Quantum lithography

As quantum lithography are photolithographic method refers to the quantum mechanical properties of the photon field exploited to achieve compared to the conventional ( "classic") methods improved performance and in particular the diffraction limit of the resolution power to overcome. The concept was proposed by Jonathan Dowling and co-workers in 1999 . So far (as of 2018), implementation has been limited to proof-of-principle experiments.

Principle of quantum lithography

The main advantage of quantum lithography processes is that they make it possible to increase the achievable resolution for light of a given wavelength . In classical photolithography, the resolution is limited by the Rayleigh criterion to structures of the order of magnitude of the wavelength. For example, the increasingly smaller structure size in the mass production of computer chips increasingly requires the use of radiation of shorter wavelengths ( ultraviolet and X-ray radiation ), which leads to large increases in the cost of imaging systems.

In order to fall below the diffraction limit while maintaining the same wavelength, quantum lithography uses a combination of two effects that do not play a role in conventional methods: on the one hand, non-classical (mostly entangled) states of the light field and, on the other hand, special photoresists that (in the observed wavelength) only show multiphoton absorption .

In photolithography, a geometric structure is transferred by means of light from a photomask onto a light-sensitive material that changes chemically due to the illumination (and then further processed, e.g. by etching or coating processes). If the light-sensitive process is based on single-photon absorption, the structures transferred in this way are determined by the intensity of the light (on the surface of the material). The intensity typically varies on length scales of the wavelength of the light used, e.g. For example, two plane waves running in opposite directions with a wavelength and a circular wave number generate an intensity pattern (“ interference stripes”) of the shape , that is, the distance between two neighboring maxima is . In contrast, the probability depends on two-photon absorption by the square of the intensity of the light field from what proportion to a pattern results, d. that is, the pattern also contains components that change with it, which could enable improved spatial resolution. However, they always occur together with terms ( ) that vary more slowly and do not allow improved resolution. To do this, one would have to eliminate the slower terms. ${\displaystyle \lambda }$ ${\displaystyle k=2\pi /\lambda }$ ${\displaystyle I(x)\propto (1+\cos(2kx)}$ ${\displaystyle \lambda /4}$ ${\displaystyle (1+\cos(2kx))^{2}}$ ${\displaystyle \cos(4kx)}$ ${\displaystyle \propto \cos(2kx)}$

In the original proposal for quantum lithography, this is done using entangled photons in the incident light field. According to the proposal by Boto and co-workers, a photon absorber and an entangled state with photons in two field modes are used, which describes a coherent superposition between the state with all photons in the first and all in the second mode ${\displaystyle N}$ ${\displaystyle N}$

${\displaystyle \vert N00N\rangle =\left(\vert N\rangle _{1}\vert 0\rangle _{2}+\vert 0\rangle _{1}\vert N\rangle _{2}\right)/{\sqrt {2}}.}$

These so-called N00N states can lead to interference patterns with a characteristic length scale of the th part of the wavelength used, as the relative phase between the interfering paths (which is decisive for the interference pattern and inversely proportional to the wavelength ) is multiplied by a factor . ${\displaystyle N}$ ${\displaystyle \lambda }$ ${\displaystyle N}$

Boto et al. considered, as described above, two modes with wave numbers propagating parallel to the plane opposite to one another . The interference pattern for -fold coincidences is then determined by a -Term, i. H. it has a resolution of . In particular, due to the use of the entangled photons, the lower-frequency terms do not occur, which enables a -fold better spatial resolution. This means that, for example, with a 10-photon N00N state and red light ( ) you could achieve the same resolution as with classic extreme UV radiation ( ). ${\displaystyle k}$ ${\displaystyle N}$ ${\displaystyle \cos ^{2}(Nkx)}$ ${\displaystyle \lambda /N}$ ${\displaystyle N}$ ${\displaystyle \lambda =650\,\mathrm {nm} }$ ${\displaystyle \lambda =65\,\mathrm {nm} }$

The quantum lithography is closely related to the fields of quantum imaging ( quantum imaging ), quantum metrology and quantum sensors .

On the practicability and efficiency of the procedure

Both the -states used and the -photon absorber materials present a major (and increasingly difficult) challenge. In addition, the question arises of how efficient the process is; That is, what fraction of the photons used is actually available for the desired processes and how the necessary exposure time is scaled. As Steueragel noted in 2003, Boto et al. also based on the assumption that the photons also propagate together, i.e. h. that when a first photon currently in place is absorbed, the rest arrive at the same place at the same time. If this assumption is not fulfilled, the super-resolution only occurs if all photons happen to arrive at the same place, which leads to an efficiency that decreases exponentially. As analyzed by Kothe and co-workers, experiments carried out to date indicate that the photons do not always arrive together. On the other hand, however, it is stated that the pessimistic analyzes of this question make highly simplified assumptions about the spatiotemporal character of the -photon wave function and a more realistic view can lead to an improved coincidence rate. This question is still the subject of current research. ${\displaystyle N00N}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle t}$ ${\displaystyle x}$ ${\displaystyle N}$ ${\displaystyle N}$

While the original proposal only looked at one-dimensional structures, it was possible to show in later work that the method basically allows any two-dimensional structures to be defined with the quantum-mechanical improved resolution.

