Large-scale quantum coherence of nearly circular wave packets

C O Reinhold^1,2, S Yoshida³, J Burgdörfer^2,3, B Wyker⁴, J J Mestayer⁴ and F B Dunning⁴

¹ Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6372, USA
² Department of Physics, University of Tennessee, Knoxville, TN 37996, USA
³ Institute for Theoretical Physics, Vienna University of Technology, A-1040 Vienna, Austria, EU
⁴ Department of Physics and Astronomy and the Rice Quantum Institute, Rice University, Houston, TX 77005-1892, USA

Received 20 March 2009, in final form 
Published 27 April 2009

Realizing macroscopic or mesoscopic systems that exhibit coherent quantum dynamics is an active field of current research. Large-scale quantum systems typically have exceedingly small level spacings and are highly susceptible to decoherence due to energy exchange with the environment. Atoms in high-lying Rydberg states provide a valuable laboratory to study the behaviour of mesoscopic systems. Their size can be controlled through the principal quantum number n (the Bohr radius of an atom scales as n²). Very-high-n Rydberg atoms are giants in the atomic world and for n > 150 have diameters ≥1 µm. This length scale is comparable both to the coherence length (or bonding length) of Cooper pairs in typical type I superconductors [1] and to the size of mesoscopic quantum dots in semiconductor heterostructures patterned by precision electron lithography [2]. However, while the typical decoherence time in quantum dots due to inelastic electron–electron and electron–phonon scattering is on the picosecond scale [3], we show here that quantum coherences in Rydberg atoms can be extended to the microsecond scale with careful control of external perturbations.

Because the energy levels in atoms are not equispaced, any observable associated with a wave packet will quickly dephase. However, as long as decoherence is suppressed, this (reversible) dephasing will be eventually `undone' leading to full or partial quantum revivals [4–7]. Such revivals have been seen in the vibrational motion of molecules on a picosecond time scale [8] and on pico to nanosecond time scales in freely evolving Rydberg wave packets in moderately high-n levels (n < 60) [9, 10]. However, quantum behaviour has so far escaped observation for very-high-n Rydberg wave packets because their quantum coherence is extremely fragile [11]. The main challenge lies in controlling stray electric fields which Stark-mix n manifolds and effectively broaden the spectrum. The critical electric field at which the extreme Stark states in two adjacent manifolds first cross scales as n^–5 and decreases by four orders of magnitude from n  =  45 to n  =  300.

Here we show that the quantum coherence of near-circular Rydberg wave packets with n ~ 305 is surprisingly robust and can be preserved on microsecond time scales. This large spatio-temporal coherence is, to our knowledge, unprecedented for electronic degrees of freedom involving a large ensemble (here ~100) of quantum states. The present coherences should be distinguished from coherences in spin systems which involve only a few levels [12, 13]. One key ingredient is that all stray electric fields are reduced to less than ~50 µV cm^–1 as described elsewhere [11]. In addition, we prepare Rydberg wave packets consisting of a partially coherent superposition of near-circular states, whose decoherence rate is dramatically reduced compared to other states. Because of the latter property, circular states are of interest in areas such as information processing and cavity quantum electrodynamics [14–17].

Revivals in the present study result from the creation of a partially coherent state of the electron, a Rydberg wave packet described by the density operator ρ(t). As the observable, we focus here on the expectation value of the z-coordinate of the electron, z(t), which is inferred from the measurements of the survival probability following application of a probe electric field along the z-axis. Initial wave packet preparation involves several steps [11]. First, Rydberg atoms are produced using an extra-cavity doubled Rh6G dye laser to photoexcite potassium atoms contained in a thermal-energy beam to the extreme redshifted states in the n_i~305 Stark manifold in a weak dc field of ~ –400 µV cm^–1 that defines the x-axis. This results in the creation of quasi-one-dimensional (quasi-1D) states oriented along the x-axis [18]. Because of Doppler broadening, photoexcitation leads to the production of an ensemble of about 36 low-lying redshifted states [18], which can be represented by the density operator,

centred at the electric quantum number with an average initial polarization x ≈ 1.25n²_i. The probabilities are taken to be constant. (A smooth variation with k_x and m_x cannot be ruled out but is not important numerically.) The degree of coherence of the ensemble c  =  Tr(ρ²)/Tr(ρ) ≈ 10^–2 is quite low. Equivalently, the participation ratio of the incoherently excited states, c^–1 ≈ 100, is high, making it all the more surprising that long-time quantum coherence can be observed in this system. Due to the ground-state hyperfine structure of potassium, ~75% of the laser-excited atoms are in n  =  305 states and ~25% are in n  =  307 states. This additional degree of incoherence is taken into account in the calculations but is not explicitly shown in equation (1) to simplify the notation.

