In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the Rabi cycle (or Rabi flop) is the cyclic behaviour of a two-level
quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of
quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
,
condensed matter
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
, atomic and molecular physics, and nuclear and
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
can be conveniently studied in terms of
two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an oscillatory driving field. The effect is important in
quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...
,
magnetic resonance
Magnetic resonance is a process by which a physical excitation (resonance) is set up via magnetism.
This process was used to develop magnetic resonance imaging and Nuclear magnetic resonance spectroscopy technology.
It is also being used to d ...
and
quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
, and is named after
Isidor Isaac Rabi
Isidor Isaac Rabi (; born Israel Isaac Rabi, July 29, 1898 – January 11, 1988) was an American physicist who won the Nobel Prize in Physics in 1944 for his discovery of nuclear magnetic resonance, which is used in magnetic resonance ima ...
.
A two-level system is one that has two possible energy levels. These two levels are a ground state with lower energy and an excited state with higher energy. If the energy levels are not degenerate (i.e. not having equal energies), the system can absorb a
quantum
In physics, a quantum (plural quanta) is the minimum amount of any physical entity ( physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizat ...
of energy and transition from the ground state to the "excited" state. When an
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, a ...
(or some other
two-level system
In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a sys ...
) is illuminated by a coherent beam of
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s, it will cyclically
absorb photons and re-emit them by
stimulated emission
Stimulated emission is the process by which an incoming photon of a specific frequency can interact with an excited atomic electron (or other excited molecular state), causing it to drop to a lower energy level. The liberated energy transfers to th ...
. One such cycle is called a Rabi cycle, and the inverse of its duration is the
Rabi frequency The Rabi frequency is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field. It is proportional to the Transition Dipole Moment of the two levels and to the amplitude (''not ...
of the photon beam. The effect can be modeled using the
Jaynes–Cummings model
The Jaynes–Cummings model (sometimes abbreviated JCM) is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field), with or without the prese ...
and the
Bloch vector formalism.
Mathematical description
A detailed mathematical description of the effect can be found on the page for the
Rabi problem
The Rabi problem concerns the response of an atom to an applied harmonic electric field, with an applied frequency very close to the atom's natural frequency. It provides a simple and generally solvable example of light–atom interactions and is ...
. For example, for a two-state atom (an atom in which an electron can either be in the excited or ground state) in an electromagnetic field with frequency tuned to the excitation energy, the probability of finding the atom in the excited state is found from the Bloch equations to be
:
where
is the Rabi frequency.
More generally, one can consider a system where the two levels under consideration are not energy
eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s. Therefore, if the system is initialized in one of these levels, time evolution will make the population of each of the levels oscillate with some characteristic frequency, whose
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
is also known as the Rabi frequency. The state of a two-state quantum system can be represented as vectors of a two-dimensional
complex Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
, which means that every
state vector is represented by good
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
coordinates:
:
where
and
are the coordinates.
If the vectors are normalized,
and
are related by
. The basis vectors will be represented as
and
.
All
observable physical quantities associated with this systems are 2 × 2
Hermitian matrices
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
, which means that the
Hamiltonian of the system is also a similar matrix.
Procedure
One can construct an
oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendul ...
experiment through the following steps:
# Prepare the system in a fixed state; for example,
# Let the state evolve freely, under a
Hamiltonian ''H'' for time ''t''
# Find the probability
, that the state is in
If
is an eigenstate of H,
and there will be no oscillations. Also if the two states
and
are degenerate, every state including
is an eigenstate of H. As a result, there will be no oscillations.
On the other hand, if H has no degenerate eigenstates, and the initial state is not an eigenstate, then there will be oscillations. The most general form of the Hamiltonian of a two-state system is given
:
here,
and
are real numbers. This matrix can be decomposed as,
:
The matrix
is the 2
2 identity matrix and the matrices
are the
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
. This decomposition simplifies the analysis of the system especially in the time-independent case where the values of
and
are constants. Consider the case of a
spin-1/2
In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one full ...
particle in a magnetic field
. The interaction Hamiltonian for this system is
:
,
where
is the magnitude of the particle's
magnetic moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
,
is the
Gyromagnetic ratio
In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol , gamma. Its SI u ...
and
is the vector of
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
. Here the eigenstates of Hamiltonian are eigenstates of
, that is
and
, with corresponding eigenvalues of
. The probability that a system in the state
can be found in the arbitrary state
is given by
.
Let the system be prepared in state
at time
. Note that
is an eigenstate of
:
:
Here the Hamiltonian is time independent. Thus by solving the stationary Schrödinger equation, the state after time t is given by
with total energy of the system
. So the state after time t is given by:
:
.
Now suppose the spin is measured in x-direction at time t. The probability of finding spin-up is given by:
where
is a characteristic
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
given by
, where it has been assumed that
. So in this case the probability of finding spin-up in x-direction is oscillatory in time
when the system's spin is initially in the
direction. Similarly, if we measure the spin in the
-direction, the probability of measuring spin as
of the system is
. In the degenerate case where
, the characteristic frequency is 0 and there is no oscillation.
Notice that if a system is in an eigenstate of a given
Hamiltonian, the system remains in that state.
This is true even for time dependent Hamiltonians. Taking for example
; if the system's initial spin state is
, then the probability that a measurement of the spin in the y-direction results in
at time
is
.
