Введение
Квантовый эффект неопределенности
Quantum noise is noise arising from the indeterminate state of matter in accordance with fundamental principles of quantum mechanics, specifically the uncertainty principle and via zero point energy fluctuations. Quantum noise is due to the apparently discrete nature of the small quantum constituents such as electrons, as well as the discrete nature of quantum effects, such as photocurrents. Quantified noise is similar to classical noise theory and will not always return an asymmetric spectral density. Shot noise as coined by J. Verdeyen is a form of quantum noise related to the statistics of photon counting, the discrete nature of electrons, and intrinsic noise generation in electronics. In contrast to shot noise, the quantum mechanical uncertainty principle sets a lower limit to a measurement. The uncertainty principle requires any amplifier or detector to have noise. A signal's noise is quantified as the Fourier transform of its autocorrelation. The autocorrelation of a signal is given as
which measures when our signal is positively, negatively or not correlated at different times and The time average, , is zero and our is a voltage signal. Its Fourier transform is
because we measure a voltage over a finite time window. The Wiener–Khinchin theorem generally states that a noise's power spectrum is given as the autocorrelation of a signal, i. e.,
The above relation is sometimes called the power spectrum or spectral density. In the above outline, we assumed that
Our noise is stationary or the probability does not change over time. Only the time difference matters. Noise is due to a very large number of fluctuating charge so that the central limit theorem applie, i. e., the noise is Gaussian or normally distributed. decays to zero rapidly over some time We sample over a sufficiently large time, , that our integral scales as a random walk So our is independent of measured time for Said in another way, as
One can show that an ideal "top hat" signal, which may correspond to a finite measurement of a voltage over some time, will produce noise across its entire spectrum as a sinc function. Even in the classical case, noise is produced.
Квантовый шум — это шум, возникающий из неопределенного состояния материи в соответствии с фундаментальными принципами квантовой механики, в частности, принципом неопределенности и через флуктуации энергии нулевой точки. Квантовый шум обусловлен, по-видимому, дискретным характером небольших квантовых составляющих, таких как электроны, а также дискретным характером квантовых эффектов, таких как фототоки. Количественный шум аналогичен классической теории шума и не всегда приводит к асимметричной спектральной плотности. Шум выстрела, введенный J. Verdeyen, является формой квантового шума, связанной со статистикой счета фотонов, дискретной природой электронов и внутренним шумообразованием в электронике. В отличие от шума выстрела, квантово-механический принцип неопределенности устанавливает нижний предел измерения. Принцип неопределенности требует, чтобы любой усилитель или детектор имел шум. Шум сигнала количественно определяется как преобразование Фурье его автокорреляции. Автокорреляция сигнала задается как
Quantum noise is noise arising from the indeterminate state of matter in accordance with fundamental principles of quantum mechanics, specifically the uncertainty principle and via zero point energy fluctuations. Quantum noise is due to the apparently discrete nature of the small quantum constituents such as electrons, as well as the discrete nature of quantum effects, such as photocurrents. Quantified noise is similar to classical noise theory and will not always return an asymmetric spectral density. Shot noise as coined by J. Verdeyen is a form of quantum noise related to the statistics of photon counting, the discrete nature of electrons, and intrinsic noise generation in electronics. In contrast to shot noise, the quantum mechanical uncertainty principle sets a lower limit to a measurement. The uncertainty principle requires any amplifier or detector to have noise. A signal's noise is quantified as the Fourier transform of its autocorrelation. The autocorrelation of a signal is given as
which measures when our signal is positively, negatively or not correlated at different times and The time average, , is zero and our is a voltage signal. Its Fourier transform is
because we measure a voltage over a finite time window. The Wiener–Khinchin theorem generally states that a noise's power spectrum is given as the autocorrelation of a signal, i. e.,
The above relation is sometimes called the power spectrum or spectral density. In the above outline, we assumed that
Our noise is stationary or the probability does not change over time. Only the time difference matters. Noise is due to a very large number of fluctuating charge so that the central limit theorem applie, i. e., the noise is Gaussian or normally distributed. decays to zero rapidly over some time We sample over a sufficiently large time, , that our integral scales as a random walk So our is independent of measured time for Said in another way, as
One can show that an ideal "top hat" signal, which may correspond to a finite measurement of a voltage over some time, will produce noise across its entire spectrum as a sinc function. Even in the classical case, noise is produced.