Following laser excitation the weak dc field is turned off slowly (t_fall ~500 ns) to prevent inhomogeneous Stark line broadening. A transverse pump electric field, F^pump_z, is then suddenly applied along the z-axis at t  =  t_on by applying a voltage pulse to a nearby electrode. The rise (and fall) times of F^pump_z (~0.3 ns) are much smaller than the classical orbital period, 4.3 ns (atomic units are used throughout). F^pump_z exerts a torque on the dipole d_x  =  –x creating a Stark wave packet that is localized in the xz plane and that undergoes periodic changes in its y component of electron angular momentum, L_y(t). The pump pulse is turned off at t_off  =  t_on  +  π/(3F^pump_z n_i) corresponding to a quarter of the Stark precession period, when L_y is near its extreme value. Further evolution in L_y ceases and the wave packet is frozen in a near-circular state. Each state populated in the initial density matrix (1) evolves into a coherent superposition:

(For simplicity, in the following we set t_off  =  0.) As will be shown, for F^pump_z  =  –5 mV cm^–1 as used here, this coherent superposition involves Δn~7 field-free n levels (302 ≤ n ≤ 308) and about 70 near maximal m_y states (relative to the L_y quantization axis) with |m_y| ≈ n_i.

After the turning-off of the pump pulse, the wave packet evolves freely. Its time evolution is described by the density matrix elements,

with

Note that, neglecting possible decoherence effects due to stray fields, the initial degree of coherence in (1) is preserved during the wave packet preparation steps. The field-free time evolution (3) is characterized by a set of Δn(Δn – 1)/2 beat frequencies, . Among these, only a few non-degenerate discrete frequencies effectively contribute. This results because the dipole matrix elements decrease rapidly with (n ' – n) for hign-n near-circular (n ≈ ℓ) states. Therefore, only beat frequencies involving adjacent manifolds (n ' ≈ n ± 1), ω_n,n±1  =  ω⁽¹⁾_n contribute and

To zeroth order in |n – n_i|/n_i all the beat frequencies coincide with the classical orbital frequency, i.e., . Following the turn-off of F^pump_z, the beat amplitude of z(t) first increases due to the creation of a transiently localized Bohr-like state, as described in detail elsewhere [19, 21]. This localization is subsequently lost due to higher-order differences in the beat frequencies on a time scale t_L ≈ π/Δω, where Δω  =  3n^–4Δn is the width of the frequency spectrum. The resulting near uniform distribution in azimuth displays only small variations in z(t) and the beats are damped. Revivals, which are associated with the smallest (adjacent) frequency differences, take place on a much longer time scale given by

when all the time-evolution phases ω⁽¹⁾_n t in (5) rephase (modulo 2π and to first order in |n – n_i|/n_i). Observation of revivals at n~300 requires the suppression of any incoherent dephasing, i.e. decoherence, over times T_R ≈ 400 ns. The wave packet must remain effectively disentangled from the environment over this interval, i.e., line broadening is sufficiently suppressed that the discreteness of the spectrum survives. Fractional revivals [7, 9] occur at earlier times but cannot be seen through measurements of z(t) for a wave packet with circular symmetry.

Quantum calculations of z(t) in three dimensions are currently feasible for quantum numbers up to n_i ~ 100 but not 300 as employed here. Thus the experimental data are discussed with a novel quantized classical trajectory Monte Carlo (QCTMC) method. To validate this method, quantum and QCTMC predictions at n  =  100 are now compared. Figure 1(a) shows the time dependence of z(t) following the application of a pump field for an initial Stark state with n_i  =  100, m_x  =  0 and k_x  =  83, i.e., a strong initial polarization n_ik_xm_x|x|n_ik_xm_x ≈ 1.25n²_i close to that achievable experimentally [18]. The quantum simulations were performed using an expansion of the wavefunction in 182 289 hydrogenic states with 92 ≤ n ≤ 109. As shown by the inset in figure 1, sudden turn-off of F^pump_z populates a distribution of n levels with Δn ~ 9 (this width is controlled with the strength of the pump field [19]). At the turn-off of F^pump_z, i.e. t  =  0, the wave packet is nearly uniformly distributed around a circle in the xz plane. Therefore, the time dependence of z(t) is initially small. The wave packet localizes transiently at t_L ≈ π/Δω leading to a strong periodic oscillation in z(t) but subsequently dephases on the same time scale. Quantum revivals, which result in pronounced oscillations of z(t), are evident near the expected first and second scaled quantum revival times and . Remarkably, the present wave packet features a double peaked revival whose origin will be discussed below. Also shown in figure 1(a) are the results of a conventional CTMC simulation that follows the dynamics of a phase space ensemble of classical orbits that initially mimics the corresponding quantum state [19, 20]. It is clear that while the predicted classical ensemble average z(t) initially reproduces the quantum-mechanical result, at late times where revivals occur a purely classical description fails.

