Introduction to quantum noise, measurement, and amplification

The topic of quantum noise has become extremely timely due to the rise of quantum information physics and the resulting interchange of ideas between the condensed matter and atomic, molecular, optical–quantum optics communities. This review gives a pedagogical introduction to the physics of quantum noise and its connections to quantum measurement and quantum amplification. After introducing quantum noise spectra and methods for their detection, the basics of weak continuous measurements are described. Particular attention is given to the treatment of the standard quantum limit on linear amplifiers and position detectors within a general linear-response framework. This approach is shown how it relates to the standard Haus-Caves quantum limit for a bosonic amplifier known in quantum optics and its application to the case of electrical circuits is illustrated, including mesoscopic detectors and resonant cavity detectors.

Figure 1

Quantum noise spectral density of voltage fluctuations across a resistor (resistance $R$) as a function of frequency at zero temperature (dashed line) and finite temperature (solid line).

Figure 2

(Color online) Schematic of position distributions of an atom in the detector plane of a Stern-Gerlach apparatus whose field gradient is in the $z$ direction. For small values of the displacement $d$ (described in the text), there is significant overlap of the distributions and the spin cannot be unambiguously inferred from the position. For large values of $d$, the spin is perfectly entangled with position and can be inferred from the position. This is the limit of strong projective measurement.

Figure 3

(Color online) Distribution of the integrated output for the cavity detector $I\left(t\right)$ for the two different qubit states. The separation of the means of the distributions grows linearly in time, while the widths of the distributions grow only as $\sqrt{t}$.

Figure 4

(Color online) Spectral density of the symmetrized output noise ${\overline{S}}_{\theta \theta }\left[\omega \right]$ of a linear position detector. The oscillator’s noise appears as a Lorentzian on top of a noise floor (i.e., the measurement imprecision). As discussed in the text, the width of the peak is proportional to the oscillator damping rate ${\gamma }_0$, while the area under the peak is proportional to temperature. This latter fact can be used to calibrate the response of the detector.

Figure 5

(Color online) Noise power of the added position noise of a linear position detector, evaluated at the oscillator’s resonance frequency $\left({\overline{S}}_{xx,\text{add}}\left[\Omega \right]\right)$ as a function of the magnitude of the back-action noise spectral density $S_{FF}$. $S_{FF}$ is proportional to the oscillator-detector coupling, and in the case of the cavity detector is also proportional to the power incident on the cavity. The optimal value of $S_{FF}$ is given by $S_{FF,\mathrm{opt}}=\hslash M\Omega {\gamma }_0∕2$ [cf. Eq. (3.59)]. We have assumed that there are no correlations between measurement-imprecision noise and back-action noise, as is appropriate for the cavity detector.

Figure 6

(Color online) Spectral density of measured position fluctuations of a harmonic oscillator ${\overline{S}}_{xx,\mathrm{tot}}\left[\omega \right]$ as a function of frequency $\omega$, for a detector which reaches the quantum limit at the oscillator frequency $\Omega$. We have assumed that without the coupling to the detector the oscillator would be in its ground state. The $y$ axis has been normalized by the zero-point position noise spectral density ${\overline{S}}_{xx}^0\left[\omega \right]$, evaluated at $\omega =\Omega$. One clearly sees that the total noise at $\Omega$ is twice the zero-point value, and that the peak of the Lorentzian rises a factor of 3 above the background. This background represents the measurement imprecision and is equal to $1∕2$ of ${\overline{S}}_{xx}^0\left(\Omega \right)$.

Figure 7

(Color online) Schematic of a generic linear-response detector.

Figure 8

(Color online) Energy level scheme of the nondegenerate (phase-preserving) parametric oscillator.

Figure 9

(Color online) Schematic of a two-port bosonic amplifier. Both the inputs and outputs of the amplifier are attached to transmission lines. The incoming and outgoing wave amplitudes in the input (output) transmission line are labeled ${\widehat{a}}_{\mathrm{in}},{\widehat{a}}_{\mathrm{out}}$ $({\widehat{b}}_{\mathrm{in}},{\widehat{b}}_{\mathrm{out}})$, respectively. The voltages at the end of the two lines $({\widehat{V}}_a,{\widehat{V}}_b)$ are linear combinations of incoming and outgoing wave amplitudes.

Figure 10

Illustration of a bosonic two-port amplifier used in the scattering mode of operation. The signal is an incoming wave in the input port of the amplifier, and does not depend on what is coming out of the amplifier. This is achieved by connecting the input line to a circulator and a “cold load” (i.e., a zero-temperature resistor): all that goes back toward the source of the input signal is vacuum noise.

Figure 11

(Color online) Schematic of a generic linear-response position detector, where an auxiliary oscillator $y$ is driven by the detector output.

Figure 12

Schematic of a linear voltage amplifier, including a reverse gain ${\lambda }_I$. $\tilde{V}$ and $\tilde{I}$ represent the standard voltage and current noises of the amplifier, as discussed in the text. The case with reverse gain is discussed in Sec. 6.

Figure 13

Schematic of a minimal two-port amplifier which reaches the quantum limit in the scattering mode of operation, but misses the quantum limit when used as a weakly coupled op-amp. See text for further description.