, которая измеряет, когда наш сигнал положительно, отрицательно или не коррелирован в разные моменты времени, а среднее по времени, , равно нулю, а наш сигнал – сигнал напряжения. Его преобразование Фурье равно
Quantum noise is noise arising from the indeterminate state of matter in accordance with fundamental principles of quantum mechanics, specifically the uncertainty principle and via zero point energy fluctuations. Quantum noise is due to the apparently discrete nature of the small quantum constituents such as electrons, as well as the discrete nature of quantum effects, such as photocurrents. Quantified noise is similar to classical noise theory and will not always return an asymmetric spectral density. Shot noise as coined by J. Verdeyen is a form of quantum noise related to the statistics of photon counting, the discrete nature of electrons, and intrinsic noise generation in electronics. In contrast to shot noise, the quantum mechanical uncertainty principle sets a lower limit to a measurement. The uncertainty principle requires any amplifier or detector to have noise. A signal's noise is quantified as the Fourier transform of its autocorrelation. The autocorrelation of a signal is given as
which measures when our signal is positively, negatively or not correlated at different times and The time average, , is zero and our is a voltage signal. Its Fourier transform is
because we measure a voltage over a finite time window. The Wiener–Khinchin theorem generally states that a noise's power spectrum is given as the autocorrelation of a signal, i. e.,
The above relation is sometimes called the power spectrum or spectral density. In the above outline, we assumed that
Our noise is stationary or the probability does not change over time. Only the time difference matters. Noise is due to a very large number of fluctuating charge so that the central limit theorem applie, i. e., the noise is Gaussian or normally distributed. decays to zero rapidly over some time We sample over a sufficiently large time, , that our integral scales as a random walk So our is independent of measured time for Said in another way, as
One can show that an ideal "top hat" signal, which may correspond to a finite measurement of a voltage over some time, will produce noise across its entire spectrum as a sinc function. Even in the classical case, noise is produced.
, поскольку мы измеряем напряжение в течение конечного временного окна. Теорема Винера — Хинчина обычно утверждает, что спектр мощности шума задается как автокорреляция сигнала, то есть
Quantum noise is noise arising from the indeterminate state of matter in accordance with fundamental principles of quantum mechanics, specifically the uncertainty principle and via zero point energy fluctuations. Quantum noise is due to the apparently discrete nature of the small quantum constituents such as electrons, as well as the discrete nature of quantum effects, such as photocurrents. Quantified noise is similar to classical noise theory and will not always return an asymmetric spectral density. Shot noise as coined by J. Verdeyen is a form of quantum noise related to the statistics of photon counting, the discrete nature of electrons, and intrinsic noise generation in electronics. In contrast to shot noise, the quantum mechanical uncertainty principle sets a lower limit to a measurement. The uncertainty principle requires any amplifier or detector to have noise. A signal's noise is quantified as the Fourier transform of its autocorrelation. The autocorrelation of a signal is given as
which measures when our signal is positively, negatively or not correlated at different times and The time average, , is zero and our is a voltage signal. Its Fourier transform is
because we measure a voltage over a finite time window. The Wiener–Khinchin theorem generally states that a noise's power spectrum is given as the autocorrelation of a signal, i. e.,
The above relation is sometimes called the power spectrum or spectral density. In the above outline, we assumed that
Our noise is stationary or the probability does not change over time. Only the time difference matters. Noise is due to a very large number of fluctuating charge so that the central limit theorem applie, i. e., the noise is Gaussian or normally distributed. decays to zero rapidly over some time We sample over a sufficiently large time, , that our integral scales as a random walk So our is independent of measured time for Said in another way, as
One can show that an ideal "top hat" signal, which may correspond to a finite measurement of a voltage over some time, will produce noise across its entire spectrum as a sinc function. Even in the classical case, noise is produced.
.
