along the a) x and b)
z-axis. The position of the atom cloud, corresponding to the peak of the density profile (black symbols), is detected by the fluorescence signal in the CCD camera with a resolution of 1 µm. The strength of the gradient is a) ∇n = 28.9 G/cm and b) ∇n = 57, 8 G/cm. Changing the current of the big coils around the vacuum chamber, the magnetic field can be tuned and the position of the atom cloud can be changed. The current is measured using a digital multimeter with a resolution of 0.1 mA. . . 29 2.11 Estimating the temperature of the atoms in the MOT. A focused
420 nm beam irradiates the atom cloud in a 1D geometry. The excitation laser pulse tex = 1 µs is applied after a variable dark time tdark during which the cloud expands. The fit function NInt, defined in the main text, has the parameters σy = 190 µm, σz = 140 µm and σblue= 7 µm fixed to the measured value. It returns a thermal velocity vth = 0.105 − 0.112 ± 0.003 µm/µs for η = 1 − 10 respectively. This estimate corresponds to a temperature of T ≈ 110 − 130 µK in good agreement with the expected value reported in [119]. . . 31
2.12 Classical potential along the z-direction. The dashed blue line represents the ∼ 1/r potential V (r) in the absence of the electric field E (red dashed line). The solid black line is the classical potential along the z-direction experienced by an electron at ~r position. One saddle point is identified at zsd < 0 where the potential reaches the ionization value. . . 33 2.13 Graphic representation of the detection system. a) Two couples
of electric plates (front and lateral) around the vacuum cell generates the electric field needed to ionize and to accelerate the ions. A grid at −1.15 kV bends the trajectories of the ions toward our detector (CEM) KKBL510. Each ion generates a voltage signal (detected by a Lecroy-waverunner 104 Mxi oscilloscope) proportional to the detector internal gain which is determined both by the model in use and the applied voltage (set, in our case, at −2.5 kV). b) Typical voltage signal generated by the detector. Our Labview program identifies the peaks above a tuneable threshold value, the number of peaks as well as the position in time, is recorded for each experimental run. Once corrected by the detection efficiency, from this analysis we get the number of ions in the cloud and the arrival time at the detector. The repetition frequency of the experiments is set to 4 Hz. This allows us to repeat the experiment hundreds of times (typically 100 − 200) and get the full distribution of the ion number [85] and the arrival times [39]. . . 34 2.14 Measurement of the ionization threshold. Number of detected ions
(blue circles) as a function of the applied potential difference between the front and the lateral plates. The ionization timing has been adjusted for each value of the potential difference to maximise the detection efficiency. An ion signal generated by the absorption of two 420 nm photons has been taken as a reference. A sigmoid Fit function (solid line) guide the eyes. . 35 3.1 Excitation scheme and two-level approximation. (Left) Two laser
beams, 420 and 1013 nm, drive the atoms from the ground state | ↓i to the excited state | ↑i. The population of the intermediate state |6P3/2i is suppressed by detuning the 420 nm coupling out of resonance. In the regime ∆420 Ω420, the intermediate state can be adiabatically elimi-nated and the three-level excitation scheme is approximated as a two-level system with the two-photon coupling strength Ω. . . 38
3.2 Dependence of incoherent dynamics on the Rabi frequency for
resonant excitation. The inset shows the mean number of excitations in a sample of 7×105atoms in a MOT of width 60 µm in the 3D configuration as a function of time for three different Rabi frequencies: Ω/2π = 81 (open diamonds), 43 (green squares), and 20 kHz (red circles). When multiplying the excitation times by Ω2/γ where γ/2π = 0.7 MHz), the three curves collapse onto each other (main figure), demonstrating the expected Ω2 scaling in the incoherent excitation regime. Representative error bars are one standard deviation of the mean (s.d.m) . . . 42
3.3 Analytical behavior of the excitation rate Γ as expressed in Eq. 3.14 for 1D geometry (n0 = 2). The excitation rate is shown for different values of the detuning ∆ from resonance as function of the average distance : in the resonant case ∆ = 0 (Blue solid line) the ex-citation rate has the maximum value Ω2/2γ for average distances larger than rb but is strongly suppressed for an average distance smaller than rb. For the off-resonant excitation case, ∆/2π = 5 MHz (red solid line), ∆/2π = 10 MHz (green solid line) and ∆/2π = 15 MHz (black solid line), the excitation rate reaches the same maximum value when the average distance is equal to rf ac(∆) and is suppressed by a factor 1/∆2 otherwise. 44 3.4 Evidence for the blockade constraint in a gas of Rydberg atoms
in 1D configuration. (a) Mean number of excitations and (b) excitation fraction as a function of time for different atom numbers Ng inside the interaction volume: 5600 (open diamonds), 715 (green squares), and 180 (red circles). These correspond to values of the parameter rb/a of 4.2, 2.1, and 1.3, respectively. The numerical simulations (solid lines) have been scaled vertically by a factor of 1.8. (c) Normalized growth rate as a function of the mean distance d between excited atoms. The solid line is obtained from the analytical result of Γ as expressed above in Eq. 3.15. (d) Mandel Q parameter as a function of mean excitation number. The deviations of the experimental data from the simulation are most likely due to technical noise and fluctuations in the atom number, laser intensity and detuning (positive Q for small values of N), and saturation effects of the detection system (values of Q < −1 for large N). The vertical axis on the left corresponds to experimental data, and that on the right to the numerical simulation (solid lines). Representative error bars are one s.d.m. 47 3.5 Evidence for the facilitation constraint in a gas of Rydberg
atoms in 3D configuration. Rubidium atoms collected in a 160 × 45 × 80 µm MOT excited with Ω/2π = 80 ± 15 kHz. (a) Mean number of exci-tations as a function of time for ∆/2π = +19 MHz (blue circles) for which rf ac = 6.4 ± 0.1 µm and δrf ac = 40 ± 1 nm, ∆/2π = −19 MHz (red dia-monds), and ∆/2π = 0 MHz (gray squares). The numerical simulations (solid lines) have been scaled vertically by a factor 0.56. Representative error bars are one s.d.m. (b) The Mandel Q parameter as a function of time. The numerical simulations (solid lines) have been scaled by a factor 2. (c) Left axis, focus on the acceleration of the mean number of excitations as a function of time, ∆/2π = +19 MHz (filled blue circles). Representative error bars are one s.d.m.. The dashed line is a polyno-mial Fit to guide the eye. On the right axis, the enhancement of the Q parameter due to the facilitation constraint. . . 49
