The coherence of a quantum state plays a central role in a wide array of research areas. These areas, e.g. the coherent control of processes, quantum information processing and quantum measurement, share common concerns and goals. They desire: 1) mechanisms for creating a specific coherent superposition state (this is the target state in coherent control applications [1, 2, 3, 4], the qubit in quantum computation [5], or the subject state for quantum measurement [6]); 2) understanding of the problem of decoherence or how to preserve the specific state from decay; and 3) mechanisms for modifying the coherent superposition (this is the feedback mechanism in coherent control applications [7]; the quantum tomographic measurement process itself, or the quantum gate in quantum computing). These concerns are interrelated, e.g. a mechanism that modifies a coherent superposition state can also be viewed as a mechanism for creating or measuring a specific coherent superposition state. The experiment in this paper shows that a process that is ordinarily considered a decay or decoherence effect can also be used as a mechanism for modifying the coherent superposition state and, in particular, the phase relationships of that state. It is also significant that this modification may be controlled by a process that is independent of the mechanism that created the initial superposition. This paper develops a modification of the standard Ramsey fringe measurement technique to allow a more complete study of this design mechanism.

Rydberg atomic systems are ideal systems for studying these concerns. Various different coherent superpositions of atomic Rydberg states have been created through the application of short pulses and external fields (see [8]). The modification of the coherent superposition state has also been demonstrated [9]. In fact, the measurement of the coherent superposition state by a coherent probe pulse modifies the superposition. However, such a direct manipulation depends strongly upon the coherence between the pump and probe pulses. Here, we offer an alternative means of modifying the coherence stored in a wave packet via the application of an independent laser source. This experiment manipulates the coherent superposition associated with an electronic Rydberg wave packet in atomic calcium (a two-electron system) by the application of a resonant optical field to the other valence electron. The entanglement or correlations between these two electrons is the physical origin of the modification.

In this experiment, one of the valence electrons of the calcium atom is excited by a short optical pulse to a superposition of highly excited states (or Rydberg states). This coherent superposition contains states with a range of principal quantum numbers (n) and is commonly known as a radially localized wave packet [10, 11, 12]. For such a wavepacket the electronic probability density oscillates with the classical period, τ_cl=2πn̄ ³, from the inner turning point to the outer turning point of the classical orbit, where n̄ is the average n of the superposition of Rydberg states. This pulsed excitation takes place in the presence of a strong cw optical field that is tuned near the resonance between the ground state and a low-lying excited state of singly ionized calcium, Ca⁺. Until the pulsed excitation takes place this cw field is far from any resonance in the atom. Once the radial wave packet is excited the states of the remaining valence electron are much like the states of Ca⁺ so that this type of excitation is described as an isolated core excitation [13]. The cw field or core laser field is then in resonance with the ground state to excited state transition of this core valence electron. This core electron undergoes Rabi oscillations between this ground and excited state. The coherent superposition state may be modified by varying the ratio of the frequency of the Rabi oscillations, Ω, to the frequency of the classical oscillations, ω_cl, of the wave packet. This ratio, κ=Ω/ω_cl, affects the overlap of the two electrons or the magnitude of the correlation between them over a given period of time.

Wavepackets in two-electron systems have become a subject of great current interest. There have been many theoretical papers discussing the shaping of wave packets via a strong core laser field [14, 15, 16, 17, 18, 19, 20, 21, 22]. Recent experiments investigating Rydberg wave packets in two-electron systems have explored quantum measurement possibilities [23], the effect of the core configuration on their evolution [24], the effect of the presence of a perturber state on their behavior[25], and the evolution of two-electron wave packets [26, 27]. The theoretical work has also predicted that it is possible to form a nondecaying wavepacket [18, 22]. Such a wave packet has been recently observed [28] in a calcium atomic system.

