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Shadows and Mirrors: Reconstructing FIYJ TODDY H P RIL 1998 'NEGATIVE PROBABILITIES' SHADOWS AND MIRRORS: RECONSTRUCTING QUANTUM STATES OF ATOM MOTION Tmagine that a pair of coins Quantum mechanics allows us only one mation about the marginal Xare tossed in a black box. probabilities of the individ- The box reports only one of incomplete glimpse of a wavefunction, ual coins.1 the following three results at but if systems can be identically prepared Although this example of random: (1) the outcome of coins in a black box is highly the first coin (heads or tails), over and over, quantum equivalents of artificial, a similar situation (2) the outcome of the second shadows and mirrors can provide the full arises in nature when we de- coin (heads or tails), or (3) picture. scribe the probability distri- whether the outcomes of the bution of a quantum me- two coins matched or were chanical particle in position- different. Our task is to con- Dietrich Leibfried, Tilman Pfau and momentum phase space. A struct a joint probability dis- classical particle occupies a tribution of the four possible Christopher Monroe single point in phase space, outcomes of the coins (HH, and an ensemble of classical TT, HT, TH) based on many observations of the black box particles can be characterized by a phase-space probability outputs. Now suppose that after many trials, the black distribution. On the other hand, the Heisenberg uncer- box reports that each coin comes up heads two-thirds of tainty relationship requires that a quantum mechanical particle be described by an area of uncertainty in phase the time when measured individually, yet the coins never space no smaller than Ax Ap = fi/2. If a particle's position match when they are compared. (Clearly the results of is known well, then its momentum is not, and vice versa. the coin tosses have been correlated—perhaps a joker in In mathematical language, the position wavefunction ^ (x) the black box flips the coins and then changes the out- x and momentum wavefunction ^p(p) are related by a comes appropriately.) We seek a distribution that both Fourier transform; thus, localized position wavefunctions reflects this correlation and obeys the marginal distribu- lead to delocalized momentum wavefunctions, and vice tions of each coin as two-thirds heads, one-third tails (see versa. A probability distribution in quantum phase space the three tables on page 23). The only way to satisfy both must somehow incorporate this feature. requirements is to force the joint probability P(TT) of getting two tails to be negative! Mathematically, this is because P(HH)+P(TT) is observed to be zero, yet we Wigner distribution and 'negative probabilities' expect P(HH) to be greater than P(TT), because the indi- In 1932, Eugene Wigner presented a convenient mathe- matical construct for visualizing quantum trajectories in vidual coins are weighted toward heads. 2 The sleight of hand giving rise to negative prob- phase space. The Wigner distribution, or Wigner function abilities is that we have attempted to reconstruct a joint W(.r,p), retains many of the features of a probability probability distribution without ever having observed in- distribution, except that it can be negative in some regions dividual joint outcomes of the coins. The only measured of phase space. In the coin example above, practical use events are described by sums of joint probabilities such of the quasi-probability distribution of table 3 is limited as P(HT) + P(TH) = 1 or P(HH) + P(HT) = %. One way to to events described by sums of any two entries. Similarly, interpret the distribution of table 3 is to note that, since when we apply the Wigner distribution to measurements individual joint outcomes of the coins are inaccessible, in quantum phase space, the probability distribution for nothing prevents us from assigning negative probabilities the outcome of a measurement is obtained essentially by to such immeasurable events. With this rule in mind, convolving W(x, p) with a distribution of possible states of this joint "quasi-probability" distribution may be a useful the measurement device, which must be distributed over bookkeeping tool, as it not only characterizes the hidden an area of order h or larger. This prescription leads to a correlations within the black box, but also retains infor- natural connection between quantum and classical phase space: As the measurement resolution is degraded away from the quantum limit so that the Heisenberg uncer- DlETRICH LEIBFRIED is a physicist at Innsbruck University in tainty relationship plays no role, localized regions of Austria. He was a guest researcher at the National Institute of W(x, p) (with possible negative values) become washed out, Standards of Technology in Boulder, Colorado, during