3.6 Sketch of the analytical model in the 1D configuration. The
excitations (red circles) are placed along the propagation axis of the laser beam which defines the interaction volume Vint(shaded region). a) In the case of negative detuning, the blockaded volumes (gray circles) enclose the excitations and fill the interaction volume. The overlap degree between the blockaded volumes is taken into account in the model by the parameter α. b) For positive detuning, the facilitation occurs within the facilitation volume (black dashed lines). . . 50
3.7 Geometrical model numerical results. Solid lines represent the num-ber of excitations N as a fucntion of time for different values of the de-tuning: ∆ = 0 (gray line), ∆ < 0 (red line) and ∆ > 0 (blue line). The dashed straight line shows a linear growth of N and highlights the acceleration of the facilitation constraint for ∆ > 0. Parameters used: Ω/2π = 70 kHz, γ/2π = 700 kHz, ∆/2π = 13 MHz and k = 1/150 µs−1. In this simple model the density is constant ρ = 6 × 1011cm−3 and the interaction volume is defined as Vint = πσ2blu× 2σx, where σblu = 10 µm is the waist of the beam and σx = 150 µm the size of the MOT. . . 52 3.8 Control the avalanche excitation process. (a) Schematic
represen-tation of the facilirepresen-tation constraint and the seeded avalanche dynamics. Initially, the off-resonant creation of single Rydberg excitations is unlikely and the total number of excitations in the system is close to zero. Af-ter creating a seed excitation in the atomic cloud, facilitation leads to a chain reaction of excitations that eventually fill the entire cloud. (b) Energy scheme of the first facilitated excitation and (c) of the successive facilitated excitations. . . 53 3.9 Triggering the avalanche process. Mean number hNobsi of Rydberg
excitations as a function of time for off-resonant excitation at ∆/2π = +75 MHz and different values of the time ttrig (market by dashed lines) at which ≈ 2 seeds are created by a short resonant pulse. The red data points correspond to ttrig = 10 µs, blue data to ttrig = 25 µs and green data to 45 µs. Interpolated curves (fine dotted lines) are drawn in the plot of experimental data to guide the eye. The experiments were carried out in a 3D geometry with 1.70±0.34×106atoms (of which around 2.15 in the interaction volume) in a MOT with dimensions 160 µm × 130 µm × 100 µm. 54 3.10 The bimodal model. Off-resonant excitation in a 3D geometry at
∆/2π = +27 MHz. The mean number hNobsi of Rydberg excitations (red circles and left axis) and the Mandel Q-parameter Qobs (grey circles and right axis) after 100 µs excitation are plotted vs the measured num-ber of seed excitations hNseedi. As hNseedi increases, the system reaches saturation. The dashed lines are fit curves of the bimodal model with N1 = 2 ± 6 and N2 = 45 ± 6. The inset reports the corresponding prob-ability distribution P ( ˜N ) of the bimodal model excluding (vertical bars) and including (stepwise line) the binomial convolution of ref. [85] taking into account the detection efficiency η. The values ˜N correspond to Nobs for the line and N for the bars. The experiments were performed in a 3D geometry with 9 ± 2 × 104 atoms (of which around 5 × 104 in the interaction volume) in a MOT with dimensions 80 µm × 40 µm × 60 µm. 55
3.11 Mean number of facilitated excitations in a quasi-1D geometry. (a) hNf aci is plotted as a function of the excitation time for different values of hNseedi. Green data points hNseedi = 1.6 ± 0.1, blue points hNseedi = 3.4 ± 0.1, black points hNseedi = 8.6 ± 0.1, and red points for hNseedi = 18 ± 1. For hNseedi = 18 the mean distance between seeds is around 6.8µm and, therefore, much smaller than 2rf ac. Parameters: ∆/2π = +24 MHz, around 4.9 ± 1 × 105 atoms (of which around 1300 in the interaction volume) in a MOT with dimensions 160 µm × 130 µm × 100 µm. (b) hNf aci is plotted as a function of hNseedi for a fixed excitation time of 70 µs with ∆/2π = +30 MHz. The uncertainties used for the mean number of the seeds are calculated from the standard error of the mean when hNseedi . 10 (non-interacting regime) while for hNseedi = 18 (interacting regime) we used the measured standard deviation. The typical error bar shown in the plot reflects the standard deviation. The grey line at hNseedi ' 10 indicates a mean distance equal to 2rf ac. Data for 8 ± 1.6 × 105 atoms (of which around 2, 100 in the interaction volume) in a MOT with the same dimensions as in (a). The inset reports the number of facilitated excitations per seed vs hNseedi. . . 56 3.12 Experimental (left) and theoretical (right) full counting
dis-tributions Nobs for a 1D geometry (with co-propagating laser beams). The histograms in each row have the same mean number of atoms in the facilitation shell (at the centre of the cloud): from the top, hNf aci = 1.3, 1.5 and 1.8, respectively (between the two sets of histograms in (a)-(c), the values of hNf aci matched to within 0.1). In the exper-iments, at t = 0 a mean number of 0.7 (observed) seed excitations is created, and hNf aci is obtained by adjusting the laser detuning and the atomic density using the relation hNg(rf ac)i ∝ ρ/∆7/6. The MOT con-tains 2.90±0.58×106atoms and has dimensions 230 µm×130 µm×170 µm. In (a)-(c), for the red bars ρ = 3.6 × 1010cm−3, ρ = 4.4 × 1010cm−3 and ρ = 5.3 × 1010cm−3 at fixed ∆/2π = +12 MHz (red bars), while for the black lines ∆/2π = +15 MHz, ∆/2π = +13.5 MHz and ∆/2π = +12 MHz at fixed ρ = 6.0 × 1010cm−3.(d)-(f) Theoretical counting distribu-tions based on the simulation described in the text for different values of hNg(rf ac)i. . . 58 4.1 Excitation/deexcitation experimental protocol. Two laser beams
at 420 nm (down gray arrow) and 1013 nm (up gray arrow) excite the atoms to the Rydberg state 70S1/2. The detuning ∆6P is set to avoid the population of the intermediate level 6 P3/2. After a variable dark time tdark, a second infrared pulse at 1013 nm (red arrow), detuned ∆ from the intermediate level, is switched on for tdeex and induces the deexcitation process. Then an electric field ionizes all the Rydberg excitations with principal quantum number n & 55. . . 61
4.2 Energy distribution of Rydberg atoms after a resonant (a) and off-resonant (b) deexcitation process. In (a), ∆ex= 0 and the atoms in the ground state are resonantly coupled to the Rydberg level, creating a quasi-1D chain of Rydberg atoms. The excitation of further Rydberg atoms at a relative distance less than rb, red zones, is slowed down by the interaction energy. In (b) the same ground state atoms are excited by an off-resonant laser beam. The detuning ∆ex defines a preferred relative distance rf acwhere the atoms are resonantly excited, blue circles. In this case, the system displays a preferred position of the Rydberg excitations as well as an energy distribution structure. . . 63 4.3 Population of 70S Rydberg state as a function of the deexcitation