Quantum noise is noise arising from the indeterminate state of matter in accordance with fundamental principles of quantum mechanics, specifically the uncertainty principle and via zero point energy fluctuations. Quantum noise is due to the apparently discrete nature of the small quantum constituents such as electrons, as well as the discrete nature of quantum effects, such as photocurrents. Quantified noise is similar to classical noise theory and will not always return an asymmetric spectral density. Shot noise as coined by J. Verdeyen is a form of quantum noise related to the statistics of photon counting, the discrete nature of electrons, and intrinsic noise generation in electronics. In contrast to shot noise, the quantum mechanical uncertainty principle sets a lower limit to a measurement. The uncertainty principle requires any amplifier or detector to have noise. A signal's noise is quantified as the Fourier transform of its autocorrelation. The autocorrelation of a signal is given as
which measures when our signal is positively, negatively or not correlated at different times and The time average, , is zero and our is a voltage signal. Its Fourier transform is
because we measure a voltage over a finite time window. The Wiener–Khinchin theorem generally states that a noise's power spectrum is given as the autocorrelation of a signal, i. e.,
The above relation is sometimes called the power spectrum or spectral density. In the above outline, we assumed that
Our noise is stationary or the probability does not change over time. Only the time difference matters. Noise is due to a very large number of fluctuating charge so that the central limit theorem applie, i. e., the noise is Gaussian or normally distributed. decays to zero rapidly over some time We sample over a sufficiently large time, , that our integral scales as a random walk So our is independent of measured time for Said in another way, as
One can show that an ideal "top hat" signal, which may correspond to a finite measurement of a voltage over some time, will produce noise across its entire spectrum as a sinc function. Even in the classical case, noise is produced.
Вышеуказанное соотношение иногда называют спектром мощности или спектральной плотностью. В приведенном выше описании мы предположили, что
наш шум стационарен, или вероятность не меняется со временем. Имеет значение только разница во времени. Шум возникает из-за очень большого числа флуктуирующих зарядов, поэтому применима центральная предельная теорема, то есть шум имеет гауссовское или нормальное распределение. быстро убывает до нуля с течением времени. Мы выбираем достаточно большое время , чтобы наши интегралы масштабировались как случайное блуждание. Таким образом, наша не зависит от измеренного времени для . Другими словами, как .
Quantum noise is noise arising from the indeterminate state of matter in accordance with fundamental principles of quantum mechanics, specifically the uncertainty principle and via zero point energy fluctuations. Quantum noise is due to the apparently discrete nature of the small quantum constituents such as electrons, as well as the discrete nature of quantum effects, such as photocurrents. Quantified noise is similar to classical noise theory and will not always return an asymmetric spectral density. Shot noise as coined by J. Verdeyen is a form of quantum noise related to the statistics of photon counting, the discrete nature of electrons, and intrinsic noise generation in electronics. In contrast to shot noise, the quantum mechanical uncertainty principle sets a lower limit to a measurement. The uncertainty principle requires any amplifier or detector to have noise. A signal's noise is quantified as the Fourier transform of its autocorrelation. The autocorrelation of a signal is given as
which measures when our signal is positively, negatively or not correlated at different times and The time average, , is zero and our is a voltage signal. Its Fourier transform is
because we measure a voltage over a finite time window. The Wiener–Khinchin theorem generally states that a noise's power spectrum is given as the autocorrelation of a signal, i. e.,
The above relation is sometimes called the power spectrum or spectral density. In the above outline, we assumed that
Our noise is stationary or the probability does not change over time. Only the time difference matters. Noise is due to a very large number of fluctuating charge so that the central limit theorem applie, i. e., the noise is Gaussian or normally distributed. decays to zero rapidly over some time We sample over a sufficiently large time, , that our integral scales as a random walk So our is independent of measured time for Said in another way, as
One can show that an ideal "top hat" signal, which may correspond to a finite measurement of a voltage over some time, will produce noise across its entire spectrum as a sinc function. Even in the classical case, noise is produced.
Можно показать, что идеальный сигнал типа "прямоугольник", который может соответствовать конечному измерению напряжения в течение определенного времени, будет генерировать шум по всему спектру в виде sinc-функции. Даже в классическом случае генерируется шум.
Quantum noise is noise arising from the indeterminate state of matter in accordance with fundamental principles of quantum mechanics, specifically the uncertainty principle and via zero point energy fluctuations. Quantum noise is due to the apparently discrete nature of the small quantum constituents such as electrons, as well as the discrete nature of quantum effects, such as photocurrents. Quantified noise is similar to classical noise theory and will not always return an asymmetric spectral density. Shot noise as coined by J. Verdeyen is a form of quantum noise related to the statistics of photon counting, the discrete nature of electrons, and intrinsic noise generation in electronics. In contrast to shot noise, the quantum mechanical uncertainty principle sets a lower limit to a measurement. The uncertainty principle requires any amplifier or detector to have noise. A signal's noise is quantified as the Fourier transform of its autocorrelation. The autocorrelation of a signal is given as
which measures when our signal is positively, negatively or not correlated at different times and The time average, , is zero and our is a voltage signal. Its Fourier transform is
because we measure a voltage over a finite time window. The Wiener–Khinchin theorem generally states that a noise's power spectrum is given as the autocorrelation of a signal, i. e.,
The above relation is sometimes called the power spectrum or spectral density. In the above outline, we assumed that
Our noise is stationary or the probability does not change over time. Only the time difference matters. Noise is due to a very large number of fluctuating charge so that the central limit theorem applie, i. e., the noise is Gaussian or normally distributed. decays to zero rapidly over some time We sample over a sufficiently large time, , that our integral scales as a random walk So our is independent of measured time for Said in another way, as
One can show that an ideal "top hat" signal, which may correspond to a finite measurement of a voltage over some time, will produce noise across its entire spectrum as a sinc function. Even in the classical case, noise is produced.
