Experimental studies of the blackbody induced population migration in dissipative Rydberg systems

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University of Pisa

Faculty of Mathematical, Physical and Natural Sciences

Doctoral Thesis in Physics

Experimental studies of the blackbody

induced population migration in dissipative

Rydberg systems

Author: Matteo Archimi Supervisor: Dr. Oliver Morsch March 1, 2019

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List of Figures

1 (a) The figure shows the plot of the mean number of Rydberg excitations in the system at a time much longer (t = 1500 µs) than the lifetime of the Ryd-berg state (in this case, 70S, whose lifetime at T = 300 µs is τ_70S ≈ 150 µs), as a function of the Rabi frequency of the off resonant driving radiation. The inset shows the log-log plot of the mean number and the fit necessary to ob-tain the experimental value of the critical exponent. (b) The figure shows the plot of the variance of the number of Rydberg excitations as a function of the Rabi frequency of the off resonant radiation, which represents the fluctuations of the number of excitations [42]. . . v

1.1 (a) A schematic representation of the blockade process with two atoms: the three lines refer respectively to zero, one and two Rydberg excitations, while in the horizontal axis is reported the distance between the two considered atoms. When d > r_b, the shift in energy due to van der Waals interaction is negligible, while for d < r_b the interactions shift the energy level and the excitation laser results out of resonance. (b) A plot of the mean number of Rydberg excitations as a function of the excitation time for three different densities, i.e., the number of atoms in the interaction volume, defined as the intersection between the laser beams and the atomic sample. The data refer respectively to 5600 (white diamonds), 715 (green squares), and 180 (red circles) atoms in the interaction volume. For each curve, the typical error is reported for one experimental point. (c) A plot of the normalized grow rate for the three different number of atoms in the interaction volume, as a function of the mean distance between excited atoms. The vertical scale is logarithmic [34]. . . 13

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LIST OF FIGURES LIST OF FIGURES 1.2 (a) A schematic representation of the facilitation process with two atoms: the

three lines refer respectively to zero, one and two Rydberg excitations, while in the horizontal axis is reported the distance between the two considered atoms. The first Rydberg excitation, represented by the yellow arrow, occurs out of resonance, with a slower rate with respect to the resonant case. When the first excitation is created, it can facilitate another excitation thanks to the energy shift caused by the interaction, which compensates the detuning of the excitation laser. This facilitated excitation is represented by the green arrow. The facilitation process can occur only at a precise distance, the facilitation radius rf ac, that depends on the value of the detuning of the

excitation laser. (b) A plot of the mean number of Rydberg excitation for three different detunings of the laser, in particular ∆/2π = +19 MHz (blue circles), ∆/2π = −19 MHz (red diamonds) and ∆/2π = 0 MHz (grey squares) which corresponds to the resonant case. For each curve, the typical error is reported for one experimental point. The continuous lines represent numerical simulations. (c) Mandel’s Q parameter as a function of the excitation time, for the three different detunings [34]. . . 15 2.1 Figure taken from [84]. Two counterpropagating, linearly polarized beams,

with the directions of the polarization orthogonal to each other are repre-sented. This configuration leads to a spatial modulation of the polarization of the light. The light shift of the ground state Zeeman sublevels is also spatially modulated. An atom moving in such a system will be continuously pumped to lower energy sublevel, losing energy and thus decreasing its speed. 21 2.2 (a) Principle of operation of the one dimensional MOT. The inhomogeneous

magnetic field induces a space-dependent shift on the Zeeman sublevels lead-ing to a space dependent force on the atoms. The figure is taken from [84]. (b) Schematic representation of the three dimensional magneto-optical trap, with the three pairs of counterpropagating laser beams with the correct po-larization, and the magnetic field quadrupole produced by two coils in anti-Helmholtz configuration. . . 22 2.3 Complete scheme of the optical paths for the trap and repump lasers in our

laboratory. The essential electronics cited on the text is also reported in the scheme. . . 23 2.4 (a) Scheme of the energy level involved in the trapping process of

Rubid-ium atoms. The trap and repump transitions are highlighted in the scheme. (b) Transmission observed in the saturated absorption imaging for Rubidium

87_{Rb. The transmission presents the typical Doppler profile, inside which}

some peaks are visible, each one corresponding to an atomic transition. The top figure refers to the absorption imaging for the trap laser, the bottom one to the absorption imaging for the repump laser. . . 24 2.5 The figure represents a scheme of the vacuum apparatus of our setup. The

main parts are indicated and explained in the figure. The reference axes which we typically use to describe the apparatus are also reported. . . 27 2.6 Scheme of the two-photon excitation of the Rubidium atoms to Rydberg

states, as used in our experiments. The 421 nm laser beam is detuned from the transition with the intermediate state in order to avoid to populate it. . . 29

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LIST OF FIGURES LIST OF FIGURES 2.7 Scheme of the optical paths of the excitation laser beams arriving at the

three dimensional MOT. We use two main configurations, one in which the 421 nm laser beam is approximately copropagating with the 1013 nm beam, and another in which the 421 nm is sent to the atoms through an optical fiber, and then is focused to a waist of 7 µm. Since the waist of the laser beam in the second case is smaller than the blockade radius for the typical Rydberg states we use in our experiments, the second configuration leads to a quasi one-dimensional geometry of the experiment, while the first configuration leads to a three-dimensional geometry. . . 30 2.8 (a) Photoionization through two photon absorption from the 421 nm laser

beam. The laser frequency must be resonant with the transition to the inter-mediate state |6P_3/2i. (b) Direct photoionization of the atoms through the absorption of a single photon from the 421 nm. In this case, the atoms must be in the |5P_3/2i in order to be ionized. . . 31 2.9 Representation of the TTL pulses produced by the delay generator. When

the signal is high, the corresponding element is on, while when the TTL signal is zero the corresponding element is off. The first pulse switches off the MOT beams at the beginning of the experiment. The second switch on the excitation beams for a few µs in order to excite atoms to Rydberg states. The last pulse sents high voltages to the electrodes around the vacuum cell and is used for the field ionization of Rydberg atoms. Depending on the experiment, other TTL pulses can be employed to control the experimental cycle. . . 32

3.1 Schematic representation of the last quartz cell, where the 3D MOT is formed, and where the flight of the ions takes place. The key elements for the ion-ization, the frontal and lateral electrodes, the rectangular electrodes and the detector, are represented in the scheme and described in the legend. The red dot represents the exact position where the 3D MOT is formed. . . 35 3.2 (a) Schematic section of the channel electron multiplier, model KBL510 [89].

