C. Ell, H. M. Gibbs, G. Khitrova,
E. S. Lee, and S. Park
Optical Sciences Center,
University of Arizona,
Tucson, AZ 85721
and
D. G. Deppe and D. L. Huffaker,
Microelectronics Research Center,
University of Texas,
Austin, TX 78712-1100
A quantum dot in a semiconductor nanocavity, exhibiting single-exciton/ single-photon strong coupling, is proposed as a solid-state system for quantum entanglement. The key is lateral confinement to minimize the cavity- mode volume.
A quantum well grown in the center of a semiconductor microcavity gives rise to a nonperturbative light-matter interaction resulting in two transmission peaks and reflection dips – the normal-mode-coupling between one cavity mode and one exciton resonance [1]. This effect is often related to vacuum-field Rabi (VRS) splitting of atoms or to a polariton describing light propagation in a medium. The regime of many-oscillator vacuum-field Rabi splitting was reached in the early 1980’s using many atoms in a cavity - after years of hard work to improve the cavity quality factor Q, to make the photon escape rate less than the many-atom VRS oscillation rate. After a decade of gradual improvements, vacuum-field Rabi splitting was seen with a single atom [2,3]; this led to the first clear demonstration of the discrete nature of the coherent exchange of energy between the atom and the quantized electromagnetic field [4]. For such a truly quantum system the optical properties are changed by the addition of a single photon or a single atom.
Present-day semiconductor microcavities showing normal-mode coupling, however, are still in the opposite limit, the limit of many atoms, where many photons are needed to change the optical properties. This system behaves semiclassically, i.e., the light-matter interaction can be described equally well on the basis of a classical field in linear and nonlinear studies [5-8]. E.g., we measured 2 ´ 105 photons to be absorbed nonresonantly from a 150-fs pulse in a 50-µm-diameter beam to reduce the peak exciton absorption coefficient by two and the VRS transmission by five [9]. Thus we are far from the quantum limit. It is the very large QW absorption coefficient, i.e., the potential for the formation of many excitons, that gives rise to the VRS, much like the many-atom case. To obtain a useful quantum gate, the number of oscillators causing the VRS must be reduced to one [10]. Then the oscillator-cavity system will be truly quantum; a second photon will feel the first-photon/ exciton entanglement. And expectation values of matter-field operators cannot be factored into a product of a matter expectation value and a field expectation value.
The following questions are addressed here. How can we realize a quantum dot that would reduce the potential of forming many excitons? Can the vacuum field be increased sufficiently by reducing the cavity-mode volume, so that a single quantum dot can exhibit VRS similar to an atom in a high-finesse cavity [11,12]? Such a quantum-dot nanocavity has two advantages: a length close to the theoretical minimum and a much larger dipole moment permitting a strong coupling without an ultra high-Q cavity even for such a short length. The single-atom VRS is g0 = µEvac. The vacuum field Evac at the oscillator site is given by e0 /Evac/2V = hw/2, with
Thus
the larger the dipole moment µ of the oscillator and the smaller the cavity-mode waist w0 and length Lcav, the larger the splitting 2g0.
While the cavity length for a 1l spacer is already close to its minimum value of Lcav » 1µm, the vacuum field can be increased by oxidation of an A|As layer to form an aperture, a technique we have already used to fabricate record-low-threshold quantum- dot VCSELs [13-16]. Figure 1 shows an example of a quantum dot microcavity. The depths of the oxidation fronts in the different layers are set by the Al composition in the starting AlGaAs layers. The fully oxidized layers are Al0.98Ga0.02As, while the partially oxidized layers have a reduced Al content, such as Al0.6Ga0.4As. The cavity spacer region contains self-organized quantum dots with a density of » 1010cm-3. A 0.5-µm-diameter mode will contain » 20 quantum dots with different resonance frequencies. The l/2 spacer combined with the high-contrast mirror can give an effective cavity length of 1l = l0/n and the minimum mode volume may be as small as 2l3. Here, l is the exciton wavelength and n = e01/2 is the refractive index, respectively.
