Testing the Randomness of Quantum Mechanics

D.J. Berkeland (P-21)

Possibly the most nonintuitive aspect of quantum mechanics is that a single particle can be put into a superposition of two distinct states. Moreover, when one makes a measurement, the particle is found in only one state, and that result is unpredictable, or random. Since its inception, quantum theory has been rigorously tested under many diverse conditions and often with extremely high precision. Surprisingly, there are very few statistically significant tests of the randomness of a quantum-mechanical process, including the transitions between quantum states. Some experiments have monitored the decays of a large sample of nuclear particles, whereas others have measured whether a photon is transmitted or reflected from a beamsplitter. However, these methods have limitations such as accounting for interactions between their nuclei or the inability to detect every decay particle or photon—only a small level of paranoia is required to imagine that the detectors are missing patterns in the directions of decaying particles or in the timing of photon transmissions.

It is important to improve these tests of the statistics of quantum-mechanical processes for several reasons. First, quantum mechanics is such a fundamental part of our view of the physical world that we must test it as carefully as possible. History is full of scientific theories that were widely accepted until precise and accurate measurements illuminated their subtle deficiencies. Second, applications such as quantum cryptography rely on the generation of strings of numbers that are as random as possible. Devices based on quantum-mechanical processes are ideal candidates for quantum cryptography. We must therefore demonstrate that the underlying processes behind these devices are indeed free of cyclic behavior and correlations between number sequences. Finally, the trapped strontium ions that we use to perform our experiments could also be used to implement a quantum computer. It is imperative that the quantum-mechanical processes that make quantum computation so powerful are not compromised by systematic effects. For all these reasons, we have developed an experimental system based on trapped strontium ions that permits us to observe spontaneous and laser-induced transitions between internal states in single ions and pairs of ions. We then statistically analyze them, searching for signs of memory in these physical systems and patterns in their behavior.

Trapping Ions to Study Quantum Effects

Our tests of quantum mechanics are, in principle, cleaner than those of previous experiments because we monitor the transitions of a single ion between two sets of its internal states. Because we use only a single ion that is suspended in space and localized to less then 100 nm by electric fields, our experiments are not susceptible to multiparticle effects. Also, because we can tell with near-unity efficiency the state of the ion, our experiments are immune to detector-efficiency loopholes. Previous researchers have used a similar trapped-ion system to analyze approximately 1,000 such transitions; we analyze 240,000 transitions in single ions and 230,000 transitions and 8,600 spontaneous decays in two simultaneously trapped ions.

To do this, we first confine ions in a trap such as that shown in Figure 2(a).¹ An rf voltage is applied to two diagonally opposite rods, while dc potentials are applied to the remaining electrodes. This creates a time-averaged potential that forces ions towards the trap’s long axis. We apply several hundred volts to the “sleeve” electrodes to keep the ions from leaking from the ends of the trap. Ions are formed inside the trap when neutral strontium vapor from a small oven intersects with an electron beam from a tungsten filament. The whole apparatus is inside a small chamber at ultra-high-vacuum conditions.

Typically, tens of ions are created inside the trap. They make a relatively hot cloud that is hundreds of microns long and about a hundred microns in diameter. The motion of the ions is forced by the trap’s rf electric field and by the Coulomb interactions between the charged ions, and individual ions cannot be distinguished. In this state, they are not useful for our experiments, so we reduce their motion by Doppler cooling them with laser light. In this process, 422-nm laser light is tuned slightly below the ions’ S_1/2 ↔ D_5/2 resonance (Figure 3). When ions travel towards the light source, they absorb a 422-nm photon, which reduces their speed due to conservation of momentum. On the other hand, if the ion is moving away from the light source and absorbs a photon, its speed increases. But the frequency of the laser light is such that an ion moving away from the light source is Doppler shifted far out of resonance with the light. So, on average, the laser light cools the ions.