The rectangular aperture allows the ions to enter the pyramidal-shaped fun-nel, where the first impact with the sensitive surface takes place. The partic-ular shape of the curved internal channel is necessary to maximize secondary impacts with the surface and to prevent ion feedback. (b) Example of the signal arriving at the oscilloscope, relative to a single repetition of the ex-perimental cycle. Each dip corresponds to a single ion arriving at the CEM. In the figure, the horizontal scale is 200 ns/div, while the vertical scale is 10.0 mV/div. The later dips are slightly less deep probably because they cor-respond to ions starting from a particular region of the MOT, thus following different trajectories, resulting in a lower detection efficiency. . . 37 3.3 Typical histogram of the arrival times of ions to the detector. The distribution

of the arrival times can be fitted with a gaussian function. Here the dashed black line represents a fit of the arrival times distribution. . . 38

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LIST OF FIGURES LIST OF FIGURES 3.4 These figures show the plot of the number of detected ions, centers and widths

of the arrival times distributions as a function of the potential applied on the two pairs of electrodes. Figure (a) shows the plot of the number of detected ions (red squares) on the left side, and the center of the arrival times dis-tributions on the right side (blue circles) as a function of the potential on the frontal electrodes, while figure (b) shows again the plot of the number of detected ions on the left (red squares) and the widths of the arrival times dis-tributions on the right (green diamonds), as a function of the potential on the frontal electrodes. Figures (c) and (d) show the plot of the same observables but as a function of the potential applied to the lateral electrodes. For each experimental point, the error bar is the standard deviation of the ion counts over 200 repetition of the experiment. . . 39 3.5 Plot of the number of detected ions, centers and widths of the arrival times

distributions as a function of the potential applied on the rectangular elec-trode. Figure (a) shows the number of detected ions on the left side (red squares), and the center of the arrival times distributions on the right side (blue circles) as a function of the potential on the rectangular electrode, figure (b) shows again the number of detected ions on the left (red squares) and the widths of the arrival times distributions on the right (green diamonds), as a function of the potential on the frontal electrodes. . . 40 3.6 (a) Number of detected ions as a function of the application time of the

poten-tials on the frontal and lateral electrodes. The three sets of data correspond to three different values of the potential on the frontal electrodes: V = 3400 V red squares, V = 3300 V dark red squares, V = 3200 V yellow squares. (b) Center of the arrival time distributions as a function of the application time. (c) Gaussian widths of the arrival time distributions as a function of the application time. . . 41 3.7 Number of detected ions (a), center of arrival times (b) and Gaussian widths

of the arrival times (c) as a function of the application time of the voltages on the frontal and lateral electrodes. The four sets of data correspond to four different positions of the MOT along the x axis, which is the axis of the science cell. The relative displacements are reported in the legend. . . 42 3.8 (a) Number of detected ions (red squares, plotted on the left) and center of

arrival times distribution (blue circles, plotted on the right) as a function of the switching time. (b) Number of detected ions (red squares, plotted on the left) and Gaussian widths (green diamonds, plotted on the right) of the arrival times distribution as a function of the switching time. (c) and (d) are the same plots as (a) and (b) but for the measurements in which we apply a compensation electric field during the entire duration of the flight of the ions. 43 3.9 Calibration of the arrival times with the positions along the three main axis

of the setup. Blue squares refers to x direction, red circles to z direction and green diamond to y direction [48]. . . 45 3.10 Map of the arrival times as a function of the positions on the xz plane. The

black dashed lines represents the isochronous planes. The color scale on the left represents the value of the center of the arrival time distribution, expressed in µs [93]. . . 46

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LIST OF FIGURES LIST OF FIGURES 3.11 Representation of the geometry of the system used in the simulation. Only

the relevant parts involved in the detection process have been included, that are the quartz cell, composed by the parallelepipedal part and the cylindrical part, the frontal, lateral and rectangular electrodes, the detector, the two multiturn coils for the magnetic quadrupole, and finally the first metallic flange that connects the quartz cell to the rest of the vacuum system. . . 47 3.12 Three dimensional plot of the magnetic field generated by the pair of coils in

anti-Helmholtz configuration. The plot is composed by an arrow volume plot (black arrows) of the magnetic field vector B and a multislice plot (colored disks) of the magnitude of the magnetic field, |B|, where the reddest points correspond to regions where the magnetic field is more intense. . . 48 3.13 Plot of the magnetic field along the z directions as a function of the z

coor-dinate (a) and the gradient of the magnetic field along the z directions as a function of the z coordinate (b). . . 49 3.14 Plot of the isosurfaces of the electric potentials in the system at t = 0 µs

(a) and t = 20 µs (b). The red surfaces corresponds to higher values of the electrostatic potentials, and blue surfaces to the lowest ones. The two plots are not in the same scale of potentials, and the colour scales are normalized to the maximum potential surface for each configuration. . . 50 3.15 Trajectories of the Rubidium ions inside the apparatus for different times,

under the action of the quadrupole magnetic and electric fields for the detec-tion. The simulations corresponds to t = 0 µs (a), t = 6 µs (b), t = 12 µs (c), t = 20 µs (d). it can be clearly seen that part of the trajectories ends inside the opening of the detector. . . 50 3.16 (a) Plot of the number of atoms in the MOT as a function of time. Each

curve corresponds to a different duration of the photoionization pulse. (b) Plot of the number of ions produced via photoionization and detected on the CEM using the standard detection process. . . 51 3.17 Plot of the detection efficiency of our setup as a function of the duration of

the photoionization pulse. . . 53 3.18 Plot of the loss rate due to the photoionization of atoms in the 5P state.

Green squares refer to the loss rate calculated from the loading of the MOT, the blue squares refer to the loss rate calculated from the direct detection on the CEM, while the red squares refer to the calculated loss rates. . . 54 4.1 Schematic representation of the combined potential obtained from the sum

of Coulomb potential, produced by the atomic nucleus and screened by the electrons of the filled electronic shells, and linear potential due to the presence of a uniform and static electric field on the atom. . . 56 4.2 Typical graph of the experiment on the ionization. In the graph is plotted

the mean number of Rydberg excitation as a function of the total applied voltage between the frontal and lateral plates. The detection settings have been optimized for each value of the applied voltage. The graph refers to the measurement for the 70S Rydberg state. . . 60 4.3 Graph of the experimental data on the ionization threshold of Rydberg atoms.

The red squares are the experimental data, while the green dashed line is the theoretical value as calculated in section 4.1.1. The red dashed line between the points is only to guide the eye. . . 60

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LIST OF FIGURES LIST OF FIGURES 4.4 Plot of the ramp of the ionization electric fields as a function of time, for the

first part of the ionization pulse. The green curve corresponds to the case without the RC circuits, for which the electric fields are switched on faster. The blue curve refers to the case with the RC circuits and thus a slower ramp. Both curves refer to a total applied voltage between the frontal and lateral electrodes of 4500 V. In the graph are also reported the calculated ionization thresholds for several Rydberg states of Rubidium, for comparison. . . 61 4.5 (a) Arrival time histogram in the case of Rydberg excitation with the MOT

al-ways on. The first peak in the histogram refers to the rubidium ions obtained by direct photoionization, while the second refers to Rubidium ions deriving from the field ionization of Rydberg atoms. (b) Arrival time histograms for three different Rydberg states, the 78S, the 80S and the 82S, using the same settings for the detection. . . 62 4.6 Calibration of the arrival time distribution centers as a function of the

exci-tation moment. Since different moments of the exciexci-tation correspond to dif-ferent values of the ionization electric field, ions produced by photoionization at different moments will follow the same trajectories of the Rydberg atoms which would be field ionized at the corresponding value of the ionization field. 63 4.7 Calibration of the arrival time distribution centers as a function of the

Ryd-berg level. This plot has been obtained from the data reported in figure 4.6 and translating each ionization moment into the corresponding Rydberg level whose ionization threshold is precisely the value of the ionization electric field in that moment. . . 64 4.8 (a) Plot of the number of detected ions as a function of the duration of the

ionization pulse, for different starting moments of the photoionization, as de-scribed in the inset. Varying the duration of the ionization pulse it is possible to recover the maximum observed value of the detection efficiency for different excitation start moments. However, this value is different for different exci-tation start moments, i.e, for different Rydberg levels. (b) Maximum number of detected ions for the different excitation start moments, normalized to the higher between the observed values. This plot shows that the maximum de-tection efficiency can be recovered by adjusting the duration of the ionization pulse. . . 65 4.9 (a) Plot of the mean number of detected ions as a function of the excitation

start, for different values of the duration of the electric field pulse, as described in the inset. (b) Plot of the centers of the arrival time distributions as a function of the excitation start moment, for different values of the duration of the electric fields. . . 66 5.1 Example of data taken during a frequency scan around the Rydberg resonance

with the MOT loss technique. In the graph is reported the stationary number of atoms in the MOT as a function of the frequency detuning. . . 73 5.2 Stationary number of atoms in the MOT as a function of the frequency.

