Resonance

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https://en.wikipedia.org/wiki/Resonance

Increase of amplitude as damping decreases and frequency approaches resonant frequency of a driven damped simple harmonic oscillator.

Resonance is the phenomenon, pertaining to oscillatory dynamical systems, wherein amplitude rises are caused by an external force with time-varying amplitude with the same frequency of variation as the natural frequency of the system. The amplitude rises that occur are a result of the fact that applied external forces at the natural frequency entail a net increase in mechanical energy of the system.

Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is often desirable in certain applications, such as musical instruments or radio receivers. However, resonance can also be detrimental, leading to excessive vibrations or even structural failure in some cases.

All systems, including molecular systems and particles, tend to vibrate at a natural frequency depending upon their structure; this frequency is known as a resonant frequency or resonance frequency. When an oscillating force, an external vibration, is applied at a resonant frequency of a dynamic system, object, or particle, the outside vibration will cause the system to oscillate at a higher amplitude (with more force) than when the same force is applied at other, non-resonant frequencies.

The resonant frequencies of a system can be identified when the response to an external vibration creates an amplitude that is a relative maximum within the system. Small periodic forces that are near a resonant frequency of the system have the ability to produce large amplitude oscillations in the system due to the storage of vibrational energy.

Resonance phenomena occur with all types of vibrations or waves: there is mechanical resonance, orbital resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and resonance of quantum wave functions. Resonant systems can be used to generate vibrations of a specific frequency (e.g., musical instruments), or pick out specific frequencies from a complex vibration containing many frequencies (e.g., filters).

The term resonance (from Latin resonantia, 'echo', from resonare, 'resound') originated from the field of acoustics, particularly the sympathetic resonance observed in musical instruments, e.g., when one string starts to vibrate and produce sound after a different one is struck.

Overview

Resonance occurs when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a simple pendulum). However, there are some losses from cycle to cycle, called damping. When damping is small, the resonant frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple, distinct, resonant frequencies.

Examples

Pushing a person in a swing is a common example of resonance. The loaded swing, a pendulum, has a natural frequency of oscillation, its resonant frequency, and resists being pushed at a faster or slower rate.

A familiar example is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interval of the swing (its resonant frequency) makes the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo produce smaller arcs. This is because the energy the swing absorbs is maximized when the pushes match the swing's natural oscillations.

Resonance occurs widely in nature, and is exploited in many devices. It is the mechanism by which virtually all sinusoidal waves and vibrations are generated. For example, when hard objects like metal, glass, or wood are struck, there are brief resonant vibrations in the object. Light and other short wavelength electromagnetic radiation is produced by resonance on an atomic scale, such as electrons in atoms. Other examples of resonance include:

• Timekeeping mechanisms of modern clocks and watches, e.g., the balance wheel in a mechanical watch and the quartz crystal in a quartz watch
• Tidal resonance of the Bay of Fundy
• Acoustic resonances of musical instruments and the human vocal tract
• Shattering of a crystal wineglass when exposed to a musical tone of the right pitch (its resonant frequency)
• Friction idiophones, such as making a glass object (glass, bottle, vase) vibrate by rubbing around its rim with a fingertip
• Electrical resonance of tuned circuits in radios and TVs that allow radio frequencies to be selectively received
• Creation of coherent light by optical resonance in a laser cavity
• Orbital resonance as exemplified by some moons of the Solar System's gas giants
• Material resonances in atomic scale are the basis of several spectroscopic techniques that are used in condensed matter physics
  • Electron spin resonance
  • Mössbauer effect
  • Nuclear magnetic resonance

Linear systems

Resonance manifests itself in many linear and nonlinear systems as oscillations around an equilibrium point. When the system is driven by a sinusoidal external input, a measured output of the system may oscillate in response. The ratio of the amplitude of the output's steady-state oscillations to the input's oscillations is called the gain, and the gain can be a function of the frequency of the sinusoidal external input. Peaks in the gain at certain frequencies correspond to resonances, where the amplitude of the measured output's oscillations are disproportionately large.

Since many linear and nonlinear systems that oscillate are modeled as harmonic oscillators near their equilibria, this section begins with a derivation of the resonant frequency for a driven, damped harmonic oscillator. The section then uses an RLC circuit to illustrate connections between resonance and a system's transfer function, frequency response, poles, and zeroes. Building off the RLC circuit example, the section then generalizes these relationships for higher-order linear systems with multiple inputs and outputs.

The driven, damped harmonic oscillator

Main article: Harmonic oscillator § Driven harmonic oscillators

Consider a damped mass on a spring driven by a sinusoidal, externally applied force. Newton's second law takes the form

$${\displaystyle m\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}=F_0\sin (\omega t)-kx-c\frac{\mathrm{d}x}{\mathrm{d}t},}$$

(1)

where m is the mass, x is the displacement of the mass from the equilibrium point, F₀ is the driving amplitude, ω is the driving angular frequency, k is the spring constant, and c is the viscous damping coefficient. This can be rewritten in the form

$${\displaystyle \frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+2\zeta {\omega }_0\frac{\mathrm{d}x}{\mathrm{d}t}+{\omega }_0^{2}x=\frac{F_0}{m}\sin (\omega t),}$$

(2)

where

• ${\omega }_0=\sqrt{k/m}$ is called the undamped angular frequency of the oscillator or the natural frequency,
• ${\displaystyle \zeta =\frac{c}{2\sqrt{mk}}}$ is called the damping ratio.

Many sources also refer to ω₀ as the resonant frequency. However, as shown below, when analyzing oscillations of the displacement x(t), the resonant frequency is close to but not the same as ω₀. In general the resonant frequency is close to but not necessarily the same as the natural frequency. The RLC circuit example in the next section gives examples of different resonant frequencies for the same system.

The general solution of Equation (2) is the sum of a transient solution that depends on initial conditions and a steady state solution that is independent of initial conditions and depends only on the driving amplitude F₀, driving frequency ω, undamped angular frequency ω₀, and the damping ratio ζ. The transient solution decays in a relatively short amount of time, so to study resonance it is sufficient to consider the steady state solution.

It is possible to write the steady-state solution for x(t) as a function proportional to the driving force with an induced phase change φ,

$${\displaystyle x(t)=\frac{F_0}{m\sqrt{{\left(2\omega {\omega }_0\zeta \right)}^2+({\omega }_0^2-{\omega }^2{)}^2}}\sin (\omega t+\phi ),}$$

(3)

where ${\displaystyle \phi =\arctan \left(\frac{2\omega {\omega }_0\zeta }{{\omega }^2-{\omega }_0^2}\right)+n\pi .}$

The phase value is usually taken to be between −180° and 0 so it represents a phase lag for both positive and negative values of the arctan argument.

Steady-state variation of amplitude with relative frequency ${\displaystyle \omega /{\omega }_0}$ and damping ${\displaystyle \zeta }$ of a driven simple harmonic oscillator

Resonance occurs when, at certain driving frequencies, the steady-state amplitude of x(t) is large compared to its amplitude at other driving frequencies. For the mass on a spring, resonance corresponds physically to the mass's oscillations having large displacements from the spring's equilibrium position at certain driving frequencies. Looking at the amplitude of x(t) as a function of the driving frequency ω, the amplitude is maximal at the driving frequency

$${\displaystyle {\omega }_r={\omega }_0\sqrt{1-2{\zeta }^2}.}$$

ω_r is the resonant frequency for this system. Again, the resonant frequency does not equal the undamped angular frequency ω₀ of the oscillator. They are proportional, and if the damping ratio goes to zero they are the same, but for non-zero damping they are not the same frequency. As shown in the figure, resonance may also occur at other frequencies near the resonant frequency, including ω₀, but the maximum response is at the resonant frequency.

Also, ω_r is only real and non-zero if $\zeta <1/\sqrt{2}$, so this system can only resonate when the harmonic oscillator is significantly underdamped. For systems with a very small damping ratio and a driving frequency near the resonant frequency, the steady state oscillations can become very large.