When the ions are cold enough, they undergo a sudden phase change, freezing into ion crystals such as that shown in Figure 2(b). This shows a crystal of five strontium ions that scattered 422-nm light into an imaging camera. The ions are forced together by the trap potential that we have applied to the sleeve electrodes, and they are forced apart because they are all positively charged. Typically, the ions are spaced tens of microns apart. We have formed linear chains of approximately 40 ions but typically experiment with only a single ion in the trap. Once an ion is trapped and laser cooled, it stays in the trap indefinitely so that we can perform experiments that were considered impossible when quantum mechanics was first conceived.

Observing Quantum Jumps in Trapped Ions

For example, we can observe quantum jumps. To begin, we can briefly drive the S_1/2 ↔ D_5/2 transition with a 674-nm laser while the 422-nm light is blocked. After the laser pulse, we can ask whether or not the ion is in the long-lived (τ = 0.4 s) D_5/2 state. To do this, we shine 422-nm and 1,092-nm light on the ion. If the ion is in the D_5/2 state, then neither of these lasers can drive a resonance in the ion; the detector that would observe 422-nm light from the atom does not register any signal. But if the 674-nm laser failed to drive the ion to the D_5/2 state, 422-nm and 1,092-nm light continually excites the atom, and the detector registers tens of thousands of blue photons in a single second. As we scan the 674-nm laser frequency, we observe a resonance such as that in Figure 4.

Instead of pulsing the red 674-nm laser light while the blue 422-nm laser light is blocked, we can leave all of the lasers on at the same time. Then it is as though the 422-nm light were continuously measuring whether or not the 674-nm laser has driven the atom into or out of the D_5/2 state. As quantum mechanics predicts, the results of such a measurement (i.e., is the atom in the D_5/2 state or not) should be unpredictable. Indeed, the 422-nm signal from the ion under these conditions is shown in Figure 5, and it randomly and suddenly switches between a large and small value. We collect such data in continuous blocks of approximately 30 minutes each, during which we monitor on the order of 10,000 quantum jumps. In total, we analyze 230,000 quantum jumps in a search for patterns or correlations in the times between jumps.²

Analyzing the Trapped-Ion Data

Although there are very many different statistical tests that have been performed on our data, we will illustrate only one in this article. We ask the following question: “If we are told the interval time between one set of quantum jumps, do we then have more information about subsequent interval times than we would otherwise?” The most direct way to answer this question is to measure the joint entropy between pairs of intervals. The entropy of a set of data tells us how many bits of data are required to describe the full data set; the more random the data, the higher the entropy. The joint entropy for two data sets tells us how many fewer bits are required to describe one data set if the other data set is known. We normalize this value so that if the data sets are completely correlated we obtain a value U = 1, and if they are completely unrelated, we obtain U = 0.

For example, if we have a stack of playing cards ordered by the face value of the cards (so the four 3s are together, the four 8s are together, etc.), if a 6 is drawn from the top of the deck then we immediately know that a 6 will be drawn from the top of the deck next. The normalized joint entropy, U, of pairs of cards drawn from this deck would be 1. If the deck of cards is shuffled well and we play this game long enough, we would find that the normalized joint entropy approaches zero.

Instead of using the values of playing cards, we use the interval times generated by the ion. Figure 6 represents a typical data set that we analyze this way. Here we have made a scatter plot of the lengths of adjacent intervals (T_i , T_i+1) during which the ion is scattering many blue photons (i.e., when it is not in the D_5/2 state). One feature we search for in such plots is asymmetry about the diagonal axis. For example, one possible result of potential memory in the ion (that is, nonrandomness) would be that a short interval, Ti, is more likely to be followed by a long interval, T_i+1, than a short interval. This would manifest itself by showing many more events in the upper left quadrant of the plot than in the lower right quadrant. We make such plots not only for consecutive intervals but also for intervals that are separated by up to 20 other intervals (i.e., we plot the frequencies of pairs {T_i , T_i+k}, where k ranges from 1 to 20). We also analyze intervals for which the ion is in the D_5/2 state and intervals between times of emitting a 674-nm photon and between times of absorbing a 674-nm photon. Qualitatively, we see no features in any of these graphs. Quantitatively, we calculate the normalized U between the two data sets comprising the first and second intervals for all the pairs of data. We find that U < 7 × 10^-4 for all of our data and does not depend on the interval spacing for any of the different types of intervals. This analysis is an order of magnitude more sensitive than those previously performed on quantum jump data, and we expect to reduce our limit on U as we collect even more statistics.