Red points refer to the case in which no electric field is applied during the excitation of Rydberg atoms, while the green points refer to the case in which an electric field is applied on the atoms during the excitation process by setting V_{f ront}= 50 V and Vlat = 50 V on the electrodes around the cell. . . 74

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LIST OF FIGURES LIST OF FIGURES 5.3 (a) Stationary number of atoms in the MOT as a function of the frequency.

Red points refer to the dip without the application of any electric fields, while green points refer to the application of a pulse of the electric fields with duration τ_{f ield} = 1 µs, applied τwait = 1 µs after the end of the excitation

process. (b) Stationary number of atoms in the MOT as a function of the applied electric field. . . 75 5.4 (a) Plot of the measured Stark shifts as a function of the applied potential for

the 70S Rydberg state. The black dashed line is a fit of the experimental data with a quadratic function. The error bars are smaller than the symbols of the data. (b) Plot of the widths of the frequency dip as a function of the applied voltage. The red dashed line is only a line that connects the experimental points. . . 77 5.5 Stark map for the 70S Rydberg state, calculated with the software ARC

[124, 125]. . . 78 5.6 (a) Schematic representation of the cell and of the electric plates with the

quantities used in the calculations. . . 78 5.7 Calculated electric field along the centre axis of the cell for different divisions

of the potential between the plates but the same difference of potential. . . . 80 5.8 Plot of the value of the electric field inside the cell. The dashed line refers

to the theoretical calculated values, while the diamonds refers to the experi-mental data. . . 80 5.9 Simulation of the electric field along the main axis of the cell with COMSOL

Multiphysics, setting a symmetric configuration of the applied potentials on the electrodes Vf ront = +20 V and Vlat = −20 V, and considering also the

dielectric screening of the quartz cell. At the position of the MOT, that is around x = 33.5 µm, we have a value of the electric field around E_simu= 56_cmV. 82 5.10 (a) Stark shift as a function of the total application time of the electric field.

Since the repetition rate of the experimental cycle is 500 Hz, an application time of t_appl = 2 ms corresponds to continuous application. (b) Stationary number of atoms as a function of time. By applying the electric fields in a continuous way, a static electric field is created in the cell which takes the atoms out of resonance. . . 83 5.11 (a) Number of detected Rydberg atoms as a function of the detuning from

the principal peak, in MHz. Due to the presence of the background elec-tric field, the target nS Rydberg state and the manifold of 82f and higher angular momentum states mix, and the new eigenstates of the system are superpositions of the zero field states of Rubidium. During excitation, we observe a series of peak, the height of which depends on the coupling of the ground state with the particular Stark state considered. (b) Stark map of the 85S Rydberg state, calculated using the software ARC. Since the value of the background electric field is higher than the Inglis-teller limit, we observe a crossing between the 85S state with the manifold of 82f . By counting the number of peaks and comparing with the previous measurement, we obtain an estimate of the background electric field. . . 85

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LIST OF FIGURES LIST OF FIGURES 5.12 Plot of the centers of the arrival times distributions of the ions as a function

of the waiting time. Using the calibration of the arrival times presented in chapter 3, we are able to obtain the background electric field in the cell. In fact, the center of the arrival times distribution moves in time due to the acceleration of the ions caused by the presence of the background electric field. 86 5.13 (a) Stark shift as a function of the applied potential on the frontal electrodes.

The value of the applied potential corresponding to the vertex of the parabola represents the optimal value for the compensation of the background electric field. Note that the Stark shift depends on the magnitude of the electric field vector, while the compensation acts only in one direction. (b) Screenshot of the oscilloscope showing the main pulses involved in the measurement: the blue pulse controls the MOT beams, and it goes to zero at the beginning of the experimental cycle; the pink pulse represents the Rydberg excitations, which in this scale is a short spike in the second division of the grid of the oscilloscope; the green pulse is the value of the electric measured directly on the frontal plates. Before the beginning of the measurement, a negative potential is applied to the frontal plates, so that the background electric field is compensated during Rydberg excitation. After the excitation, field ionization occurs, and it is represented by the short positive spike at the beginning of the third division of the oscilloscope scale. The negative potential lasts for a small fraction of the experimental cycle in order to avoid screening effects as we observed in the previous section. . . 88

6.1 (a) Graph of the transition rates from Rydberg state 30S due to spontaneous emission, in red, and to blackbody radiation, in green. These rates refer to Rybidium at 300 K [43]. (b) Schematic representation of the migration of atoms from an initially excited Rydberg states to the neighbouring ones due to the presence of blackbody radiation. . . 92 6.2 Figure (a) represents the mean number of Rydberg excitations < N > as a

function of excitation time texc, represented by red diamonds, as well as the

Mandel Q parameter, grey circles. The distinction between the non interact-ing and interactinteract-ing regime is determined by the change in the slope of < N > as a function of texc, and confirmed by the Q value approaching −1, meaning

that the statistics of the excitation is becoming sub-Poissonian, as expected in the interaction regime. Figure (b) represents a scheme of the quasi-one dimensional system in which some Rydberg excitations, represented by red dots, are created. The red disks around each excitation represent the blockade volume, inside which the excitation of other atoms is strongly suppressed. In this case, we expect to see only one deexcitation resonance at ∆ = 0 MHz. Figure (c) represents a similar situation but Rydberg atoms are excited with off resonant radiation through the facilitation mechanism. The produced sys-tem is represented by a quasi one-dimensional chain of Rydberg excitations at distance rf ac, and we expect to see two deexcitation resonances at ∆ = ∆exc

and ∆ = 2∆_exc, depending on whether the Rydberg atoms have one or two neighbours respectively. [39] . . . 95

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LIST OF FIGURES LIST OF FIGURES 6.3 deexcitation process following resonant excitation for different < Ninit >, in

particular 25 (blue circles), 34 (green triangles) and 50 (white squares). In (a) is plotted the remaining fraction as a function of the detuning ∆, and tdark = 0.5 µs, tdeex = 2 µs. By increasing the number of initial excitations,

the remaining fraction of Rydberg atoms is higher at resonance due to the presence of the interactions. In (b) is plotted the remaining fraction of Ry-dberg atoms as a function of < Ninit >, where the gradual crossing between

non interacting to interacting regime can be observed. In (c) is plotted the dynamics of the deexcitation process. The continuous lines are numerical simulations [39]. . . 96 6.4 Deexcitation process following off-resonant excitation in the facilitation regime.