The pendulum

For other driven, damped harmonic oscillators whose equations of motion do not look exactly like the mass on a spring example, the resonant frequency remains

$${\displaystyle {\omega }_r={\omega }_0\sqrt{1-2{\zeta }^2},}$$

but the definitions of ω₀ and ζ change based on the physics of the system. For a pendulum of length ℓ and small displacement angle θ, Equation (1) becomes

$${\displaystyle m\ell \frac{{\mathrm{d}}^2\theta }{\mathrm{d}{t}^2}=F_0\sin (\omega t)-mg\theta -c\ell \frac{\mathrm{d}\theta }{\mathrm{d}t}}$$

and therefore

• ${\displaystyle {\omega }_0=\sqrt{\frac{g}{\ell }},}$
• ${\displaystyle \zeta =\frac{c}{2m}\sqrt{\frac{\ell }{g}}.}$

RLC series circuits

See also: RLC Circuit § Series circuit

An RLC series circuit

Consider a circuit consisting of a resistor with resistance R, an inductor with inductance L, and a capacitor with capacitance C connected in series with current i(t) and driven by a voltage source with voltage v_in(t). The voltage drop around the circuit is

${\displaystyle L\frac{di(t)}{dt}+Ri(t)+V(0)+\frac{1}{C}{\int }_0^{t}i(\tau )d\tau =v_{\text{in}}(t).}$

(4)

Rather than analyzing a candidate solution to this equation like in the mass on a spring example above, this section will analyze the frequency response of this circuit. Taking the Laplace transform of Equation (4),

$${\displaystyle sLI(s)+RI(s)+\frac{1}{sC}I(s)=V_{\text{in}}(s),}$$

where I(s) and V_in(s) are the Laplace transform of the current and input voltage, respectively, and s is a complex frequency parameter in the Laplace domain. Rearranging terms,

$${\displaystyle I(s)=\frac{s}{s^{2}L+Rs+\frac{1}{C}}{V}_{\text{in}}(s).}$$

Voltage across the capacitor

An RLC circuit in series presents several options for where to measure an output voltage. Suppose the output voltage of interest is the voltage drop across the capacitor. As shown above, in the Laplace domain this voltage is

$${\displaystyle V_{\text{out}}(s)=\frac{1}{sC}I(s)}$$

or

$${\displaystyle V_{\text{out}}=\frac{1}{LC(s^2+\frac{R}{L}s+\frac{1}{LC})}{V}_{\text{in}}(s).}$$

Define for this circuit a natural frequency and a damping ratio,

$${\displaystyle {\omega }_0=\frac{1}{\sqrt{LC}},}$$

$${\displaystyle \zeta =\frac{R}{2}\sqrt{\frac{C}{L}}.}$$

The ratio of the output voltage to the input voltage becomes

$${\displaystyle H(s)\triangleq \frac{V_{\text{out}}(s)}{V_{\text{in}}(s)}=\frac{{\omega }_0^2}{s^2+2\zeta {\omega }_{0}s+{\omega }_0^2}}$$

H(s) is the transfer function between the input voltage and the output voltage. This transfer function has two poles–roots of the polynomial in the transfer function's denominator–at

$${\displaystyle s=-\zeta {\omega }_0\pm i{\omega }_0\sqrt{1-{\zeta }^2}}$$

(5)

and no zeros–roots of the polynomial in the transfer function's numerator. Moreover, for ζ ≤ 1, the magnitude of these poles is the natural frequency ω₀ and that for ζ < 1/${\displaystyle \sqrt{2}}$, our condition for resonance in the harmonic oscillator example, the poles are closer to the imaginary axis than to the real axis.

Evaluating H(s) along the imaginary axis s = iω, the transfer function describes the frequency response of this circuit. Equivalently, the frequency response can be analyzed by taking the Fourier transform of Equation (4) instead of the Laplace transform. The transfer function, which is also complex, can be written as a gain and phase,

$${\displaystyle H(i\omega )=G(\omega )e^{i\mathrm{\Phi }(\omega )}.}$$

Bode magnitude plot for the voltage across the elements of an RLC series circuit. Natural frequency ω₀ = 1 rad/s, damping ratio ζ = 0.4. The capacitor voltage peaks below the circuit's natural frequency, the inductor voltage peaks above the natural frequency, and the resistor voltage peaks at the natural frequency with a peak gain of one. The gain for the voltage across the capacitor and inductor combined in series shows antiresonance, with gain going to zero at the natural frequency.

A sinusoidal input voltage at frequency ω results in an output voltage at the same frequency that has been scaled by G(ω) and has a phase shift Φ(ω). The gain and phase can be plotted versus frequency on a Bode plot. For the RLC circuit's capacitor voltage, the gain of the transfer function H(iω) is

$${\displaystyle G(\omega )=\frac{{\omega }_0^2}{\sqrt{{\left(2\omega {\omega }_0\zeta \right)}^2+({\omega }_0^2-{\omega }^2{)}^2}}.}$$

(6)

Note the similarity between the gain here and the amplitude in Equation (3). Once again, the gain is maximized at the resonant frequency

$${\displaystyle {\omega }_r={\omega }_0\sqrt{1-2{\zeta }^2}.}$$

Here, the resonance corresponds physically to having a relatively large amplitude for the steady state oscillations of the voltage across the capacitor compared to its amplitude at other driving frequencies.

Voltage across the inductor

The resonant frequency need not always take the form given in the examples above. For the RLC circuit, suppose instead that the output voltage of interest is the voltage across the inductor. As shown above, in the Laplace domain the voltage across the inductor is

$${\displaystyle V_{\text{out}}(s)=sLI(s),}$$

$${\displaystyle V_{\text{out}}(s)=\frac{s^2}{s^2+\frac{R}{L}s+\frac{1}{LC}}{V}_{\text{in}}(s),}$$

$${\displaystyle V_{\text{out}}(s)=\frac{s^2}{s^2+2\zeta {\omega }_{0}s+{\omega }_0^2}{V}_{\text{in}}(s),}$$

using the same definitions for ω₀ and ζ as in the previous example. The transfer function between V_in(s) and this new V_out(s) across the inductor is

$${\displaystyle H(s)=\frac{s^2}{s^2+2\zeta {\omega }_{0}s+{\omega }_0^2}.}$$

This transfer function has the same poles as the transfer function in the previous example, but it also has two zeroes in the numerator at s = 0. Evaluating H(s) along the imaginary axis, its gain becomes

$${\displaystyle G(\omega )=\frac{{\omega }^2}{\sqrt{{\left(2\omega {\omega }_0\zeta \right)}^2+({\omega }_0^2-{\omega }^2{)}^2}}.}$$

Compared to the gain in Equation (6) using the capacitor voltage as the output, this gain has a factor of ω² in the numerator and will therefore have a different resonant frequency that maximizes the gain. That frequency is

$${\displaystyle {\omega }_r=\frac{{\omega }_0}{\sqrt{1-2{\zeta }^2}},}$$

So for the same RLC circuit but with the voltage across the inductor as the output, the resonant frequency is now larger than the natural frequency, though it still tends towards the natural frequency as the damping ratio goes to zero. That the same circuit can have different resonant frequencies for different choices of output is not contradictory. As shown in Equation (4), the voltage drop across the circuit is divided among the three circuit elements, and each element has different dynamics. The capacitor's voltage grows slowly by integrating the current over time and is therefore more sensitive to lower frequencies, whereas the inductor's voltage grows when the current changes rapidly and is therefore more sensitive to higher frequencies. While the circuit as a whole has a natural frequency where it tends to oscillate, the different dynamics of each circuit element make each element resonate at a slightly different frequency.