time tdeex with different deexcitation Rabi frequency Ωdeex. Data normalised at tdeex = 0. N = 1.4 × 106 87Rb atoms are collected in a 220 µm×160 µm×190 µm magneto optical trap. Damped Rabi oscillations can be seen for Ωdeex ≈ 2π × 4 MHz (red diamonds), they are reduced for Ωdeex ≈ 2π × 2 MHz (black squares), whereas they are almost negligible for Ωdeex≈ 2π ×1.4 MHz (blue circles). The data are well reproduced by a simple damped-oscillation model (solid lines) based on our measurement of the Rabi frequency. The solid lines are time shifted by 60 ns to take into account the finite rise time of the deexcitation laser intensity (gray zone). . . 64 4.4 Deexcitation process following resonant excitation for different
initial mean numbers of Rydberg atoms. hNini is equal to: 20 ± 1 (blues circles), 34 ± 3 (green triangles) and 50 ± 3 (white squares). The different values of hNini (ranging from the weakly interacting to the strongly interacting regime) are obtained by varying tex between 0.5 µs and 5 µs. In (a), the remaining fraction of Rydberg atoms hN i/hNini is plotted as a function of the deexcitation detuning ∆. Here, tdark = 0.5 µs and tdeex = 2 µs. The solid lines are Lorentzian fits to guide the eye. The expected shift in the deexcitation detuning is visible mainly as an increase in the remaining fraction at ∆ = 0, shown systematically in (c). Here, the remaining fraction after a deexcitation pulse of duration tdeex = 1 µs is plotted as a function of hNini. Note that the linewidths in (a) are larger than in the simulation due to long-term experimental jitter in the laser frequency of around 0.7 MHz. In (e), the deexcitation dynamics is shown for ∆ = 0. In (a), (c), and (e), error bars are one standard deviation of the mean. Panels (b), (d) and (f) show the results of the corresponding numerical simulations (see main text). In (b) and (d) we have adjusted tdeex to 0.5 µs to obtain quantitative agreement. . . 66
4.5 Deexcitation process following off-resonant excitation in the fa-cilitation regime. In (a) the remaining fraction hN i/hNini after excita-tion of hNini = 20 ± 1 excitations (tex = 5 µs) at ∆ex= 2π × (16 ± 1) MHz is plotted as a function of ∆. The blue circles correspond to deexcitation (tdeex = 2 µs) after a dark time tdark = 0.5 µs, whereas the green diamonds are obtained for tdark = 5 µs. The solid lines are triple-Lorentzian fits to guide the eye. The results of the corresponding numerical simulations are shown in (b), where the solid lines correspond to deexcitation of an artifi-cial state with 70 atoms placed on a line at distance of rf ac( blue line) or αrf ac ( green line) as described in the main text. Plots (c)-(e) illustrate the experimental data (symbols) and the simulation results (solid lines ) of the deexcitation dynamics (tdark = 0.5 µs) at ∆ = 0 (c), ∆ = ∆ex (d) and ∆ = 2 × ∆ex (e) following resonant excitation (red symbols) and off-resonant excitation at ∆ex = 2π × (16 ± 1) MHz (blue symbols). In the experiments, the same values hNini = 20 ± 1 are obtained in both cases for tex= 1 µs by adjusting Ω. . . 68 5.1 Graphic representation of our apparatus. Cold atoms in a MOT are
excited to Rydberg states in an elongated interaction volume (continuous ellipse), and they subsequently expand due to the van der Waals repulsion. After a variable free expansion time, they are field ionized by switching on the field plates. Finally, the ions (blue dots) are electrically accelerated through a grid towards a detector. The isochronous planes defined in the text are schematically represented. The lower part of the figure reports a typical signal of the ions (each marked peak correspond to one ion) detected by the detector. . . 76 5.2 Atom cloud displacement and arrival times of the ions. a)-b)The
z-position of a small atom cloud (150 µm has been shifted by around 700 µm keeping constant the y and x-positions. c) The centre of the arrival times distribution of the ions is given by a Gaussian fit: 9.740 ± 0.001 µm (red bars) and 9.654 ± 0.001 µm (blue bars). . . 77 5.3 Coil current calibration along the y-axis. The number of ions signal
is proportional to the atoms in the interaction volume defined by the overlap between the beam, with a fixed position, and the atom cloud. The current of the coils that generate the magnetic field along the y-axis is tuned as in the x and z-axes and changes the position of the cloud as well as the overlap with the beam (right panel). The number of ions (dark squares) is plotted as a function of the coil current in the left panel. . . . 78
5.4 Calibration of the arrival time distributions at the detector. The center of the arrival time distributions is plotted as a function of the MOT position in the x- (blue squares), y- (green diamonds) and z-directions (red circles); in each case, the MOT was centered at 0 in the other two directions. The error bars (corresponding to the standard deviations of Gaussian fits of the MOT images and the arrival time distributions) are smaller than the size of the data symbols. The dashed lines are linear fits to the experimental data within the range (-0.4,0.4) mm. The red circles with black borders correspond to a variation in the z-position for a fixed x = 460 ± 5 µm where the arrival times were corrected using the x-axis calibration. Their agreement with the x = 0 data demonstrates that the arrival times are linearly dependent on the x and z positions without cross terms. . . 78 5.5 Contour plot of the arrival times as a function of the cloud
position. The position of the cloud of width ∼ 40 µm (black stars) has been changed along the x-z plane. For each x-y position a focused 420 nm beam has been systematically realigned on the cloud. The focused beam, with the MOT trap beams kept on, ionizes the atoms whose arrival times have been detected. The experiment has been repeated 200 times and a Gaussian Fit on the distribution of the arrival times gives the averaged arrival time for each position of the cloud (color panel). . . 79 5.6 Histogram of the probabilities P of ion arrival times as a
func-tion of τarr. Data for ∆ = 80 MHz for two different expansion times: 130 µs (light blue bars) and 1000 µs (dark blue bars). SD’s of P (τarr) are calculated from Gaussian fits (dashed lines) to the histograms. The slight asymmetry in the histogram for 130 µs was consistently observed for small spatial distributions and might be due to the intrinsic response of the detector. . . 81 5.7 Plot of the standard deviation (SD) of the ion arrival times.
