Quantum noise is any noise arising from quantum mechanical phenomena such as field quantization or the uncertainty principle. Quantum noise differs from classical noise fundamentally not only in the kinds of phenomena from which it arises, but also in key characteristic features such as its spectral density and temperature dependence.
For example, the uncertainty principle says that some groups of observables cannot simultaneously be known with arbitrary precision. As a result, measuring one observable to some precision can actually impose a limit on how precisely another observable can be known. Even for a system in its ground state -- at zero temperature -- this quantum indeterminacy can cause fluctuations in measured observable quantities. Such fluctuations are known as zero-point energy fluctuations.
Quantum noise can also come from the discrete nature of the components of flowing currents approximated as continuous, such as electrons and photons. An example of this form of quantum noise is shot noise as coined by J. Verdeyen which comes from the discrete arrival of photons or electrons in a detector. Because these quanta arrive randomly in time, even a perfectly steady current or light beam exhibits fluctuations in the detected signal.
In most systems, classical noise dominates over quantum noise. Under everyday environmental conditions, classical fluctuations are typically several orders of magnitude larger than quantum fluctuations, and thus mask their effect. Quantum noise generally only becomes visible after suppressing the effects of conventional noise sources such as thermal fluctuations, mechanical vibrations, and industrial noise by mechanically isolating a system, cooling it to a millikelvin range, and using extremely low-noise electronics for control and readout. This is why quantum noise is a major engineering problem in superconducting circuits and in the LIGO gravitational wave observatory, but not in many conventional settings. Even if all classical noise is eliminated, devices such as detectors and amplifiers will still be affected by quantum noise. As a result, experimental physicists define an "ideal" or "quantum-limited" amplifier or detector as one with only that noise which arises from quantum sources.
The term "quantum noise" is sometimes used in the fields of quantum information and quantum computing as an umbrella term for unwanted environmental disturbances that affect quantum systems and cause decoherence. An isolated quantum system, such as a qubit, has a state that will evolve deterministically. But in an open system, such as those found in nature, the qubit interacts with uncontrolled degrees of freedom in its environment, introducing fluctuations which are commonly referred to as quantum noise. This is distinct from the above definition, which specifically concerns intrinsic noise due to the nature of quantum mechanics, not all environmental sources of noise and decoherence. In practice, however, definitions of quantum noise often include environmental or external disturbances affecting quantum systems.
History
Principles
Noise theory A signal's noise is quantified as the Fourier transform of its autocorrelation. The autocorrelation of a signal is given as
which measures when our signal is positively, negatively or not correlated at different times and . The time average, , is zero and our is a voltage signal. Its Fourier transform is
because we measure a voltage over a finite time window. The Wiener–Khinchin theorem generally states that a noise's power spectrum is given as the autocorrelation of a signal, i.e., The above relation is sometimes called the power spectrum or spectral density. In the above outline, we assumed that Our noise is stationary or the probability does not change over time. Only the time difference matters. Noise is due to a very large number of fluctuating charge so that the central limit theorem applied, i.e., the noise is Gaussian or normally distributed. decays to zero rapidly over some time . We sample over a sufficiently large time, , that our integral scales as a random walk . So our is independent of measured time for . Said in another way, as .
One can show that an ideal "top-hat" signal, which may correspond to a finite measurement of a voltage over some time, will produce noise across its entire spectrum as a sinc function. Even in the classical case, noise is produced.
Classical to quantum noise To study quantum noise, one replaces the corresponding classical measurements with quantum operators, e.g.,
where are the quantum statistical average using the density matrix in the Heisenberg picture.
Heisenberg microscope
Quantum noise can be illustrated by considering a Heisenberg microscope where an atom's position is measured from the scattering of photons. The uncertainty principle is given as,
Where the is the uncertainty in an atom's position, and the is the uncertainty of the momentum or sometimes called the backaction (momentum transferred to the atom) when near the quantum limit. The precision of the position measurement can be increased at the expense of knowing the atom's momentum. When the position is precisely known enough backaction begins to affect the measurement in two ways. First, it will impart momentum back onto the measuring devices in extreme cases. Secondly, we have decreasing future knowledge of the atom's future position. Precise and sensitive instrumentation will approach the uncertainty principle at sufficiently control environments.
Heisenberg uncertainty and noise The Heisenberg uncertainty implies the existence of noise. An operator with a hermitian conjugate follows the relationship, . Define as where is real. The and are the quantum operators. We can show the following,
where the are the averages over the wavefunction and other statistical properties. The left terms are the uncertainty in and , the second term on the right is to covariance or which arises from coupling to an external source or quantum effects. The first term on the right corresponds to the Commutator relation and would cancel out if the x and y commuted. That is the origin of our quantum noise.
