Navigation between quantum states by quantum mirrors

P. A. Ivanov, B. T. Torosov, and N. V. Vitanov

Phys. Rev. A 75, 012323 – Published 22 January 2007

Abstract

We introduce a technique that allows one to connect any two arbitrary (pure or mixed) superposition states of an $N$ -state quantum system. The proposed solution to this inverse quantum mechanical problem is analytical, exact, and very compact. The technique uses standard and generalized quantum Householder reflections (QHRs), which require external pulses of precise areas and frequencies. We show that any two pure states can be linked by just a single generalized QHR. The transfer between any two mixed states with the same dynamic invariants (e.g., the same density matrix eigenvalues) requires in general $N$ QHRs. Moreover, we propose recipes for synthesis of arbitrary preselected mixed states using a combination of QHRs and incoherent processes (pure dephasing or spontaneous emission).

Received 11 October 2006

DOI:https://doi.org/10.1103/PhysRevA.75.012323

Authors & Affiliations

P. A. Ivanov¹, B. T. Torosov¹, and N. V. Vitanov^1,2

¹Department of Physics, Sofia University, James Bourchier 5 boulevard, 1164 Sofia, Bulgaria
²Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko chaussée 72, 1784 Sofia, Bulgaria

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Issue

Vol. 75, Iss. 1 — January 2007

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Images

Figure 1
Physical realization of the quantum Householder reflection: $N$ degenerate [in the rotating-wave approximation (RWA) sense] ground states, forming the qunit, coherently coupled via a common excited state by pulsed external fields of the same time dependence and the same detuning, but different amplitudes and phases.Reuse & Permissions
Figure 2
(Color online) Time evolution of the pulsed fields (top) and the populations and the state mismatch $D$ (bottom) for the transition (23) in a qutrit. We have assumed sech pulses, $f (t) = sech (t ∕ T)$ , and rms pulse area $A = 4 π$ $(χ T = 4)$ . The individual couplings $χ_{n}$ $(n = 1, 2, 3)$ are given by the components of the QHR vector (24), each multiplied by $χ$ . The detuning is $Δ_{0} T = 1.732$ (which gives $φ = π$ ).Reuse & Permissions
Figure 3
(Color online) Time evolution of the pulsed fields (top) and the populations and the state mismatch $D$ (bottom) for the qutrit transition (26). We have assumed sech pulse shapes with rms pulse area $A = 2 π$ $(χ T = 2)$ and detuning $Δ = 0$ . The individual couplings $χ_{n}$ $(n = 1, 2, 3)$ are given by the components of the standard QHR vectors (27), each multiplied by $χ$ .Reuse & Permissions
Figure 4
(Color online) Time evolution of the pulsed fields (top) and the populations and the state mismatch $D$ (bottom) for the qutrit transition (26). We have assumed sech pulse shapes with rms pulse area $A = 2 π$ $(χ T = 2)$ . The individual couplings $χ_{n} (n = 1, 2, 3)$ are given by the components of the generalized QHR vector (28), each multiplied by $χ$ . The detuning is $Δ T = 0.791$ , which produces the desired phase $φ = 0.574 π$ .Reuse & Permissions
Figure 5
(Color online) Time evolution of the pulsed fields (top) and the populations and the state mismatch (bottom) for the transition between states (34a, 34b) in a qutrit. We have assumed sech pulse shapes and rms pulse area $A = 2 π$ $(χ T = 2)$ . The individual couplings $χ_{n}$ $(n = 1, 2, 3)$ are given by the components of the generalized QHR vectors (36) each multiplied by $χ$ . The detunings are $Δ_{1} T = - 0.255$ , $Δ_{2} T = 0.049$ , and $Δ_{3} T = - 4.918$ , which produce the phases (36d).Reuse & Permissions
Figure 6
(Color online) Time evolution of the pulsed fields (top) and the populations and the state mismatch (25) (bottom) for mixed-state engineering in a qutrit. The qutrit starts in state ∣1⟩ and the target final state is given by Eq. (34b). We have assumed sech pulse shapes and rms pulse area $A = 2 π$ $(χ T = 2)$ . The individual couplings $χ_{n}$ $(n = 1, 2, 3)$ are given by the components of the generalized QHR vectors (37) and (40), each multiplied by $χ$ . The detunings are $Δ = 0$ , $Δ_{1} T = 0.072$ and $Δ_{2} T = - 0.396$ , which produce the desired QHR phases (40c). The dephasing rate is $Γ = 2 ∕ T$ .Reuse & Permissions

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