The points reflect the width of the ion cloud vs the expansion time for different detunings ∆: 0 (green triangles), 55 MHz (blue squares) and 80 MHz (red circles). On the l.h.s axis, the SD’s of the arrival times are shown, and on the r.h.s axis the SD’s of the spatial positions, obtained from the previous SD’s by using the inversion of Eq. (5.2), are shown. The dashed lines report numerical simulations of the van der Waals explosion . 82 5.8 Graphic representation of the experimental procedure for the
lifetime measurement. (a) Excitation of the 70S1/2 Rydberg state and migration to other states. The electric field ionizes all the excitations with n > 55. (b) Before the ionization field, a resonant de-excitation beam removes the 70S1/2 Rydberg atoms. The detected signal comes from the excitation in other Rydberg states. . . 88 5.9 A graphic representation of the analytical model. Different
Ryd-berg states |ri, labelled with the index j, decay to the ground state |gi with a rate γj. Those states are coupled, so that the Rydberg excitation migrates from an initial state |ji to a final state |j + 1i with a coupling rate γj+1,j. The lifetime of a state is define by the inverse sum of all the rates driving the excitation out of that state. . . 89
5.10 Lifetime measurement. (a) Simple model results: Free decays of the target population Nr (green solid line),PjNj with j 6= t (blue solid line) and the variable Ntot (red solid line). The coupling γk,jused in this model has been calculated from the typical lifetime at zero temperature τj(0) and room temperature τj(T ) for T = 300 K of states close to the target state 70S1/2: (70P3/2, 70D3/2, 65P3/2and 65D3/2). Lifetime values taken from [15]. For example: γj+1,j= γj−1,j = 1/2(1/τj(T )−1/τj(0)). The de-cay rate used in the model for a state j is the inverse of the lifetimes at zero temperature, γj = τj(0)−1. (b) Experimental results: (2.40 ± 0.48) × 105 87Rb atoms collected in a MOT 130 × 70 × 90 µm. The initial excitations are created in 5 µs, red squares are related to the total Rydberg popu-lation decaying signal. The blue circles points represent the remaining Rydberg population as a function of the dark time, starting from the same initial excitation number as in the red squares data points, after 5 µs of deexcitation pulse. In this case the dark time tdark reported on the graph includes the 5 µs of deexcitation pulse. The green diamonds are the difference between the red and the blue data sets and represent the population of the target Rydberg state. . . 92 5.11 Numerical solutions for the model with two states |0i and |1i
and symmetric coupling γ1,0 = γ0,1. The two states have different decay rates, γ0 γ1. Exponential decay with γ0 (black dashed line), γ1 (black dot-dashed line) and (γ0+ γ1)/2 (solid red curve). Empty red diamonds mark the numerical solution of the total population decay in two regimes: a) low coupling regime γ1,0 γ1, γ0 and b) high coupling regime γ1,0 γ1, γ0 . . . 93 5.12 Overall population lifetime measurement in a 1D
configura-tion. The sample is formed by 7.6 × 105 ground state atoms col-lected in 150×100 µm MOT. (a) Free decays of Rydberg excitations start-ing from different initial mean excitation numbers: light blue diamonds hNini = 32.6, red squares hNini = 25, light blue circles hNini = 22.4, green diamonds hNini = 18.2, blue diamonds hNini = 12.35, gray squares hNini = 6.69. (b) By an exponential fit we get the timescale τef f reported as a function of the initial number hNini in different configurations: 1D small MOT 130×70×90 µm changing the number hNini by the excitation beams intensity (red diamonds) or by the excitation time (blue squares), 1D bigger MOT 260 × 150 × 200 µm (gray diamonds) and 3D in a small MOT (white circles). . . 95 6.1 Schematic representation of percolation networks from isotropic
to directed in space and in time a) network sites bound with proba-bility p and form an open cluster whose the size d can be large enough to percolate through the entire network. b) If no preferred direction influ-ences the probability p the percolation is isotropic, otherwise it is directed. A preferred spatial direction can be identified, for example, by gravity in the directed bond percolation model c) or a preferred time direction in the infection spreading model d). . . 97
6.2 Processes of DP used to describe our experiments (a) Correlated processes: facilitated excitation (offspring production) or de-excitation (coalescence) changes the number of active sites with a resonant rate Γ0. (b) Uncorrelated processes: atoms can be excited off resonantly (self-activation) with a rate Γspon. Once excited to the Rydberg state each atom can decay spontaneously to the ground state (self-destruction) with a rate k. . . 98
6.3 Threshold Transfer Process (TTP) model graphic
representa-tion and numerical results a) Particles are created in each site with probability p and annihilate with probability 1 − p. Doubly occupied sites are defined as active whereas singly occupied or empty sites are inactive. Any configuration without active sites defines the absorbing state. A per-turbation in the model is implemented by allowing motion through the network of single particles with probability h. They can, eventually, move to a site already occupied and form an active site. b) Numerical results of (2 + 1) dimensions TTP model simulations [83]. Different symbols refer to different numbers of network sites L and mark the behavior at h = 0. As the motion probability h increases, the crossover becomes smoother and, as shown in the inset, the divergence of the fluctuations ∆ρ at the critical point reduces to a peak. . . 100 6.4 Level schemes and stationary fraction in the two-level and in the
three-level mean-field model. Graphs are taken from [54]. (a) Two level scheme: the ground state |gi and the Rydberg state |ri are coupled by a driving laser with Rabi frequency Ω. Atoms in |ri decay to the ground state with a rate k. (b) An additional state |ai is included in the level scheme. The migration from |ri → |ai occurs with a rate η while atoms in |ai decay to the ground state with a rate k0. The stationary fraction nss (indicated in this picture with nmf) is plotted as a function of the coupling strength in two-level (c) or three-level model (d) for different values of the detuning 2π∆. The red dashed lines show the solution in absence of perturbation, i.e., Γspon= 0. . . 103 6.5 The motion of atoms in the DP model. (a) intuitive picture of
ground state atoms moving in a 1D chain. The facilitation mechanism stops after few steps in a chain with fixed atom position (up), moving atoms that pass at the facilitation distance are excited to the Rydberg state and the facilitation mechanism proceeds (down). Numerical results from [54] based on N = 450 atoms lying on a 1D chain with L = 1500 sites. The detuning is fixed at 2π∆ = 10, MHz. (b) Mean number of Rydberg atoms and (c) fluctuations as a function of 2πΩ for different values of the mobility parameter λ = 0, 0.2, 1, 2 and 20 MHz. In the inset, the mean number of Rydberg atoms is shown in a logarithmic plot vs (Ω − Ωc)/2π where Ωc is the value at which the fluctuations have a peak. . . 104
6.6 Time evolution of the mean number of excitations with a high number of seeds a) and the corresponding Mandel Q factor b). Atom cloud parameters: N = (1.90 ± 0.38) × 106 rubidium atoms, gaus-sian density distribution σx,y,z = 200 × 160 × 180 µm and peak density ρ0 = 6 × 1010cm−3. (a) Starting from hNseedi = 20 ± 2 seeds, the mean number of Rydberg excitations hN i is reported as a function of the ex-citation time for different values of the Rabi frequency Ω/2π and the detuning ∆/2π. The Rabi frequency ranges 0 from to 130 ± 25 kHz: Ω/2π = 130 ± 25 kHz (dark blue circles), Ω/2π = 85 ± 15 kHz (light blue diamonds), Ω/2π = 0 kHz (white squares), Ω/2π = 130 ± 25 kHz (red cir-cles) and Ω/2π = 85 ± 15 kHz (light red diamonds). Light and dark blue symbols represent excitation with ∆ > 0. Light and dark red symbols are related to excitation with ∆ < 0. The two dashed lines guide the eye and highlight, for similar dynamics of the mean number, the difference in the Q parameter between the negative and positive detuning case. . . 107
