### Operation of an Electrical Amplifier Close to the Quantum Limit

**Dartmouth researchers (L to R) Joel Stettenheim, Alex Rimberg, and Weiwei Xue**

[This is an invited article based on recent works of the author and his team members -- 2Physics.com]

**Author:**

**Alex Rimberg**

**Affiliation:**

**Dept of Physics and Astronomy, Dartmouth College**

**, USA**

Link to Rimberg Group >>

Classically, it is possible to imagine purely passive measurements in which an instrument collects information from some measured system without disturbing it in any way. Measurements of quantum mechanical systems, in contrast, must always be active. A measuring device, no matter how sophisticated, must influence what it is being used to measure; such influence is commonly referred to as backaction. Since the backaction associated with the measurement randomly changes the behavior of the system, the act of measurement must always introduce additional noise. The result is a strict lower bound on the minimal noise an amplifier can introduce for a given sensitivity [1]. This bound on amplifier performance is essentially a manifestation of the uncertainty principle, and implies that there is a well-defined limit on how "good' an amplifier can be.

In a paper recently published in Nature Physics, researchers at Dartmouth College have operated an electrical amplifier that very nearly approaches this quantum limit [2]. The amplifier in question is a superconducting single electron transistor (S-SET), which is well-known to be one of the world's most sensitive detectors of electrical charge. It has been suggested for sometime that the SET can be closely approach the quantum limit [1,3]. However, technical limitations have prevented researchers from approaching the limit by closer than a factor of roughly 20.

To understand the difficulties researchers have faced, it is necessary to have some understanding of how the SET is usually operated as a charge detector. The most sensitive approaches are currently based on the radio-frequency SET technique (RF-SET) [4], in which a radio-frequency wave is reflected off the SET,and the reflected wave is amplified by a secondary classical amplifier. When a charge moves near the SET, its conductance changes -- causing changes in the amplitude of the reflected wave.

The SET is a high-impedance device (about 25 kOhm) while coaxial cable is relatively low impedance (usually 50 Ohm). To make energy transfer from the SET to later classical amplifiers more efficient, an LC matching network is used to impedance match the SET to the coaxial line. In principle, it is possible to make the impedance matching and power transfer nearly perfect. In practice, however, unless great care is taken, the matching network will be imperfect, and some power coming from the SET will be lost. The loss occurs either by having the outgoing power reflected back toward the SET, or lost in the matching network, or both.

Why is this a problem? A quantum-limited amplifier disturbs the system it is measuring, collects all possible information based on the disturbance, and transmits the information, via a chain of classical amplifiers, to the laboratory (the macroscopic world). If any information is lost, either by dissipation or through being buried in the inevitable classical noise of the amplifier chain, the result is to move the measurement away from the quantum limit: the same disturbance occurs but the measurement uncertainty is higher. In the case of the RF-SET, if the matching network is lossy, or impedance matching is imperfect, the result will necessarily be less than quantum limited performance. Worse, in most cases the impedance matching is imperfect enough that noise from the classical amplifier chain dominates the measurement. Note however, that even if the classical amplifiers introduced no noise of their own, imperfect matching necessarily implies a departure from the quantum limit.

In order to optimize the matching network, the Dartmouth researchers developed fully superconducting on-chip matching networks consisting of a superconducting spiral and a parasitic capacitance [5]. The resulting networks are nearly lossless, and due to their very small parasitic capacitance, provide excellent impedance matching at their resonant frequency of 1 GHz. As a result, the power transfer from the S-SET to the subsequent amplifiers is vastly improved, allowing the Dartmouth team to measure the quantum noise of the S-SET near a particularly useful operating point for the first time.

The particular operating point chosen was a feature known as the Double-Josephson-quasiparticle (DJQP) resonance that occurs at bias voltages too small to break Cooper pairs at both junctions. Instead, charge is transferred through the S-SET by means of a complex cycle of Cooper pair and quasiparticle tunneling. A special characteristic of the DJQP cycle is that when operated here, the S-SET has been predicted to have a combination of charge sensitivity and backaction that will allow it to closely approach the quantum limit [3].

By measuring the quantum noise of the S-SET near this feature, it was possible to demonstrate that the S-SET can either emit or absorb energy from the resonator, depending on its precise bias conditions. Classical amplifiers are characterized by a singlenoise parameter because they are equally likely to emit or absorb energy. Quantum mechanically, however, an amplifier may be much more likely to emit than absorb, or vice versa, depending on its precise operating conditions. As a result, two parameters are required to describe the noise. Here, the noise was described by a damping rate that described the S-SET's net tendency to emit or absorb energy from the LC tank circuit, and an effective temperature that describes the degree of asymmetry between emission and absorption. The resulting values of the effective temperature and damping, shown in Fig. 1, constitute the first complete and quantitative characterization of the quantum noise of the S-SET near the DJQP resonance.

**Fig. 1: (a) S-SET damping rate and (b) S-SET effective temperature. Together, these give a complete and quantitative description of the S-SET quantum noise.**

In addition, the charge sensitivity of the S-SET near the DJQP resonance was shown to be excellent, approaching the world record for RF-SET operation. By estimating the charge fluctuations on the S-SET island, it was possible to determine the backaction the S-SET would likely have on a system such as a quantum dot. Ignoring the noise of the classical amplifiers, the S-SET operated within a factor of 3.6 of the quantum limit, a factor of five improvement over the nearest previous results.

Near quantum limited amplifiers such as this one could have a host of applications in the fields of quantum computation and quantum measurement. They would allow fast, efficient measurement of qubits, might lead the way to direct observation of quantum charge oscillations, and could potentially be used in the preparation of exotic squeezed quantum states.

**References**

[1]"Amplifying Quantum Signals with the Single-Electron Transistor,"

[1]

M. H. Devoret and R. J. Schoelkopf, Nature 406, 1039(2000). Abstract.

**[2]**"Measurement of Quantum Noise in a Single-Electron Transistor near the Quantum Limit," W. W. Xue, Z. Ji, Feng Pan, Joel Stettenheim, M. P. Blencowe, A. J. Rimberg, Nature Phys. 5, 660(2009). Abstract.

**[3]**"Resonant Cooper Pair Tunneling: Quantum Noise and Measurement Characteristics,"

A. A. Clerk, S. M. Girvin, A. K. Nguyen and A. D. Stone, Phys. Rev. Lett. 89, 176804 (2002). Abstract.

**[4]**"The Radio-Frequency Single-Electron Transistor (RF-SET): A Fast and Ultrasensitive Electrometer," R. J. Schoelkopf, P. Wahlgren, A. A. Kozhevnikov, P. Delsing and D. E. Prober, Science, 280, 1238 (1998). Abstract.

**[5]**"On-Chip Matching Networks for Radio-Frequency Single-Electron Transistors," W. W. Xue, B. Davis, F. Pan, J. Stettenheim, T. J. Gilheart, A. J. Rimberg and Z. Ji, Appl. Phys.Lett. 91, 093511 (2007). Abstract.

Labels: Quantum Computation and Communication 2, Squeezed State

## 1 Comments:

A nice and cute work indeed! I'll certainly look forward to hearing about the squeezed states of such amplifiers. That'll open paths for numerous applications.

Hope Prof. Rimberg and his team have started work in that direction...

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