:
Derivation using nonperturbative procedure by means of the Pauli matrices
Consider a Hamiltonian of the form
The eigenvalues of this matrix are given by
where
and
, so we can take
.
Now, eigenvectors for
can be found from equation
So
Applying the normalization condition on the eigenvectors,
. So
Let
and
. So
.
So we get
. That is
, using the identity
.
The phase of
releative to
should be
.
Choosing
to be real, the eigenvector for the eigenvalue
is given by
Similarly, the eigenvector for eigenenergy
is
From these two equations, we can write
Suppose the system starts in state
at time
; that is,
For a time-independent Hamiltonian, after time ''t'', the state evolves as
If the system is in one of the eigenstates
or
, it will remain the same state. However, for a time-dependent Hamiltonian and a general initial state as shown above, the time evolution is non trivial. The resulting formula for the Rabi oscillation is valid because the state of the spin may be viewed in a reference frame that rotates along with the field.
The probability amplitude of finding the system at time t in the state
is given by
.
Now the probability that a system in the state
will be found to be in the state
is given by
This can be simplified to
This shows that there is a finite probability of finding the system in state
when the system is originally in the state
. The probability is oscillatory with angular frequency
, which is simply unique Bohr frequency of the system and also called
Rabi frequency The Rabi frequency is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field. It is proportional to the Transition Dipole Moment of the two levels and to the amplitude (''not ...
. The formula () is known as
Rabi Rabi may refer to:
Places
* Rábí, a castle in the Czech Republic
* Rabí, a village in the Czech Republic
* Räbi, a village in Estonia
* Rabi, Panchthar, a village development committee in Nepal
* Rabi Island, a volcanic island in northern ...
formula. Now after time t the probability that the system in state
is given by
, which is also oscillatory.
These types of oscillations of two-level systems are called Rabi oscillations, which arise in many problems such as
Neutrino oscillation, the
ionized Hydrogen molecule,
Quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
,
Ammonia maser
A maser (, an acronym for microwave amplification by stimulated emission of radiation) is a device that produces coherent electromagnetic waves through amplification by stimulated emission. The first maser was built by Charles H. Townes, Jame ...
etc.
In quantum computing
Any two-state quantum system can be used to model a
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
. Consider a
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...
-
system with magnetic moment
placed in a classical magnetic field
. Let
be the
gyromagnetic ratio
In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol , gamma. Its SI u ...
for the system. The magnetic moment is thus
. The Hamiltonian of this system is then given by
where
and
. One can find the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s and
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of this Hamiltonian by the above-mentioned procedure. Now, let the qubit be in state
at time
. Then, at time
, the probability of it being found in state
is given by
where
. This phenomenon is called Rabi oscillation. Thus, the qubit oscillates between the
and
states. The maximum amplitude for oscillation is achieved at
, which is the condition for
resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscil ...
. At resonance, the transition probability is given by
. To go from state
to state
it is sufficient to adjust the time
during which the rotating field acts such that
or
. This is called a
pulse. If a time intermediate between 0 and
is chosen, we obtain a superposition of
and
. In particular for
, we have a
pulse, which acts as:
. This operation has crucial importance in quantum computing. The equations are essentially identical in the case of a two level atom in the field of a laser when the generally well satisfied rotating wave approximation is made. Then
is the energy difference between the two atomic levels,
is the frequency of laser wave and
Rabi frequency The Rabi frequency is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field. It is proportional to the Transition Dipole Moment of the two levels and to the amplitude (''not ...
is proportional to the product of the transition electric dipole moment of atom
and electric field
of the laser wave that is
. In summary, Rabi oscillations are the basic process used to manipulate qubits. These oscillations are obtained by exposing qubits to periodic electric or magnetic fields during suitably adjusted time intervals.
[''A Short Introduction to Quantum Information and Quantum Computation'' by Michel Le Bellac, ]
See also
*
Atomic coherence
Atomic may refer to:
* Of or relating to the atom, the smallest particle of a chemical element that retains its chemical properties
* Atomic physics, the study of the atom
* Atomic Age, also known as the "Atomic Era"
* Atomic scale, distances com ...
*
Bloch sphere
*
Laser pumping
*
Optical pumping
Optical pumping is a process in which light is used to raise (or "pump") electrons from a lower energy level in an atom or molecule to a higher one. It is commonly used in laser construction to pump the active laser medium so as to achieve pop ...
*
Rabi problem
The Rabi problem concerns the response of an atom to an applied harmonic electric field, with an applied frequency very close to the atom's natural frequency. It provides a simple and generally solvable example of light–atom interactions and is ...
*
Vacuum Rabi oscillation
A vacuum Rabi oscillation is a damped oscillation of an initially excited atom coupled to an electromagnetic resonator or cavity in which the atom alternately emits photon(s) into a single-mode electromagnetic cavity and reabsorbs them. The atom ...
*
Neutral particle oscillation In particle physics, neutral particle oscillation is the transmutation of a particle with zero electric charge into another neutral particle due to a change of a non-zero internal quantum number, via an interaction that does not conserve that quantu ...
References
* ''Quantum Mechanics'' Volume 1 by C. Cohen-Tannoudji, Bernard Diu, Frank Laloe,
* ''A Short Introduction to Quantum Information and Quantum Computation'' by Michel Le Bellac,
The Feynman Lectures on Physics, Volume III* ''Modern Approach To Quantum Mechanics'' by John S Townsend, {{ISBN, 9788130913148
Quantum optics
Atomic physics