As of 2018, it is not clear whether quantum lithography can be developed into a practically relevant technology, but the potential of this quantum technology is also seen by industry.

Experimental realization

The first experiments on quantum lithography have demonstrated that the predicted super-resolution actually exists, but so far these have been pure demonstration experiments that have not been carried out with photon absorbers, but simulated such processes using photon coincidence counters . ${\displaystyle N}$

Quantum lithography with classical light

New proposals for realizing quantum lithography can also be realized with classical light if the atoms in the material have sufficiently long coherence times. The theoretical proposals are based on multiphoton absorption and emission processes and show that it is possible to put the atoms into an excited internal state with a location-dependent probability , with the probability over distances that correspond to only a fraction of the optical wavelengths used, varies between 0 and 1. In this way, conventional lithographic processes, e.g. B. affect only atoms in the internal state , achieve a structure size determined by the location dependence of . ${\displaystyle P(x)}$ ${\displaystyle e}$ ${\displaystyle P_{e}(x)}$ ${\displaystyle e}$ ${\displaystyle P_{e}(x)}$

According to a proposal from 2006, atoms that have four energy levels relevant to the process (energy eigenstates) are driven with off-resonant light of the four frequencies and in such a way that the two leading resonant processes are the absorption of two photons of the frequency and the emission of one photon corresponds to the frequency or of two photons to the frequency and the emission of one photon to the frequency . Since the absorption takes place in pairs, can be as structures of the half to or write associated wavelength. The method can be generalized to processes that are based on photon absorption and thus have a correspondingly higher resolution. ${\displaystyle \nu _{\pm }}$ ${\displaystyle \omega _{\pm }}$ ${\displaystyle \nu _{+}}$ ${\displaystyle \omega _{+}}$ ${\displaystyle \nu _{-}}$ ${\displaystyle \omega _{-}}$ ${\displaystyle \nu _{+}}$ ${\displaystyle \nu _{-}}$ ${\displaystyle N}$

A 2010 proposal only uses two internal states. The transition between the ground state and the excited state is driven resonantly by coherent light , which leads to Rabi oscillations between the two states. The frequency of this oscillation (the "Rabi frequency" ) is proportional to the electric field strength and thus dependent on location and time, e.g. B. for a standing wave. If the field strength of the light is large enough (or the duration of the pulse long enough) that strongly coupled atoms perform several complete Rabi oscillations (i.e. if the angle is much larger than ), structures below the diffraction limit can again be written. In the case of the standing wave one finds and thus the excitation probability of an atom at the location , which is given by , is strongly non-linearly dependent on the location (via the function ). The achievable resolution increases linearly with the maximum angle , i. i.e., with the number of complete Rabi oscillations. One allows a resolution of a tenth of the wavelength. First demonstration experiments in a gas of cold atoms show a resolution below one ninth of the Rayleigh limit. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \Omega (x,t)}$ ${\displaystyle \propto \cos(kx)}$ ${\displaystyle \textstyle \Theta (x)=\int _{0}^{\infty }\Omega (x,s)ds}$ ${\displaystyle \pi }$ ${\displaystyle \Theta (x)=\Theta _{0}\cos(kx)}$ ${\displaystyle x}$ ${\displaystyle (1-\cos[\Theta (x)])/2}$ ${\displaystyle \cos[\Theta _{0}\cos(kx)]}$ ${\displaystyle \Theta _{0}}$ ${\displaystyle \Theta _{0}=400}$

Decoherence processes , such as the spontaneous decay of the involved excited states, limit the achievable resolution in the two methods presented here.

literature

• Christian Kothe, Gunnar Björk, Shuichiro Inoue, Mohamed Bourennane: On the efficiency of quantum lithography . In: New J. Phys. tape 13 , 2011, p. 043028 , doi : 10.1088 / 1367-2630 / 13/4/043028 , arxiv : 1006.2250 (English).
• RW Boyd and JP Dowling: Quantum lithography: status of the field . In: Quant. Inf. Process. tape 11 , 2012, p. 891 , doi : 10.1007 / s11128-011-0253-y (English).

Web links

• Phillip F. Schewe and Ben Stein: Entangled Photons can defeat the diffraction limit. (No longer available online.) In: AIP Physics News Update. September 22, 2000, archived from the original on August 6, 2007 . 
• Pieter Kok: Introduction to Quantum Lithography. In: Sheffield University. Accessed September 4, 2018 . 
• Anne Eisenberg: WHAT'S NEXT; Quantum Theory Could Expand the Limits of Computer Chips. In: New York Times. Accessed September 4, 2018 . 
• Wolfgang Stieler: With quantum lithography to smaller chip structures. In: heise online. September 22, 2001 . 
• Chris Lee: Is quantum lithography dead on arrival? In: Ars Technica. May 18, 2011 (English). 

