User:Kirby2u2/EIT

The effect of EIT on a typical absorption line. A weak probe normally experiences absorption shown in blue. A second coupling beam induces EIT and creates a "window" in the absorption region (red). This plot is a computer simulation of EIT in an InAs/GaAs quantum dot

Electromagnetically induced transparency (EIT) is a coherent optical nonlinearity which renders a medium transparent over a narrow spectral range within an absorption line. Extreme dispersion is also created within this transparency "window" which leads to "slow light", described below. Basically it "is a quantum interference effect that permits the propagation of light through an otherwise opaque atomic medium".

Observation of EIT involves two optical fields (highly coherent light sources, such as lasers) which are tuned to interact with three quantum states of a material. The "probe" field is tuned near resonance between two of the states and measures the absorption spectrum of the transition. A much stronger "coupling" field is tuned near resonance at a different transition. If the states are selected properly, the presence of the coupling field will create a spectral "window" of transparency which will be detected by the probe. The coupling laser is sometimes referred to as the "control" or "pump", the latter in analogy to incoherent optical nonlinearities such as spectral hole burning or saturation.

EIT is based on the destructive interference of the transition probability amplitude between atomic states. Closely related to EIT are coherent population trapping (CPT) phenomena.

The quantum interference in EIT can be exploited to laser cool atomic particles, even down to the quantum mechanical ground state of motion.

Medium requirements

EIT level schemes can be sorted into three categories; vee, ladder, and lambda.

There are specific restrictions on the configuration of the three states. Two of the three possible transitions between the states must be "dipole allowed", i.e. the transitions can be induced by an oscillating electric field. The third transition must be "dipole forbidden." One of the three states is connected to the other two by the two optical fields. The three types of EIT schemes are differentiated by the energy differences between this state and the other two. The schemes are the ladder, vee, and lambda. Any real material system may contain many triplets of states which could theoretically support EIT, but there are several practical limitations on which levels can actually be used.

Also important are the dephasing rates of the individual states. In any real system at finite temperature there are processes which cause a scrambling of the phase of the quantum states. In the gas phase, this means usually collisions. In solids, dephasing is due to interaction of the electronic states with the host lattice. The dephasing of state $|3\rangle$ is especially important; ideally $|3\rangle$ should be a robust, metastable state.

Current EIT research uses atomic systems in dilute gases, solid solutions, or more exotic states such as Bose–Einstein condensate. EIT has been demonstrated in electromechanical and optomechanical systems, where it is known as optomechanically induced transparency. Work is also being done in semiconductor nanostructures such as quantum wells, quantum wires and quantum dots.

Theory

EIT was first proposed theoretically by professor Jakob Khanin and graduate student Olga Kocharovskaya at Gorky State University (present: Nizhny Novgorod), Russia; there are now several different approaches to a theoretical treatment of EIT.

One approach is to extend the density matrix treatment used to derive Rabi oscillation of a two-state, single field system. In this picture the probability amplitude for the system to transfer between states can interfere destructively, preventing absorption. In this context, "interference" refers to interference between quantum events (transitions) and not optical interference of any kind. As a specific example, consider the lambda scheme shown above. Absorption of the probe is defined by transition from $|1\rangle$ to $|2\rangle$. The fields can drive population from $|1\rangle$-$|2\rangle$ directly or from $|1\rangle$-$|2\rangle$-$|3\rangle$-$|2\rangle$. The probability amplitudes for the different paths interfere destructively. If $|3\rangle$ has a comparatively long lifetime, then the result will be a transparent window completely inside of the $|1\rangle$-$|2\rangle$ absorption line.

Another approach is the "dressed state" picture, wherein the system + coupling field Hamiltonian is diagonalized and the effect on the probe is calculated in the new basis. In this picture EIT resembles a combination of Autler-Townes splitting and Fano interference between the dressed states. Between the doublet peaks, in the center of the transparency window, the quantum probability amplitudes for the probe to cause a transition to either state cancel. To see this, we begin with the assumption that the Rabi frequency $\Omega _{c}$ of the coupling field is much stronger than that of the probe field $\Omega _{p}$ and consider that only the coupling field induces a non-negligible light-shift on the excited state. In this case, the excited state is split into two energy levels, ${\textstyle |+\rangle =1/{\sqrt {2}}(|3\rangle +|2\rangle )}$ and ${\textstyle |-\rangle =1/{\sqrt {2}}(-|3\rangle +|2\rangle )}$. The effect of the coupling field having been considered, the system is now reduced to a three-level system with only an incident probe field. The Hamiltonian for this system is ${\textstyle H_{tot}=H_{bare}+H_{L-int}}$, where