Figure 1. Time evolution of the expectation value of z, z(t), following the application of a pump field with scaled strength n4i Fpumpz =  0.017 to a ni  =  100, kx  =  83, mx  =  0 Stark state. The quantum result is compared with CTMC (a) and quantized CTMC (b) simulations. Fpumpz was turned on and off suddenly and had a duration equal to one quarter of the Stark precession period, π/(3Fpumpz ni). The inset shows the frequency spectrum calculated from the Fourier transform of z(t). The vertical lines indicate the revival times calculated as TRA ≈ 2πn4A/3 and TRB ≈ 2πn4B/3 (see the text). The surface plots at the top are the QCTMC electron probability densities in the xz plane at ~ 30, 33.2 and 36.

The failure of the CTMC method can be corrected by accounting for the discreteness of the excitation spectrum using a quantized CTMC (QCTMC) approach. To this end, we discretize the energy spectrum that governs the free evolution immediately after the turn-off of F^pump_z. The principal classical action of each trajectory is calculated, , and is quantized such that the discretized classical action becomes n '_c  =  n  +  1/2 where n ( = n_c) is the integer part of n_c. Similarly, we rescale the phase space coordinates, r → (n '_c/n_c)²r and p → (n_c/n '_c)p. The new classical orbital frequency, Ω_n + 1/2  =  n '^–3_c  =  (n  +  1/2)^–3, agrees with the quantum frequency ω⁽¹⁾_n up to order n^–4, i.e., the order governing the revivals. Figure 1(b) shows that the time evolution of z(t) predicted by quantum theory agrees well with the QCTMC result:

where P_n is the occupation probability of a given n level, i.e., of a bin of classical action n ≤ n_c ≤ n  +  1, and z(t)_n is the ensemble average over trajectories within the same n level. Equation (7) uses the fact that for near-circular orbits z(t)_n ≈ (n  +  1/2)²sin(Ω_n + 1/2 t  +  (0))_n, where is the initial azimuthal angle. Both the quantum-mechanical (equation (5)) and the QCTMC (equation (7)) expectation values are given by a superposition of about Δn sinusoidal terms with frequencies Ω_n + 1/2 ≈ ω⁽¹⁾_n. The QCTMC simulation reproduces the quantum predictions very well.

The additional modulation of z(t) near the first revival, i.e., the minimum at the revival time T_R, results because (as discussed later) the wave packet consists of a superposition of two wave packets that tend to be located on opposite sides of the nucleus near the revival time (see the electron probability density in figure 1). This is directly related to the frequency spectrum of z(t). The inset of figure 1 shows this spectrum as a function of n  =  ω^–1/3 – 1/2 such that the n levels governing the dynamics can be identified. The spectrum exhibits a minimum at n ≈ n_i  =  100 with well-separated peaks centred at n_A  =  97 and n_B  =  102. As discussed elsewhere [19], these are associated with the sub-ensembles of circular trajectories that are localized, respectively, at z ~  + n²_i and z ~ –n²_i immediately after the pump field is turned off. The destructive interference near the revival can be understood by separating the initial wave packet into two components A and B associated with the lower and upper halves of the frequency spectrum, as shown in figure 2. The expectation value of z for each wave packet can be calculated by splitting the sum in (7) into two sums for n < n_i and n > n_i. The mean azimuthal angles of the component wave packets are illustrated by the diagrams at the top of figure 2 (the wave packets are not completely localized at each of the times indicated by the vertical lines). Wave packet A (B) is centred at n_A  =  97 (n_B  =  102) with an average orbital period ( ). Initially, wave packet A (B) is in the (0) > 0 ((0) < 0) region such that z(0)_A > 0 (z(0)_B < 0) and z(0)  =  z(0)_A  +  z(0)_B ≈ 0. Following a time t ~ 3  =  2.5 , wave packet B catches up with wave packet A and the oscillations in z  =  z_A  +  z_B reach their maximum amplitude. Subsequently the wave packets move out of step and dephase. At later times each wave packet rephases but at different revival times T_RA ≈ 2πn⁴_A/3 and T_RB ≈ 2πn⁴_B/3. At the revival time estimated using n_i (2πn⁴_i/3) the wave packets A and B are out of phase (more precisely, on the opposite sides of the nucleus). This leads to destructive temporal interference between z(t)_A and z(t)_B. It is noteworthy that this phenomenon is specific to the present protocol for producing nearly circular wave packets because initially A and B are on the opposite sides of the nucleus. The present wave packet consisting of a superposition of two wave packets, which periodically separate and recombine in space, can be viewed as a realization of a Schrödinger cat state.

