Research Update: December '96 - September '97
Overview:
This phase of the research program is principally experimental in character, with overall objectives that can be placed into two broad categories. In the first instance, the goal is to investigate quantum logic with single-photon pulses, including applications to both "digital" and "analog" quantum computation as well as to quantum communication. This work will build on our recent experimental demonstration of conditional logic at the single photon level with a quantum phase gate [a]. Here the individual quantum bits (or "qubits") are optical photons, with the internal state specified by polarization and with interactions between qubits proceeding via strong coupling to an atom in an optical cavity. This experiment represents the first realization of conditional dynamics suitable for quantum logic with single photons.
The second major aspect of the research will be directed toward the realization of the scheme for quantum computation proposed by Pellizzari, Gardiner, Cirac, and Zoller [4]. In this scheme, a set of atoms (or ions) is placed inside an optical resonator, with each atom at a distinct site. Individual atoms serve as qubits with information encoded in the internal atomic state. Computation proceeds by a series of one and two bit interactions between the atoms. In contrast to the original Cirac-Zoller proposal for computation with a linear array of ions where the interaction proceeds via a vibrational coupling [5], here the interactions between qubits are accomplished via single photons in the intracavity field (with which all atoms interact).
Our investigation of optical systems is motivated in large measure by realistic prospects for the laboratory implementation of elementary quantum logic operations. Moreover, the systems that we propose to investigate should provide a powerful meeting ground for theory and experiment in terms of assessing the impact of dissipation, for testing various error correction protocols, and for implementing realistic simulations. A major advantage of our atom-cavity systems compared to other proposals for quantum logic is the ability to observe directly and efficiently the principal dissipative channel, namely the decay of the cavity field. In the longer term, optical systems offer potentially high "clock" rates as set by the fundamental atom-field coupling coefficient g /2¹ ~ 100MHz.
(a) Progress to date in the demonstration of a quantum-phase gate
Although numerous physical systems have been proposed for the implementation of a quantum gate [4-6], thus far only two schemes have successfully demonstrated conditional dynamics at the single quantum level that are suitable for quantum logic [a, 3]. The scheme that we have pursued for implementing quantum logic in cavity QED [a] employs an atom-cavity system as a two-bit gate, processing qubits in the form of two monophotonic pulses (a, b) either propagating through or reflecting from an optical cavity. To make the two qubits distinguishable, a small frequency offset between the two fields is introduced. As computational basis we chose the single photon ± circular polarization states (|1±>) for channels a and b.
As described in the attached reprint (Section III), for the particular configuration of our atom-cavity system, the relevant transformation of input fields to output fields is assumed to be of the following form:
(1)
where the states of the basis are {|1->a|1->b,
|1+>a|1->b, |1->a|1+>b,
|1+>a|1+>b} for the qubits a and b. Here, (phia,phib)
are the phase shifts for either of the qubits (a,b) coupled individually
to the strong transition. The quantity delta parametrizes conditional dynamics;
for nonzero delta, the total phase shift for both fields coupled to the
strong transition is different from the sum of the individual phase shifts.
Indeed, as one of us has shown [b], delta not identically zero is precisely
the condition for the gate to serve as a universal component for quantum
logic. Transformation (1) is the quantum version of a classical truth table,
specifying the output as a function of the input. As U only changes the
overall phases of the basis states, the corresponding gate is called a
quantum phase gate (QPG).
To achieve conditional dynamics, i.e., to make the phase shift of the probe beam a dependent on the polarization of the control beam b, the intensity of the control beam must be increased beyond the critical photon number (mo = 0.02 photons for our system for zero detuning) into the nonlinear regime. Here, the control beam begins to saturate the atom in the cavity so that cross-phase modulation of the probe beam is achieved. For example, for detunings Wa = 30 MHz and Wb = 40 MHz from the common atom-cavity resonance, we observe a 30% reduction of the probe phase |phia| for an intracavity control photon number mb ~ 0.2 photon and an average atom number N ~ 1, demonstrating conditional dynamics at the single-quantum level. It should be noted that in the experiments performed so far, each input beam is in a coherent state and therefore not in an eigenstate of transformation (1) which is based on one-photon Fock states. Nevertheless, we are able to extract quantitatively the single-photon phase shifts defined in Eq.(1). Explicitly we find that delta may be determined directly from measurements of the initial slope d(phia )/d(mb ) in a plot of the phase phia of the probe field versus control intensity mb. By way of this technique, we obtain one-photon phase shifts (phia,phib) ~ (8, 11)° and a conditional phase shift delta ~ (17±3)° and confirm that delta is invariant under the exchange of the frequencies of control and probe, corresponding to the symmetry of the last diagonal element in Eq. (1) under the exchange of qubits a and b.
b. Program of research
Our experimental demonstration of a non-zero delta for single atoms and sub-photon intracavity fields shows that our cavity QED-scheme is quite promising for performing quantum logic using polarization-dependent conditional phase shifts. By choosing a different set of basis states, other types of gates may be realized. For example, transforming the probe field basis from circular to linear polarization, the gate would act as a Controlled-NOT for delta = pi. Although a significant first step has been taken, there remain a number of important questions of both a fundamental and practical nature which we propose to address. Our program of research is to proceed along two main avenues, namely (1) an effort to quantify the efficacy of single-photon pulses for quantum computation and (2) a program to investigate and implement the scheme of Pellizzari et al. for a cavity-QED quantum computer.