6.7 Time evolution of the mean number of excitations with a low
number of seeds N = (1.60 ± 0.32) × 106 rubidium atoms, cloud with gaussian density distribution σx,y,z = 210×150×180 µm and peak density ρ0 = 5 × 1010cm−3. (a) Initially, the system is prepared with hNseedi = 3±1, after which the excitation coupling is detuned with ∆/2π = 10 MHz. Different symbols refer to different values of the Rabi frequency Ω/2π: 0 (white squares), 100 ± 20 kHz (light blue diamonds) and 140 ± 30 kHz (dark blue circles). (b) The Mandel Q parameter is also reported as a function of the excitation time with the same color code. . . 108 6.8 Crossover dependence on the initial number of seed. Rubidium
atom cloud parameters: N = (635 ± 120) × 103 ground state atoms and x, y, z gaussian distributions with σx,y,z = 130 × 100 × 110 µm and peak density ρ0 = 9 × 1010cm−3. (a) Mean number crossover after 1.5 ms of positive detuned coupling ∆/2π = 20 MHz for different numbers of initial seeds: no seeds (orange triangles), few seeds hNseedi = 4 ± 1 (blue circles) and a high number of seeds hNseedi = 20 ± 2 (green diamonds). (b) The corresponding empty symbols represent the standard deviation as a function of the Rabi frequency. . . 109 6.9 Symmetric detuned excitation coupling. N = (1.1 ± 0.2) × 106
87Rb ground state atoms are collected in a σ
x,y,z = 180 × 120 × 150 µm
MOT leading to a peak density of ρ0 = 6 × 1010cm−3. The system is prepared with hNseedi = 3 ± 1 coupled for 1.5 ms with a positive detuning ∆/2π = +13 MHz (blue circles) or ∆/2π = −13 MHz (red diamonds). One representative error bar is shown in both cases. (a) Crossover of the mean excitation value for both cases as a function of the Rabi frequency. (b) Counting distribution for positive (blue bars) and negative (red bars) detuning after 1.5 ms of excitation coupling and maximum Rabi frequency Ω/2π = 180 kHz. . . 110
6.10 Crossover detuning dependence. Mean number of excitations mea-sured after 1.5 ms of off resonant coupling. Cold cloud with N = (1.40 ± 0.28) × 106 ground state atoms, widths σ
x,y,z = 230 × 150 × 180 µm and
peak density ρ0 = 4 × 1010cm−3. The detuning ranges from 0 (green circles) to +15 MHz (black circles). One representative error bar, cor-responding to one SD, is shown on the mean number of excitations in (a) while the behavior of the standard deviations is reported in (b) with the same color code. Solid lines are reported to guide the eye in the fluctuations plot. . . 111 6.11 Mean number of excitations crossover. (left axis) The mean
num-ber of excitations (red circles) is reported as a function of the coupling strength Ω. The excitation coupling is detuned by ∆/2π = +10 MHz and drives the dynamics for 1.5 ms starting from hNseedi ' 3. One represen-tative error bar is shown, corresponding to one standard deviation. (right axis) Mean number fluctuations (SD, black diamonds) have a peak at the critical point Ωc (grey dashed line). The inset shows the same data in a lin-log plot for Ω > Ωc= 67 ± 5 kHz and the power law fit (red solid line). 112
AMO Atomic Molecular Optical system
AOM Acusto Optical Modulator
BBR Black Body Radiation
BEC Bose Einstein Condensation
FPI Fabry P`erot Interferometer
MOPA Master Oscillator Power Amplifier
MOT Magneto Optical Trap
RF Radio Frequency
TA Tapered Amplifier
VCO Voltage Controlled Oscillator
Constant Name Symbol = Constant Value (with units)
Speed of Light c = 2.997 924 58 × 108 ms−s (exact)
Magnetic constant µ0 = 4π × 10−7 NA−2 (exact)
Electric constant 1/µ0c2 0 = 8.854 187 817... × 10−12 F m−1
Planck constant h = 6.626 070 040(81) × 10−34 J s
~ = 1.054 571 800(13) × 10−34 J s
Elementary charge e = 1.602 176 6208(98) × 10−19 C
Fine-structure constant e2/4π0~c α = 7.297 352 5664(17) × 10−3 inverse fine-structure constant α−1 = 137.035 999 139(31) Rydberg constant α2m
ec/2h Ry = 10 973 731.568 508(65) m−1
Bohr radius α/4πRy a0 = 0.529 177 210 67(12) × 10−10m
Boltzmann constant kB = 1,380 6488 (13) × 10−23 J K−1
Electron mass me = 9.109 383 56(11) × 10−31kg
Proton mass mp = 1.672 621 898(21) × 10−27kg
Rubidium mass mRb = 1.443 160 648(72) × 10−25kg
Symbol Name Unit
a Distance m
P Power W (Js−1)
T Temperature K
TD Doppler-limit temperature µK
ω Angular frequency rad s−1
f (v) Maxwell Boltzmann distribution (m/s)−1
vth Thermal velocity (m/s)
∆ Detuning excitation rad s−1
∆6P Detuning Intermediate state rad s−1
Vint Interaction volume µm3
Vf ac Facilitation volume µm3
Ω Two-photon excitation Rabi frequency rad s−1
Ω420 First transition excitation Rabi frequency rad s−1 Ω1013 Second transition excitation Rabi frequency rad s−1
Γ Single atom excitation rate rad s−1
Γ0 Single atom resonant excitation rate rad s−1
Γspon Single atom spontaneous excitation rate rad s−1
k Decay Rate rad s−1
λth de Broglie thermal wavelength m
ρ Atom density cm−3
ρ0 Peak atom density cm−3
tex Excitation time µs
tdark Dark time µs
tdeex Deexcitation time µs
Introduction
In this Thesis, I will report on the experiments performed during my Ph.D. in Pisa. The aim of the thesis is to explore the evolution of a many-body system in a dissipa-tive regime. This can be done by the means of a cold ensemble of Rydberg atoms that, thanks to their peculiar properties, provides a versatile experimental platform to observe non-trivial dynamics of correlated excitations and new phase transitions. In this chapter (Ch. 1), after a brief introduction of the many-body problem and the latest achievements of cold atom systems, the basic properties of atoms excited to high lying energy states, i.e., Rydberg states, are briefly outlined. An overview of the experimental apparatus is given in Ch. 2 focusing on the devices and calibrations used in the experiments. In the first part of this Thesis (chapters 1 to 3), I will describe the interactions between Rydberg atoms and the mechanisms responsible for the kinetic constraints. The many-body nature of the interacting system under investigation leads to correlated dynamics theoretically well described in [78, 79] in the non-dissipative regime. The experimen-tal results reported in the first part of the Thesis have been taken from experiments performed during my Master Thesis and thoroughly described in Maria Martinez Val-ado’s P h.D. Thesis [126]. The original contributions are reported in the second part, dedicated to the dissipative regime. In Ch. 4, a deexcitation technique is described and tested experimentally, giving the opportunity to implement a controllable loss channel and to observe the effect of the interactions in the Rydberg atom system from a com-plementary point of view. In the dissipative regime the correlated dynamics is now affected by the spontaneous decay to the ground state, making our Rydberg atoms sys-tem suitable for simulating the behavior of an open many-body syssys-tem. In Ch. 5, the internal, as well as the external, degrees of freedom are investigated experimentally for timescales much longer than the lifetime of the Rydberg state. The Rydberg-Rydberg mechanical interaction is measured and the deexcitation technique is used to implement
a state selective experimental protocol aimed at the measurement of the lifetime. Fi-nally, we use our Rydberg atom system in the dissipative regime to implement a second order absorbing-state phase transition, which arises from the competition between two opposing processes: the correlated excitation and the spontaneous decay to the ground state. Clear signatures of the undergoing phase transition have been measured. These results pave the road for future investigations into the largely unexplored physics of nonequilibrium phase transitions in open many-body quantum systems.