the and the convolved Wigner distribution approaches the writing of this article. TlLMAN PFAU is a physicist at the usual classical phase-space probability distribution. University ofKonstanz in Germany. CHRISTOPHER Similar to the quasi-distribution of the coins above, the MONROE is a staff physicist at the National Institute of Wigner distribution is not a bona fide probability distri- Standards and Technology in Boulder, Colorado. bution, but can be a useful bookkeeping tool that high- APRIL 1998 PHYSICS TODAY V?n American Institute of Physics, S-OO3l>jL28-<)S04-010-l momentum FIGURE 1: WIGNER FUNCTION FOR THE DOUBLE-SLIT EXPERIMENT, visualized in phase space, a: The initial Wigner distribution representing the superposition of two Gaussian lobes directly behind the slits. The oscillating part in the center is due to the spatial coherence between the two lobes. The "spacehke shadow" (on the orange screen, at left) shows the spatial marginal distribution l^fx)!2 of the state, obtained by ignoring the momentum information. The pale burgundy shadow at rear shows the corresponding : "momentum-like shadow" |^f,(jp)| . With a position-sensitive detector measuring the spacelike shadow, we can view the initial Wigner distribution from different angles by either rotating it (b) or shearing it (c). lights the inherent anticorrelation of position and momen- tum uncertainty. TABLES: Joint probability distributions of the outcomes For a pure quantum state, the Wigner distribution is of tossing two coins. (1) Probability distribution given related to the position or momentum wavefunction by that the two coins are tossed independently, both weighted toward heads with P(H) = 2A and P(T) = \'\. The marginal probabilities of the outcomes of either coin (obtained by W(x,p) = -±- Jx +1 d.s adding the entries vertically or horizontally) result in two-thirds heads and one-third tails. (2) Probability distribution given that the coins are tossed in a black box that reports that the coins never match—that is, P(HH) + ATT) = 0. The off-diagonals add to 1 as required, (1) with p arbitrary. The marginal probabilities can no longer be two-thirds heads for both coins. (3) "Quasi-probability" distribution under same conditions as (2) and also where we have set ft = 1. The Wigner distribution of a exhibiting marginal probabilities of each coin as two-thirds mixed quantum state is a weighted sum of either expres- l h d h f h sion over the appropriate wavefunctions. These expres- neaus i u oiie-uni u 1 lit piice paiu li iiiai u me sions may not be very illuminating, and the equivalent of entries is negative! the Schrodinger equation describing the time evolution of W(x,p) is even less so. (However, Wolfgang Schleich and H T H T H Georg Stissmann discussed a physical interpretation of 2 H 0 H y this form of the Wigner distribution in PHYSICS TODAY, H % /9 P '/„ 2 October 1991, page 146.) Nevertheless, Wigner showed T 2 T \-p 0 T that W(x,p) is indeed the closest thing we have to a prob- A '/, '/: ability distribution in quantum phase space, as it corresponds (I) (2) (3) to the phase-space probability distribution in the classical APRIL 1998 PHYSICS TODAY 23 pulsed source screen momentum-sensitive detector yields the "momentum-like 50 urn shadow" l^pfp)!2. As shown HD*-<[ f )))) in figure la, we can observe 1.2 - a ( <• shadows across different an- gles in phase space either by a l.o- rotating the detector's point u 10umi of view or by rotating the - 0.8- 0.2- 0.2- Wigner distribution and keep- o ing the detector fixed. For CO 0.6- example, figure lb shows the E3Q Wigner distribution rotated r3 0.4- by 60° and measured with O 0.1- 0.1 a position detector. The spacelike shadow on the 0.2 screen now contains informa- tion about both x and p of the initial distribution. The POSITION c POSITION d POSITION Wigner distribution can be FIGURE 2: DOUBLE-SLIT ATOM INTERFEROIVIETRY. a: The experimental arrangement. sheared in phase space as Atoms from a colhmated source propagate through a double slit with 8 fim separation and shown in figure lc by allow- strike a position-sensitive detector screen. The source produces fast-moving atoms and a range ing the particle to evolve of slower atoms, b: Diffraction pattern of the atomic matter waves plotted as a function of freely. A shear rotates the position and of the propagation time tj for the atoms to travel from the double slit to the spacelike shadow and gives detector. The distance from double slit to detector is d = 195 cm, for which the slow atoms an additional stretching, propagate long enough to produce a Fraunhofer (far-field) diffraction pattern. The fast atoms which can easily be compen- produce the near-field shadow of the slits at the bottom. (This shadow is magnified because of sated for.
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