Logic gates using high Rydberg states
 ^{*}The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel; ^{†}Département de Chimie, B6, Université de Liège, B 4000 Liège, Belgium; ^{‡}Institut für Physikalische Chemie, TUM D85747 Garching, Germany; ^{§}Max Planck Institute of Quantum Optics, D85740 Garching, Germany; ^{¶}School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel; and ^{**}Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095
See allHide authors and affiliations

Contributed by R. D. Levine
Abstract
Connected logic gates can be operated on the levels of one molecule by making use of the special properties of high Rydberg states. Explicit experimental results for the NO molecule are provided as an example. A number of other options, including that of several gates concatenated so as to operate as a full adder, are discussed. Specific properties of high Rydberg states that are used are: their autoionization is delayed so that they can be distinguished from direct multiphoton ionization, during their long life such states also can decay by energy transfer to the molecular core in a way that can be controlled by the judicious application of very weak external electrical fields, and the Rydberg states can be detected by the application of an ionizing electrical field. The combination of two (or three) color photons with and without external weak fields allows the construction of quite elaborate logic circuit diagrams and shows that taking advantage of the different intramolecular dynamics of levels that differ by their excitation enables the compounding of logic operations on one molecular frame.
The dynamics of high molecular Rydberg states, of principal quantum number n of 100 or more, has received much recent interest (1–6). The essential findings that are relevant to this paper are mentioned briefly in the abstract and are summarized further below. The studies of the dynamics were motivated both by the applications to ZEKE (zero electron kinetic energy) spectroscopy and by the inherent interest in the unusual dynamical regime exhibited by such states (7). We show here how this new understanding and the technological capabilities developed for these experiments allow a novel type of application: the concatenation of several logic gates by using only one molecular frame (8).
Computing with molecules is becoming an active area of research. Singlemoleculebased rectifiers (9) and switches (10, 11) have been reported recently and offer a quite different approach (12) to moleculebased computing. The switches can be turned on or off by both redox and chemical inputs (13). The switch allows an electrical current to flow or not by causing a conformational change in [2]catananetype molecules. The mechanochemical mechanism is that a pair of interlocking rings can be switched between two conformations in an oxidation/reduction cycle. The molecule was incorporated in a solidstate device and could be recycled many times (11). Chemical inputs and optical outputs also have been discussed (14, 15) for molecules in solution. To be clear, we emphasize that in this paper the logical response is due to one molecule but that the observed signal is due to a finite but not enormous number of molecules (as few as 100 per laser shot and even 10 molecules per shot if one is working at the peak of the spectrum). These molecules are acting independently of one another, and, because of the unusually large size of a high Rydberg state, they need to be kept as far apart as possible. This is why working the least possible number of molecules is experimentally advantageous.
This paper discusses the integration of reasonably large logic circuits on one molecule. We believe this is a worthwhile goal in its own right and, furthermore, that it is an essential first stage of developments that would become practical only if means can be devised to communicate the logical output of one molecule as an input to another.
In the background to this work is the suggestion that molecular spectroscopy can be thought of in terms of logic gates and that the intramolecular dynamics are logically equivalent to the concatenation of several gates (8). Here we apply this idea to ZEKE spectroscopy and provide experimental results showing that such integration indeed is possible. Not discussed in this paper is the integration of different molecules into even larger circuits.
Many of the special dynamical properties of high Rydberg states are a result of the very high density of quantum states of the system (6, 16, 17). [There are n^{3} states of n per unit (au) of energy. Each value of n corresponds to n^{2} quantum states, and n is in the range of 100. Compounding this are the states of the molecular core.] This high density and the ability to take fine spectral slices (see, e.g., refs. 18 and 19) suggest that Rydberg states could be used to perform quantum computations (20–22). In this paper we do not take advantage of quantum parallelism (23) and confine ourselves to classical twovalued logic. What this means is that the logic gates compute the value of a Boolean function (24). This is a function of variables that take on only one of two possible values, say “on” and “off.” The variables are connected by the familiar logical operations of Boolean algebra and its axioms. The complete representation of such a function is by a truth table, where the value of the function is given for every possible combination of the values of the variables. An equivalent representation is by means of switches (25). These switches can be grouped into slightly larger units, known as logical gates, and we also will use the representation in terms of such gates. The point of our paper is that more than one such gate, in succession, can be operated on the molecule so that the molecular response is equivalent to quite a number of switches.