The effect of the strong core laser field on the wave packet is sensitive to the synchronization of the Rabi oscillations with the classical oscillation of the wave packet [18]. This insight can be understood with a semiclassical picture. Consider the situation in which the goal is to suppress the autoionization of the excited calcium atom. If one of the valence electrons is in a highly excited state(s) and the other is in the 4p excited state of the isolated core, then it is energetically possible for the atom to autoionize. For a single highly excited state, the autoionization rate is quite large (typically on the order of (1/τ_cl). However, if a wave packet is formed, then the overlap of the Rydberg wave packet with the core is time dependent. The core electron also oscillates as the strong core laser field causes it to undergo Rabi oscillations between the ground state and the excited state. If the core laser field is of sufficient strength so that the two oscillations have the same frequency, then it is possible to synchronize the wave packet’s motion so that it is localized away from the core when the core is in the excited state and near the core when the core is in the ground state. In the first instance, autoionization is still energetically possible but the lack of overlap limits the autoionization rate. In the second, the core is not excited so that despite the overlap the autoionization does not proceed [18, 22]. This simple physical picture leads to the formation of a nondecaying wave packet. Note the synchronization occurs naturally since the core laser field is nonresonant until the wavepacket is excited.

This semiclassical picture does not tell the whole story. A more complete analysis [18, 22] shows that the autoionization process itself plays a significant role in the evolution of the wave packet. The channel interaction between the wave packet states and the continuum modifies the distribution of Rydberg states. In the nondecaying situation (described above), when the Rabi frequency (Ω) is equal to the classical frequency (2π/τ_cl) the effect of the channel interaction is minimized. However, it is also possible to enhance the interaction when Ω≠2π/τ_cl. Consider the case when the ratio κ=Ω/(2π/τ_cl)=0.5. With this strength of the core laser field, the wave packet executes one full orbit in the time it takes for the core electron to excecute half a Rabi cycle. Therefore, the wave packet returns to the core when the core is excited. This optimal overlap greatly enhances the interaction with the continuum. Population is lost to the continuum, but a Raman-like process via the continuum allows the states within the superposition to interact and be modified [22]. The result is that the wave packet is strongly modified in the vicinity of κ=0.5 and other sub-multiples.

The current experiment explores this modification of the superposition as κ is varied. Precisely, we are concerned with the response of the wave packet system to a coherent probe. The experiment establishes certain phase information in the wave packet system and studies how this phase information is modified by the independent control mechanism of core laser field. This modification is observed with a coherent probe at a fixed delay time. The coherent superposition may be described in terms of two distinct phases. There is a global phase, ϕ, between the ground state and the Rydberg states and there are the individual phase relationships between the Rydberg states. When there is no external field present, each of these phases evolve in a well-determined manner as the states freely precess. The application of a strong isolated core excitation can modify this evolution due to the channel interaction with the autoionization continuum. It is quite common that an applied field can change the rates at which these phases evolve. For example, an applied DC electric field may Stark shift the states so that these rates are changed and therefore the phases are different at the fixed delay time. What is of special interest, in the present experiment, is that the modification of the global phase displays resonance-like behavior as κ is varied. The observed resonances are tied directly to synchronization of the two oscillations and specifically to simple fractional values of κ, e.g. 1/2, 1/4. This resonant behavior serves as an sensitive indicator of the modification of the wave packet and provides a sensitive means of modifying this global phase and by inference the relative phases of the individual states.

The experiment uses a modified Ramsey fringe technique to extract the global phase. The interaction of a coherent probe pulse with the wave packet can greatly modify the population in the Rydberg states [29]. The probe pulse can either coherently add (constructive interference) or coherently subtract (destructive interference) from the population in the Rydberg states. Varying the optical phase from 0 to 2π, i.e. a path delay of a wavelength of the probe pulse, yields a sinusoidal variation of the population of the Rydberg states or Ramsey fringes in the population. Observing the effect of κ on the global phase of the wave packet is made possible by the incorporation of a local oscillator into the standard technique. The phase of the Ramsey fringe for a given value of κ is measured against the phase of the Ramsey fringes of a radial wave packet that evolves in the presence of an undriven core electron (κ=0). The difference between these phases is directly the global phase, ϕ described above.