$H_{bare}=\hbar \left({\begin{matrix}\omega _{g}&0&0\\0&\omega _{+}&0\\0&0&\omega _{-}\\\end{matrix}}\right)$ is the "bare-atom" Hamiltonian (including the light shift from the coupling field), and

$H_{L-int}=\hbar \left({\begin{matrix}0&\Omega _{p}-\delta /2&\Omega _{p}+\delta /2\\\Omega _{p}-\delta /2&0&0\\\Omega _{p}+\delta /2&0&0\\\end{matrix}}\right)$ is the light-atom interaction of the probe field with the new system, where ${\textstyle \delta }$ is the detuning of the probe from resonance with the two excited states.

The evolution of the population in the excited states, then, is given by Liouville's equation to be

${\dot {\rho }}_{++}={\frac {i}{2}}(\rho _{g+}-\rho _{+g})(\delta -2\Omega _{p})$

${\dot {\rho }}_{--}={\frac {i}{2}}(\rho _{g-}-\rho _{-g})(\delta +2\Omega _{p})$

Returning to the original basis,

${\dot {\rho }}_{22}={\frac {1}{\sqrt {2}}}\left[(\rho _{g+}-\rho _{+g})(\delta -2\Omega _{p})+(\rho _{g-}-\rho _{-g})(\delta +2\Omega _{p})\right]|2\rangle$

${\dot {\rho }}_{22}=\left[{\frac {2\Omega _{p}}{\sqrt {2}}}\left((\rho _{g-}-\rho _{-g})-(\rho _{g+}-\rho _{+g})\right)\quad +\quad {\frac {\delta }{\sqrt {2}}}((\rho _{g-}-\rho _{-g})+(\rho _{g+}-\rho _{+g}))\right]|2\rangle \qquad$.

From this, we can see a destructive interference of the coherences between the ground and excited state.

A polariton picture is particularly important in describing stopped light schemes. Here, the photons of the probe are coherently "transformed" into "dark state polaritons" which are excitations of the medium. These excitations exist (or can be "stored") for a length of time dependent only on the dephasing rates.

Zeeman EIT

F=1> ground state undergoes Zeeman Splitting in the presence of a magnetic field. $\sigma _{+}$ and $\sigma _{-}$ pump and probe beams couple magnetic sub-levels to the excited state.

Zeeman Electromagnetically Induced Transparency (Zeeman EIT) is a form of EIT in which the incident optical fields excite transitions between two Zeeman sublevels of a given ground state and a common excited state.  If the probability amplitudes associated with these transitions are equal, they interfere destructively and induce a transparency in the sample, as discussed below. Transitions in the system are addressed by different circular polarizations within the optical field; for instance, in the picture on the right, the $\sigma _{+}$ field excites a transition between the m=-1 ground state and the m=0 excited state, whereas the $\sigma _{-}$ field excites a transition between the m=+1 ground state and the m=0 excited state.  There are accordingly two different paths to the same final state, such that the system undergoes Fano-type interference.

The total Hamiltonian of the system includes the Hamiltonian of the bare atom $H_{bare}$, that of the light-atom interaction $H_{L-int}$, and that of the magnetic field-atom interaction $H_{B-int}$.

Populations of energy levels in a Zeeman EIT scheme. As the probe detuning changes, the population levels shift. Near zero detuning, the coherences associated with the excited state interfere, and the population remains in the ground state.

$H_{tot}=H_{bare}+H_{L-int}+H_{B-int}$

$H_{bare}/\hbar =\omega _{g}\left[|a\rangle \langle a|+|b\rangle \langle b|+|c\rangle \langle c|\right]+\omega _{e}|d\rangle \langle d|$

$H_{L-int}/\hbar =\Omega _{1}\left[|a\rangle \langle d|+|d\rangle \langle a|\right]+(\delta -\Omega _{2})[|c\rangle \langle d|+|d\rangle \langle c|]$

$H_{B-int}/\hbar =\Omega _{L}\left[|a\rangle \langle a|-|c\rangle \langle c|\right]$

which yields

$H_{tot}=\hbar \left({\begin{matrix}\omega _{g}+\Omega _{L}&0&0&\Omega _{1}\\0&\omega _{g}&0&0\\0&0&\omega _{g}-\Omega _{L}&\delta -\Omega _{2}\\\Omega _{1}&0&\delta -\Omega _{2}&\omega _{e}\end{matrix}}\right)$

In the above equations, ${\textstyle \omega _{g}}$ is the energy of the un-perturbed ground state, ${\textstyle \omega _{e}}$ is the energy of the excited state, ${\textstyle \Omega _{1}}$ and ${\textstyle \Omega _{2}}$ are the Rabi frequencies of the pump and probe fields, ${\textstyle \delta }$ is the detuning of the probe from resonance, and ${\textstyle \Omega _{L}}$ is the Larmor frequency associated with the applied magnetic field.