High Rydberg states exhibit unusual dynamics. Some familiarity with the recent findings is needed for the development below. The essential background is discussed first. Next, we show the gates that can be realized by using the twocolor, twofield experiments on NO (26). Finally, we explore what can be done with experiments by using three colors (27), including a socalled full adder, which typically is realized with five logic gates, two of which are the already nontrivial ExclusiveOR gates (24). The paper concludes by pointing out the many options for further concatenation that are open when one realizes that it is possible to combine all of the already available statepreparation options with the more detailed probing that also is currently practical.
Spectroscopy and Dynamics of High Rydberg States
Rydberg states of molecules often are prepared by a twocolor excitation (see Figs. 1 and 2). The first photon brings the molecule to its first excited (S_{1}) electronic state. The combined energy of the two photons is just below a threshold for direct ionization of the molecule. One of the essential points of ZEKE spectroscopy is that this need not be the lowest threshold. If it is not, the second photon can induce direct ionization or it can reach a Rydberg electron that is revolving around an excited ionic core. By waiting for tens of nanoseconds, the prompt electrons depart, pushed away by a weak DC electrical field. The kinetic energy of the direct electrons can be measured, and this is the basis for photo electron spectroscopy. The neutral high Rydberg states that are left can be ionized later by the rapid application of an electrical field.
The high Rydberg states are longlived because most of the time the electron is away from the positively charged molecular core. The electron initially is promoted into a Bohr–Sommerfeld orbit of low angular momentum, ℓ. Once per its (long) period such an orbit brings the electron close to the core, where it may gain or lose energy. The period of the orbit scales with the principal quantum number n as n^{3}, so that low ℓ and high but not very high n states do interact with the core. To stabilize the Rydberg electron, it should be made to move into a circularlike orbit of high angular momentum, where it is never near the core. The presence of a static electricalfield F_{ℓ} breaks the spherical symmetry and allows the angular momentum to increase, which stabilizes the orbit (28–31). However, the angular momentum ℓ increases and then decreases (16) so that the electron is shielded from the core only for part of the time. To fully isolate the electron, one must make the magnetic quantum number m increase because ℓ ≥ m. This is possible (26) by the application of a second electrical field, F_{m}, which is in an orthogonal direction in space to that of the first field, whose role is to change ℓ. We will emphasize the on/off control made possible by the second field, but one also could use the first field as a Boolean variable.
Experimental Realization of Concatenated Gates
Fig. 1 shows the experimental ZEKE spectrum that forms the basis for the discussion. The plot is vs. the frequency of the second photon. (In our proposed logic circuit, this frequency would be fixed.) Fig. 1 b and c shows that the spectrum without and with the m changing field, F_{m}, being off and on, is readily detectable. The point is that when m is higher, the condition ℓ ≥ m stabilizes Rydberg states of lower n and, therefore, allows their detection, which occurs for a lower frequency of the second photon. The Fig. 1 a and b shows that the condition that the ℓ changing field is reapplied or not is also experimentally detectable. Either field or both therefore can be used as Boolean variables.
The construction in Fig. 2 begins with the result that a twophoton transition, pumped with two photons of different color, already is an “AND” gate (8). This is because only if both lasers are “on” does one observe a ZEKE signal. The truth table for that part of the experiment is shown. In terms of Boolean variables, the output of the AND gate is w_{0}w_{1}, where w_{i} is 0 or 1 depending on whether the laser is off or on. The output of the AND gate is one input into a second AND gate. The other input to the second gate is the Boolean variable F_{m} that indicates whether the m changing field is off or on. The complete gate evaluates ω_{0}ω_{1}F_{m}, as seen also from the truth table included in Fig. 3.
The higher frequency ω_{0} could lead to a twophoton direct ionization. Because the absorption crosssection to S_{1} is typically high, this is minimized by using an intensity that is low enough. In any case, direct ions are discriminated against by the weak DC field that is already present at the ionization stage as shown in Fig. 1. The lower frequency needs to be low enough (say, 30 cm^{−1} below the ionization potential) that one is in the spectral regime in which one can discern whether the control field F_{m} is on or off. In the same spectral range, one could also distinguish whether an ℓ changing field, F_{ℓ}, is or is not imposed for a second time (see Fig. 1). Therefore, one even has a threestage gate that evaluates w_{0}w_{1}F_{m}F_{ℓ}. Even w_{0}w_{1}F_{ℓ}F_{m}F_{ℓ} is possible, where the order of terms indicates the sequence of physical operations.