The general experimental setup is shown in Fig. 1. The excitation of the isolated core transition (4s _1/2 to 4p _1/2; 397nm) of calcium is accomplished by a resonant cw laser field. This field is derived from a modified Coherent 899 Ti:Sapphire laser producing 2W at 794nm. This infrared field is frequency doubled to the ultraviolet in a stabilized traveling wave cavity. The design is similar to that described in [30] and produces 200 mW of light at 397 nm. Initially this ultraviolet field, is far from any resonance in the Ca atom. However, once the pulsed laser excites the wave packet the cw laser becomes resonant and so the classical oscillations of the wave packet and the Rabi oscillations of the core transition are synchronized. The pulsed field is the frequency-doubled output of a passively, mode-locked Ti:Sapphire laser (pulsewidth=2ps, energy=1nJ, wavelength=~406nm). This pulse is split into pump and probe pulse by a Michelson interferometer setup. The cw field and the pulsed field are collimated and travel parallel paths. The overlap of the focal regions of the two fields are such that the atoms excited by the pulsed laser field see a nearly uniform intensity cw field.

 

Figure 1. Experimental setup. The pump and probe pulses interact with the calcium atoms in the thermal beam in the presence of the core laser field. The interference in the total population of the Rydberg states is observed by ionizing those states and counting the resulting electrons.

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A preliminary experiment finds the resonance between the isolated core’s ground and excited states. The frequency of the core laser is then maintained at this value by locking to a stabilized Fabry-Perot interferometer. The frequency of the core laser changes by less than 500kHz over the duration of the experiment.

In the primary experiment, the pump pulse excites a wave packet, via a two-photon process (in the presence of the core laser). The resulting wavepacket consists predominantly of nd states in the vicinity of n̄=67. For such wave packets, we may attain values of κ greater than unity. We have excited wave packets with values of κ ranging from 0.2 to 1.2. These wave packets are probed at a fixed time delay of 86 ps. This time is chosen because it corresponds to 2 classical periods of the wave packet. This choice of time serves two purposes. First, the wave packet is near the core at the time of the probe which enhances the modulation depth of the Ramsey fringes. Second, it allows sufficient time for the wave packet to be significantly modified by its interaction with the core. The Michelson interferometer used to generate the pump and probe pulses is stabilized and locked to a frequency-stabilized HeNe laser. The lock level may be varied so that delays of nearly half a wavelength of the HeNe laser may be set continuously. This allows the phase delay of the probe laser to be varied by more than 2π. The total population of the Rydberg states is measured after the probe pulse interaction by electric field ionization of the Rydberg states. This population is measured for eight different values of the phase delay of the probe pulse spanning 2π(steps of π/4). The phase of the resulting Ramsey fringe is found by Fourier analysis of this data. The same measurement is made on a wave packet without the core laser excitation, i.e. the cw laser is alternately blocked and unblocked at each different value of κ). This phase of the Ramsey fringe without the core laser is the local oscillator against which the phase as a function of κ is measured. Therefore, the difference between these phases is the global phase ϕ. If κ is zero, then ϕ=0. This technique reduces changes in ϕ due to drifts in the system that are not associated with the cw laser field.

The experimental results are shown in Fig. 2(a). A theoretical description of this system was developed in [22] and is used to model this experiment. The qualitative agreement is good (Fig. 2(b)). The phase ϕ undergoes rapid change in the vicinity of κ=0.25 and 0.5. It is at these values that the coupling of the autoionizing Rydberg series to the continuum is enhanced by the overlap of the wave packet with the excited core electron. The resonant behavior in the vicinity of these fractional values indicate that the modification is not a simple level shift. The states are undergoing some strong coupling, i.e. Raman-like transition via the continuum. In an equivalent picture, these resonances may be tied to avoided crossings in a dressed-energy level map of the system as a function of κ [19]. The plateau region extending from κ~0.7 to 1.2 is the nonresonant contribution to the phase shift. This contribution can be thought of as an AC Stark effect due to the core laser field. In this region, no resonances are expected since the wave packet does not encounter a fully excited core.