We also consider the relaxation ${\textstyle R}$ of the excited state due to spontaneous emission ${\textstyle \Gamma }$, and the corresponding repopulation ${\textstyle \Lambda _{\Gamma }}$ of the ground state from the excited state: ${\textstyle R=\Gamma |d\rangle \langle d|}$ , ${\textstyle \Lambda _{\Gamma }=\Gamma \rho _{dd}/3*(|a\rangle \langle a|+|b\rangle \langle b|+|c\rangle \langle c|)}$

Using Liouville's Equation ${\textstyle {\dot {\rho }}={\frac {1}{{i}\hbar }}[H_{tot},\rho ]-{\frac {1}{2}}[R,\rho ]+\Lambda _{\Gamma }}$, we can obtain the time derivative of the population density matrix.

We focus on the time rate of changes of the populations in each of the four energy levels.

${\dot {\rho }}_{aa}=\Gamma \rho _{dd}/3+{i}(\rho _{ad}-\rho _{da})\Omega _{1}$

${\dot {\rho }}_{bb}=\Gamma \rho _{dd}/3$

${\dot {\rho }}_{cc}=\Gamma \rho _{dd}/3+{i}(\rho _{cd}-\rho _{dc})(\delta -\Omega _{2})$

${\dot {\rho }}_{dd}=-\Gamma \rho _{dd}-{i}((\rho _{ad}-\rho _{da})\Omega _{1}+(\rho _{cd}-\rho _{dc})(\delta -\Omega _{2}))$

By adjusting the field parameters $\Omega _{1}$, $\Omega _{2}$, and $\delta$, the coherences associated with the excited state destructively interfere; the system is no longer driven to the excited state by either field, and any population in the excited state eventually decays to the ground state (i.e., ${\textstyle \rho _{dd}\Rightarrow 0}$). This can also be seen by considering the complex susceptibility of the EIT medium.

Compared with traditional EIT, Zeeman EIT provides many advantages for experiment; e.g, as the frequencies $\omega _{c}$and $\omega _{p}$ differ only slightly (by twice the Zeeman splitting), only one optical source is typically required. This relaxes requirements for phase-locking two different optical sources or employing optics for each frequency used, etc. However, theoretical considerations for typical atomic systems are often complicated by the presence of other Zeeman sub-levels; for instance, given a system with $|F_{g}=3\rangle$ and $|F_{e}=2\rangle$, if the probe field is resonant with the transition between $|m_{g}=-1\rangle$ $\Rightarrow$ $|m_{e}=-2\rangle$, then it is also resonant with $|m_{g}=0\rangle$ $\Rightarrow$ $|m_{e}=-1\rangle$, with $|m_{g}=+1\rangle$ $\Rightarrow$$|m_{e}=0\rangle$, etc., and similarly the pump field is also resonant with many transitions. This typically requires theoretical treatments to consider all of the energy levels in the system, making density matrix approaches very time-consuming.

Two common experimental treatments of Zeeman EIT are given by X. Feng et al, wherein the authors observe probe transmission spectra through warm Rubidium vapor.

In the first treatment, they split the single-frequency output of an external cavity diode laser into two orthogonal linear polarizations; the frequencies of these different polarizations were independently adjusted using double-pass accousto-optic modulators (AOMs) to closely match the desired transitions. The two beams of orthogonal linear polarization were made co-linear, and then passed through a quarter wave plate to transform the beams into $\sigma _{+}$ and $\sigma _{-}$ polarizations. Transmission of the probe beam (independent of the pump) through a Rb gas cell was measured by first converting the circularly polarized pump and probe beams back to linear polarization, and subsequently polarization-separating the two beams using a polarizing beamsplitter. The absorption spectrum was obtained by varying the detuning ${\textstyle \delta }$ using its AOM, and measuring the resultant probe transmission on a photodetector. In this configuration, the magnetic field applied to the Rb gas cell was held constant using magnetic shielding in the presence of the Earth's magnetic field. The above theoretical treatment Zeeman EIT closely matches the results obtained by this method.