The figure shows the plot of the remaining number of Rydberg excitations with < N_init >≈ 20 and a detuning ∆ex = 2π × 16 MHz. Blue circles

corre-spond to a t_dark = 0.5 µs, while green diamonds refers to tdark = 5 µs. The

data set represented by the blue circles shows three peaks, one at resonance, one at ∆ = ∆_ex and ∆ = 2 × ∆_ex, corresponding to the deexcitation of Ry-dberg atoms without neighbours, only one neighbour, or two neighbours in the quasi-one dimensional chain respectively. On the other hand, the data set represented by the green diamonds shows only one peak at ∆ = 0 MHz, since after some time the Rydberg cloud expands and reciprocal interactions play a minor role. The continuous lines are numerical simulations [39]. . . 97 6.5 Schematic representation of the measurement of the lifetime of Rydberg atoms.

The measurement is composed of two parts. In the first part (measurement 1 ) few atoms are initially excited to the target Rydberg state in the atomic cloud; after a variable amount of time t_dark, during which a redistribution of the Rydberg population and radiative decay occur, a deexcitation pulse is applied to empty the target state, and the remaining Rydberg atoms are field ionized. In the second part (measurement 2 ), the experimental cycle is similar but we do not apply the deexcitation pulse, and field ionize all the Rydberg atoms in the system instead. . . 99 6.6 Typical graph of the measurement described above, for the 70S Rydberg state.

The red squares are the total number of Rydberg atoms in the system, the blue circles are only the atoms in the support states, and the green diamonds are the atoms in the target state. By fitting with an exponential function the green diamonds we obtain the target state lifetime, while by fitting the red squares we obtain the ensemble lifetime. . . 100 6.7 Plot of the lifetimes measured for several nS Rydberg state of Rubidium.

The green solid diamonds refer to the measured target state lifetimes when a compensation voltage is applied in order to cancel out the background electric field in the cell during the entire duration of the experiment, while the green empty diamonds refer to measurements without any compensation of the background field. The blue solid diamonds represent the calculated values of the target state lifetime @300K, considering blackbody induced population transfer to other Rydberg states and also the transfer back to the initial state [49]. . . 102

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LIST OF FIGURES LIST OF FIGURES 6.8 Plot of the measured ensemble lifetime for several nS Rydberg state of

Rubid-ium, together with the theoretical prediction for a sample @ 300K. The red solid circles represent the ensemble lifetime measured using a compensation potential to erase the background electric field during the entire duration of the measurement, while the orange empty circles refer to measurements without any compensation of the background electric field. The blue circles represent the value calculated from the theory, considering the blackbody induced population transfer between Rydberg states [49]. . . 103 6.9 Plot of the ensemble lifetime of nS Rydberg states at T = 300 K (red squares)

along with the calculated values of the lifetime of the same states at T = 0 K (black circles). The values of the lifetimes at T = 0 K are taken from [43]. . . 104 6.10 Plot of the deexcitation efficiency at the shortest waiting time. The

deexci-tation process is still effective when the measured target state lifetime starts to deviate from the theoretical value. For higher principal quantum number, above around n = 90, also the deexcitation efficiency starts to drop. . . 105 6.11 Plot of the target state lifetime (green diamonds), in this case the 85S_1/2and

the corresponding ensemble lifetime (red circles) for different values of the electric field acting on the atoms. The compensation potential is applied on the frontal electrodes only and lasts for the entire duration of the measure-ment. The dashed line represents the Ingli-Teller limit for the 85S_1/2, which is around 125 mV/cm. . . 107 6.12 Target state lifetime and ensemble lifetime of 72S Rydberg state as a function

of the initial number of excitations in the system. Red circles represents the measured value of the ensemble lifetime, while the green diamonds represents the target state lifetime. . . 108 6.13 Rydberg excitation dynamics to the state 80S. The excitation is divided in

two parts, a first excitation that lasts texc1 = 0.2 µs in (a) and texc1 = 1 µs

in (b), while the rest of the excitation is made through a second pulse after a variable amount of time, as shown in the inset. Different curves refer to different dark times between the two excitation pulses. In the horizontal axis is reported the total excitation time texc1+ texc2. . . 110

6.14 Total number of Rydberg atoms in the system as a function of time. The initially excited state in this measurement is 85S. In this graph, it is possible to observe a first part of the excitation which is much faster than the rest of the decay, with a lifetime of the order of tens of µs, while the final part of the curve corresponds to a lifetime of the order of one ms. We did not use any compensation of the background electric field in this measurement, and we observe this behaviour only for around 10 excitation or more and only for the states whose measured lifetime is shorter than the predicted value, due to the presence of the background field. . . 110 6.15 (a) Number of Rydberg excitations in the system as a function of time. The

initial excited state is 85S, and the data are taken until 8 ms, that is much longer than the target state and ensemble lifetimes of the state under investi-gation. (b) Local lifetime as a function of the waiting time. The local lifetime has been obtained by a line fit of around 5 experimental points in the log-lin graph of data reported in (a), for each time interval considered. . . 111 6.16 Histograms of the arrival times for different times, with initial state 85S. The

detection settings have been optimized for the 85S Rydberg state. . . 112 16

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LIST OF FIGURES LIST OF FIGURES 6.17 Widths of the arrival times distributions for 65S (a) and 85S (b) initial

Ry-dberg state respectively, as a function of time. . . 113 6.18 Early calculation of the blackbody spectrum when the finite dimensions of

the cell are taken into account. The red dots represent the number of elec-tromagnetic modes at each frequency, while the continuous blue line is the best fit of the red dots to a parabola. The range of frequencies reported in this graph is that of the typical transitions between Rydberg states in the range of principal quantum number we are investigating. This calculation shows the difference from the spectrum obtained with the Plank formula, and in particular we observe that some frequencies are suppressed, while other frequencies are enhanced by the geometric constraints of the setup [162]. . . . 115 A.1 (a) Plot of the widths of the Rydberg resonance lines as a function of the

applied electric field. The widths are obtained through Gaussian fits. (b) Schematic representation of the quadratic Stark broadening, taken from [114]. 123 B.1 Time dependence of the population of Rydberg states when 70S state is

initially excited [151]. . . 126

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LIST OF FIGURES LIST OF FIGURES

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Introduction

In this thesis I report the main results of the studies I carried out during my three year PhD course. My work concerns the experimental study of many body physics using ultracold atoms. The complexity of many body systems relies in the large number of interacting components, which prevent in the majority of situations to obtain analytical solutions, and adequate approximations must be applied in order to solve this kind of problems. One different approach to tackle these problems is using a simulator, which is a system that behaves similarly to the investigated one, but over which we have a large level of control and have access to a larger ensemble of parameters [1, 2, 3, 4]. The simulator reproduces the complete quantum dynamics of the system under investigation, since it is described by the same hamiltonian, and we can read the output of the system as we change each parameter. For this purpose, ultracold atoms represent a suitable framework for quantum simulators, due to the high level of control over both the internal end external degrees of freedom of the atoms, and the possibility to have tunable interactions, allowing to span a large range of regimes, from the nearly non interacting to the strongly interacting one.