Conclusion

Our experimental work has increased the sensitivity of our power to observe quantum effects and reduced the uncertainty in the randomness of those effects by over an order of magnitude. In addition to collecting more data with a recently improved laser system, we are developing the capability to coherently control the external and internal states of the ion. We do this by driving the S_1/2 ↔ D_5/2 transition with our narrow-bandwidth 674-nm laser, which can also cleanly couple specific quantized motional states of the trapped ion.
This work opens up the possibility of performing many other experiments. The ion can be laser-cooled to the ground state of its external motion where its temperature is nearly absolute zero. From this point, we can manipulate every physical aspect of the ion, tailoring its quantum-mechanical wavefunction as we see fit. We can control the interactions of the ion with the laser light to put it into quantum mechanical superpositions of states and observe their behavior and interactions with the environment. Or we can build a quantum logic gate for a quantum computer. And this, of course, is one of the motivations for testing the randomness of quantum mechanics as we have done.

Figure Captions

Figure 1. A time series of two ions simultaneously undergoing quantum jumps.

Figure 2. (a) A rendering of the linear rf trap. Current traveling through the tungsten filament heats it to produce electrons, which are directed towards the trap by the bias grid. The strontium oven is heated so that the neutral atoms flow through the trapping region and collide with the electrons, making ions. To trap the ions, we apply potentials to the trap electrodes, V_rf ~ 100 to 200 V, Ω/2π ~ 7.1 MHz and U₀ ~ 50 to 500 V. The trapped ions are immediately cooled to several mK by lasers propagating through the trap openings. In addition, the trap is placed in a vacuum chamber with pressure < 10^-10 torr. The crystallized ions are depicted lying along the trap axis and (b) as imaged by our intensified CCD camera.

Figure 3. A partial energy level diagram of Sr⁺. A frequency-doubled Ti:S laser drives the 422-nm transition to Doppler cool the ions, and we detect this scattered light to monitor the ions. A fiber laser drives the 1,092-nm transition to optically pump the ions out of the D_3/2 state. A diode laser with a bandwidth of < 2 kHz drives the 674-nm transition to induce quantum jumps and to coherently manipulate the ions.

Figure 4. Resonance curve of the S_1/2 ↔ D_5/2 transition. At each frequency step, the 422-nm light is blocked and a 3-ms pulse of 674-nm laser light interacts with the ion. After each pulse, the state of the ion is measured by returning the 422-nm light to the ion. By repeating this process 100 times, we determine the average probability of exciting the ion from the S_1/2 to D_5/2 state. After accounting for broadening caused by laser intensity, we conclude that the laser linewidth is 1.3 kHz. This corresponds to jitter in the length of the 674-nm laser cavity of only 0.4 pm (the radius of a hydrogen atom in its ground state is 53 pm).

Figure 5. Quantum jumps in a single ion. Times at which the count rate is relatively6 high correspond to the atom being in a superposition of the S_1/2 and P_1/2 states. Times at which the count rate is very low are when the atom is in the D_5/2 state. Transitions between these two conditions indicate either the absorption or emission of a 674-nm photon.

Figure 6. Scatter plot of the lengths of adjacent intervals during which the ion continually scattered 422-nm light.

References

1. D.J. Berkeland, “Linear Paul trap for strontium ions,” Review of Scientific Instruments 73(8), 2856–2860 (2002).
2. D.J. Berkeland, D.A. Raymondson, and V.M. Tassin, “Tests for non-randomness in quantum jumps,” Los Alamos National Laboratory report (submitted 2003).

Acknowledgment

I am grateful for the efforts of Véronique Tassin and Daisy Raymondson (currently in the graduate program at the University of Colorado, Boulder) in the analysis and collection of statistics of transitions with multiple ions. This work was funded through the LANL LDRD program as part of 20020052DR, “Applied Quantum Technologies.”

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For more information, contact Dana Berkeland at djb@lanl.gov.