1.1
The many-body problem
One very famous quote attributed to Albert Einstein is commonly cited as ”Everything should be made as simple as possible, but no simpler”. This concept guides our approach in the description of physical systems: to make a model with as few assumptions as possible and, at the same time, to achieve a realistic representation of the system under investigation. As a first step, the system is isolated from its environment and then reduced to a sum of smaller parts (Fig. 1.1). Once the reduction has reached its lower limit, meaningful for our description, the system will appear as an ensemble of single components. At that point, the system will be in its easiest form to study the physic that governs its evolution. However, even if the evolution of the single component is well known, a reliable description of the system as an ensemble of elements is far more complex.
Figure 1.1: Graphic representation of the system and the environment. The system (light blue circle) is coupled to the environment (green blob). If the coupling is weak and the environment relaxes quickly to equilibrium (so that the Born approx-imation and Markov approxapprox-imation can be applied) the evolution of the system is described by the master equation in Lindblad form [30]. If the system is made up of a large number of interacting components (blue circles) its evolution is a many body
When the components of the system interact with each other, the special case where the evolution of the ensemble is well described by the behaviour of the individual compo-nents of the system is no longer applicable. Our description should, in general, take into account the evolution of all the ensemble simultaneously. Very often this is a difficult task, even if the physics behind the single component evolution is simple. The behavior of the system, then, becomes rapidly unsolvable as the number of the components in-creases. This is the essence of the Many Body Problem. In practice, the vast majority of physical systems in nature are made up of a large number of components, are coupled to the environment and are not well described by a single component treatment. As pointed out by Phil Warren Anderson in [6] ”More is different”, the physics behind the many-body system contains the answer to fundamental solid-state questions such as superconductivity at high temperature [7] or new exotic quantum phase transitions (such as MI-SF [51] or Dicke quantum phase transition [12]).
Different approaches have been developed in recent years to tackle with the many-body problem. The simplest one is the mean field Theory (MFT) which appears at the begin-ning of the XX century in the article on phase transitions by Pierre Weiss [136]. The idea is simple: the interaction effect of all the components is averaged and represented by a single mean field. Hence, the many-body problem is reduced again to a single body problem in which the i-component interacts with the field generated by all the N others (labeled by the index k = 1...N )
Vi = X
k6=i
V (ri, rk) ≈ VM F T. (1.1)
As we will see in the following chapters, we use this approach to describe the evolution of our Rydberg atom system. Other approaches, are based on the description of the many-body system with a high number of components in terms of few quasiparticles or elementary excitations [139], e.g. phonons in a crystal structure. The problem is to find the effective mass of those quasiparticles and the potential under which they evolve. For example, the Hartree and Hartree-Fock Theory have been developed to describe the behavior of an electron in solids that interacts with all the other electrons. These ap-proaches are based on the assumption about the form of the many-electron wavefunction and the possibility to study the evolution of the single electron wavefunction under the effect of a new potential, the Hartree Potential
VH =
Z n(r0) |r − r0|dr
0 (1.2)
which depends on the electron charge density distribution n(r). These two approaches aim to find the correct expression of the many-electron wavefunction in terms of single
electron ones (the simple product for Hartree and the antisymmetric combination in the Hartree-Fock case) and then find the equation of motion using the variational principle. In general, the description of the behavior of a single component is easier if the pertur-bation on the potential given by all the others is small compared to the energy scales associated with the evolution of the single component. If this condition is fulfilled, the evolution of the interacting system can be described as a perturbation, to some degree, of the noninteracting case. Another approach to the many-body system is, instead of calculating the solution of the Schr¨odinger equation to find the wavefunction evolution, to simulate the system in the laboratory by a well-controlled experimental platform. This was the approach used by Vannervar Bush in 1924 to study the behavior of a large alternating current network. Even if the Maxwell equations were already well known, the computation of the solution of a large coupled network was impossible at the time and its evolution was an open question. The answer came from a device, a classical simulator of the network called AC network analyzers, built in 1929 and in use for more than twenty years (till 1953). Nowadays, modern computers can quickly solve a large number of coupled differential equation and calculate the classical evolution of a many-body system. However, nature is not classical. In 1982 [42] Richard Feynman questioned the possibility of simulating a quantum system by a classical Turing machine, i.e., a classical computer. The dimensions of the Hilbert space that describes a quantum system scale exponentially with the number of components, and the computation of the quantum evolution becomes impracticable even for as few as a hundred components (which is very small compared to the typical number of components in a real physical system). As Vannervar Bush did in 1924, many research groups now make use of cold atoms to simulate a many-body system to test theoretical predictions [33, 82, 85, 110, 114, 122, 129, 131] and explore quantum phase transitions [51, 70, 76, 89]. In the next section, a brief description of advantages provided by cold atoms and a few examples of experimental results are reported.
1.2
The cold atoms opportunity
In the last few years, Atomic Molecular and Optical (AMO) systems such as cold atoms, molecules and trapped ions have been intensely used for studying strongly interacting many-body systems. At very low temperatures, the thermal de Broglie wavelength, expressed as λth= h √ 2πmkBT , (1.3)
increases as the temperature decreases. When it becomes equal to (or larger than) the mean distance between the atoms, the wave-like behavior makes them interfere with each other, which leads to non-classical phenomena (Fig 1.2). One of the most remarkable
examples of these is the achievement with rubidium and sodium atoms of the Bose Einstein condensation (BEC) in 1995 [5, 18, 31], where an ensemble of atoms behaves as a macroscopic quantum object.
Figure 1.2: Cold atoms system. a) At high temperature T each atom of the en-semble moves isotropically with a finite thermal velocity vth(T ). b) As the temperature
decreases, the motion of the atoms is also reduced and the thermal de Broglie wave-length λth increases. c) When the wavelength λthis higher than the relative distances
between two atoms of the ensemble, their wavefunctions overlap and form a single macroscopic quantum object.
Even if the temperature of the system is above the critical temperature Tc at which the atom wave-like behavior is dominant, at low temperatures their motion can be neglected on the time scale of the experiment (Frozen gas approximation), simplifying the theoretical treatment. Advanced numerical techniques such as quantum trajectories [30, 64, 139] have been developed to efficiently solve the master equation. Enhanced light-matter manipulation techniques, as well as new single atom imaging systems [55, 137, 138], allow for an unprecedented high degree of control on the experimental platform so that AMO systems are used to engineer microscopic Hamiltonian for the purpose of quantum simulation. The flexibility provided by such an experimental platform makes cold atoms optimal candidates to study many-body physics. Remarkable examples are the studies of the Bose-Einstein condensate BEC-BCS crossover [94], the superfluid-Mott insulator transition [51] and Ising Model [116].
