Quantum noise

Basics of noise theory

Noise is of practical concern for precision engineering and engineered systems approaching the standard quantum limit. Typical engineered consideration of quantum noise is for quantum nondemolition measurement and quantum point contact. So quantifying noise is useful. A signal's noise is quantified as the Fourier transform of its autocorrelation. The autocorrelation of a signal is given as $${\displaystyle G_{vv}(t-t^{\prime })=⟨V(t)V(t^{\prime })⟩,}$$ which measures when our signal is positively, negatively or not correlated at different times ${\displaystyle t}$ and ${\displaystyle t^{\prime }}$. The time average, ${\displaystyle ⟨V(t)⟩}$, is zero and our ${\displaystyle V(t)}$ is a voltage signal. Its Fourier transform is $${\displaystyle V(\omega )=\frac{1}{\sqrt{T}}{\int }_0^{T}V(t)e^{i\omega t}dt}$$ 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., $${\displaystyle S_{vv}(\omega )={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{e}^{i\omega t}{G}_{vv}dt={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{e}^{i\omega t}⟨|V(\omega ){|}^2⟩dt}$$

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.
• ${\displaystyle G_{vv}}$ decays to zero rapidly over some time ${\displaystyle {\tau }_c}$.
• We sample over a sufficiently large time, ${\displaystyle T}$, that our integral scales as a random walk ${\displaystyle \sqrt{T}}$. So our ${\displaystyle V(\omega )}$ is independent of measured time for ${\displaystyle T\gg {\tau }_c}$. Said in another way, ${\displaystyle G_{vv}(t-t^{\prime })\to 0}$ as ${\displaystyle |t-t^{\prime }|\gg {\tau }_c}$.

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., $${\displaystyle S_{xx}(\omega )={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{e}^{i\omega t}⟨\hat{x}(t)\hat{x}(0)⟩dt,}$$ where ${\displaystyle ⟨\cdot ⟩}$ are the quantum statistical average using the density matrix in the Heisenberg picture.

Quantum noise and the uncertainty principle

The Heisenberg uncertainty implies the existence of noise. An operator with a hermitian conjugate follows the relationship, ${\displaystyle A{A}^{†}\ge 0}$. Define ${\displaystyle A}$ as ${\displaystyle A=\delta x+\lambda e^{i\theta }\delta y}$ where ${\displaystyle \lambda }$ is real. The ${\displaystyle x}$ and ${\displaystyle y}$ are the quantum operators. We can show the following,

$${\displaystyle ⟨\delta x^2⟩⟨\delta y^2⟩\ge \frac{1}{4}|⟨[\delta x,\delta y]⟩{|}^2+|⟨[\delta x,\delta y{]}_{+}⟩{|}^2}$$ where the ${\displaystyle ⟨\cdot ⟩}$ are the averages over the wavefunction and other statistical properties. The left terms are the uncertainty in ${\displaystyle x}$ and ${\displaystyle y}$, the second term on the right is to covariance or ${\displaystyle ⟨\delta x\delta y+\delta y\delta x⟩}$ 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 ${\displaystyle x}$ and ${\displaystyle y}$ correspond to position and momentum that meets the well known commutator relation, ${\displaystyle [x,p]=i\hslash }$. Then our new expression is,

$${\displaystyle \mathrm{\Delta }x\mathrm{\Delta }y\ge \sqrt{\frac{1}{4}{\hslash }^2+{\sigma }_{xy}^2}}$$

Where the ${\displaystyle {\sigma }_{xy}}$ 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, ${\displaystyle M}$, and frequency, ${\displaystyle \mathrm{\Omega }}$, coupled to some heat bath which keeps the system in equilibrium. The equations of motion are given as,

$${\displaystyle x(t)=x(0)\cos (\mathrm{\Omega }t)+p(0)\frac{1}{M\mathrm{\Omega }}\sin (\mathrm{\Omega }t)}$$

The quantum autocorrelation is then,

$${\displaystyle \begin{array}{cc}{G}_{xx}& =⟨\hat{x}(t)\hat{x}(0)⟩\\ & =⟨\hat{x}(0)\hat{x}(0)⟩\cos (\mathrm{\Omega }t)+⟨\hat{p}(0)\hat{x}(0)⟩\sin (\mathrm{\Omega }t)\end{array}}$$