It is demonstrative to let and correspond to position and momentum that meets the well known commutator relation, . Then our new expression is,
Where the is the correlation. If the second term on the right vanishes, then we recover the Heisenberg uncertainty principle.
Harmonic motion and weakly coupled heat bath
Consider the motion of a simple harmonic oscillator with mass, , and frequency, , coupled to some heat bath which keeps the system in equilibrium. The equations of motion are given as,
The quantum autocorrelation is then,
Classically, there is no correlation between position and momentum. The uncertainty principle requires the second term to be nonzero. It goes to . We can take the equipartition theorem or the fact that in equilibrium the energy is equally shared among a molecule/atoms degrees of freedom in thermal equilibrium, i.e.,
In the classical autocorrelation, we have
while in the quantum autocorrelation we have
Where the fraction terms in parentheses is the zero-point energy uncertainty. The is the Bose-Einstein population distribution. Notice that the quantum is asymmetric in the due to the imaginary autocorrelation. As we increase to higher temperature that corresponds to taking the limit of . One can show that the quantum approaches the classical . This allows
Physical interpretation of spectral density Typically, the positive frequency of the spectral density corresponds to the flow of energy into the oscillator (for example, the photons' quantized field), while the negative frequency corresponds to the emitted of energy from the oscillator. Physically, an asymmetric spectral density would correspond to either the net flow of energy from or to our oscillator model.
Linear gain and quantum uncertainty Most optical communications use amplitude modulation where the quantum noise is predominantly the shot noise. A laser's quantum noise, when not considering shot noise, is the uncertainty of its electric field's amplitude and phase. That uncertainty becomes observable when a quantum amplifier preserves phase. The phase noise becomes important when the energy of the frequency modulation or phase modulation is comparable to the energy of the signal (frequency modulation is more robust than amplitude modulation due to the additive noise intrinsic to amplitude modulation).
Linear amplification An ideal noiseless gain cannot exist. Consider the amplification of stream of photons, an ideal linear noiseless gain, and the Energy-Time uncertainty relation.
The photons, ignoring the uncertainty in frequency, will have an uncertainty in its overall phase and number, and assume a known frequency, i.e., and . We can substitute these relations into our energy-time uncertainty equation to find the number-phase uncertainty relation or the uncertainty in the phase and photon numbers.
Let an ideal linear noiseless gain, , act on the photon stream. We also assume a unity quantum efficiency, or every photon is converted to a photocurrent. The output will be following with no noise added.
The phase will be modified too,
where the is the overall accumulated phase as the photons traveled through the gain medium. Substituting our output gain and phase uncertainties, gives us
Our gain is , which is a contradiction to our uncertainty principles. So a linear noiseless amplifier cannot increase its signal without noise. A deeper analysis done by H. Heffner showed the minimum noise power output required to meet the Heisenberg uncertainty principle is given as
where is half of the full width at half max, the frequency of the photons, and is the Planck constant. The term with is sometimes called quantum noise
Examples
Shot noise The quantum principle of wave-particle duality means that flows that seem continuous at a macroscopic scale are actually made of discrete components when examined at small enough scales. For example, electrical current is made of individual electrons and light is made of individual photons. As a result, the number of constituent particles received in any given time interval from even an ideal source with constant average flow will be random with a Poisson distribution.
An experiment by S. Saraf, et .al.
demonstrated shot noise limited measurements as a demonstration of quantum noise measurements. Generally speaking, they amplified a Nd:YAG free space laser with minimal noise addition as it transitioned from linear to nonlinear amplification. The experiment required Fabry-Perot for filtering laser mode noises and selecting frequencies, two separate but identical probe and saturating beams to ensure uncorrelated beams, a zigzag slab gain medium, and a balanced detector for measuring quantum noise or shot-noise limited noise.
Shot Noise Power The theory behind noise analysis of photon statistics (sometimes called the forward Kolmogorov equation) starts from the Masters equation from Shimoda et al.
where corresponds to the emission cross section and upper population number product , and the is the absorption cross section . The above relation is describing the probability of finding photons in radiation mode . The dynamic only considers neighboring modes and as the photons travel through a medium of excited and ground state atoms from position to . This gives us a total of 4 photon transitions associated to one photon energy level. Two photon number adding to the field and leaving an atom, and and two photons leaving a field to the atom and . Its noise power is given as,
Where, is the power at the detector, is the power limited shot noise, the unsaturated gain and is also true for saturated gain, is the efficiency factor. That is the product of transmission window efficiency to our photodetector, and quantum efficiency.