Cold atoms provide a versatile experimental tool for studying many-body physics also in the semi-classical regime, in which quantum coherences are negligible. Our experi-ments are performed using rubidium atoms confined in a magneto-optical trap (MOT) and cooled at temperatures of around 150 µK well above the condensation threshold but allowing for a description, on the experimental timescale, within the frozen gas approx-imation. Hence, we are not interested in quantum effects on the external (motional) degree of freedom of the atoms, but instead, we focus on the many-body behavior of the internal states of an ensemble of Rydberg atoms. Although the single component
evolution is treated classically, the dynamics of the system is highly nontrivial due to the strong spatially dependent Rydberg-Rydberg interaction. This approximation is as-sumed for the experiments described in Ch. 3, where the atomic motion is negligible and the basic mechanisms of the kinetic constraints are described, then in Ch. 5, the evolu-tion of the system beyond this approximaevolu-tion are is investigated. Both with or without spatial motion of the atoms, the many-body behavior of our system is investigated by well-controlled and strongly interacting components. In the next section, we will see how Rydberg atoms can fulfil this fundamental requirement.
1.3
A general introduction to Rydberg atoms
In recent years, thanks to their extreme properties, cold atoms excited to high-lying energy states, i.e. Rydberg atoms, have been used as a powerful tool in quantum sim-ulation. Here, we report a brief overview of the basic properties needed in our ex-periments, focusing on the energy level structure and on the interactions effect on the excitation dynamics. A comprehensive description of all the other Rydberg atom prop-erties, the calculation of the wavefunction, excitation and ionization techniques can be found in [45]. The most important Rydberg atom properties used in this thesis are the very long lifetime ∼ n3, which can reach hundreds of microseconds, large sizes ∼ n2, and strong interactions ∼ n12. A detailed calculation of those properties requires the knowledge of the wavefunction and, in general, in a complex task. However, alkali atoms excited to high energy states allow a simple description in terms of the quantum defect theory (QDT) due to their hydrogen-like energy structure. One can derive the correct dependence on the principal quantum number n of general properties, such as the size the Rydberg atoms as well as the ionization field, simply as a correction of the corre-sponding results for the Hydrogen atom case where the wavefunction is well known. For the sake of completeness, some basic properties, their n dependence and approximated values are reported in the following table (Tab. 1.1).
As shown schematically in Fig. 1.3, a H atom excited to the Rydberg state compared to a87Rb atom excited to the same Rydberg state looks very similar, except for the size of the atom core: 1 proton for H and Z = 37 protons for the rubidium atom. When the valence electron spends most of its time far from the nucleus, such as for high l states, it is sensitive only to the net charge. Hence, the differences between the H atom and87Rb are expected to be small. On the other hand, for low l states, the electron could pass through the inner electron charge distribution and polarize it. In this case, we expect to see a correction in the87Rb atom wavefunction, as well as in its properties, compared to the H case.
Property Symbol Dependence Rb(5S) Ground-70S1/2 state
Binding energy En n−2 4.18 eV - 3.04 meV
Energy difference En− En+1 n−3 2.50 eV - 87.7 µeV
Orbital radius hri n2 5.632- 6706 a0
Lifetime τ n3 26.2ns ( 5P3/2− 5S1/2)- 151.55 µs
Van der Waals coefficient C6 n11 4707- 601 ×1019 au
Polarizzability α n7 -79.4 - 660.8 MHz (V cm−1)−2
Table 1.1: Main properties and principal quantum number dependence. The approx-imated values are reported for Rb in the ground state and in the 70S1/2Rydberg state.
Lifetime and C6values are taken form [15, 135] whereas the others from [82] rescaling
to n = 70.
Figure 1.3: Rydberg atoms of a) H and b) Rb. The electron e orbits around the nuclear charge. In the H atom the electron orbits around the point charge of the proton. In the Rb atom, the electron orbits around the nuclear charge of Z = 37 protons and the Z − 1 inner electrons. For high l states, the electron orbits away from the nucleus and the Rb atom behaves identically to the H. For low l states, the electron penetrates
We can formalize the above idea as follows. The hamiltonian of a neutral atom with Z protons and electrons is
H = Z X i=1 p2 2me −Ze 2 ri +1 2 X i,k e2 rik (1.4)
where the spin and relativistic terms are neglected. In an alkali atom, we can distinguish the outer electron, i.e. the valence electron, that is responsible for the optical transitions, from the electrons in the inner shells. We can rewrite the hamiltonian as
Htot= Hinner+ Hvalence+ 1 2 X i,k e2 rik (1.5)
in which Hinner refers to the inner electrons and Hvalence to the valence electron. The term 12P
i,k e
2
rik contains both the valence and the inner electrons, hence we cannot
separate the hamiltonian in two parts. For this reason the state of the atom includes all the electrons and not only the valence electron. However, we can describe the coupling of the valence electron with all the electrons in the inner shell as a perturbation. We add in the hamiltonian the potential generated by the inner electrons V (r)
Htot= Hinner+ Hvalence+ V (r) − V (r) + 1 2 X i,k e2 rik . (1.6)
We assume that the perturbation of the inner electron state due to the coupling with the valence electron is small for the atoms under investigation (that is in general true for Rydberg atoms). Hence, the term V0 =
V (r) + 12P i,k e2 rik is assumed to be a perturbation of the unperturbed Hamiltonian
Htot0 = Hinner+ Hvalence+ V (r). (1.7)
This hamiltonian can be separated into two parts: one related only to the inner electrons and the other (Hvalence+ V (r)) depends only on the valence electron coordinates. In our description, we are interested only in the optical transition of the valence electron, so we can consider the Hamiltonian
H = p 2 2me −Ze 2 r + V (r). (1.8)
We define R0 as the classical radius of the inner electron shell. This radius corresponds to the size of the ionized atom, of the order of an ˚A. The coulomb potential seen by the
valence electron can be estimated using the Gauss theorem. U (r) = −Ze 2 r + V (r) ≈ −e2 r r & R0 −Zer2 r → 0 (1.9)
This description shows why at high principal quantum number n the hydrogen-like description of a Rydberg atom is, in general, a good starting point. For high l-states the valence electron likely orbits far from the nucleus and never crosses the inner electron shell. For l < 3, however, a correction is needed. For those states, known as defect states, the electron passes through the inner electron shell and the potential is perturbed. To take into account this effect the principal quantum number n is replaced by n∗ = n−δn,l. The correction δn,l is known as Quantum defect. It decreases as l increases (δn,l≈ 0 for l > 3) and removes the l-degeneracy. The value of this correction is determined for each atomic species experimentally by spectroscopy measurements [56] and approximated by the empirical Rydberg-Ritz formula [45]
δn,l = δ0+ δ2 (n − δ0)2 + δ4 (n − δ0)4 + ... (1.10)
in which the δi values for i = 0, 2, 4... have to be determined for the atomic species under investigation [80]. In table (Tab. 1.2), for completeness the values of δ0 and δ2 from [80] are reported.