Classically, there is no correlation between position and momentum. The uncertainty principle requires the second term to be nonzero. It goes to ${\displaystyle i\hslash /2}$. 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.,

$${\displaystyle \frac{1}{2}M{\mathrm{\Omega }}^2⟨x^2⟩=\frac{1}{2}{k}_{\text{B}}T}$$

In the classical autocorrelation, we have

$${\displaystyle G_{xx}=\frac{k_{\text{B}}T}{M{\mathrm{\Omega }}^2}\cos (\mathrm{\Omega }t)\to S_{xx}(\omega )=\pi \frac{k_{\text{B}}T}{M{\mathrm{\Omega }}^2}[\delta (\omega -\mathrm{\Omega })+\delta (\omega +\mathrm{\Omega })]}$$

while in the quantum autocorrelation we have

$${\displaystyle G_{xx}=\left(\frac{\hslash }{2M\mathrm{\Omega }}\right)\left\{n_{BE}(\hslash \mathrm{\Omega })e^{i\mathrm{\Omega }t}+[n_{BE}(\hslash \mathrm{\Omega })+1]e^{-i\mathrm{\Omega }t}\right\}\to S_{xx}(\omega )=2\pi \left(\frac{\hslash }{2M\mathrm{\Omega }}\right)[n_{BE}(\hslash \mathrm{\Omega })\delta (\omega -\mathrm{\Omega })+[n_{BE}(\hslash \mathrm{\Omega })+1]\delta (\omega +\mathrm{\Omega })]}$$

Where the fraction terms in parentheses is the zero-point energy uncertainty. The ${\displaystyle n_{BE}}$ is the Bose-Einstein population distribution. Notice that the quantum ${\displaystyle S_{xx}}$ is asymmetric in the due to the imaginary autocorrelation. As we increase to higher temperature that corresponds to taking the limit of ${\displaystyle k_{B}T\gg \hslash \mathrm{\Omega }}$. One can show that the quantum approaches the classical ${\displaystyle S_{xx}}$. This allows $n_{BE}\approx n_{BE}+1\approx \frac{k_{\text{B}}T}{\hslash \mathrm{\Omega }}$

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 exit. Consider the amplification of stream of photons, an ideal linear noiseless gain, and the Energy-Time uncertainty relation.

$${\displaystyle \mathrm{\Delta }E\mathrm{\Delta }t\gtrsim \hslash /2}$$

The photons, ignoring the uncertainty in frequency, will have an uncertainty in its overall phase and number, and assume a known frequency, i.e., ${\displaystyle \mathrm{\Delta }\varphi =2\pi \nu \mathrm{\Delta }t}$ and ${\displaystyle \mathrm{\Delta }E=h\nu \mathrm{\Delta }n}$. 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. $${\displaystyle \mathrm{\Delta }n\mathrm{\Delta }\varphi >1/2}$$

Let an ideal linear noiseless gain, ${\displaystyle G}$, 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.

$${\displaystyle n_0\pm \mathrm{\Delta }{n}_0\to G{n}_0\pm G\mathrm{\Delta }{n}_0}$$

The phase will be modified too,

$${\displaystyle {\varphi }_0\pm \mathrm{\Delta }{\varphi }_0\to {\varphi }_0+\theta +\mathrm{\Delta }{\varphi }_0,}$$ where the ${\displaystyle \theta }$ is the overall accumulated phase as the photons traveled through the gain medium. Substituting our output gain and phase uncertainties, gives us $${\displaystyle \mathrm{\Delta }{n}_0\mathrm{\Delta }{\varphi }_0>1/2G.}$$

Our gain is ${\displaystyle G>1}$, 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 $${\displaystyle P_n=h\nu B(G-1)}$$ where ${\displaystyle B}$ is half of the full width at half max, the ${\displaystyle \nu }$ frequency of the photons, and ${\displaystyle h}$ is the Planck constant. The term ${\displaystyle h\nu B_0/2}$ with ${\displaystyle B_0=2B}$ is sometimes called quantum noise

Shot noise and instrumentation

In precision optics with highly stabilized lasers and efficient detectors, quantum noise refers to the fluctuations of signal.

The random error of interferometric measurements of position, due to the discrete character of photons measurement, is another quantum noise. The uncertainty of position of a probe in probe microscopy may also attributable to quantum noise; but not the dominant mechanism governing resolution.