Классический шум к квантовому
Для изучения квантового шума соответствующие классические измерения заменяются квантовыми операторами, например,
где – квантовое статистическое среднее, вычисленное с использованием матрицы плотности в картине Гейзенберга.
Квантовый шум и принцип неопределенности
Неопределенность Гейзенберга подразумевает существование шума. Оператор с эрмитовым сопряженным удовлетворяет соотношению, определите как , где – вещественное число. и – квантовые операторы. Мы можем показать следующее:
где – это усреднения по волновой функции и другим статистическим свойствам. Левые члены – это неопределенность в и , второй член справа – ковариация или , возникающая в результате связи с внешним источником или квантовыми эффектами. Первый член справа соответствует коммутационному соотношению и аннулируется, если x и y коммутируют. Именно в этом и заключается источник нашего квантового шума. Для наглядности рассмотрим случай, когда и соответствуют координате и импульсу, удовлетворяющим известному коммутационному соотношению. Тогда наше новое выражение будет:
где – корреляция. Если второй член справа обращается в ноль, то мы восстанавливаем принцип неопределенности Гейзенберга.
Физическая интерпретация спектральной плотности
Как правило, положительная частота спектральной плотности соответствует притоку энергии к осциллятору (например, к квантованному полю фотонов), а отрицательная частота – оттоку энергии от осциллятора. Физически, асимметричная спектральная плотность соответствует чистому потоку энергии либо от, либо к нашей модели осциллятора.
Линейный прирост и квантовая неопределенность
Большинство оптических систем связи используют амплитудную модуляцию, где квантовый шум в основном обусловлен шумом выстрела. Квантовый шум лазера, не считая шума выстрела, представляет собой неопределенность амплитуды и фазы его электрического поля. Эта неопределенность становится наблюдаемой, когда квантовый усилитель сохраняет фазу. Фазовый шум становится значимым, когда энергия частотной или фазовой модуляции сопоставима с энергией сигнала (частотная модуляция более устойчива к воздействию шума, чем амплитудная модуляция из-за аддитивного шума, характерного для амплитудной модуляции).
Флуктуации нулевой точки
Существование энергетических флуктуаций нулевой точки хорошо установлено в теории квантованного электромагнитного поля. В общем случае, при наименьшем энергетическом возбуждении квантованного поля, пронизывающего все пространство (то есть, когда мода поля находится в вакуумном состоянии), среднеквадратичное отклонение напряженности поля не равно нулю. Это объясняет вакуумные флуктуации, пронизывающие все пространство. Эти вакуумные флуктуации, или квантовый шум, оказывают влияние на классические системы. Это проявляется как квантовая декогеренция в запутанной системе, обычно приписываемая тепловым различиям в условиях, окружающих каждую запутанную частицу. Поскольку запутанность интенсивно изучается на примере простых пар запутанных фотонов, наблюдаемая в экспериментах декогеренция вполне может быть синонимом "квантового шума" как источника этой декогеренции. Вакуумные флуктуации могут быть причиной спонтанного появления квантов энергии в данном поле или пространстве-времени, и в этом случае с этим событием должны быть связаны тепловые различия. Следовательно, это вызовет декогеренцию в запутанной системе вблизи места возникновения флуктуации.
Когерентные состояния и шум квантового усилителя
Лазер описывается когерентным состоянием света или суперпозицией собственных состояний гармонических осцилляторов. Эрвин Шрёдингер впервые получил когерентное состояние для уравнения Шрёдингера, чтобы соответствовать принципу соответствия, в 1926 году. Квантовое усиление можно представить с помощью унитарного оператора, , как указано в работе Д. Кузнецова 1995 года.