## Building Super-Amplifiers in Nano-Electric Systems using Strange Weak Values

*February 5, 2008*

In a recent Physical Review Letters (PRL 100, 026804) article, Assistant Professor Andrew Jordan and third-year PhD student Nathan Williams describe how to implement one of the most bizarre predictions in quantum mechanics: a strange weak value in a nano-electric system. For a quantum system, their proposed method could provide an electrical current that exceeds the current supplied by the analogous classical system by factors of hundreds or thousands; that is, their device could boost a nano-amp to one amp or even to ten amps. This new method could also be used to determine whether an experimental system is a quantum mechanical device.

So what is a strange weak value, and why does it correspond to such a dramatic boost in current? The answers are key to understanding just how remarkable their nano-electric system is and just how peculiar strange weak values are.

When a strong measurement is made on a spin 1/2 system, the outcome is always one of two things: either the spin is plus 1/2 (pointing up) or minus 1/2 (pointing down).

However, with a weak measurement in the same system, the outcome becomes continuous and the average is somewhere between plus 1/2 (pointing up) or minus 1/2 (pointing down). So in a very simple system, we might see something like this:

**Figure 1:**

Here X is the average of the weak measurements. A subsequent strong measurement would force the spin into its up or down state. A weak value is the average of the weak measurement results, where the later measurement gives (only) spin up.

Notice that we've been discussing a weak value as opposed to a strange weak value. Jordan and Williams find that, after averaging a large sample of weak measurement results, the value of X is nowhere between minus 1/2 and plus 1/2. Nor is the value at minus 1/2 or plus 1/2. Indeed, the value of X -- the strange weak value -- can be a hundred or even a thousand times higher than plus 1/2 or minus 1/2. So if the spin is measured in units of Planck's constant, X might be 100 or 1,000 times Planck's constant.

The proposed device takes two measurements, the first being a weak measurement at X, the second being the post-selected strong measurement. "Using this method of pre-selection, weak measurement, and post-selection strong measurement," explains Jordan, "the resulting value of X is huge, and this phenomenon only exists quantum mechanically. Quantum mechanics can't possibly be explained using classical physics. If we perform the same procedure in classical physics, the value of X would always lie between plus 1/2 and minus 1/2."

The design of the nano-electric quantum device is shown below. The detector consists of a double quantum dot, denoted by DQD, which functions as the spin 1/2 system. Next to the qubit is a detector, the quantum point contact (QPC), which is pulsed with a tiny amount of voltage through a wire width on the scale of the electron wavelength. One electron can pass through the QPC at any given time. The position of the electron in the DQD affects the transmission of electrons through the QPC, where the flow is translated into current. For example, if the electron is sitting at state 1, then 1 nano-amp flows through the QPC, and if the electron is at state 2, then 2 nano-amps are flowing: there are two distinct currents.

**Figure 2:**

The system is set to an initial state, a weak measurement is made at X, and then a post-selection is made with a strong measurement, as shown in Figure 1. After running a large sample, the mechanism produces an average amount of current for the total of all weak measurements. It is this value that exceeds anything within the classical system's normal range, which could be anywhere from 1 nano-amp to 2 nano-amps, denoted by the two horizontal lines in Figure 2. Rather than produce between 1 and 2 nano-amps, the device could produce current that lies in the range shown by the blue curve in the upper right corner of the graph. Any value above the two horizontal lines flies into non-classical regime. When asked how it is possible for a value measured between plus 1/2 and minus 1/2 can reach a thousand times past those spin values, Nathan Williams replies, "The math proves the possibility. Yet it confounds our intuition, doesn't it? And that's why we call these phenomena, strange weak values. Because they are incredibly strange."

A final point to note is that the Leggett-Garg criteria are actually a disguised form of the strange weak value criteria. The Leggett-Garg criteria, proposed in 1985, test a system to determine if it is classical or quantum mechanic. With Leggett-Garg, when a correlation function exceeds a classical bound the system, is shown to be quantum mechanical. When Jordan and Williams' detector produces an incredibly large current, it violates the Leggett-Garg criteria for classical systems. Hence, when the correlation function is written in terms of post-selection, the Leggett-Garg test is basically the same as that for the strange weak value. The Leggett-Garg criteria and the technique proposed by Jordan and Williams are different expressions of the same physics. (lhg)