In an electric circuit, the random fluctuations of a signal due to the discrete character of electrons can be called quantum noise. 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.

$${\displaystyle \frac{d{P}_n}{dx}=a[n{P}_{n-1}-(n+1)P_n]+b[(n+1)P_{n+1}-n{P}_n]}$$

where ${\displaystyle a}$ corresponds to the emission cross section and upper population number product ${\displaystyle {\sigma }_e{N}_2}$, and the ${\displaystyle b}$ is the absorption cross section ${\displaystyle {\sigma }_a{N}_1}$. The above relation is describing the probability of finding ${\displaystyle n}$ photons in radiation mode ${\displaystyle |n⟩}$. The dynamic only considers neighboring modes ${\displaystyle |n+1⟩}$ and ${\displaystyle |n-1⟩}$ as the photons travel through a medium of excited and ground state atoms from position ${\displaystyle x}$ to ${\displaystyle x+dx}$. 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, ${\displaystyle |n-1⟩\to |n⟩}$ and ${\displaystyle |n⟩\to |n+1⟩}$ and two photons leaving a field to the atom ${\displaystyle |n+1⟩\to |n⟩}$ and ${\displaystyle |n⟩\to |n-1⟩}$. Its noise power is given as,

$${\displaystyle P_d^2=P_{\text{shot}}^2[1+2f_{sp}\eta (G-1)]}$$

Where,

• ${\displaystyle P_d}$ is the power at the detector,
• ${\displaystyle P_{\text{shot}}}$ is the power limited shot noise,
• ${\displaystyle G}$ the unsaturated gain and is also true for saturated gain,
• ${\displaystyle \eta }$ is the efficiency factor. That is the product of transmission window efficiency to our photodetector, and quantum efficiency.
• ${\displaystyle f_{sp}}$ is the spontaneous emission factor that typically corresponds relative strength of spontaneous emission to stimulated emission. A value of unity would mean all doped ions are in the excited state.

Sarif, et al. demonstrated quantum noise or shot noise limited measurements over a wide range of power gain that agreed with theory.

Zero-point fluctuations

The existence of zero-point energy fluctuations is well-established in the theory of the quantised electromagnetic field. Generally speaking, at the lowest energy excitation of a quantized field that permeates all space (i.e. the field mode being in the vacuum state), the root-mean-square fluctuation of field strength is non-zero. This accounts for vacuum fluctuations that permeate all space.

This vacuum fluctuation or quantum noise will effect classical systems. This manifest as quantum decoherence in an entangled system, normally attributed to thermal differences in the conditions surrounding each entangled particle. Because entanglement is studied intensely in simple pairs of entangled photons, for example, decoherence observed in experiments could well be synonymous with "quantum noise" as to the source of the decoherence. Vacuum fluctuation is a possible causes for a quanta of energy to spontaneously appear in a given field or spacetime, then thermal differences must be associated with this event. Hence, it would cause decoherence in an entangled system in proximity of the event.

Coherent states and noise of a quantum amplifier

A laser is described by the coherent state of light, or the superposition of harmonic oscillators eigenstates. Erwin Schrödinger first derived the coherent state for the Schrödinger equation to meet the correspondence principle in 1926.

The laser is a quantum mechanical phenomena (see Maxwell–Bloch equations, rotating wave approximation, and semi-classical model of a two level atom). The Einstein coefficients and the laser rate equations are adequate if one is interested in the population levels and one does not need to account for population quantum coherences (the off diagonal terms in a density matrix). Photons of the order of 10⁸ corresponds to a moderate energy. The relative error of measurement of the intensity due to the quantum noise is on the order of 10⁻⁵. This is considered to be of good precision for most of applications.

Quantum amplifier

A quantum amplifier is an amplifier which operates close to the quantum limit. Quantum noise becomes important when a small signal is amplified. A small signal's quantum uncertainties in its quadrature are also amplified; this sets a lower limit to the amplifier. A quantum amplifier's noise is its output amplitude and phase. Generally, a laser is amplified across a spread of wavelengths around a central wavelength, some mode distribution, and polarization spread. But one can consider a single mode amplification and generalize to many different modes. A phase-invariant amplifier preserves the phase of the input gain without drastic changes to the output phase mode.

Quantum amplification can be represented with a unitary operator, ${\displaystyle A_{\text{out}}=U^{†}{A}_{\text{in}}U}$ , as stated in D. Kouznetsov 1995 paper.