Voltage across the resistor

Suppose that the output voltage of interest is the voltage across the resistor. In the Laplace domain the voltage across the resistor is

$${\displaystyle V_{\text{out}}(s)=RI(s),}$$

$${\displaystyle V_{\text{out}}(s)=\frac{Rs}{L\left(s^2+\frac{R}{L}s+\frac{1}{LC}\right)}{V}_{\text{in}}(s),}$$

and using the same natural frequency and damping ratio as in the capacitor example the transfer function is

$${\displaystyle H(s)=\frac{2\zeta {\omega }_{0}s}{s^2+2\zeta {\omega }_{0}s+{\omega }_0^2}.}$$

This transfer function also has the same poles as the previous RLC circuit examples, but it only has one zero in the numerator at s = 0. For this transfer function, its gain is

$${\displaystyle G(\omega )=\frac{2\zeta {\omega }_0\omega }{\sqrt{{\left(2\omega {\omega }_0\zeta \right)}^2+({\omega }_0^2-{\omega }^2{)}^2}}.}$$

The resonant frequency that maximizes this gain is

$${\displaystyle {\omega }_r={\omega }_0,}$$

and the gain is one at this frequency, so the voltage across the resistor resonates at the circuit's natural frequency and at this frequency the amplitude of the voltage across the resistor equals the input voltage's amplitude.

Antiresonance

Main article: Antiresonance

Some systems exhibit antiresonance that can be analyzed in the same way as resonance. For antiresonance, the amplitude of the response of the system at certain frequencies is disproportionately small rather than being disproportionately large. In the RLC circuit example, this phenomenon can be observed by analyzing both the inductor and the capacitor combined.

Suppose that the output voltage of interest in the RLC circuit is the voltage across the inductor and the capacitor combined in series. Equation (4) showed that the sum of the voltages across the three circuit elements sums to the input voltage, so measuring the output voltage as the sum of the inductor and capacitor voltages combined is the same as v_in minus the voltage drop across the resistor. The previous example showed that at the natural frequency of the system, the amplitude of the voltage drop across the resistor equals the amplitude of v_in, and therefore the voltage across the inductor and capacitor combined has zero amplitude. We can show this with the transfer function.

The sum of the inductor and capacitor voltages is

$${\displaystyle V_{\text{out}}(s)=(sL+\frac{1}{sC})I(s),}$$

$${\displaystyle V_{\text{out}}(s)=\frac{s^2+\frac{1}{LC}}{s^2+\frac{R}{L}s+\frac{1}{LC}}{V}_{\text{in}}(s).}$$

Using the same natural frequency and damping ratios as the previous examples, the transfer function is

$${\displaystyle H(s)=\frac{s^2+{\omega }_0^2}{s^2+2\zeta {\omega }_{0}s+{\omega }_0^2}.}$$

This transfer has the same poles as the previous examples but has zeroes at

$${\displaystyle s=\pm i{\omega }_0.}$$

(7)

Evaluating the transfer function along the imaginary axis, its gain is

$${\displaystyle G(\omega )=\frac{{\omega }_0^2-{\omega }^2}{\sqrt{{\left(2\omega {\omega }_0\zeta \right)}^2+({\omega }_0^2-{\omega }^2{)}^2}}.}$$

Rather than look for resonance, i.e., peaks of the gain, notice that the gain goes to zero at ω = ω₀, which complements our analysis of the resistor's voltage. This is called antiresonance, which has the opposite effect of resonance. Rather than result in outputs that are disproportionately large at this frequency, this circuit with this choice of output has no response at all at this frequency. The frequency that is filtered out corresponds exactly to the zeroes of the transfer function, which were shown in Equation (7) and were on the imaginary axis.

Relationships between resonance and frequency response in the RLC series circuit example

These RLC circuit examples illustrate how resonance is related to the frequency response of the system. Specifically, these examples illustrate:

• How resonant frequencies can be found by looking for peaks in the gain of the transfer function between the input and output of the system, for example in a Bode magnitude plot
• How the resonant frequency for a single system can be different for different choices of system output
• The connection between the system's natural frequency, the system's damping ratio, and the system's resonant frequency
• The connection between the system's natural frequency and the magnitude of the transfer function's poles, pointed out in Equation (5), and therefore a connection between the poles and the resonant frequency
• A connection between the transfer function's zeroes and the shape of the gain as a function of frequency, and therefore a connection between the zeroes and the resonant frequency that maximizes gain
• A connection between the transfer function's zeroes and antiresonance

The next section extends these concepts to resonance in a general linear system.

Generalizing resonance and antiresonance for linear systems

Next consider an arbitrary linear system with multiple inputs and outputs. For example, in state-space representation a third order linear time-invariant system with three inputs and two outputs might be written as

$${\displaystyle \left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\end{array}\right]=A\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]+B\left[\begin{array}{c}{u}_1(t)\\ u_2(t)\\ u_3(t)\end{array}\right],}$$

$${\displaystyle \left[\begin{array}{c}{y}_1(t)\\ y_2(t)\end{array}\right]=C\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]+D\left[\begin{array}{c}{u}_1(t)\\ u_2(t)\\ u_3(t)\end{array}\right],}$$

where u_i(t) are the inputs, x_i(t) are the state variables, y_i(t) are the outputs, and A, B, C, and D are matrices describing the dynamics between the variables.

This system has a transfer function matrix whose elements are the transfer functions between the various inputs and outputs. For example,

$${\displaystyle \left[\begin{array}{c}{Y}_1(s)\\ Y_2(s)\end{array}\right]=\left[\begin{array}{ccc}{H}_{11}(s)& H_{12}(s)& H_{13}(s)\\ H_{21}(s)& H_{22}(s)& H_{23}(s)\end{array}\right]\left[\begin{array}{c}{U}_1(s)\\ U_2(s)\\ U_3(s)\end{array}\right].}$$

Each H_ij(s) is a scalar transfer function linking one of the inputs to one of the outputs. The RLC circuit examples above had one input voltage and showed four possible output voltages–across the capacitor, across the inductor, across the resistor, and across the capacitor and inductor combined in series–each with its own transfer function. If the RLC circuit were set up to measure all four of these output voltages, that system would have a 4×1 transfer function matrix linking the single input to each of the four outputs.

Evaluated along the imaginary axis, each H_ij(iω) can be written as a gain and phase shift,

$${\displaystyle H_{ij}(i\omega )=G_{ij}(\omega )e^{i{\mathrm{\Phi }}_{ij}(\omega )}.}$$

Peaks in the gain at certain frequencies correspond to resonances between that transfer function's input and output, assuming the system is stable.

Each transfer function H_ij(s) can also be written as a fraction whose numerator and denominator are polynomials of s.

$${\displaystyle H_{ij}(s)=\frac{N_{ij}(s)}{D_{ij}(s)}.}$$

The complex roots of the numerator are called zeroes, and the complex roots of the denominator are called poles. For a stable system, the positions of these poles and zeroes on the complex plane give some indication of whether the system can resonate or antiresonate and at which frequencies. In particular, any stable or marginally stable, complex conjugate pair of poles with imaginary components can be written in terms of a natural frequency and a damping ratio as

$${\displaystyle s=-\zeta {\omega }_0\pm i{\omega }_0\sqrt{1-{\zeta }^2},}$$

as in Equation (5). The natural frequency ω₀ of that pole is the magnitude of the position of the pole on the complex plane and the damping ratio of that pole determines how quickly that oscillation decays. In general,

• Complex conjugate pairs of poles near the imaginary axis correspond to a peak or resonance in the frequency response in the vicinity of the pole's natural frequency. If the pair of poles is on the imaginary axis, the gain is infinite at that frequency.
• Complex conjugate pairs of zeroes near the imaginary axis correspond to a notch or antiresonance in the frequency response in the vicinity of the zero's frequency, i.e., the frequency equal to the magnitude of the zero. If the pair of zeroes is on the imaginary axis, the gain is zero at that frequency.

In the RLC circuit example, the first generalization relating poles to resonance is observed in Equation (5). The second generalization relating zeroes to antiresonance is observed in Equation (7). In the examples of the harmonic oscillator, the RLC circuit capacitor voltage, and the RLC circuit inductor voltage, "poles near the imaginary axis" corresponds to the significantly underdamped condition ζ < 1/${\displaystyle \sqrt{2}}$.