One can also use the experimental arrangements to define a controlled gate, as shown in Fig. 3. The role of the control field, F_{m}, is to determine which one of the two possible ZEKE spectra is obtained as the output.
The gates based on delayed pulsedfield ionization are not fast (see Fig. 1 legend for the time scales, which are tens of nanoseconds). We have used them in illustrating the logical circuits because of ease of presentation. One can realize far faster gates [e.g., about 1 ps (32) or even faster (8)]. The Rydbergbased gates are also one time and irreversible because the molecule is destroyed. These problems can be avoided by making use of the already demonstrated experimental ability to slice the ZEKE spectrum by the application of a stepwise ionizing field (18, 19) or by proceeding via different vibronic states of the electronically excited intermediate state [B^{1}Π_{u} in the case of Ag_{2} (33); A^{2}Σ^{+} for NO]. Other options include applying crossed electric and magnetic fields (34) so as to break the ℓ and m quantum numbers or to delay the second frequency with respect to the first. Timeresolved ZEKE spectroscopy (35) revealed that there is more than one time scale in the dynamics of high Rydberg states. This separation of time scales (6, 36), which is due to several bottlenecks in the time evolution of the optically accessed state (37), allows one further to add columns to the truth tables. A particular but important special case is the chemical transformations that are possible in the ionic core, about which the Rydberg electron revolves. These changes, such as dissociation, can be spontaneous (38) or lightinduced (39). It is a testimonial to the remarkable stability of the Rydberg electron that one of the two fragments of the dissociation is, itself, a high Rydberg state. It too can be probed by ZEKE spectroscopy, and mass resolution (40) can distinguish it from the parent. Another option that we will discuss below is the predissociation of the neutral excited state that is used as an intermediate in reaching the Rydberg state (31). An option that we do not use here is to induce transitions among the high Rydberg levels by millimeter radiation (41). This is a very promising direction because it makes a large number of states addressable by controlled fields. Instead of outlining all of the many options, we discuss next how to design a full adder.
The Full Adder
A full adder carries out a process somewhat more complicated than addition modulo 2. It must also add any carry digit brought from the previous adder and deliver any required carry digit to the next adder. Addition modulo 2, denoted by ⊕, is implemented by a socalled “halfadder.” As shown in Fig. 4, a halfadder does not require a carry input bit. To implement a halfadder one needs to monitor S_{n}, n low (42). This provides the sum, as S_{n} lights up if either photon is present. Detecting the signal S_{n} functions as an XOR gate, because absence of the signal indicates either absence of both photons or the presence of both. The presence of one photon but not the other produces signal S_{n}. The forward carry bit is generated when both photons are present, which allows the molecule to reach a high Rydberg state, and this provides the needed signature.
The halfadder uses only two photons (here, of different colors) and the molecule. In Fig. 5, we increase the concatenation by adding the role of one field. This is a twophoton, onefield scheme.
A full adder is a halfadder supplemented by the ability to allow for an input carry bit. This input bit, if present, of course, contributes to the sum. Unlike the halfadder, the logical evaluation of the sum is now more complicated because there are three ways for the sum to be unity. Fig. 6 indicates how this can be realized by using three photons. Experimental ZEKE spectra with three photons have been reported by Held et al. (27). The third frequency needed for the scheme proposed in Fig. 6 would have to be somewhat lower than that used in the Held experiment, but there is no reason the method should not work. The logic circuit corresponding to Fig. 6 is shown in Fig. 7.
The circuit diagram in Fig. 7 is quite concatenated. This is seen from the vertical lines that are drawn to show what the inputs are to each gate. In fact, the circuit is even more concatenated than shown because we used seven AND gates, each with three inputs. Such a gate is really two concatenated AND gates of two inputs each. (A circuit showing how to concatenate two such AND gates to get a threeinput AND gate is shown in Fig. 2.) There are also two OR gates with three inputs each. So the circuit contains 3 NOT gates, shown as triangles, 4 OR gates, and 14 AND gates. Of course, we do not propose to actually build such a circuit, just to indicate that the threephoton spectroscopy of a single molecule can be thought of as logically equivalent to such a circuit.