Individual evidence

1. ↑ AGEDI N. Boto, Pieter Kok, Daniel S. Abrams, Samuel L. Braunstein, Colin P. Williams and Jonathan P. Dowling: Quantum Interferometric Optical Lithography: Exploiting Entanglement to beat the diffraction limit . In: Phys. Rev. Lett. tape 85 , 2000, pp. 2733 , doi : 10.1103 / PhysRevLett.85.2733 , arxiv : quant-ph / 9912052 .
2. ↑ For the sake of simplicity we consider the limit case of parallel incidence. The angle of incidence is taken into account in a somewhat more general way and replaced in the formula by the component of the wave vector in the plane ; here then describes perpendicularly incident light, which leads to a spatially constant time-averaged intensity, and the limit case considered above. ${\displaystyle \theta }$ ${\displaystyle k}$ ${\displaystyle k_{x}=k\cos \theta )}$ ${\displaystyle \theta =0}$ ${\displaystyle \theta =90^{\circ }}$
3. ^ Eli Yablonovitch and Rutger B. Vrijen: Optical projection lithography at half the Rayleigh resolution limit by two-photon exposure . In: Opt. Eng. tape 38 , no. 2 , p. 334-338 , doi : 10.1117 / 1.602092 .
4. ↑ The relevant object for lithography based on -fold absorption is the exposure pattern , which is determined by the expected value of the operator , with the creation and annihilation operators of a photon at the location . ${\displaystyle N}$ ${\displaystyle (a_{x}^{\dagger })^{N}(a_{x})^{N}}$ ${\displaystyle a_{x}^{\dagger },a_{x}}$ ${\displaystyle x}$
5. ↑ Pieter Kok, Samuel L. Braunstein and Jonathan P. Dowling: Quantum lithography, entanglement and Heisenberg-limited parameter estimation . In: J. Opt. B: Quantum Semiclass. Opt. Band 6 , 2004, p. S811-S815 , doi : 10.1088 / 1464-4266 / 6/8/029 .
6. ↑ ^{a b} Christian Kothe, Gunnar Björk, Shuichiro Inoue, Mohamed Bourennane: On the efficiency of quantum lithography . In: New J. Phys. tape 13 , 2011, p. 043028 , doi : 10.1088 / 1367-2630 / 13/4/043028 , arxiv : 1006.2250 (English).
7. ↑ Olesteueragel: Comment on "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit" . arxiv : quant-ph / 0305042 . ; Olesteueragel : On the concentration behavior of entangled photons . In: J. Opt. B: Quantum Semiclass. Opt. Band 6 , 2004, p. S606 , doi : 10.1088 / 1464-4266 / 6/6/021 .
8. ↑ WH Peeters, JJ Renema and MP van Exter: Engineering of two-photon spatial quantum correlations behind a double slit . In: Phys. Rev. A . tape 79 , 2009, p. 043817 .
9. ↑ RW Boyd and JP Dowling: Quantum lithography: status of the field . In: Quantum Inf Process . tape 11 , 2012, p. 891 , doi : 10.1007 / s11128-011-0253-y (English).
10. ^ Gunnar Björk, Luis L. Sánchez-Soto and Jonas Söderholm: Entangled-State Lithography: Tailoring Any Pattern with a Single State . In: Phys. Rev. Lett. tape 86 , 2001, p. 4516 , doi : 10.1103 / PhysRevLett.86.4516 , arxiv : quant-ph / 0011075 .
11. ↑ Promotion of quantum technologies. Position paper of the German industry. (PDF) VDI Technologiezentrum GmbH, January 2017, p. 13 , accessed on September 4, 2018 . 
12. ↑ Milena D'Angelo, Maria V. Chekhova and Yanhua Shih: Two-Photon Diffraction and Quantum Lithography . In: Phys. Rev. Lett. tape 87 , 2001, p. 013602 , doi : 10.1103 / PhysRevLett.87.013602 , arxiv : quant-ph / 0103035 .
13. ↑ PR inhibitors, A. Muthukrishnan, MO Scully and MS Zubairy: Quantum Lithography with Classical Light . In: Phys. Rev. Lett. tape 96 , 2006, p. 163603 , doi : 10.1103 / PhysRevLett.96.163603 .
14. ↑ ^{a b} Z. Liao, M. Al-Amri, and MS Zubairy: Quantum Lithography beyond the Diffraction Limit via Rabi Oscillations . In: Phys. Rev. Lett. tape 105 , October 25, 2010, p. 183601 , doi : 10.1103 / PhysRevLett.105.183601 .
15. ↑ Jun Rui, Yan Jiang, Guo-Peng Lu, Bo Zhao, Xiao-Hui Bao and Jian-Wei Pan : Experimental demonstration of quantum lithography beyond diffraction limit via Rabi oscillations . In: Phys. Rev. A . tape 93 , 2016, p. 033837 , doi : 10.1103 / PhysRevA.93.033837 , arxiv : 1501.06707 .