Standing waves

Main article: Standing wave

A mass on a spring has one natural frequency, as it has a single degree of freedom

A physical system can have as many natural frequencies as it has degrees of freedom and can resonate near each of those natural frequencies. A mass on a spring, which has one degree of freedom, has one natural frequency. A double pendulum, which has two degrees of freedom, can have two natural frequencies. As the number of coupled harmonic oscillators increases, the time it takes to transfer energy from one to the next becomes significant. Systems with very large numbers of degrees of freedom can be thought of as continuous rather than as having discrete oscillators.

Energy transfers from one oscillator to the next in the form of waves. For example, the string of a guitar or the surface of water in a bowl can be modeled as a continuum of small coupled oscillators and waves can travel along them. In many cases these systems have the potential to resonate at certain frequencies, forming standing waves with large-amplitude oscillations at fixed positions. Resonance in the form of standing waves underlies many familiar phenomena, such as the sound produced by musical instruments, electromagnetic cavities used in lasers and microwave ovens, and energy levels of atoms.

Standing waves on a string

A standing wave (in black), created when two waves moving from left and right meet and superimpose

When a string of fixed length is driven at a particular frequency, a wave propagates along the string at the same frequency. The waves reflect off the ends of the string, and eventually a steady state is reached with waves traveling in both directions. The waveform is the superposition of the waves.

At certain frequencies, the steady state waveform does not appear to travel along the string. At fixed positions called nodes, the string is never displaced. Between the nodes the string oscillates and exactly halfway between the nodes–at positions called anti-nodes–the oscillations have their largest amplitude.

Standing waves in a string – the fundamental mode and the first 5 harmonics.

For a string of length ${\displaystyle L}$ with fixed ends, the displacement ${\displaystyle y(x,t)}$ of the string perpendicular to the ${\displaystyle x}$-axis at time ${\displaystyle t}$ is

$${\displaystyle y(x,t)=2y_{\text{max}}\sin (kx)\cos (2\pi ft),}$$

where

• ${\displaystyle y_{\text{max}}}$ is the amplitude of the left- and right-traveling waves interfering to form the standing wave,
• ${\displaystyle k}$ is the wave number,
• ${\displaystyle f}$ is the frequency.

The frequencies that resonate and form standing waves relate to the length of the string as

$${\displaystyle f=\frac{nv}{2L},}$$

$${\displaystyle n=1,2,3,\dots }$$

where ${\displaystyle v}$ is the speed of the wave and the integer ${\displaystyle n}$ denotes different modes or harmonics. The standing wave with n = 1 oscillates at the fundamental frequency and has a wavelength that is twice the length of the string. The possible modes of oscillation form a harmonic series.

Resonance in complex networks

A generalization to complex networks of coupled harmonic oscillators shows that such systems have a finite number of natural resonant frequencies, related to the topological structure of the network itself. In particular, such frequencies result related to the eigenvalues of the network's Laplacian matrix. Let ${\displaystyle \mathbf{A}}$ be the adjacency matrix describing the topological structure of the network and ${\displaystyle \mathbf{L}=\mathbf{K}-\mathbf{A}}$ the corresponding Laplacian matrix, where ${\displaystyle \mathbf{K}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{k_i\}}$ is the diagonal matrix of the degrees of the network's nodes. Then, for a network of classical and identical harmonic oscillators, when a sinusoidal driving force ${\displaystyle f(t)=F_0\sin \omega t}$ is applied to a specific node, the global resonant frequencies of the network are given by ${\displaystyle {\omega }_i=\sqrt{1+{\mu }_i}}$ where ${\displaystyle {\mu }_i}$ are the eigenvalues of the Laplacian ${\displaystyle \mathbf{L}}$.

Types

Mechanical and acoustic

Main articles: Mechanical resonance, Acoustic resonance, and String resonance

School resonating mass experiment

Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, trains, and aircraft. When designing objects, engineers must ensure the mechanical resonance frequencies of the component parts do not match driving vibrational frequencies of motors or other oscillating parts, a phenomenon known as resonance disaster.

Avoiding resonance disasters is a major concern in every building, tower, and bridge construction project. As a countermeasure, shock mounts can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The Taipei 101 building relies on a 660-tonne pendulum (730-short-ton)—a tuned mass damper—to cancel resonance. Furthermore, the structure is designed to resonate at a frequency that does not typically occur. Buildings in seismic zones are often constructed to take into account the oscillating frequencies of expected ground motion. In addition, engineers designing objects having engines must ensure that the mechanical resonant frequencies of the component parts do not match driving vibrational frequencies of the motors or other strongly oscillating parts.

Clocks keep time by mechanical resonance in a balance wheel, pendulum, or quartz crystal.

The cadence of runners has been hypothesized to be energetically favorable due to resonance between the elastic energy stored in the lower limb and the mass of the runner.

Acoustic resonance is a branch of mechanical resonance that is concerned with the mechanical vibrations across the frequency range of human hearing, in other words sound. For humans, hearing is normally limited to frequencies between about 20 Hz and 20,000 Hz (20 kHz), Many objects and materials act as resonators with resonant frequencies within this range, and when struck vibrate mechanically, pushing on the surrounding air to create sound waves. This is the source of many percussive sounds we hear.

Acoustic resonance is an important consideration for instrument builders, as most acoustic instruments use resonators, such as the strings and body of a violin, the length of tube in a flute, and the shape of, and tension on, a drum membrane.

Like mechanical resonance, acoustic resonance can result in catastrophic failure of the object at resonance. The classic example of this is breaking a wine glass with sound at the precise resonant frequency of the glass, although this is difficult in practice.

International Space Station

The rocket engines for the International Space Station (ISS) are controlled by an autopilot. Ordinarily, uploaded parameters for controlling the engine control system for the Zvezda module make the rocket engines boost the International Space Station to a higher orbit. The rocket engines are hinge-mounted, and ordinarily the crew does not notice the operation. On January 14, 2009, however, the uploaded parameters made the autopilot swing the rocket engines in larger and larger oscillations, at a frequency of 0.5 Hz. These oscillations were captured on video, and lasted for 142 seconds.

Electrical

Main article: Electrical resonance

Animation illustrating electrical resonance in a tuned circuit, consisting of a capacitor (C) and an inductor (L) connected together. Charge flows back and forth between the capacitor plates through the inductor. Energy oscillates back and forth between the capacitor's electric field (E) and the inductor's magnetic field (B).

Electrical resonance occurs in an electric circuit at a particular resonant frequency when the impedance of the circuit is at a minimum in a series circuit or at maximum in a parallel circuit (usually when the transfer function peaks in absolute value). Resonance in circuits are used for both transmitting and receiving wireless communications such as television, cell phones and radio.

Optical

Main article: Optical cavity

An optical cavity, also called an optical resonator, is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric oscillators and some interferometers. Light confined in the cavity reflects multiple times producing standing waves for certain resonant frequencies. The standing wave patterns produced are called "modes". Longitudinal modes differ only in frequency while transverse modes differ for different frequencies and have different intensity patterns across the cross-section of the beam. Ring resonators and whispering galleries are examples of optical resonators that do not form standing waves.

Different resonator types are distinguished by the focal lengths of the two mirrors and the distance between them; flat mirrors are not often used because of the difficulty of aligning them precisely. The geometry (resonator type) must be chosen so the beam remains stable, i.e., the beam size does not continue to grow with each reflection. Resonator types are also designed to meet other criteria such as minimum beam waist or having no focal point (and therefore intense light at that point) inside the cavity.

Optical cavities are designed to have a very large Q factor. A beam reflects a large number of times with little attenuation—therefore the frequency line width of the beam is small compared to the frequency of the laser.

Additional optical resonances are guided-mode resonances and surface plasmon resonance, which result in anomalous reflection and high evanescent fields at resonance. In this case, the resonant modes are guided modes of a waveguide or surface plasmon modes of a dielectric-metallic interface. These modes are usually excited by a subwavelength grating.

