We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values $(x,y)$ to $(x,x \oplus y)$) is universal in the sense that all unitary operations on arbitrarily many bits $n$ (U($2^n$)) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two-and three-bit quantum gates, the asymptotic number required for $n$-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary $n$-bit unitary operations.Comment: 31 pages, plain latex, no separate figures, submitted to Phys. Rev. A. Related information on http://vesta.physics.ucla.edu:7777

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We report the development of an atom interferometer that uses optical standing waves as phase gratings and operates in the time domain. The observed signal is entirely caused by the wave nature of the atomic center-of-mass motion. The opportunities to measure recoil frequency and gravity acceleration are demonstrated. [S0031-9007(97) PACS numbers: 03.75. Dg, 06.20.Jr, 32.80.Pj, 42.50.Md Atom optics and interferometry is a field that has undergone considerable development in recent years [1]. Atom interferometric techniques have been used to make a number of new and high precision measurements. Examples are measurements of the atomic index of refraction of a gas [2], the loss of atomic coherence in spontaneous emission [3], the Earth's gravitational acceleration g [4],h͞m [5], and precise values of atomic level spacings [6,7]. Atom interferometers (AI) fall into two classes, "microfabricated structure AI" [8-10], which use microfabricated structures as beam splitters, and "optical field AI" [4][5][6][7][11][12][13][14], which use optical fields as beam splitters. The atoms in different arms of microfabricated structure AI are in the same internal state, while in optical field AI they can be in different [4][5][6][7]11], or the same [12 -14] internal states.In this paper we report the development of an optical field atom interferometer in which pulsed standing wave optical fields act as phase gratings on an "uncollimated" cloud of cold atoms. As in Refs. [8-10,12 -14] this interferometer is a "de Broglie wave" interferometer in that the beam splitters do not alter the atomic internal state; the interference occurs between different paths of the atomic center-of-mass. In contrast to optical field AI using atomic beams [12,13], all the atoms in our interferometer interact with the light fields for the same amount of time, eliminating broadening effects due to velocity dispersion, and the time delay between the standing wave fields can easily be varied. The operation of the interferometer depends critically on a mechanism similar to that occurring in photon echo formation. As in other echolike interferometers [6,7,10,11], collimation of the transverse atomic velocity distribution to better than a photon recoil momentum is not necessary. As is discussed below, a time-domain atom interferometer of this type offers a unique combination of features that are well suited to high precision measurements and complement those of other atom interferometers. We have used this interferometer to measure the recoil frequency of a 85 Rb atom and the acceleration due to gravity ("little g").In our experiments two off-resonant standing wave pulses separated by a time T are applied to a sample of cold (150 mK) 85 Rb atoms [15]. The first laser pulse imposes a spatial phase modulation on the initial atomic state, which, due to the dispersion of de Broglie waves in free space, evolves into an amplitude modulation (representing an atomic population grating). For short times this population grating can be explained as the focusing of ...

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The spectral density of Nyquist noise current in a tuned circuit coupled to a sample of nuclear spins has been measured at 4 He temperatures with a dc SQUID used as a rf amplifier. When the sample is in thermal equilibrium, a dip is observed in the spectral density at the Larmor frequency. For zero spin polarization, on the other hand, a bump in the spectral density is observed. This bump is due to temperature-independent fluctuations in the transverse component of magnetization, and represents spontaneous emission from the spins into the circuit.PACS numbers: 33.25.Fs, 05.40. + j, 74.50. + r, 78.90. + t In his pioneering paper 1 on nuclear induction, Bloch noted that in the absence of any external radiofrequency (rf) driving field a sample of N spins of magnetic moment fi contained in a pickup coil would induce very small voltage fluctuations proportional to N 1/2 fi.In this Letter, we report the observation of these temperature-independent fluctuations at liquid-4 He temperatures arising from the 35 C1 nuclei in NaC10 3 at the nuclear quadrupole resonance (NQR) frequency of about 30 MHz.In the experiment, a sample of nuclear spins is placed in the inductor L p of a tuned LCR circuit and the spectral density of the current fluctuations is measured over the bandwidth of the circuit. The circuit resistance R; produces a Nyquist voltage noise and therefore a current noise that, in the absence of a sample, has a Lorentzian spectral density. The presence of the sample is found to modify the shape of this noise-power spectrum in the region of the NQF frequency. The influence of the sample is determined from its complex spin susceptibility 2 X(co)==X'(co)~jX"(ca), where X' and X" are the dispersion and absorption. The complex impedance of the coil in the presence of the sample is written aswhere £=2^/3^ is the sample filling factor; 7T S and f c are the volumes of the sample and the pickup coil. The added spin inductance L S =4TT%L P X' shifts the circuit resonant frequency, while the added spin resistance R s =4TrgcoL p X" modifies the damping of the circuit and acts as a source of Nyquist noise. This noise is due to spin fluctuations in the transverse direction. To observe these fluctuations in a reasonable averaging time one requires, first, that R s /R t be not too small, and second, that the noise current be measured by an amplifier with a noise temperature comparable with or smaller than the bath temperature T.We can compute the Nyquist noise generated by the spins in terms of the microscopic parameters of the sample. Since the NQR sample is equivalent to a two-level system, 3 we take as a model an ensemble of spins in an external magnetic field H z z with spin / = y, spin density n =N/Tly and Larmor frequency CO S /2TT=YH Z /2 9 where y is the gyromagnetic ratio. The axis of the pickup coil is along the x direction. We ascribe a spin temperature T s to the magnetizationWe assume that Bloch's equations apply, so that X" =X'/AcoT 2 is given bywhere Aco=co s -co, and the linewidth is given by Af s = l/TrT 2 . Th...