Figure 2. Same as figure 1 but the full QCTMC wave packet (c) is split into two wave packets A (a) and B (b). The diagrams at the top illustrate the mean azimuthal position of wave packets A and B at the times indicated by the vertical lines (see the text). Note that these wave packets are not well localized at all times. The degree of localization can be inferred from the amplitude of z(t).

In order to test the theoretical predictions, electronic wave packets were created experimentally using the multi-step protocol discussed above (see [19, 21] for details). Figure 3(a) displays measurements of the time evolution of a wave packet created by a pump field F^pump_z  =  –5 mV cm^–1 as monitored using a probe field applied following the turn-off of the pump pulse along the z-axis after a variable time delay t_d. The fraction of surviving atoms, P_S(t_d), was determined by selective field ionization in which a slowly varying ramped electric field was applied to the atoms and the liberated electrons detected by a particle multiplier. The probe field had short rise and fall times of ~0.3 ns, a duration of 6 ns ( ) and an amplitude of F^probe_z≈ 100 mV cm^–1, sufficient to ionize ~50% of the initial Rydberg atoms. The survival probability provides an indirect measure of the average electron position coordinate z(t_d) since those electrons with energies E  +  z(t_d)F^probe_z that lie above the top of the barrier in the electron potential can be ionized [11, 22]. The structure in the measured and calculated survival probabilities agrees well in both magnitude and phase up to the first quantum revival (see figure 3(c)) where the predicted destructive interference at the centre of the first quantum revival is clearly seen. The predicted structure in the frequency distribution is also evident (see the inset). A second quantum revival is present in the data at ~900 ns but it is somewhat damped because atoms move out of the true zero field region and begin to experience field inhomogeneities. Nonetheless, the data reveal a hint of a third revival (not shown) at ~1.3 µs.

The observation of quantum revivals at times around ~900 ns (equivalent to ~200 orbital periods) is remarkable. To our knowledge, these are the first quantum revivals seen in very-high-n atoms and correspond to the longest quantum coherence times yet observed for mesoscopic electronic wave packets. One key ingredient in this success is the use of nearly circular wave packets [21], which are quite robust against noise because the electron remains far from the nucleus thereby reducing the likelihood of energy transfer in interactions with spurious fields. Moreover, the strength of the revival signal is surprisingly strong considering the low initial coherence (c ~ 10^–2) of the ensemble. One important feature is that for Δn/n 1 the dominant frequencies within the incoherent ensemble match each other closely.

In summary, we have shown that with careful minimization of stray electric fields, quantum coherence of n ~ 305 near-circular wave packets can be preserved for very long times. The observed quantum revivals provide a valuable tool to study the transition from quantum to classical dynamics induced by interactions with the environment [23], i.e., by spurious electrical noise or by collisions [24–26]. Such interactions effectively broaden the beat frequency spectrum, quantum revivals being destroyed when this broadening becomes comparable to the difference between adjacent frequency components. The present protocol creates a wave packet that initially (and at the centre of revivals) comprises a superposition of two wave packets of different n localized on the opposite sides of the nucleus. It thus resembles a Schrödinger cat state [27] allowing temporal interferences to be observed. For n~305, the two localized wave packets are intermittently ~10 µm apart and it would be interesting if these mesoscopic degrees of freedom could be entangled with microscopic degrees of freedom, such as spin.

Research supported by the NSF under grant no 0650732, the Robert A Welch foundation under grant no C-0734, the OBES, US DOE to ORNL, which is managed by the UT-Batelle LLC under contract no AC05–00OR22725 and by the FWF (Austria) under SFB016.

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