Yesterday at the Centre for Quantum Technologies, H. T. Ng from RIKEN in Japan gave a talk on “Quantum Phase Measurement in a Superconducting Circuit”. His goal is to develop a procedure for measuring the relative phase in a superposition of states of the electromagnetic field. Consider a single vibrational mode of the electromagnetic field in some cavity. Take a superposition of a state with 0 photons in this mode and one with N photons in this mode:

|0> + exp(-iθ)|N>

How can we measure the phase θ?

The approach is to let the electromagnetic field interact with a Josephson junction. The dream is to use this as part of a recipe for factoring integers, using the mathematics of so-called "Gauss sums":

• H. T. Ng, Franco Nori, Quantum phase measurement and Gauss sum factorization of large integers in a superconducting circuit.

The math of Gauss sums is an important branch of number theory, but I don’t understand it, so I’ll focus on the physics.

The trick is to use a Josephson junction. A Josephson junction consists of two superconductors separated by a very thin layer of insulating material – 3 nanometers or less – or a possibly thicker layer of conducting but not superconducting material. Electrons can tunnel through this barrier, so current can flow through it. As I mentioned here earlier, the London-Landau-Ginzburg theory says a superconductor is characterized by a ‘macroscopic wave function’ – a complex function with a phase that depends on position and time. This phase is different at the two sides of the barrier, and this phase difference is very important!

If we call this phase difference φ, the basic equations governing a Josephson junction say that:

• The voltage V across the junction is proportional to dφ/dt.

• The current I across the junction is proportional to sin φ.

I’ve never studied Josephson junctions before, but here’s the impression I got from the talk together with some superficial web browsing. Please correct me if I’m wrong…

We can treat the phase difference φ and the voltage V as quantum observables that are canonically conjugate, up to some constant factor. In other words, φ is analogous to ‘position’, while V is analogous to ‘momentum’.

(The phase difference really lives on a circle – in other words, exp(iφ) is what really matters, not φ. So, the voltage should take on discrete evenly spaced values. Right? In math jargon: the dual of the circle group is the group of integers.)

This analogy is useful for understanding the dynamics of the Josephson junction. The junction acts like a quantum particle running around a circle, with the phase difference φ acting like the particle’s position. We can set up our Josephson junction so this particle moves in a potential. The potential is a function on the circle.

Clever experimentalists can make sure this potential has a nice deep local minimum. How? I don’t know. There was some questions from the audience about how the potential arises — what causes it, physically. But I’m still ignorant about this, so I’d appreciate help.

Anyway: if the phase difference φ stays near this local minimum, we can approximate the behavior of the Josephson junction by a harmonic oscillator. The discrete energy levels only become apparent at very low temperatures – less than 1 degree above absolute zero.

In physics, approximations reign supreme! If the potential is deep enough, we simplify the problem further and restrict attention to the two lowest-energy states of our harmonic oscillator, say (the ground state) and (the first excited state). Then the Josephson junction acts like a two-state system… or in modern jargon, a qubit!

I believe this is called a **phase qubit**. You can learn more here:

• Wikipedia, Phase qubit.

It’s been shown that by adjusting a coupling between two systems of this sort, we can get a iSWAP gate:

Also, Andreas Wallraff and coworkers have shown how to couple a phase qubit to a single vibrational mode of the electromagnetic field in a superconducting resonator!

• A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin and R. J. Schoelkopf, Circuit quantum electrodynamics: coherent coupling of a single photon to a Cooper pair box, extended version of *Nature* (London) **431** (2004), 162.

This system can be described by the Jaynes-Cummings model. A single mode of the electromagnetic field is nicely described by a quantum harmonic oscillator, and the Jaynes-Cummings model describes a quantum harmonic oscillator coupled to a qubit.

Back in 1996, Law and Eberly proposed a method to use such a coupling to create arbitrary states of a single mode of the electromagnetic field:

• C. K. Law and J. H. Eberly, Arbitrary control of a quantum electromagnetic field, *Phys. Rev. Lett.* **76** (1996), 1055–1058.

At U.C. Santa Barbara, Hofheinz and coworkers carried out this idea experimentally:

• Max Hofheinz, H. Wang, M. Ansmann, Radoslaw C. Bialczak, Erik Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, John M. Martinis, and A. N. Cleland, Synthesizing arbitrary quantum states in a superconducting resonator, *Nature* **459** (28 May 2009), 546-549.

They constructed quite general superpositions of multi-photon states of a single mode of the electromagnetic field in a superconducting resonator – a box full of microwave radiation. They used a phase quibit to pump photons into the resonator. Then they measured the state of these photons using another phase qubit, via a technique called “Wigner tomography”. They can measure the state of the qubit with 98% fidelity!

The challenge discussed by Ng is to start with a Fock state of this form:

|0> + exp(-iθ)|N>

and encode the phase information to a qubit. The goal is to do ‘state transfer’, letting the photon field interact with the qubit so that the above state winds up making the qubit have the state

|g> + exp(-iθ)|e>

He described a strategy for doing this. It gets harder when gets bigger. Then he spoke about using this to factor integers…

I’m just beginning to learn about various ways to use superconductors to hold qubits. This looks like a good place to start:

• M. H. Devoret, A. Wallraff, and J. M. Martinis, Superconducting qubits: a short review.

Abstract:Superconducting qubits are solid state electrical circuits fabricated using techniques borrowed from conventional integrated circuits. They are based on the Josephson tunnel junction, the only non-dissipative, strongly non-linear circuit element available at low temperature. In contrast to microscopic entities such as spins or atoms, they tend to be well coupled to other circuits, which make them appealing from the point of view of readout and gate implementation. Very recently, new designs of superconducting qubits based on multi-junction circuits have solved the problem of isolation from unwanted extrinsic electromagnetic perturbations. We discuss in this review how qubit decoherence is affected by the intrinsic noise of the junction and what can be done to improve it.

You’ll note that this post was *A Tale of Two Phases*: the relative phase in a superposition of quantum states, and the phase difference across a Josephson junction. They’re quite different in character: the former is just a number, while we’re treating the latter as an *operator!* This may seem weird, so I thought I should emphasize it. I want to ponder the appearance of ‘phase operators’ in quantum optics and elsewhere… there should be some good math in here.

Wow, it is the same Jaynes as was so influential in Bayesian inference!

That’s right — good old E. T. “Extraterrestrial” Jaynes.

I removed this from an earlier draft because I thought it was too distracting, but now you’ve given me an excuse to mention it:

Cummings was also Jack Sarfatti’s thesis advisor — a deed that almost cancelled out all his positive achievements.