After this brief introduction of the H-like model, we start our classical description of a Rydberg atom with an infinitely heavy nucleus of positive charge e. The orbital radius of the outer electron follows from Newton’s law for circular motion and the quantization of angular momentum mv2 r = 1 4π0 e2 r2 mevr = n~ (1.11)
where me is the electron mass, v the velocity of the electron and 0 the permittivity of free space. From the previous two equation, we can derive the orbital radius of the outer electron r = n 2 ~24π0 e2m e = n2a0 (1.12) where a0= 0h 2 πe2m
e is the Bohr radius. To have an estimate of the Rydberg atom size for
n = 70, this expression gives an orbital radius of around 260 nm.
The energy of the states can be expressed by the sum of the kinetic term and the potential energy E(n, l) = mev 2 r − e2 4π0 = − e 44m e 2(4π0)2n2~2 = − Ry∗ n∗2 (1.13)
Series δ0 δ2 nS1/2 3.1311804(10) 0.1784(6) nP1/2 2.6548849(10) 0.2900(6) nP3/2 2.6416737(10) 0.2950(7) nD3/2 1.348 091 71 -0.60286(26) nD5/2 1.346 465 72(30) -0.59600(18)
Table 1.2: Reported Quantum Defect δ0 values and their first corrections δ2 for the
states S, P and D.
Here, to extend the formula also for an alkali atom, in the last right-hand side of the expression the Ry∗ is the generalized Rydberg constant corrected by the nuclear mass M
Ry∗ = Ry
1 + me/M
(1.14) and n∗ = (n − δn,l) is the principal quantum number corrected by the screening effect of the inner electron shells through the quantum defect term δn,l.
The energy difference of two states leads to an important formula of spectroscopy, known as Rydberg’s formula, expressed as follows
E(n2, l) − E(n1, l) = Ry 1 n2 1 − 1 n2 2 = 1 λvac (1.15)
where λvac is the wavelength of the light emitted in the vacuum, and n1 < n2 are the principal quantum number of the two states. From this simple derivation, we already get the correct n dependence of the bound energy ∼ n∗−2, the orbital radius ∼ n∗2 and the scattering cross section ∼ n∗4.
To gain some insight into the difference between a ground state atom compared to the Rydberg atom, we can compare the two cases for the H atom. The electron binding energy in the ground state is 1 Ry (13.6eV) and the orbital radius is one Bohr radius a0 = 4π0~
2
mee2 . Suppose that we now excite the H atom to a Rydberg state n = 70. The
electron binding energy, in this case, is around 2 × 10−4Ry and the orbital radius is ≈ 5 × 103 larger than Bohr radius. So, the electron of a H Rydberg atom has a very large orbital radius and is weakly bound to the nucleus. The corresponding values for rubidium atoms are reported in Tab. 1.1. The large displacement of the electron charge from the nucleus leads to a high polarizability of the Rydberg atom which scales as ∼ n∗7. This brings us to another set of peculiar properties of Rydberg atoms: the high sensitivity to an electric field, the Stark effect, and the very low ionization threshold ∼ n∗−4. We will use this last property in our experiments to easily ionize the Rydberg atoms, as described in the next chapter (Ch. 2), and thus to detect the number of excitations in the cloud. But the most important reason for using cold Rydberg atoms
to study many-body physics is based on the long lifetime of the Rydberg state and the strong interactions between them. These two last properties allow one to investigate experimentally the many-body problem both in the non-dissipative regime, where the decay to the ground is negligible, and in the dissipative regime, where the decay to the ground is important. In the following, we report a simple derivation of the principal quantum number n dependence of the lifetime and the interaction term.
1.3.1 The radiative lifetime
The knowledge of the wavefunction of an excited atom allows the calculation of the decay rates to the low-lying energy states and the lifetime of the excited state. Starting from the simple case of the H atom, the radiative properties of the alkali atom can be calculated using the quantum defect theory. In the following section, we derive the expression of the lifetime τ in terms of the transition strengths towards low-lying energy states [45].
The oscillator strength from the state |nlmi to the state |n0l0m0i is expressed as fn0l0m0,nlm = 2
m ~ωn
0l0,nl|hn0l0m|r|nlmi|2, (1.16)
where ωn0l0,nl = (E(n0, l0) − E(n, l)) /~. As underlined in [45], the radiative decay in free
space, in the absence of a quantization axis, cannot depend upon m. So we introduce the average oscillator strength
¯ fn0l0,nl = 2 3ωn0l0,nl lmax 2l + 1|hn 0l0|r|nli|2, (1.17)
where lmax = max[l, l0]. The oscillator strengths fulfil useful relations such as the Thomas-Reiche-Kuhn sum rule
X
n0l0m0
fn0l0m0,nlm = Z, (1.18)
and, valid for one electron in a central potential, the sum rules X n0 ¯ fn0l−1,nl= −1 3 l(2l − 1) 2l + 1 (1.19) and X n0 ¯ fn0l+1,nl = 1 3 (l + 1)(2l + 3) 2l + 1 , (1.20)
from the sum of which we derive
X
n0l0
¯
fn0l0,nl = 1, (1.21)
where l0 = l ± 1. The spontaneous decay from a state |nli to a lower state |n0l0i is, usually, defined by the Einstein A coefficient
An0l0,nl = 4e2ω3n0l0,nl 3~c3 lmax 2l + 1|hn 0 l0|r|nli|2 (1.22)
where ωn0l0,nl is the transition frequency and |hn0l0|r|nli| is the radial matrix element.
The Einstein coefficient can be written in terms of the average oscillator strength as
An0l0,nl = −
2e2ω2n0l0,nl
~c3 ¯
fn0l0,nl. (1.23)
In Ch. 5, we will include the additional effect of the black body radiation coupling in the definition of the lifetime but, in this chapter, we consider only the spontaneous decay of a Rydberg atoms at T = 0K. Hence, the lifetime is expressed as the inverse of the sum
τnl = " X n0l0 An0l0,nl #−1 (1.24)
of the decay constants to all the lower lying energy states |n0l0i. From this expression, we conclude that due to the ∼ ω3 dependence, the transition with the highest frequency contributes the most to the radiative decay [15, 82]. With the exception of the S states, the highest frequency decay is to the lowest lying state |n0l − 1i. Nevertheless, even if the highest transition frequency ω increases as the principal quantum number n, it does not mean that the lifetime decreases for high n. Indeed, as n → ∞ the highest transition frequency approaches a constant and the value of the A coefficient depends only on the radial matrix element |hn0l0|r|nli|. Hence, the Rydberg atom lifetime at zero temperature is determined mostly by the overlap between the wavefunctions of the Rydberg state and the low lying states |n0l0i. As a result, for the Rydberg state 70S1/2 used in most of the experiments reported in this thesis and due to the normalization of the Rydberg wavefunction [45], the lifetime exhibits a principal quantum number dependence
τnl ∝ n3. (1.25)
with a predicted value at zero temperature of around 410 µs [15]. As mentioned above, this value changes as a function of the temperature, due to the blackbody radiation coupling, and it is, therefore, quite different at room temperature (around 150 µs) where we perform our experiments. This value, even if smaller, allows the investigation of the