The variety of systems that can be simulated depends on the peculiar characteristics of the simulator. In order to produce long-range interactions between the atoms, there are several possibilities such as dipolar forces, both magnetic ones between atoms with large magnetic momentum [5, 6], or electric ones between polar molecules [7, 8, 9], or tuning the interaction by changing the scattering length using Feshback resonances [10, 11]. Another important approach, that is the one I used in the experiments reported in this thesis, is to use atoms excited to high-lying states, called Rydberg atoms, which thanks to their large electric polarizabilities, allow long-range van der Waals interactions between excited atoms [12, 13, 14, 15].

The simulator represents a valuable tool of investigation of a large variety of physical phenomena, and many systems have already been simulated, such as superfluid phases, crystalline-like phases described by the hamiltonian of the Hubbard model [16], quantum phase transitions such as BEC to BCS [17], superfluid to Mott insulators [18, 19] and many others. Quantum simulators could be used also to unveil the working principles of high-temperature superconductors [20], which still remains unexplained by the accepted theory of standard superconductivity. Another particular phenomenon is the absorbing state phase transition [21] in open many body quantum systems, that belongs to one of the simplest universality class of non-equilibrium phase transitions. In such open systems, dissipation is a key feature necessary to simulate the dynamics, and the possibility to control it opens up to a large number of applications.

When simulating this kind of system with Rydberg atoms, we must consider the spon-taneous decay as well as the migration to neighbouring Rydberg states due to blackbody induced transitions as part of the total dissipation of the system. The major contribution of my work is the characterization of the complex dynamics between Rydberg states during

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ii CHAPTER 0. INTRODUCTION the evolution of systems in the dissipative regime, which is of paramount importance for every quantum simulator based on Rydberg atoms.

In the following, I present a general introduction of my work by contextualizing it in the framework of many body physics and quantum simulators. I make a general overview of the main experiments I contributed to, specifying my original contribution. The measurements reported in this thesis have been performed in the laboratory of cold atoms of the Department of Physics of the University of Pisa, Italy.

The many body problem

The common approach of the scientific method to study a physical system is by analyzing it, in the sense of its ancient greek root that means breaking up, from ana-, "throughout" and lysis, "a loosening" [22]. A complex system composed of several elements is thus studied by dividing it into single components, which have well-defined properties independently from the rest of the system, and then defining the interactions between them. This analysis can be iterated many times, and at each step the fundamental particles and forces become the bound states and the effective interactions of a more fundamental theory.

However even knowing the underlying theory, this is not sufficient to solve every problem in nature, especially those pertaining to systems composed by many interacting elements. The difficulty in describing these systems does not rely on the computational complexity of deriving their properties from the properties of their components but is a more fundamental difference. In fact, as stated by P.W. Anderson, More is different [23], in the sense that systems composed by a large number of components are not only quantitatively, but also qualitatively different from a simple ensemble of their parts, and many properties of a system as a whole cannot be easily deduced from the underlying physics of its components. The real challenge is thus to explain the behaviour of complex systems starting from a simple underlying theory, and this is commonly referred to as many-body problem [24].

The peculiarity of many body systems is the appearance of emergent phenomena [25], which are completely new properties of a system resulting from the collective behaviour of its constituents, but that cannot be derived from the properties of its single elements. The complexity of a system is also tamed with the concept of universality, according to which the properties of a complex system result from the correlations between its components rather than from the specific details of the microscopic interactions.

Finding a solution for many body problems is highly non-trivial. Even if the microscopic dynamics is known, and even if the only interactions we consider are two body interactions, in order to have a complete description of the dynamics of the system we need to take into account the quantum correlations arising from the repeated interactions between the particles. Moreover, owing to the large number of particles involved, which is of the order of the Avogadro number for many realistic systems, it is often impractical or even impossible to carry out analytical calculations. In fact, the Hilbert space necessary to describe the complete quantum dynamics of a system of particles scales exponentially with the number of components [26], and even considering relatively small systems, composed by hundreds of particles, would require an unrealistic amount of computational power.

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QUANTUM SIMULATION WITH ULTRACOLD ATOMS iii

Quantum simulation with ultracold atoms

A completely different approach was proposed by R.P. Feynman in 1982, who stated "Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy" [27]. If it is necessary to simulate a system that is governed by the laws of quantum mechanics, we could use another system, that intrinsically behaves quantum mechanically, but over which we have a high degree of control. By creating a quantum simulator, we can reproduce the physics of the system under investigation and study the response of the system by looking at its output state starting from different initial conditions. The high level of control allows one to explore a wide region of the parameter space by changing the parameters on the simulator, and since during the simulation the exact dynamics is taken into account, this could allow one to obtain new information beyond that obtained with measurements on the system itself or with numerical calculations.

Among the large variety of systems that have been proposed to implement such a quan-tum simulator, a remarkable success was achieved by ultracold atoms. In fact, using laser sources it is possible to control both the internal degrees of freedom, by selectively exciting a single atomic level at a time, and external, by applying mechanical forces to the atoms that allow to cool and trap them, or finely move them with optical tweezers or using optical lattices to reproduce periodic structures. In order to implement the tunable long-range in-teractions to simulate the physics of many-body systems there are several possibilities such as magnetic dipole forces between atoms with large ground state magnetic dipole momen-tum, electric dipole forces between polar molecules, Coulomb interactions between trapped ions [28, 29], and van der Waals interactions between Rydberg excited atoms.

Using Rydberg atoms it is possible to have long-range interactions in an ultracold atomic sample. In the non-dissipative regime, where the dynamics of the Rydberg atoms are studied at times much shorter than their lifetime, the excitation dynamics is influenced by two important mechanisms, namely the Rydberg blockade [30, 31, 32, 33] and the facilitation mechanism [34, 35, 36]. Once an atom is excited to a Rydberg state, the energy levels of the neighbouring atoms are shifted due to the van der Vaals interaction, which prevents the excitation of other atoms inside a finite volume around the initially excited Rydberg atom. This mechanism is known as Rydberg blockade, and its effect is to slow down the excitation dynamics of Rydberg atoms when the mean distance between the atoms becomes shorter than the blockade radius, that is the distance below which the interactions shift the energy level more than a linewidth of the excitation laser.

Using off resonant radiation it is still possible to excite atoms to Rydberg states, but with a lower rate due to the detuning. On the other hand, if an atom is already excited, by using off resonant radiation is possible to excite an atom nearby if the detuning of the excitation laser compensates the shift in energy due to the interaction at that distance. This process is called facilitation mechanism, and once the detuning of the excitation radiation is set, only atoms inside a spherical shell around the Rydberg atoms can be excited as well. Moreover, the mechanism of facilitation occurs as an avalanche-like process, since after the first facilitation, the last excited atom can facilitate an excitation with another atom, and so on.

Several experiments on the non-dissipative dynamics of Rydberg atoms have been per-formed in our laboratory and were reported in the PhD thesis of M. M. Valado [37]. I report some results on these mechanisms in chapter 1 of my thesis since they are functional to the understanding of my work.

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iv CHAPTER 0. INTRODUCTION

Simulating absorbing phase state transitions with Rydberg atoms

Rydberg atoms are also suitable for simulating non-equilibrium physics by considering the regime in which the timescale of the experiments is comparable or longer than the lifetime of Rydberg atoms. In this case, the dissipation consists in any process that depletes the population of the initial Rydberg state: the spontaneous decay towards ground state and low excited levels, and the blackbody induced migration, the rate of which becomes of the same order of magnitude as that of the spontaneous decay for high-lying Rydberg states [38].