 

Figure 2. The global phase of the superposition as a function of κ. In (a) experimental measurements are shown. In (b) the results of a theoretical model are shown. The change in this phase probes the modification of the coherence due to the core laser field. The importance of synchronization is seen by the resonance-like structure near κ=0.5 and 0.25. The error bars are derived from the noise in the measured Rydberg atom population.

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The coherence is modified with the aid of the strong coupling to the continuum. At the points where the phase undergoes rapid change, the overlap between the wave packet and the core electron is maximized. There is a significant loss of the population to the continuum at these points. For example, the total population for κ=0.5 is four times smaller than the total population at κ=1.0. This loss of population is in agreement with the theoretical predictions. However, there is still coherence between the ground state and the remaining superposition of Rydberg states; the phase relationship is modified but still exists.

Full control of the coherence of a quantum state requires the manipulation of the individual states. Therefore, it is interesting to explore the change in the relative phases between individual states of the superposition. It has recently become possible to extract such information for a Rydberg atomic wave packet [31, 6]. Extension of these tomographic measurements to two-electron systems have not yet been achieved. However, the good qualitative agreement between the experiment and the theoretical model suggests that the theoretical model can provide insight into this question. We can theoretically explore the ability of this core laser field to manipulate the individual phases of states of the wave packet.

The phase of the superposition was affected most significantly when the Rabi frequency was equal to half the classical frequency, so we will focus on this region. In Fig. 3, the phase difference between two neighboring states (n=66 and 67) is shown. In the vicinity of κ=0.5, the phase rapidly undergoes nearly a 180 degree shift. This suggests a high degree of control over the individual phases of the states of the superposition in this system.

 

Figure 3. The change in relative phase of neighboring states of the superposition as a function of κ. The theoretical model that produced Fig. 2(b) yields this curve. Note the rapid change in phase near κ=0.5.

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This rapid phase change is producing striking changes in the resulting wave packets (see Fig. 4). The slight changes in κ do not significantly a ect the total popula-tion in these wave packet states but the distribution of the amplitudes and phases of the states of the superposition are strongly affected. For values of κ below the point where the rapid change occurs, the wave packet has a significant component near the core. This is also true for wave packets above this transition point. However, for a value of κ=0.5 that lies in the center of the transition, a wave packet is produced that is well-localized at the outer turning point. For values of κ near 0.5 the wave packet approaches the core when the core is excited. The semiclassical picture predicts a loss population due to autoionization. This occurs for all of these wave packets (the population is only 25% of the original population). However, as mentioned above, this loss is accompanied by a coupling between the states of the wave packet via the continuum. This strong coupling rearranges the amplitudes and phases of the states so that the wave packet is away from the core avoiding further strong coupling to the continuum. This effect is seen clearly for the wave packet with κ=0.5.

 

Figure 4. The radial probability distribution of three wave packet for three values of κ in the vicinity of κ=0.50. The three values are κ=0.48 (blue curve, circle), κ=0.50 (red curve, square), and κ=0.52 (green curve, triangle). The wave packets are shown at the same fixed delay time of 86 ps. Note the localization of the ‘red’ wave packet at the outer turning point.

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Creation, manipulation and measurement of coherent superposition states are central topics in a broad range of research areas. The observations reported in this paper point to a means of modifying such coherent superposition states via an independent laser source. The strong electron-electron correlations of the two valence electrons in the calcium atom allows an independent laser source to generate an isolated core excitation that modifies the superposition. This core laser bears no phase relationship with the pump laser and so is an independent knob that exerts control over the coherent superposition. Also, this experiment shows that what is commonly considered a decay or decoherence effect can be used as a mechanism for modifying the coherent superposition state.

This work was supported by the National Science Foundation under Grant No. PHY-9733643.