Orbital

Main article: Orbital resonance

In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa, and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to clear the neighborhood around their orbits by ejecting nearly everything else around them; this effect is used in the current definition of a planet.

Atomic, particle, and molecular

Main articles: Nuclear magnetic resonance and Resonance (particle physics)

NMR Magnet at HWB-NMR, Birmingham, UK. In its strong 21.2-tesla field, the proton resonance is at 900 MHz.

Nuclear magnetic resonance (NMR) is the name given to a physical resonance phenomenon involving the observation of specific quantum mechanical magnetic properties of an atomic nucleus in the presence of an applied, external magnetic field. Many scientific techniques exploit NMR phenomena to study molecular physics, crystals, and non-crystalline materials through NMR spectroscopy. NMR is also routinely used in advanced medical imaging techniques, such as in magnetic resonance imaging (MRI).

All nuclei containing odd numbers of nucleons have an intrinsic magnetic moment and angular momentum. A key feature of NMR is that the resonant frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this feature that is exploited in imaging techniques; if a sample is placed in a non-uniform magnetic field then the resonant frequencies of the sample's nuclei depend on where in the field they are located. Therefore, the particle can be located quite precisely by its resonant frequency.

Electron paramagnetic resonance, otherwise known as electron spin resonance (ESR), is a spectroscopic technique similar to NMR, but uses unpaired electrons instead. Materials for which this can be applied are much more limited since the material needs to both have an unpaired spin and be paramagnetic.

The Mössbauer effect is the resonant and recoil-free emission and absorption of gamma ray photons by atoms bound in a solid form.

Resonance in particle physics appears in similar circumstances to classical physics at the level of quantum mechanics and quantum field theory. Resonances can also be thought of as unstable particles, with the formula in the Universal resonance curve section of this article applying if Γ is the particle's decay rate and Ω is the particle's mass M. In that case, the formula comes from the particle's propagator, with its mass replaced by the complex number M + iΓ. The formula is further related to the particle's decay rate by the optical theorem.

Disadvantages

A column of soldiers marching in regular step on a narrow and structurally flexible bridge can set it into dangerously large amplitude oscillations. On April 12, 1831, the Broughton Suspension Bridge near Salford, England collapsed while a group of British soldiers were marching across. Since then, the British Army has had a standing order for soldiers to break stride when marching across bridges, to avoid resonance from their regular marching pattern affecting the bridge.

Vibrations of a motor or engine can induce resonant vibration in its supporting structures if their natural frequency is close to that of the vibrations of the engine. A common example is the rattling sound of a bus body when the engine is left idling.

Structural resonance of a suspension bridge induced by winds can lead to its catastrophic collapse. Several early suspension bridges in Europe and United States were destroyed by structural resonance induced by modest winds. The collapse of the Tacoma Narrows Bridge on 7 November 1940 is characterized in physics as a classic example of resonance. It has been argued by Robert H. Scanlan and others that the destruction was instead caused by aeroelastic flutter, a complicated interaction between the bridge and the winds passing through it—an example of a self oscillation, or a kind of "self-sustaining vibration" as referred to in the nonlinear theory of vibrations.

Q factor

Main article: Q factor

High and low Q factor

The Q factor or quality factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, and characterizes the bandwidth of a resonator relative to its center frequency. A high value for Q indicates a lower rate of energy loss relative to the stored energy, i.e., the system is lightly damped. The parameter is defined by the equation:

$${\displaystyle Q=2\pi \frac{\text{ maximum energy stored}}{\text{total energy lost per cycle at resonance}}}$$

The higher the Q factor, the greater the amplitude at the resonant frequency, and the smaller the bandwidth, or range of frequencies around resonance occurs. In electrical resonance, a high-Q circuit in a radio receiver is more difficult to tune, but has greater selectivity, and so would be better at filtering out signals from other stations. High Q oscillators are more stable.

Examples that normally have a low Q factor include door closers (Q=0.5). Systems with high Q factors include tuning forks (Q=1000), atomic clocks and lasers (Q≈10¹¹).

Universal resonance curve

"Universal Resonance Curve", a symmetric approximation to the normalized response of a resonant circuit; abscissa values are deviation from center frequency, in units of center frequency divided by 2Q; ordinate is relative amplitude, and phase in cycles; dashed curves compare the range of responses of real two-pole circuits for a Q value of 5; for higher Q values, there is less deviation from the universal curve. Crosses mark the edges of the 3 dB bandwidth (gain 0.707, phase shift 45° or 0.125 cycle).

The exact response of a resonance, especially for frequencies far from the resonant frequency, depends on the details of the physical system, and is usually not exactly symmetric about the resonant frequency, as illustrated for the simple harmonic oscillator above. For a lightly damped linear oscillator with a resonance frequency Ω, the intensity of oscillations I when the system is driven with a driving frequency ω is typically approximated by a formula that is symmetric about the resonance frequency:

$${\displaystyle I(\omega )\equiv |\chi {|}^2\propto \frac{1}{(\omega -\mathrm{\Omega }{)}^2+{\left(\frac{\mathrm{\Gamma }}{2}\right)}^2}.}$$

Where the susceptibility ${\displaystyle \chi (\omega )}$ links the amplitude of the oscillator to the driving force in frequency space:

$${\displaystyle x(\omega )=\chi (\omega )F(\omega )}$$

The intensity is defined as the square of the amplitude of the oscillations. This is a Lorentzian function, or Cauchy distribution, and this response is found in many physical situations involving resonant systems. Γ is a parameter dependent on the damping of the oscillator, and is known as the linewidth of the resonance. Heavily damped oscillators tend to have broad linewidths, and respond to a wider range of driving frequencies around the resonant frequency. The linewidth is inversely proportional to the Q factor, which is a measure of the sharpness of the resonance.

In radio engineering and electronics engineering, this approximate symmetric response is known as the universal resonance curve, a concept introduced by Frederick E. Terman in 1932 to simplify the approximate analysis of radio circuits with a range of center frequencies and Q values.

Quantum memory

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_memory

In quantum computing, quantum memory is the quantum-mechanical version of ordinary computer memory. Whereas ordinary memory stores information as binary states (represented by "1"s and "0"s), quantum memory stores a quantum state for later retrieval. These states hold useful computational information known as qubits. Unlike the classical memory of everyday computers, the states stored in quantum memory can be in a quantum superposition, giving much more practical flexibility in quantum algorithms than classical information storage.

Quantum memory is essential for the development of many devices in quantum information processing, including a synchronization tool that can match the various processes in a quantum computer, a quantum gate that maintains the identity of any state, and a mechanism for converting predetermined photons into on-demand photons. Quantum memory can be used in many aspects, such as quantum computing and quantum communication. Continuous research and experiments have enabled quantum memory to realize the storage of qubits.

Background and history

The interaction of quantum radiation with multiple particles has sparked scientific interest over the past decade. Quantum memory is one such field, mapping the quantum state of light onto a group of atoms and then restoring it to its original shape. Quantum memory is a key element in information processing, such as optical quantum computing and quantum communication, while opening a new way for the foundation of light-atom interaction. However, restoring the quantum state of light is no easy task. While impressive progress has been made, researchers are still working to make it happen.

Quantum memory based on the quantum exchange to store photon qubits has been demonstrated to be possible. Kessel and Moiseev discussed quantum storage in the single photon state in 1993. The experiment was analyzed in 1998 and demonstrated in 2003. In summary, the study of quantum storage in the single photon state can be regarded as the product of the classical optical data storage technology proposed in 1979 and 1982, an idea inspired by the high density of data storage in the mid-1970s. Optical data storage can be achieved by using absorbers to absorb different frequencies of light, which are then directed to beam space points and stored.