The dissipation process can be enhanced by selectively de-exciting initially excited Ry-dberg atoms to the ground state [39]. However, the de-excitation process, as well as the excitation process, is extremely sensitive to the interactions between Rydberg atoms, which can shift the energy levels out of the resonance with the de-excitation radiation. I con-tributed to the experimental study of the de-excitation dynamics of Rydberg atoms during my PhD thesis, and the description of the technique and some important experimental re-sults are briefly reported in chapter 6. The study of the de-excitation technique is part of Cristiano Simonelli’s PhD thesis [40].

The introduction of a dissipation mechanism allows one to simulate a large variety of systems out of equilibrium. One of the simplest universality classes of non-equilibrium systems experiencing phase transitions is the absorbing state phase transition, which can be employed to explain a number of phenomena in nature, such as the spreading of a disease in a population of individuals or the evolution of wildfires in forests [41]. In the case of the epidemic spreading, if the system is initially supplied with an infected person, the "patient zero", the evolution of the system relies on the competition between two mechanisms, namely infection and healing. In the infection mechanism, an infected person could infect a healthy person, while the healing process consists in the spontaneous healing of a previously infected person. Depending on the ratio between these two rates, the system can end up in a state without any infected person, which we call absorbing state since the disease cannot spread anymore, or in an active state, in which people are continuously infected and heal spontaneously. The crossing between these two states is described by a phase transition, and the order parameter is represented by the mean number of infected people.

This process has been simulated in another experiment to which I contributed during my PhD, and is the main topic of Cristiano Simonelli’s PhD Thesis [40]. In this experiment [42], we simulated the infected people with Rydberg excited atoms and the healthy ones with ground state atoms. The infection mechanism is simulated through the facilitated excitation of Rydberg atoms, using an off resonant driving radiation during the dynamics, while spontaneous healing is represented by spontaneous radiative decay and blackbody induced transitions to neighbouring Rydberg states. Finally, the patient zero is supplied to the system through a short resonant Rydberg excitation.

The results are reported in figure 1(a). Here are reported the mean number of Rydberg excitations in the system at a time much longer than the lifetime of the considered Rydberg state, as a function of the Rabi frequency of the driving radiation. When the infection rate is lower than the healing rate, the mean number of excitations is compatible with zero, and it corresponds to an absorbing state. As the infection rate is increased, the system contains a finite number of Rydberg excitations at the end of the experiment, and this represents the active state. Figure 1(b) shows a graph of the variance of the number of Rydberg excitations as a function of the Rabi frequency of the driving radiation, and shows a peak in correspondence of the critical point. From the experiment, we also obtained a value of the

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SIMULATING ABSORBING PHASE STATE TRANSITIONS WITH RYDBERG

ATOMS v

critical exponent in agreement with the theoretically predicted value for one dimensional directed percolation, which is the simplest phenomenon in the universality class of systems exhibiting absorbing phase state transitions.

Figure 1: (a) The figure shows the plot of the mean number of Rydberg excitations in the system at a time much longer (t = 1500 µs) than the lifetime of the Rydberg state (in this case, 70S, whose lifetime at T = 300 µs is τ70S ≈ 150 µs), as a function of the Rabi frequency

of the off resonant driving radiation. The inset shows the log-log plot of the mean number and the fit necessary to obtain the experimental value of the critical exponent. (b) The figure shows the plot of the variance of the number of Rydberg excitations as a function of the Rabi frequency of the off resonant radiation, which represents the fluctuations of the number of excitations [42].

In the dissipative regime, a fundamental role is played by dissipation which, as stated above, must account for any process that depletes the initially excited Rydberg state. When dealing with high-lying Rydberg states, the blackbody induced migrations can have even higher rates with respect to spontaneous decay to lower states, and once several Rydberg states are populated from the initial one, the dynamics of the entire ensemble of Rydberg atoms can become difficult to describe [43]. Moreover, the lifetime of the total ensemble of Rydberg atoms is much longer than the lifetime of the initially populated state, due mainly to the migration to long living high angular momentum states, and this could change drastically the dynamics of the system.

My original contribution consisted in the study of this dissipation process in experi-ments involving Rydberg atoms, on one hand by introducing a novel technique to measure the lifetime of high-lying Rydberg state, where other techniques are not adequate, and on the other hand by characterizing the migration to other Rydberg states due to blackbody radiation. The measurement of Rydberg states lifetimes is often obtained through a time-resolved selective state field ionization [44, 45, 46, 47], which consists in applying a slowly rising ramp of the electric field that subsequently ionizes different Rydberg states which can be distinguished and identified at the detector. However, for high-lying Rydberg states this technique is difficult to apply since the ionization thresholds of neighbouring Rydberg states are very close to each other. We developed a new technique based on the selective de-excitation of Rydberg states. By de-exciting the initially excited Rydberg atoms at dif-ferent points of the evolution of the system, we can distinguish the part of the total Rydberg population that remains in the initial, or target state, with respect to all the other populated

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vi CHAPTER 0. INTRODUCTION states. Moreover, through this measurement we can retrieve information on the lifetime of the ensemble of Rydberg atoms.

In the first two chapters of my thesis, I present a general theoretical introduction to the physics of Rydberg atoms (chapter 1) and to our experimental apparatus and the most important experimental techniques (chapter 2). My original contributions are reported in chapters 3-6. Part of my work was to fully characterize the detection system in our appara-tus. In fact, even if the same setup has been characterized in the past and used for many years for different experiments, the necessity of exploring a larger range of Rydberg states with an optimized detection efficiency for every state required a new, more detailed, characteri-zation of the apparatus. Chapter 3 is dedicated to the description and the charactericharacteri-zation of the detection system of our apparatus, while chapter 4 deals with the optimization of the detection for Rydberg atoms. In chapter 5 I characterize the behaviour of Rydberg atoms under the action of electric fields and I present a method to compensate the stray electric field in the setup, which could otherwise affect other measurements. The main results of my thesis are reported in chapter 6, which is dedicated to the study of the lifetime of high-lying Rydberg states through the new technique based on the selective de-excitation and the study of blackbody induced transitions between Rydberg states.

The results obtained during my PhD have led to the following scientific publications: • Van der Waals explosion of cold Rydberg clusters R. Faoro, C. Simonelli, M. Archimi,

G. Masella, M. M. Valado, E. Arimondo, R. Mannella, D. Ciampini, O. Morsch, Phys. Rev. A 93, 030701(R) (2016) [48];

• De-excitation spectroscopy of strongly interacting Rydberg gases C. Simonelli, M. Archimi, L. Asteria, D. Capecchi, G. Masella, E. Arimondo, D. Ciampini, O. Morsch, Phys. Rev. A 96, 043411 (2017) [39];

• Experimental signatures of an absorbing-state phase transition in an open driven many-body quantum system Ricardo Gutierrez, Cristiano Simonelli, Matteo Archimi, Francesco Castellucci, Ennio Arimondo, Donatella Ciampini, Matteo Marcuzzi, Igor Lesanovsky, Oliver Morsch, Phys. Rev. A 96, 041602(R), (2017) [42].