To see vacuum-Rabi splitting, g0 must exceed the photon decay rate dc and the oscillator’s dephasing rate g. The VRS of N identical oscillators is W0
= 
. If but
< dc, g but g0
< dc, g,
N-atom VRS is much easier to see than single-atom VRS needed for a quantum gate [11]. Our
planar semiconductor normal-mode-coupling microcavities, each containing one or more
narrow-linewidth quantum wells (QW) in the antinodes of the electromagnetic field, exhibit
record many-oscillator vacuum-field Rabi splitting [5-8]. They show a single QW splitting
of > 3meV. We even see VRS at 300K [17,18]. The question arises, whether these
VRS samples can also be modified such that they may operate in the single-oscillator VRS
regime [4]? Since the QW and cavity-mode areas (AQW,Acav) are equal
in the usual VRS microcavity, W0 does not change
with the waist w0 of the cavity mode volume. But N decreases
as 
; so the
key is to reduce w0. Extrapolating N »
105 for w0 » 50mm gives N » 1 for a QW
diameter of dQW = 158nm. Since an optical-cavity diameter can be as
small as l0/n, or 115nm for l0 = 830nm, the quantum limit cannot be ruled
out. With present technology, w0 » 1µm
seems more realistic. In general, the QW diameter could be less than w0,
so
where µ2/AQW is a constant. For a QW with dQW = 158nm in a 1-µm-diameter nanocavity, this extrapolation gives a vacuum-field splitting of
Since this value exceeds VRS linewidths already achieved in present-day quantum-well microcavities [8], such a small-diameter QW, that can be considered as a big “quantum dot”, should reach the quantum limit. Finally, we show that the regime of true strong coupling is also conceivable using the well-studied monolayer-fluctuation quantum dot of Gammon et al. [19] in a nanocavity with cavity-mode volume of (1µm)3. Already Gutbrod et al. [20] report a 2-meV VRS splitting for a QW in a 1.2- µm -diameter “photon dot.” And an aluminum-native-oxide aperture around quantum dots has led to a record-low-threshold VCSEL [14]. In a monolayer-fluctuation dot, fluctuations by one monolayer in the height of the GaAs surface result in several-meV potential wells for lateral confinement. Nonlinear optical experiments [21] on such a 40-nm dot conclude that it behaves like an atom; this would seem to imply that dephasing is dominated by radiative decay. However, Fan et al. [22] studied the same dots and concluded that the radiative recombination time » 4´ the dephasing time. Since the radiative rate in a dielectric medium is
where
is the free-space radiative rate [23,24],
The cavity photon decay rate is given by
.
Compare parameters of a quantum-dot [19] nanocavity, with mirror reflectivity of 99.8%, with those in parentheses for a cold Cs atom [12,25]: l0 = 830nm (852.4nm); t = 4 × 29ps (32ns); e0 = 13 (1); w0 = 1µm (13.5µm); Lcav = 1µm (10.1µm), 2g0 = 70µeV (»1µeV); dc = 55µeV (165neV); g = 11.5µeV (10.7neV); Q = 1.4 × 104 (4.4 × 106). Since the quantum-dot nanocavity VRS FWHM linewidth is (g + dc) » 66µeV, comparable to 2g0 = 70µeV, vacuum-Rabi splitting may be resolvable. The quantum-dot nanocavity VRS is much larger than that of Cs because its dipole moment and vacuum field are » 6 ´ and » 12 ´ larger, respectively. Only moderate Q is needed for a quantum-dot nanocavity. These estimates show that single-oscillator VRS in a quantum-dot nanocavity is difficult but not inconceivable.
Support is acknowledged in Tucson from AFOSR/DARPA, JSOP (AFOSR, ARO) and NSF (AMOP, EPDT) and in Austin from the University of New Mexico DARPA sponsored Optoelectronic Materials Center.
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Toward Quantum Entanglement In A Quantum-Dot Nanocavity
Epitaxial long wavelength DBRs on InP: AlAsSb or lateral oxidation
High Temperature 1300 nm VCSELs for single-mode fiber-optic communication
Manufacturing of Oxide VCSEL at Hewlett Packard