Types

Atomic Gas Quantum Memory

Normal, classical optical signals are transmitted by varying the amplitude of light. In this case, a piece of paper, or a computer hard disk, can be used to store information on the lamp. In the quantum information scenario, however, the information may be encoded according to the amplitude and phase of the light. For some signals, you cannot measure both the amplitude and phase of the light without interfering with the signal. To store quantum information, light itself needs to be stored without being measured. An atomic gas quantum memory is recording the state of light into the atomic cloud. When light's information is stored by atoms, relative amplitude and phase of light is mapped to atoms and can be retrieved on-demand.

Solid Quantum Memory

In classical computing, memory is a trivial resource that can be replicated in long-lived memory hardware and retrieved later for further processing. In quantum computing, this is forbidden because, according to the no clone theorem, any quantum state cannot be reproduced completely. Therefore, in the absence of quantum error correction, the storage of qubits is limited by the internal coherence time of the physical qubits holding the information. "Quantum memory" beyond the given physical qubit storage limits will be a quantum information transmission to "storing qubits" not easily affected by environmental noise and other factors. The information would later be transferred back to the preferred "process qubits" to allow rapid operations or reads.

Discovery

Optical quantum memory is usually used to detect and store single photon quantum states. However, producing efficient memory of this kind is still a huge challenge for current science. A single photon is so low in energy as to be lost in a complex light background. These problems have long kept quantum storage rates below 50%. A team led by professor Du Shengwang of the department of physics at the Hong Kong University of science and technology and William Mong Institute of Nano Science and Technology at HKUST has found a way to increase the efficiency of optical quantum memory to more than 85 percent. The discovery also brings the popularity of quantum computers closer to reality. At the same time, the quantum memory can also be used as a repeater in the quantum network, which lays the foundation for the quantum Internet.

Research and application

Quantum memory is an important component of quantum information processing applications such as quantum network, quantum repeater, linear optical quantum computation or long-distance quantum communication.

Optical data storage has been an important research topic for many years. Its most interesting function is the use of the laws of quantum physics to protect data from theft, through quantum computing and quantum cryptography unconditionally guaranteed communication security.

They allow particles to be superimposed and in a superposition state, which means they can represent multiple combinations at the same time. These particles are called quantum bits, or qubits. From a cybersecurity perspective, the magic of qubits is that if a hacker tries to observe them in transit, their fragile quantum states shatter. This means it is impossible for hackers to tamper with network data without leaving a trace. Now, many companies are taking advantage of this feature to create networks that transmit highly sensitive data. In theory, these networks are secure.

Microwave storage and light learning microwave conversion

The nitrogen-vacancy center in diamond has attracted a lot of research in the past decade due to its excellent performance in optical nanophotonic devices. In a recent experiment, electromagnetically induced transparency was implemented on a multi-pass diamond chip to achieve full photoelectric magnetic field sensing. Despite these closely related experiments, optical storage has yet to be implemented in practice. The existing nitrogen-vacancy center (negative charge and neutral nitrogen-vacancy center) energy level structure makes the optical storage of the diamond nitrogen-vacancy center possible.

The coupling between the nitrogen-vacancy spin ensemble and superconducting qubits provides the possibility for microwave storage of superconducting qubits. Optical storage combines the coupling of electron spin state and superconducting quantum bits, which enables the nitrogen-vacancy center in diamond to play a role in the hybrid quantum system of the mutual conversion of coherent light and microwave.

Orbital angular momentum is stored in alkali vapor

Large resonant light depth is the premise of constructing efficient quantum-optical memory. Alkali metal vapor isotopes of a large number of near-infrared wavelength optical depth, because they are relatively narrow spectrum line and the number of high density in the warm temperature of 50-100 ∘ C. Alkali vapors have been used in some of the most important memory developments, from early research to the latest results we are discussing, due to their high optical depth, long coherent time and easy near-infrared optical transition.

Because of its high information transmission ability, people are more and more interested in its application in the field of quantum information. Structured light can carry orbital angular momentum, which must be stored in the memory to faithfully reproduce the stored structural photons. An atomic vapor quantum memory is ideal for storing such beams because the orbital angular momentum of photons can be mapped to the phase and amplitude of the distributed integration excitation. Diffusion is a major limitation of this technique because the motion of hot atoms destroys the spatial coherence of the storage excitation. Early successes included storing weakly coherent pulses of spatial structure in a warm, ultracold atomic whole. In one experiment, the same group of scientists in a caesium magneto-optical trap was able to store and retrieve vector beams at the single-photon level. The memory preserves the rotation invariance of the vector beam, making it possible to use it in conjunction with qubits encoded for maladjusted immune quantum communication.

The first storage structure, a real single photon, was achieved with electromagnetically induced transparency in rubidium magneto-optical trap. The predicted single photon generated by spontaneous four-wave mixing in one magneto-optical trap is prepared by an orbital angular momentum unit using spiral phase plates, stored in the second magneto-optical trap and recovered. The dual-orbit setup also proves coherence in multimode memory, where a preannounced single photon stores the orbital angular momentum superposition state for 100 nanoseconds.

Optical Quantum

GEM

GEM (Gradient Echo Memory) is a protocol for storing optical information and it can be applied to both atomic gas and solid-state memories. The idea was first demonstrated by researchers at ANU. The experiment in a three-level system based on hot atomic vapor resulted in demonstration of coherent storage with efficiency up to 87%.

Electromagnetically induced transparency

See also: Electromagnetically induced transparency

Electromagnetically induced transparency (EIT) was first introduced by Harris and his colleagues at Stanford University in 1990. The work showed that when a laser beam causes a quantum interference between the excitation paths, the optical response of the atomic medium is modified to eliminate absorption and refraction at the resonant frequencies of atomic transitions. Slow light, optical storage, and quantum memories can be achieved based on EIT. In contrast to other approaches, EIT has a long storage time and is a relatively easy and inexpensive solution to implement. For example, electromagnetically induced transparency does not require the very high power control beams usually needed for Raman quantum memories, nor does it require the use of liquid helium temperatures. In addition, photon echo can read EIT while the spin coherence survives due to the time delay of readout pulse caused by a spin recovery in non-uniformly broadened media. Although there are some limitations on operating wavelength, bandwidth, and mode capacity, techniques have been developed to make EIT-based quantum memories a valuable tool in the development of quantum telecommunication systems. In 2018, a highly efficient EIT-based optical memory in cold atom demonstrated a 92% storage-and-retrieval efficiency in the classical regime with coherent beams and a 70% storage-and-retrieval efficiency was demonstrated for polarization qubits encoded in weak coherent states, beating any classical benchmark. Following these demonstrations, single-photon polarization qubits were then stored via EIT in a ⁸⁵Rb cold atomic ensemble and retrieved with an 85% efficiency and entanglement between two cesium-based quantum memories was also achieved with an overall transfer efficiency close to 90%.

Crystals doped with rare earth

The mutual transformation of quantum information between light and matter is the focus of quantum informatics. The interaction between a single photon and a cooled crystal doped with rare earth ions is investigated. Crystals doped with rare earth have broad application prospects in the field of quantum storage because they provide a unique application system. Li Chengfeng from the quantum information laboratory of the Chinese Academy of Sciences developed a solid-state quantum memory and demonstrated the photon computing function using time and frequency. Based on this research, a large-scale quantum network based on quantum repeater can be constructed by utilizing the storage and coherence of quantum states in the material system. Researchers have shown for the first time in rare-earth ion-doped crystals. By combining the three-dimensional space with two-dimensional time and two-dimensional spectrum, a kind of memory that is different from the general one is created. It has the multimode capacity and can also be used as a high fidelity quantum converter. Experimental results show that in all these operations, the fidelity of the three-dimensional quantum state carried by the photon can be maintained at around 89%.

Raman scattering in solids

Diamond has very high Raman gain in optical phonon mode of 40 THz and has a wide transient window in a visible and near-infrared band, which makes it suitable for being an optical memory with a very wide band. After the Raman storage interaction, the optical phonon decays into a pair of photons through the channel, and the decay lifetime is 3.5 ps, which makes the diamond memory unsuitable for communication protocol.