Furthermore, the main results of my thesis, which are reported in chapter 6, will be divided in two publications, which are currently in preparation:

• Measurement of the target state lifetime and ensemble lifetime of high lying Rydberg states in cold atomic system M. Archimi, C. Simonelli, L. Di Virgilio, A. Greco, M. Ceccanti, I. I. Beterov, I. I. Ryabtsev, E. Arimondo, D. Ciampini, O. Morsch, in preparation (2019) [49];

• Direct measurement of the deviation of the blackbody spectrum due to geometry using Rydberg atoms M. Archimi, C. Simonelli, L. Di Virgilio, A. Greco, M. Ceccanti, L. Schachter, E. Arimondo, D. Ciampini, O. Morsch, in preparation (2019) [50];

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Chapter 1

Rydberg atoms

1.1 Introduction to Rydberg atoms

When an atom is excited to a quantum state with a high principal quantum number n, it is called a Rydberg atom [38, 51]. These atoms are particularly interesting because they have exaggerated properties with respect to their ground state counterparts, and many effects that are completely negligible for the latter ones, become extremely important for Rydberg atoms. To be considered in this class, an atom has to be excited to a level usually equal or greater than about n=10, but far higher levels can be achieved, the only limitation being the vicinity of the levels to the continuum that limits their practical usefulness. Although Rydberg atoms were already known and theoretically predicted at the birth of atomic physics [52, 53, 54], a systematic study could not take place before the invention of the laser, which allowed to access specific Rydberg states, and the development of spectroscopy techniques through which it was possible to investigate their properties [55].

Other than being an interesting field of study in their own right, these properties allow a high level of control on the atoms and open the path to a number of important applications, from the research on fundamental physics to technological applications. In particular, while the interaction between ground state atoms is typically contact-like, for Rydberg atoms long range interactions arise and this makes them particularly suitable for a number of applications in cold atoms for the study of many body physics and, for example, are used for the realization of quantum simulators [12, 13, 14, 15]. Exciting and controlling Rydberg atoms also makes it possible to study the interaction between atoms and electromagnetic fields, both in the weak field and strong field regimes [59]. Rydberg atoms have also been proposed for the implementation of quantum gates and quantum memories, thanks to their suitability to realize qubits and the possibility to control them with radiation [56, 57, 58].

Rydberg atoms also appear naturally in some physical systems, such as in the radiative recombination in interstellar media, or as a step of the radiative recombination between low energy electrons and ions in plasmas [60, 61].

1.2 Wavefunction of Rydberg states

The properties of Rydberg atoms can be derived through the calculation of the Rydberg state wavefunctions. The standard approach to this calculation is that of quantum defect theory [66, 67, 68], which uses the similarities between single valence electron Rydberg state, such as alkali atoms, and hydrogen atoms for which the calculation of the wavefunction is

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2 CHAPTER 1. RYDBERG ATOMS straightforward. The first part of this section is dedicated to the derivation of the eigen-functions and eigenenergies of the hydrogen atom, which can be found in standard quantum mechanics book [62, 63, 64, 65], while the part concerning Rydberg atoms follows chapter 1 of the book Rydberg atoms by T.F.Gallagher [38]. If we consider Rydberg states of H and N a for example, we can state that the Coulomb potential felt by the Rydberg electron, i.e., the electron in the outermost shell, is similar in the two cases. In particular, when the elec-tron is in high orbital angular momentum (high l) states, it spends most of its time outside the ionic core and the Coulomb potential is similar in the two cases since the electron is sensitive to the net charge, thus we expect Rydberg state properties to be similar as well. By contrast, when the electron is in low l states it can penetrate and polarize the ionic core of N a+ and is exposed to the bare nuclear potential, so the properties of Rydberg states are different in the two cases. When the Rydberg electron penetrates the N a+ core, its binding energy is increased and the total energy decreased.

For this reasons, we start by briefly recalling the calculation of the wavefunction for H atom. The stationary Schrödinger equation of the system is:

−~ 2 2µ∇ 2₋ e2 4π0r ψ(r, θ, φ) = Eψ(r, θ, φ) where µ = meMnucl

me+Mnucl is the reduced mass of the system, me the mass of the electron and

Mnucl the mass of the nucleus. By writing the Laplacian in spherical coordinates:

∇2= 1 r2 ∂ ∂r r2 ∂ ∂r + 1 r2_{sin θ} ∂ ∂θ sin θ ∂ ∂θ + 1 r2_sin2_θ ∂ ∂2_φ2

we obtain the equation: −~ 2 2µ 1 r2 ∂ ∂r r2∂ψ ∂r + 1 r2_{sin θ} ∂ ∂θ sin θ∂ψ ∂θ + 1 r2_sin2_θ ∂2_ψ ∂φ2 − e 2 4π0r ψ = Eψ (1.1) ~2 2µ 1 r2 ∂ ∂r r2∂ψ ∂r + 1 r2_{sin θ} ∂ ∂θ sin θ∂ψ ∂θ + 1 r2_sin2_θ ∂2_ψ ∂φ2 + 2µ ~2 E + e 2 4π0r ψ = 0 (1.2) Since the potential depends only on the radial coordinate, we can express the solution as a product of a radial and an angular wavefunction, ψ(r, θ, φ) = R(r)Y (θ, φ), and ∂ψ_∂r =

∂

∂rRY = Y ∂

∂rR. Substituting the expression for ψ and multiplying by r2 RY we obtain: 1 R d dr r2dR dr +2µr 2 ~2 E + e 2 4π0r + 1 Y sin θ ∂ ∂θ sin θ∂Y ∂θ + 1 Y sin2θ ∂2Y ∂φ2 = 0 (1.3)

Since the first two terms depend only on the radial coordinates, and the second two terms depend only on angular coordinates, in order to obey this equation they must be equal to the same constant λ but with opposite signs. We thus obtain two equations:

d dr r2dR dr +2µr 2 ~2 E + e 2 4π0r R − λR = 0 (1.4) 1 sin θ ∂ ∂θ sin θ∂Y ∂θ + 1 sin2θ ∂2Y ∂φ2 + λY = 0 (1.5)

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1.2. WAVEFUNCTION OF RYDBERG STATES 3 The second equation depends only on angular coordinates and can be solved by defining Y (θ, φ) = Θ(θ)Φ(φ), which obeys the equation:

Φ sin θ d dθ sin θdΘ dθ + Θ sin2θ ∂2Φ ∂φ2 + λΘΦ = 0 (1.6) sin θ Θ d dθ sin θdΘ dθ + 1 Φ ∂2Φ ∂φ2 + λ sin 2_{θ = 0} _(1.7)

Here again we have the first and last term which depend on the angle θ and the second term that depends on φ, so the must be separately equal to a constant ζ but with opposite values:

sin θ Θ d dθ sin θdΘ dθ + λ sin2θ − ζ = 0 (1.8) 1 Φ ∂2Φ ∂φ2 + ζ = 0 (1.9)

The solution of the equation for Θ(θ) are the spherical harmonics, while the solution for Φ(φ) are plane waves. Combining the two we have the solution for the angular part of the Schrödinger equation for the hydrogen atom:

Θ(θ)Φ(φ) = Ylm(θ, φ) = (−1) (m+|m|) 2 s (2l + 1)(l − |m|) 4π(l + |m|) P |m| l (cos θ)e imφ _(1.10)

where P_l|m| are the Legendre polynomials, l an integer ≥ 0, and −l ≤ m ≤ +l integer, and λ = l(l + 1), where λ is the constant defined in the above equations. Substituting the expression for λ and rearranging the term we can now write the radial equation as:

d2R dr2 + 2 r dR dr + 2µ ~2 E + e 2 4π0r −l(l + 1) r2 R = 0 (1.11) that is a one dimensional equation in the effective potential Vef f(r) = − e

2

4π0r +

l(l+1)~2

2µr2 . A

known solution of the radial equation is the one associated with the Laguerre polynomials Ll_n, with which we can write the solution:

Rn,l = Cn,lρle−

ρ 2L2l+1

n+l (ρ) (1.12)

where we have defined the new variable ρ = _nr2r

B and where rB =

(4π0)~2

µe2 is the Bohr radius.