To summarize, just two photons suffice to implement an AND gate. The logic for this gate is done by the photons. The molecule is used for its resonance condition. Two photons plus a monitoring of the intermediate electronically excited state suffice for a connected circuit known as a halfadder (Fig. 4). For this scheme the key is the intramolecular dynamics. Two photons and a field is the experiment shown in Fig. 1 and discussed in Figs. 2 and 3. Here, too, the dynamics of the molecule play a key role. In the experiment of Fig. 1, the intermediate state is not monitored. If it were, the logical circuit is that shown in Fig. 5. With three photons (27), one already can do a full adder (Figs. 6 and 7). A circuit using as inputs three photons and one field is already beyond our graphical ability to display logic circuits on one page, so this is a good place to stop.‡‡ Before closing the discussion, we note that three photons and two fields (even three photons and three fields) is still within current experimental capabilities of what can be implemented by using the levels of a single molecule.
Fig. 7 shows a well connected circuit, so a clear comment on what we have not shown is also in order. We have demonstrated concatenation in the sense of logical equivalence between the dynamics of one molecule and a connected logical circuit. We have not shown how logical circuits of different molecules can be connected. Real integration ultimately will require that the logical output of one molecule is the input to another.
Concluding Remarks
The experiment on NO that is discussed and the many other options that are allowed by the existing technology demonstrate that several logic gates can be concatenated by using but one molecular frame. In other words, even relatively simple Hamiltonians can have a dynamical behavior that is rich enough for implementing nontrivial logic operations. These gates were shown to be possible even with the crudest experimental probes that are currently available. Refined detection methods allow the evaluation of even more complex Boolean functions. Some of the logical operations are implemented by the interaction of light with matter, and, so, it can be said that they are as much a result of the photons as they are of the molecule. Other operations require both the molecule and the external field. Other logical operations, such as the quenching of a ZEKE spectrum of Rydberg states of high but not very high values of n (see Fig. 1b) or the predissociation of low electronically excited states, are strictly a result of the nature of the dynamics of the isolated molecule itself.
Acknowledgments
The work of F.R. is supported by the Actions de Recherche Concertée (Direction de la Recherche Scientifique–Communauté Française de Belgique). This work was carried out under the auspices of the James Franck Program for Laser–Matter Interaction and by SFB 377. We thank Profs. E. E. B. Campbell and J. L. Kinsey for their extensive and useful comments on the manuscript.
Footnotes
 Accepted January 11, 2001.
 Copyright © 2001, The National Academy of Sciences
References
 ↵
 Schlag E W,
 Levine R D

 Sandorfy C

 Schlag E W

 Bixon M,
 Jortner J
 ↵
 ↵
 ↵
 Kompa K L,
 Levine R D
 ↵
 ↵
 Collier C P,
 Wong E W,
 Belohradsk M,
 Raymo F M,
 Stoddart J F,
 Kuekes P J,
 Williams R S,
 Heath J R
 ↵
 Collier C P,
 Mattersteig G,
 Wong E W,
 Luo Y,
 Beverly K,
 Sampaio J,
 Raymo F M,
 Stoddart J F,
 Heath J R
 ↵
 ↵
 ↵
 deSilva A P,
 Gunaratne H Q N,
 McCoy C P
 ↵
 deSilva A P,
 Dixon I M,
 Gunaratne H Q N,
 Gunnlaugsson T,
 Maxwell P R S,
 Rice T E
 ↵
 ↵
 ↵
 ↵
 ↵
 Ahn J,
 Weinacht T C,
 Bucksbaum P H

 Kwiat P G,
 Hugues R J
 ↵
 Meyer D A
 ↵
 ↵
 Tanenbaum A S
 ↵
 Shannon C E
 ↵
 Held A,
 Baranov L Y,
 Selzle H L,
 Schlag E W
 ↵
 ↵

 Merkt F,
 Fielding H H,
 Softley T P
 ↵
 ↵
Witte, T., Bucher, C., Remacle, F., Proch, D., Kompa, K. L. & Levine, R. D. (2001) Angew. Chem., in press.
 ↵
 Nemeth G I,
 Ungar H,
 Yeretzian C,
 Selzle H L,
 Schlag E W
 ↵
 ↵
 ↵
 Levine R D
 ↵
 Remacle F,
 Levine R D
 ↵
 Grebner T L,
 Unold P v,
 Neusser H J
 ↵
 ↵
 ↵
 ↵