Nevertheless, diamond memory has allowed some revealing studies of the interactions between light and matter at the quantum level: optical phonons in a diamond can be used to demonstrate emission quantum memory, macroscopic entanglement, pre-predicted single-photon storage, and single-photon frequency manipulation.

Future development

For quantum memory, quantum communication and cryptography are the future research directions. However, there are many challenges to building a global quantum network. One of the most important challenges is to create memories that can store the quantum information carried by light. Researchers at the University of Geneva in Switzerland working with France's CNRS have discovered a new material in which an element called ytterbium can store and protect quantum information, even at high frequencies. This makes ytterbium an ideal candidate for future quantum networks. Because signals cannot be replicated, scientists are now studying how quantum memories can be made to travel farther and farther by capturing photons to synchronize them. In order to do this, it becomes important to find the right materials for making quantum memories. Ytterbium is a good insulator and works at high frequencies so that photons can be stored and quickly restored.

Time crystal

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Time_crystal

 

In condensed matter physics, a time crystal is a quantum system of particles whose lowest-energy state is one in which the particles are in repetitive motion. The system cannot lose energy to the environment and come to rest because it is already in its quantum ground state. Time crystals were first proposed theoretically by Frank Wilczek in 2012 as a time-based analogue to common crystals – whereas the atoms in crystals are arranged periodically in space, the atoms in a time crystal are arranged periodically in both space and time. Several different groups have demonstrated matter with stable periodic evolution in systems that are periodically driven. In terms of practical use, time crystals may one day be used as quantum computer memory.

The existence of crystals in nature is a manifestation of spontaneous symmetry breaking, which occurs when the lowest-energy state of a system is less symmetrical than the equations governing the system. In the crystal ground state, the continuous translational symmetry in space is broken and replaced by the lower discrete symmetry of the periodic crystal. As the laws of physics are symmetrical under continuous translations in time as well as space, the question arose in 2012 as to whether it is possible to break symmetry temporally, and thus create a "time crystal" that is resistant to entropy.

If a discrete time-translation symmetry is broken (which may be realized in periodically driven systems), then the system is referred to as a discrete time crystal. A discrete time crystal never reaches thermal equilibrium, as it is a type (or phase) of non-equilibrium matter. Breaking of time symmetry can only occur in non-equilibrium systems. Discrete time crystals have in fact been observed in physics laboratories as early as 2016 (published in 2017). One example of a time crystal, which demonstrates non-equilibrium, broken time symmetry is a constantly rotating ring of charged ions in an otherwise lowest-energy state.

Concept

Ordinary (non-time) crystals form through spontaneous symmetry breaking related to a spatial symmetry. Such processes can produce materials with interesting properties, such as diamonds, salt crystals, and ferromagnetic metals. By analogy, a time crystal arises through the spontaneous breaking of a time-translation symmetry. A time crystal can be informally defined as a time-periodic self-organizing structure. While an ordinary crystal is periodic (has a repeating structure) in space, a time crystal has a repeating structure in time. A time crystal is periodic in time in the same sense that the pendulum in a pendulum-driven clock is periodic in time. Unlike a pendulum, a time crystal "spontaneously" self-organizes into robust periodic motion (breaking a temporal symmetry).

Time-translation symmetry

Main article: Time-translation symmetry

Symmetries in nature lead directly to conservation laws, something which is precisely formulated by Noether's theorem.

The basic idea of time-translation symmetry is that a translation in time has no effect on physical laws, i.e. that the laws of nature that apply today were the same in the past and will be the same in the future. This symmetry implies the conservation of energy.

Broken symmetry in normal crystals

Main articles: Crystal symmetry and spontaneous symmetry breaking

Normal process (N-process) and Umklapp process (U-process). While the N-process conserves total phonon momentum, the U-process changes phonon momentum.

Common crystals exhibit broken translation symmetry: they have repeated patterns in space and are not invariant under arbitrary translations or rotations. The laws of physics are unchanged by arbitrary translations and rotations. However, if we hold fixed the atoms of a crystal, the dynamics of an electron or other particle in the crystal depend on how it moves relative to the crystal, and particle momentum can change by interacting with the atoms of a crystal — for example in Umklapp processes. Quasimomentum, however, is conserved in a perfect crystal.

Time crystals show a broken symmetry analogous to a discrete space-translation symmetry breaking. For example,^{[citation needed]} the molecules of a liquid freezing on the surface of a crystal can align with the molecules of the crystal, but with a pattern less symmetric than the crystal: it breaks the initial symmetry. This broken symmetry exhibits three important characteristics:

• the system has a lower symmetry than the underlying arrangement of the crystal,
• the system exhibits spatial and temporal long-range order (unlike a local and intermittent order in a liquid near the surface of a crystal),
• it is the result of interactions between the constituents of the system, which align themselves relative to each other.

Broken symmetry in discrete time crystals (DTC)

Time crystals seem to break time-translation symmetry and have repeated patterns in time even if the laws of the system are invariant by translation of time. The time crystals that are experimentally realized show discrete time-translation symmetry breaking, not the continuous one: they are periodically driven systems oscillating at a fraction of the frequency of the driving force. (According to Philip Ball, DTC are so-called because "their periodicity is a discrete, integer multiple of the driving period".)

The initial symmetry, which is the discrete time-translation symmetry (${\displaystyle t\to t+nT}$) with ${\displaystyle n=1}$, is spontaneously broken to the lower discrete time-translation symmetry with ${\displaystyle n>1}$, where ${\displaystyle t}$ is time, ${\displaystyle T}$ the driving period, ${\displaystyle n}$ an integer.

Many systems can show behaviors of spontaneous time-translation symmetry breaking but may not be discrete (or Floquet) time crystals: convection cells, oscillating chemical reactions, aerodynamic flutter, and subharmonic response to a periodic driving force such as the Faraday instability, NMR spin echos, parametric down-conversion, and period-doubled nonlinear dynamical systems.

However, discrete (or Floquet) time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking:

• it is a broken symmetry – the system shows oscillations with a period longer than the driving force,
• the system is in crypto-equilibrium – these oscillations generate no entropy, and a time-dependent frame can be found in which the system is indistinguishable from an equilibrium when measured stroboscopically (which is not the case of convection cells, oscillating chemical reactions and aerodynamic flutter),
• the system exhibits long-range order – the oscillations are in phase (synchronized) over arbitrarily long distances and time.

Moreover, the broken symmetry in time crystals is the result of many-body interactions: the order is the consequence of a collective process, just like in spatial crystals. This is not the case for NMR spin echos.

These characteristics makes discrete time crystals analogous to spatial crystals as described above and may be considered a novel type or phase of nonequilibrium matter.

Thermodynamics

Time crystals do not violate the laws of thermodynamics: energy in the overall system is conserved, such a crystal does not spontaneously convert thermal energy into mechanical work, and it cannot serve as a perpetual store of work. But it may change perpetually in a fixed pattern in time for as long as the system can be maintained. They possess "motion without energy"—their apparent motion does not represent conventional kinetic energy. Recent experimental advances in probing discrete time crystals in their periodically driven nonequilibrium states have led to the beginning exploration of novel phases of nonequilibrium matter.

Time crystals do not evade the Second Law of Thermodynamics, although they spontaneously break "time-translation symmetry", the usual rule that a stable object will remain the same throughout time. In thermodynamics, a time crystal's entropy, understood as a measure of disorder in the system, remains stationary over time, marginally satisfying the second law of thermodynamics by not decreasing.

History

Nobel laureate Frank Wilczek at University of Paris-Saclay

The idea of a quantized time crystal was theorized in 2012 by Frank Wilczek, a Nobel laureate and professor at MIT. In 2013, Xiang Zhang, a nanoengineer at University of California, Berkeley, and his team proposed creating a time crystal in the form of a constantly rotating ring of charged ions.

In response to Wilczek and Zhang, Patrick Bruno (European Synchrotron Radiation Facility) and Masaki Oshikawa (University of Tokyo) published several articles stating that space–time crystals were impossible.