The normalization constant can be written as: Cn,l = − 2 n2r −3/2 B s (n − l − 1)! ((n + l)!)3 (1.13)

The complete solution for the wavefunction of the hydrogen atom can be written thus as ψ(r, θ, φ) = Rn,l(r)Yl,m(θ, φ), with n, l, and m integer numbers, 0 ≤ l ≤ n and −l ≤ m ≤ +l,

and the eigenenergies associated with the eigenstates are given by: En= − µe4 2(4π0)2~2n2 = − e2 2(4π0)rB 1 n2. (1.14)

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4 CHAPTER 1. RYDBERG ATOMS The above treatment can be exploited to find the wavefunction of single valence electron atoms outside the ionic core. We can use a simplified hamiltonian for these systems that are similar to the hydrogen one but with a potential Vcore(r). Since there still is the spherical

symmetry, the orbital part of the wavefunction will be equal to that of the hydrogen atom. As regards the radial part, while for large orbital radii r > r0, the radius of the ionic core,

the potential of the shielded atomic core and that of the hydrogen atom are equal, for shorter orbital radii the electron can penetrate the core and feel the unshielded electrostatic potential, and we expect the wavefunction to be different. The radial wavefunction in the region of interest, that is for r > r0, can be described by the same radial wavefunction for the

hydrogen atom, but with a phase shift ϕ due to the different core potential. The magnitude of the shift can be evaluated with the difference of the momentum of the electron with energy E in the case of the hydrogen and that of a single valence electron atom, such as alkali atoms. Remembering that an electron with energy E has linear momentum p =p2m(E − V ) in the non-relativistic regime, we have:

ϕ =p2µ Z r0 0 (E − Vcore(r))1/2− (E + e2 4π0r )1/2 dr (1.15) We can define a new principal quantum number for the single valence electron case, which differs from the integer values of the hydrogen by an amount ϕ_π = δl, where δl is called

quantum defect [66, 67]. To express the shifted radial wavefunction we can use the regular and irregular Coulomb functions f and g. These two functions are two independent solutions of the second order differential equation for radial coordinates expressed above, and in the classical allowed regions are two oscillatory functions shifted by π/2, while in the classically forbidden regions they are increasing and decreasing exponentials. A complete expression of these function can be found in [68, 69]. Combining the shifted radial wavefunction with the orbital wavefunction found for the hydrogen atom, we find an expression for the bound state of a single valence electron atom:

ψ(r, θ, φ) = Yl,m(θ, φ) (f (E, l, r) cos πδl− g(E, l, r) sin πδl)

r (1.16)

We note that the radial wavefunction is not normalized. To estimate the normalization of the radial wavefunction for bound states E < 0, we take the semiclassical WKB radial wavefunction, which is a good approximation in the classically allowed region:

R(r) = N sin Rr

rikdr 0

r√k (1.17) where we define ri and ro the inner and outer classical turning points of the electron orbit

respectively, and k = s 2 E + e 2 4π0r − ~2l(l + 1) 2µr2 (1.18) that is the square of twice the kinetic energy of the electron in the classical allowed region. Imposing the standard normalization condition, we obtain an approximated value of the normalization for the bound state wavefunction, N =

q

2

πn3, and at small r, this is the only

dependence on energy of the wavefunction. The eigenenergies can be expressed as: En,l = −

Ry∗ (n − δl)2

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1.2. WAVEFUNCTION OF RYDBERG STATES 5

Property Scaling law ground state 5S Rydberg state 70S Binding Energy (n∗)−2 4.18 eV 3.04 meV

Orbital Radius (n∗)2 5.632 a0 6706 a0

Lifetime (@T = 300 K) (n∗)3 (5P-5S) 26.2 ns 151 µs [43]

Polarizability (n∗)7 −79.4 mHz(Vcm−1₎−2 _{−557.4 MHz(Vcm}−1₎−2 _[72]

VdW coeff (C6) (n∗)11 7.1 × 10−15MHzµm6 h × 867.6 MHzµm6 [73]

Table 1.1: The table shows the scaling laws of some properties of Rydberg atoms with the principal quantum number n∗ and the corresponding values for the 70S Rydberg state of Rubidium. The values of the ground state 5S are also reported [74]. The binding energy and the orbital radius for the 70S Rydberg have been obtained rescaling the relative value of the 5S state.

where Ry∗ = Ry/(1 + me

Mnucl) and Ry = 13.605693009(84) eV. The quantum defect

intro-duced above can be approximated with the Rydberg-Ritz formula:

δl= δ0+ δ2 (n − δ0)2 + δ4 (n − δ0)4 + · · · (1.20)

The wavefunctions found using the approach above, the quantum defect theory [66], are suitable to describe the electron outside the ionic core in highly excited states, and using them we can calculate the matrix elements of several operators for bound states, allowing us to predict some important properties of atoms in Rydberg states, in particular their scaling with the effective quantum number n∗ = (n − δl). For example, if we consider the

simple model of the Bohr atom, imposing the orbit condition and the quantization of the angular momentum, we obtain that the radius of the classical orbit scales with (n∗)2, and this represents well the spatial spread of the exact wavefunction of the electron for high n states. The diamagnetic energy shift, which depends on the orbital angular momentum l and on the geometric cross-section, scales as (n∗)4, while the electric dipole moment and the polarizability scale respectively as (n∗)2 and (n∗)7. These properties show that the Rydberg atoms are extremely sensitive to electric and magnetic fields compared to ground state atoms. The large polarizability leads also to a strong van der Waals interaction between atoms in these states, which can lead to mechanical forces that are negligible in neutral atoms [48]. Another important feature of Rydberg atoms is their radiative lifetime, which scales as (n∗)3 and increases by several orders of magnitude for high n with respect to the ground state. This is a fundamental feature for many applications because it allows longer access to the Rydberg state during an experiment or during quantum computation operations. As will be shown in the next section, blackbody radiation plays an important role in the determination of the lifetime of Rydberg atoms.

Furthermore, Rydberg levels have a low ionization threshold because the electron in the most external orbital is loosely bound to the ionic core and its energy level lies near to the continuum. This feature is used for the detection of Rydberg atoms since small electric fields are sufficient to ionize them. The table 1.2 summarizes some of the scaling laws of the main properties and gives the example of the 70S Rydberg state of Rubidium, which was used in many experiments reported in this thesis.