In the second treatment, both the pump and the probe beams undergo identical frequency shifts in the same double-pass AOM, and the frequency of both beams is held constant. In order to generate the desired absorption spectrum, the magnetic field applied to the Rb vapor cell is varied using a solenoid, and the probe transmission is observed on a detector as before. Although the above theoretical treatment qualitatively agrees with the results of this experimental scheme, it does not adequately explain the changes to the shape of the observed EIT feature. Scanning the magnetic field not only changes the Zeeman splitting ${\textstyle \Omega _{L}}$ for each energy level, but also the detuning of both the pump ${\textstyle \delta _{pump}}$ and the probe ${\textstyle \delta _{probe}}$ beams. This also changes the ($\omega$-dependent) Rabi frequencies of the pump ${\textstyle \Omega _{1}}$ and probe ${\textstyle \Omega _{2}}$ at different rates, introducing an asymmetry into the observed spectrum. These factors present a considerable challenge to quantitatively interpreting results in terms of an energy spectrum, such that some studies using this experimental setup report the width of observed spectral features simply in terms of magnetic field strength.

Slow light and stopped light

Rapid change of index of refraction (blue) in a region of rapidly changing absorption (gray) associated with EIT. The steep and positive linear region of the refractive index in the center of the transparency window gives rise to slow light

It is important to realize that EIT is only one of many diverse mechanisms which can produce slow light. The Kramers–Kronig relations dictate that a change in absorption (or gain) over a narrow spectral range must be accompanied by a change in refractive index over a similarly narrow region. This rapid and positive change in refractive index produces an extremely low group velocity. The first experimental observation of the low group velocity produced by EIT was by Boller, Imamoglu, and Harris at Stanford University in 1991 in strontium. In 1999 Lene Hau reported slowing light in a medium of ultracold sodium atoms, achieving this by using quantum interference effects responsible for electromagnetically induced transparency (EIT). Her group performed copious research regarding EIT with Stephen E. Harris. "Using detailed numerical simulations, and analytical theory, we study properties of micro-cavities which incorporate materials that exhibit Electro-magnetically Induced Transparency (EIT) or Ultra Slow Light (USL). We find that such systems, while being miniature in size (order wavelength), and integrable, can have some outstanding properties. In particular, they could have lifetimes orders of magnitude longer than other existing systems, and could exhibit non-linear all-optical switching at single photon power levels. Potential applications include miniature atomic clocks, and all-optical quantum information processing." The current record for slow light in an EIT medium is held by Budker, Kimball, Rochester, and Yashchuk at U.C. Berkeley in 1999. Group velocities as low as 8 m/s were measured in a warm thermal rubidium vapor.

Stopped light, in the context of an EIT medium, refers to the coherent transfer of photons to the quantum system and back again. In principle, this involves switching off the coupling beam in an adiabatic fashion while the probe pulse is still inside of the EIT medium. There is experimental evidence of trapped pulses in EIT medium. In authors created a stationary light pulse inside the atomic coherent media. In 2009 researchers from Harvard University and MIT demonstrated a few-photon optical switch for quantum optics based on the slow light ideas. Lene Hau and a team from Harvard University were the first to demonstrate stopped light.

See also

• Atomic coherence
• Electromagnetically Induced Grating

References

Primary work

• O.Kocharovskaya, Ya.I.Khanin, Sov. Phys. JETP, 63, p945 (1986)
• K.J. Boller, A. Imamoglu, S. E. Harris, Physical Review Letters 66, p2593 (1991)
• Eberly, J. H., M. L. Pons, and H. R. Haq, Phys. Rev. Lett. 72, 56 (1994)
• D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, Physical Review Letters, 83, p1767 (1999)
• Lene Vestergaard Hau, S.E. Harris, Zachary Dutton, Cyrus H. Behroozi, Nature v.397, p594 (1999)
• D.F. Phillips, A. Fleischhauer, A. Mair, R.L. Walsworth, M.D. Lukin, Physical Review Letters 86, p783 (2001)
• Naomi S. Ginsberg, Sean R. Garner, Lene Vestergaard Hau, Nature 445, 623 (2007)

Review

• Harris, Steve (July, 1997). Electromagnetically Induced Transparency. Physics Today, 50 (7), pp. 36–42 (PDF Format)
• Zachary Dutton, Naomi S. Ginsberg, Christopher Slowe, and Lene Vestergaard Hau (2004) The art of taming light: ultra-slow and stopped light. Europhysics News Vol. 35 No. 2
• M. Fleischhauer, A. Imamoglu, and J. P. Marangos (2005), "Electromagnetically induced transparency: Optics in Coherent Media", Reviews Modern Physics, 77, 633

Category:Wave mechanics Category:Quantum mechanics Category:Optics