Subsequent work developed more precise definitions of time-translation symmetry-breaking, which ultimately led to the Watanabe–Oshikawa "no-go" statement that quantum space–time crystals in equilibrium are not possible. Later work restricted the scope of Watanabe and Oshikawa: strictly speaking, they showed that long-range order in both space and time is not possible in equilibrium, but breaking of time-translation symmetry alone is still possible.

Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed. In 2014 Krzysztof Sacha at Jagiellonian University in Kraków predicted the behaviour of discrete time crystals in a periodically driven system with "an ultracold atomic cloud bouncing on an oscillating mirror".

In 2016, research groups at Princeton and at Santa Barbara independently suggested that periodically driven quantum spin systems could show similar behaviour. Also in 2016, Norman Yao at Berkeley and colleagues proposed a different way to create discrete time crystals in spin systems. These ideas were successful and independently realized by two experimental teams: a group led by Harvard's Mikhail Lukin and a group led by Christopher Monroe at University of Maryland. Both experiments were published in the same issue of Nature in March 2017.

Later, time crystals in open systems, so called dissipative time crystals, were proposed in several platforms breaking a discrete and a continuous time-translation symmetry. A dissipative time crystal was experimentally realized for the first time in 2021 by the group of Andreas Hemmerich at the Institute of Laser Physics at the University of Hamburg. The researchers used a Bose–Einstein condensate strongly coupled to a dissipative optical cavity and the time crystal was demonstrated to spontaneously break discrete time-translation symmetry by periodically switching between two atomic density patterns. In an earlier experiment in the group of Tilman Esslinger at ETH Zurich, limit cycle dynamics was observed in 2019, but evidence of robustness against perturbations and the spontaneous character of the time-translation symmetry breaking were not addressed.

In 2019, physicists Valerii Kozin and Oleksandr Kyriienko proved that, in theory, a permanent quantum time crystal can exist as an isolated system if the system contains unusual long-range multiparticle interactions. The original "no-go" argument only holds in the presence of typical short-range fields that decay as quickly as r^−α for some α > 0. Kozin and Kyriienko instead analyzed a spin-1/2 many-body Hamiltonian with long-range multispin interactions, and showed it broke continuous time-translational symmetry. Certain spin correlations in the system oscillate in time, despite the system being closed and in a ground energy state. However, demonstrating such a system in practice might be prohibitively difficult, and concerns about the physicality of the long-range nature of the model have been raised.

In 2022, the Hamburg research team, supervised by Hans Keßler and Andreas Hemmerich, demonstrated, for the first time, a continuous dissipative time crystal exhibiting spontaneous breaking of continuous time-translation symmetry.

In February 2024, a team from Dortmund University in Germany built a time crystal from indium gallium arsenide that lasted for 40 minutes, nearly 10 million times longer than the previous record of around 5 milliseconds. In addition, the lack of any decay suggest the crystal have lasted even longer, stating that it could last "at least a few hours, perhaps even longer".

Experiments

In October 2016, Christopher Monroe at the University of Maryland claimed to have created the world's first discrete time crystal. Using the ideas proposed by Yao et al., his team trapped a chain of ¹⁷¹Yb⁺ ions in a Paul trap, confined by radio-frequency electromagnetic fields. One of the two spin states was selected by a pair of laser beams. The lasers were pulsed, with the shape of the pulse controlled by an acousto-optic modulator, using the Tukey window to avoid too much energy at the wrong optical frequency. The hyperfine electron states in that setup, ²S_1/2 |F = 0, m_F = 0⟩ and |F = 1, m_F = 0⟩, have very close energy levels, separated by 12.642831 GHz. Ten Doppler-cooled ions were placed in a line 0.025 mm long and coupled together.

The researchers observed a subharmonic oscillation of the drive. The experiment showed "rigidity" of the time crystal, where the oscillation frequency remained unchanged even when the time crystal was perturbed, and that it gained a frequency of its own and vibrated according to it (rather than only the frequency of the drive). However, once the perturbation or frequency of vibration grew too strong, the time crystal "melted" and lost this subharmonic oscillation, and it returned to the same state as before where it moved only with the induced frequency.

Also in 2016, Mikhail Lukin at Harvard also reported the creation of a driven time crystal. His group used a diamond crystal doped with a high concentration of nitrogen-vacancy centers, which have strong dipole–dipole coupling and relatively long-lived spin coherence. This strongly interacting dipolar spin system was driven with microwave fields, and the ensemble spin state was determined with an optical (laser) field. It was observed that the spin polarization evolved at half the frequency of the microwave drive. The oscillations persisted for over 100 cycles. This subharmonic response to the drive frequency is seen as a signature of time-crystalline order.

In May 2018, a group in Aalto University reported that they had observed the formation of a time quasicrystal and its phase transition to a continuous time crystal in a Helium-3 superfluid cooled to within one ten thousandth of a kelvin from absolute zero (0.0001 K).  On August 17, 2020 Nature Materials published a letter from the same group saying that for the first time they were able to observe interactions and the flow of constituent particles between two time crystals. 

In February 2021 a team at Max Planck Institute for Intelligent Systems described the creation of time crystal consisting of magnons and probed them under scanning transmission X-ray microscopy to capture the recurring periodic magnetization structure in the first known video record of such type.

In July 2021, a team led by Andreas Hemmerich at the Institute of Laser Physics at the University of Hamburg presented the first realization of a time crystal in an open system, a so-called dissipative time crystal using ultracold atoms coupled to an optical cavity. The main achievement of this work is a positive application of dissipation – actually helping to stabilise the system's dynamics.

In November 2021, a collaboration between Google and physicists from multiple universities reported the observation of a discrete time crystal on Google's Sycamore processor, a quantum computing device. A chip of 20 qubits was used to obtain a many-body localization configuration of up and down spins and then stimulated with a laser to achieve a periodically driven "Floquet" system where all up spins are flipped for down and vice-versa in periodic cycles which are multiples of the laser's frequency. While the laser is necessary to maintain the necessary environmental conditions, no energy is absorbed from the laser, so the system remains in a protected eigenstate order.

Previously in June and November 2021 other teams had obtained virtual time crystals based on floquet systems under similar principles to those of the Google experiment, but on quantum simulators rather than quantum processors: first a group at the University of Maryland obtained time crystals on trapped-ions qubits using high frequency driving rather than many-body localization and then a collaboration between TU Delft and TNO in the Netherlands called Qutech created time crystals from nuclear spins in carbon-13 nitrogen-vacancy (NV) centers on a diamond, attaining longer times but fewer qubits.

In February 2022, a scientist at UC Riverside reported a dissipative time crystal akin to the system of July 2021 but all-optical, which allowed the scientist to operate it at room temperature. In this experiment injection locking was used to direct lasers at a specific frequency inside a microresonator creating a lattice trap for solitons at subharmonic frequencies.

In March 2022, a new experiment studying time crystals on a quantum processor was performed by two physicists at the university of Melbourne, this time using IBM's Manhattan and Brooklyn quantum processors observing a total of 57 qubits.

In June 2022, the observation of a continuous time crystal was reported by a team at the Institute of Laser Physics at the University of Hamburg, supervised by Hans Keßler and Andreas Hemmerich. In periodically driven systems, time-translation symmetry is broken into a discrete time-translation symmetry due to the drive. Discrete time crystals break this discrete time-translation symmetry by oscillating at a multiple of the drive frequency. In the new experiment, the drive (pump laser) was operated continuously, thus respecting the continuous time-translation symmetry. Instead of a subharmonic response, the system showed an oscillation with an intrinsic frequency and a time phase taking random values between 0 and 2π, as expected for spontaneous breaking of continuous time-translation symmetry. Moreover, the observed limit cycle oscillations were shown to be robust against perturbations of technical or fundamental character, such as quantum noise and, due to the openness of the system, fluctuations associated with dissipation. The system consisted of a Bose–Einstein condensate in an optical cavity, which was pumped with an optical standing wave oriented perpendicularly with regard to the cavity axis and was in a superradiant phase localizing at two bistable ground states between which it oscillated.