With regard to employing single-photon pulses as quantum bits, a critical issue is to develop a fully quantized theory that adequately describes the input, interaction, and output of single-photon wave packets in the setting of cavity Q.E.D. This is a nontrivial exercise in quantum field theory since the interaction of the monophotonic pulses in the quantum-phase gate is necessarily strong (nonperturbative) and involves some degree of dissipation. However, we believe that it is essential to make progress on this front in order to understand the potential utility of single-photon pulses for quantum logic and computation.
On the theoretical front, we are attempting to formulate a full quantum theory based on the master equation for a sequence of cascaded quantum systems [7]. Here, a pair of monophotonic pulses are generated as the output of a suitable cavity system, either by direct decay of an initial state or by active generation as in the quantum state synthesis scheme of Ref. [8]. These pulses then propagate and interact with a cavity QED system containing a single atom, as described in the preceding section. We propose to explore theoretically the issues surrounding the generation of quantum-state entanglement by the quantum-phase gate, which becomes a nontrivial undertaking for even two qubits in the (inevitable) presence of dissipation. On the experimental front, we will pursue a measurement program related to the response of our quantum-phase gate to inputs composed of single photons. We will investigate effects related to timing and pulse reshaping for single-photon wavepackets, such as the impact of nonlinear dispersion on pulse shapes, the generation of field frequencies other than those of the incident pulses, and the overall efficiency of input to output. A major theme of the research program is to establish necessary and sufficient testing procedures (i.e., measurement strategies) for the direct experimental determination of the "fidelity" of a given "black box" laboratory system with respect to the implementation of quantum computation. Note that no such operational criteria currently exist, although the preservation of coherence and the generation of entanglement must surely be necessary conditions for calling some candidate device a quantum gate. Hence, an important objective will be to demonstrate quantum-state entanglement for the fields emerging from the QPG and to quantify the role of decoherence in the gate transformation.
From a technical perspective, this research requires nontrivial advances on several fronts. First and foremost is the need to eliminate the fluctuations inherent in an atomic beam and to work instead with a single atom trapped within the mode of a high finesse optical resonator. We are currently exploring two avenues toward this end, with the first being the use of light forces to confine a single neutral atom and the second being the localization of an ion with an RF Paul trap. Note that we have already made reasonable progress as documented in our report of an individual Cesium atom trapped in free-space with a magneto-optical trap [9]. In collaboration with our group, Dr. A. S. Parkins (University of Auckland) has carried out numerical simulations which indicate that it should be possible to trap a Cesium atom in a high finesse optical cavity with an average intracavity field of less than one photon. The ion trapping research is being carried out in collaboration with the group of Dr. L. Maleki at JPL, with the initial effort being an assessment of the operation of an RF Paul trap located in the vicinity of dielectric surfaces (i.e., the mirrors forming the optical cavity). The principal problem that we are exploring is the charging of the dielectric surfaces and techniques to mitigate this problem. In the longer term, we are endeavoring to isolate a linear array of atoms within an optical cavity. Such capabilities would enable us to explore the implementation of a quantum computer in cavity QED as suggested by Pellizzari et al., and to which we now turn our attention. In the scheme of Pellizzari et al. [4], the "wiring" or interaction of one atom i with a second atom j is accomplished via interactions of atoms (i, j) which involve single photons in the intracavity field. More specifically, the interaction proceeds via a process of adiabatic passage whereby internal state information from atom i is mapped first into a photon in the intracavity field and thereby into atom j. The process of two-atom adiabatic passage is an extension of the quantum-state mapping problem described in Ref.[8] for a single intracavity atom. Note that neither the one-atom nor the more ambitious two-atom state mapping process has ever been carried out with quantum fields, although the former has received considerable attention for the case of coherent-state (classical) fields.
Beyond the particular setting of the quantum computer in cavity QED as proposed by Pellizzari et al. [4], we have been exploring the question of whether adiabatic passage could be employed to map readily generated superpositions in a Hilbert space of internal atomic states into entanglements in the joint Hilbert space of two distinct cavity modes. If such were possible, we would be able to synthesize diverse quantum states of interest for quantum computation and communication, such as so called Bell and GHZ (Greenberger-Horne-Zeilinger) states of the field. Furthermore, if we could find a way to implement general two-channel state mappings, we should be able to provide a compact and efficient scheme for quantum error correction for the quantum computer of Pellizzari et al. [4], along the lines suggested by Mabuchi and Zoller [g].
Hence, another major theme of our experimental research program will be an investigation of quantum state synthesis via adiabatic passage. The initial work will involve a single intracavity atom and will attempt to generate one and two photon Fock states for a single mode, as well as entanglements for two-mode fields. Although the technical challenges associated with this aspect of the proposal are nontrivial, we have nonetheless already made some headway in the design and construction of an apparatus for preliminary studies of quantum-state generation via adiabatic passage. In the longer term, we propose to explore two-atom adiabatic passage as would be required for the scheme of Pellizzari et al., with the technical approach involving the localization of a pair of atoms inside a high-finesse optical cavity as described above.
Finally, we note that the two principal thrusts of the experimental program have a large degree of overlap, in both conceptual and practical terms. For example, apart from the fact that monophotonic pulses can be used to implement quantum logic (in conjunction with quantum-phase gates), we view the investigation of the input and output of single-photon wavepackets as fundamental to the question of distributed quantum computing whereby individual "quantum processors" communicate via monophotonic pulses, including possibilities for the distribution of quantum-state entanglement. Similarly, one likely scenario by which these monophotonic pulses could be generated is that of adiabatic passage, which is the same basic avenue as that along which quantum computations might proceed in the individual processors.
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